Fundamentals of Gas Dynamics 9811690847, 9789811690846

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Mrinal Kaushik

Fundamentals of Gas Dynamics

Fundamentals of Gas Dynamics

Mrinal Kaushik

Fundamentals of Gas Dynamics

Mrinal Kaushik Department of Aerospace Engineering Indian Institute of Technology Kharagpur Kharagpur, West Bengal, India

ISBN 978-981-16-9084-6 ISBN 978-981-16-9085-3 (eBook) https://doi.org/10.1007/978-981-16-9085-3 © Springer Nature Singapore Pte Ltd. 2022 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

The current book results from the author’s supersonic and hypersonic aerodynamic courses taught at the Indian Institute of Technology, Kharagpur, and the Defence Institute of Advanced Technology, Pune, over the past decade. The goal is to serve as a complete textbook for an undergraduate course in gas dynamics, and it is prepared to study the fundamental aspects of graduate courses. It is created to provide students, engineers, and researchers with a comprehensive text covering basic and advanced compressible flow concepts. This book offers subjects that any beginner can easily follow. It is believed that the compressible flow phenomena are complicated enough to require the reader’s entire attention without being diverted by the demands of newly learned mathematical tools that get in the way of acquiring the physical feel of the subject. Therefore, the amount of mathematics in this text is kept to a minimum level. The book’s contents are ordered logically, beginning with a brief overview of gas kinetic theory in Chap. 1. This chapter is provided to create a solid foundation for developing gas dynamic principles in later chapters. Numerical examples are used throughout the work to make the key mathematical entities more understandable rather than intimidating. Following that, in Chap. 2, the conservation laws are thoroughly examined. At this stage, the thermodynamic rules as they apply to the flow of an inviscid compressible fluid and the equation of state of a perfect gas are introduced. Chapter 3 is concerned with sound wave propagation incompressible fluids, notably air. Chapter 4 investigates steady one-dimensional isentropic flow processes and derives area-velocity and area-Mach number relations. Following that, Chap. 5 delves deeply into the flow processes through shock and expansion waves. The first principle is applied to the well-known Prandtl’s velocity equations and the Rankine-Hugoniot equations for normal and oblique shock waves. The steady quasione-dimensional flows are next discussed in Chap. 6, with a focus on the flow through a convergent-divergent nozzle and a convergent-divergent diffuser. Chapters 7 and 8 are devoted to the one-dimensional analysis of Fanno and Rayleigh flow processes. Chapter 9 describes the general characteristics of the steady multi-dimensional adiabatic flow of a compressible fluid. Chapter 10 elaborates on the small perturbation theory and its application to various flow situations. Finally, Chap. 11 describes the v

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fundamental concepts of the hypersonic flow of an inviscid compressible fluid. Many tables and a chart are provided in the appendices to assist in solving the numerical problems. The author expresses the Almighty’s gratitude for his generosity and graces, especially during difficult times. The author wishes to express his sincere gratitude to his students at the Indian Institute of Technology Kharagpur for their direct or indirect contributions to completing this work. The author wishes to thank his doctoral students Tamal Jana and Gopinath Shanmugaraj, for their invaluable assistance in completing the manuscript. The author dedicates this book to his children Talin and Tanvi, who are a constant motivation for him to do something meaningful with his life. Last but not least, the author wishes to express his gratitude to Ms. Swati Meherishi and Ms. Rini Christy of Springer for their generous support in finishing the work. Any reader who has constructive suggestions for potential contradictions, inaccuracies, or flaws in the book is welcome to forward them to the author. Kharagpur, India September 2021

Mrinal Kaushik

Contents

1

Gas Kinetics—Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The Fundamental Postulates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 The Maxwell-Boltzmann Distribution Law . . . . . . . . . . . . . . . . . . . 1.3.1 The Characteristic Molecular Speeds . . . . . . . . . . . . . . . . 1.4 Molecular Collision Theory and the Mean Free Path . . . . . . . . . . . 1.5 Kinetic Interpretation of Pressure and the Ideal Gas Law . . . . . . . 1.6 Kinetic Interpretation of Temperature . . . . . . . . . . . . . . . . . . . . . . . 1.7 Concept of Internal Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.1 The Equipartition of Energy and the Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Specific Heats of an Ideal Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.1 The Ratio of Specific Heats for a Monoatomic Gas . . . . 1.8.2 The Ratio of Specific Heats for a Diatomic Gas . . . . . . . 1.8.3 The Ratio of Specific Heats for a Polyatomic Gas . . . . . 1.9 The Continuum Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9.1 The Characteristic Molecular Length and the Knudsen Number . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9.2 Limitations of the Continuum Hypothesis . . . . . . . . . . . . 1.10 The Perfect Substance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10.1 The Perfect Solid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10.2 The Perfect Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11 Transport Properties of Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11.1 Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11.2 Thermal Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11.3 Self-diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.12 Real Fluid Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.12.1 Comparison of the Perfect and Real Fluid Flow Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 3 7 12 15 20 21 22 25 27 28 28 29 30 32 35 37 38 39 39 41 42 43 43 45 45 vii

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Governing Equations and Thermodynamics of Compressible Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Mathematical Description of a Continuous Media . . . . . . . . . . . . . 2.2.1 System, Control Volume, and Control Surface . . . . . . . . 2.2.2 Extensive and Intensive Properties . . . . . . . . . . . . . . . . . . 2.2.3 Property Field and Flow Description Methods . . . . . . . . 2.2.4 Correlation Between System and Control Volume Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Conservation of Mass: The Continuity Equation . . . . . . . . . . . . . . 2.3.1 Mass and Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Integral Form of Continuity Equation for a Finite Control Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Conservative Partial Differential Form of Continuity Equation for a Differential Control Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Non-conservative Partial Differential Form of Continuity Equation for a Moving Fluid Element . . . . 2.4 Conservation of Momentum (Newton’s Second Law of Motion for Fluid Flows) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Integral Form of Momentum Equation for a Finite Control Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Conservative Partial Differential Form of Momentum Equation for a Differential Control Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Non-conservative Partial Differential Form of Momentum Equation for a Moving Fluid Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Euler’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Bernoulli’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Integration of Euler’s Equation in Steady Rotational Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Integration of Euler’s Equation in Unsteady Irrotational Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Conservation of Energy: The First Law of Thermodynamics . . . . 2.7.1 Concept of Enthalpy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.2 Integral Form of Energy Equation for a Finite Control Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.3 Conservative Partial Differential Form of Energy Equation for a Differential Control Volume . . . . . . . . . . . 2.7.4 Non-conservative Partial Differential Form of Energy Equation for a Moving Fluid Element . . . . . . . 2.8 Increase of Entropy Principle: The Second Law of Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.1 Concept of Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.2 Integral Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.8.3 Conservative Partial Differential Form . . . . . . . . . . . . . . . 2.8.4 Non-conservative Partial Differential Form . . . . . . . . . . . 2.9 Combined Expressions of the First and Second Laws of Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 Thermal and Calorical Properties of a Perfect Gas . . . . . . . . . . . . . 2.10.1 Thermal Equation of State . . . . . . . . . . . . . . . . . . . . . . . . . 2.10.2 Calorical Equation of State . . . . . . . . . . . . . . . . . . . . . . . . . 2.10.3 Thermal and Calorical Perfectness of a Gas . . . . . . . . . . . 2.10.4 Specific Heat Relationships . . . . . . . . . . . . . . . . . . . . . . . . 2.10.5 Change in Entropy of a Perfect Gas . . . . . . . . . . . . . . . . . . 2.10.6 Isentropic Relations for a Perfect Gas . . . . . . . . . . . . . . . . 2.10.7 Thermal and Calorical Imperfectness of the Air . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Sound Wave Propagation in Compressible Fluids and Flow Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Speed of Propagation of Small Disturbances in a Compressible Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 The Speed of Sound in a Perfect Gas . . . . . . . . . . . . . . . . 3.3 The Compressibility Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 The Speed of Sound in a Real Gas . . . . . . . . . . . . . . . . . . . 3.4 Two-Dimensional Propagation of Sound Waves and the Signaling Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Von Karman’s Rules for Supersonic Flows and the Activity Envelope . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 The Mach Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Similarity Parameters in the Compressible Flows . . . . . . . . . . . . . 3.5.1 Mach Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Reynolds Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.3 Prandtl Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.4 Ratio of Specific Heats . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Flow Regime Classification Using Mach Number . . . . . . . . . . . . . 3.6.1 Incompressible Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2 Subsonic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.3 Transonic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.4 Supersonic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.5 Hypersonic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.6 Hypervelocity Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Steady One-Dimensional Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Concept of One-Dimensional Flows . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Governing Equations of the Steady One-Dimensional Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Equations Governing the Steady One-Dimensional Isentropic Flow of a Perfect Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Stagnation or Reservoir Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Stagnation Enthalpy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Stagnation Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Stagnation Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.4 Stagnation Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.5 Stagnation Speed of Sound . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.6 Entropy Change in Terms of Stagnation Properties . . . . . 4.5 Characteristic Speeds in Gas Dynamics . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Maximum Isentropic Discharge Speed . . . . . . . . . . . . . . . 4.5.2 Critical Speed of Sound . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Thermodynamic Properties in Terms of Critical Speed of Sound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.4 Relationships Between the Characteristic Speeds . . . . . . 4.5.5 Kinematic Forms of the Energy Equation for Steady One-Dimensional Adiabatic Flow of a Perfect Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.6 Dimensionless Speed M∗ . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Steady, Adiabatic Flow Ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 Incompressible Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Compressible Subsonic Flow . . . . . . . . . . . . . . . . . . . . . . . 4.6.3 Transonic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.4 Supersonic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.5 Hypersonic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Continuity Equation for Steady One-dimensional Flow of a Perfect Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.1 Maximum Isentropic Mass Flow Rate per Unit Cross-Sectional Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.2 Physical Interpretation of Flow at Throat . . . . . . . . . . . . . 4.7.3 Critical Area Ratio in Terms of Mach Number . . . . . . . . 4.8 Stream Thrust and Impulse Function . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Dynamic Pressure and Compressibility Correction Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10 Pressure Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Wave Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Shock Wave Formation in Supersonic Flows . . . . . . . . . . . . . . . . . 5.3 Steady One-Dimensional Flow Through a Stationary Normal Shock Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Governing Equations for a General Fluid . . . . . . . . . . . . . 5.3.2 Governing Equations for a Perfect Gas . . . . . . . . . . . . . . . 5.3.3 Working Formulae for a Perfect Gas . . . . . . . . . . . . . . . . . 5.4 Strong and Weak Normal Shock Waves . . . . . . . . . . . . . . . . . . . . . . 5.5 Prandtl’s Velocity Equation for a Normal Shock Wave . . . . . . . . . 5.6 The Rankine-Hugoniot Equation for a Normal Shock Wave . . . . 5.7 Supersonic Pitot Probe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.1 Rayleigh Supersonic Pitot Probe Formula . . . . . . . . . . . . 5.8 Fanno and Rayleigh Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.1 Fanno Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.2 Rayleigh Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.3 Normal Shock Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9 The Plane Oblique Shock Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9.1 Governing Equations for a General Fluid . . . . . . . . . . . . . 5.9.2 Working Formulae for a Perfect Gas . . . . . . . . . . . . . . . . . 5.9.3 Minimum and Maximum Wave Angles . . . . . . . . . . . . . . 5.9.4 Relationship Between θ and β . . . . . . . . . . . . . . . . . . . . . . 5.10 Prandtl’s Velocity Equation for an Oblique Shock Wave . . . . . . . . 5.11 The Rankine-Hugoniot Equation for an Oblique Shock Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.12 Weak Oblique Shock Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.13 The Hodograph Shock Polar Diagram . . . . . . . . . . . . . . . . . . . . . . . 5.14 The Pressure-Deflection Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.15 Compression and Expansion Waves . . . . . . . . . . . . . . . . . . . . . . . . . 5.15.1 Supersonic Flow Around a Concave Corner: Isentropic Compression by Turning . . . . . . . . . . . . . . . . . . 5.15.2 Supersonic Flow Around a Convex Corner: Isentropic Expansion by Turning . . . . . . . . . . . . . . . . . . . . 5.16 The Prandtl-Meyer Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.16.1 Important Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.17 Simple and Non-simple Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.18 Supersonic Flow Past Wedges and Conical Bodies . . . . . . . . . . . . 5.18.1 Flow over a Planar Wedge: The Two-Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.18.2 Flow over a Conical Body: The Three-Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.19 Shock-Expansion Theory: The Supersonic Airfoils . . . . . . . . . . . . 5.19.1 A Flat Plate Supersonic Airfoil . . . . . . . . . . . . . . . . . . . . . 5.19.2 A Diamond-Shaped Supersonic Airfoil . . . . . . . . . . . . . . 5.20 Supersonic Thin Airfoil Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5.21 Reflection and Interaction of Oblique Shock Waves . . . . . . . . . . . 5.21.1 Reflection from a Solid Boundary . . . . . . . . . . . . . . . . . . . 5.21.2 Reflection from a Free Pressure Boundary . . . . . . . . . . . . 5.21.3 Neutralization or Cancellation of an Oblique Shock Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.21.4 Interactions of Shock-Shock and Shock-Expansion Waves of the Same Family . . . . . . . . . . . . . . . . . . . . . . . . . 5.21.5 Interaction of the Oblique Shock Waves of Opposite Family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.22 Deviation from the Theoretical Pressure Distribution . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

7

Steady One-Dimensional Isentropic Flow in a Variable-Area Duct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Steady One-Dimensional Isentropic Flow Through Ducts . . . . . . 6.2.1 Area-Velocity Relationship for the Flow Through a Duct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Flow Through a Convergent Nozzle . . . . . . . . . . . . . . . . . 6.2.3 Flow Through a Convergent-Divergent (de Laval) Nozzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 Performance of a Real Nozzle . . . . . . . . . . . . . . . . . . . . . . 6.2.5 Flow Through a Convergent-Divergent Diffuser: The Supersonic Wind Tunnel Diffuser . . . . . . . . . . . . . . . 6.3 Flow Through an Air-Breathing Intake . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Flow Through a Subsonic Intake . . . . . . . . . . . . . . . . . . . . 6.3.2 Performance Criteria of a Subsonic Intake . . . . . . . . . . . . 6.3.3 Flow Through a Supersonic Intake . . . . . . . . . . . . . . . . . . 6.3.4 Flow Through a Hypersonic Intake . . . . . . . . . . . . . . . . . . 6.4 Supersonic Combustion and Scramjet . . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adiabatic Flow Through Constant-Area Ducts with Friction . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 The D’Arcy-Weisbach Equation . . . . . . . . . . . . . . . . . . . . 7.2.2 The Fanning Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 The Friction Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Relationships Among the Thermodynamic Properties for a Fanno Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 The Fanno Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Frictional Choking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Effects of Wall Friction on Thermodynamic and Flow Properties of a Perfect Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

275 276 277 283 284 285 286 292 293 297 297 298 298 302 304 306 310 316 318 325 328 333 337 338 339 341 341 342 344 344 345 346 353 356 356

Contents

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7.6 Working Formulae for a Perfect Gas . . . . . . . . . . . . . . . . . . . . . . . . 358 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 Exercise Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 8

9

Frictionless Flow Through Constant-Area Ducts with Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 The Rayleigh Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Thermal Choking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Governing Equations for a Flow with Heat Addition . . . . . . . . . . . 8.3.1 Effects of Heat Addition on Thermodynamic and Flow Properties of a Perfect Gas . . . . . . . . . . . . . . . . 8.4 Relationships Among the Thermodynamic Properties for a Rayleigh Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Working Formulae for a Perfect Gas . . . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Steady, Inviscid, and Adiabatic Multi-dimensional Compressible Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Governing Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Momentum Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.3 Energy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.4 Equation of State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Circulation, Rotation, and Vorticity . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Circulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Fluid Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.3 Vorticity and the Vortex Tubes . . . . . . . . . . . . . . . . . . . . . . 9.4 Kelvin’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Crocco’s Theorem and Shock-Induced Vorticity . . . . . . . . . . . . . . 9.6 Irrotational Flows and the Velocity Potential Function (φ) . . . . . . 9.6.1 The General Governing Equation in Terms of φ . . . . . . . 9.6.2 The General Characteristics of the Velocity Potential Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7 The Stream Function (ψ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7.1 The General Governing Equation in Terms of ψ . . . . . . . 9.7.2 Relationship Between φ and ψ . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

375 375 376 378 379 383 384 387 397 398 399 399 400 400 402 403 406 406 406 408 412 414 419 425 427 429 430 431 433 435 435

10 Linearized Potential Flow: The Small-Perturbation Theory . . . . . . . 437 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 10.2 The Linearized Velocity Potential Equation for Steady, Two-Dimensional, Irrotational Flow with Small Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438

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Contents

10.2.1 The General Behavior of the Perturbation Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 Linearized Pressure Coefficient . . . . . . . . . . . . . . . . . . . . . 10.3 Steady, Two-Dimensional, Irrotational, and Isentropic Flow Past an Infinite Wave-Shaped Wall . . . . . . . . . . . . . . . . . . . . . 10.3.1 Uniform Subsonic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 Uniform Supersonic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Similarity Laws for the Steady, Two-Dimensional, Irrotational, and Isentropic Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.1 Subsonic Case: The Prandtl-Glauert Rule . . . . . . . . . . . . 10.4.2 Supersonic Case: The Ackeret’s Formula . . . . . . . . . . . . . 10.5 Supersonic Thin Airfoil Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Elementary Concepts of Inviscid Hypersonic Flows . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Thin Shock Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Entropy Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.3 Viscous Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.4 High-Temperature Effects . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Hypersonic Oblique Shock Wave Relations . . . . . . . . . . . . . . . . . . 11.3 Mach Number Independence Principle . . . . . . . . . . . . . . . . . . . . . . 11.4 Hypersonic Expansion Wave Relations . . . . . . . . . . . . . . . . . . . . . . 11.5 Hypersonic Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6 Estimation of Aerodynamic Forces: The Newtonian Theory . . . . 11.6.1 Lift and Drag Coefficients for an Inclined Flat Plate . . . . 11.6.2 Some Observations on Newtonian Sine-Squared Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.7 Modified Newtonian Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

442 443 444 446 450 454 455 460 462 464 464 467 467 468 468 469 471 474 478 479 481 483 485 487 492 493 493

Appendix A: The Standard Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495 Appendix B: Isentropic Table (γ = 1.4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539 Appendix C: Normal Shock Table (γ = 1.4) . . . . . . . . . . . . . . . . . . . . . . . . . 551 Appendix D: Oblique Shock Chart (γ = 1.4) . . . . . . . . . . . . . . . . . . . . . . . . . 561 Appendix E: One-Dimensional Flow with Friction (γ = 1.4) . . . . . . . . . . . 563 Appendix F: One-Dimensional Flow with Heat Transfer (γ = 1.4) . . . . . 569 Appendix G: Letter of Admittance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579

About the Author

Dr. Mrinal Kaushik is an Associate Professor in the Department of Aerospace Engineering at the Indian Institute of Technology, Kharagpur. Before joining this institute in 2013, he worked at the Defense Institute of Advanced Technology (Pune), General Motors Technical Center (Bangalore), Indian Space Research Organization (Trivandrum), and Tata Consultancy Services (Mumbai). Dr. Kaushik received his Ph.D., M.Tech, and B.Tech degrees in Aerospace Engineering from the Indian Institute of Technology, Kanpur in 2012, 2003, and 1999, respectively. His research interests include Shock-Wave and Boundary-Layer Interactions, and Jet Control and Base Flows. He has published several research articles in peer-reviewed international journals and refereed conference proceedings. He is a member of many national and international technical societies. He has written four other books for the engineering students, namely, Instrumentation and Measurements in Compressible Flows (CRC Press, Taylor & Francis Group, Boca Raton, USA, In Progress); Theoretical and Experimental Aerodynamics (Springer Nature, Singapore, 2019); Essentials of Aircraft Armaments (Springer Nature, Singapore, 2016); and Innovative Passive Control Techniques for Supersonic Jet Mixing (Lap LAMBERT Academic Publishing, Germany, 2012).

xv

Nomenclature

Cy Cz α β η ηD γ ∞ κ λ  m max 4f LD Th  s  τ Amax A∞ Ai Cpw s μ − → ζ θmax A∗ a0 a

scalar component of the velocity vector in y-direction scalar component of the velocity vector in z-direction number of degrees of freedom shock angle nozzle efficiency diffuser efficiency ratio of specific heats freestream conditions Boltzmann constant mean free path temporal average or ensemble average mass flow rate friction parameter increase in the component of thrust that acts on the front external surface of the intake circulation mass flow rate crossing from the sides of the control surface characteristic length scale shear stress maximum area of the intake streamtube cross-sectional area intake lip area surface pressure coefficient entropy coefficient of viscosity vorticity vector maximum deflection angle critical area stagnation speed of sound speed of sound

xvii

xviii

A B c∀ Cx CD CL Cpw Cp cp D d Erot Etrans Evib ff F fD h h0 KT KCM Kn K M∗ Mw No p0 pb pcr pr Pr p q rd Re T0 Tcr Tr T t u Uinter Uintra U v

Nomenclature

cross-sectional area arbitrary extensive property specific heat at constant volume scalar component of the velocity vector in x-direction drag coefficient lift coefficient pressure coefficient at the wall pressure coefficient specific heat at constant pressure wave drag per unit span effective molecular diameter rotational kinetic energy of a gas molecule translational kinetic energy of a molecule vibrational kinetic energy of a gas molecule fanning friction factor impulse function D’Arcy-Weisbach friction factor amplitude of the wavy wall stagnation enthalpy isothermal bulk modulus kinetic energy due to the motion of the center of mass of the molecules Knudsen number correction coefficient for dynamic pressure dimensionless speed molecular weight number density stagnation pressure back pressure critical pressure reduced pressure Prandtl number static pressure dynamic pressure ratio of stagnation pressures across a diffuser or an intake Reynolds number stagnation temperature critical temperature reduced temperature static temperature airfoil thickness at the shoulder perturbation-velocity in the x-direction potential energy due to intermolecular interactions potential energy due to intramolecular interactions internal energy perturbation-velocity in the y-direction

Nomenclature

vavg vmp vrms vi vrel v w xyzZ μ ν − → q φ φ ψ ρ ρ0 τd θ Φ Ks m N NA Ru

average speed most probable speed root mean square speed flow velocity at the intake lip relative velocity flow velocity perturbation-velocity in the z-direction along x-axis along y-axis along z-axis compressibility factor Mach angle kinematic viscosity perturbation-velocity vector perturbation-velocity potential velocity potential function stream function static density stagnation density ratio of stagnation temperatures across an intake deflection angle total velocity potential isentropic bulk modulus mass of the molecule number of molecules Avogadro’s number universal gas constant

xix

Chapter 1

Gas Kinetics—Basic Concepts

Abstract Kinetic theory of gases, also known as gas kinetics, is a theory that focuses on a simplified molecular or particle description of a gas from which many of its gross properties (macroscopic properties) such as volume, pressure, and temperature can be derived, as well as transport properties such as viscosity, thermal conductivity, and mass diffusivity. In this chapter, we will describe briefly how the principles of gas kinetics (microscopic description of characteristics) are related to the continuum theory of gases or gas dynamics (macroscopic description of properties), which will be covered in later chapters.

1.1 Introduction We know that both liquids and gases together are termed as fluids. A fluid of constant density is termed incompressible. If the density variation is negligibly small in real fluids, they can be assumed incompressible with reasonable accuracy. It is essentially true for liquids where the density varies marginally with the temperature and moderately with the pressure over a wide range of operating conditions. Thus, the liquids are invariably considered incompressible. On the other hand, a gas may be either incompressible or compressible, depending on its operating conditions. In inflow processes, where the gas experiences a very small density change, the flow can be considered incompressible; however, if the flow occurs at relatively higher speeds, the change in density will be quite significant and cannot be neglected. Thus, the high-speed flows compressible. As a rule   are invariably ≥ 5% is significant enough to treat of thumb, a density change of more than 5% ρ ρ the flow as compressible. At standard conditions, a 5% change in density corresponds to a flow velocity of about 360 km h, corresponding to Mach 0.3. Thus, the fluid flows with Mach 0.3 and above are called compressible flows. For an incompressible flow, temperature and density fluctuations are negligible, whereas temperature variations may be negligible for compressible flows, but density variations are always finite in magnitude. Similarly, a 5% change in flow temperature essentially corresponds to the flow velocity of 650 km h (or Mach 0.5) at standard sea-level conditions, which may be © Springer Nature Singapore Pte Ltd. 2022 M. Kaushik, Fundamentals of Gas Dynamics, https://doi.org/10.1007/978-981-16-9085-3_1

1

2

1 Gas Kinetics—Basic Concepts

considered the lower gas dynamics limit. Thus, gas dynamics refers to the realm of compressible flow in which both compressibility and heat effects are significant. Waves, in this case, dominate the whole flow field, and any change in flow property or direction is produced solely by these waves. Subsonic flow fields, for example, are dominated by Mach waves,1 whereas supersonic flow fields are dominated by Mach waves, shock waves, and expansion waves. Among these, Mach waves are isentropic and have no effect on flow characteristics or direction; however, shock waves, due to their non-isentropic behavior, have a considerable effect on flow properties and direction. Thus, unlike subsonic flows, changes in flow properties or direction in supersonic flows are induced by waves with both isentropic and non-isentropic characteristics. As can be seen from the preceding discussion, studying gas dynamics necessitates a thorough understanding of thermodynamics. The thermodynamic changes are measured in terms of properties, which are macroscopically averaged quantities over many gas molecules and atoms. These properties are normally approximated using thermodynamic relations; however, this is not practicable when the gas density becomes extremely low (i.e., in a rarefied medium), as in the upper layer of the atmosphere. In a rarefied gas, it is preferable to examine the behavior and movement of each molecule microscopically. The kinetic theory of gases (or gas kinetics) is a branch of science that studies the behavior of gas molecules at the microscopic level, which is primarily impacted by temperature. Therefore, before we get started on compressible flows, let us go over some important concepts from gas kinetics.

1.2 The Fundamental Postulates The kinetic theory of gases, often known as gas kinetics, is a basic theoretical model that can describe the properties of gases. The macroscopic properties of substances are explained by this hypothesis, which is based on molecular motion. Its fundamental assumptions or postulates are as follows: 1. The gases are made up of many molecules that behave like hard spherical objects in a state of continual random motion. The diameters of gas molecules are supposed to be very small compared to the distance between them, implying that the gas is sufficiently dilute. Thus, the volume occupied by molecules is quite small in comparison to the overall volume of the gas. 2. Molecules do not interact with one another until during collisions. Hence, when the molecules are not colliding, they do not exert intermolecular forces on one other or the wall. 3. Molecules travel with constant velocity and in straight lines before and after colliding with other molecules or the container’s wall. Their motion is consistent with Newton’s law of motion. A molecule alters its direction of motion only when 1

The Mach waves are the weakest waves across which the changes in flow properties are negligibly small and hence they are essentially isentropic waves.

1.2 The Fundamental Postulates

3

it collides with another molecule or the container’s wall. The collisions with the wall do account for the pressure of the gas. 4. During the molecular collisions, no kinetic energy or momentum is lost, and hence these collisions are said to be perfectly elastic.2 The total kinetic energy and the momentum of molecules remain constant unless there is some external interference. Moreover, the duration of molecular collisions is very small compared to the time spent by the molecules in between the collisions. 5. An elastic collision reflects that the angle of incidence and reflection made by the attacking molecule to the normal drawn at the impact point are equal. In other words, unless there is some external intervention, the total kinetic energy and momentum of molecules remain constant. Moreover, molecular collisions’ duration is quite short compared to the time spent by molecules in between collisions.

1.3 The Maxwell-Boltzmann Distribution Law In 1859, James Clerk Maxwell, a Scottish physicist, established his kinetic theory of gases and formulated the functions reflecting the velocity distribution among dilute gas molecules. Ludwig Boltzmann, a German physicist, generalized this theory in 1871. Boltzmann derived the energy distribution function among gas molecules from Maxwell’s velocity distribution. As a result, Maxwell’s distribution function is frequently referred to as the Maxwell-Boltzmann distribution law. It is based on statistical averages that predict whether macro-states can be derived from microstates. The kinetic theory of gases is used to determine the motion of gas molecules under certain conditions. However, when the number of molecules is very large, such as in a mole of an ideal gas, it is difficult to determine the velocity of each molecule at each time instant. In such cases, the Maxwell-Boltzmann distribution is used to predict how many molecules are traveling with velocities between v and v + dv. It is assumed here that one-dimensional velocity distributions are independent of each other, that is, the velocity distribution in the x-direction is independent of the velocity distributions in the y- and z-directions. For example, the Maxwell-Boltzmann velocity distribution function for a spatially uniform gas at equilibrium conditions is given by dN  m 3/2 −( m )v2 = e 2κT dv N 2π κT

(1.1)

where dN is the fraction of molecules traveling at velocities ranging from v to v + dv; N m is the mass of molecule; κ is the Boltzmann constant; and T is the absolute temperature of the gas.

2

A perfectly elastic collision reflects that the angle of incidence and angle of reflection made by attacking molecule with respect to the normal drawn at the impact point are equal.

4

1 Gas Kinetics—Basic Concepts

Let Cx , Cy , and Cz be the scalar components of the velocity vector v in the x-, y-, and z-directions, respectively, in Cartesian space. The Maxwell-Boltzmann velocity distribution function for gas molecules is as follows (Vincenti W.G. and Kruger C.H., Introduction to Physical Gas Dynamics, Krieger Publishing, FL, USA, 1975):   m 3/2 − m (C2 +C2 +C2 )  e 2κT x y z F Cx , Cy , Cz = 2π κT

(1.2)

Note that Eq. (1.2)does not involve any preferred direction of velocity in its  derivation, and thus F Cx , Cy , Cz is independent of any axes rotation and depends solely on C2x + C2y + C2z . The probability associated with a randomly chosen molecule having speeds in the intervals Cx , Cy , Cz to Cx + dCx , Cy + dCy , Cz + dCz , respectively, is ˚

  F Cx , Cy , Cz dCx dCy dCz

(1.3)

Because the probability of a molecule having a component of speed in a specified range in one coordinate direction is independent of the probabilities associated with other coordinate directions, Eq. (1.3) can be expressed as the product of three independent probabilities; each represents the probability that a molecule will have the speed in a given interval in one specific coordinate direction. ˚

  F Cx , Cy , Cz dCx dCy dCz =

ˆ

 ˆ  ˆ     Cy dCy ×  (Cz ) dCz  (Cx ) dCx ×

(1.4)

Consider the following functions:  m  m 1/2 C2x exp − 2π κT 2κT  m    m 1/2  Cy = exp − C2y 2π κT 2κT  m  m 1/2  (Cz ) = C2z exp − 2π κT 2κT

 (Cx ) =

(1.5) (1.6) (1.7)

Substituting in Eq. (1.4) yields ˚

  F Cx , Cy , Cz dCx dCy dCz ˚    m  m 3/2 C2x + C2y + C2z dCx dCy dCz = exp − 2π κT 2κT

(1.8)

Because the speed components Cx , Cy , and Cz can all be anywhere between −∞ to +∞, the preceding integral should be calculated for all possible values of speeds, i.e., from −∞ to ∞. Thus,

1.3 The Maxwell-Boltzmann Distribution Law ∞ ˚ −∞

5

⎧ ⎫ ⎬  m 3/2 ⎨ ˆ∞  m 2 C dCx F Cx , Cy , Cz dCx dCy dCz = exp − ⎩ ⎭ 2π κT 2κT x 



×

⎧ ∞ ⎨ˆ ⎩

−∞

−∞

⎫ ⎧ ∞ ⎫ ⎬ ⎨ˆ ⎬  m m 2 Cy dCy × C2z dCz exp − exp − ⎭ ⎩ ⎭ 2κT 2κT 

−∞

(1.9)   Here,  (Cx ),  Cy , and  (Cz ) are all even functions, and we can use the property3 of definite integrals to find them. ∞ ˚ −∞

⎧ ⎫ ⎬  m  m 3/2 ⎨ˆ∞   C2x dCx F Cx , Cy , Cz dCx dCy dCz = 23 exp − ⎩ ⎭ 2π κT 2κT ×

⎧∞ ⎨ˆ ⎩

0

⎫ ⎧∞ ⎫ ⎬ ⎨ˆ ⎬  m  m 2 2 C dCy × C dCz exp − exp − ⎭ ⎩ ⎭ 2κT y 2κT z

0

0

(1.10) The right-hand side of Eq. (1.10) contains three Gaussian integrals of the form ˆ∞

  1  π 1/2 exp −k  α 2 dα = 2 k

0

Hence, ∞ ˚

−∞

 m 3/2   F Cx , Cy , Cz dCx dCy dCz = 23 2π κT

   1 3 1 2π κT /2 = 1 (1.11) 2 m

Despite the fact that the probability of finding molecules in a certain direction is always less than one, the total probability of finding molecules with magnitudes of velocities in the interval (−∞, ∞) is always unity.

3

Let f be a continuous function on the interval [−a, a]. If f is an even function, i.e., f (−x) = f (x) Then, ˆa

ˆa f (x) dx = 2

−a

f (x) dx 0

6

1 Gas Kinetics—Basic Concepts

Fig. 1.1 The effect of temperature on the Maxwell-Boltzmann distribution of a given gas

The Maxwell-Boltzmann distribution function (Eq. (1.1)) can alternatively be → expressed in terms of speed C rather than velocity vector − v . We can rewrite Eq. (1.8) as ˚  ˚  m m 3/2 C2 d3 C exp − (1.12) F (C) d3 C = 2π κT 2κT  where d3 C = dCx dCy dCz and C = C2x + C2y + C2z . Recognizing the symmetry of F (C), the integration of Eq. (1.12) over the solid angle yields the following speed distribution function (Muller-Kirsten H.J.W., Basics of Statistical Physics, 2nd ed., World Scientific Publishing, 2013, Singapore): ξ (C)= 4π C2

 m 3/2 2 m e−( /2κT)C 2π κT

(1.13)

which is also the probability of finding a molecule with a speed ranging from C to C + dC, regardless of its direction of motion. Figure 1.1 depicts the effect of temperature on the Maxwell-Boltzmann velocity distribution of a specific gas. When the gas temperature is low, the molecules have less kinetic energy and travel at a slower rate. Thus, the speed distribution curve (curve A) has a narrower range. However, as the temperature rises, the distribution curves (curves B and C) widen out and shift to the right. Because of the high temperature, the gas molecules have more kinetic energy and so move faster. Consequently, the distributions have widened and shifted towards higher speed zones. Moreover, the meeting of all the curves at the origin indicates that there will always be some molecules that are at rest. For a given number of molecules, the area occupied by all the curves is the same.

1.3 The Maxwell-Boltzmann Distribution Law

7

Fig. 1.2 Dependence of Maxwell-Boltzmann distribution on the molecular mass

For a given gas at a given temperature, all molecules have the same kinetic energy consequently, the fraction of molecules traveling faster will increase only if the mass of the molecule m and thus the molecular weight Mw of the gas decrease. Figure 1.2 depicts the effects of molecular weight on the Maxwell-Boltzmann velocity distribution. The figure indicates that the molecules of heavier gases travel at a slower rate, whereas the molecules of lighter gases move at a faster rate. Therefore, for heavy gases, the speed distribution curves have a narrow range, but for lighter gases, they spread out. 1 mv2 ; 2

1.3.1 The Characteristic Molecular Speeds The Maxwell-Boltzmann distribution function is fundamental to the kinetic theory of gases. It characterizes the distribution of molecular speeds at a given temperature. This function can be used to calculate the root mean square (rms) speed vrms , average speed vavg , and the most probable speed vmp . In the following sections, a brief description of these characteristic speeds will be provided.

1.3.1.1

The Root Mean Square Speed

The measured instantaneous velocity for all gas molecules at a given temperature provides a wide range of values ranging from zero to very high velocities. However, in addition to having velocities at these extreme ends, majority of molecules will

8

1 Gas Kinetics—Basic Concepts

have velocities that lie between a more or less clearly defined range. One would be interested in calculating the average of all of these velocities; nevertheless, we must proceed with caution. Because molecules in a gas are always in random thermal motion, there will be the same number of molecules traveling in either direction. Clearly, the normal average of these velocity vectors will be zero, because two velocities with opposite signs cancel each other out. Because we are more interested in the magnitude of the velocities than their directions, the average velocity should be calculated in a somewhat different manner. The best method is to first calculate the average of the squares of the molecular velocities and then take the square root of the resultant value. The resulting   amount is known as the root mean square (rms) speed, indicated by vrms or v2 ,

vrms

   ΣC2 = v2 = N

(1.14)

where  is the temporal average or ensemble average, and N is the number of molecules in the gas. The equation for root mean square speed derived from the Maxwell-Boltzmann distribution function is

vrms

⎡∞ ⎤ 21 ˆ   = v2 = ⎣ C2 ξ (C)dC⎦

(1.15)

0

Substituting Eq. (1.13) into the vrms equation yields ˆ∞ 2 vrms

=

4π C2 0

 m 3/2 2 m C2 e−( /2κT)C dC 2π κT

 m 3/2 ˆ∞ 2 m = 4π C4 e−( /2κT)C dC 2π κT 0

Because the above integral has the following form: ˆ∞ 0

  3  π 1/2 α 4 exp −k  α 2 dα = 8 k 5

(1.16)

1.3 The Maxwell-Boltzmann Distribution Law

9

Thus, 2 vrms

 5  m 3/2 3  π 1/2 3  m 3/2 2κT /2 = 4π = 2π κT 8 (m/2κT)5 2 2κT m  3κT vrms = m

(1.17)

Equation (1.17) expresses the root mean square speed of the gas molecules. The average or mean translational kinetic energy of a molecule can thus be represented as Etrans =

1 2 3 mvrms = κT 2 2

(1.18)

This demonstrates that the average or mean kinetic energy of a molecule is proportional to the absolute temperature of the gas. Example 1.1 From the concepts of gas kinetics, calculate the root mean square speed of a nitrogen molecule at room temperature. The mass of a nitrogen molecule is about 4.68 × 10−26 kg. Solution Given, T = 20 ◦ C + 273 = 293 K mN2 = 4.68 × 10−26 kg Therefore from Eq. (1.17), we have  vrms =

 3κT = m

3 × 1.38 × 1023 × 293 = 509.11 m/s 4.68 × 10−26

Example 1.2 The escape velocity on the moon is about 2.37 × 103 m/s. At certain spots on the moon, the temperature of surfacematerial reaches 500 K. Suppose the molecule of hydrogen mH2 = 3.33 × 10−27 kg collides with the surface of the moon and attains the speed equal to the root mean square speed at this temperature. What would happen to this molecule? Solution Given, ve =2.37 × 103 m/s

T = 500 K

κ = 1.38 × 10−23 J/K

mH2 = 3.33 × 10−27 kg

10

1 Gas Kinetics—Basic Concepts

The root mean square speed is  vrms = =



3κT m 3 × 1.38 × 10−23 × 500 3.33 × 10−27

= 2.49 × 103 m/s Since vrms > ve , the hydrogen molecule will escape from the moon’s surface. It is worth noting that the heavier molecules are not easily capable of reaching this speed, but the majority of them are, and, as a result, there is no atmosphere on the moon. 238 Example 1.3 Fragments of U92 atoms, which have encountered nuclear fission, have an average kinetic energy of 1.1 × 10−11 J. What would be the estimated temperature of a gas made up of such fragments of nuclear fission?

Solution Given, Etrans = 1.1 × 10−11 J The average translational kinetic energy of a molecule is given by 3 κT 2 2Etrans T= 3κ 2 × 1.1 × 10−11 = 3 × 1.38 × 10−23 = 5.314 × 1011 K Etrans =

(1.18)

Thus, the temperature of the gas consisting of fission fragments is 5.314×1011 K.

1.3.1.2

The Average Speed

The average or mean speed is the sum of the speeds of all the molecules divided by the number of molecules. vavg =

Σ|C| N

(1.19)

where N is the number of molecules in the gas. The average speed can be calculated using the Maxwell-Boltzmann velocity distribution as follows:

1.3 The Maxwell-Boltzmann Distribution Law

11

ˆ∞ vavg = v =

Cξ (C)dC

(1.20)

0

Substituting ξ (C) from Eq. (1.13) into the above equation, we get vavg = 4π

 m 3/2 ˆ∞ 2 m C3 e−( /2κT)C dC 2π κT

(1.21)

0

Because the above integral has the following form: ˆ∞

  1 α 3 exp −k  α 2 dα = 2 2k

0

Thus,  m 3/2 1  m 2 2π κT 2 2κT  −1/2 1 4×4 m (κT) /2 = 3/2 1/2 2 ×2 π

vavg = 4π

(1.22)

Therefore, the average speed is  vavg ≈

8κT πm

(1.23)

Furthermore, we can observe from Eqs. (1.17) and (1.23) that 2 vrms = 1.12vavg

1.3.1.3

(1.24)

The Most Probable Speed

The molecular speed, which corresponds to the highest point on the MaxwellBoltzmann distribution curve, is the most probable speed. It is possible that only a small percentage of molecules have this speed, yet it is far more likely than any other speed. Mathematically, the most probable speed is derived by equating the first derivative of ξ (C) to zero. Let us rewrite Eq. (1.13) as ξ (C)= βC2 e−αC where β = 4π



m 2πκT

3/2

and α =

m . 2κT

2

12

1 Gas Kinetics—Basic Concepts

The differentiation of ξ (C) with respect to C gives ξ  (C) =

 dξ (C) 2 2 =β (2C) e−αC + C2 (−α) 2Ce−αC = 0 dC  2  e−αC 1 − αC2 = 0

(1.25) (1.26)

Since e−αC = 0, hence 2



 1 − αC2 = 0 1 C= √ α

(1.27) (1.28)

Differentiating ξ  (C) with respect to C again and substituting in the expression for C yield ξ  (C)|C= √1 = −4 α

  β 0). In this instance, because the molecules have enough energy to overcome the intermolecular interactions, the substance transforms into a gas.

1.7.1 The Equipartition of Energy and the Degrees of Freedom In gas kinetics, the distribution of internal energy among different molecules is governed by a law called the equipartition of energy. This law assumes that in a system in thermal equilibrium, an equal amount of energy is associated with each independent energy state (degree of freedom), and that the energy is 21 κT6 (per molecule) or 21 RT7 (per mole). Thus, if α is the number of degrees of freedom and N is the number of molecules in a gas, the total internal energy of the gas will be U=

α NκT 2

(1.62)

In this case, the degree of freedom is essentially an independent part of the energy of a gas molecule, and in order to count the degrees of freedom, we must first model the molecule. Consider the atoms to be point masses, and the bonds that connect them to be stiff springs. Because each gas molecule is free to move in any direction, the molecules of a monoatomic gas can spin around any axis. The rotational motions of diatomic and polyatomic gas molecules, on the other hand, are determined by their chemical configurations. Furthermore, intermolecular vibrational movements may arise as a result of intermolecular interactions operating between the atoms in a molecule. Internal energy may be increased further by the internal structure of an individual atom. Indeed, each of these modes of motion, whether translational, rotational, or vibrational, is a degree of freedom and hence constitutes an independent part of 6 7

Here, κ is the Boltzmann constant, which is equal to 1.38 × 10−23 JK−1 . In which, R is the specific gas constant.

1.7 Concept of Internal Energy

23

a molecule’s energy. Therefore, the total energy (or total internal energy) of a gas molecule is the sum of its translational, rotational, and vibrational kinetic energies. Etotal = Etrans + Erot + Evib

(1.63)

Because a gas molecule has three degrees of freedom in translational motion (along the x-, y- and z-axes), the translational kinetic energy of a molecule can be described as Etrans =

1 2 1 2 1 2 mv + mv + mv 2 x 2 y 2 z

(1.64)

It has three independent translational kinetic energies in the x-, y-, and zdirections. These three quadratic terms are essentially the three translational degrees of freedom. In rotating motion, a gas molecule may have three degrees of freedom along three coordinate axes. Hence, the rotational kinetic energy of a gas molecule can be represented as Erot =

1 2 1 2 1 2 Iω + Iω + Iω 2 1 2 2 2 3

(1.65)

Moreover, when considering vibrational motions, the potential energy should be considered in addition to the kinetic energy, as in a harmonic oscillator, i.e., the term  2 1 kx2 due to position should be added to 21 m dx . Therefore, the vibrational kinetic 2 dt energy of a gas molecule becomes Evib =

 2 dx 1 1 m + kx2 2 dt 2

(1.66)

where x denotes the vibrational coordinate and k denotes the force constant. It is worth noting that the potential energy term here provides an additional degree of freedom. The fact that vibrational motions are highly quantized means that at room temperature, most molecules remain in their ground state, making higher levels thermally inaccessible. For this reason, the equipartition contributions from higher vibrational states should be considered only at extremely high temperatures. From Eqs. (1.64)–(1.66), it is clear that any type of motion that contributes a quadratic term to the internal energy in some generalized coordinate is a degree of freedom, and the law of energy equipartition governs the distribution of total energy of a gas molecule among its degrees of freedom. The equal distribution of total energy among all energetically available modes of motion is a positive phenomenon since it attempts to maximize the entropy of the system.

24

1.7.1.1

1 Gas Kinetics—Basic Concepts

Internal Energy of a Monoatomic Gas

Because a rigid monoatomic gas molecule has no internal structure and is regarded similarly to point masses, there are no conceivable contributions to the energy owing to rotational motions of the molecules for a monoatomic ideal gas (such as helium, neon, or argon). In monoatomic gases, all three translational motions of the molecules contribute to energy. Thus, substituting α = 3 into Eq. (1.62) yields the total internal energy of n moles (or N number of molecules) of an ideal monoatomic gas (one atom per molecule). U=

3 3 NκT = nRu T 2 2

(1.67)

It means that the internal energy of n moles of an ideal monoatomic gas is equal to the average kinetic energy per molecule multiplied by the number of molecules, N. Equation (1.67) is known as the thermal equation of state for a monoatomic gas.

1.7.1.2

Internal Energy of a Diatomic Gas

If the gas molecule has more than one atom, the rotational kinetic energy, in addition to the three translational directions, contributes to the total energy of the molecule, but only for rotations about two of the three perpendicular axes. That is, a rigid diatomic gas molecule will have five degrees of freedom (three translational and two rotational). Thus, by substituting α = 5 into Eq. (1.62), the expression for total internal energy of n moles (or N number of molecules) of a diatomic gas can be obtained. U=

5 5 NκT = nRu T 2 2

(1.68)

This is the thermal equation of state for a diatomic gas. Equation (1.68) is simply an approximation and only applies at intermediate temperatures. When the temperature is exceedingly low, only the translational motions of the molecules contribute to the total energy. At higher temperatures, however, molecular vibrations provide two additional inputs (kinetic and potential energies). It should be noted that the internal energy of a diatomic gas at a given temperature is greater than that of a monoatomic gas, although it is always a function of temperature for an ideal gas. The internal energy of a real gas also relies upon generally on the temperature, but it additionally depends somewhat at the pressure and volume. At extremely low pressures, all real gases approach the ideal state. Because the molecules are far away at low pressures, they do not interact with each other on a regular basis.

1.8 Specific Heats of an Ideal Gas

25

Fig. 1.4 An ideal gas enclosed in a cylinder-piston assembly

1.8 Specific Heats of an Ideal Gas The specific heat of a substance is defined as the amount of heat necessary to increase its temperature by one degree for a unit mass of the substance. Temperature is measured on either the Celsius or Kelvin scales, but most commonly on the Kelvin scale. Because the mass of a substance is frequently expressed in moles, the specific heat is referred to as molar-specific heat. The mass in SI units is typically expressed in J grams or kilograms. Hence, the most widely used SI units of specific heat are (mol.K) J and (kg.K) . Because a slight change in temperature generates a negligible change in pressure and volume in both solids and liquids, the external work of solids and liquids is negligibly small. As a result, all of the heat provided is mostly used to raise the temperature of solids and liquids. For this reason, both solids and liquids have the same specific heat value. When heat is introduced to the gas, a slight change in temperature generates a significant change in both pressure and volume. Thus, a gas can have any specific heat value within the range (0, ∞), and in order to predict a specific value, either the pressure or the volume must be held constant. Consequently, the gases have two specific heat values: specific heat at constant pressure and specific heat at constant volume. Consider a specific amount of ideal gas inside a cylinder-piston arrangement, as illustrated in Fig. 1.4. If dQ amount of heat is added to the gas (system), then the first law of thermodynamics states that a portion of dQ increases its internal energy by

26

1 Gas Kinetics—Basic Concepts

dU, while the remaining amount does dW of work on the surroundings. dQ = dU + dW

(1.69)

Assume that heat is added to the gas in a constant volume. This signifies that the gas is not permitted to expand and so cannot perform any work, i.e., dW = 0. All of the heat delivered will increase the internal energy of the gas. Thus, Eq. (1.69) is reduced to dQ = dU

(1.70)

Because the increase in internal energy of an ideal gas is solely a function of temperature, the molar-specific heat at constant volume can be expressed as  C∀ =

dQ dT





∀

=

dU dT

 (1.71) ∀

dEint = C∀ dT

(1.72)

Hence, the first law of thermodynamics (Eq. (1.69)) has become dQ = C∀ dT + dW

(1.73)

Now, if heat is added to the gas at constant pressure and the volume of the gas can be modified, the thermal energy supplied to the gas will be used to perform external work as well as change its internal energy. At constant pressure, the work done by the gas on its surroundings can be expressed as dW=pd∀

(1.74)

dQ = C∀ dT + pd∀

(1.75)

Thus, Eq. (1.73) becomes

The molar-specific heat at constant pressure is denoted as  Cp =

dQ dT

 (1.76) p

We get from Eq. (1.75) Cp =

C∀ dT + pd∀ dT

The ideal gas law (Eq. (1.56)) for one mole of gas equals

(1.77)

1.8 Specific Heats of an Ideal Gas

27

p∀ = Ru T

(1.78)

Differentiating above at constant pressure, we find pd∀ = Ru dT

(1.79)

Introducing this equation into Eq. (1.77), we get Cp = C∀ + Ru

(1.80)

It is known as Mayer’s relation. Note that Cp > C∀

(1.81)

That is, under constant pressure, the molar-specific heat is always greater than the molar-specific heat at constant volume. This is because the temperature increase at constant pressure is smaller for the same amount of heat provided than when the volume of the gas is held constant. To put it another way, for the same temperature increase, the amount of heat added required at constant pressure is always more than at constant volume. The ratio of the specific heat at constant pressure to the specific heat at constant volume is commonly denoted by γ . Thus, γ =

Cp C∀

(1.82)

1.8.1 The Ratio of Specific Heats for a Monoatomic Gas Recall that the average or mean internal energy per mole of a monoatomic gas is U=

3 Ru T 2

(1.67)

However, for the ideal gas, Ei = C∀ T. So, 3 Ru 2

(1.83)

3 5 Ru + Ru = Ru 2 2

(1.84)

C∀ = From Mayer’s relation, we get Cp = Thus, the ratio of specific heat is

28

1 Gas Kinetics—Basic Concepts

γ =

5 Ru Cp 5 = 23 = = 1.67 C∀ 3 R 2 u

(1.85)

The ratio of specific heats of a monoatomic gas is thus 1.67.

1.8.2 The Ratio of Specific Heats for a Diatomic Gas The average internal energy per mole of a diatomic gas is 5 Ru T 2

(1.68)

C∀ =

5 Ru 2

(1.86)

Cp =

7 Ru 2

(1.87)

7 Ru Cp 7 = 25 = = 1.4 C∀ 5 R 2 u

(1.88)

U= Also, for an ideal diatomic gas

and from Mayer’s relation

Thus, the ratio of the specific heat is γ =

That is, the ratio of specific heat of a diatomic gas is 1.4.

1.8.3 The Ratio of Specific Heats for a Polyatomic Gas The average internal energy per mole of a polyatomic ideal gas with alpha degrees of freedom is given by Eq. (1.62). Hence, the specific heat at constant volume is C∀ =

α Ru 2

(1.89)

From Mayer’s relation Cp = C∀ + Ru  α + 1 Ru = 2

(1.90) (1.91)

1.8 Specific Heats of an Ideal Gas

29

Therefore, the ratio of specific heats is Cp γ = C∀ α  + 1 Ru 2 = α Ru  2  2 = 1+ α

(1.92) (1.93) (1.94)

A rigid polyatomic gas molecule has six degrees of freedom at room temperature (α = 6). Thus, the specific heat ratio is 4 3 = 1.33

γ =

(1.95) (1.96)

1.9 The Continuum Hypothesis Any substance that appears to be continuous is actually made up of a vast number of molecules that are constantly moving and colliding. The motion of substances is addressed in the kinetic theory of gases (also known as statistical mechanics) by defining the governing equations for each molecule (or statistical group) with respect to particular specified initial conditions. Although this approach has benefits in its own right, it is unsuitable for practical use. Indeed, engineering challenges are not concerned with investigating the motion of individual molecules, but rather with the overall behavior of the fluid as a continuous material. Although the assumption of continuous fluid (continuum) is merely a convenient assumption, it turns out to be a valid approach to many real situations using only macroscopic information. It is also inherent in this assumption that the fluid element can be subdivided indefinitely into smaller and smaller particles, with the properties of these subdivided particles remaining constant. The validity of the continuum assumption, on the other hand, is based on the fact that statistical averages are relevant whenever the smallest fluid element has a sufficient number of molecules. Moreover, changes in properties are expected to be gradual over time. Consequently, any property can be defined as continuous function of both geometrical positions in space and time. Theoretical fluid dynamics and gas dynamics are areas of continuum mechanics that do not seek to describe the molecular structure of the media or the motion of individual molecules. Rather, the continuum hypothesis simulates a substance dense enough to allow a meaningful specification of properties (macroscopic quantities) to be averaged over a very large number of molecules. It should be noted that these properties can also be estimated using Boltzmann’s gas kinetics equations, which assume fluid flow as the motion of the aggregation of molecules (microscopic view).

30

1 Gas Kinetics—Basic Concepts

However, the value of the properties will be essentially the same in any case. Thus, the continuum hypothesis simplifies the difficult or nearly impossible process of dealing with a huge number of molecules at the same time, instead allowing for a significant meaning of the macroscopic quantity. In other words, rather than looking into the instantaneous states of a large number of molecules, we must focus on a few properties that describe the overall behavior of matter. In fluids, for example, the important properties are the pressure p, the temperature T, the density ρ, the velocity v, the shear stress τ , the coefficient of viscosity μ, the internal energy U, the entropy s, and the coefficient of thermal conductivity κ.

1.9.1 The Characteristic Molecular Length and the Knudsen Number Consider fluid flow past a rigid body, such as the sphere as depicted in Fig. 1.5, and try to determine the density of the flowing fluid. For a small spherical volume δ∀, containing the fluid of mass δm, the density ρ at a point P is defined as ρ = lim

δ∀→δ∀0

δm δ∀

(1.97)

Note that if there were no variations in density throughout the flow field (i.e., fluid flow is not compressible), the control volume δ∀ could be chosen arbitrarily. That is, for incompressible flows, the value of ρ, as predicted by Eq. (1.97), will be essentially the same at all points in the flow field. However, all fluids of practical interest are compressible in nature and thus experience a change in density as they move. For example, if the compressible fluid flows past the sphere, as sketched in Fig. 1.5, it experiences deceleration near the upstream side of the sphere and thus undergoes a process of compression. As the flow continues to progress around the sphere, there is a process of expansion due to fluid acceleration. This is followed by a second compression process that occurs as the fluid decelerates close to the back of the sphere. The characteristic length scale corresponding to these compression and expansion processes coincides with the diameter D of the sphere. In addition, let the point P at which the flow density is to be observed be indicated → by the vector radius − r . This point must then be enclosed by an elemental volume δ∀ (shown by R), whose length scale  is smaller than D, i.e.,   D. →characteristic  The density at point P − r , t can be written in the form →  δm ρ − r , t = lim →0 δ∀ where  → 0 is when δ∀ is made smaller and smaller. The variation of respect to  (or δ∀) is shown in Fig. 1.5.

(1.98) δm δ∀

with

1.9 The Continuum Hypothesis

31

Fig. 1.5 Concept of density

When the characteristics length scale  of the elemental volume R is comparable to depends not only on δ∀, but also on its location. the body scale D ( ∼ D), then δm δ∀ If the elemental volume is chosen upstream of the sphere (such as the volume R depicted by curve C1 in Fig. 1.5), then Eq. (1.98) will actually overestimate the true density at P; however, if the elemental volume is selected in the region where the fluid experiences the expansion process, then Eq. (1.98) will underestimate the density (the volume R shown by curve C2 ). But, if the elemental volume is sufficiently dense then regardless of its location, Eq. (1.98) will always measure the true density. This region is called the continuum region. However, if δ∀ is so small that the value of δm is

32

1 Gas Kinetics—Basic Concepts

strongly dependent on the number of molecules existing in R at any time t, the fluctuations will also be a function of time. Therefore, in order to define a constant density value, the volume of the fluid must have a sufficient number of molecules at any time t. Under standard pressure and temperature conditions, δ∀ is as small as δ∀0 = 10−9 mm3 contains about 3 × 107 molecules, which is assumed to be sufficient to have the average density ρ constant over δ∀. But for δ∀ < δ∀0 , the volume of the fluid is small enough that the length characteristics of the scale  are comparable to the mean free path λ. Here, λ is defined as the average distance traveled by a molecule before it collided with another molecule. The continuum approach is therefore valid only if the characteristics of the length scale  are small compared to the body scale D but much larger than λ. That is, λ<  < D

(1.99)

In this case, the molecules collide the surface of the body so often that the body cannot distinguish between the individual molecular collisions and experiences the fluid as a continuous medium. This type of flow is called the continuum flow. The Knudsen number is defined as Kn =

λ 

(1.100)

and it is immediately clear from Eq. (1.99) that Kn must be less than one. On the basis of the Knudsen number, the flow regimes can be classified as follows: 1. 2. 3. 4.

Continuum flow regime (Kn < 0.01). Slip flow regime (0.01 < Kn < 0.1). Transition flow regime (0.1 < Kn < 3). Low-density and free molecular flow regime (Kn > 3). A gas is made up of a large number of discrete atoms and molecules which are in continuous random motion with frequent collisions. However, the properties such as pressure, temperature, density, viscosity, etc. are based on overall (macroscopic) behavior where the continuum hypothesis holds. In some situations, where the mean free path λ is large enough to be comparable in the same order of magnitude as that of the characteristic dimension l, the flow is termed as low density (rarefied) flow. Under extreme conditions, when λ is much larger than l, the flows are defined as the free molecular flows. Such flows essentially occur in flight at very high altitudes (beyond 100 km) and some laboratory devices like electron beams.

1.9.2 Limitations of the Continuum Hypothesis At higher altitudes, the continuum considerations do not hold true due to the lower number of molecules present in an arbitrary elemental volume δ∀. These molecules get further and further apart and begin to have their own identities, eventually leading

1.9 The Continuum Hypothesis

33

to a large λ with respect to . This is the other extreme where the mean free path of the molecules is of the same order or greater than the characteristic dimension of the problem. In this situation, molecular collisions with the body surface occur only intermittently and each molecular impact can be clearly felt on the surface. Since the density at these altitudes is no longer constant, the continuum hypothesis is essentially broken down. This type of flow is called free molecular flow or rarefied flow, where the flow is analyzed at the microscopic level. Such a flow is encountered by the space shuttle at the time of re-entry into the outer layer of the earth’s atmosphere. Here, the air density is so low that the mean free path becomes the order of the length of the shuttle. The Knudsen number, Kn, can be expressed in terms of the Mach number, M, and the Reynolds number, Re, as follows: √ M Kn=1.26 γ Re

(1.101)

M where γ denotes the ratio of the specific heats of a gas. When Re ≥ 1, the flow is called the rarefied gas flow, which is indeed a low Reynolds number flow. However, when both Reynolds number and Mach number are large, the Knudsen number is given by

M Kn = √ Re

(1.102)

Remark At sea level, we feel the warmth when several gas molecules collide with our skin simultaneously, and the skin is unable to distinguish between the individual collisions; the air behaves as a continuous medium. We may feel individual molecular impacts as we travel through space, where the continuum hypothesis essentially breaks down; the air no longer behaves like a continuous medium. When successive collisions are long, we feel cold even though all of a molecule’s kinetic energy is converted into heat after hitting with our skin. Example 1.7 A small rocket which probes the atmosphere has a length of 3 m. It is shot vertically upwards through the atmosphere at an average speed of 1000 m/s. Consider the flow past the rocket at this average speed at altitudes of 30,000 and 80,000 m. Can the airflow over the rocket be assumed to be continuous at these two altitudes? At an altitude of 30,000 m, the air has a temperature, pressure, and viscosity of −55 ◦ C, 120 kPa, and 1.5 × 10−5 kg/(m.s), respectively, while at an altitude of 80,000 m, the air has a temperature, pressure, and viscosity of −34 ◦ C, 0.013 Pa, and 1.7 × 10−5 kg/(m.s), respectively. Solution Given, l = 3m At an altitude of 30,000 m

v = 1000 m/s

34

1 Gas Kinetics—Basic Concepts

T = − 55 ◦ C + 273 = 218 K p =120 kPa μ = 1.5 × 10−5 kg/(m.s) Thus, the speed of sound is #

γ RT √ = 1.4 × 287 × 218 m = 296 s

a=

and the corresponding Mach number is 1000 296 = 3.38

M=

In addition, the Reynolds number is ρvl Re = μ  =

120×103 287×218



× 1000 × 3

1.5 × 10−5 = 3.835 × 108

The Knudsen number therefore becomes   M √ Kn=1.26 γ Re  √ = 1.26 × 1.4 ×

3.38 3.835 × 108



= 1.313 × 10−8  0.01 Hence, the airflow over the rocket at an altitude of 30,000 m can therefore be presumed to be continuous. At an altitude of 80,000 m, we have T = − 34 ◦ C + 273 = 239 K p = 0.013 Pa kg μ = 1.7 × 10−5 (m.s)

1.9 The Continuum Hypothesis

35

Hence, the speed of sound is #

γ RT √ = 1.4 × 287 × 239

a=

= 309.88 m/s The Mach number is therefore 1000 309.88 = 3.23

M=

Besides, the Reynolds number is

=

ρvl Re = μ  0.013  × 1000 × 3 287×239 1.7 × 10−5 = 33.44

Thus, the Knudsen number will be √ Kn = 1.26 γ = 1.26 ×



M Re



√ 1.4 ×



3.23 33.44



= 0.144 > 0.01 It indicates that the continuum theory is no longer valid at an altitude of 80,000 m and therefore the flow over the rocket cannot be considered to be continuous. In fact, the airflow over the rocket is classified under the transitional flow regime.

1.10 The Perfect Substance The materials encountered in a variety of applications undergo unique physical and chemical changes. The actual mathematical description of these materials and accompanying processes lead to mathematical equations that are either impossible or exceedingly difficult to solve. Thus, in order to facilitate mathematical analysis, the actual material is replaced by an ideal substance that obeys some simple laws and has a simple mathematical description.

36

1 Gas Kinetics—Basic Concepts

Fig. 1.6 Surface forces acting on a small cubic fluid element

Despite the fact that the perfect substance is an idealized situation, its analysis yields derived equations that can be further changed by inserting correction factors from experimental data to be applied to the actual gas. The perfect substance is physically and chemically homogeneous across its entire mass. It is also isotropic, which means that its elastic properties are the same in all directions. Moreover, the stress generated in the ideal substance is solely determined by the magnitude of the strains. The elastic properties of the perfect substance are defined by two elastic moduli: the rigidity or shear modulus N and the bulk modulus B. Consider a small cubic element of unity dimensions as illustrated in Fig. 1.6a. If the shear stress τ is applied to the upper surface so that the upper surface DC has moved a distance L with respect to the lower surface AB. Thus, the shear modulus or rigidity is defined as N= = =

shear stress shear strain τ L L

τ tan α

(1.103)

But in the case of small angular deformations, tan α ∼ =α N=

τ α

(1.104)

Consider again the unit cube of a perfect substance, which is subjected to the hydrostatic pressure p normal to the cube surfaces as shown in Fig. 1.6b. If the hydrostatic pressure is increased from p to p + dp, the volume of the cube is reduced from ∀ to ∀ − d∀. Thus, the fractional change in the volume of the cube will be − d∀ ∀ (the negative sign shows a decrease in volume with an increase in pressure). The

1.10 The Perfect Substance

37

bulk modulus of elasticity is therefore defined as dp K =  d∀  −∀   dp = −∀ d∀

(1.105) (1.106)

The reciprocal of bulk modulus is called the compressibility. Thus, κ=− Since ρ =

1 ∀

  1 d∀ ∀ dp

(1.107)

and therefore Eq. (1.107) can also be written as   1 dρ κ= ρ dp

(1.108)

The compression depends on the actual process by which it is carried out. Consequently, for the isothermal compression process, the bulk modulus is called the isothermal bulk modulus KT while for the isentropic compression process it is known as the isentropic bulk modulus Ks . That is,  dp KT = ρ dρ  T dp Ks = ρ dρ s 

(1.109) (1.110)

1.10.1 The Perfect Solid The perfect solid is a substance that has both non-zero rigidity (N = 0) and non-zero bulk modulus (K = 0). When subjected to external stress, the perfect solid creates internal forces owing to strain to counterbalance the external forces. Because the perfect solid can resist shear stress, the resulting stress at any location on an arbitrary plane drawn through that point can be in any direction with respect to that plane. Moreover, the perfect solid obeys Hooke’s law, which states that stress is directly proportional to strain within the elastic limit.

38

1 Gas Kinetics—Basic Concepts

1.10.2 The Perfect Fluid The perfect fluid is a substance with a non-zero bulk modulus (K = 0) but a zero shear modulus (N = 0). Perfect fluid can only withstand normal force (i.e., pressure force) and cannot withstand shear stress. Thus, the resulting force on any arbitrary plane drawn in fluid is always normal to the plane. The magnitude of the pressure force exerted at a location, on the other hand, is independent of the orientation of the plane traced across that point. Therefore, the perfect fluid is defined as the perfect substance that can only transmit pressure force (which is the same in all directions) and cannot sustain shear force. The resulting force at any point on any arbitrary plane drawn in perfect fluid is always normal to the plane. The magnitude of the pressure force is thus independent of the orientation of the plane. Perfect fluids are further subdivided into ideal liquids and perfect gases. The Perfect Liquid The perfect liquid is the perfect fluid that is incompressible. The bulk modulus of this substance is theoretically infinite, i.e.,  K = −∀

dp d∀

 =∞

(1.111)

and therefore the compressibility is close to zero. κ=−

  1 d∀ =0 ∀ dp

(1.112)

In reality, no real liquid is totally incompressible, but for most liquids, the value of K is quite large, thus it is acceptable to presume that the liquids are incompressible, especially when the pressure changes involved are small. Additionally, given a limited temperature range, the value of K remains nearly constant before and after the compression process. The Perfect Gas In terms of elastic moduli, a perfect gas is defined as a fluid with zero rigidity but an isothermal bulk modulus equal to its static pressure p, i.e., 

dp KT = −∀ d∀

 =p

(1.113)

T

The pressure p and the volume ∀ of an isentropic change of state of a perfect gas are related as p∀γ = constant. Thus, the isentropic bulk modulus will be given as follows:

1.10 The Perfect Substance

39

 Ks = −∀

dp d∀

 = γp

(1.114)

s

The ratio of specific heat γ is assumed to be constant in Eq. (1.114). Also, from Eqs. (1.113) and (1.114), the ratio KKTs is Cp Ks γp =γ = = = constant KT p C∀

(1.115)

Thus, γ is also the ratio of the slopes of isentropic and isothermal process curves drawn in a p − ∀ diagram.

1.11 Transport Properties of Gases The kinetic theory of gases is concerned not only with gases that are in thermodynamic equilibrium, but also with gases that are not. In other words, it takes into account the molecular transport properties, such as viscosity, thermal conductivity, and the diffusion coefficient, which generally flow in such a way that an equilibrium condition is achieved.

1.11.1 Viscosity Absolute or dynamic viscosity is the molecular transport property by which the layers of the real fluid resist the shearing stress while in motion. This is in contrast to the motion of the ideal fluid, which cannot withstand shear stress while being inviscid. Viscosity basically describes the rate of linear momentum transported through a fluid. In gases, the viscosity arises due to molecular exchange of momentum between the adjacent fluid layers. From kinetic theory of gases, it can be shown that for a perfect gas the viscosity μ is given by μ=

1 ρλvavg 2

(1.116)

where ρ is the density of the gas, λ is the mean free path, and vavg is the mean molecular speed. In this case, if the density of the gas ρ decreases, the mean free path λ grows until the point where the product of ρ and λ remains constant. The viscosity μ is thus exactly proportional to vavg , which is entirely dependent on the gas temperature T. As a result, the viscosity of the ideal gas is exclusively determined by T and increases with it. The motion of a two-dimensional viscous fluid is governed by a relationship known as Newton’s law of viscosity, which states that the shearing stress τ operating

40

1 Gas Kinetics—Basic Concepts

between the fluid layers is proportional to the shearing strain rate, i.e., 

∂u ∂v + τ=μ ∂y ∂x

 (1.117)

where μ is dynamic viscosity, generally represented in N.s . In addition, the kinematic m2 viscosity is defined as the ratio of dynamic viscosity μ to gas density ρ. kinematic viscosity ν, i.e., ν = μρ . A Newtonian fluid is a real fluid that obeys Newton’s law of viscosity (Eq. (1.117)). The temperature has a large influence on the viscosity of the fluid, whereas pressure has a minor influence. In fact, the viscosity of gases and most liquids progressively increases with pressure, but the change in viscosity owing to pressure is negligibly small, and therefore pressure effects are usually neglected. The viscosity of the liquid reduces as the temperature rises. This dependency can be stated mathematically as  ln

μ μ0



 =a+b

T0 T



 +c

T0 T

2 (1.118)

where T0 denotes the reference temperature and μ0 denotes the reference viscosity at T0 . The viscosity of the liquid obviously varies inversely with temperature. In contrast, when the temperature rises, the viscosity of the gas rises. The following two relationships are commonly used to calculate the viscosity of a gas. ⎧  n ⎨ TT Power law μ 0 =   23   ⎩ T T0 +S μ0 Sutherland’s law T0 T+S

(1.119)

where μ0 is a known viscosity at a known absolute temperature T0 . For air, the exponent is n = 0.7 and S is the Sutherland constant, which is equivalent to 110 K. A more useful variant of Sutherland’s formula for air Sutherland’s formula is written as follows: ⎛ μ = 1.46 × 10−6 ⎝

⎞ 3 ( T2 ⎠ ; for 0.01 atm < pstatic < 100 atm; 0 K < T < 3000 K T + 111

(1.120)

It is clear from Eqs. (1.118) and (1.120) that increasing the temperature decreases the viscosity of the oil while increasing the viscosity of the gas. Why? Let us try to analyze this occurrence as follows. In the motion of liquids and gases, molecules are free to migrate from one layer to the next at varying speeds. A molecule must be accelerated when it moves from a lower velocity layer to a higher velocity layer. Similarly, a molecule moving from a higher velocity layer to a lower velocity layer must be slowed. Consequently, the net energy is carried by molecules as they scatter

1.11 Transport Properties of Gases

41

through the fluid layers, gradually contributing to shear stress between the layers. If the layers move at the same pace or the fluid is at rest, shear stresses would be insignificant. Furthermore, it is believed that the binding force of cohesion operates between the fluid molecules, and that shear stress must overcome these coherent forces in order for relative motion to occur. Thus, the total shear stress in the fluid is the sum of the two above-mentioned factors: shear stress owing to molecular diffusion and shear stress due to the overcoming of coherent forces between molecules. Shear stress in liquids is caused more by the overcoming of cohesive forces than by diffusion due to decreased molecular velocity. Shear stresses in gases, on the other hand, are predominantly induced by the transfer of momentum through the layers since the velocity of the gas molecules is much higher, but the cohesive forces are very small. Hence, as the temperature rises, the intermolecular spacing increases and the strength of the cohesive force diminishes. Thus, as temperature rises, the viscosity of the liquid lowers. In the case of a gas, viscosity is primarily related to momentum transfer. As the temperature rises, additional kinetic energy is imparted to gas molecules, increasing the net change in momentum between the layers. As a result, the viscosity of the gas increases. Sutherland’s relationship closely represents the variation of μ with air temperature over a wide temperature range, but the performance of the equation representing the variation of μ with T for many gases does not determine the validity of Sutherland’s gas molecular model. In other words, the equation is insufficient to characterize the center of the molecule as a sphere or to consider solely firstorder molecular attraction. The increase in exponential of μ with T, when compared to the non-attractive rigid sphere, must be explained partly by the softness of the repulsive field at a small distance and partly by the attractive force of the first-order effect. Therefore, Sutherland’s relationship may only be viewed as a rudimentary interpolation method for limited temperatures. At temperatures less than 3000 K, air viscosity is independent of pressure. The collision cross section of atom-atom and atom-molecule collisions should take into account the transport properties at high air temperatures.

1.11.2 Thermal Conductivity Thermal conduction is a method of heat transfer in a material that does not require the material to move as a whole. Rather, it occurs as a result of the exchange of energy between the molecules that make up matter. The fundamental law governing the mechanism of heat conduction is known as Fourier’s law of heat conduction. It asserts that the rate of heat transfer per unit area in an isotropic8 medium is directly proportional to the temperature gradient. q˙ = −k∇T 8

(1.121)

When the physical and mechanical properties of a substance are same in all directions, the substance is known as isotropic, e.g., metal and glass.

42

1 Gas Kinetics—Basic Concepts

where the proportionality constant k is referred to as thermal conductivity. The minus sign indicates that the temperature is falling in the direction of heat transfer and that the temperature gradient is a negative quantity. Thermal conductivity in gases is caused by the exchange of energy between gas molecules, just as viscosity is caused by momentum transport and diffusion through mass transfer. Therefore, all of these transport properties are determined by the mean free path of the gas molecules. According to the kinetic theory of gases, the thermal conductivity of a gas is determined by k=

5 ρvavg λC∀ 4 5 ≈ μC∀ 2

(1.122) (1.123)

where ρ denotes gas density, vavg denotes average molecular speed, λ denotes mean free path, and C∀ denotes specific heat at constant volume.

1.11.3 Self-diffusion Consider the gas contained within the container and examine the motion of the given molecule. We can see that the molecule travels in straight lines, disrupted only by collisions with other molecules and the wall of the container. As a result, the direction of motion of the stated molecule is altered and, after a few collisions, the molecule completely forgets its initial velocity. Therefore, the specified molecule can be regarded to have de-correlated from its initial state. It begins with a random walk, in which the molecule picks a direction at random and begins traveling in that direction. This is referred to as gas self-diffusion. If the movements of several other known molecules are detected due to random walking, we can see that they all flow into the system and interact with each other in a homogeneous manner. The rate of completion of this process is expressed in terms of the self-diffusion coefficient D, which is defined according to the kinetic theory of gases as D=

μ 1 λvavg = 3 ρ

where μ is the dynamic viscosity and ρ is the density of gas.

(1.124)

1.12 Real Fluid Flows

43

1.12 Real Fluid Flows The perfect fluid discussed in the previous section is an idealization that was supposed to be incompressible and inviscid. Mechanical energy is conserved in the absence of viscosity, which is the essence of the Bernoulli equation. However, in order to comprehend one or more important properties of a fluid, such as its ability to drag objects, we must go beyond the inviscid assumption and add the effects of viscosity into our description of fluid flows. This section discusses the general characteristics of real fluids.

1.12.1 Comparison of the Perfect and Real Fluid Flow Properties A perfect fluid is incompressible and viscous, whereas a real fluid can be both compressible and viscous. Shear stresses are generated in real fluids whenever there is relative motion between fluid layers and continue until such motion ceases. If a real fluid is at rest, there will clearly be no shear stress between its layers, which is analogous to the condition of a stationary ideal fluid. Thus, neither perfect nor real fluids can sustain shear stress at rest. Even when there is a relative movement between the fluid and the body, the body does not experience drag when submerged in the perfect fluid. For example, if the infinite circular cylinder is held in a steady flow of velocity U∞ and density ρ, 2 the forces experienced by the body are inertia forces equivalent to ρU2∞ and static pressure forces p. The Bernoulli equation connects these two forms of forces.  U∞ 2 = constant 2   U∞ 2 =0 dp + ρd 2 

p+ρ

(1.125) (1.126)

In other words, changes in inertia force are balanced by changes in static pressure along the streamline. From a mathematical perspective, the inviscid and incompressible flow of fluid is known as potential flow, implying that the flow field is irrotational (free of vorticity). The velocity field is described by potential flow as the gradient of a scalar function termed the velocity potential potential for an   φ. The velocity incompressible flow solves the Laplace equation ∇ 2 φ = 0 . Potential flows, on the other hand, have been utilized to characterize compressible flows. Consider a circular cylinder in a cross-flow arrangement in an incompressible and inviscid freestream, as shown in Fig. 1.7a. According to potential flow theory, the drag experienced by a cylinder when a perfect fluid flows past it is zero. This is referred to as d’Alembert’s paradox. If a viscous flow passes through the cylinder, no

44

1 Gas Kinetics—Basic Concepts

Fig. 1.7 Inviscid and viscous flows past a circular cylinder a without trailing wake b with trailing wake

1.12 Real Fluid Flows

45

matter how small the fluid viscosity, vorticity will always appear in a thin boundary layer proximal to the cylinder. A boundary layer separation generates a trailing wake behind the cylinder somewhere on the downstream side of the cylinder (Fig. 1.7b).

Concluding Remarks In this chapter, we have pointed out the fundamental postulates of the kinetic theory of gases. We have derived the Maxwell-Boltzmann distribution function and employed it to obtain the expressions for the characteristic molecular speeds, namely, the root mean square speed, the average speed, and the most probable speed. The average total distance covered by a molecule between two successive collisions is called the mean free path. It is shown that the mean free path must be much smaller than the minimum characteristic dimension of the medium in order to hold the continuum hypothesis. The ratio of the mean free path to the characteristic dimension of the medium is called the Knudsen number. It is shown that for continuous media, the Knudsen number should be much less than one. The kinetic interpretations of pressure and temperature are also covered. Finally, we have discussed a few transport properties: viscosity, thermal conductivity, and self-diffusion.

Exercise Problems Exercise 1.1 If N represents the number of molecules of a perfect gas of volume ∀, then prove that the number of pairs having the intermolecular distances between  2 2 and  + d would be 2πN∀ d . Exercise 1.2 Calculate the root mean square speed, average speed, and the most probable speed of hydrogen molecules at 0 ◦ C. Exercise 1.3 The root mean square speed of oxygen molecules at 0 ◦ C is 450 m/s. Find the root mean square speed of helium molecules at 40 ◦ C. The molecular weights g , respectively. of helium and oxygen are 4 g/mol and 32 mol Exercise 1.4 Calculate the mean free path of air molecules at standard temperature and pressure of 0 ◦ C and 1 atm, respectively. Suppose the diameters of the oxygen ˚ and nitrogen molecules are approximately 3 A. Exercise 1.5 The escape velocity on the Jupiter is 60 km/s. At certain spots on the Jupiter, the temperature of the surface material is found to be −150 ◦ C. Calculate the root mean square speed for CO2 molecules at this temperature. Exercise 1.6 There is one hydrogen atom per cubic centimeter in outer space where the temperature is about 4 K. Calculate the root mean square speed of the hydrogen atoms in the outer space and the pressure they exert.

46

1 Gas Kinetics—Basic Concepts

Exercise 1.7 The isentropic bulk modulus of a fluid is usually expressed as  Ks = −ρ

dp dρ

 s

Show that the speed of sound is  a=

Ks ρ

Exercise 1.8 A gas obeys the van der Waal’s thermal equation of state as per the equation  p = ρRT

aρ 1 − 1 − bρ RT

where a and b are the constants. Derive the expression for the isothermal bulk of elasticity for this gas.

Chapter 2

Governing Equations and Thermodynamics of Compressible Flows

Abstract Before delving into the topic of gas dynamics, it is assumed that the reader has a solid foundation in mathematics (especially calculus) as well as a course in introductory thermodynamics. Furthermore, exposure to fundamental fluid mechanics would be quite advantageous. While this chapter is not intended to be a rigorous review of these courses, it should be viewed as a collection of fundamental concepts and facts relevant to our discussions in the next chapters.

2.1 Introduction The study of compressible fluids that are mobile in nature is the focus of gas dynamics. The governing equations are the mathematical formulae that describe the motion of such fluids. In some cases, the one-dimensional form of these equations is sufficient to explain the phenomenon, but in others, such as the unbounded fluid flow past an airfoil, a multi-dimensional analysis is required. Similar to the flow of an incompressible fluid, the flow of a compressible fluid is governed by one or more of the following conservation laws: 1. The continuity equation (mass conservation) ensures that total mass is always conserved in fluid flow processes. 2. The conservation of momentum (the momentum equation) is basically Newton’s second law for fluid motion. According to which the total momentum in a fluid flow process is invariant. 3. The conservation of energy (the energy equation), often known as the first law of thermodynamics, stipulates that in a fluid flow process, energy is neither created nor destroyed and remains constant. 4. The direction of a process is determined by the second law of thermodynamics (the entropy equation). It states that an irreversible or spontaneous process can occur only in the direction of increasing the entropy of the universe or of an isolated system. A process can never happen in the opposite direction of entropy decrease.

© Springer Nature Singapore Pte Ltd. 2022 M. Kaushik, Fundamentals of Gas Dynamics, https://doi.org/10.1007/978-981-16-9085-3_2

47

48

2 Governing Equations and Thermodynamics of Compressible Flows

These conservation laws are supplemented by constitutive relationships such as the ideal gas law (p = ρRT) or the isentropic relation between pressure and density  p = constant . ργ It is observed that a control mass approach is more convenient for examining a system with defined limits, such as a rigid body. Nonetheless, the fluid has a high degree of fluidity, and it is impossible to identify the limits of the fluid system over a significant period of time. As a result, using the volume control technique, it is simple to analyze the flow process. The control volume is the given volume in space through which the fluid moves. When the properties are observed as fluid flows across defined spatial boundaries, the analysis is greatly simplified. The control volume might be stationary or moving at a constant speed with respect to the inertial coordinate system. The relevant flow equations for the finite control volume are deduced in the following sections. Moreover, we are well aware that, in the event of a minor change in temperature, the energy equation can be conveniently eliminated from the flow analysis. However, if the corresponding temperature change is significant (∼5% or higher), it is necessary to investigate the heat transport characteristics. Thermodynamic considerations for low-speed flows (M < 0.5) are not provided due to their enormous heat capacity compared to their kinetic energies; the temperature will remain constant even if the full kinetic energy is transferred to heat. Consequently, the static and stagnation1 temperatures of the fluid medium are nearly equal. For   high-speed 2 flows (M > 0.5), on the other hand, the kinetic energy per unit mass V2 is quite enormous. As the flow goes through solid bodies or through ducts (such as a nozzle or a diffuser), a significant change in flow velocity is seen, and hence the flow experiences a large change in kinetic energy. Thus, there is a significant difference between the static and stagnation temperatures of the fluid medium, requiring the consideration of thermodynamic concepts2 in the flow process.

2.2 Mathematical Description of a Continuous Media Before defining the governing equations for compressible fluid flow, it is useful to define several key mathematical concepts. For example, the system, the control volume, the control surface, the extensive and intensive properties, the Lagrangian and Eulerian views of the flows, and the substantial (material or Eulerian) derivatives.

1

A stagnation state is achieved when a fluid is brought to rest traversing through a reversible and adiabatic (i.e., isentropic) process. 2 Entropy and temperature are the two such fundamental thermodynamic concepts.

2.2 Mathematical Description of a Continuous Media

49

Fig. 2.1 System approach

2.2.1 System, Control Volume, and Control Surface The term system refers to a fixed mass with defined boundaries. While the boundary of a system can alter, the enclosed mass remains constant. Consider a cylinder-piston assembly having a specific amount of gas, as shown in Fig. 2.1a. In this case, gas can be thought of as our system. The piston can be pushed or pulled in order to compress or expand the gas. Though the boundary of our system changes in both compression and expansion processes, there is no mass transfer across the boundary. This is due to the fact that the system is, by definition, a fixed-mass entity. The concept of a system, however, does not preclude work and heat passage across the boundary. The system approach is quite beneficial for studying system configurations that are very simple, such as the gas present in the cylinder-piston assembly. However, where the system configuration is not simple, such as the flow surrounding a flying

50

2 Governing Equations and Thermodynamics of Compressible Flows

Fig. 2.2 Control volume approach

aircraft or the flow through a turbomachinery, the system undergoes rapid changes and uncontrollable distortions at its boundary (Fig. 2.1b). The system approach is nearly completely ruled out. Using the control volume approach, these flow processes are easily analyzed. The concept of control volume is essentially a mathematical representation of three-dimensional space. The control volume in an inertial frame of reference is either fixed in space or moving at a constant velocity through which the fluid flows (Fig. 2.2). The control surface is the surface that encloses the control volume. The control surface allows for the transmission of mass, momentum, and energy. The properties of the control volume can also be modified over time. The control volume technique is useful for studying fluid flow through turbines, compressors, nozzles, diffusers, pumps, heat exchangers, and other similar devices.

2.2.2 Extensive and Intensive Properties Extensive properties are those that are dependent on the mass under consideration, such as volume, mass, momentum, and energy. An extensive property is commonly indicated with the letter B. In contrast, an intensive property is one that is independent of the mass of the fluid system. There are two types of intensive properties. The first set of properties, such as pressure and temperature, is independent of mass and indicates the overall condition of the system. The other collection consists primarily of the specific values of extensive properties. For example, the particular internal energy, u, is calculated by dividing the internal energy (extensive property) by the mass of the system. Similarly, the specific entropy, s, and the specific enthalpy, h, are the specific values of entropy, S, and enthalpy, H. It is worth noting that the lowercase letter, b, denotes a general intensive property. Thus, a general extensive property of a system is given by

2.2 Mathematical Description of a Continuous Media

51

ˆ B=

bdm

(2.1)

sys

Because the density ρ of a fluid is represented as ρ =

dm , d∀

ˆ B=

bρd∀

(2.2)

sys

where

´ sys

is evaluated over the volume ∀ occupied by the system.

• For mass, B = m and b = 1, and thus ˆ m=

ρd∀

(2.3)

∀

• For momentum, B = mv = P and b = v, and thus ˆ P=

ρvd∀

(2.4)

eρd∀

(2.5)

sρd∀

(2.6)

∀

• For energy, B = E and b = e, and thus ˆ E= ∀

• For entropy, B = S and b = s, and thus ˆ S= ∀

The governing equations for the flow of a fluid through the control volume will be derived using the above relationships.

52

2 Governing Equations and Thermodynamics of Compressible Flows

2.2.3 Property Field and Flow Description Methods The concept of a property field makes it simple to characterize fluid flow properties. The properties of each individual particle or body are explained as a function of time in solid mechanics. Such an approach, however, is cumbersome for flowing fluids since they are made up of a relatively large number of discrete fluid particles that move relative to one another in a given mass of fluid. Thus, the fluid flow properties, also known as the flow properties, are characterized in terms of a position in space, resulting in the establishment of a property field. Because a spatial point is essentially a geometric location, the property at a point is the property of the fluid elements flowing through that location. Nevertheless, the fictitious concept of a property field is highly useful in generating the mathematical expressions that govern the flow → → v i (x, y, z) for the velocity of phenomena. Instead of writing equations like − vi=− an arbitrary fluid particle (say i), the concept of a property field allows you to assign → → a value of − v =− v (x, y, z, t) to every spatial location in the flow field.

2.2.3.1

Lagrangian Description of Fluid Flows (System Approach)

One method for studying fluid motion is to consider one fluid particle at a time and follow its trajectory through the flow field, as shown in Fig. 2.3. It is worth noting that a fluid particle in this context is essentially made up of a large number of discrete particles with fixed identities that adhere to the continuum hypothesis. The strategy focused on following the trajectories of the individual particles of the fluid is known as the system or the Lagrangian approach. However, due of the difficulties involved in keeping track of the enormously huge number of particles that comprise a body of fluid, the Lagrangian technique is not applicable in many practical situations.

Fig. 2.3 System or Lagrangian description of fluid motion

2.2 Mathematical Description of a Continuous Media

53

Fig. 2.4 Control volume or Eulerian description of fluid motion

2.2.3.2

Eulerian Description of Fluid Flows (Control Volume Approach)

The limitations of the Lagrangian description for evaluating fluid flows have been identified. Another approach, known as Eulerian description, focuses on a fixed volume in space and evaluates the properties of the fluid instantaneously occupying that volume, as seen in Fig. 2.4. This method reduces the requirement to identify individual fluid particles. The properties of fluid flow are entirely explained in terms of the field description. → → For example, − v =− v (x, y, z, t) denotes the velocity of a fluid particle at point P (x, y, z) at time t. Individual particles are not followed in this approach, and so their properties are not obtained separately. Nonetheless, once the velocity field is known, the track of each particle may be traced through the known velocity field, allowing the properties of the individual particles to be determined. In most flow conditions, precise information about individual particles is not necessary, hence the Eulerian description of the flow is perfectly appropriate. This method will be used throughout the book.

2.2.4 Correlation Between System and Control Volume Approaches Remember that the conversation laws (governing equations) for fluid flows are usually expressed in terms of the variables relevant to a system. To transform these

54

2 Governing Equations and Thermodynamics of Compressible Flows

Fig. 2.5 A fluid particle traveling in three-dimensional Lagrangian space

equations from a system (Lagrangian) analysis to a control volume (Eulerian) analysis, we must first conceptually and technically relate both of them. The Reynolds transport theorem is used to convert them through differential analysis, whereas the substantive or material derivative is used to convert them through integral analysis.

2.2.4.1

Substantial or Material Derivative (Differential Analysis)

Through the differentiation process, the governing equations for a system can be expressed in terms of field description (control volume approach). The time derivative (rate of change) of a flow property at a point in space is known as a substantial or material derivative. Although the material derivative is a Lagrangian concept, the property under investigation is an Eulerian quantity. Thus, the Lagrangian time derivative of an Eulerian quantity leads to a material derivative, which is commonly , where () is an extensive property under consideration. represented as D() Dt Consider an arbitrary extensive property B; in a field representation (Eulerian →  view) B is written as B − x , t . The Lagrangian time derivative yields →    DB − x ,t dB = Dt dt fixed w.r.t. P

(2.7)

→ Because the fluid particle P is moving, the spatial coordinate − x changes with time t and follows the particle. That is, the particle P has an Eulerian velocity of . Using calculus, DB can be written as v = δx δt Dt  −  →  →  → B → x +− v δt, t + δt − B − x ,t DB − x ,t = lim δt→0 Dt δt

(2.8)

2.2 Mathematical Description of a Continuous Media

55

→ −  A three-dimensional Taylor series expansion on B − x +→ v δt, t + δt yields →  →  −  −  ∂B − x ,t → → − → → + δ− x .∇B − x ,t B x + v δt, t + δt = B x , t + δt ∂t

(2.9)

The substantial derivative is calculated using Eqs. (2.9) and (2.8). →  →  →  ∂B − x ,t DB − x ,t → = +− v .∇B − x ,t Dt ∂t

(2.10)

In generalized notations, D () Dt

 Lagrangian term

=

∂ () − +→ v .∇ () ∂t



(2.11)

Eulerian term

→  →  → Considering the extensive property B − x , t as Eulerian velocity − v − x ,t , Eq. (2.11) yields → D− v Dt

 Material or Lagrangian acceleration

2.2.4.2

→ ∂− v ∂t



=

+

− → − →

v .∇  v

(2.12)

Convective acceleration

Local or Eulerian acceleration

Reynolds Transport Theorem (Integral Analysis)

The Reynolds transport theorem, also known as the Leibniz-Reynolds transport theorem, is used in integral analysis to transfer the governing equations of fluid motion from the system approach to the control volume approach. In fact, it is a threedimensional expansion of Leibniz’s integral theorem3 that serves as the foundation for all conservation laws. One can choose a fixed, moving, or deformable control volume here, however we shall only derive the theorem for a fixed control volume. We will now define the same for a moving or deformable control volume. → → Consider an arbitrary velocity field − v =− v (x, y, z, t) as seen from some reference xyz, where we witness a system of finite mass at times t and t + t, as depicted in Fig. 2.6 by the curves ABCM and ANCD. The streamlines correspond to those found in time t. Consider an arbitrary extensive property of the fluid, B, for the pur3 According to Leibniz’s integral theorem, the time derivative of the integral of f (x, t) between the limits a (t) and b (t) is given as

d dt

ˆb(t) ˆb(t) ∂f db da f (x, t) dx = dx + f (b, t) − f (a, t) . ∂t dt dt

a(t)

a(t)

56

2 Governing Equations and Thermodynamics of Compressible Flows

Fig. 2.6 A moving system at times t and t + t

pose of relating the rate of change of this property for the control system to variations in this property associated with the control volume, i.e., ˚ B=

βρd∀

(2.13)

  where β denotes the distribution of B per unit mass β = mB . It should be noted that the volume in space occupied by the control system ABCM at time t is also the control volume at time t.   Bsys t = (BC∀ )t

(2.14)

That is, both control system and control volume are the same at time t. At time t + t, the control surface exits the control volume and assumes the new location ANCD. As illustrated in Fig. 2.6, we have divided our system at time t and the system at time t + t into three regions: I, II, and III. Thus, the fluid property B associated with the control system at time t + t comprises region II and region III. We have,   Bsys t+t = (BII )t+t + (BIII )t+t

(2.15)

  Adding and subtracting the property associated with the region I (BI )t+t , we have  But,

Bsys

 t+t

= (BII )t+t + (BI )t+t + (BIII )t+t − (BI )t+t

(2.16)

2.2 Mathematical Description of a Continuous Media

57

(BII )t+t + (BI )t+t = (Bcontrol volume )t+t

(2.17)

Hence, 

Bsys

 t+t

= (BC∀ )t+t + (BIII )t+t − (BI )t+t

(2.18)

  Subtracting Bsys t from both sides       Bsys t+t − Bsys t = (BC∀ )t+t − Bsys t + (BIII )t+t − (BI )t+t     Bsys t+t − Bsys t = (BC∀ )t+t − (BC∀ )t + (BIII )t+t − (BI )t+t

(2.19) (2.20)

Dividing by the differential time t gives     Bsys t+t − Bsys t t

=

(BC∀ )t+t − (BC∀ )t (BIII )t+t − (BI )t+t + t t

(2.21)

In the limiting case, when t → 0  lim

 lim

t→0

Bsys

 t+t

  − Bsys t

t

t→0

=

(BIII )t+t − (BI )t+t (BC∀ )t+t − (BC∀ )t + t t

(2.22) 

But, 

dB dt

 sys

DB = lim = Dt t→0

    Bsys t+t − Bsys t t

Thus,     DB (BIII )t+t − (BI )t+t (BC∀ )t+t − (BC∀ )t + = lim Dt sys t→0 t t

(2.23)

(2.24)

Using sum4 law of limits, we can rearrange the above equation as 

 (BC∀ )t+t − (BC∀ )t = lim t→0 t sys     (BIII )t+t (BI )t+t − lim + lim t→0 t→0 t t

4

DB Dt





(2.25)

The Sum Law states that the limit of the sum of two functions is the sum of the limits. That is, if lim x→a f (x) = L, and lim x→a g (x) = M. Further, if both L and M exist then lim x→a [ f (x) + g (x)] = L + M.

58

2 Governing Equations and Thermodynamics of Compressible Flows

From Eq. (2.13), 

DB Dt

 ˝



+ lim

t→0

= lim sys

  ˝   βρd∀ C∀ t+t − βρd∀ C∀ t t  ˝

t→0

 ˝

  III βρd∀ t+t t

− lim

t→0

I

  βρd∀ t+t t

(2.26)

Let us consider each term the limiting process ˝  ˝ofβρd∀  above separately. The first term in ( )t+t −( βρd∀)t denotes the partial derivative with Eq. (2.26), i.e., limt→0 t respect to time. We have,  ˝ lim

t→0

  ˝   ˚ βρd∀ C∀ t+t − βρd∀ C∀ t ∂ βρd∀ = t ∂t

(2.27)

C∀

 ˝ βρd∀  ( III )t+t essentially represents the In Eq. (2.26), the second term limt→0 t amount of property moving out of the control volume through the control sur ˝ βρd∀  ( III )t+t face, ABC. Thus, the term limt→0 approximates the average rate of t efflux of B ˝ across ABC  during the interval t. Similarly, considering the last term ( I βρd∀)t+t limt→0 in Eq. (2.26) which approximates the amount of B that has t passed into the control volume during t through the control surface AMC. Hence, the last two integrals of Eq. (2.26) give the net rate of efflux of B from the control volume at time t. Thus, net rate of efflux of B is  ˝  ˝     III βρd∀ t+t I βρd∀ t+t = lim − lim (2.28) t→0 t→0 t t We will now compute a more compact and workable form of Eq. (2.28). Consider Fig. 2.7, where a portion of the control surface, ds, is in the steady-state velocity field − → v . The elemental area ds is also a fluid interface that is sometimes just touching the control surface, t and t + t, as shown in Fig. 2.7a, b, respectively. It is worth noting that the elemental area (interface) has traveled a distance of ds in the direction tangent to the streamline at that moment in the differential time period dt (Fig. 2.7a). Consequently, the volume of fluid that occupies the region swept away by ds in time dt, producing a streamtube, is d∀= vdt (ds cos α) d∀ = v (ds cos α) dt

(2.29) (2.30)

2.2 Mathematical Description of a Continuous Media

59

Fig. 2.7 Control surface at different moments in time

When the above value is multiplied by ρ, the instantaneous mass flow rate of fluid going out of the control volume through the designated area is given by ds. We find that   dm d∀ = ρv (ds cos α) (2.31) =ρ dt CS dt In vector notation, 

dm dt



− → → = ρ− v . ds

(2.32)

CS

− → where d S = nˆ ds. The efflux rate of property B through the control surface ABC at time t + dt can be calculated as

60

2 Governing Equations and Thermodynamics of Compressible Flows

¨ Befflux rate =

− → → βρ − v . ds

(2.33)

ANC

Similarly, the influx rate of property B through the control surface AMC at time t is ¨ − → → Binflux rate = − βρ − v . ds (2.34) ANC

The negative sign in the preceding expression implies that at the control surface → AMC, the velocity vector − v and the unit normal nˆ to ds are pointing in opposing directions, as shown in Fig. 2.7b. Thus, the net efflux rate of B throughout the entire control surface is (Bnet efflux rate )CS =Befflux rate − Binflux rate ⎛ ⎞ ¨ − → − → → → βρ − v . ds − ⎝− βρ − v . ds ⎠

¨ (Bnet efflux rate )CS = ANC

(2.35)

(2.36)

ANC

¨

− → → βρ − v . ds +

(Bnet efflux rate )CS = ANC

¨

− → → βρ − v . ds

(2.37)

ANC

The above relation becomes exact in the limit as t → 0, and therefore the terms on the right side can be represented as " (Bnet efflux rate )CS =

 − → → β ρ− v . ds

(2.38)

CS

where the integral is a closed surface integral across the entire control surface. It is also worth noting that, while we assumed a steady flow while deriving Eq. (2.38), it is equally applicable to unsteady flows. Now, by combining Eqs. (2.27), (2.38), and (2.26), we obtain 

DB Dt



∂ = ∂t system

"

˚ (βρ) d∀+ C∀

 − → → β ρ− v . ds

(2.39)

CS

Equation (2.39) is the well-known Reynolds transport theorem. This theorem allows for the transfer of mass, momentum, and energy conservation laws established from the Lagrangian (system) point of view to the Eulerian (control volume) point of view. These two perspectives were introduced in Sect. 2.2.3; the Eulerian approach

2.2 Mathematical Description of a Continuous Media

61

allows us to observe fluid particles traveling through a fixed point in space, whereas the Lagrangian approach allows us to follow a single particle. It is worth noting in this context that when deriving the Reynolds transport theorem, we assumed a fixed control volume in an arbitrary frame of reference and measured the velocity field relative to this frame of reference. Moreover, the fluid velocities must be expressed relative to the control volume when determining the rates of entering and departing flow properties. Thus, Eq. (2.39) is more aptly described as   ˚ ∂ DB = + (βρ) d∀ Dt system ∂t C∀





Time rate of change of property within a system " CS



Time rate of change of property within the control volume

 − → → β ρ− v rel . ds 

(2.40)

Net flux of property through the control surface

→ In Eq. (2.40), − v rel is the fluid velocity relative to the control volume, and is given by → → − → v −− v C∀ v rel = −

(2.41)

→ → where − v and − v C∀ are the fluid and control volume velocities as experienced in a fixed reference frame, respectively.

2.3 Conservation of Mass: The Continuity Equation According to the conservation of mass principle, mass cannot be generated or destroyed and always remains constant for an isolated system of objects. The volume filled by a fluid and its density inside a system may change, but the total mass of the system remains constant. The equation of the conservation of mass principle, often known as the continuity equation, is developed in this section. The term continuity indicates that the fluid is always a continuum, that is, a continually distributed matter. We derive the integral and differential forms of the conservation of mass principle using both finite Eulerian control volume and infinitesimal fluid element models of fluid flows. These two modeling methodologies are used specifically to discuss their physical characteristics. We will analyze a control volume that is fixed in an inertial frame, whereas the fluid element is supposed to be moving with the fluid. This arrangement will allow us to distinguish between the conservation and non-conservation forms of the continuity equation.

62

2 Governing Equations and Thermodynamics of Compressible Flows

Fig. 2.8 An infinitesimal fluid element in motion

2.3.1 Mass and Density We know that the mass of an object or collection of objects never changes, no matter how the constituent elements rearrange themselves. The mass, a non-negative scalar, is regarded as the measure of an object’s inertia in response to forces applied to it. Consider a small fluid element of volume ∀ with a mass in m, as depicted schematically in Fig. 2.8. The average density of this fluid element is expressed as ρavg =

m ∀

(2.42)

Let P represent a particular point in the elemental ∀. The spatial density at P is indeed the limiting value of Eq. (2.42), causing the elemental volume to drop to zero. ρ (x, t) = lim

∀→0

dm m = ∀ d∀

(2.43)

In Eq. (2.43), ∀ is permitted to decrease up to a certain minimum value until it contains a sufficient number of molecules to make any average relevant. The spatial mass density is a representative average produced by having ∀ large in comparison to the atomic scale but small in comparison to the physical dimensions of the region under consideration, such as an aircraft wing or a pipe in a hydraulic system. Similar to displacement and velocity, density is defined as a function of space and time for a specific set of fluid elements in a continuum, ρ = ρ (x, t). Mass, on the other hand, is not defined in this way and is typically represented for an infinitesimal fluid element—a mass element. dm = ρ (x, t) d∀

(2.44)

Thus, for the mass m occupying a volume ∀ at time t ˆ ρ (x, t) d∀

m= ∀

(2.45)

2.3 Conservation of Mass: The Continuity Equation

63

2.3.2 Integral Form of Continuity Equation for a Finite Control Volume We apply the Reynolds transport theorem to deduce the integral form of the continuity equation. We substitute the extensive property (B) equal to the total mass (m) of the system of fluid elements, i.e., B = m and consequently β = mB = 1 in Eq. (2.39). 

Dm Dt

 = sys

∂ ∂t

 Dm 

For a fixed-mass system,

Dt

∂ ∂t

sys

"

˚ ρd∀ +

(2.46)

CS

C∀

= 0. Thus, Eq. (2.46) reduces to "

˚ ρd∀ +

− → → ρ− v . ds = 0

(2.47)

CS

C∀

∂ ∂t

− → → ρ− v . ds

˚

" − → → ρd∀ = − ρ− v . ds

(2.48)

CS

C∀

Equation (2.47) is the well-known integral form of the equation of conservation of mass, also known as the continuity equation. Equation (2.48) demonstrates that the rate of mass increase within the control volume must be equal to the net rate of mass flux into ˝the control volume. Assume the flow within the control volume is steady, then ∂t∂ C∀ ρd∀ = 0. Therefore, Eq. (2.48) becomes "

− → → ρ− v . ds = 0

(2.49)

CS

That is, in steady flow, the mass flows entering and exiting the control volume must exactly balance. Alternative Approach Consider an arbitrarily drawn Eulerian control volume C∀ fixed in coordinate space, as depicted in Fig. 2.9. The boundary of control volume is called the control surface − → (CS). We take a small vector segment of that surface ds , where the magnitude of the vector is the scalar area of the segment ds and the unit vector nˆ is the outward normal − → to the surface at that point; thus, ds = dsˆn. First, we will evaluate the net mass flow → rate out of the control volume. If − v is the velocity of the fluid flow at the surface − → − → area ds , then the mass flow rate leaving through ds is

64

2 Governing Equations and Thermodynamics of Compressible Flows

Fig. 2.9 An Eulerian control volume

 → − → → ρ− v . ds = ρ nˆ .− v ds

(2.50)

and integrating the above expression over the entire CS, the net mass flow rate leaving through C∀ becomes "

− → → ρ− v . ds =

CS

"

 → ρ nˆ .− v ds

(2.51)

CS

Now, we will evaluate the time rate of change of fluid mass inside the control volume. Since the mass contained ˝within the elemental volume d∀ is ρd∀, the total mass inside the control volume is C∀ ρd∀. Thus, the time rate of change of mass within ˝ the control volume will be − ∂t∂ C∀ ρd∀, where the negative sign in above expression reflects the decrease of mass within the control volume. Since our fundamental physical law that mass is conserved means {Net mass flow out of control volume through control surface} = {Time rate of decrease of mass inside control volume} Finally, the integral form of continuity equation is ∂ ∂t

"

˚ ρd∀ + C∀

− → → ρ− v . ds = 0

(2.52)

CS

2.3.3 Conservative Partial Differential Form of Continuity Equation for a Differential Control Volume The integral form of the continuity equation (Eq. (2.52)) can be expressed in another useful form by transforming the surface integral to a volume integral using

2.3 Conservation of Mass: The Continuity Equation

65

Gauss’ theorem5 "

− → → ρ− v . ds =

˚

CS

 → ∇. ρ − v d∀

(2.53)

C∀

Substituting Eq. (2.53) into Eq. (2.52) leads to ∂ ∂t

˚

˚ ρd∀ + C∀

 → ∇. ρ − v d∀ = 0

(2.54)

C∀

Using Leibniz’s theorem6 for a constant control volume, switching the order of integration and differentiation in the first term yields ˚ C∀

∂ρ d∀ + ∂t ˚  C∀

˚

 → ∇. ρ − v d∀ = 0

(2.55)

C∀

  → ∂ρ + ∇. ρ − v d∀ = 0 ∂t

(2.56)

Because the preceding expression is true for any finite control volume arbitrarily drawn in space, the only condition that it may be satisfied is if the integrand vanishes identically. Thus, 5

The divergence theorem, commonly known as Gauss’s theorem and also known as the GaussOstrogradsky theorem, is a theorem in vector calculus that can be defined as follows. If C∀ is the control volume in space with its control surface CS, then the volume integral of the divergence of f over C∀ and the surface integral of f over CS are related by ˆ  ˆ − → − →− → ∇. f d∀ = f . ds . CS

C∀ 6

According to Leibniz’s theorem, the substantial time derivative of an integral with time-dependent limits equals the integral of the partial derivative of the integrand plus a term that accounts for the movement of the integration boundary. That is, ˆ ˆ ˆ ∂f (x, t) d → f (x, t) d∀ = f (x, t) − d∀+ v .ˆnds dt ∂t ∀(t)

∀(t)

A(t)

However, if the integration volume ∀ (t) is fixed in space then the motion of the integration → boundary ceases, i.e., − v = 0. The above equation therefore reduces to ˆ ˆ ∂f (x, t) d f (x, t) d∀ = d∀. dt ∂t ∀(t)

∀(t)

66

2 Governing Equations and Thermodynamics of Compressible Flows

 → ∂ρ + ∇. ρ − v =0 ∂t

(2.57)

This is the requisite partial differential form of the continuity equation. We can see that Eq. (2.57) is the conservation form of the continuity equation; it is just a relationship between velocity and density fields. A kinematic relation is one that does not involve any dynamic quantities such as forces and pressures. Thus, the continuity equation is a purely kinematic relation that holds true for the motion of all fluids. For steady flow, ∂ρ ≡ 0; thus Eq. (2.57) reduces to ∂t  → ∇. ρ − v =0

(2.58)

The density is constant in very low-speed flows, whether steady or unsteady; therefore → ∇.− v =0

(2.59)

2.3.4 Non-conservative Partial Differential Form of Continuity Equation for a Moving Fluid Element We can easily deduce the non-conservative form of the continuity equation using its conservation form and subsequently invoking the following vector identity:  → −  → ∇. ρ − v =→ v .∇ρ + ρ ∇.− v

(2.60)

Substituting Eq. (2.60) into Eq. (2.57) gives   → ∂ρ − → + v .∇ρ +ρ ∇.− v =0 ∂t





(2.61)

Dρ Dt

is the material or substantial derivative of the density. Therefore, Eq. (2.61) where Dρ Dt becomes  → Dρ + ρ ∇.− v =0 Dt

(2.62)

This is the non-conservation form of the continuity equation. Equation (2.62) is obtained from the point of view of an infinitesimal fluid element which is moving along with the flow.

2.4 Conservation of Momentum (Newton’s Second Law of Motion for Fluid Flows)

67

2.4 Conservation of Momentum (Newton’s Second Law of Motion for Fluid Flows) The conservation of momentum principle states that a quantity termed momentum, which characterizes motion, stays invariant for an isolated system of objects. That is, the total momentum of an isolated system is always conserved. In this section, we will develop the momentum conservation equation in both integral and differential forms, utilizing the finite control volume and infinitesimal fluid element models of fluid flows. For the reasons described in Sect. 2.3, the control volume is stationary in space, but the fluid element moves with the fluid.

2.4.1 Integral Form of Momentum Equation for a Finite Control Volume Consider a finite control volume that is fixed in space, as shown in Fig. 2.9. We shall use the linear momentum P as the extensive property B in the Reynolds transport equation (Eq. (2.40)). The linear momentum P is defined as ˚ P=

− → v ρd∀ =

C∀

˚

− → v dm

(2.63)

C∀

and thus → β=− v

(2.64)

Hence, Eq. (2.40) becomes 

DP Dt

 = system

∂ ∂t

˚ C∀

→ ρ− v d∀+

"

 − → → → ρ − v . ds − v

(2.65)

CS

Since the mass of the system7 is fixed, rewriting the above expression as  ˆ  − "  ˚ D→ v ∂ − → → → → dm = ρ− v d∀+ ρ − v . ds − v Dt ∂t

sys

C∀

(2.66)

CS

− → → → But, DDtv = − a , where − a is the acceleration of the fluid element. Therefore, from Newton’s second law of motion

7

A system in fluid mechanics is analogous to a control mass in thermodynamics.

68

2 Governing Equations and Thermodynamics of Compressible Flows

ˆ − → → a dm F sys = −

(2.67)

sys

" ˚  − ∂ − → → → − → → ρ− v d∀+ v ρ− v . ds F sys = ∂t

(2.68)

CS

C∀

− → The net force on the system, F sys , is made up of two types of forces: − → 1. Surface forces F s —these forces act on the boundary of the system and are given as the force per unit area on the boundary surfaces, i.e., they are represented as the stress-tensor, σij . Thus, " − − → → σij ds Fs=

(2.69)

CS

− → 2. Body forces F b —these forces act on the material inside the boundary and are − → given as the force per unit mass at a point. Let f be the body force per unit mass. Hence, ˚ − − → → ρ f d∀ (2.70) Fb= C∀

Thus, Eq. (2.68) can be rewritten as ˚ " − − − − → − → → → → ρ f d∀ + σij . ds F sys = Fb+ Fs=

(2.71)

CS

C∀

Combining Eqs. (2.68) and (2.71) yields " " ˚ ˚  − − − ∂ − → − → → → − → → → − → → Fb+ Fs= ρ− v d∀ + ρ f d∀ + σij . ds v ρ− v . ds = F sys = ∂t C∀

CS

C∀

(2.72)

CS

The stress-tensor σij in Eq. (2.72) for a moving fluid is basically the sum of an isotropic part −pδij , which is equivalent to the stress-tensor in a static fluid, and a remaining non-isotropic part τij , which consists of any shear stresses with diagonal components equal to zero, i.e., τij = 0. Thus, σij = − pδi j +

 Isotropic part

τij

 Non-isotropic part

(2.73)

2.4 Conservation of Momentum (Newton’s Second Law of Motion for Fluid Flows)

69

  where p is the static pressure and τij is a symmetric stress-tensor τij = τji , called the deviatoric stress, which is a function of the rate of deformation of fluid element under consideration. For an isotropic and homogeneous fluid in motion, the linear variation of deviatoric stress with the deformation rate tensor is given by 

∂vj ∂vi + τij = μ ∂xj ∂xi





∂vk +λ ∂xk

 δij

(2.74)

where μ is the dynamic viscosity and λ is the bulk viscosity; i, j, k = 1, 2, 3; and δij is the Kronecker delta, defined as  1 for i = j δij = (2.75) 0 for i = j   Following Stokes hypothesis λ = − 23 μ , Eq. (2.74) becomes     ∂vj 2 ∂vk ∂vi − δij + ∂xj ∂xi 3 ∂xk     ∂vj 2 − ∂vi τij = μ − ∇.→ v + ∂xj ∂xi 3 

τij = μ

(2.76) (2.77)

Interestingly, we can observe that for a static fluid, τij = 0. That is, the deviatoric stress is a consequence of fluid motion and vanishes for a fluid at rest. Now, from Eqs. (2.77) and (2.73), the stress-tensor is finally written as  σij = −pδij + μ

∂vj ∂vi + ∂xj ∂xi

 −

  2 − ∇.→ v 3

(2.78)

Substituting Eq. (2.78) into Eq. (2.72), we obtain ∂ ∂t

∂ ∂t

˚

→ ρ− v d∀ +

C∀

˚ C∀

→ ρ− v d∀ +

"



"  − → − → → v ρ− v . ds =



−pδij + μ

CS

CS

"

" "  − → − → − → → p ds + μ v ρ− v . ds = −

CS

CS

CS

∂vj ∂vi + ∂xj ∂xi



 −

∂vj ∂vi + ∂xj ∂xi

 ˚  2 − − → − → ∇.→ v . ds + ρ f d∀ 3

(2.79)

C∀

 −

˚  − 2 − → − → ∇.→ v . ds + ρ f d∀ 3

(2.80)

C∀

Equation (2.80) is the momentum conservation equation in an integral form. Note that this is a general equation which applies to the unsteady three-dimensional flow of any fluid, compressible or incompressible, viscous or inviscid. → For incompressible flows, ∇.− v = 0, so Eq. (2.81) becomes

70

2 Governing Equations and Thermodynamics of Compressible Flows ∂ ∂t

˚ C∀

→ ρ− v d∀ +

"

" "  − → − → − → → v ρ− v . ds = − p ds +

CS

CS

  μ

CS

∂vj ∂vi + ∂xj ∂xi



− → . ds +

˚

− → ρ f d∀

C∀

(2.81) −      → For a steady ∂t∂ = 0 , inviscid τij = 0 flow with no body forces f = 0 , Eq. (2.81) reduces to "

"  − → − → − → → p ds v ρ− v . ds = −

CS

(2.82)

CS

2.4.2 Conservative Partial Differential Form of Momentum Equation for a Differential Control Volume We may simplify Eq. (2.72) by transforming the surface integral to a volume integral using Gauss’ divergence (Sect. 2.3.3). In addition, according to Leibniz’s theorem, swapping the order of integration and differentiation in the first term yields ˚ C∀

 → ˚ ˚ ˚  →−  ∂ ρ− v − → d∀+ ∇. ρ − v→ v d∀ = ρ f d∀ + ∇.σij d∀ ∂t C∀

C∀

(2.83)

C∀

 ˚   −  −  − ∂ ρ→ v → → − → +∇. ρ v v −ρ f − ∇.σij d∀ = 0 ∂t

(2.84)

C∀

The above integral is correct for a finite C∀ arbitrarily drawn in space only if the integrand itself vanishes identically. Thus,  →  →−  − ∂ ρ− v → +∇. ρ − v→ v −ρ f − ∇.σij = 0 ∂t

(2.85)

 →  →−  ∂ ρ− v − → +∇. ρ − v→ v =∇.σij + ρ f ∂t

(2.86)

This is the conservation form of the momentum equation, also called Cauchy’s → → → → → equation. The dyadic ρ − v− v , where the term − v− v denotes the outer product of − v with itself, is a second-order velocity tensor, called the momentum flux tensor. Substituting in σij from Eq. (2.78) into Eq. (2.86) yields the conservative partial differential form of the following set of momentum equations:

2.4 Conservation of Momentum (Newton’s Second Law of Motion for Fluid Flows)

71

 →         →−  ∂ ρ− v ∂vj ∂vi 2 − ∂ ∂p − → δij + μ − ∇.→ v + ρ f (2.87) + +∇. ρ − v→ v =− ∂t ∂xj ∂xj ∂xj ∂xi 3

 →       →−  ∂ ρ− v ∂vj ∂vi 2 − ∂ − → ∇.→ v + μ − +∇. ρ − v→ v = − ∇p + +ρ f ∂t ∂xj ∂xj ∂xi 3

(2.88)

They are the well-known Navier-Stokes equations, named after their inventors, Claude-Louis Navier and George Gabriel Stokes. Equation (2.88) describes the unsteady three-dimensional flow of any fluid, whether compressible or incompressible, viscous or inviscid. If we take a three-dimensional flow domain, we can see that Eq. (2.88) is a set of three second-order partial differential equations. Thus, in the Cartesian coordinate system, the Navier-Stokes equations in conservation form are         ∂ ρu2  ∂ ∂ (ρu) ∂ (ρuv) ∂ (ρuw) ∂p ∂ ∂u 2 − ∂v ∂u ∇.→ v + + + + =− + μ 2 − μ + ∂t ∂x ∂y ∂z ∂x ∂x ∂x 3 ∂y ∂x ∂y    ∂u ∂ ∂w μ + ρfx (2.89) + + ∂z ∂z ∂x         ∂ ρv2  ∂v ∂ ∂v ∂ (ρv) ∂ (ρuv) ∂ (ρvw) ∂p ∂ ∂u 2 − μ + μ 2 ∇.→ v + + + =− + + − ∂t ∂x ∂y ∂z ∂y ∂x ∂x ∂y ∂y ∂y 3    ∂ ∂w ∂v + (2.90) μ + + ρfy ∂z ∂y ∂z         ∂ ρw2 ∂ (ρw) ∂ (ρuw) ∂ (ρvw) ∂p ∂ ∂u ∂w ∂ ∂w ∂v + + + =− + μ + + μ + ∂t ∂x ∂y ∂z ∂z ∂x ∂z ∂x ∂y ∂y ∂z     ∂w ∂ 2 − μ 2 ∇.→ v + ρfz (2.91) + − ∂z ∂z 3

where u, v, and w, respectively, are flow velocity components and fx , fy , and fz , respectively, are body force components along the x-, y-, and z-axes. → For incompressible fluids, ∇.− v = 0; thus Eq. (2.88) reduces to      − −  ∂vj ∂vi ∂ ∂→ v − → → − → μ +ρ f +∇. v v = −∇p + + ρ ∂t ∂xj ∂xj ∂xi

(2.92)

If the fluid has constant viscosity, then Eq. (2.92) becomes → →−  ∂− v ∇p μ ∂ +∇. − v→ v =− + ∂t ρ ρ ∂xj In Cartesian notations,



∂vj ∂vi + ∂xj ∂xi



− → + f

(2.93)

72

2 Governing Equations and Thermodynamics of Compressible Flows

  ∂u ∂u ∂u 1 ∂p μ ∂ 2 u ∂ 2 u ∂ 2 u ∂u +u +v +w =− + + 2 + 2 + fx (2.94) ∂t ∂x ∂y ∂z ρ ∂x ρ ∂x2 ∂y ∂z   2 2 ∂v ∂v ∂v ∂v 1 ∂p μ ∂ v ∂ v ∂ 2 v + 2 + 2 + fy (2.95) +u +v +w =− + ∂t ∂x ∂y ∂z ρ ∂y ρ ∂x2 ∂y ∂z   2 2 ∂w ∂w ∂w ∂w 1 ∂p μ ∂ w ∂ w ∂ 2 w +u +v +w =− + + 2 + 2 + fz (2.96) ∂t ∂x ∂y ∂z ρ ∂z ρ ∂x2 ∂y ∂z

2.4.3 Non-conservative Partial Differential Form of Momentum Equation for a Moving Fluid Element By utilizing the chain rule to extend the terms on the left side of Eq. (2.88) and then substituting the vector identity (Eq. (2.60)), we get ρ

→  → →  − ∂− v ∂ρ − ∂ → +− v +→ v ∇. ρ − v +ρ − v .∇ → v = −∇p + ∂t ∂t ∂xj

  μ

∂vj ∂vi + ∂xj ∂xi

 −

 2 − ∇.→ v 3



− → +ρ f

(2.97)

Using the continuity equation (Eq. (2.57)) and combining the second and third terms on the left side, we obtain ρ

     → ∂vj  →  −  → ∂vi ∂− v ∂ρ 2 − ∂ − → → ∇.→ v + +− v +∇. ρ − v +ρ − +ρ f μ − v .∇ → v = −∇p + ∂t ∂t ∂xj ∂xj ∂xi 3



(2.98)

0

    −  ∂vj  →  − ∂vi 2 − ∂→ v ∂ − → ∇.→ v +ρ f μ − ρ + + − v .∇ → v = −∇p + ∂t ∂xj ∂xj ∂xi 3 



(2.99)

→ D− v Dt

Thus, ρ

    →  ∂vj ∂vi 2 − ∂ D− v − → μ − = −∇p + ∇.→ v +ρ f + Dt ∂xj ∂xj ∂xi 3

(2.100)

This is the Navier-Stokes equation in its non-conservation form, and it applies to a fluid element that moves with the flow. In Cartesian coordinates, Eq. (2.100) can be expressed as           ∂u ∂ ∂v ∂ ∂u Du ∂p ∂ 2 − ∂u ∂w μ 2 ∇.→ v + μ + μ + ρfx =− + − + + Dt ∂x ∂x ∂x 3 ∂y ∂x ∂y ∂z ∂z ∂x           ∂ Dv ∂p ∂ ∂v ∂u ∂ ∂v 2 − ∂w ∂v ∇.→ v + ρ =− + μ + + μ 2 − μ + + ρfy Dt ∂y ∂x ∂x ∂y ∂y ∂y 3 ∂z ∂y ∂z           Dw ∂p ∂ ∂u ∂w ∂ ∂w ∂v ∂ ∂w 2 − ∇.→ v + ρfz ρ =− + μ + + μ + + μ 2 − Dt ∂z ∂x ∂z ∂x ∂y ∂y ∂z ∂z ∂z 3 ρ

(2.101) (2.102) (2.103)

2.4 Conservation of Momentum (Newton’s Second Law of Motion for Fluid Flows)

73

→ For incompressible flows, ∇.− v = 0; hence Eq. (2.100) becomes    → ∂vj ∂vi D− v ∂ − → μ +ρ f + ρ = −∇p + Dt ∂xj ∂xj ∂xi

(2.104)

and for constant viscosity fluids,   → ∂vj D− v ∂vi ∂ − → ρ +ρ f = −∇p + μ + Dt ∂xj ∂xj ∂xi

(2.105)

Equation (2.105) is written in Cartesian notation as   2 ∂p ∂ u ∂ 2u ∂ 2u Du =− +μ ρ + 2 + 2 + ρfx Dt ∂x ∂x2 ∂y ∂z   2 Dv ∂p ∂ v ∂ 2v ∂ 2v ρ + ρfy =− +μ + + Dt ∂y ∂x2 ∂y2 ∂z2   2 ∂p ∂ w ∂ 2w ∂ 2w Dw + ρfz =− +μ + + ρ Dt ∂z ∂x2 ∂y2 ∂z2

(2.106) (2.107) (2.108)

2.5 Euler’s Equation It is sometimes convenient to ignore viscous stresses in fluid flows, such as the flow through a nozzle or diffuser. In such instances, by eliminating the viscous term from the non-conservation form of the Navier-Stokes equations (2.100), (2.106)–(2.108), we obtain → ∇p − D− v → =− + f Dt ρ

(2.109)

and in Cartesian notations, Du ∂p =− + ρfx Dt ∂x ∂p Dv =− + ρfy ρ Dt ∂y ∂p Dw =− + ρfz ρ Dt ∂z ρ

(2.110) (2.111) (2.112)

Equations (2.109)–(2.112) are known as Euler’s equations for inviscid fluid flows. While these equations have been substantially simplified in comparison to the NavierStokes equations, they cannot be solved analytically due to the existence of nonlinear

74

2 Governing Equations and Thermodynamics of Compressible Flows

 →−  terms such as ∇. ρ − v→ v . Therefore, Euler’s equations are generally solved solely using computational means.

2.6 Bernoulli’s Equation Euler’s equations can be integrated to yield the well-known Bernoulli’s equation. The following are potential integration conditions: 1. For the steady and rotational flows in the presence of conservative body forces, the integration of Eq. (2.109) is possible only along a streamline. 2. For the irrotational flows under conservative body forces, Eq. (2.109) can be integrated in any direction regardless of whether the flow is steady or unsteady.

2.6.1 Integration of Euler’s Equation in Steady Rotational Flows For steady flows, Euler’s equation (Eq. (2.109)) reduces to − → ∇p − → → v .∇ − v =− + f ρ

(2.113)

If the flow is also barotropic,8 then ρ = ρ (p). Thus, for incompressible flows, Eq. (2.113) is rewritten as − → → v .∇ − v = −∇

  p − → + f ρ

It can be seen that integration of Eq. (2.113) is conceivable if performed along a streamline and the body force is considered to be conservative (irrotational). Now, by using the identity  −   →  →  − → − → → − − → − →  → → ∇ − a.b = − a .∇ b + b .∇ − a +→ a × ∇ × b + b × ∇ ×− a (2.114) →  − − → → where − a and b vectors are the functions of position, the term − v .∇ → v in Eq. (2.113) can be expressed as

8

Constant density, isothermal, and isentropic flows are barotropic.

2.6 Bernoulli’s Equation

75

→ −  →  − →  −   →   → → → ∇ − v .→ v = − v .∇ → v + − v .∇ → v +− v × ∇ ×− v +− v × ∇ ×− v (2.115)  →  −   1 − → → → ∇ → v .− v = − v .∇ → v +− v × ∇ ×− v 2  2   →  − v → → −− v × ∇ ×− v = − v .∇ → v ∇ 2

(2.116) (2.117)

→  − Eliminating − v .∇ → v from Eq. (2.113) by means of Eq. (2.117), we get  ∇

v2 2



  → → −− v × ∇ ×− v = −∇

  p − → + f ρ

(2.118)

Since ∇ operator has only spatial derivatives, Eq. (2.118) becomes  ∇

   − v2 p → → → + −− v × ∇ ×− v = f ρ 2

(2.119)

  → → → Note that the term − v × ∇ ×− v is a vector normal to the velocity − v and, by − → definition, v is tangential to the streamline. Therefore, the dot (scalar) product of   − → − → → v × ∇ ×− v with a small element dr along the streamline is equal to zero. Hence, − → by taking the scalar product of Eq. (2.119) with dr , we get  ∇

 p v2 − → − →− → . dr = f . dr + ρ 2

(2.120)

Since the body forces are assumed to be irrotational (as is usually the case), they can be represented as the gradient of a scalar function such as . Thus, − → f = ∇

(2.121)

This allows us to rewrite Eq. (2.120) as  ∇ which shows that ∇ (2.122) integrates to

p ρ

+

v2 2

 v2 p − → + − . dr = 0 ρ 2

(2.122)

! + is a vector normal to the streamline. Equation

v2 p + − = constant ρ 2

(2.123)

76

2 Governing Equations and Thermodynamics of Compressible Flows

In the fluid flow problems, the body force consists of only gravitational force and this is irrotational. Hence, can be expressed in terms of the uniform gravitational force per unit mass g pointing in the negative z-direction as = −gz

(2.124)

Combining Eqs. (2.124) and (2.123), we get v2 p + + gz = B1 = constant along a streamline ρ 2

(2.125)

which is Bernoulli’s equation along a streamline. The Bernoulli constant B1 in Eq. (2.123) will have different values along different streamlines in a multi-dimensional flow.

2.6.2 Integration of Euler’s Equation in Unsteady Irrotational Flows An unsteady form of Bernoulli’s equation can be derived only if the flow is irrotational. Consider an unsteady, irrotational flow of a barotropic fluid under conservative − → body forces. For unsteady conditions, the local acceleration term ∂∂tv in Eq. (2.109) must be retained. However, if the flow is assumed to be irrotational the curl of velocity → → is zero, i.e., ∇ × − v = 0, and thus − v can be expressed as →  − → v = ∇φ − r ,t

(2.126)

→  where φ = φ − r , t is a scalar function of position and time.

− → Because the body forces are considered to be conservative, f can be represented in the same way as Sect. 2.6.1. − → f = −∇

(2.121)

where is a scalar function of position. Substituting Eqs. (2.126) and (2.121) into Eq. (2.109), we find that ∂ (∇φ) ∇p + +∇ ∂t ρ



v2 2

 + ∇ = 0

(2.127)

For the barotropic flows with constant density, Eq. (2.127) can be expressed as

2.6 Bernoulli’s Equation

77

    2 p v ∂φ +∇ +∇ + ∇ = 0 ∇ ∂t ρ 2   ∂φ p v2 ∇ + + + =0 ∂t ρ 2 

(2.128) (2.129)

This readily integrates to p v2 ∂φ + + + =0 ∂t ρ 2

(2.130)

If the gravity is the only body force in the flow, then can be eliminated from Eq. (2.130) by means of Eq. (2.124). This gives ∂φ p v2 + + + gz = B2 (t) = constant ∂t ρ 2

(2.131)

Equation (2.131) is the generic Bernoulli’s equation for unsteady flows and is therefore referred to as the unsteady Bernoulli’s equation. Since the integration is performed with respect to spatial coordinates, Bernoulli’s constant B2 in Eq. (2.131) will be space independent but will still be time dependent. So, it is represented as B2 (t). At any moment, B2 (t) will have a constant value throughout the flow. If the flow is steady, the flow properties are time independent, and Eq. (2.131) becomes v2 p + + gz = B3 = constant throughout flow field ρ 2

(2.132)

Equation (2.132) is the famous Bernoulli’s equation. It was derived by the Swiss mathematician and physicist Daniel Bernoulli, long before the invention of Euler’s equation, using a method similar to the modern energy conservation law. The constant B3 is now a real constant, i.e., independent of both space and time. It has the same value all the time throughout the flow.

2.7 Conservation of Energy: The First Law of Thermodynamics The law of conservation of energy holds that energy cannot be generated or destroyed, but that different types of energy can be interchanged. When applied to a system (control mass), this law leads to the first law of thermodynamics, which states that the net balance of heat and work conducted across the system boundaries equals the total amount of energy to be stored in the system. The first law of thermodynamics is written as follows:

78

2 Governing Equations and Thermodynamics of Compressible Flows

˚ −W ˚ E˚ = Q

(2.133)

˚ and W ˚ are the heat and work transfer rates, respectively. The total quantity where Q of energy stored within the system, often known as the system energy, is represented ˚ The heat and work transfer sign conventions are as follows: when heat is by E. added/extracted from the system, Q is positive/negative, and when work is performed by/on the system, W is positive/negative. Equation (2.133) can be written in differential form as dE = δQ − δW

(2.134)

where δQ and δW are the amounts of heat and work transfers across the system boundaries, respectively. The symbol δ is used to indicate that they are not exact differentials but rather depend on the path followed. The system energy, denoted by E in Eq. (2.134), is composed of the internal energy U, the kinetic energy Ek , the potential energy Ep , and other forms of energy such as those produced by chemical reactions, nuclear reactions, and electrostatic or magnetic field effects. However, in our analysis, we ignore the other types of energy and emphasize solely on the first three types of system energy: E = U + Ek + E p

(2.135)

Let z be defined as the elevation of the system from a reference datum; Eq. (2.135) can be represented as 1 E = U + mv2 + mgz 2

(2.136)

This, when expressed in terms of specific quantities, becomes 1 e = u + v2 + gz 2

(2.137)

In many practical applications, the potential energy, kinetic energy, and chemical energy of a thermodynamic system are constant or insignificant. In these cases, the first law is represented in terms of specific quantities as du = δq − δw

(2.138)

If a thermodynamic system changes from an initial state 1 to a final state 2 at constant pressure (isobaric process), the mechanical work associated with the change d∀ in specific volume is simply pd∀. If no other work is performed, the total work per unit mass δw is given by

2.7 Conservation of Energy: The First Law of Thermodynamics

79

δw =pd∀

(2.139)

  1 δq = du + pd∀ = du + pd ρ

(2.140)

Hence, Eq. (2.138) becomes

Because it is appropriate for managing systems of a fixed mass, the expression of the first law given by Eq. (2.140) is often known as the control mass form.

2.7.1 Concept of Enthalpy Consider a thermodynamic system containing a fixed amount of a gas; heat addition or extraction at constant volume changes the internal energy of the gas. However, in the real world, most processes do not occur at constant volume, but rather at constant pressure. Thus, a mathematical function that relates the quantity of heat transfer across the boundary of the system under constant pressure is required. This mathematical function, known as enthalpy, is expressed in terms of specific quantities as h = u + p∀ (2.141) That is, the enthalpy is essentially the sum of internal energy and flow work—the energy held by a gas to cause the flow across the system boundary. Differentiating Eq. (2.141) yields dh = du + pd∀ + ∀dp

(2.142)

δq = dh − ∀dp dp δq = dh − ρ

(2.143)

Thus, Eq. (2.140) becomes

(2.144)

For the constant pressure process, dp = 0; therefore, δq = dh

(2.145)

Thus, the change in enthalpy at constant pressure is equal to the amount of heat added to the system.

80

2 Governing Equations and Thermodynamics of Compressible Flows

2.7.2 Integral Form of Energy Equation for a Finite Control Volume If we utilize the dummy variables equal to total energy, i.e., B = E, and energy per dE = e in Eq. (2.39). This provides unit mass β = dm 

DE Dt

 = sys

∂ ∂t

"

˚ ρed∀+

 − → → e ρ− v . ds

(2.146)

CS

C∀

Substituting Eq. (2.133) into the above equation yields ˚ −W ˚ = ∂ Q ∂t

"

˚ ρed∀+

 − → → e ρ− v . ds

(2.147)

CS

C∀

˚ and W ˚ in detail. The heat Let us now examine the heat and work transfer terms Q ˚ can be divided into conduction, convention, and radiation modes of heat transfer Q transfer; however, their individual discussions are lengthy and beyond the scope of ˚ unbroken and consider it the present text. For this reason, we will leave the term Q only occasionally. If the volumetric rate of heat addition per unit mass is q˚ the rate of heat addition to an elemental fluid volume d∀ inside the control volume will be dq˚ = q˚ ρd∀

(2.148)

Thus, the total heat transfer across the system boundary is ˚ q˙ ρd∀

˚ = Q

(2.149)

C∀

˚ across the control surface is divided into following The rate of work transfer W parts: ˚ =W ˚ p+W ˚ s+W ˚ b+W ˚ sh W

(2.150)

It should be noted that the work done by gravitational forces is not included here because it is already mentioned in Eq. (2.137). The rate of work done by pressure forces, also known as flow work, is only performed at the control surface and not inside the control volume; all work done on the fluid inside the control volume − → by equal and opposite pressure forces is effectively zero. Let ds represent a small surface element on the control surface. This fluid element is subjected to the following − → − → pressure force: dF = −p ds . Thus, the work performed by the pressure force is

2.7 Conservation of Energy: The First Law of Thermodynamics

 − → → ˚p=p − dW v . ds

81

(2.151)

Integrating across the whole CS surrounding the fluid body in C∀ produces ‹ " ˚p= W

 − → → p − v . ds

(2.152)

CS

"

p − − → ρ→ v . ds ρ

˚p= W CS

" ˚p= W

 − → → p∀ ρ − v . ds

(2.153)

(2.154)

CS

˚ s , which is comThe rate of work due to viscous forces at the control surface, W posed of the products of tangential and normal stresses with their corresponding velocity components, is " −  → → ˚s=− σ .− v ds W

(2.155)

CS

˚ b denotes the rate of work done by the body forces. Consider an The term W − → elemental volume d∀ within the control volume to get its expression. If f is the body force per unit mass, the rate of work done on the elemental volume owing to body force is →  → ˚ b = −ρ − dW f .− v d∀ Summing over the entire CS encompassing the fluid body in C∀ yields ˚ ˚b=− W

ρ

−  →− f .→ v d∀

(2.156)

C∀

˚ sh is the work carried out by an external turboFinally, the rate of shaft work W machinery on the control volume, protruding through the control surface. If the rate ˙ sh , then of shaft work per unit volume is denoted by W ˚ ˚ sh ρd∀ W

˚ sh = − W C∀

(2.157)

82

2 Governing Equations and Thermodynamics of Compressible Flows

˚ essentially comprises Thus, the net rate of work W " ˚ = W

˚  " ˚  −  − → − → → → → → ˚ sh ρd∀ σ .− v ds − W p∀ ρ − v . ds − ρ f .− v d∀ −

CS

CS

C∀

C∀

(2.158) By combining all of these terms in Eq. (2.147), we get ˚ −W ˚ s−W ˚ b−W ˚ sh = ∂ Q ∂t ˚ b −W ˚ sh = ∂ ˚ −W ˚ s−W Q ∂t

˚

˚ C∀

 "  p  − − → e+ ρ→ v . ds ρed∀ + ρ

(2.159)

CS

   "  1 1 − → → u + p∀ + v2 + gz ρ − ρ u + v2 + gz d∀ + v . ds 2 2

C∀

CS

(2.160) The term u + p∀ indicates the specific enthalpy h; therefore, the energy equation (i.e., the first law of thermodynamics) for the flow through a control volume is given by ˚ b−W ˚ sh = ∂ ˚ −W ˚ s−W Q ∂t

˚ C∀

   "  1 1 − → → h + v2 + gz ρ − ρ u + v2 + gz d∀ + v . ds 2 2 CS

(2.161) Equation (2.161) is the integral form of the energy equation. It is, in fact, the application of the fundamental law of thermodynamics to a fluid flow. As illustrated in the surface integral term of Eq. (2.161), during a flow process through the control volume, the internal energy u is always coupled with flow work p∀. Therefore, the specific enthalpy of the fluid, given by Eq. (2.161), is useful. Moreover, the volume integral term Eq. (2.161) indicates the unsteady stored energy term. Because it lacks p∀, just u appears in the integral.       ˚ = 0 , inviscid W ˚ s = 0 , without If the flow is steady ∂t∂ = 0 , adiabatic Q     ˚ sh = 0 , then Eq. (2.161) reduces to ˚ b = 0 , and zero shaft work W body forces W  "  1 2 − → → h + v + gz ρ − v . ds = 0 2 CS

(2.162)

2.7 Conservation of Energy: The First Law of Thermodynamics

83

2.7.3 Conservative Partial Differential Form of Energy Equation for a Differential Control Volume In Eq. (2.161), by applying Gauss’ divergence theorem to transform the surface integrals to volume integrals, and then applying Leibniz’s theorem in the first term on the right-hand side, we obtain ˚

˚ ρ q˚ d∀ + C∀

C∀

→ −  ∇. − σ .→ v d∀ +

˚ ρ C∀

   ˚ ˚ −  1 ∂ →− ρ u + v2 + gz d∀ f .→ v d∀+ ρ W˚ sh d∀ = ∂t 2 C∀

C∀

    ˚ 1 → ∇. ρ h + v2 + gz − + v d∀ 2

(2.163)

C∀

Collecting all the terms inside the same volume integral, we get ˚  C∀

        − → −  1 ∂ 1 → → → v − ρ q˚ − ρ W˚ sh − ρ f .− v − ∇. − σ .→ v ρ u + v2 + gz + ∇. ρ h + v2 + gz − =0 ∂t 2 2

(2.164) Because the control volume is drawn arbitrarily in space, the above integral holds only if the integrand itself is zero. Thus,        − → −  1 ∂ 1 → → → ˚ v = ρ q˚ + ρ f .− v + ∇. − σ .→ v + ρW ρ u + v2 + gz + ∇. ρ h + v2 + gz − sh ∂t 2 2

(2.165)

  Since the total energy per unit mass is defined as, e = u + 21 v2 + gz , and specific   enthalpy is defined as, h = u + ρp , Eq. (2.165) becomes     − → −  ∂ p − → → → ˚ sh v = ρ q˚ + ρ f .− v + ∇. − σ .→ v + ρW [ρe] + ∇. ρ e + ∂t ρ (2.166) −   −      ∂ → → → → → ˚ sh v = ρ q˚ − ∇. p− v + ρ f .− v + ∇. − σ .− v + ρW [ρe] + ∇. ρe→ ∂t (2.167) Equation (2.167) is the energy equation in conservation form. In the absence of shaft work, Eq. (2.167) in the condition of steady adiabatic inviscid fluid flow without body forces becomes  →  → ∇. ρe− v = −∇. p− v      → 1 → ∇. ρ u + v2 + gz − v = −∇. p− v 2

(2.168) (2.169)

84

2 Governing Equations and Thermodynamics of Compressible Flows

Beginning with Chap. 4, Eqs. (2.162) and (2.169) are thoroughly examined and applied.

2.7.4 Non-conservative Partial Differential Form of Energy Equation for a Moving Fluid Element Using the chain rule to expand the first term on the left side of Eq. (2.167) and applying the vector identity (Eq. (2.60)) in the second term yield ρ

∂ ∂t

         → 1 ∂ρ 1 1 1 → v + ρ− v .∇ h + v2 + gz u + v2 + gz + u + v2 + gz + h + v2 + gz ∇. ρ − 2 2 ∂t 2 2 − → −  → → (2.170) = ρq ˙ + ρ f .− v + ∇. − σ .→ v + ρ W˚ sh

By definition, h = u + p∀; hence,         → 1 ∂ρ 1 1 → v + ρ− v .∇ u + u + v2 + gz + u + v2 + gz + u + v2 + gz ∇. ρ − 2 2 ∂t 2   − → −  p p  → → → → v + ρ− v .∇ = ρ q˚ + ρ f .− v + ∇. − v + ρ W˚ sh σ .→ + ∇. ρ − ρ ρ         → ∂ 1 ∂ρ 1 1 → ρ v + ρ− v .∇ u + u + v2 + gz + u + v2 + gz + u + v2 + gz ∇. ρ − ∂t 2 2 ∂t 2 −  → → −  → → + ∇. p− v = ρ q˚ + ρ f .− v + ∇. − σ .→ v + ρ W˚ sh ρ

∂ ∂t

1 2 v + gz 2



(2.171) 1 2 v + gz 2



(2.172) (2.173)

Combining the second and third terms on the left side and substituting Eq. (2.57) result in ρ

∂ ∂t

        → 1 ∂ρ 1 1 → v .∇ u + v2 + gz u + v2 + gz + u + v2 + gz + ∇. ρ − v + ρ− 2 2 ∂t 2



0

−  → → −  → → = ρ q˚ − ∇. p− v + ρ f .− v + ∇. − σ .→ v + ρ W˚ sh      −  → → −  ∂ 1 1 → → → ρ v .∇ u + v2 + gz = ρ q˚ − ∇. p− v + ρ f .− v + ∇. − σ .→ v + ρ W˚ sh u + v2 + gz + − ∂t 2 2





D 1 2 Dt u+ 2 v +gz

Therefore,



(2.174) (2.175)

2.7 Conservation of Energy: The First Law of Thermodynamics

ρ

85

  −  → → −  1 D → → ˚ sh u + v2 + gz = ρ q˚ − ∇. p− v + ρ f .− v + ∇. − σ .→ v + ρW Dt 2 (2.176)       De − →→ → → → ˚ sh ρ = ρ q˚ − ∇. p− v + ρ f .− v + ∇. − σ .− v + ρW Dt (2.177)

This is the energy equation in its non-conservation (convective) form. Example 2.1 If the air is expanded from an initial state (p1 , ∀1 ) to the final state (p2 , ∀2 ), following a polytropic process p∀n = constant. Show that the work done is p1 ∀1 − p2 ∀2 n−1 Solution Assume the air expansion process to be reversible. Thus, the work done is ˆ2 W=

pd∀ 1

The process equation is p∀n = constant = C and its logarithmic differentiation yields d∀ dp +n =0 p ∀ ∀ dp d∀ = − n p Substituting in the work expression gives ˆ2 W=− 1

But ∀ =

  n1 C p

; thus,

∀ dp n

86

2 Governing Equations and Thermodynamics of Compressible Flows

ˆ2 W=−

1 n

  n1 C dp p

1 1

Cn =− n

ˆ2   n1 1 dp p 1

1

= −C n 1

= −C n

n−1 p( n )

!2 1

(n − 1) ! ( n−1 ) ( n−1 ) p2 n − p1 n (n − 1)

Hence,

W=

p1 ∀1 − p2 ∀2 n−1

2.8 Increase of Entropy Principle: The Second Law of Thermodynamics We know from experience that when a hot body interacts with a cold body, the hot body cools while the cold body warms up. However, Eq. (2.134) does not imply that this will always be the case. Yes, the first law needs the cold body to cool and the hot body to heat up as long as energy is conserved during the process. We know from experience, however, that a hot body can never take heat from a cold body on its own. The second law of thermodynamics is the fundamental law that specifies the direction in which a process will occur.

2.8.1 Concept of Entropy The second law of thermodynamics specifies a phenomenon known as entropy, which starts with the concept of a reversible process. A reversible process is one that can be reversed without leaving any trace on the environment. It means that at the completion of the reverse process, both the system and its surroundings are returned to their initial states. To compensate for gradual changes in thermodynamic properties, a reversible process occurs very slowly. A reversible process is thus a quasi-static process.9 An 9

A quasi-static process is the process that occurs slowly enough that the system remains at thermodynamic equilibrium at all times, in spite of the fact that the system changes over time.

2.8 Increase of Entropy Principle: The Second Law of Thermodynamics

87

actual process, on the other hand, is always irreversible and cannot be reversed without affecting the system or its surroundings. We know from experience that not all of the heat given to a heat engine can be converted into useful work. This is due to the fact that the conversion of heat to work is frequently followed by a degraded portion of the energy given, which is less useful. Furthermore, as previously stated, the first law does not apply to the portion of heat that can be transformed into work. Rather, it is concerned with the amount of heat that is transformed to work. The second law, in fact, is concerned with the proportion of heat that may be transformed into useful work. According to the second law, any natural system left alone will change instantly and attain a state of equilibrium in an irreversible process. When a change or process pushes a system towards equilibrium, the system loses the ability to make immediate changes. The property associated with the potential of a system for instantaneous change is known as its entropy, which is symbolized by the symbol s. Entropy is essentially a measure of the degradation of the capacity to produce work. For a closed system that undergoes a cyclic change, where the heat per unit mass δqR is added reversibly at the absolute temperature T, we can write ˛

δqR =0 T

(2.178)

In Eq. (2.178), Clausius defines the term δqTR in terms of a change in a thermodynamic property known as the entropy (1850). It is commonly represented by the symbol s. Thus, the change in entropy of a system can be represented as ds =

δqR T

(2.179)

where δqR is the incremental quantity of heat added reversibly to the system. Because entropy is a state variable, it can be utilized in combination with any form of process, reversible or irreversible, that connects the same initial and final states. Thus, for a given beginning and final state, if we replace a reversible process with an irreversible one, an alternative and more explicit relation for entropy change can be written as ds =

δq + dsir T

(2.180)

In this equation, δq represents the actual quantity of heat added in an irreversible process, and dsir is the entropy generated by dissipative phenomena occurring inside the system, such as fluid viscosity, thermal conductivity, and mass diffusion. Because these phenomena are irreversible in nature, they constantly increase the entropy of the system, i.e., dsir  0

(2.181)

88

2 Governing Equations and Thermodynamics of Compressible Flows

Finally, the entropy equation for a system is obtained by combining Eqs. (2.179) and (2.180): ds 

δq T

(2.182)

The equality in Eq. (2.182) applies to reversible processes and the inequality to irreversible (natural) processes. Thus, if the process is adiabatic δq = 0, Eq. (2.182) is reduced to ds  0

(2.183)

Equations (2.183) and (2.182) are two distinct variations of the second law of thermodynamics. These equations demonstrate that all natural processes in isolated closed systems frequently proceed in a direction that increases entropy. This is known as the increase of entropy principle. Thus, for an adiabatic process, the entropy of the universe (system plus surroundings) can never be decreased. When ds > 0, the process is said to be irreversible, and whends = 0, the process is said to be reversible. The isentropic process is a reversible and adiabatic process.

2.8.2 Integral Form Consider a system of mass m; if the entropy per unit mass of the system is s, then the total entropy is S = ms. Hence, Eq. (2.182) becomes dS 

δQ T

(2.184)

It should be noted that the aforementioned equation is true for a fluid element, and thus the time rate of change in entropy of the system can be stated in terms of particle or substantial derivative of S. Therefore, ˙ DS δQ  Dt T

(2.185)

˝ In terms of the properties B and β, they are as follows: B = S = C∀ sρd∀ and β = s. Invoking S and s in Eq. (2.39) and using the integral theorem of Leibniz yields DS = Dt

˚ C∀

∂ (ρs) d∀ + ∂t

" CS

Combining Eqs. (2.185) and (2.186) provides

 − → → s ρ− v . ds

(2.186)

2.8 Increase of Entropy Principle: The Second Law of Thermodynamics

˚ C∀

∂ (ρs) d∀ + ∂t

‹ "

 ˙ − → δ Q → s ρ− v . ds  T

89

(2.187)

CS

This is known as the entropy equation, and it is the integral form of the second law of thermodynamics for a control volume.

2.8.3 Conservative Partial Differential Form In this section, we derive the conservative partial differential form of the entropy equation by converting the surface integral in Eq. (2.187) to a volume integral using the Gauss divergence theorem, and then applying the result to an infinitesimal fluid element (i.e., a control volume of differential size), which yields ˚ C∀

˙ = Let δQ

˝ C∀

∂ (ρs) d∀ + ∂t

‹ ˚

˙  → Q ∇. ρs− v d∀  T

(2.188)

C∀

δ q˙ d∀, where δ q˙ is the rate of heat transfer per unit volume; thus, ˚  C∀

∂ (ρs) + ∂t

‹

  → δ q˙ d∀  0 ∇. ρs− v − T

(2.189)

For any given control volume drawn in space, the integral will vanish only if the integrand vanishes identically. Thus, ∂ (ρs) + ∂t

‹

∂ (ρs) + ∂t

 → δ q˙ ∇. ρs− 0 v − T ‹

 → δ q˙ ∇. ρs− v  T

(2.190)

(2.191)

This is the conservation form of the entropy equation.

2.8.4 Non-conservative Partial Differential Form The second term in Eq. (2.191) is expanded using the vector identity (Eq. (2.60)). s

∂ρ ∂s +ρ + ∂t ∂t

‹

 → δ q˙ → s∇. ρ − v + ρ− v .∇s  T

(2.192)

90

2 Governing Equations and Thermodynamics of Compressible Flows

 s

∂ρ + ∂t

‹

    → ∂s − δ q˙ ∇. ρ − v +ρ +→ v .∇s  ∂t T

(2.193)

However, the first two terms constitute the continuity equation (Eq. (2.57)) and hence vanish     ‹  → ∂s − ∂ρ δ q˙ s + ∇. ρ − v +ρ +→ v .∇s  (2.194) ∂t ∂t T 

0

 ∂s − δ q˙ +→ v .∇s  ∂t T 



ρ

(2.195)

Ds Dt

Therefore, we have ρ

δ q˙ Ds  Dt T

(2.196)

Equation (2.196) is the desired non-conservation form of the entropy equation.

2.9 Combined Expressions of the First and Second Laws of Thermodynamics Consider the first law of thermodynamics for a closed system: du = dq − dw

(2.138)

Work transfer at the boundary of a closed system that is undergoing a reversible process can be represented as dw = pd∀

(2.197)

and the second law of thermodynamics can be used to express heat transfer as dq = Tds

(2.198)

Substituting Eqs. (2.197) and (2.198) into Eq. (2.138) results in the following equation, which is only valid for reversible processes:

2.9 Combined Expressions of the First and Second Laws of Thermodynamics

Tds = du + pd∀ = du + pd

  1 ρ

91

(2.199)

This is a combination of the first and second laws of thermodynamics. Equation (2.199) is also known as the Gibbs equation or the first Tds. If the other forms of work are also performed, Eq. (2.199) can have the form Tds = du + pd∀ − XdY

(2.200)

where X represents a pressure-like number and Y represents a volume-like quantity. Additionally, we must note that, while Eq. (2.199) is derived for a reversible process, it is also valid for irreversible processes. This is due to the fact that Eq. (2.199) only contains state variables, and so their change is independent of the path taken. Equation (2.199) is sometimes more conveniently written in terms of enthalpy or specific enthalpy, h = u + pd∀; when this equation is differentiated, the result is dh = du + pd∀ + ∀dp

(2.201)

Substituting Eq. (2.199) in the preceding expression and rearranging terms result in the second combined form of first and second laws: Tds = dh − ∀dp

(2.202)

Since ∀ = ρ1 , where ρ is the density, Eq. (2.202) becomes Tds = dh −

dp ρ

(2.203)

Equations (2.199), (2.202), and (2.203) are known as the Gibbs equations or Tds equations.

2.10 Thermal and Calorical Properties of a Perfect Gas A certain number of thermodynamic properties can be used to fix the state of a given system in thermodynamics. The equation of state expresses the relation between the properties that determine the thermodynamic state. The equation of state can be used to estimate the third property if any two are known. For a perfect gas, pressure p, volume ∀, density ρ, and temperature T are the referred thermal properties, and internal energy u, enthalpy h, and entropy s are the commonly used calorical properties. The equation that relates thermal properties is known as the thermal equation of state, and any relationship between thermal properties and calorical properties is known as the calorical equation of state.

92

2 Governing Equations and Thermodynamics of Compressible Flows

2.10.1 Thermal Equation of State A perfect gas, often called as an ideal gas, is a gas that follows Boyle’s and Charles’ laws. They are described for any thermodynamic process as follows: Boyle’s Law Boyle’s law, also known as Mariotte’s law, states that the pressure p exerted by a given gas is inversely proportional to its volume ∀ , if the temperature remains constant while the gas compresses or expands. Thus,   p∀ T = constant

(2.204)

At low pressure (less than 1 atm) and high temperature, real gases obey Boyle’s law. The Charles Law If the pressure is held constant, this law states that the volume ∀ occupied by a given amount of gas is directly proportional to its absolute temperature T. The Charles law applies to real gases with sufficiently low pressure and high temperature. Hence, 

∀ T

 = constant

(2.205)

p

Combining Eq. (2.204) with Eq. (2.205) yields the perfect gas equation (or ideal gas law): p∀= RT

(2.206)

where ∀ denotes the specific volume and R denotes the specific gas constant. The following expression is used to calculate the value of R. R=

R¯ u Mw

(2.207)

J where Ru =8314 (kg.K) is the universal gas constant and Mw is the molecular weight. 1 Because ρ = ∀ , where ρ is the density and ∀ is the specific volume of the gas, Eq. (2.206) can alternatively be written as

p = ρRT

(2.208)

We can observe from the above discussion that Eqs. (2.206) and (2.208) are the two forms of thermal equation of state for a perfect gas.

2.10 Thermal and Calorical Properties of a Perfect Gas

93

2.10.2 Calorical Equation of State We can see from the state principle that by specifying any two independent thermodynamic properties, all other properties are fixed. For any gas, let us represent the internal energy as a function of temperature and specific volume, often known as the calorical equation of state: u= u (T, ∀)

(2.209)

Differentiating the above yields  du=

∂u ∂T





∀

dT +

∂u ∂∀

 d∀

(2.210)

T

It may be demonstrated that, for a thermally ideal gas, 

∂u ∂∀

 =0

(2.211)

T

This demonstrates that the internal energy of a thermally perfect gas is determined solely by temperature. Thus, u = u (T)

(2.212)

Thus, Eq. (2.210) reduces to  du=

∂u ∂T

 dT

(2.213)

∀

Combining Eqs. (1.71) and (2.213) yields du= c∀ dT

(2.214)

where c∀ represents the specific heat at constant volume. The integration of Eq. (2.214) yields ˆ

u

uref

ˆ du =

T

c∀ dT

(2.215)

Tref

where Tref is an arbitrary reference temperature where u = uref . For a constant c∀ , the above integration becomes u − uref = c∀ (T − Tref )

(2.216)

94

2 Governing Equations and Thermodynamics of Compressible Flows

Assume that at zero reference temperature, the reference internal energy is zero; Eq. (2.216) reduces to u = c∀ T

(2.217)

Consider the enthalpy of a general gas as a function of temperature and pressure: h= h (T, p) In terms of exact differentials,     ∂h ∂h dT + dp dh= ∂T p ∂p T

(2.218)

(2.219)

However, it can be demonstrated for a perfect gas that 

∂h ∂p

 =0

(2.220)

T

Invoking Eq. (2.220) into Eq. (2.219) and then combining with Eq. (1.76) result in dh= cp dT

(2.221)

where cp is the specific heat at constant pressure. The integration of Eq. (2.221) yields ˆ

h

href

ˆ dh =

T

c∀ dT Tref

where h = href at an arbitrary reference temperature Tref . For a constant cp , the above expression yields h − href = cp (T − Tref )

(2.222)

If both href and Tref are arbitrarily chosen zero, we eventually get h = cp T

(2.223)

It should be noted that Eqs. (2.214) and (2.221) are two different forms of the calorical equation of state for a perfect gas.

2.10 Thermal and Calorical Properties of a Perfect Gas

95

2.10.3 Thermal and Calorical Perfectness of a Gas A thermally perfect gas is one that obeys the thermal equation of state (Eq. (2.206)). A perfect gas is essentially a gas with a large intermolecular distance and, as a result, small intermolecular forces between the gas molecules. In addition, for a perfect gas   ∂h  ∂u and c = are constant and hence independent of temperature. both cp = ∂T ∀ ∂T ∀ p cp = cp (T) ⇒ cp = constant

(2.224)

c∀ = c (T) ⇒ c∀ = constant

(2.225)

This type of gas is known as a calorically perfect gas. A perfect gas is always thermally and calorically perfect, that is, it satisfies both the thermal and caloric equations of state. It is important to note that a calorically perfect gas is always thermally perfect, but a thermally perfect gas may or may not be calorically perfect. Thus, thermal perfection is a prerequisite for caloric perfection.

2.10.4 Specific Heat Relationships The equation of state for a thermally perfect gas is given by P∀= RT

(2.206)

Pd∀ + ∀dp= RdT

(2.226)

Differentiating the above yields

Because enthalpy is defined as h = u + p∀, differentiation produces dh= du + pd∀ + ∀dp dh − du= pd∀ + ∀dp

(2.227)

Combining Eqs. (2.226) and (2.227), we obtain dh − du= RdT

(2.228)

Finally, substituting Eqs. (2.214) and (2.221) into the preceding equation yields cp − c∀ = R

(2.229)

This is referred to as Mayer’s relation. Some intriguing inferences can be drawn from Eq. (2.229). For a thermally perfect gas, cp and c∀ may be functions of

96

2 Governing Equations and Thermodynamics of Compressible Flows

temperature, but their difference is always constant. Moreover, it can be seen from Eq. (2.229) that cp > c∀

(2.230)

That is, the specific heat at constant pressure is always greater than the specific heat at constant volume. This is because the temperature rise in a constant pressure process is always less than that in a constant volume process. In other words, for an equivalent rise in temperature, the amount of heat addition required at constant pressure is invariably greater than at constant volume. c Let the ratio of specific heats be denoted by γ = c∀p , then Eq. (2.229) shows that γR (γ − 1) R c∀ = (γ − 1) cp =

(2.231) (2.232)

The ratio of specific heats can also be expressed in terms of the number of degrees of freedom α in the gas molecules, as shown by Eq. (1.94). Thus, γ= 1 +

2 α

(2.233)

• For a monoatomic gas, α = 3, hence γ =

5 = 1.67 3

• For a rigid diatomic gas, α = 5, hence γ =

7 = 1.4 5

• For a polyatomic gas, α = 6, hence γ =

8 = 1.33 6

The above results are most accurate for monoatomic gases, less accurate for diatomic gases, and least accurate for polyatomic gases. Because some polyatomic gases, such as gaseous compounds of Uranium, have high α values, the value of γ is slightly more than unity. Hence, in general, γ is found in the range 1 γ 1.67

(2.234)

2.10 Thermal and Calorical Properties of a Perfect Gas

97

Example 2.2 Consider an ideal gas (molecular weight 28.97 g/mol) in which the speed of sound is found to be 400 m/s at 100 ◦ C. Determine cp , c∀ , and γ . Solution The molecular weight of the ideal gas is Mw = 28.97 Thus, the specific gas constant is Ru Mw 8314 = 28.97 ≈ 287 J/(kg.K) R=

From the given value of speed of sound,

=

"

a=

"

γ RT

γ × 287 × 373 = 400 m/s

Hence, the ratio of specific heats is γ ≈ 1.5 Therefore, γR (γ − 1) 1.5 × 287 = 0.5 = 861 J/(kg.K)

cp =

and R (γ − 1) 287 = 0.5 = 574 J/(kg.K)

c∀ =

98

2 Governing Equations and Thermodynamics of Compressible Flows

2.10.5 Change in Entropy of a Perfect Gas The Gibbs equations, as obtained in Sect. 2.9, can be used to calculate the expressions for the change in entropy of a perfect gas in a reversible process in terms of thermodynamic properties. Thus, writing the first Tds Tds = du + pd∀

(2.199)

For a perfect gas, p∀ = RT, du = c∀ dT, and dh = cp dT. Thus, RT d∀ ∀ d∀ dT +R ds = c∀ T ∀

Tds = c∀ dT +

(2.235) (2.236)

The integration of Eq. (2.236) between an initial state 1 and a final state 2 results in ˆ s =

ˆT2 ds =

dT +R c∀ T

T1

ˆ∀2 ∀1

d∀ ∀

(2.237)

Assuming that the gas is also calorically perfect, ˆT2

(2.238)

T2 ∀2 s = c∀ ln + R ln T1 ∀1

(2.239)

s T2 ∀2 = ln + (γ − 1) ln c∀ T1 ∀1

(2.240)

T1

However, c∀ =

R , (γ −1)

ˆ∀2

d∀ ∀

s = c∀

dT +R T

∀1

thus

Using the ideal gas law and substituting ∀ = becomes ds = c∀

1 ρ

dρ dT −R T ρ

By integrating between the states 1 and 2, we get

into Eq. (2.199), Eq. (2.236)

(2.241)

2.10 Thermal and Calorical Properties of a Perfect Gas

ˆT2 s =

dT c∀ −R T

T1

99

ˆρ2 ρ1

dρ ρ

(2.242)

For a perfect gas with constant specific heats, Eq. (2.242) yields T2 ρ2 − R ln T1 ρ1 s T2 ρ2 = ln − (γ − 1) ln c∀ T1 ρ1 s = c∀ ln

(2.243) (2.244)

Similarly, by repeating the preceding steps, the second Gibbs equation (Eq. (2.202)) can be expressed as RT dp p dp dT −R ds = cp T p

Tds = cp dT −

(2.245) (2.246)

which, when integrated between the states 1 and 2, provides ˆT2

(2.247)

T2 p2 s = cp ln − R ln T1 p1

(2.248)

T1

For the perfect gas, cp =

ˆp2

dp p

s =

γR ; (γ −1)

dT −R cp T

p1

Eq. (2.248) becomes

s T2 γ − 1 p2 ln = ln − cp T1 γ p1

(2.249)

Example 2.3 A perfect gas, enclosed by an insulated (upright) cylinder and piston assembly, is at equilibrium at conditions p1 , ∀1 , T1 . A weight w is placed on the piston and after a number of oscillations, the motion subsides and the gas reaches a new equilibrium at conditions p1 , ∀1 , T1 . Find the temperature ratio TT21 in terms of the pressure ratio λ = pp21 and show that the change of entropy is given by    γ  1 1 + (γ − 1) λ γ −1 s = R ln λ γ If the initial disturbance is small, so that λ = 1 + ,  1, then show that

100

2 Governing Equations and Thermodynamics of Compressible Flows

Fig. 2.10 The insulated cylinder-piston assembly with a mass on the piston

2 s = R 2γ Solution Consider a certain amount of the perfect gas enclosed in an insulated cylinder-piston (upright) assembly, shown in Fig. 2.10. When the piston is pushed downwards, the internal energy of the system increases. Thus, the first law of thermodynamics in terms of specific quantities is q = u + w

(2.138)

Since the cylinder-piston assembly is insulated, q = 0 and if the compression process is reversible, then we have ˆ u = w =

− →− → F . ds =

ˆ ˆ # ## # →# → # #− #− # F # # ds # cos 0 = Fds

Note that the work is done on the system (gas) and thus w is negative in u expression. At final equilibrium position, if A is the piston face area, then F = p2 A. Also, the 2) . Therefore, the change in piston displacement between states 1 and 2 is ds = (∀1 −∀ A internal energy of the gas is

2.10 Thermal and Calorical Properties of a Perfect Gas

u = p2 (∀1 − ∀2 ) where ∀1 and ∀2 are the initial and final volumes, respectively. For a perfect gas p = ρRT, du = c∀ dT and c∀ = (γ R−1) . Thus, ˆ u =

ˆ du =

c∀ dT = p2 (∀1 − ∀2 )

R (T1 − T2 ) = p2 (∀1 − ∀2 ) (γ − 1)   T1 T2 (T1 − T2 ) = p2 − p1 p2 (γ − 1)   p2 (T1 − T2 ) = T1 − T2 p1 (γ − 1) (T1 − T2 ) = (λT1 − T2 ) (γ − 1)     T2 T2 = (γ − 1) λ − 1− T1 T1   1 + (γ − 1) λ T2 = T1 γ Now from isentropic relation    p2 T2 − R ln T1 p1 ⎧  γ ⎫ γ −1 ⎪ ⎪ ⎨ TT2 ⎬ s 1 = ln ⎪ ⎪ R λ ⎩ ⎭ 

s = cp ln

Substituting in

T2 T1

expression    γ  1 1 + (γ − 1) λ γ −1 s = ln R λ γ

For small initial disturbance, substituting in λ = (1 + )

101

102

2 Governing Equations and Thermodynamics of Compressible Flows

Fig. 2.11 An insulated tank divided into two halves

  γ   1 s 1 + (γ − 1) (1 + ) γ −1 = ln R γ (1 + )   γ 1 + (γ − 1) (1 + ) = ln − ln (1 + ) γ −1 γ     γ γ −1 = ln 1 +  − ln (1 + ) γ −1 γ From the expansion of logarithmic series, ln (1 + x) = x −

!

3 4 x2 + x3 − x4 +......∞ 2

,

      γ γ −1 γ − 1 2 2 2 s = − + ......∞ −  − + ......∞ R γ −1 γ γ 2 2      2  γ − 1 2 2  s = − −− + ......∞ = + ......∞ R γ 2 2 2γ s 2 ≈ R 2γ Example 2.4 An insulated tank is separated by a diaphragm into two equal volume compartments. One of the compartments is kept empty in the vacuum and the other contains a unit mass gas. Once the diaphragm is ruptured, both compartments are filled with the gas. Assuming the gas to be ideal, show that the final temperature of the gas is the same as its temperature and that the entropy increases by R ln 2. Solution Consider an insulated tank as shown in Fig. 2.11. Once the diaphragm ruptures, shown in Fig. 2.11b, the gas fills both the compartments. However, no work is carried out in the expansion of the gas, i.e., w ˙ = 0. Since the tank is insulated q˙ = 0, the first law of thermodynamics gives u = (uf − ui ) = 0 uf = ui

2.10 Thermal and Calorical Properties of a Perfect Gas

103

For a perfect gas, u = c∀ T. Hence, T2 = T1 From isentropic relation,    pf Tf − R ln s = cp ln Ti pi     pf Ti − R ln s = cp ln Ti pi   pf s = −R ln pi 

From thermal equation of state for the perfect gas,  s = −R ln

∀i ∀f



However, the volume occupied by the gas after rupturing of diaphragm is twice the initial volume, ∀f = 2∀i . Therefore,   1 s = −R ln 2 s = R ln 2

2.10.6 Isentropic Relations for a Perfect Gas We know that an adiabatic and reversible process is also known as an isentropic process. For an adiabatic process, δq = 0, and for a reversible process, the corresponding irreversibilities are zero, (dsir = 0). Thus, Eq. (2.180) indicates that for an isentropic process, ds = 0, that is, the entropy of a system remains constant if the state of the system is changed through an adiabatic and quasi-static process. The fundamental relations for an isentropic process can be determined directly from Eqs. (2.240) and (2.249) by setting s = 0. Thus, Eq. (2.240) gives    ∀2 T2 + (γ − 1) ln T1 ∀1    (γ −1) T2 ∀1 ln = ln T1 ∀2

 0 = ln

Taking anti-log on both sides, we obtain

(2.250) (2.251)

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2 Governing Equations and Thermodynamics of Compressible Flows

T2 = T1



∀1 ∀2

(γ −1) (2.252)

Because ∀ = ρ1 , Eq. (2.252) can also be written as T2 = T1



ρ2 ρ1

(γ −1) (2.253)

Similarly, from Eq. (2.249), we have 

T2 ln T1



  p2 (γ − 1) − =0 ln γ p1     (γ γ−1) T2 p2 ln =ln T1 p1

(2.254) (2.255)

Taking anti-log on both sides, we have   (γ γ−1) p2 T2 = p1 T1

(2.256)

We can deduce from Eqs. (2.253) and (2.256) that p2 = p1



ρ2 ρ1

γ

 =

T2 T1

 (γ γ−1) (2.257)

Equation (2.257) is a useful relationship that is frequently utilized in the analysis of compressible flows. Remark Despite the fact that real flow processes are not truly isentropic, they can be approximated to the perfect gas due to insignificant irreversible effects. Consider the movement of a jet engine through the nozzle. The development of entropy (i.e., the amount of disorder) is significant at the nozzle wall, where the irreversible dissipative processes of viscosity, thermal conductivity, and diffusion are substantial. Fluid particles outside the boundary layer, on the other hand, have minor dissipative effects and no heat transfer in or out of the fluid in the outer region. Thus, the flow outside the boundary layer undergoes reversible and adiabatic processes, resulting in an isentropic flow. Large spans of flow fields are frequently isentropic because the boundary layer is usually thin. For this reason, a study of the isentropic process is directly applicable to many different types of flow problems.

2.10 Thermal and Calorical Properties of a Perfect Gas

105

2.10.7 Thermal and Calorical Imperfectness of the Air Atmospheric air is a gas mixture composed primarily of nitrogen (78.09 percent by volume), oxygen (20.5 percent by volume), argon (0.93 percent by volume), and trace amounts of other gases. Because the majority of air is made up of simply nitrogen and oxygen, it is classified as a diatomic gas. Air is regarded the ideal gas since it obeys the thermal equation of state and exhibits constant specific heats at room temperature. The thermal and caloric properties of air, on the other hand, vary with temperature, as shown below: • When the temperature is less than 500 K, air obeys the equation of state (p= ρRT) and its specific heats cp and c∀ remain constant, with a ratio γ of 1.4. That is, air is both thermally and calorically perfect for T < 500 K. • When the temperature is greater than 500 K but less than 2000 K, air maintains its thermal perfectness because it obeys the equation of state. However, it becomes calorically imperfect because both cp and c∀ are now functions of temperature; cp = cp (T) and c∀ = c∀ (T). Nevertheless, the specific heat ratio, γ , remains constant. • When the temperature of air exceeds 2000 K but remains less than 4000 K, the air is no longer thermally  begins to dissociate into oxygen ions  perfect. The oxygen and free electrons O2 ⇐⇒ 2O+ + 2e− . Hence, T > 2000 K, the air becomes both thermally and calorically imperfect. • Around 4000 K, nitrogen  begins to dissociate into nitrogen ions and free electrons  N2 ⇐⇒ 2N+ + 2e− and is entirely dissociated at 9000 K. Free electrons and ions created during the process of dissociation of oxygen and nitrogen create an electron cloud, which is responsible for the communication blackout, during the re-entry phase of a spacecraft at an altitude of nearly 70 km and around 20 min before landing. This process, which often lasts several minutes, is known as radio blackout, ionization blackout, or re-entry blackout. The spacecraft re-enters the earth’s atmosphere at high speeds during this phase, and due to the extreme compression, ionized air is found to be in the plasma state, which blocks radio communications. Unfortunately, this is the most critical phase of a space mission, because a loss of communication disables earth-based diagnostic telemetry. On February 1, 2003, a catastrophic malfunction occurred during the re-entry of the space shuttle Colombia (USA).

Concluding Remarks The conservation laws of mass, momentum, and energy for a control volume are derived in integral and differential forms in this chapter. Their conservation and non-conservation forms, useful in computational fluid dynamics, are also described.

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2 Governing Equations and Thermodynamics of Compressible Flows

The first and second laws of thermodynamics are thoroughly explored, and their combined expressions, known as the Gibbs equations for a perfect gas, are derived. The direction of flow processes is dictated by the second law of thermodynamics, commonly known as the increase of entropy principle. The thermal and calorical properties of perfect gas and the latter’s deviation from ideal behavior with increasing temperature are all discussed.

Exercise Problems Exercise 2.1 The enthalpy of air is measured considering the temperature 350 K as the datum. If the enthalpy of air at 450 K is 101.87 KJ/(kg.K), calculate the average value of the specific heat at constant pressure cp . Assume air to be a perfect gas. Exercise 2.2 The air is expanded in a cylinder attached with a frictionless piston. If the initial temperature of air is 1300 K and its original volume is 81th of the final volume, then determine (i) the change in temperature and temperature ratio, (ii) the pressure ratio, and (iii) the work done by the gas. Exercise 2.3 A thermally perfect but calorically imperfect gas obeys the following relationship: cp = co + αT A certain quantity of this gas at the temperature T1 and pressure p1 is first heated isobarically to the temperature T2 and then expanded isochorically to the temperature T3 . Calculate the net change in the internal energy (u3 − u1 ) of the gas. Exercise 2.4 For a general intensive thermodynamic property β, show that d dt

˚

˚ ρβd∀ = C∀

ρ

dβ d∀ dt

C∀

Exercise 2.5 Air is heated isobarically from 289 to 1000 K. Calculate (i) the change in enthalpy and (ii) the change in entropy. Assume air to be a perfect gas. Exercise 2.6 Air is heated by combustion gases in a cross-tube boiler from 305 to 530 K. The air enters at a velocity of of 50 m/s and leaves at a velocity of 15 m/s. Calculate (i) the stagnation enthalpies of air at the inlet and exit, (ii) the heat absorbed by unit mass of air, and (iii) error when kinetic energy effects are ignored. Exercise 2.7 Consider an adiabatic flow of air through a duct. If the properties of air at two different locations are v1 =200 m/s, T1 =350 K, p1 =250 kPa, v2 =250 m/s, and p1 =300 kPa, then show that the total loss in stagnation pressure is about 35.3 kPa. Also, calculate the corresponding increase in the entropy.

Exercise Problems

107

Exercise 2.8 The internal energy of a fictitious gas obeys the following relationship: u = αT /2 + β 1

where α and β are constants. Derive the expressions for cp , c∀ , and γ .

Chapter 3

Sound Wave Propagation in Compressible Fluids and Flow Regimes

Abstract This chapter briefly discusses the mechanism and speed of sound wave propagation in compressible fluids. Small perturbations cause waves to spread at the speed of sound. The propagation mechanism determines how disturbances are perceived and consequently characterizes the signaling mechanism. The speed of sound is determined by the macroscopic properties of the fluid, despite the fact that the signaling process has a microscopic origin.

3.1 Introduction Sound waves are, in general, small fluctuations in pressure, and sound speed, also known as acoustic speed or speed of sound, is the rate at which these waves travel in a medium. In reality, the phenomenon of density fluctuations caused by pressure variations between two places in the flow field is immediately observed in a compressible flow. The variation in density caused by pressure changes has a substantial impact on the propagation of sound waves. In the sections that follow, we will look into the characteristics and phenomena associated with the propagation of sound waves in both ideal and actual gases.

3.2 Speed of Propagation of Small Disturbances in a Compressible Media The equation for the speed of sound for a plane infinitesimal pressure wave propagating along a duct with uniform cross section is derived in this section. The onedimensional form of conservation laws deduced in Chap. 2 will be utilized to demonstrate that small pressure disturbances propagate at the speed of sound a. Consider a constant-area duct containing a gas that is initially at rest and a piston that can slide without friction in the duct, as shown in Fig. 3.1a. The pressure and density ρ of the gas are initially uniform throughout the duct. When a small impulse is applied to the piston, the gas layer immediately close to the piston experiences © Springer Nature Singapore Pte Ltd. 2022 M. Kaushik, Fundamentals of Gas Dynamics, https://doi.org/10.1007/978-981-16-9085-3_3

109

110

3 Sound Wave Propagation in Compressible Fluids and Flow Regimes

Fig. 3.1 The propagation of a planar pressure disturbance in a cylinder-piston assembly

an infinitesimal pressure increase dp and a corresponding density increase dρ. Thus, if the piston moves at an infinitesimally small velocity dv, the pressure wave corresponding to the piston movement begins to propagate towards right with a velocity a, leaving behind the flow field with increased pressure (p + dp) and density (ρ + dρ). However, due to the stationary fluid, the pressure and density ahead of the pressure pulse are still p and ρ, respectively. If the coordinate system is chosen in such a way that it moves with the pressure wave, the pressure wave appears to be stationary, and the gas to the right of the pressure wave moves towards the left (towards the wave) with a velocity a, while the gas to the left of the wave moves with a velocity a − dv (Fig. 3.1b). We write this using the continuity equation: m ˚ = ρAa = (ρ + dρ) A (a − dv) dv dρ = ρ a

(3.1) (3.2)

After neglecting the frictional and body forces, the momentum equation can be expressed as follows: A (p + dp − p) = m ˙ (a − dv − a)

(3.3)

Substituting in m ˙ = ρAa produces dp = −ρadv

(4.26)

3.2 Speed of Propagation of Small Disturbances in a Compressible Media

111

It is Euler’s equation for the steady flow of any fluid. Substituting dv from Eq. (4.26) with Eq. (3.2) yields  dp (3.4) a= dρ According to Eq. (4.36), the energy equation can be applied across the wave front, yielding the equation: cp T+

a2 (a − dv)2 = cp (T + dT) + 2 2 (dv)2 cp dT + = adv 2

(3.5) (3.6)

The higher order term can be discarded for a small impulse applied to the piston. Thus, (3.7) cp dT = adv R Because the gas is perfect, substituting p = ρRT and cp = γγ−1 into Eq. (3.7) and then combining the resulting expression with Eq. (3.4) yields

  p dp γ d = γ −1 ρ ρ

(3.8)

It should be noted that Eq. (3.8) is only valid if the change of state is an isentropic process. Consequently, the change of state across a small pressure wave front, as shown in Fig. 3.1b, can be assumed to be isentropic. Therefore, instead of Eq. (3.4), it is appropriate to use partial differential notation, i.e.,  a=

∂p ∂ρ

 (3.9) s

where the subscript s indicates that the partial derivative is evaluated at constant entropy. It can be seen that the finiteness of the velocity of sound is attributable to the compressibility of the gas. Because there is no adjustment in volume for an incompressible fluid, the velocity of sound would be infinite, i.e., if ρ = constant then dρ → 0, and any pressure change would be immediately conveyed and felt throughout the field of gas. It should be noted that the identical conclusions for the speed of propagation of an infinitesimal pressure wave would be obtained for cylindrical and spherical wave spreading from a point source. Finally, for historical purposes, Sir Isaac Newton was the first to discuss the propagation of pressure waves in air. He discovered that the speed of propagation of a pressure wave is based on the assumed isothermal process in a perfect gas.

112

3 Sound Wave Propagation in Compressible Fluids and Flow Regimes

 a=

p ρ

(3.10)

Because thermodynamics was not recognized as a science at the time of Newton, the 19% gap between his theory and experiments was never justified. Marquis de Laplace revised Newton’s calculations over a century later. The fundamental distinction between Laplace’s theory and Newton’s theory is that pressure propagation occurs in an adiabatic manner. Because the compression generated by pressure propagation causes a relatively small temperature gradient, heat owing to compression cannot be transported to the surrounding region. Based on the adiabatic process, Laplace’s correction for the speed of sound is to multiply Newton’s formula by a √ factor γ .

3.2.1 The Speed of Sound in a Perfect Gas In an isentropic process, the relation between pressure and density for an ideal gas is represented as p = constant (3.11) ργ where γ is the specific heat ratio. Now, by logarithmic differentiation of Eq. (3.11), we get dp dρ −γ =0 p ρ

(3.12)

Because the process is assumed to be isentropic, 

dp dρ

 = s

γp = γ RT ρ

(3.13)

Substituting Eq. (3.13) into Eq. (3.13) results in a=

 γ RT

(3.14)

Equation (3.14) denotes the speed of sound in a perfect gas.

3.2.1.1

Effects of Molecular Weight on the Acoustic Speed

The specificgas constant R for a gas is calculated by dividing the universal gas J with the molecular weight Mw of the gas. constant Ru = 8.314 (mol.K)

3.2 Speed of Propagation of Small Disturbances in a Compressible Media

 a=

γ Ru T Mw

113

(3.15)

This relationship demonstrates that the speed of sound is affected by both the specific heat ratio and the molecular weight of the gas. As we all know, γ is also an unaltered specification, with a value ranging from 1 to 1.67, resulting in lower molecular weight gases having a higher acoustic speed than higher molecular weight gases. Let’s have a look at the sound speed in several different gases. • In the case of monoatomic gases, such as hydrogen, the specific heat ratio is 1.67. Because the molecular weight of hydrogen is 2 g/mol, the sound speed under ordinary atmospheric conditions is about 1200 m/s. • In the case of general diatomic gas, the specific heat ratio is 1.4. Air, for example, belongs to the diatomic gas group and has a molecular weight of about 28.96 g/mol. Thus, the speed of sound in the air at standard sea level is approximately 330 m/s. • The specific heat ratio of polyatomic gases, such as Freon-22 (refrigeration gas), is 1.18. Because the molecular weight of Freon-22 is 86.47 g/mol, the speed of sound is approximately 110 m/s. Taking into account the foregoing observations, it is now time to put all of the elements together. It is our primary concern to avoid unnecessary stress generated by rotation for the sake of long durability of turbomachines, the rotor speed is regulated from 270 m/s to 450 m/s. In addition, studies reveal that when the rotor speed is impacted by the sonic condition, the efficiency loss increases. Therefore, both stress and compressibility considerations are the limiting design factors for air compressors. Furthermore, fluid compressibility is never considered in the design of hydrogen compressors, whereas compressibility is always prominent in the design of compressors that use Feron-22 as their working fluid. Example 3.1 A weak pressure pulse (a sound wave), through which the pressure rise is 0.05 kPa, travels down a duct into air at a temperature of 30 ◦ C and a pressure of 105 kPa. Evaluate the velocity of the airstream behind the wave. Solution. Consider the analysis of a sound wave presented in Sect. 3.2. We have seen that the conservation of momentum for the flow through the control volume, shown in Fig. 3.1b, is given by dp = −ρadv Thus, the magnitude of the velocity downstream of the sound wave is dp |dv| = ρa From the given values, the density of air is

(4.26)

114

3 Sound Wave Propagation in Compressible Fluids and Flow Regimes p RT 105×103 = 287×303 = 1.207 kg/m3

ρ=

and the speed of sound is √ √ a = γ RT = 1.4 × 287 × 303 = 348.7 m/s Given that, the pressure rise across the wave is dp = 0.05 × 103 kPa Thus, 3

0.05×10 |dv| = 1.207×348.7 = 0.1188 m/s

Therefore, the velocity of the airstream behind the sound wave is 0.1188 m/s.

3.3 The Compressibility Factor The equation of state given by Eq. (2.208) for a perfect gas is based on the following assumptions. • The volume of the gas molecules is small in comparison to the volume of the gas. • The intermolecular forces are assumed to be negligible. Consequently, the potential energy between molecules is likewise low. • The molecules behave independently of one another in the absence of intermolecular forces. Real gases, on the other hand, depart from ideal gas behavior and are hence referred to as non-ideal gases. This variation at the specified temperature, T, and pressure, p, is adjusted by using a correction factor known as the compressibility factor, indicated by Z. Z=

p∀ RT

(3.16)

where ∀ is the specific volume of the gas and R is the specific gas constant. The compressibility factor is also known as Z=

∀real 1 ∀ideal

(3.17)

3.3 The Compressibility Factor

115

where Z = 1 denotes the ideal gas

Listing 3.1 A MATLAB program for plotting non-ideal gas compressibility factors. % V e r s i o n 1.1 C o p y r i g h t M r i n a l Kaushik , IIT Kharagpur , 2 9 / 0 7 / 2 0 2 1 function [ Z_calc ] = compressibility ( T_Kelvin , P_bar ) T = T_Kelvin ; P = P_bar ; Tc = 1 3 2 . 6 ; Pc = 3 7 . 7 1 ; Tr = T / Tc ; Pr = P / Pc ; Z = .01:.0001:5; nm = max ([ l e n g t h ( Tr ) l e n g t h ( Pr ) ]) ; if l e n g t h ( Tr ) ~= l e n g t h ( Pr ) && min ([ l e n g t h ( Tr ) l e n g t h ( Pr ) ]) ~= 1 w a r n i n g ( â T e m p e r a t u r e and p r e s s u r e a r r a y l e n g t s must match . â ) end if l e n g t h ( Tr ) ==1 Tr = Tr â ones ( size ( Pr ) ) ; end if l e n g t h ( Pr ) ==1 Pr = Pr â ones ( size ( Tr ) ) ; end Z _ c a l c = NaN ( nm ,1) ; for n = 1: nm V a n D e r W a a l = [( Z + 27 â Pr ( n ) ./ (64 â (( Tr ( n ) ) ^2) . â Z ) ) . â (1 â Pr ( n ) . / ( 8 . â Tr ( n ) . â Z ) ) ]; Z _ c a l c ( n ) = i n t e r p 1 q ( V a n D e r W a a l , 'Z â ,1) ; end

The generalized compressibility chart for actual gases, illustrated in Fig. 3.2, is generated by plotting the compressibility factors against reduced pressures at various reduced temperatures. The figure indicates that when the pressure and temperature are decreased, the general characteristics of the gases are nearly same. Here, the reduced pressure pr is acquired by dividing the pressure of the gas by its critical pressure pcr , and the reduced temperature Tr is derived by dividing the temperature of the gas by its critical temperature Tcr . p pcr T Tr = Tcr pr =

(3.18) (3.19)

Figure 3.2 allows us to draw the following significant conclusions.

 • Regardless of temperature, real gases at very low pressures, pr  1 , can be considered ideal gases. However, they tend to deviate from ideal gas behavior at high pressures. • At sufficiently high temperatures, real gases are assumed to be the ideal gas (Tr  2), but at very low temperatures, they depart from the ideal gas behavior. • Real gases exhibit the greatest deviation in the vicinity of the critical point.

116

3 Sound Wave Propagation in Compressible Fluids and Flow Regimes

Fig. 3.2 The generalized compressibility chart for non-ideal gases

Example 3.2 Suppose the steam density at 100 ◦ C and 1.01325 bar is 0.5974 kg/m3 . Determine (i) the compressibility factor and (ii) the steam volume deviation from the ideal behavior. Solution. Given, ρ = 0.5974 kg/m3 T = 100 ◦ C + 273 = 373 K p = 1.01325 × 105 Pa Mw = 18.0152 × 10−3 kg (i) The compressibility factor is Z= Substituting ∀ =

1 ρ

and R =

Ru Mw

p∀ RT

into Z expression yields

pMw ρ Ru T 1.01325 × 105 × 18.0152 × 10−3 = 0.5974 × 8.314 × 373 = 0.985 Z=

(ii) The deviation of the volume from ideal behavior can be expressed as % Deviation =

∀ideal − ∀real ∀ideal

(3.16)

3.3 The Compressibility Factor

117

Also, from Eq. (3.17) ∀real = Z∀ideal Therefore,   −∀real × 100 % Deviation = ∀ideal∀ideal = (1 − Z) × 100 = (1 − 0.985) × 100 = 1.5% Hence, the steam shows 1.5% deviation from the ideal gas behavior under stipulated conditions.

3.3.1 The Speed of Sound in a Real Gas It was found in Sect. 3.3 that real or non-ideal gases only obey the ideal gas law at low pressures and high temperatures. However, at high pressures and very low temperatures, a real gas tends to deviate from ideal gas characteristics for the following reasons. • In real gases, the space occupied by molecules (molecular volume) is not insignificant. As a result, individual gas molecules are not free to travel in the total volume of the gas, but only in a part determined after subtracting the molecular volume. • In contrast to a perfect gas, the attraction and long-range intermolecular interactions between the molecules of a real gas are active. Furthermore, the molecules of a real gas effectively attract one other, hence actual gases are more compressible than ideal gases. Thus, the ideal gas law or the equation of state for the perfect gas is no longer enough to represent the behavior of real gases. Furthermore, the sound speed or acoustic speed calculated from the ideal gas law with constant specific heat assumptions is not applicable for real gases. Several state equations for non-ideal gases have been developed, the most prominent of which is Van der Waal’s equation of state, named after Johannes D. Van der Waal in 1873. He incorporated corrections for finite molecular volume and cohesive forces (also known as Van der Waal’s forces of attraction) existing between molecules at sufficiently small distances to produce a more realistic equation of state. If ∀ is the volume of the gas, then ∀ − b is the volume accessible for free molecular motion, where b is the number of molecules per mole of gas. In order to account for the effects of cohesive forces, Van der Waal proposed that the molecule adjacent to the wall of the container is similarly drawn in all directions by the molecules surrounding it. As a result, the molecule in close proximity to the wall receives a net inward force as it moves towards the wall. Because pressure is defined as the number of collisions per unit area per unit time in the kinetic theory of gases, this inward force results in fewer molecules colliding with the wall. As can

118

3 Sound Wave Propagation in Compressible Fluids and Flow Regimes

be seen, the pressure encountered by the real gas is lower than that of the ideal gas, which lacks certain intermolecular forces. Therefore, the equation of state requires that the pressure p be increased by the term ∀a2 , which is proportionate to density or inversely proportional to the specific volume of the gas. Incorporating the aforementioned corrections into the ideal gas law yields the well-known form of Van der Waal’s state equation shown below.    a

p + 2 ∀ − b = RT ∀

(3.20)

where the constants a and b have positive values and represent characteristics of a specific gas Substituting ∀ = ρ1 into Eq. (3.20) yields



 p + aρ 2 1 − bρ = ρRT

ρRT 2  − aρ p=

1 − bρ

(3.21) (3.22)

The speed of sound in a compressible medium that is not a perfect gas can be approximated using its isentropic compressibility κs . As a result of Eqs. (1.108) and (3.9), it follows that  1 (3.23) a= ρκs Because the compressibility κ is defined as the reciprocal of the bulk modulus of elasticity K,  Ks (3.24) a= ρ Both Eqs. (3.9) and (3.23) can still be used to compute the speed of sound in a general gas, but κs is a matter-specific property that must be known ahead of time. Now, let us develop a generalized expression for acoustic speed in real gas. Assume that entropy varies with pressure and density. Thus, s = s (p, ρ)

(3.25)

In terms of partial derivatives  ds =

∂s ∂p



 ρ

dp +

∂s ∂ρ

 dρ p

(3.26)

3.3 The Compressibility Factor

119

For an isentropic process ds = 0, hence 

dp dρ

 



∂s ∂ρ

= − 

p

(3.27)

∂s ∂p ρ

s

The chain rule can be used to expand the phrases on the right side as follows: 



dp dρ

dp dρ

 s

 

∂s   ∂T 



= − s

∂T p

∂ρ

p



∂s   ∂T  ∂T ρ

∂p

(3.28)

ρ

∂s      ∂p ∂T ∂T p = − ∂s  ∂T ∂ρ p ρ ∂T ρ

(3.29)

Assume f (p, T, ρ) = 0; therefore, using the cyclic rule of partial derivatives, we may write 

     ∂p ∂T ∂p = −1 ∂T ρ ∂ρ p ∂ρ T       ∂p ∂T ∂p =− ∂T ρ ∂ρ p ∂ρ T

(3.30) (3.31)

Substituting Eq. (3.31) into Eq. (3.29) yields 

Since

∂s 

∂T p

=

cp T

and

∂s 

∂T ρ

dp dρ =

 s c∀ , T

∂s    ∂p ∂T p = ∂s  ∂ρ T ∂T ρ

(3.32)

thus

  cp  ∂p dp T = c∀ dρ s ∂ρ T T     dp ∂p =γ dρ s ∂ρ T



(3.33) (3.34)

The speed of sound in a real gas is thus calculated using Eq. (3.9).    ∂p a= γ ∂ρ T

(3.35)

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3 Sound Wave Propagation in Compressible Fluids and Flow Regimes

By differentiating Eq. (3.22) with respect to ρ while keeping the temperature constant, we get

  RT ∂p =γ

(3.36) 2 − 2aρ ∂ρ T 1 − bρ Substituting this into Eq. (3.35) results in  a=

γ RT

2 − 2γ aρ 1 − bρ

(3.37)

Modifying the Van der Waal’s equation of state with Bethelot’s correction improves the speed of sound, especially at low temperatures: aρ 2 ρRT − p=

T 1 − bρ

(3.38)

Therefore, the speed of sound in a real gas eventually takes the following form:  a=

γ RT 2γ aρ

2 − T 1 − bρ

Example 3.3 Show that, 

∂p ∂ρ



 =γ

s

∂p ∂ρ

 T

Solution. Let entropy be a function of pressure and density. s = s (p, ρ) Writing in terms of partial derivatives, we get     ∂s ∂s ds = dp + dρ ∂p ρ ∂ρ p For an isentropic process ds = 0, hence 

dp dρ



  ∂s ∂ρ

= −  s

p

∂s ∂p ρ

(3.39)

3.3 The Compressibility Factor

121

By the chain rule, 

dp dρ

 

∂s   ∂T 

 s

∂s      ∂p ∂T ∂T p  = − ∂s  = −

∂s   ∂T  ∂T ∂ρ p ρ ∂T ρ ∂T p

∂ρ

p

∂T ρ

∂p

ρ

and if f (p, T, ρ) = 0 then the cyclic rule of partial derivatives gives      

∂p ∂p ∂T = −1 ∂T ρ ∂ρ p ∂ρ T      ∂p ∂p ∂T = − ∂ρ ∂T ρ ∂ρ p T

  Substituting in

dp dρ

 s

expression yields

∂s      ∂p dp ∂T p = ∂s  dρ s ∂ρ T ∂T ρ

But,

∂s  =

∂T p ∂s = ∂T ρ

cp T c∀ T

Therefore, 



dp = dρ s dp = dρ s

cp T c∀ T

γ

  ∂p

 ∂ρ T ∂p ∂ρ T

3.4 Two-Dimensional Propagation of Sound Waves and the Signaling Mechanism It will be observed later in Sect. 3.6 that compressible flows are classified into distinct regimes, notably three, incompressible, subsonic, and supersonic. Because of the compressibility of the medium, the flow patterns and overall behavior of subsonic and supersonic flows change significantly. The physical distinction between subsonic and supersonic flow patterns is explored qualitatively in this section by displaying the pressure field produced by the moving point source in the still fluid. When an object moves through a stationary fluid or the fluid flows past a stationary object, each element of the solid surface tends to deflect the air away from the path

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Fig. 3.3 Propagation of sound waves from a point source

it would otherwise go. These local disturbances operate as a point source, emitting spherically propagating pressure waves into the surrounding air. At any given time, these waves can be represented as a superposition of all previous waves released from the same source. However, depending on whether the source is stationary or moving, the resulting wave pattern may be symmetrical or asymmetrical; and the speed of a moving source is in contrast to the speed of sound in the fluid. Consider the following situations. 1. When the point source that emits the sound waves in a compressible fluid is stationary (M = 0), the pressure disturbance wave fronts appear to radiate in the form of concentric spheres at varied time intervals. Figure 3.3(i) depicts the intersection of these wave fronts with a plain containing the source at integer multiples of t. 2. The wave fronts are considerably different when the point source is moving because each pressure disturbance (sound wave) is emitted from the source at different points in the fluid. For example, if the source is moving faster than the speed of sound (M < 1), the wave fronts will appear like Fig. 3.3(ii). In this situation, because the pressure disturbance moves faster than the source, a sequence of circles are produced, one inside the other, but with different centers. That is, the wave fronts do not intersect because U∞ < a. Similarly to the stationary source situation, the wave fronts propagate horizontally upstream and downstream, as well as vertically upward and downward. Consequently, an object moving at subsonic speeds influences the entire flow field.

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3. When a point source moves at the speed of sound (M = 1), the resulting wave fronts are identical to those shown in Fig. 3.3(iii). In this case, each pressure disturbance strengthens the previous one, resulting in the formation of a planar wave front. Because this is a sound wave front, it is, by definition, isentropic. 4. When the point source moves faster than the speed of sound (M > 1), the emitted pressure disturbances are contained within a cone, with its vertex at the body at that time. Upstream of the cone vertex, the presence of the disturbance is not sensed; disturbances are only conveyed downstream within the cone. The cone within which the disturbances are confined is known as the Mach cone, and the half angle of the cone is known as the Mach angle. An examination of Fig. 3.3(iv) reveals that the Mach angle of the Mach cone is given by sin μ =

Distance traveled by the point source in a particular time interval t Distance traveled by the sound wave in the same time interval t (3.40) U∞ t 1 sin μ = = (3.41) at M 1 (3.42) μ = sin−1 M

It is worth noting that when the Mach number decreases, the Mach cone widens and forms a planar wave front (μ = 90◦ ) at M = 1, i.e., the object moves at sonic speed. Hence, point 3 is effectively a subset of point 4.

3.4.1 Von Karman’s Rules for Supersonic Flows and the Activity Envelope The point source considered in Sect. 3.4 could be a solid body that emits pressure disturbances as it moves through the fluid. The envelope they describe is the true significance of the propagation of the sound wave relative to its speed. Figure 3.3(iii) and Fig. 3.3(iv) show that when the Mach number is equal to or greater than one, pressure disturbances or sound waves do not propagate upstream. The wave pattern seen in Fig. 3.3(iv) reveals the three supersonic flow rules presented in 1947 by Hungarian-American mathematician and aerospace engineer Theodore Von Karman. Despite the fact that these rules are predicated on the assumption of small pressure disturbances, they can be applied qualitatively to large disturbances as well.

3.4.1.1

The Rule of Forbidden Signals

Pressure disturbances caused by a moving object at supersonic Mach numbers cannot reach ahead of the body.

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The Zone of Action and Zone of Silence

Pressure disturbances caused by an object moving at supersonic Mach numbers are contained inside the activity zone defined by the Mach cone trailing behind the object. The zone of action is the activity zone where the effects of pressure disturbances can be felt at any arbitrary location. The zone of silence is the region outside the zone of activity where pressure signals cannot reach.

3.4.1.3

The Rule of Concentrated Action

The effects of pressure disruptions caused by a moving object at supersonic speeds are concentrated along the Mach waves. We can still see its qualitative application if we extrapolate this rule to large disturbances. For example, consider the concentration of effects along a shock wave accompanying a body moving at supersonic speeds. Therefore, because the pressure signal emitted by a stationary body propagates uniformly in all directions, the resulting wave fronts are symmetrical (Fig. 3.3(i)). When a body moves faster than the speed of sound, the pressure signal created by the moving body will always be ahead of the latter, generating a wave pattern similar to Fig. 3.3(ii). In this example, the body moves steadily into a fluid that has recently changed due to the motion of the body. Consequently, if a body moves at subsonic speed in a compressible fluid, the fluid ahead of the body may be said to become aware of the presence of the body because the latter propagates pressure signals ahead of itself. In contrast, if the body moves at sonic speed, all of the pressure signals it generates condense into a single wave front (Fig. 3.3(iii)). The region of fluid upstream of the wave front would be unaware of the presence of the body in this situation, and so this region is known as the zone of silence. The zone of action is the region of fluid downstream of the wave front where the presence of the body can be felt everywhere. The rule of concentrated action applies to a body traveling faster than the speed of sound. Accordingly, the pressure signals released by a supersonic body are predominantly concentrated in the vicinity of the Mach cone, marking the outer boundary of the zone of action (Fig. 3.3(iv)), and an observer outside the Mach cone would not perceive or hear these signals. Thus, the disturbance caused by a supersonic aircraft passing overhead is not felt by a ground observer until its Mach cone reaches the observer. If the body is at rest and supersonic air flows by it, all disturbances created by the body are swept downstream and fall within the Mach cone illustrated in Fig. 3.4. When the flow reaches the cone, the flow characteristics will change. This cone is known as the conical Mach cone. Similarly, in two-dimensional flows, all waves emanating from a weak line source of disturbance will all lie behind a plane wave inclined at the Mach angle to the flow direction, as illustrated in Fig. 3.4. Interestingly, this method is occasionally used to calculate the Mach number of a fluid flow. A suitable optical flow visualization technology makes apparent a small irregularity specifically configured on the wall or the roughness on the wall, as well as the Mach wave created at this source of disturbances. The Mach number can be calculated using

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Fig. 3.4 Mach waves in two and three dimensions

Eq. (3.42) by measuring the angle subtended by the Mach wave at the disturbance to the flow direction. Example 3.4 The so-called sonic boom of a jet aircraft, flying at Mach 3.0, is heard by a stationary observer at the ground when the aircraft has traveled 1 km ahead of the observer. Assuming a level flight operation and neglecting the air temperature variation with altitude, determine the flight altitude of the aircraft with respect to the ground. Solution. Let the positions of the observer and the aircraft are given by O and B, respectively, as shown in Fig. 3.5. From the definition of Mach angle,

 μ = sin−1 M1 = sin−1 13 ≈ 19.47◦

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Fig. 3.5 An aircraft in a level flight and a stationary observer on the ground

From ABC, we have = tan α = tan 19.47 = 0.35355 h d

Therefore, h = 1000 × 0.35355 = 353.55 m Hence, the flying altitude of aircraft is 353.55 m above the ground.

3.4.2 The Mach Line If the source of disturbance discussed above is very small, or if there is no change in direction of the air as it passes over a body, such as the sharp lip of a flat plate maintained at a zero degree angle of attack to the freestream, the line of disturbance created along the envelope of individual pressure disturbances is known as a Mach line or, in mathematical treatment of the problem, as a characteristic line. Because the flow through a Mach line is isentropic, the pressure, temperature, density, and other properties remain constant. A Mach line is analogous to an indefinitely faint shock wave. The Mach line can be used to determine the region of influence of certain disturbances in a supersonic flow field, as well as to investigate pressure in such a field.

3.5 Similarity Parameters in the Compressible Flows Similarity between two problems in fluid flows is achieved by establishing geometric, kinematic, and dynamic similarity. When the ratio of one characteristic length of one system to the equivalent length of the other system is maintained, geometric similarity

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is achieved. Similarly, kinematic similarity is attained when the velocity of a fluid particle in one system and the velocity of the equivalent particle in the other system are found to be in a constant ratio. Moreover, dynamic similarity is maintained when the ratio of forces acting at a point in one system equals the ratio of forces acting at the same point in the other system. Therefore, geometric similarity corresponds to shape similarity, kinematic similarity to motion similarity, and dynamic similarity to force similarity between two systems. When all of the physical equations that govern the flow of a compressible fluid are considered, namely, the continuity equation, the momentum equation, the energy equation, and the equation of state, four non-dimensional parameters can be deduced, namely, Mach number M, Reynolds number Re, Prandtl number Pr, and ratio of specific heats γ . In order to achieve flow similarity between the model and the prototype, these non-dimensional parameters must be the same.

3.5.1 Mach Number The Mach number, or simply Mach, is a prominent similarity parameter in compressible flows, implying that any vehicle flying at a certain Mach number will always have identical aerodynamic characteristics regardless of altitude or dimensional speed. When the compressibility of either the fluid or the flow becomes significant, the elastic forces, along with pressure and inertial forces, must be incorporated in the analysis. The ratio of inertial to elastic forces is basically characterized as the Mach number. − → F i inertial forces (3.43) = − M≈ → elastic forces F e − → Using Newton’s second law, the inertial force F i acting on a fluid element of mass m may be represented as, − → → a (3.44) F i = m− In other words, the amount of the inertial force is equal to the mass of the fluid element multiplied by its acceleration. Now, if  is the geometric dimension (length scale) of the fluid element and ρ is the fluid density, the mass of the element will be → a , of a fluid element in proportional to ρ3 . Besides that, because the acceleration, − any direction is the rate at which its velocity in that direction varies with time, it is proportional → to vthe speed (magnitude of the velocity) divided by some time interval, that is, − a ∼ t. It is worth noting that the time scale t is proportional to the length scale divided → 2 a is proportional to v , and hence by the velocity, t ∼ v . Thus, the acceleration − the magnitude of the inertial force is proportional to ρ2 v2 .

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− → F i ∝ ρ2 v2

(3.45)

According to Eq. (1.106), for a given fractional change in volume, the increase in pressure is equal to the bulk modulus of elasticity K and gives rise to a force known − → as the compressibility or elastic force F e . − → F e ∝ K2

(3.46)

The ratio of inertial force to elastic force is calculated using Eqs. (3.45) and (3.46): − → F i ρ2 v2 − = → K2 F e =

(3.47)

ρv2 K

(3.48)

The expression ρvK is known as the Cauchy number, after its originator, French mathematician A. L. Cauchy. If the flow is assumed to be isentropic, 2

− → F i ρv2 − = → Ks F e

(3.49)

However, we saw in Sect. 3.3.1 that  the velocity with which acoustic waves prop-

agate in a fluid medium is equal to Kρs . Hence, the term ρv will be shortened to va2 , Ks v where a represents the speed of sound. The ratio a is known as the Mach number, after Austrian physicist Earnest Mach. 2

M=

v a

2

(3.50)

Equation (3.50) demonstrates that the Mach number in the given fluid is the ratio of flow speed (or the speed of an object in a quiescent fluid) to the speed of sound (an important measure of compressibility effects). When the Mach number is calculated using the local speed of sound, it is referred to as the local Mach number. According to the studies, if the Mach number exceeds 0.3, as in the flow past an airborne high-speed aircraft, a missile, or a spaceship, the compressibility effects in the flow become important and must be accounted for. The equity of Mach number is a crucial requirement for dynamic similarity in such instances. Finally, let us rewrite Eq. (3.50) in terms of M2 and substitute a2 = γ RT. This results in v2 (3.51) M2 = γ RT

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In Eq. (3.51), the term v2 in the numerator is a measure of the directed kinetic energy of the fluid, and the term γ RT in the denominator is a measure of the random thermal energy of the fluid. While there are various advantages to utilizing the nondimensional Mach number instead of the dimensional speed, there are two major drawbacks. 1. Because the speed of sound is variable, the Mach number is not directly proportional to the flow speed. 2. At extremely low temperatures, the Mach number becomes infinitely large. These limitations become obvious when dealing with flows at very high speeds or at extremely high altitudes, where the continuum hypothesis fails.

3.5.2 Reynolds Number The Reynolds number, denoted by Re, is named after the British engineer and physicist Osborne Reynolds. It is defined as the ratio of the convective term to the viscous diffusion term, which eventually becomes the ratio of inertial to viscous forces: − → F i inertial forces convection (3.52) ≈ = − Re = → diffusion viscous forces F μ The viscous force, which occurs as a result of shearing action between fluid layers due to viscosity, can be written as − → F μ = τS

(3.53)

where τ and S denote the shear stress and the surface area on which the shear stresses apply, respectively. The shear stress is rewritten as,   ∂v (3.54) τ = μ× ∂y where μ and Thus,

∂u ∂y

represent the viscosity and shear strain rate, respectively.  −  ∂v → F = μ × S μ ∂y

(3.55)

If  is the characteristics dimension of a particular flow domain, then ∂y ∼  and − → S ∼ 2 follow. Considering ∂v ∼ v, the viscous force F μ may be represented as

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−   v → F μ ∝ μ × × 2  − → F μ ∝ μv

(3.56) (3.57)

Substituting Eqs. (3.45) and (3.57) into Eq. (3.52) yields ρv2 2 μv ρv Re = μ

Re ∝

(3.58) (3.59)

Equation (3.59) shows that, 1. If Re  1, the viscous forces (i.e., friction) are quite large. Consequently, when compared to the viscous diffusion term, the convective term can be neglected. 2. If Re  1, the convective forces (i.e., inertial forces) are much greater than viscous diffusion forces. However, we must exercise caution in making such a conclusion: viscous forces cannot be ignored anywhere in the flow, including away from interfaces. Because of the occurrence of significant gradients in the region of interfaces, viscous forces are generally of the same order as inertial forces in the vicinity of boundary layers and interfaces.

3.5.3 Prandtl Number The Prandtl number, named after German physicist Ludwig Prandtl, is another dimensionless number. It is defined as the ratio of momentum diffusivity to thermal diffusivity.

Pr ≈

μ/ρ momentum diffusivity (ν) = K thermal diffusivity (α) /ρCp μCp Pr = K

(3.60) (3.61)

where, μ is the dynamic viscosity, Cp is the specific heat, and K is the thermal conductivity of the fluid medium. It is obvious from Eq. (3.61) that the Prandtl number is not reliant on the length scale and so is independent of the geometric configuration under consideration. However, the value of the Prandtl number is solely determined by the fluid properties. Equation (3.60) demonstrates that the Prandtl number is the ratio of momentum diffusivity to thermal diffusivity. Velocity gradients, particularly in boundary layers, cause momentum diffusion in a fluid. Thus, momentum diffusion is also referred to as viscous diffusion. Thermal diffusion is the process through which heat moves

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from one point to another in a fluid. Similarly to the momentum boundary layer, a thermal boundary layer emerges as a result of temperature gradients at the surface of an object, which in turn causes thermal diffusion. The Prandtl number is, in fact, proportional to the ratio of momentum boundary layer thickness δ to thermal boundary layer thickness δt . δ (3.62) Pr ∼ δt Thus, if Pr  1 then δ  δt , implying that the rate of heat diffusion is greater than the rate of momentum diffusion. So, the distribution of heat will be greater than the momentum. However, if Pr  1, then δ  δt , indicating that momentum spreads faster than heat. In terms of heat transfer modes, Pr  1 implies that conduction is more dominant than convection, but Pr  1 indicates that convection is more dominant than conduction. The Prandtl number for air is around 0.715 at standard sea level conditions. Therefore, the rate of heat spread in air is somewhat greater than the rate of momentum spread.

3.5.4 Ratio of Specific Heats In addition to the three dimensionless numbers, the Mach number, the Reynolds number, and the Prandtl number, the ratio of specific heats of gases is an essential nondimensional similarity parameter used in compressible flow theory. As mentioned in Sect. 1.8, γ denotes the ratio of specific heat at constant pressure to specific heat at constant volume and is defined as γ =

cp c∀

However, for a perfect gas in terms of specific quantities, cp = so γ is proportional to h (h/T) γ ∼ = u u ( /T)

(1.82) h T

and c∀ = Tu , and (3.63)

where h denotes the specific enthalpy and u denotes the specific internal energy. Equation (3.63) demonstrates that the ratio of specific heats of a perfect gas is also the ratio of its enthalpy and internal energy. Because air is a diatomic gas under typical conditions, the value of gamma is 1.4. Remark Only M and γ are adequate to assure similarity in the flow region outside the boundary layer, where the viscosity and heat conduction effects are rather small. The similarity of M is significantly more important than that of γ , as the latter has a relatively small influence on the flow pattern. Additionally, the viscous and heatconduction effects are significant within the boundary layer or in the core of the shock waves, therefore Re and Pr should be matched in the similarity analysis. It is

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worth noting that the value of Pr for the majority of gases is nearly the same and only slightly varies with temperature. Example 3.5 An aircraft is flying at an altitude of 10,000 m above the sea level at a velocity of 360 km/h. Calculate the Mach number of the aircraft relative to air at this altitude. (At 10,000 m above sea level, the air temperature is 224 K) Solution. Given, v = 360 km/h T = 224 K The speed of sound is √ √ a = γ RT = 1.4 × 287 × 224 = 300 m/s Thus, the Mach number becomes v a 360 × 1000 = 300 × 3600 = 0.33

M=

3.6 Flow Regime Classification Using Mach Number The Mach number, as stated in Sect. 3.5.1, is the ratio of flow speed to sound speed in a given fluid medium. The amount of the compressibility effects is determined by the Mach number, which allows us to classify the flow regimes in which these effects vary. The Mach number can theoretically have any value between 0 and ∞. However, as illustrated in Fig. 3.6, this large range of Mach numbers is separated into several groups, which in turn classify the flow regimes as subsonic flow, transonic flow, supersonic flow, hypersonic flow, and hypervelocity flow. Consider the flow past an aerodynamic body, as seen in Fig. 3.7. If the freestream velocity is U∞ and the speed of sound in air is a∞ , the freestream Mach number is M∞ =

U∞ a∞

(3.64)

Figure 3.7 depicts the streamline patterns over various bodies. The local Mach number over the body surface is allowed to vary along the streamlines in these cases.

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Fig. 3.6 Flow regime spectrum based on Mach number

Fig. 3.7 Flow past an aerodynamic body in various flow regimes

3.6.1 Incompressible Flow When the freestream Mach number is extremely small (M∞  1), the effect of compressibility can be neglected, and flow can be treated as incompressible without contributing considerable inaccuracy. In reality, this condition arises in two ways. The first is with a very small flow speed, for which the local flow speed is negligibly small compared to the speed of sound. The second instance corresponds to flows with extremely high speed of sound. In both circumstances, the flow approaches incompressibility, and the associated flow is said to as incompressible.

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3.6.2 Subsonic Flow When the freestream velocity is less than the speed of sound to the (U∞ < a∞ ), but of comparable magnitude, the local Mach number is less than one everywhere in the flow field. This flow is known as subsonic flow, and it is characterized by smooth streamlines with continuously varying properties. The streamlines, which are initially straight, bend gradually as they approach the body. Figure 3.7a displays the flow past an airfoil, where the flow extends over the surface of the body, deflecting the streamlines.

3.6.3 Transonic Flow When the freestream Mach number rises but remains in the subsonic range near one, the flow accelerates across the body, causing supersonic flow to occur locally on its surface. Consequently, large portions of the surface become mixed flows in which the local Mach number is either less than or greater than one. These are known as sonic pockets, and they develop on the body surface when the local Mach number is near to but less than one (Fig. 3.7b). These pockets are eventually terminated by a compression front (referred to as the shock wave), across which the flow characteristics change and the flow becomes subsonic again. When the mach number is slightly greater than one (Fig. 3.7c), the shock structure shifts towards the trailing edge and a curved shock (also known as bow-shock) forms at the leading edge. The flow becomes subsonic near the freestream value when it passes through the bowshock. After passing over the trailing edge shock, this local subsonic flow accelerates to supersonic Mach values before reverting to subsonic flow.

3.6.4 Supersonic Flow Supersonic flow occurs when the Mach number is greater than one everywhere in the flow field. In contrast to subsonic flow, a significant amount of total drag in supersonic flow is mostly due to wave generation. In order to reduce this drag, aerodynamic bodies in supersonic flows typically feature sharp-edged tips. In this case, here, oblique shock waves characterize the flow field (Fig. 3.7d). It is worth noting that the flow is both supersonic upstream and downstream of the oblique shock. However, in rare circumstances where strong oblique shock waves form, the flow downstream of the shock may be subsonic.

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135

3.6.5 Hypersonic Flow When a supersonic flow is expanded to high Mach numbers, the oblique shock wave approaches the body (Fig. 3.7e). Hypersonic flow refers to flows with exceptionally high velocity. Because the shock waves in a hypersonic flow are extremely strong, the pressure, temperature, and density across the shock increase greatly. The flow field between the surface and the shock heats up sufficiently to ionize the gas. Hypersonic flows have typical properties such as a thin shock layer, ionization and dissociation of gases, shock wave and boundary layer interactions, and so on. Although this flow regime begins at Mach 5.0, some of its properties may occur at Mach values as low as 3.5.

3.6.6 Hypervelocity Flow The hypervelocity flows are defined as flows having speeds comparable to planetary atmospheric entry from orbit, ranging from a few km/s to some tens of km/s. In this regime, the speed of sound is so low that it is mathematically ignored once again. Conclusion. Based on the above discussion, we can classify fluid flows depending on their local Mach number as follows: • • • • •

If 0 < M < 0.3, the flow is said to be incompressible. If 0.3 < M < 1.0, the flow is said to be subsonic. If 0.8 < M < 1.2, the flow is said to be transonic. If 1.2 < M < 5.0, the flow is referred to as supersonic. If M > 5.0, the flow is referred to as hypersonic.

Concluding Remarks This chapter discusses some general properties of the propagation of small pressure disturbances caused by a source in a compressible fluid. It is demonstrated that a small pressure disturbance in a fluid moves at the local speed of sound. When the source is at rest or moving at subsonic speed (M < 1) in a stationary compressible fluid, its presence can be sensed throughout the flow field. If, on the other hand, the source is moving at supersonic speed (M > 1) in a stagnant fluid, it must pass through each of the pressure disturbances it emits. In this situation, a Mach cone is attached

 to the source, and the semi-vertex angle of the cone, α, is given by α = sin−1 M1 , where M is the Mach number defined as the ratio of the flow speed and the speed of sound, i.e., M = va . The flow regimes are classified as: incompressible, subsonic, transonic, supersonic, hypersonic, or hypervelocity based on the Mach number.

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Exercise Problems Exercise 3.1 Using the one-dimensional continuity, momentum and energy equations show that the expression for the speed of sound in a gas is    p a= γ ρ Exercise 3.2 A supersonic aircraft flying at Mach 3 is in the level flight at an altitude of 5 km above the sea level. Estimate the time for a stationary observer on the ground to experience the noise from the moment when the aircraft was vertically overhead. Exercise 3.3 The maximum possible Mach number that an aircraft can achieve is 0.9 at sea level. Calculate the maximum speed at which this aircraft can operate at sea level if the air temperature is (i) 5 ◦ C and (ii) 55 ◦ C. Exercise 3.4 The molecular weight of a gas is 44 and its specific heat ratio is 1.3. Calculate the speed of sound in this gas if the gas temperature is −30 ◦ C. If the gas flows at a speed of 450 m/s, find the Mach number and the Mach angle. Exercise 3.5 Find the speed of sound at 15 ◦ C in (i) H2 , (ii) He, and (iii) N2 . Under what conditions will the speed of sound in H2 be equal to that in He. Exercise 3.6 An aircraft is flying at 1000 kmph at an altitude where the air temperature is −57 ◦ C. Calculate the Mach number at which the aircraft is flying. Exercise 3.7 In the Mach 3.5 supersonic wind tunnel, the cross-sectional area of the test-section is 1.2 m × 1.2 m. When the tunnel is operating under design conditions, the static pressure and static temperature in the test-section are 25 kPa and −110 ◦ C, respectively. Calculate the mass flow rate of air through the test-section. Exercise 3.8 A gas at a temperature of 35 ◦ C flows through the test-section of a supersonic wind tunnel. The optical flow visualization images confirm the generation of weak waves at the roughness on the wall. These waves subtend an angle of 50◦ to the flow direction. If the specific heat ratio of the gas is 1.4, determine the flow speed in the test-section.

Chapter 4

Steady One-Dimensional Flows

Abstract The governing equations, derived in Chap. 2, for the study of fluid flows, contain four independent variables, three spatial coordinates, and time. However, due to the mathematical complexities involved in deriving the general solutions to these equations, basic assumptions are added to achieve a more suitable physical model for mathematical treatment. The most important and widely used assumption is that the flow is considered to be one-dimensional. In this chapter, the governing equations of fluid motion are formulated using the one-dimensional and quasi-one-dimensional approximations that apply to any fluid flow.

4.1 Introduction Various engineering fluid flow problems can be evaluated by treating steady, onedimensional, and isentropic. Because the fundamental equations for steady, onedimensional flows are derived in Chap. 2, they will not be addressed further. Although these equations are simpler than the original three-dimensional formulations, they still involve too many independent driving potentials, such as area change, heat transfer, friction, and body forces, to allow closed-form solutions. However, in many instances, one of them predominates to the point where other elements can be completely ignored, allowing for even more simple governing equations. In this chapter, we analyze the steady and one-dimensional flow of a fluid. The solely derived potential accountable for the change in flow properties is the variation in the cross-sectional area of the flow passage (duct). We investigate the steady, one-dimensional fluid flow in a duct using the general fluid relations through our discussion. We also obtain the equations in a form that applies only to a perfect gas. When we consider the flow to be steady and onedimensional, we assume that all quantities relating to the state of the fluid are uniform across any cross section of the duct and with time. More precisely, we mean a flow in which the rate of change of fluid characteristics normal to the streamline direction is negligibly small compared to the rate of change along the streamline. This assumption is, however, not valid for real fluids but gives satisfactory solutions to many problems in which the walls of the duct do not change the direction abruptly, and the © Springer Nature Singapore Pte Ltd. 2022 M. Kaushik, Fundamentals of Gas Dynamics, https://doi.org/10.1007/978-981-16-9085-3_4

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cross-sectional area and shape change only slowly along the fluid path, provided the true mean values are taken over each cross section. Nevertheless, suppose rapid changes in the wall direction do take place, as at a sudden enlargement. In that case, these methods cannot be used near the change but can often be applied to the flow between two planes, one well upstream and the other well downstream of the disturbance. The laws of conservation of mass, momentum, and energy must be followed in all duct flows. However, if the area changes are quite large, as in a nozzle, the effect of friction may be negligible, and the flow might be considered reversible. Furthermore, this chapter deals with steady and one-dimensional flows in which heat transfer, friction, and body forces are ignored, implying that the flow is considered isentropic. By definition, it is an adiabatic flow (flow with no heat exchange with the surroundings) with negligible viscous dissipation. In a nutshell, an isentropic flow is both adiabatic and frictionless. This chapter established the equations that explain such flows. It is important to understand that a real flow can never be completely isentropic, but a considerable portion of it can be assumed. In internal duct flows, such as nozzles and diffusers, the effects of viscosity and heat transfer, for example, are contained within the boundary layer near the wall. Hence, the remainder of the flow is considered isentropic. Similarly, in external flows, such as the flow past a wing, the effects of viscosity and heat transfer are confined to boundary layers and shock waves, and the remainder of the flow is assumed to be isentropic. It is often possible to determine flow properties in real flows by first assuming them to be isentropic and then applying an appropriate correction coefficient to the resulting solutions to account for non-isentropic effects.

4.2 Concept of One-Dimensional Flows The term one-dimensional refers to variations in the properties of the matter under investigation occurring only in one direction. In fluids, for example, one-dimensional flow denotes the fact that all fluid properties are uniform throughout any cross section of the flow but may vary along the streamline direction. As a result, the rate of change of properties in a one-dimensional flow normal to the streamline direction is much lower than the rate of change in the streamline direction. A curved channel flow, for example, can be considered one-dimensional as long as the turning of the flow is small in comparison to the length of the channel segment. The channel needs not to be constant in area as long as the divergence or convergence within the channel is small compared to the distance along the channel. However, the channel may either be bounded by physical boundaries such as the walls of a nozzle or diffuser or by streamlines such as those surrounding the wings of a flying aircraft. The one-dimensional flow approximation is more pronounced in the case of fluid flows through a streamtube, where no assumptions are involved. In fact, in limiting condition, the flow through an elementary streamtube is strictly one-dimensional.

4.2 Concept of One-Dimensional Flows

139

When one-dimensional approximation is applied to the flows in ducts, where the properties of the fluid actually change over each cross section, it is usual to deal with some forms of average of these properties. However, in doing so, the errors involved in estimating the rate of change of properties along the ducts can be minimized, if 1. The variation of the fractional rate of change in the cross-sectional area with dA respect to the distance along the duct-axis is very small, i.e., dx/A  1. 2. The radius of curvature of the duct-axis, R, is very large compared to the diameter, d, of the cross section of the duct (R  d). 3. The velocity and temperature profiles in the streamline direction are almost selfsimilar. One-dimensional flow approximation is a powerful tool for dealing with a wide range of practical engineering problems due to its inherent simplicity, which gives rise to faster calculation methods. In fact, due to the usefulness and high reliability of the information obtained, the one-dimensional point of view is considered one of the most elegant approaches. At the same time, however, we must realize that the information collected from a one-dimensional analysis is solely for the variation of the average flow properties over each cross section. It gives no details of the changes in properties normal to streamlines. In many cases, as this is of major concern, a onedimensional analysis must complement a two or even three-dimensional treatment. The one-dimensional steady-flow approximation is mostly used in the following two categories of flows. Fluid Flows in Imaginary Streamtubes There are infinitesimal streamtubes outside the boundary layer (in the inviscid flow region) in almost all internal flows and many external flows. Flows in such elementary streamtubes are both adiabatic and reversible; therefore, isentropic relations may be considered accurate unless some discontinuity, such as shock waves, appears in the flow field. Fluid Flows in Channels or Ducts The flow in the channel or duct is usually adiabatic but may occur in the presence or absence of friction. When the duct length is very short, for example, in nozzles and diffusers, the flow takes place almost in a frictionless environment. Therefore, the flow process can be assumed to be reversible. The fluid flows in short nozzles, and diffusers are thus both adiabatic and reversible and therefore isentropic. Since these ducts are used to accelerate and decelerate the flow as efficiently as possible, the isentropic process is used as a reference for comparing the performance of the actual nozzles and diffusers.

140

4 Steady One-Dimensional Flows

Fig. 4.1 Steady uniform flow in a one-dimensional region

4.2.1 Governing Equations of the Steady One-Dimensional Flows The equations for mass, momentum, and energy conservation for steady onedimensional flows will be derived in this section using their respective integral forms, as derived in Chap. 2. Consider the shaded area in Fig. 4.1, which depicts a constant and uniform flow through a one-dimensional region. This region could be a typical shock wave, or it could be a heat-added region. As the gas travels through the region, the flow characteristics vary as a function of x. Upstream of the shaded region, the flow velocity, pressure, temperature, density, and enthalpy are v1 , p1 , T1 , ρ1 , and h1 , respectively; while downstream of this region, the properties are v2 , p2 , T2 , ρ2 , and h2 . It is assumed that the cross-sectional areas at the inlet and exit surfaces are equal to A. It is also assumed that the body forces are not present.

4.2.1.1

Continuity Equation

Let us rewrite the integral form of the continuity equation below for convenience. ∂ ∂t For a steady flow,

∂

∂t

"

˚ ρd∀ +

− → → ρ− v . ds = 0

(2.52)

CS

C∀

 = 0 . Thus, "

− → → ρ− v . ds = 0

(4.1)

CS

By applying Eq. (4.1) to the flow through one-dimensional region depicted in Fig. 4.1, we get

4.2 Concept of One-Dimensional Flows

141

m ˚ = ρ1 v1 A = ρ2 v2 A

(4.2)

m ˚ = ρvA = constant

(4.3)

Equation (4.3) is the continuity equation for the steady one-dimensional flow of any fluid, where m ˚ is the mass flow rate through the control volume shown in Fig. 4.1. Because the locations 1 and 2 are arbitrary, Eq. (4.3) follows from Eq. (4.2).

4.2.1.2

Momentum Equation

Let us recall the momentum equation in vector notation as derived in Sect. 2.4.1 for the steady inviscid flow without body forces. "

"  − → → − → → ρ − v . ds − v =− p ds

CS

(2.82)

CS

Because we are concerned with one-dimensional flow, only the x-component of Eq. (2.82) is taken into account for the analysis: "

"   − − → → → p ds ρ − v . ds v = −

x

CS

(4.4)

CS

where v denotes the scalar component of flow velocity in the x-direction. When Eq. (4.4) is applied to a flow across a one-dimensional region (Fig. 4.1) between 1 and 2, the result is ρ1 (−v1 A)v1 + ρ2 (v2 A) v2 = −(−p1 A + p2 A) p1 +

ρ1 v12 2

= p2 +

ρ2 v22

p + ρv = constant

(4.5) (4.6) (4.7)

Equation (4.7) is the momentum equation for steady one-dimensional inviscid flow of a fluid. The momentum equation in this form is useful for analyzing a onedimensional flow through a normal shock or a constant-area duct with friction or heat transfer, as discussed later in Chaps. 7 and 8.

4.2.1.3

Dynamic Equation and Euler’s Equation

We will now derive the momentum equation in a somewhat different manner than in Sect. 4.2.1.2. In this derivation, we will incorporate, in addition to external forces, internal forces such as wall friction fw and drag fd experienced by the flow as a result

142

4 Steady One-Dimensional Flows

Fig. 4.2 Diverging flow passage

of some hindrance. Consider the small divergent flow passage depicted in Fig. 4.2. It is worth noting that fw and fd are supposed to act along the flow in the x-direction. The resultant force operating on a small fluid element in the flow passage in the x-direction can be represented as Fx = pA − (p + dp) (A + dA) + Rx + fw + fd

(4.8)

where Rx is the component of wall reaction Rw in the x-direction. If α is the angle of divergence and dx is the length of the flow passage, the equation for Rx is (4.9) Rx = Rw sin α Let the annular circumferential area of the flow passage be dAw . The pressure on dAw can be considered as the mean of the pressures at locations 1 and 2 (Fig. 4.2), and assumed to operate perpendicularly at the mid-point of dAw . Therefore, the wall reaction Rw is   p + (p + dp) dAw (4.10) Rw = 2

4.2 Concept of One-Dimensional Flows

143

Thus, Eq. (4.9) becomes  dp dAw sin α Rx = p + 2  dp dA Rx = p + 2

(4.11) (4.12)

By substituting the term Rx in Eq. (4.8) with Eq. (4.12), we get  dp dA + fw + fd Fx = pA − (p + dp) (A + dA) + p + 2

(4.13)

By simplifying and neglecting higher order terms, Fx = −Adp + fw + fd

(4.14)

Using Newton’s second law of motion, Fx may be represented as Fx =rate of change of momentum = mass × acceleration = density × volume × acceleration

(4.15)

 dv dA dx Fx = ρ A + 2 dt

(4.16)

That is,

The velocity of a one-dimensional flow is a function of both position x and time t. That is, v = v (x, t) (4.17) The total differential of v is written as dv = vx (x, t) dx + vt (x, t) dt ∂v ∂v dx + dt dv = ∂x ∂t dv ∂v ∂v dx = + dt ∂t ∂x dt

(4.18) (4.19) (4.20)

and dv dt

 Total acceleration

=

∂v ∂t



+

Local acceleration

For steady flow, the local acceleration term,

∂v ∂t

∂v v ∂x



(4.21)

Spatial acceleration

= 0. Thus, Eq. (4.21) reduces to

144

4 Steady One-Dimensional Flows

∂v dv =v dt ∂x

(4.22)

By substituting Eq. (4.22) into Eq. (4.16) and discarding higher order differentials, we have Fx = ρAvdv (4.23) Combining Eqs. (4.14) and (4.23) yields ρAvdv = −Adp + fw + fd (fw + fd ) dp + ρvdv = A

(4.24) (4.25)

Equation (4.25) is a dynamic equation that is essentially a momentum equation. If the flow is inviscid and does not encounter drag, Eq. (4.25) reduces to dp = −ρvdv

(4.26)

This is the most common form of Euler’s equation for steady one-dimensional flows with no losses. It should be noted that Eq. (4.26) only applies to inviscid flows, whether compressible or incompressible.

4.2.1.4

Bernoulli’s Equation

The well-known Bernoulli’s Equation for constant one-dimensional flows can be found directly by integrating Euler’s equation (Eq. (4.26)) with the density assumed to be constant. This results in v2 p + = constant ρ 2

(4.27)

Alternatively by integrating between any two points 1 and 2, p1 v2 p2 v2 + 1 = + 2 ρ1 2 ρ2 2

(4.28)

For compressible flows, Eq. (4.26) can be integrated if density is only a function of pressure, i.e., ρ = ρ (p). Flows with these properties are referred to as barotropic flows. Thus, in the case of compressible flows, a relationship between p and ρ must be known or anticipated in advance in order to integrate Euler’s equation. For the perfect gas undergoing an isentropic process, pressure and density are related as follows: p = constant = C (4.29) ργ

4.2 Concept of One-Dimensional Flows

145

Differentiating this with respect to p yields dp = Cγρ γ −1 dρ

(4.30)

By substituting this in Eq. (4.26) and integrating, we get C

v2 γ ρ γ −1 + = constant γ −1 2

(4.31)

Simplifying this produces the Bernoulli’s Equation for inviscid compressible flows as

v2 γ p + = constant γ −1ρ 2

(4.32)

Alternatively, when the integration is performed between any two points 1 and 2, γ p2 γ p1 v2 v2 + 1 = + 2 γ − 1 ρ1 2 γ − 1 ρ2 2

(4.33)

It can be seen that we did not specify whether the flow is rotational or irrotational while deriving Eqs. (4.27) and (4.32), therefore, these equations are valid in any situation. Because the integration constants in a rotational flow have different values along different streamlines, pressure and velocity cannot be directly coupled for these streamlines. Thus, in a rotational flow, the Bernoulli’s equation can only be applied along the streamline. In an irrotational flow, on the other hand, the integration constants have a single value for the entire flow, and so the Bernoulli’s equation is applicable between any two points in the flow, not necessarily along a streamline.

4.2.1.5

Energy Equation

In the absence of body forces, the energy equation for the steady, one-dimensional, adiabatic flow of an inviscid fluid is  "  1 2 − → → h + v + gz ρ − v . ds = 0 (2.162) 2 CS

In gas dynamics, we mostly deal with low-density fluids, such as gases, where the potential energy term gz is relatively small in comparison to other terms and may thus be ignored. This results in "  CS

1 2  − − → h+ v ρ→ v . ds = 0 2

(4.34)

146

4 Steady One-Dimensional Flows

Applying Eq. (4.34) to a one-dimensional flow in the region depicted in Fig. 4.1 between 1 and 2 yields 1 1 h1 + v12 =h2 + v22 2 2 1 h + v2 = constant 2

(4.35) (4.36)

Equation (4.36) is the energy equation for the steady one-dimensional adiabatic flow of any fluid. As we will see in the following chapters, Eq. (4.35) is a very useful relationship for solving compressible fluid flow problems. Example 4.1 Prove that a steady, one-dimensional, adiabatic, inviscid and nonconducting flow is an isentropic flow. Solution For the steady, one-dimensional, adiabatic, and non-conducting flow, the energy equation is 1 (4.36) h + v2 = constant 2 Differentiating Eq. (4.36) with respect to v gives dh + vdv = 0

(i)

If the flow is inviscid then by Euler’s equation, dp + vdv = 0 ρ

(4.26)

Subtracting Eq. (4.26) from (i) gives dh −

dp =0 ρ

(ii)

Substituting (ii) into Eq. (2.203) results in ds = 0 =⇒ s = constant Therefore, a steady one-dimensional adiabatic inviscid and non-conducting flow is essentially an isentropic flow.

4.3

Equations Governing the Steady One-Dimensional Isentropic Flow …

147

4.3 Equations Governing the Steady One-Dimensional Isentropic Flow of a Perfect Gas By definition, the entropy in an isentropic process remains constant. It was demonstrated in Chap. 3 that the pressure and density in an isentropic process are related as p = constant ργ

(4.37)

Consider any two points in a steady one-dimensional isentropic flow of perfect gas through a duct, such as 1 and 2 in Fig. 4.3. It follows from Eq. (2.257) that p2 = p1



ρ2 ρ1

γ

 =

T2 T1

(γ γ−1) (2.257)

Using Eq. (4.36) for a flow through the duct between locations 1 and 2, we get 1 1 h1 + v12 =h2 + v22 2 2 √ R and a = γ RT. Thus, For a perfect gas, h = cp T, cp = γγ−1

Fig. 4.3 Steady one-dimensional isentropic flow of a perfect gas

(4.38)

148

4 Steady One-Dimensional Flows

1 1 cp T1 + v12 =cp T2 + v22 2  2

(4.39)

v2

1 + 2c1p T2  = v2 T1 1 + 2c2p    2  v1 1 + γ −1 2 γ RT1   2  = v2 1 + γ −1 2 γ RT2  2   v1 1 + γ −1 2 a2   1 = γ −1 v22 1 + 2 a2

(4.40)

(4.41)

(4.42)

2

Thus,   ⎡ ⎤ 1 + γ −1 M12 2 T2 ⎦   =⎣ T1 1 + γ −1 M2

(4.43)

2

2

This is the required temperature ratio for the steady one-dimensional flow of a perfect gas. It should be noted that when calculating Eq. (4.43), we just used the adiabatic flow condition and did not examine whether the flow process between locations 1 and 2 was reversible or irreversible. Thus, Eq. (4.43), derived from the adiabatic energy equation, applies to both reversible and irreversible flows. √ The local speed of sound is defined as a = γ RT, thus Eq. (4.43) can be represented as   ⎡ ⎤ M12 1 + γ −1 2 a22 T2 ⎦   = =⎣ (4.44) γ −1 2 T1 a12 M 1+ 2

2

However, because we assumed that the flow between locations 1 and 2 is isentropic (Fig. 4.3), Eq. (4.43) in conjunction with Eq. (2.257) yields the pressure ratio as ⎡

1+



γ −1 2



M12

⎤ γ γ−1

p2 ⎦   =⎣ p1 1 + γ −1 M2

(4.45)

2

2

with the density ratio as ⎡

1+



γ −1 2



M12

1 ⎤ γ −1

ρ2 ⎦   =⎣ ρ1 1 + γ −1 M2 2

2

(4.46)

4.3

Equations Governing the Steady One-Dimensional Isentropic Flow …

149

Furthermore, using the continuity equation (Eq. (4.3)) to the flow between 1 and 2 yields 

ρ2 ρ1



v1 v2

=

A1 A2

(4.47)

Equations (4.43)–(4.47) are sufficient to calculate all the properties of a onedimensional isentropic flow of a perfect gas. Example 4.2 In the adiabatic flow of a compressible fluid, show that the local flow velocity v and the local temperature T are related to their corresponding freestream values as   v 2 T (γ − 1) 2 M∞ =1− −1 T∞ 2 v∞ Solution The local flow properties may be related to the freestream properties by using an adiabatic flow energy equation. Thus, 1 1 2 h + v2 =h∞ + v∞ 2 2 For a perfect gas h = cp T and cp =

γR . γ −1

Hence,

1 1 2 γ RT γ RT∞ + v2 = + v∞ γ −1 2 γ −1 2 Dividing the above expression by T∞ throughout yields 2 T (γ − 1) v2 (γ − 1) v∞ + =1 + T∞ 2 γ RT∞ 2 γ RT∞ 2 But a∞ = γ RT∞ and M∞ =

v∞ ; a∞

hence

T (γ − 1) v2 (γ − 1) 2 M∞ + =1 + 2 T∞ 2 a∞ 2 2 T (γ − 1) v2 v∞ (γ − 1) 2 M∞ + =1 + 2 2 T∞ 2 v∞ a∞ 2 T (γ − 1) v2 2 (γ − 1) 2 M∞ + M∞ =1 + 2 T∞ 2 v∞ 2 Therefore,

150

4 Steady One-Dimensional Flows

T γ −1 2 M∞ =1− T∞ 2



v v∞

2

 −1

4.4 Stagnation or Reservoir Conditions The stagnation condition is an exceptionally helpful reference state since it facilitates the analysis of the flow and its solutions. It is the condition in which a moving fluid comes to a complete stop, i.e., a static fluid is in the state of stagnation. An adiabatic flow process or an isentropic flow process can bring the moving fluid element to rest. The flow deceleration in the adiabatic flow process occurs adiabatically (without any heat exchange with the surroundings); however, the process can be carried out reversibly or irreversibly. The flow deceleration in the isentropic flow process, on the other hand, occurs reversibly (without friction) and adiabatically. The concept of stagnation is independent of the fluid being investigated. Therefore, a local stagnation condition exists at every point along the flow path, including the actual flow, which entails work, heat, body forces, friction, etc. These local stagnation states, specified at each position along the flow path, are essentially the characteristics of the point in the real flow of fluid and may differ from one another.

4.4.1 Stagnation Enthalpy For the steady, one-dimensional, adiabatic, and non-conducting flow of an inviscid fluid, the energy equation is v2 = constant (4.36) h+ 2 Consider the case in which the fluid is brought to rest adiabatically from the initial speed v. By inserting v = 0 in Eq. (4.36), the value of the constant is derived as h = h0 , where h0 is referred to as the total or stagnation enthalpy that would occur at a point when the flow is brought to rest adiabatically. Thus, Eq. (4.36) can be written as h+

v2 = h0 2

(4.48)

i.e., the stagnation enthalpy is the sum of the static enthalpy h and the kinetic energy 2 per unit mass v2 at a given point in the flow field. Equation (4.48) shows a schematic representation of Fig. 4.4. Equation (4.48) is a form of energy equation that is often employed in gas dynamics and applies to both irreversible and reversible adiabatic (isentropic) flow processes.

4.4

Stagnation or Reservoir Conditions

151

Fig. 4.4 The enthalpy-entropy diagram for a general fluid flow

Finally, keep in mind that the constant in Eq. (4.48), h0 , has the same value at all points along the flow path (streamline) for a steady one-dimensional adiabatic flow. In other words, along a streamline, the total enthalpy remains constant. This type of flow is known as isenthalpic or isoenthalpic flow. If all of the streamlines are derived from a single uniform freestream, then h0 is the same for all of them. Therefore, for such a steady adiabatic flow, h0 remains constant throughout the flow field and is equal to the freestream value. Example 4.3 Consider a thin circular duct through which the adiabatic flow of air is drawn from a large reservoir with constant stagnation conditions and released into the atmosphere at the exit of the duct. If the reservoir pressure p0 is n times the ambient pressure pa , show the flow velocity at the exit is  v=

 2γ p  γ γ−1 n −1 (γ − 1) ρ

Solution The adiabatic energy equation applied to the inlet and exit of the duct gives v2 cp T0 = cp Ta + 2  v = 2cp (T0 − Ta )   T0 = 2cp Ta −1 Ta For the perfect gas p = ρRT and cp =

γR . γ −1

 2γ p v= (γ − 1) ρ



Thus, T0 −1 Ta

152

4 Steady One-Dimensional Flows

From the isentropic relation T0 = Ta



p0 p

γ γ−1

Substituting in v expression gives    v =

2γ p (γ − 1) ρ



p0 p

γ γ−1

 −1

Given that, p0 = npa Therefore,  v=

 2γ p  γ γ−1 n −1 (γ − 1) ρ

4.4.2 Stagnation Temperature The stagnation temperature at a point in the flow field is defined as the temperature reached when the flow speed is reduced to zero adiabatically. Consider the case when the flow of a perfect gas is adiabatically decelerated to zero speed (zero Mach number) from an initial speed of v (or M). Accordingly, it follows from Eq. (4.43) that γ −1 2 T0 =1+ M T 2

(4.49)

where T0 and T denote the stagnation and static temperatures, respectively. It is worth noting that because Eq. (4.49) is derived from Eq. (4.36), it applies to both reversible and irreversible adiabatic flows. As discussed in Sect. 4.4.1, if the actual flow process is adiabatic, the stagnation enthalpy has a constant value along the streamline, and if all streamlines originate from the same uniform freestream, h0 , is constant throughout the entire flow and equal to the freestream value. Because the stagnation enthalpy of a calorically perfect gas is h0 = cp T0 , these results suggest that T0 remains constant in the steady adiabatic inviscid flow of a calorically perfect gas. Accordingly, Eq. (4.49) results in

4.4

Stagnation or Reservoir Conditions

153

  γ −1 2 M = constant T0 = T 1 + 2 For a perfect gas, cp =

γR γ −1

and a =

√

(4.50)

γ RT. Hence, rewriting Eq. (4.49) as

T0 =T +

v2 2cp

(4.51)

2

v The term 2c in Eq. (4.51) is commonly referred to as the impact temperature rise p and is indicated by Tim . Therefore, Eq. (4.51) is transformed as

T0 = T+Tim The expression for the relative change in stagnation temperature, found by logarithmic differentiation of Eq. (4.49) as   γ −1 dM2 dT0 dT 2 = + γ −1 T0 T 1 + 2 M2 M2

(4.52) dT0 , T0

may be

(4.53)

4.4.3 Stagnation Pressure As explained in Sect. 4.4.2, the temperature obtained by a flow moving with an initial velocity of v1 when brought to rest adiabatically (or isentropically) is known as the stagnation temperature T0 . If the flow deceleration is done isentropically, the corresponding pressure is known as the stagnation or total pressure p0 . Because the static and stagnation states are connected by an isentropic process, the relationship between T0 and p0 for a perfect gas may be derived using Eq. (2.257). p0 = p Substituting Eq. (4.50) for

T0 T



T0 T

(γ γ−1) (4.54)

into Eq. (4.54) results in

 γ  p0 γ − 1 2 γ −1 = 1+ M p 2

(4.55)

Equation (4.55) is the stagnation pressure ratio for a perfect gas. It is worth noting that, unlike Eq. (4.49), which applies to both reversible and irreversible adiabatic flow processes, Eq. (4.55) only applies to an isentropic flow process. 0 , can be derived The expression for the relative change in stagnation pressure, dp p0 by logarithmic differentiation of Eq. (4.55) as

154

4 Steady One-Dimensional Flows

  γ M2 dM2 dp0 dp 2 + = p0 p 1 + γ −1 M2 M2 2

(4.56)

Example 4.4 A body moves through the standard atmospheric air at 200 m/s. Calculate the pressure at that point where the body experiences the flow at rest. Assume the flow to be (i) incompressible (ii) compressible. (For standard air, γ = 1.4, R = 287 J/(kg.K), T∞ =288 K, and p∞ =101 kPa.) Solution Given, p∞ =101 kPa T∞ =288 K v = 200 m/s (i) The pressure at the stagnation point in an incompressible flow with no body forces can be calculated by the incompressible Bernoulli’s equation. Thus, 1 2 p0 = p∞ + ρv∞ 2 101 1 × (200)2 p0 = 101 + × 2 287 × 288 = 125.5 kPa (ii) The speed of sound in air is a=



γ RT∞ =

√ 1.4 × 287 × 288 = 340.17 m/s

and therefore the Mach number becomes M=

200 v = = 0.59 a 340.17

Now, by Eq. (4.55)  (γ γ−1)   p0 γ −1 M2 = 1+ p∞ 2 1.4  (1.4−1)   1.4 − 1 = 1+ (1.5)2 2 = 3.67 Therefore,

4.4

Stagnation or Reservoir Conditions

155

p0 = 101 × 3.67 = 370.77 kPa Problem 4.5 What is the angle at which the uniform airflow at Mach 1.5 must be turned so that its static pressure is halved? Solution Given M1 = 1.5 p1 p2 = 2 By Eq. (4.45) ⎡

1+



γ −1 2



M12

⎤ γ γ−1

p2 ⎦   =⎣ p1 1 + γ −1 M2 1 = 2



1+

2

2

 1.4−1 

2

1+

(1.5) 2  1.4−1  2 M2 2

1.4  (1.4−1)

1 + 0.2M22 = 1.767 M2 = 1.96 Now, by Eq. (3.42) the flow turning angle θ will be   1 1 − sin−1 θ =μ1 − μ2 = sin−1 M1 M2   1 1 − sin−1 θ =sin−1 1.5 1.96 θ =41.81◦ − 30.68◦ ≈11◦ Therefore, the airflow at Mach 1.5 should then be turned by 11◦ so that its static pressure is halved.

4.4.4 Stagnation Density The stagnation density ρ0 is the flow density that corresponds to the stagnation temperature T0 and the stagnation pressure p0 . The stagnation density can be calculated using the thermal equation of state of an ideal gas:

156

4 Steady One-Dimensional Flows

ρ0 =

p0 RT0

(4.57)

Combining Eqs. (4.55) and (4.49) with Eq. (4.57) yields 1    (γ −1) ρ0 γ −1 2 = 1+ M ρ 2

(4.58)

4.4.5 Stagnation Speed of Sound The speed of sound (or acoustic speed) in a gas at its stagnation state is known as the stagnation speed of sound (or stagnation acoustic speed) and is denoted by a0 . a0 =

 γ RT0

(4.59)

The stagnation speed of the sound is a property of gas and is constant in an adiabatic flow. Combining Eqs. (4.59), (3.14) and (4.49) yields the following relation between a0 and a. a02 T0 =1+ = a2 T



γ −1 M2 2

(4.60)

4.4.6 Entropy Change in Terms of Stagnation Properties Unlike the change in enthalpy of a gas between the static and corresponding stagnation states, the change in entropy between these two states is independent of the flow velocity. Therefore, there is no such thing as static or stationary entropy. The following relationship developed in Sect. 2.10.5 can be used to express the change in entropy of a perfect gas between two arbitrary states 1 and 2. s = s2 − s1 = cp ln

T2 p2 − R ln T1 p1

(2.249)

Assume that both 1 and 2 are stagnation states. Accordingly, Eq. (2.248) yields s0 = s02 − s01 = cp ln

T02 p02 − R ln T01 p01

(4.61)

Because the stagnation temperature remains constant in a steady, adiabatic flow, i.e., T01 = T02 . Thus, Eq. (4.61) reduces to

4.4

Stagnation or Reservoir Conditions

157

s0 = s02 − s01 = −R ln

p02 p01

(4.62)

By definition, s = s0

(4.63)

Therefore, the change in entropy of a perfect gas in terms of stagnation pressures is s = s2 − s1 = R ln

p01 p02

(4.64)

The entropy remains constant in the frictionless and adiabatic flow process, often known as the isentropic process, i.e., s = s2 − s1 = 0

(4.65)

It follows from Eq. (4.64) that p01 = p02

(4.66)

This demonstrates that in an isentropic process, the stagnation pressure remains constant. However, if the process proceeds in the presence of friction, the entropy of the gas must increase, i.e., s > 0, in accordance with the second law of thermodynamics (Eq. (2.183)). Consequently, Eq. (4.64) necessitates that, p02 < p01 , that is, the stagnation pressure decreases in the presence of friction. Conversely, a decrease in the stagnation pressure of the gas can be seen as an increase in its specific entropy (due to irreversibility associated with the adiabatic flow process). Remark In fluid flows, the stagnation properties, in general, can vary throughout the flow. However, if the flow is adiabatic (but not necessarily isentropic), the stagnation enthalpy h0 , and, thus, the stagnation temperature for the perfect gas T0 remains constant for the entire flow. It follows from Eqs. (4.48), (4.49) and (4.59) that h0 , T0 and a0 are constant in adiabatic flow even though friction is present. On the other hand, the stagnation pressure p0 and stagnation density ρ0 decrease due to friction. This can be seen by considering entropy change relationship, given by Eq. (4.64). Thus, In general, the stagnation properties of fluid flows can change throughout the flow. However, if the flow is adiabatic (but not necessarily isentropic), the stagnation enthalpy h0 and hence the stagnation temperature for the perfect gas T0 stay constant throughout the flow. It follows from Eqs. (4.48), (4.49), and (4.59) that h0 , T0 , and a0 are constant in adiabatic flow even when friction exists. In the presence of friction, however, the stagnation pressure p0 and stagnation density ρ0 decrease. Consider the following entropy change relationship to demonstrate this:

158

4 Steady One-Dimensional Flows

Fig. 4.5 High-pressure air is discharged from a large reservoir through a small opening



p01 s = s2 − s1 = R ln p02

(4.64)

According to the second law of thermodynamics, a process in the presence of friction is only possible if s2 > s1 ; hence, Eq. (4.64) shows that p01 > p02

(4.67)

This demonstrates that in the presence of friction, the stagnation pressure drops. It also follows from the equation of state, p0 = ρ0 RT0 , that a drop in p0 eventually reduces ρ0 for the constant T0 .

4.5 Characteristic Speeds in Gas Dynamics There are many different characteristic speeds that represent the overall characteristics of the flow in the compressible flow analysis. This section derives some of these characteristic speeds.

4.5.1 Maximum Isentropic Discharge Speed Consider the case of a perfect gas being expanded under adiabatic conditions in a frictionless duct (streamtube), as shown in Fig. 4.5. Assume the gas is drawn into the duct from a large reservoir with constant stagnation conditions of p0 and T0 . It is needed to compute the speed obtained by the gas at some downstream position where the pressure and temperature are p and T, respectively.

4.5

Characteristic Speeds in Gas Dynamics

159

Because the flow inside the duct is frictionless (reversible) and adiabatic, the speed attained by the gas is effectively the isentropic speed. The one-dimensional adiabatic energy equation (Eq. (4.48)) applied to the flow between 1 and 2 under steady circumstances provides 

 T v = 2cp T0 1 − T0   T 2γ = RT0 1 − T0 (γ − 1)

(4.68)

(4.69)

According to Eq. (2.257), the stagnation temperature ratio is given by T = T0



p p0

γ γ−1 (4.70)

Substituting Eq. (4.70) into Eq. (4.69) yields    v=    =

  γ γ−1  2γ p RT0 1 − p0 (γ − 1)   γ γ−1  2γ p0 p 1− p0 (γ − 1) ρ0

(4.71)

(4.72)

In 1839, Saint Venant and Wantzel independently determined the isentropic discharge speed, which is given by Eq. (4.72), and hence it is commonly referred to as the Saint Venant-Wantzel formula. It is worth noting that v reaches a maximum when p is equal to zero, implying that the flow reaches its maximum isentropic speed when discharged into the vacuum. Therefore, Eq. (4.72) yields  2γ RT0 γ −1  2 = a0 γ −1

vmax =

(4.73)

vmax

(4.74)

where a0 is the stagnation speed of sound. This is the maximum speed attained by an inviscid compressible flow when discharged into a vacuum. It is the speed that corresponds to the complete transformation of the kinetic energy associated with random molecular motions into directed kinetic energy. However, vmax is a hypothetical speed that cannot be attained even by pure isentropic flow. Nevertheless,

160

4 Steady One-Dimensional Flows

because of its defined value under a specific set of conditions, it has significance and is thus considered to be the characteristic speed of the flow. Furthermore, Eq. (4.74) shows that vmax is independent of reservoir pressure p0 but dependent on reservoir temperature T0 . When the flow is not discharged into the vacuum, (p = 0), i.e., for non-zero pressure at location 2, Eq. (4.72) can be expressed by dividing it by Eq. (4.59) as     γ γ−1   2γ v p  = 1− a0 p0 (γ − 1)

(4.75)

Example 4.6 What is the maximum isentropic discharge speed of air from a reservoir at 800 K? Solution Given, T0 = 800 K By using Eq. (4.74), the maximum isentropic discharge speed is:  vmax = a0  = γ RT0 =



2 γ −1 2 γ −1 

√ 1.4 × 287 × 800 ×

2 1.4 − 1

= 1267.7 m/s

4.5.2 Critical Speed of Sound The critical state of a flow is the condition that occurs when the flow is isentropically accelerated or decelerated until the Mach number equals one. For example, if the Mach number was changed from M to 1, these are the situations that would arise. The critical condition, also known as the sonic state, is commonly marked by the asterisk (∗), and so the flow variables at point 2 are represented as p∗ , T∗ , ρ ∗ , v∗ , and A∗ . By definition, let a2 = a∗ and v2 = a∗ , and the corresponding value of the local Mach number M2 is one. In this case, a∗ is referred to as the critical speed of sound and is the characteristic speed of the flow. Substituting a2 = a∗ and M2 = 1 into Eq. (4.44), and dropping the subscript 1 results in

4.5

Characteristic Speeds in Gas Dynamics

161

Fig. 4.6 Critical state in an isentropic flow

  γ −1 2 γ + 1 ∗2 a a2 1 + M = 2 2

(4.76)

According to Eq. (4.60), the stagnation speed of sound is related to the critical speed of sound as a02 =

(γ + 1) ∗2 a 2

(4.77)

For air (γ = 1.4), Eq. (4.77) gives a∗ = 0.913 a0

(4.78)

Example 4.7 Air at 80 kPa and 450 K is flowing at Mach 2.5 through a passage. A portion of the flow seeps through a small opening in the passage wall to a still environment with a pressure of 40 kPa. Determine the maximum Mach number attained by the discharge. Solution Given, p∞ = 80 kPa

T∞ = 450 K

For M∞ = 2.5, from isentropic table, p∞ = 0.0585 p0

M∞ = 2.5

162

4 Steady One-Dimensional Flows

80 × 103 0.0585 = 1367.52 kPa

p0 =

At the exit of the opening, the pressure ratio is pa 80 = p0 1367.52 = 0.0585 p∗ < = 0.528 p0 The flow will be choked at the opening because pp0a is less than the critical pressure ratio for air, p∗ . The Mach number attained by the discharge would, therefore, be p0 one.

4.5.3 Thermodynamic Properties in Terms of Critical Speed of Sound The relationships for critical temperature T∗ , critical pressure p∗ , and critical density ρ ∗ may be derived from Eqs. (4.43)–(4.46) by setting M2 = 1. This gives    2 γ −1 T∗ = + M2 T γ +1 γ +1    γ γ−1 p∗ 2 γ −1 = + M2 p γ +1 γ +1 1    γ −1 ρ∗ 2 γ −1 2 = + M ρ γ +1 γ +1

(4.79) (4.80) (4.81)

By setting M = 0 in Eqs. (4.79)–(4.81) yields the relationships between the thermal properties at critical and stagnation conditions. This gives   2 T∗ = T0 γ +1  γ γ−1  p∗ 2 = p0 γ +1

(4.82) (4.83)

4.5

Characteristic Speeds in Gas Dynamics

163

1   γ −1 2 ρ∗ = ρ0 γ +1

(4.84)

For air (γ = 1.4), the equations above reduce to T∗ = 0.833 T0 p∗ = 0.528 p0 ρ∗ = 0.634 ρ0

(4.85) (4.86) (4.87)

Example 4.8 The perfect gas flows through an insulated duct. Show that, under critical conditions, the static pressure is given by  2RT0 γ (γ + 1)

m ˚ p = A ∗

Solution The mass flow rate at critical condition through the duct is m ˚ = ρ ∗ Aa∗ For the perfect gas p∗ = ρ ∗ RT∗ and a∗ =

√ γ RT∗ . Hence,

p∗  A γ RT∗ RT∗ p∗ √ =√ A γ ∗ RT

m ˚ =

But,

  T∗ 2 = T0 γ +1

Substituting in m ˚ expression gives  m ˚ p∗ = A

2RT0 γ (γ + 1)

(4.82)

164

4 Steady One-Dimensional Flows

4.5.4 Relationships Between the Characteristic Speeds So far in our discussion, we have noticed that the critical speed of sound a∗ and the stagnation speed of sound a0 are the characteristic speeds of the compressible flow. The relation between a0 and a∗ is given by a02 =

(γ + 1) ∗2 a 2

(4.77)

Combining Eqs. (4.74) and (4.77) yields 2 a2 (γ − 1) 0 (γ + 1) ∗2 a = (γ − 1)

2 = vmax

(4.88)

2 vmax

(4.89)

For air (γ = 1.4), we have a0 = 1.095a∗ vmax = 2.236a0

(4.90) (4.91)

vmax = 2.448a∗

(4.92)

4.5.5 Kinematic Forms of the Energy Equation for Steady One-Dimensional Adiabatic Flow of a Perfect Gas In this section, we shall derive the kinematic forms of the energy equation for the steady one-dimensional adiabatic flow of a perfect gas. In an adiabatic flow, the stagnation enthalpy h0 remains constant. Therefore, Eq. (4.48) is rewritten as 1 h0 = h + v2 =constant 2 For a perfect gas, h = cp T, cp = written as

γR γ −1

(4.93)

and a2 = γ RT. Hence, Eq. (4.93) can be

a2 v2 + =constant γ −1 2

(4.94)

The value of the constant in Eq. (4.94) can be determined under the following reference conditions.

4.5

Characteristic Speeds in Gas Dynamics

165

1. The flow comes to a rest at the specified location. 2. The flow temperature is zero at the specified location. 3. The flow speed at the specified location is equal to the speed of sound (sonic state). When the values of constants derived at these reference conditions are substituted back into Eq. (4.94), the kinematic forms of the energy equation are as follows: v2 a02 a2 + = γ −1 2 γ −1 2 2 a v v2 + = max γ −1 2 2  2 2 a v γ + 1 a∗2 + = γ −1 2 γ −1 2

(4.95) (4.96) (4.97)

4.5.6 Dimensionless Speed M∗ We have clearly seen throughout our discussion that the Mach number is a very important parameter in compressible flow analysis, but it has two major drawbacks. First, the Mach number is affected by more than just the flow speed; it is also affected by the state of the flow, particularly the temperature. Second, at extremely high speeds, the Mach number tends to become infinitely large. Because of this, another nondimensional parameter known as dimensionless speed, indicated by M∗ , is usually found to be beneficial in gas dynamics. The ratio of the flow speed v to the critical speed of sound a∗ is defined as the M∗ . M∗ =

v a∗

(4.98)

The relationship between the M∗ and M can be deduced as follows. Rewriting Eq. (4.98) as  v 2  v 2  a 2 × ∗ M∗2 = ∗ = a a a  ∗ 2   M a 2 = ∗ M a Combining Eqs. (4.97) and (4.100), then rearranging the terms, results in

(4.99) (4.100)

166

4 Steady One-Dimensional Flows

M∗2 2 (γ + 1) M∗2 + = (γ − 1) M2 (γ − 1)   2 (γ − 1) M + 2 (γ + 1) M∗2 = 2 (γ − 1) M (γ − 1) 2 + 1) M (γ M∗2 = 2 + (γ − 1) M2

(4.101) (4.102) (4.103)

or M2 =

2M∗2 (γ + 1) − (γ − 1) M∗2

(4.104)

Equation (4.103) demonstrates that the M∗ is more than simply a basic function of M. They are related in such a way that • • • • •

If M = 0 then M∗ = 0 if M < 1 then M∗ < 1 If M = 1 then M∗ = 1 if M > 1 then M∗ > 1  If M → ∞ then M∗ → γγ +1 −1

Therefore, unlike theMach number, the dimensionless speed is a bounded parameter with a range of 0 to

γ +1 . γ −1

The expressions for the stagnation property ratios, TT0 , pp0 , and by combining Eqs. (4.104), (4.49), and (2.257). This results in    γ −1 T M∗2 = 1− T0 γ +1  γ γ−1   p γ −1 ∗2 M = 1− p0 γ +1 1  γ −1   T γ −1 ∗2 M = 1− T0 γ +1

ρ , ρ0

may be derived

(4.105) (4.106) (4.107)

Example 4.9 If the p is the static pressure and p∗ is the corresponding critical value for the isentropic flow of a perfect gas, then show that  γ γ−1  p (γ + 1) = p∗ 2 + (γ − 1) M2 Solution Let us express

p p∗

as p p p0 = × ∗ ∗ p p0 p

4.5 Characteristic Speeds in Gas Dynamics

167

From the isentropic relation,  γ  p0 (γ − 1) 2 γ −1 = 1+ M p 2

(4.55)

For M = 1, we have p0 = p∗ Substituting in

p p∗



γ +1 2

γ γ−1

expression gives

p = p∗



 γ γ  γ + 1 γ −1 (γ − 1) 2 γ −1 1+ M × 2 2   γ γ−1 γ +1 = 2 + (γ − 1) M2

4.6 Steady, Adiabatic Flow Ellipse We classified the flow regimes in Sect. 3.6 based on the Mach number. Flow regimes can also be characterized using the local flow velocity and the local speed of sound. Consider a streamtube that does not exchange any heat with its neighboring streamtubes. The energy equation for steady adiabatic flow through such a streamtube can be written as  2 a v2 v2 + = max (4.96) 2 γ −1 2 2 Throughout, dividing by vmax yields

v2 + 2 vmax



2 γ −1



a2 =1 2 vmax

(4.108)

By substituting Eq. (4.88) into Eq. (4.108), we get a2 v2 + =1 2 vmax a02

(4.109)

It is essentially an ellipse equation known as the steady adiabatic flow ellipse. Equation (4.109) is depicted schematically in Fig. 4.7. Rewriting Eq. (4.109) as

168

4 Steady One-Dimensional Flows

Fig. 4.7 A typical steady adiabatic flow ellipse

 a2 = a02 −

v2 2 vmax

a02

(4.110)

This, when differentiated with respect to v, yields  γ − 1 v da =− dv 2 a  γ −1 M =− 2

(4.111) (4.112)

or  M= −

2 γ −1



da dv

(4.113)

Equation (4.113) represents the change in the slope of the curve (Fig. 4.7) from one point to another, which is equivalent to the change in the Mach number. The following conclusions can be drawn from Eq. (4.113). • At higher flow speeds, the change in speed of sound essentially results in a change in the Mach number. • At smaller flow speeds, the change in flow speed primarily results in a change in the Mach number. • When the Mach number is less than 0.3, the change in the speed of sound is negligibly small. Based on the observations above, the steady adiabatic flow ellipse is categorized into different zones with varied physical properties, reflecting different flow regimes. The stagnation condition is a useful reference state since it simplifies the flow study and its solutions. It is the state at which a moving fluid comes to a stop, i.e., a static fluid attains the stagnation state. An adiabatic flow process or an isentropic flow process can bring a moving fluid element to rest. The flow deceleration in the adiabatic flow process occurs adiabatically (i.e., without any heat exchange with the

4.6

Steady, Adiabatic Flow Ellipse

169

surroundings); however, the process can be carried out reversibly or irreversibly. The flow deceleration in the isentropic flow process, on the other hand, occurs reversibly (without friction) and adiabatically.

4.6.1 Incompressible Flow In this regime, the flow speed v is very small compared to the speed of sound a and the changes in a are very small compared to the changes in v. Consequently, the variation in the Mach number is almost entirely due to the speed of the flow.

4.6.2 Compressible Subsonic Flow In this regime, the flow speed and the speed of sound are of comparable magnitudes, yet v < a. The change in Mach number occurs primarily due to a change in v and only secondary to changes in a.

4.6.3 Transonic Flow It is the flow regime in which the difference between the flow speed and the speed of sound is small compared to either v or a. In addition, the changes in v and a are of comparable magnitude.

4.6.4 Supersonic Flow In this regime, the flow speed and the speed of sound are of comparable magnitude, but v > a. The change in Mach Number is due to significant variations in both v and a.

4.6.5 Hypersonic Flow In this realm of compressible flow, the flow speed is very high relative to the speed of sound, i.e., v  a. The change in v is very small, so the change in Mach number is almost exclusively the consequence of the change in a.

170

4 Steady One-Dimensional Flows

Example 4.10 For a perfect gas, show that the steady one-dimensional adiabatic energy equation may be expressed as a∗2 (γ − 1) cos2 μ + sin2 μ = 2 v (γ + 1) where μ is the Mach angle and sin μ =

1 . M

Solution Consider the kinematic form (Eq. (4.97)) of the adiabatic energy equation. Therefore,  a2 v2 γ + 1 a∗2 + = γ −1 2 γ −1 2 2 2a (γ − 1) 2 + v = a∗2 (γ + 1) (γ + 1) 2a2 (γ − 1) a∗2 = + (γ + 1) v2 (γ + 1) v2 1 2 (γ − 1) a∗2 = + (γ + 1) M2 (γ + 1) v2  a∗2 2 (γ − 1)  2 sin2 μ + sin μ + cos2 μ = 2 v (γ + 1) (γ + 1)   ∗2 γ − 1 2 a (γ − 1) + sin2 μ= 2 cos2 μ + γ +1 γ +1 v (γ + 1) Hence, a∗2 (γ − 1) cos2 μ+sin2 μ = 2 v (γ + 1)

4.7 Continuity Equation for Steady One-dimensional Flow of a Perfect Gas The continuity equation for the steady one-dimensional flow of a general fluid is given by m ˚ = ρvA = constant (4.3) For the perfect gas p = ρRT, hence m ˚ =

p vA = constant RT

√ Substituting v = M γ RT into this gives

(4.114)

4.7

Continuity Equation for Steady One-dimensional Flow of a Perfect Gas

 γ m ˚ =p MA = constant RT

171

(4.115)

Assume that the real flow arriving at any cross section A is essentially generated from a large reservoir at p0 and T0 from which the gas is expanded to the actual values at v, p and T. By combining Eqs. (4.49), (4.55), and (4.115), we get m ˚ = Mp0 A



γ RT0

 (γ +1) γ − 1 2 − 2(γ −1) 1+ M = constant 2

(4.116)

This is the mass flow rate per unit area for the steady one-dimensional flow of a perfect gas in terms of the Mach number. Equation (4.116) applies to each cross section of a duct as long as the mass flow remains constant. It also demonstrates that the mass flow rate per unit area √at a given Mach number is directly proportional to p0 and inversely proportional to T0 . That is, for a given Mach number, increasing the stagnation pressure increases the mass flow rate, whereas increasing the stagnation temperature decreases it.

4.7.1 Maximum Isentropic Mass Flow Rate per Unit Cross-Sectional Area Let us examine the condition for the maximum mass flow rate inside a streamtube. Assume a perfect gas is drawn into the streamtube by lowering the back pressure (pressure downstream of the streamtube exit) from a vast reservoir under constant stagnation conditions. For a given isentropic flow through the streamtube, the stagnation pressure p0 and stagnation temperature T0 remain constant. Thus, by differentiating Eq. (4.116) with respect to M and equating the derivative to zero, the condition for the maximum mass flow rate per unit area may be derived. This results in   d

m ˚ A

dM

=0

(4.117)

or   d

m ˚ A

dM  p0

 =p0

⎧ $ %⎫ (γ +1)  − 2(γ ⎨ ⎬ −1) γ −1 2 γ 1+ M − ⎭ RT0 ⎩ 2

(4.118)

⎧ % ⎫ $ (γ +1) − 2(γ  −1 ⎬ ⎨ −1) γ −1 γ −1 2 γ (γ + 1) × × 2M × 1 + M M× =0 ⎭ RT0 ⎩ 2 (γ − 1) 2 2 (4.119)

172

4 Steady One-Dimensional Flows

This reduces to 

 (γ + 1) M2 =0 2 + (γ − 1) M2

(4.120)

M2 − 1 = 0 =⇒ M= ±1

(4.121)

1−

Note that, M = −1 is an unrealistic solution, therefore, we have M= 1 at the point in the duct where the local area is either maximum or minimum. However, the ˚ and the substitution of M= 1 demonstrates evaluation of the second derivative of m A that ⎡  ⎤ ˚ d2 m A ⎣ ⎦ 1). When the Mach number is equal to one, the area ratio is one. However, for all other Mach numbers, the area ratio is always greater than one. Equations (4.49), (4.58), (4.75), and (4.133) can be used to plot the performance curves illustrated in Fig. 4.11 for a given stagnation condition. The curves of nonfor incompressible and compressible dimensional mass flow rate per unit area ρρv 0 a0 flows are the most noteworthy aspects of this figure. The fundamental distinction between them is that the density of a compressible fluid varies with its flow speed,

176

4 Steady One-Dimensional Flows

which introduces considerations that are not present in the case of incompressible fluids. It is worthwhile to explore the influence of compressibility on the mass flow rate m, ˚ assuming that the flow is isentropic. Thus, by writing the continuity equation (Eq. (4.3)) in the form G=

m ˚ = ρv A

(4.135)

where G denotes the mass flux (mass flow rate per unit area). The magnitudes of ρ and v in Eq. (4.135) depend on the pressure ratio pp0 that governs the flow process. There are two extreme operating conditions: (a) when pp0 equals one, and (b) when p equals zero. In the first operating condition, pp0 = 1, the flow is at rest (stagnation p0 condition), hence ρρ0 = 1 and av0 = 0, and so by Eq. (4.135), ρρv = 0. However, when 0 a0 p = 0, i.e., the pressure into which the flow discharges is zero, the specific volume of p0 the gas tends to grow infinitely large, resulting in ρρ0 = 0, and av0 = vamax , and ρρv = 0. 0 0 a0 ρv The function ρ0 a0 must have at least one maximum or one minimum point between these two extremes. The investigation of Fig. 4.11a demonstrates the existence of one maximum point between two extreme values of pp0 = 1 and pp0 = 0. The pressure ratio at which the non-dimensional mass∗ flow rate per unit area becomes maximum is known as the critical pressure ratio, pp0 , and it has a value of roughly 0.528 for air. Also, at this point of maximum ρρv , the flow speed v is equal to the critical speed of 0 a0 sound a∗ . As explained in Sect. 4.7.1, when v = a∗ , the duct discharges the maximum ˚ ∗ . When the mass mass flow rate m ˚ max , also known as the critical mass flow rate m ∗ ∗ ∗ flow rate is m ˚ , the duct area is A . Finally, based on the variations in M and AA with the pressure ratio pp0 shown in Fig. 4.11b, it can be deduced that the accelerated flow must first decrease in cross section and subsequently increase in cross section. According to the preceding discussion, when a reservoir with constant stagnation conditions is connected to a duct, the flow velocity at the exit of the duct will change depending on the back pressure (the ambient pressure at which the flow from the reservoir is discharged through the duct). Back pressure is generally denoted by pb . Thus, if pb > p∗ , the flow speed at the duct exit will be subsonic; if pb = p∗ , the flow speed will be sonic; and if pb < p∗ , the flow speed will be supersonic. Exercise 4.11 Using the Area-Mach number relationship, show that the local area A in a flow passage can never be less than the critical flow area A∗ . Solution Recall, the Area-Mach number relationship  (γ +1)   A 2 γ − 1 2 2(γ −1) 1 M 1+ = A∗ M (γ + 1) 2

(4.133)

4.7

Continuity Equation for Steady One-dimensional Flow of a Perfect Gas

177

Fig. 4.11 Performance curves for compressible flow relations

To prove that A ≮ A∗ , we can differentiate AA∗ with respect to M and equate the derivative to zero. Thus, by taking the log of both sides of Eq. (4.133)    2 γ −1 2 A (γ + 1) ln M = − ln M 1 + A∗ 2 (γ − 1) 2 (γ + 1)      2 γ −1 2 A (γ + 1) = − ln M ln M + ln 1 + ln A∗ 2 (γ − 1) 2 (γ + 1) 

ln

Differentiating with respect to M, we get ⎧ ⎫ ⎡ ⎤ ⎨ d A∗ A 1⎬ (γ + 1) ⎣ (γ − 1) M ⎦   − = ∗ dM A ⎩ 2 (γ − 1) M⎭ 1 + γ −1 M2 A

2

Substituting in

A A∗

expression,

  ⎧ ⎫⎧ ⎫ ⎡ ⎤   (γ +1) ⎨1  2 d AA∗ γ − 1 2 2(γ −1) ⎬ ⎨ (γ + 1) ⎣ (γ − 1) M ⎦ 1⎬   − 1+ = M ⎭ ⎩ 2 (γ − 1) ⎩ M (γ + 1) dM 2 M⎭ 2 1 + γ −1 2 M d AA∗ ) For the maximum or minimum value of the area ratio, (dM = 0, hence

178

4 Steady One-Dimensional Flows

⎧ ⎫ ⎡ ⎤  (γ +1)   2 1 1⎬ γ − 1 2 2(γ −1) ⎨ (γ + 1) ⎣ (γ − 1) M ⎦   − M =0 1+ ⎩ 2 (γ − 1) M (γ + 1) 2 M⎭ 1 + γ −1 M2 2 ⎫ ⎧ ⎡ ⎤ ⎬ ⎨ (γ + 1) − 1) M 1 (γ ⎣ ⎦ − =0 ⎩ 2 (γ − 1) M⎭ 1 + γ −1 M2 2

γ −1 2 (γ + 1) 2 M =1+ M 2 2 M2 = 1 M = ±1 Since M = −1 is a physically impossible solution, therefore M = 1 at the location in the duct where the local area may be either maximum or minimum. Now, d A differentiating ( A∗ ) again with respect M, dM

  ⎫⎡ ⎧⎧ ⎤⎫    (γ +1) (γ +1) ⎬ d2 AA∗ M γ − 1 2 2(γ −1) ⎬ ⎣ 2 1 d ⎨⎨ 1 2  − ⎦ 1+ M = 2 ⎭ dM ⎩⎩ M (γ + 1) 2 M ⎭ dM 2 1 + γ −1 2 M ⎫ ⎧ ⎡ ⎤   (γ +1) (γ +1) ⎨1  2 γ − 1 2 2(γ −1) ⎬ 2 M  − 1 ⎦ d = ⎣ 1+ M ⎭ M dM ⎩ M (γ + 1) 2 1 + γ −1 M2 2

⎫ ⎧ ⎡ ⎤   (γ +1) (γ +1) ⎨1  2 M γ − 1 2 2(γ −1) ⎬ d ⎣ 1 2  − ⎦ 1+ + M ⎭ dM ⎩ M (γ + 1) 2 M 2 1 + γ −1 2 M ⎤ ⎡ γ +1 2 M  − 1 ⎦ = ⎣ M 1 + γ −1 M2 2

⎫    3−γ   (γ +1) γ − 1 2 2(γ −1) γ − 1 2 2(γ −1) ⎬ 2 1 2 1+ 1+ M M − 2 ⎭ ⎩ (γ + 1) 2 2 (γ + 1) M ⎤ ⎫⎡ ⎧  (γ +1) γ +1 ⎨1  2  γ − 1 2 2(γ −1) ⎬ ⎢ 1 ⎥ 2 1+ M + ⎣  + 2⎦ ⎭ ⎩M γ + 1 2 M γ −1 2 2 1+ 2 M ⎧ ⎨

For M = 1, this equation reduces to    d2 AA∗ γ +3 >0 = dM2 γ +1 which shows that AA∗ will attain its minimum value at M = 1. Substituting M = 1 in Eq. (4.133), we have

4.7

Continuity Equation for Steady One-dimensional Flow of a Perfect Gas



A A∗

179

=1 min

This is a striking result of the gas dynamics indicating that the minimum value of A is A∗ . That is, the local area of the flow passage can never be less than the critical area. Example 4.12 A ramjet flies at Mach 0.9 at the Geo-potential altitude of 11000 m. The flow is decelerated to almost zero velocity in intake before it enters into the combustion chamber. The combustion takes place at constant pressure, however, a temperature rise of 1500 K results. Subsequently, the combustion products are exhausted through the nozzle. Calculate (i) the stagnation pressure and stagnation temperature and (ii) the nozzle exit velocity. Solution From the standard atmosphere table, we have p∞ = 22632 Pa T∞ = 216.65 K (i) From Eq. (4.49) T0∞ =1+ T∞

 γ −1 1.4 − 1 2 M∞ =1+ (0.9)2 = 1.162 2 2 T0∞ = 1.162 × 216.65 = 251.74 K



and from Eq. (4.55) 1.4  (1.4−1)  (γ γ−1)     p0∞ γ −1 1.4 − 1 2 2 M = 1+ = 1+ (0.9) p∞ 2 2 1.4

p0∞ = (1.162) 0.4 × 22632 = 38277.57 Pa (ii) As combustion raises the temperature by 1500 K, the temperature of the combustion products at the exit of the combustor will be T0c = T0∞ + 1500 = 251.74 + 1500 = 1751.74 K Now, from isentropic relation T0c = Te



p0c pe

γ γ−1

 =

p0∞ p∞

γ γ−1

180

4 Steady One-Dimensional Flows

1.4−1  1751.74 38277.57 1.4 = Te 22632 = 1.16 =⇒ Te =1510.12 K Let us assume the combustion products as the mixture of perfect gases with γ = 1.4 and R = 287 J/(kg.K). From the energy equation (Eq. (4.93)), we have 1 h0c = h + ve2 2 1 cp T0c =cp Te + ve2 2  ve = 2cp (T0c − Te )  γ = 2 R (T0c − Te ) γ −1 Therefore,  ve =

2×

1.4 × 287 × (1751.74 − 1510.12) 1.4 − 1 =696.72 m/s

Listing 4.1 A MATLAB program for calculating isentropic flow properties. % V e r s i o n 1.1 C o p y r i g h t M r i n a l Kaushik , IIT Kharagpur , 2 9 / 0 7 / 2 0 2 1 % This p r o g r a m c a l c u l a t e s the s t a g n a t i o n pressure , s t a g n a t i o n t e m p e r a t u r e , s t a g n a t i o n density , c r i t i c a l pressure , c r i t i c a l t e m p e r a t u r e , c r i t i c a l d e n s i t y and c r i t i c a l area . clc c l e a r all gamma =1.4; % s p e c i f i c heat ratio R = 2 8 7 . 0 ; % in J /( kg . K ) % s p e c i f i c gas c o n s t a n t for air M T p A

= = = =

input ( ' Enter input ( ' Enter input ( ' Enter input ( ' Enter

the the the the

Mach number : '); s t a t i c t e m p e r a t u r e ( in K ) : ' ) ; s t a t i c p r e s s u r e ( in Pa ) : ' ) ; cross - s e c t i o n a l area ( in m ^2) : ' ) ;

clc Rho = p /( R * T ) ; p0 = p / ( ( 1 + ( gamma -1) /2*( M ^2) ) ^( - g a m m a /( gamma -1) ) ) ; Rho0 = Rho / ( ( 1 + ( gamma -1) /2*( M ^2) ) ^( -1/( gamma -1) ) ) ; T0 = T / ( ( 1 + ( gamma -1) /2*( M ^2) ) ^( -1) ) ; p s t a r = p / ( ( 1 + ( gamma -1) /2*( M ^2) ) ^( - g a m m a /( gamma -1) ) *(( g a m m a +1) /2) ^( g a m m a /( gamma -1) ) ) ; R h o s t a r = Rho / ( ( ( 1 + ( gamma -1) /2*( M ^2) ) ^( -1/( gamma -1) ) ) *(( g a m m a +1) /2) ^ ( 1 / ( gamma -1) ) ) ; T s t a r = T / ( ( ( 1 + ( gamma -1) /2*( M ^2) ) ^( -1) ) *(( g a m m a +1) /2) ) ; A s t a r = A / ( ( 1 / ( ( ( 1 + ( gamma -1) /2*( M ^2) ) ^( -1/( gamma -1) ) ) *(( g a m m a +1) /2) ^ ( 1 / ( gamma -1) ) ) ) * sqrt ( 1 / ( ( ( 1 + ( gamma -1) /2*( M ^2) ) ^( -1) ) *(( g a m m a +1) /2) ) ) / M ) ;

4.8

Stream Thrust and Impulse Function

181

msg = [ ' S t a g n a t i o n p r e s s u r e = ' n u m 2 s t r ( p0 ) ' Pa ' ]; disp ( msg ) msg = [ ' S t a g n a t i o n d e n s i t y = ' n u m 2 s t r ( Rho0 ) ' kg / m ^3 ' ]; disp ( msg ) msg = [ ' S t a g n a t i o n t e m p e r a t u r e = ' n u m 2 s t r ( T0 ) ' K ' ]; disp ( msg ) msg = [ ' C r i t i c a l p r e s s u r e = ' n u m 2 s t r ( p s t a r ) ' Pa ' ]; disp ( msg ) msg = [ ' C r i t i c a l d e n s i t y = ' n u m 2 s t r ( R h o s t a r ) ' kg / m ^3 ' ]; disp ( msg ) msg = [ ' C r i t i c a l t e m p e r a t u r e = ' n u m 2 s t r ( T s t a r ) ' K ' ]; disp ( msg ) msg = [ ' C r i t i c a l cross - s e c t i o n a l area = ' n u m 2 s t r ( A s t a r ) ' m ^2 ' ]; disp ( msg )

4.8 Stream Thrust and Impulse Function In the case of difficulties involving the calculation of reaction forces induced by a one-dimensional flow of fluid in the duct, it is occasionally useful to define a parameter known as the impulse function, which is represented as F = pA + mv ˚

(4.136)

For the one-dimensional flow of a perfect gas p = ρRT and m ˚ = ρAv. Therefore,  ρv2 F = pA 1 + p

(4.137)

Consider the flow through a duct, for example, a nozzle or diffuser. The inlet section of the duct is labeled 1 and the exit section is labeled 2. The unit vectors at 1 and 2 for the one-dimensional flow in the positive x-direction are nˆ 1 = −ˆi and nˆ 2 = ˆi, respectively. The net force R exerted to the fluid by the duct wall is thus R = F1 − F2

(4.138)

Hence, according to Newton’s principle, the reaction force T applied by the fluid on the wall is T = F2 − F1

(4.139)

182

4 Steady One-Dimensional Flows

The stream thrust produced by the flow between sections 1 and 2 is denoted by the reaction force T. It is occasionally beneficial to express the impulse function of a perfect gas in 2 terms of the Mach number. Therefore, the term ρvp in Eq. (4.137) can be rewritten as ρv2 γ v2 γ v2 = = 2 = γ M2 p γ RT a By substituting

ρv2 p

(4.140)

in Eq. (4.137) using Eq. (4.140), we get   F = pA 1 + γ M2

(4.141)

The impulse function F for the isentropic flow of a perfect gas is usually expressed in dimensionless form by dividing it with F∗ at M = 1. Hence,     1 + γ M2 1 + γ M2 p p A p0 A F = = × × × × × F∗ p∗ A∗ p0 p∗ A∗ (1 + γ ) (1 + γ )

(4.142)

Equation (4.142) may be represented in terms of the flow Mach number M by A p eliminating the terms , pp∗0 , and ∗ using Eqs. (4.55), (4.83), and (4.133), respecp0 A tively. This gives ⎫ ⎧   ⎪ ⎪ ⎬ ⎨ 2 1 + γM F 1 = 1    F∗ M⎪ ⎭ ⎩ 2 (γ + 1) 1 + γ −1 M2 2 ⎪ 2

(4.143)

Equation (4.143) applies to a steady one-dimensional isentropic flow. Appendix E lists the values of FF∗ at various Mach numbers for air (γ = 1.4).

4.9 Dynamic Pressure and Compressibility Correction Coefficient The most important concept in fluid dynamics is pressure. Pressure is commonly described in physics as the force per unit area. However, when discussing fluid pressure, we must be more clear about its importance. Consider a solid body, such as a test-sphere, immersed in a fluid container at rest. The pressure exerted by the fluid on the surface of the sphere shall be the normal force per unit area. Now, let’s start decreasing the sphere till it becomes a point. The compressive force per unit area, therefore, acts inward from both directions to the point along all lines passing through it. This compressive force per unit area is the same along each line. This

4.9

Dynamic Pressure and Compressibility Correction Coefficient

183

is the essence of the concept of pressure: a compressive force acting equally in all directions at a point in a fluid, whether or not there is a solid surface at that location. If there is no solid surface at the location of interest, the pressure can be viewed as a compressive force exerted by one component of the fluid continuum on the other. It is worth noting that the concept of pressure discussed above for stationary fluid in a container also applies to moving fluid in a duct. In any flow passage, there are generally three pressures that are of interest to the flow analysis. They are (i) total or stagnation pressure, (ii) static pressure, and (iii) dynamic pressure. 1. The total or stagnation pressure at a location in the fluid flow, given by p0 , is the pressure that might occur if the fluid is brought to rest isentropically. That is, if the fluid velocity is reduced to zero as a result of a reversible and adiabatic process (isentropic process), the entire kinetic energy of the fluid particles is used to increase the pressure. 2. The static pressure, indicated by p, is the pressure encountered while the fluid is at rest or when the measurement is carried out while moving with the flow. It is the force that acts equally in all directions on a fluid particle. 3. The difference between stagnation and static pressure is referred to as the dynamic pressure, q. It essentially represents the kinetic energy per unit volume of a flowing fluid. The Bernoulli equations, derived in Sect. 4.2.1.4, can be used to define the relationship between p0 , p, and q. Consider a steady inviscid flow past the blunt body, as seen in Fig. 4.12. Let us first suppose that the flow is incompressible. Thus, by solving Eq. (4.28) for points 1 and 0 on the streamline S, we get 1 1 p1 + ρ1 v12 =p0 + ρv02 = constant 2 2

(4.144)

On the streamline S, however, the flow is brought to rest at point 0; consequently, v0 = 0. The point of zero velocity is referred to as the stagnation point. The pressure at point 0 is thus known as the stagnation pressure (or total pressure), and it may be calculated using Eq. (4.144) by dropping the subscript, as 1 p0 = p + ρv2 2

(4.145)

Therefore, for an incompressible inviscid flow with negligible body forces, the stagnation pressure is equal to the sum of the static and dynamic pressures and is constant throughout the flow. It is worth noting that the term 21 ρv2 in Eq. (4.145) has the same units as pressure. This pressure, however, can exist only if the fluid is in motion. We define this term as qinc =

1 2 ρv (incompressible) 2

(4.146)

184

4 Steady One-Dimensional Flows

Fig. 4.12 Inviscid flow around a blunt body

where qinc denotes the dynamic pressure of incompressible fluid. Equation (4.145) shows that the relative magnitudes of the pressure force (static pressure) and the inertia force (dynamic pressure) acting on the fluid element influence its motion. Because p0 is constant in inviscid incompressible flows, a drop in p is accompanied by an increase in v, and vice versa. Equation (4.146) can be used to express Eq. (4.145) as qinc = (p0 − p)

(4.147)

Thus, dynamic pressure can also be defined as the amount of pressure rise that occurs when the inviscid incompressible flow is brought to rest. Assume that the inviscid flow past the blunt body, seen in Fig. 4.12, is compressible. By applying Eq. (4.33) between the points 1 and 0 on the streamline S, we obtain γ p0 v2 v2 γ p1 + 1 = + 0 γ − 1 ρ1 2 γ − 1 ρ0 2

(4.148)

Because the flow on the streamline S comes to a halt at point 0, v0 = 0. The point of zero velocity is referred to as the stagnation point. The pressure and density at point 0 are referred to as stagnation pressure and stagnation density, respectively, by definition, as stated in Sect. 4.4. Thus, Eq. (4.148) becomes v2 γ p p0 γ + = 2 (γ − 1) ρ (γ − 1) ρ0

(4.149)

This is Bernoulli’s equation for an inviscid compressible flow with negligible body forces. Although Eq. (4.149) has the same form as Eq. (4.145), the dynamic pressure for the compressible flow is greater than the dynamic pressure for the incompressible flow:

4.9

Dynamic Pressure and Compressibility Correction Coefficient

185

qcomp > qinc

(4.150)

This is due to the fact that the dynamic pressure in the compressible flow is no longer merely the difference between stagnation and static pressure as compared to incompressible flows. In reality, qcomp is affected by both the Mach number and the static pressure. The relation between the stagnation pressure and the static pressure in compressible flows is defined by γ  γ − 1 2 (γ −1) M (4.55) p0 = p 1 + 2 Adding and subtracting the term p on the right side result in  p0 = p + p

γ −1 2 1+ M 2

(γ γ−1)

 −1

(4.151)

By writing Eq. (4.151) in an analogous form to Eq. (4.147), we have qcomp = (p0 − p) ,  where qcomp = p 1 +

γ −1 2 M 2

 (γ γ−1)

(4.152)

− 1 , denotes the compressible dynamic

pressure or impact pressure, which is a function of both M and p. The compressibility correction coefficient, denoted by the symbol K, is defined as the ratio of qcomp to qinc :

K=

qcomp = qinc

p

,   (γ γ−1)  2 M 1 + γ −1 − 1 2 1 ρv2 2

(4.153)

It should be noted that the Mach number is more significant than the velocity itself in the supersonic realm of compressible flow. Thus, the term 21 ρv2 in Eq. (4.153) can be represented in terms of M as follows: qinc =

1 2 1 p 2 1 γp 2 1 γp 2 1 ρv = v = v = v = γ pM2 2 2 RT 2 γ RT 2 a2 2

(4.154)

Substituting this into Eq. (4.153), we obtain 2 K= γ M2

 1+



 (γ γ−1) γ −1 M2 −1 2

(4.155)

186

4 Steady One-Dimensional Flows

Using the Taylor series to expand the term in parentheses, we get qcomp 2 = K= qinc γ M2

,

 γ M2 γ M4 1+ + + · · · higher order terms − 1 2 8 (4.156)

or 

M4 M2 + + · · · higher order terms K = 1+ 4 40

 (4.157)

Equation (4.157) is the correction coefficient for dynamic pressure in compressible flow. It is applicable where the Mach number is low. Indeed, K is a measure of error that would have occurred if the compressible flow had been considered to be incompressible. If a compressible flow is treated as incompressible, the associated error is given by 

qcomp − qinc % error = qinc

× 100 = (K − 1) × 100

(4.158)

Substituting Eq. (4.157) into Eq. (4.158), we get  M4 M2 + + · · · higher order terms × 100 % error = 4 40 

(4.159)

Equation (4.159) shows that the assumption of incompressibility introduces sufficient error when the Mach number exceeds 0.3 or 0.4. Example 4.13 The air at 1 bar and 298 K flows at a speed of 185 m/s. Determine whether the flow is compressible or not? Justify the answer. What is the percentage error if the flow is treated as incompressible? Solution Given, p∞ = 1 bar T∞ = 298 K v∞ = 185 m/s The speed of sound in air is a∞ =

√ 1.4 × 287 × 298 = 346.02 m/s

4.9 Dynamic Pressure and Compressibility Correction Coefficient

187

and the Mach number is v∞ a∞ 185 = 346.02 = 0.53

M∞ =

Since the flow Mach number is more than 0.3 (limiting Mach number beyond which the fluid is considered as compressible), the airflow under consideration is, therefore, compressible. However, if the flow is still considered as incompressible, the associated error would be   2 M M4 + + · · · × 100 % error = 4 40   (0.53)4 (0.53)2 + + · · · × 100 = 4 40 ≈ 7.2% Therefore, the error encountered in considering the flow to be incompressible is approximately 7.2%.

4.10 Pressure Coefficient When a solid body, such as an airfoil, is immersed in a flow, the interaction of fluid particles with the body produces the so-called aerodynamic forces, which are lift and drag. In fluid flows, however, it is preferable to state the results in terms of nondimensional coefficients. The pressure coefficient Cp , the lift coefficient CL , and the drag coefficient CD are the three most essential dimensionless coefficients used in flow analysis. These parameters are defined in this section. The freestream properties allow pressure to be made non-dimensional anywhere in a fluid flow. The resulting parameter is known as the pressure coefficient Cp . If p∞ and ρ∞ are the pressure and density of a uniform freestream, respectively, and v∞ is the corresponding freestream velocity, then Cp is defined by Cp =

p − p∞ q∞

where q∞ denotes the freestream dynamic pressure.

(4.160)

188

4 Steady One-Dimensional Flows

For an incompressible flow, Eq. (4.160) is Cp =

(p − p∞ ) 1 ρv∞ 2 2

(4.161)

p − p∞ p0 − p∞

(4.162)

and in the case of a compressible flow, Cp =

It is interesting to highlight that, even for compressible flows, the pressure coefficient is always defined by Eq. (4.161). The definition proposed in Eq. (4.162) is practically never used. Since qinc < qcomp , the value of Cp based on Eq. (4.162) may be more than unity for a compressible flow. This is in contrast to an incompressible flow where Cp at the stagnation point is always one. Additionally, where the local speed exceeds the freestream speed, Cp is negative. Therefore, according to Bernoulli’s equation, the local pressure p at these points eventually becomes smaller than the freestream pressure p∞ . Rewriting Eq. (4.161) as follows yields an alternative form of the pressure coefficient that is useful for compressible flows. p − p∞ 1 ρv∞ 2 2  p 2 −1 = 2 γ M∞ p∞

Cp =

(4.163) (4.164)

The ratio pp∞ in Eq. (4.164) can be obtained for the isentropic flow of a compressible fluid by applying the stagnation pressure ratio relation given by Eq. (4.55). This results in p p p0 = × = p∞ p0 p∞



γ −1 2 M∞ 2 γ −1 2 + 2 M

1+ 1

 γ γ−1 (4.165)

or γ  2  γ −1 p 2 + (γ − 1) M∞ = p∞ 2 + (γ − 1) M2

By eliminating

p p∞

(4.166)

in Eq. (4.164) using Eq. (4.166), we have

2 Cp = 2 γ M∞



2 2 + (γ − 1) M∞ 2 + (γ − 1) M2

 γ γ−1

−1

(4.167)

4.10

Pressure Coefficient

189 v2

2

2 Substituting M2 = va2 and M∞ = a∞ 2 into Eq. (4.167) and combining the results ∞ with the energy equation for a perfect gas:

v2 a2 v2 a2 + = ∞ + ∞ γ −1 2 γ −1 2

(4.168)

yields 2 Cp = 2 γ M∞



  γ γ−1 γ −1 2 v2 1+ M∞ 1 − 2 −1 2 v∞

(4.169)

Equation (4.169) is the pressure coefficient for an inviscid compressible flow of a perfect gas. It indicates that Cp for a compressible flow strongly depends on the freestream Mach number, M∞ . The script below computes the variation of isentropic flow properties with Mach number for air (γ = 1.4), and the results are presented in Appendix B.

Listing 4.2 A program in "C" for calculating isentropic flow properties. The results are tabulated in Appendix B. // V e r s i o n 1.1 C o p y r i g h t M r i n a l Kaushik , IIT Kharagpur , 2 9 / 0 7 / 2 0 2 1 # include < s t d i o . h > # include < s t d l i b . h > # include < math . h > # d e f i n e pi acos ( -1.0) void main () { f l o a t M1 , y , v , u , conv , rho , pr , temp , area , s o u n d _ s p e e d , M _ s t a r ; conv = 180/ pi ; y = 1.4; FILE * m1 ,* d ,* p ,* t ,* a , * ar , * Mstar , * Mu , * Nu ; m1 = ( f o p e n (" C :\\ M a c h 1 . txt " , " w ") ) ; d = ( f o p e n (" C :\\ D e n s i t y . txt " , " w ") ) ; p = ( f o p e n (" C :\\ P r e s s u r e . txt " , " w ") ) ; t = ( f o p e n (" C :\\ T e m p r a t u r e . txt " , " w ") ) ; a = ( f o p e n (" C :\\ S o u n d _ v e l o c i t y . txt " , " w ") ) ; ar = ( f o p e n (" C :\\ A r e a _ r a t i o . txt " , " w ") ) ; M s t a r = ( f o p e n (" C :\\ M _ s t a r . txt " , " w ") ) ; Mu = ( f o p e n (" C :\\ mu . txt " , " w ") ) ; Nu = ( f o p e n (" C :\\ nu . txt " , " w ") ) ; p r i n t f ("\ n

M \t \t A/A* \t \t

p / p0 \ t T / T0 \ t \ t a / a0 \ t \ t M* ") ;

Rho / R h o 0 \ t

for ( M1 = 0 ; M1 < . 9 9 0 1 ; M1 = M1 + 0 . 0 1 ) { pr = pow ((1 + (( y -1) /2) *( M1 * M1 ) ) , y /(1 - y ) ) ; rho = pow ((1 + (( y -1) /2) *( M1 * M1 ) ) , 1/(1 - y ) ) ; temp = pow ((1 + (( y -1) /2) *( M1 * M1 ) ) , ( -1) ) ; area = (1/ M1 ) * pow (((5+ M1 * M1 ) /6) ,3) ; s o u n d _ s p e e d = sqrt ( temp ) ; M _ s t a r = s q r t ( ( ( ( y +1) /2) * M1 * M1 ) / ( 1 + ( ( y -1) /2) * M1 * M1 ) ) ;

190

4 Steady One-Dimensional Flows

p r i n t f ("\ n % .2 f \ t %.4 f \ t %.4 f \ t %.4 f \ t %.4 f \ t %.4 f \t %.4 f " , M1 , pr , temp , rho , area , s o u n d _ s p e e d , M _ s t a r ) ; f p r i n t f ( m1 ,"\ n % .2 f " , M1 ) ; f p r i n t f ( p ,"\ n % f " , pr ) ; f p r i n t f ( t ,"\ n % f " , temp ) ; f p r i n t f ( d ,"\ n % f " , rho ) ; f p r i n t f ( ar ,"\ n % f " , area ) ; f p r i n t f ( a ,"\ n % f " , s o u n d _ s p e e d ) ; f p r i n t f ( Mstar ,"\ n % f " , M _ s t a r ) ; } p r i n t f ("\ n \ n M \t \t t A/A* \t \t Nu \ n ") ;

p / p0 \ t T / T0 \ t \ t a / a0 \ t \ t M*

Rho / R h o 0 \ Mu

for ( M1 = 1 ; M1 < 5 . 0 0 1 ; M1 = M1 + 0 . 0 1 ) { pr = pow ((1 + (( y -1) /2) *( M1 * M1 ) ) , y /(1 - y ) ) ; rho = pow ((1 + (( y -1) /2) *( M1 * M1 ) ) , 1/(1 - y ) ) ; temp = pow ((1 + (( y -1) /2) *( M1 * M1 ) ) , ( -1) ) ; area = (1/ M1 ) * pow (((5+ M1 * M1 ) /6) ,3) ; s o u n d _ s p e e d = sqrt ( temp ) ; M _ s t a r = s q r t ( ( ( ( y +1) /2) * M1 * M1 ) / ( 1 + ( ( y -1) /2) * M1 * M1 ) ) ; u = asin (1/ M1 ) * conv ; v = sqrt (6) * atan ( sqrt (( M1 * M1 - 1) /6) ) * conv - atan ( sqrt ( M1 * M1 -1) ) * conv ; p r i n t f ("\ n % .2 f \ t %.4 f \ t %.4 f \ t %.4 f \ t %.4 f \ t %.4 f \t %.4 f \ t %.4 f \ t %.4 f " , M1 , pr , temp , rho , area , s o u n d _ s p e e d , M_star , u , v ) ; f p r i n t f ( m1 ,"\ n % .2 f " , M1 ) ; f p r i n t f ( p ,"\ n % f " , pr ) ; f p r i n t f ( t ,"\ n % f " , temp ) ; f p r i n t f ( d ,"\ n % f " , rho ) ; f p r i n t f ( d ,"\ n % f " , area ) ; f p r i n t f ( a ,"\ n % f " , s o u n d _ s p e e d ) ; f p r i n t f ( Mstar ,"\ n % f " , M _ s t a r ) ; f p r i n t f ( Mu ,"\ n % f " , u ) ; f p r i n t f ( Nu ,"\ n % f " , v ) ; } f c l o s e ( m1 ) ; fclose (p); fclose (d); fclose (t); fclose (a); f c l o s e ( ar ) ; fclose ( Mstar ); f c l o s e ( Mu ) ; f c l o s e ( Nu ) ; r e t u r n 0; }

Concluding Remark

191

Concluding Remarks This chapter has described the steady, one-dimensional isentropic flow without friction, heat transfer, and body forces. Of course, no flow is truly isentropic in reality; but, the essential properties of practically significant flows can be predicted using the equations presented in this chapter. The concepts of stagnation and critical conditions are also examined. The formulae for the characteristic speeds and the nondimensional velocity are derived. The use of an isentropic table in determining the flow properties of a steady, one-dimensional, isentropic flow is presented. In spite of the fact that such tables are very useful and practical, it is important to keep in mind that they are designed for a certain value for the ratio of specific heats, γ .

Exercise Problems Exercise 4.1 The speed indicator of the aircraft, which is calibrated without taking into account the effect of compressibility, measured a speed of 800 km/h at 6000 m altitude. Find the true airspeed. Also, determine the stagnation temperature and stagnation density at the nose of the Pitot probe. Exercise 4.2 A supersonic wind tunnel is designed for Mach 2.5 at the test-section. If the air in the tunnel is drawn from a reservoir at the stagnation state of 1.4 bar and 300 K, calculate the (i) mass flow rate, (ii) area of the test-section, and (iii) pressure, temperature and density at the throat and test-section. The area of the throat is 1 m2 . Exercise 4.3 Air flows from a reservoir at the stagnation state of 7 bar and 311 K in a convergent-divergent nozzle with the throat diameter of 1.93 × 10−3 m2 and a maximum Mach number of 0.8. Calculate the mass flow rate, nozzle diameter, flow speed, pressure, and temperature at the nozzle exit, where the Mach number is 0.5. Exercise 4.4 A Mach 2 uniform airstream enters a convergent duct. If the exit area is 70 of the inlet area, evaluate the Mach number at the exit of the duct and treat the flow as one-dimensional and isentropic. Justify the answer. Exercise 4.5 Air at 400 K and 1 atm enters a convergent duct at a speed of 150 m/s and expands isentropically to a pressure of 101 kPa at the nozzle exit. The inlet cross-sectional area of the duct is 4 × 10−3 m2 . Find (i) p0 , T0 and h0 , (ii) inlet Mach number, (iii) static temperature, Mach number, and cross-sectional area at the duct exit. Exercise 4.6 For the steady isentropic flow of a highly compressible gas, the relationship between the pressure and the density is defined by  ρ

∂p ∂ρ

=K s

192

4 Steady One-Dimensional Flows

where K is a constant. Prove that  p K M2 =1+ ln 1 − p0 p0 2 and  m ˚ M2 =M 1− √ 2 A Kρ Exercise 4.7 For an arbitrary gas, show that  F + ρTds ρdh0 + vdG = d A where G (= ρv) is the mass flux, h0 is the stagnation enthalpy, and F is the impulse function. Also prove that  dv  2 dG dh0 ds =− + 2 − 2 M −1 v G a a



∂h ∂s

ρ

Exercise 4.8 The local pressures in the vicinity of a body kept in a uniform , and M∞ are expressed in terms of the nonfreestream with conditions p∞ , v∞  ∞ . Show that the pressure coefficient dimensional pressure coefficient Cp = 1p−p 2 2 ρ∞ v∞ at the onset of critical flow speed over the body surface can be expressed as , Cp =

2 2+(γ −1)M∞ (γ +1)

 (γ γ−1)

−1

γ 2 M∞ 2

Exercise 4.9 Consider the adiabatic flow through a convergent-divergent nozzle. If a∗ is the critical speed of sound at the throat, show that the maximum discharge speed is given by  vmax =

γ +1 ∗ a γ −1

Exercise 4.10 For a perfect gas, show that   v (γ − 1) v2 1− M= a0 2 a02 where a0 is the stagnation speed of sound.

Chapter 5

Wave Phenomena

Abstract Shock and expansion waves occur in nature on occasion, such as during explosions, in pipe flows, inside a nozzle, and so on. However, wave phenomena are more closely correlated with aerospace engineering, specifically supersonic and hypersonic flights. These waves occur when a body moves in a stationary fluid or when a fluid approaches a stationary body with a velocity greater than the local speed of sound. A supersonic aircraft, in reality, experiences a shock around itself. Some of the characteristics associated with shock and expansion waves include sudden changes in pressure, temperature, and density, rapid changes in flow velocity, and flow turning. Many questions arise in the minds of readers when they hear the word “shock.” For example, what is so important and fascinating about shock and expansion waves? Why do they appear in nature? What happens to the medium in which these waves travel? We will try to answer some of these intriguing questions in this chapter.

5.1 Introduction It has been observed that a compressible fluid may undergo a sudden change of state under some conditions. Common examples are the phenomenon associated with detonation waves, explosions, and the wavefield generated at the nose of a body traveling at supersonic Mach numbers. The wavefront is very steep in both situations, and the flow experiences a strong rise in pressure when traversing the wave, called the shock wave. The gas experiences a significant increase in density due to the shock wave’s large pressure gradient, with a corresponding change in refractive index. The shock wave serves as a compression front, and thus the flow through a shock cannot be a reversible process. The energy required to compress the gas flowing through the shock wave is extracted from the kinetic energy of the flow upstream of the shock. Since the shock process is irreversible, the kinetic energy of the gas downstream of the shock is less than that of isentropic flow compression between the same pressure limits. The reduction in kinetic energy caused by the shock wave manifests itself as a gas heating to a static temperature higher than that corresponding to the isentropic compression value. Consequently, the gas passing © Springer Nature Singapore Pte Ltd. 2022 M. Kaushik, Fundamentals of Gas Dynamics, https://doi.org/10.1007/978-981-16-9085-3_5

193

194

5 Wave Phenomena

through the shock wave experiences a decrease in available energy and, as a result, an increase in entropy. A shock is a very thin compression front with a thickness comparable to the mean free path of the gas molecules in the flow field. Since the flow crosses the shock at very high Mach numbers, the combination of high velocity of the flow and extremely small thickness of the shock wave causes the fluid elements to pass through the wave in an infinitesimal time, ruling out any appreciable exchange of energy between fluid elements and the surroundings, making the shock process adiabatic. Shock waves appear in a variety of shapes and sizes, each with its own set of characteristics. A normal shock to the flow direction is called the normal shock, and the shock at an angle to the flow direction is known as the oblique shock.

5.2 Shock Wave Formation in Supersonic Flows So far in our discussion, it has been observed that the speed of sound is a very important entity because it describes the physical nature of a compressible flow. A small pressure change (weak disturbance) in a continuous medium propagates at the speed of sound through molecular collisions. When a solid body is introduced into a flow field, the fluid molecules near the body’s surface collide and experience a change in momentum. Disturbances in a subsonic flow can propagate well upstream, informing the upstream flow of the presence of a body or a sudden change in area in the downstream direction. Consequently, the upstream flow organizes itself in a normal pattern to flow smoothly past the body, as shown in Fig. 5.1a. In a supersonic flow, on the other hand, disturbances cannot spread upstream; instead, they pile up and coalesce to form a shock wave in front of the body (Fig. 5.1b). Since the upstream flow would not be aware of the presence of the body, the flow through the shock wave would suddenly change direction. A shock wave is a thin region of the flow field with large gradients of flow properties, including pressure, temperature, density, and so on. It is an area of high viscous dissipation due to high gradients of velocity and temperature. The gradients in fluid properties (discontinuities) cannot occur for finite periods in a real flow because the effects of viscosity and heat conduction of the fluid appear to smooth out such discontinuities. Theoretical measurements and experimental studies, on the other hand, indicate that the thickness of a shock wave is of the same order of magnitude as the mean free path of the molecules. As a result, we should ignore the viscous and heat conduction effects when studying shock waves. Another premise in shock wave research is that the perfect gas law governs gas flow both upstream and downstream of the shock waves. Shock waves may occur either internally or externally in a supersonic flow field. They may be stationary or moving in a flow field. This can also occur inside a pipe or a nozzle.

5.3 Steady One-Dimensional Flow Through a Stationary Normal Shock Wave

195

Fig. 5.1 Propagation of disturbances in the subsonic and the supersonic flows

5.3 Steady One-Dimensional Flow Through a Stationary Normal Shock Wave The explicit relationships for the property change of a perfect gas moving through a stationary normal shock wave in terms of the upstream Mach number are derived in this section. A steady one-dimensional flow through a streamtube may be used to analyze the flow through a normal shock.

5.3.1 Governing Equations for a General Fluid In order to analyze the steady one-dimensional flow through a normal shock, consider a control volume of the form as indicated in Fig. 5.2. The control volume comprises of a normal shock and a small amount of fluid both upstream and downstream of the shock wave. Subscripts 1 and 2 denote the conditions upstream and downstream of the shock, respectively (Fig. 5.2). The following assumptions are taken into account when developing the governing equations. • • • • • • •

The flow is steady, one-dimensional, adiabatic, and frictionless. The change in properties through the shock wave is irreversible. The shock wave is perpendicular to the flow. The shock wave has a very small thickness. The cross-sectional area just upstream and downstream of the shock is the same. There is no external work done on the control volume. The body forces are insignificant.

By applying the steady one-dimensional continuity equation (Eq. (4.3)) for the fluid flow between the locations 1 and 2, we get m ˚ = ρ1 v1 = ρ2 v2 A

(5.1)

196

5 Wave Phenomena

Fig. 5.2 One-dimensional flow through a normal shock

and neglecting the body forces, the momentum equation (Eq. (4.7)) gives m ˚ (v2 − v1 ) = ρ2 v22 − ρ1 v12 A p1 + ρ1 v12 = p2 + ρ2 v22

p1 − p2 =

(5.2) (5.3)

  where the term p + ρv2 A is the impulse function. Now, since the flow is adiabatic and there is no external work is done on the control volume, thus the total enthalpy h0 remains constant. h0 = h01 = h02

(5.4)

By applying the steady one-dimensional energy equation (Eq. (4.36)), we have h1 +

v12 v2 = h2 + 2 2 2

(5.5)

It is worth noting that, for the given flow conditions upstream of the normal shock, the three Eqs. (5.1), (5.3), and (5.5) have four unknowns: p2 , v2 , h2 and p2 . As a result, the equation of state can be used as an alternative equation to obtain a closed form solution. The equation of state for a perfect gas is p = ρRT

(5.6)

The expression for the change in entropy of a fluid across the shock wave can be determined by using dp (2.230) Tds = dh − ρ For a perfect gas, this equation becomes ds = cp

dp dT −R T p

5.3 Steady One-Dimensional Flow Through a Stationary Normal Shock Wave

197

which on integration gives s2 − s1 = cp ln

T2 p2 − R ln T1 p1

(5.7)

Equations (5.1)–(5.7) are the governing equations for the flow of a perfect gas through the normal shock wave.

5.3.2 Governing Equations for a Perfect Gas For the perfect gas, h = cp T; the energy equation (Eq. (5.4)) can be expressed as v12 v2 = cp T2 + 2 2 2 T01 = T02

cp T1 +

(5.8) (5.9)

which reveals that the stagnation temperature for the perfect gas remains constant across the shock wave. Since the flow upstream and downstream of the shock are considered to be inviscid, combining the isentropic relation (Eq. (4.49)) and Eq. (5.9) results in T2 1 + = T1 1 +

γ −1 2 M1 2 γ −1 2 M2 2

(5.10)

By using Eqs. (5.6) and (5.1), we can write T2 p2 ρ1 p2 v2 = = T1 p1 ρ2 p1 v1   p1 p2 M1 T1 = M2 T2 T1 T2   2  T2 M2 2 p2 = T1 p1 M1

(5.11) (5.12) (5.13)

Substituting Eq. (5.10) into Eq. (5.13) and solving for p2 = p1



M1 M2



⎛ ⎝

1+

γ −1 2 M1 2

1+

γ −1 2 M2 2

p2 p1

gives

⎞ ⎠

Rewriting Eq. (5.3) in terms of the Mach number and solving for

(5.14)

p2 p1

yields

198

5 Wave Phenomena

  1 + γ M12 p2  = p1 1 + γ M22 Eliminating

p2 p1

(5.15)

from Eqs. (5.14) and (5.15) provides



1

1 M1 γ −1 2 2 γ −1 2 2 M2 1+ 1+ M1 = M2 2 2 1 + γ M12 1 + γ M22

(5.16)

We get two solutions by solving Eq. (5.16) for M2 . They are: M2 = M1 and M2 =

(γ − 1) M12 + 2 2γ M12 − (γ − 1)

1/2 (5.17)

Since M2 = M1 is a trivial solution, it denotes no normal shock in the control volume and is thus discarded. As a result, the solution for M2 will be

(γ − 1) M12 + 2 M2 = 2γ M12 − (γ − 1)

1/2



or

M22

(γ − 1) M12 + 2 = 2γ M12 − (γ − 1)

(5.18)

Equation (5.18) expresses the Mach number on the downstream of the shock in terms of the upstream Mach number. Note that, for M1 = 1 Eq. (5.18) gives M2 = 1. As a result, Eqs. (5.15) and (5.10) indicate that there will be no pressure or temperature increase through the shock wave. In this condition, the shock wave behaves similarly to a sound wave, with negligible pressure and temperature changes. Thus, the sound wave can be defined as a degenerated normal shock wave.

5.3.3 Working Formulae for a Perfect Gas We have seen that both M1 and M2 are involved in the standard shock relations for a perfect gas derived in Sect. 5.3.2. To obtain the working formulae, we will use Eq. (5.18) to eliminate M2 from these equations.

5.3.3.1

Static Pressure Ratio

The static pressure ratio across the normal shock wave is given by either Eq. (5.15) or Eq. (5.18). Let us consider Eq. (5.15) and eliminate M2 by using Eq. (5.18). This gives

5.3 Steady One-Dimensional Flow Through a Stationary Normal Shock Wave

p2 1 + γ M12   = p1 1 + γ (γ −1)M12 +2 2γ M2 −(γ −1)  1    2 1 + γ M12 γ2γ M − 1 1 −1 = γ +1 γ +1 + γ M12 γ −1 γ −1 =

2γ M2 − γ −1 1 γ +1 γ −1

1

199

(5.19)

(5.20)

(5.21)

Rearranging and simplifying this equation result in  p2 2γ  2 M1 − 1 =1+ p1 γ +1 p2 2γ γ −1 = M2 − p1 γ +1 1 γ +1

(5.22) (5.23)

Equation (5.23) is the static pressure ratio across a normal shock wave.

5.3.3.2

Static Temperature Ratio

The static temperature ratio across the normal shock may be obtained by eliminating M2 in Eq. (5.10) using Eq. (5.18). Thus, 1 + γ −1 M12 T2  2  = T1 1 + γ −1 (γ −1)M12 +2 2 2 2γ M −(γ −1)   1  γ −1 2 1 + 2 M1 (γ2γ M2 − 1 −1) 1   = 2γ γ −1 M12 + γ −1 2

(5.24)

(5.25)

By rearranging and simplifying terms, we obtain the static pressure ratio as



2γ T2 2 + (γ − 1) M12 γ −1 2 M − = T1 γ +1 1 γ +1 (γ + 1) M12 5.3.3.3

(5.26)

Stagnation Pressure Ratio

The following relation can be used to calculate the stagnation pressure ratio across the normal shock. p02 p02 p2 p1 = × × p01 p2 p1 p01

(5.27)

200

5 Wave Phenomena

Now, the terms pp022 and derive the expressions:

p1 p01

in Eq. (5.27) can be eliminated by using Eq. (4.49) to

γ p02 γ − 1 2 (γ −1) M2 = 1+ p2 2

γ p1 γ − 1 2 − (γ −1) M1 = 1+ p01 2

(5.28) (5.29)

Therefore, we have p02 = p01



1+ 1+

γ −1 2 M2 2 γ −1 2 M1 2

 (γ γ−1)

 2γ  2 M1 − 1 1+ γ +1

(5.30)

Eliminating M22 in this equation by using Eq. (5.18) yields ⎡ p02 ⎣ = p01 ⎡

1+

γ −1 2



1+

(γ −1)M12 +2 2γ M12 −(γ −1) γ −1 2 M1 2

 ⎤ (γ γ−1) ⎦

(γ + 1) M12 = ⎣  2 + (γ − 1) M12 γ2γ M2 − +1 1



 2γ  2 1+ M1 − 1 γ +1

⎤ (γ γ−1) γ −1 γ +1

⎦



2γ γ −1 M12 − γ +1 γ +1

(5.31)

(5.32)

Rearranging and simplifying the terms yield

(γ γ−1)

1 p02 2γ γ − 1 − (γ −1) (γ + 1) M12 2 M = − p01 γ +1 1 γ +1 2 + (γ − 1) M12

(5.33)

Equation (5.33) represents the stagnation pressure ratio across a normal shock wave.

5.3.3.4

Static Density Ratio

By using the equation of state for a perfect gas, we can write ρ2 = ρ1



p2 p1



T1 T2

 (5.34)

Now, by introducing the pressure ratio pp21 from Eq. (5.23) and the temperature ratio TT21 from Eq. (5.26) into the above equation, we find

5.3 Steady One-Dimensional Flow Through a Stationary Normal Shock Wave

201

−1



2γ ρ2 2γ γ − 1 2 + (γ − 1) M12 γ − 1 −1 2 M12 − M = − ρ1 γ +1 γ +1 γ +1 1 γ +1 (γ + 1) M12 ρ2 (γ + 1) M12 = ρ1 2 + (γ − 1) M12

(5.35)

This is the static density ratio of a normal shock wave. It is worth noting that for large Mach numbers, (M1 → ∞), Eq. (5.35) gives ρρ21 → 6. Thus, at very high Mach   numbers, density is a small range bounded parameter 0 < ρρ21 < 6 in comparison to pressure and temperature, which become unbounded across the shock wave. Combining Eq. (5.35) with Eq. (5.1) yields an alternative expression. It provides v1 (γ + 1) M12 = v2 2 + (γ − 1) M12 5.3.3.5

(5.36)

Entropy Change

The change in entropy of a perfect gas through a normal shock can be determined by substituting TT21 and pp21 in Eq. (5.7) from Eqs. (5.26) and (5.23), respectively. This gives ⎧ ⎫  2γ ⎪ 2 2 −1 ⎪

⎬ ⎨ 1 + γ −1 M M 2γ 1 2 (γ −1) 1 2 − γ −1 − R ln s = s2 − s1 = cp ln (5.37) M ⎪ ⎪ (γ +1)2 2 γ +1 1 γ +1 ⎭ ⎩ 2(γ −1) M1  γ

 (γ − 1) M12 + 2 2γ s 1 γ −1 (s2 − s1 ) ln = = M12 − where 1 ≤ γ ≤ 1.67 R R γ +1 γ +1 (γ − 1) (γ + 1) M2 1

(5.38)

This equation expresses the entropy change of a perfect gas as it passes through a normal shock wave. According to Eq. (5.38), there is a decrease in entropy across a normal shock if the Mach number in front of it is less than unity. When an adiabatic process is considered, this phenomenon directly contradicts the second law of thermodynamics. As a result, in a perfect gas, an abrupt expansion cannot occur. Furthermore, Eq. (5.38) shows that the state change caused by a shock wave is irreversible since entropy always increases. The entropy change in terms of the Mach number before the shock can be seen in Fig. 5.3 for γ = 1.4. The figure clearly shows that shock waves are only possible in supersonic flows. Example 5.1 The flow conditions of air just upstream of a normal shock are v∞ = 750 m/s, p∞ = 100 kPa, and T∞ = 350 K. Calculate the flow conditions downstream of the shock. Also, determine the change in entropy across the shock.

202

5 Wave Phenomena

1 Fig. 5.3 s2 −s R variation for a perfect gas passing through a normal shock with upstream Mach number

Solution Given, v∞ = 750 m/s,

p∞ = 100 kPa,

T∞ = 350 K

The Mach number upstream of the normal shock is: v1 γ RT 750

M1 = √

=√ 1.4 × 287 × 350 ≈ 2.0 For M1 = 2.0, from the normal shock table, we have M2 = 0.577,

p2 = 4.5, p1

T2 = 1.6875 T1

Therefore, p2 = T2 =

p2 p1 = 4.5 × 100 = 450 kPa p1

T2 T1 = 1.6875 × 350 = 590.63 K T1

and  v2 = M2 γ RT √ = 0.577 × 1.4 × 287 × 590.63 ≈ 281.08 m/s

5.3 Steady One-Dimensional Flow Through a Stationary Normal Shock Wave

203

The change in entropy across the shock is: !

γ "

2γ R γ − 1 (γ − 1) M12 + 2 2 s = M − ln γ +1 1 γ +1 (γ − 1) (γ + 1) M12 

1.4

2 × 1.4 2 1.4 − 1 (1.4 − 1) 22 + 2 287 2 − = ln 1.4 + 1 1.4 + 1 (1.4 − 1) (1.4 + 1) 22 = 93.93

J (kg.K)

Example 5.2 The Mach number downstream of a normal shock is 0.8. Find the static pressure ratio, static temperature ratio, static density ratio, and the entropy increase for air across the shock. Solution Given, M2 = 0.8,

γ = 1.4

The Mach number, M1 , upstream of the shock can be calculated using the relation: M22 =

(γ − 1) M12 + 2 2γ M12 − (γ − 1)

0.64 =

0.4M12 + 2 2.8M12 − 0.4

⇒ M1 = 1.27 The static pressure ratio is:   p2 2γ  2 2 × 1.4  M1 − 1 = 1 + 1.272 − 1 = 1.715 =1+ p1 γ +1 2.4 and the static density ratio is: ρ2 3.87 (1.4 + 1) (1.27)2 (γ + 1) M12 = 1.46 = = = 2 ρ1 (γ − 1) M1 + 2 (1.4 − 1) (1.27)2 + 2 2.645 Thus, the static temperature ratio becomes: p2/p1 T2 1.715 = 1.17 = = ρ 2 T1 /ρ1 1.46

The entropy increase across the normal shock is:

204

5 Wave Phenomena

γ "

! 2γ R γ − 1 (γ − 1) M12 + 2 2 M − ln s = γ +1 1 γ +1 (γ − 1) (γ + 1) M12 

1.4

2 × 1.4 287 (1.4 − 1) (1.4 − 1) (1.27)2 + 2 2 = ln (1.27) − 0.4 (1.4 + 1) (1.4 + 1) (1.4 + 1) (1.27)2 = 113.83

J (kg.K)

5.4 Strong and Weak Normal Shock Waves The preceding discussion shows that the strength of a normal shock wave is dependent on the upstream Mach number. Thus, it is appropriate to examine two limiting cases of a normal shock wave: a strong normal shock and a weak normal shock. A strong normal shock is one for which M1 is very large, i.e., M  1. Now, for M  1, the governing equations derived above for a normal shock wave reduce to: 2γ M12 p2 = p1 γ +1 T2 2γ (γ − 1) M12 = T1 (γ + 1)2 ρ2 (γ + 1) = ρ1 (γ − 1) # (γ − 1) M2 = 2γ

(5.39) (5.40) (5.41) (5.42)

Therefore, if M1 tends to infinity (M1 → ∞), then both pp21 and TT21 will tend to +1) −1) and M2 tends to (γ2γ . In other words, for very large infinity, but ρρ21 tends to (γ (γ −1) Mach numbers, the static pressure and static temperature ratios become unbounded, while the density ratio and downstream Mach numbers remain closed range parameters. Indeed, the assumptions on which the preceding analysis of the changes across a normal shock wave is based, namely, that the gas remains thermally and calorically perfect, cease to exist when the shock is very strong since very high temperatures would normally exist behind the shock. The preceding discussion concentrated on the flow analysis of a very strong normal shock wave. Another limiting case is a weak normal shock wave. Now, the entropy change analysis demonstrates that the flow process through a weak normal shock is basically isentropic. As a result, the static pressure and static density ratio relationships established in Chap. 4 apply to such weak normal shocks. Furthermore, whether the flow is isentropic or not, the continuity equation for a continuous onedimensional flow applies across the shock. Therefore, in the limiting case of a weak

5.4 Strong and Weak Normal Shock Waves

205

normal shock, Eq. (4.2) can be written as follows: ρ1 v1 = ρ2 v2 ρ2 a2 M1 = ρ1 a1 M2

(5.43) (5.44)

Using the isentropic relationship, ρ2 = ρ1



T2 T1

1  (γ −1)

(5.45)

and noting that, a2 = a1



T2 T1

 21 (5.46)

Equation (5.44) becomes M2 = M1



T1 T2

(γ +1)  2(γ −1)

(5.47)

Moreover, the steady one-dimensional energy equation for an adiabatic and thus isentropic flow provides: ⎡





⎤ M12 T2 ⎦   =⎣ T1 1 + γ −1 M2 1+

γ −1 2

(4.43)

2

2

By substituting this into Eq. (5.47), we obtain ⎡

1+



γ −1 2



M22

(γ +1) ⎤ 2(γ −1)

M2 ⎦   =⎣ γ −1 2 M1 M 1+ 2

(5.48)

1

For a weak normal shock wave, Eq. (5.48) determines the downstream Mach number M2 for a given upstream Mach number M1 . After calculating M2 , the isentropic relations can be used to find the changes in pressure, temperature, and density. In order to compare the strong shock, weak shock, and actual normal shock relations, Fig. 5.4 demonstrates the variation of M2 with M1 provided by these relations for the case of γ = 1.4. The figure reveals that the weak normal shock relations apply when M1 is less than around 1.1, while the strong normal shock relations apply only when M1 is very high.

206

5 Wave Phenomena

Fig. 5.4 Variations of M2 with M1 for strong shock, weak shock, and actual normal shock

5.5 Prandtl’s Velocity Equation for a Normal Shock Wave We may derive a relationship between the flow velocities v1 and v2 upstream and downstream of a normal shock wave. This relationship can be obtained for a perfect gas by substituting M1 and M2 in Eq. (5.18) with their corresponding nondimensional velocities M1∗ and M2∗ from Eq. (4.103). This results in:

2M1∗2 +2 (γ − 1) ∗2 (γ +1)−(γ −1)M1

= ∗2 ∗2 2M1 (γ + 1) − (γ − 1) M2 2γ ∗2 − (γ − 1) 2M2∗2

(γ +1)−(γ −1)M1

(5.49)

M2∗2

(γ + 1) + (γ − 1) M1∗2 − (γ − 1) M1∗2 = ∗2 4γ M1∗2 − (γ + 1) (γ − 1) + (γ − 1)2 M1∗2 (γ + 1) − (γ − 1) M2 





(5.50)

M2∗2 4γ M1∗2 − (γ + 1) (γ − 1) + (γ − 1)2 M1∗2 = (γ + 1) (γ + 1) − (γ − 1) M2∗2



(5.51) 4γ M1∗2 M2∗2 − (γ + 1) (γ − 1) M2∗2 + (γ − 1)2 M1∗2 M2∗2 = (γ + 1)2 − (γ + 1) (γ − 1) M2∗2

(5.52) 4γ M1∗2 M2∗2 + (γ − 1)2 M1∗2 M2∗2 = (γ + 1)2 (γ + 1)2 M1∗2 M2∗2 = (γ + 1)2

(5.53)

M1∗ M2∗ = ±1

(5.54)

5.5 Prandtl’s Velocity Equation for a Normal Shock Wave

207

Discarding the impossible solution M1∗ M2∗ = −1, we have M1∗ M2∗ = 1 Since M∗ =

v , a∗

(5.55)

thus Eq. (5.55) can be expressed as v2 v1 = a∗2

(5.56)

This is the famous Prandtl’s velocity relation for a normal shock wave. Equation (5.56) shows that the product of upstream and downstream flow velocities across a normal shock wave is equal to the square of the critical speed of sound a∗2 ; where a∗2 remains constant for an adiabatic flow. From Eq. (5.56), it can also be seen that the normal shock approaches a sound wave when the former’s strength becomes vanishingly small. Moreover, if the flow upstream of a normal shock is supersonic, then the flow downstream of the shock would be subsonic, according to Eq. (5.56). It is worth noting that the Prandtl’s velocity relation does not prevent a subsonic flow upstream of the shock from being supersonic downstream, though this is restricted by thermodynamic considerations.

5.6 The Rankine-Hugoniot Equation for a Normal Shock Wave The flow process through a shock wave is known to be adiabatic and irreversible. As a result, the shock wave can be thought of as an irreversible and adiabatic compressor. The work involved in this compression method will now be calculated using the shock wave. By rewriting the steady one-dimensional continuity equation (Eq. (4.2)) as,  v1 =

ρ2 ρ1

 v2

(5.57)

and introducing it into the momentum equation (Eq. (4.6)), yield 

 ρ2 2 2 v2 = p2 + ρ2 v22 ρ1 ρ2 p1 + 2 v22 = p2 + ρ2 v22 ρ1 2 ρ2 2 v −ρ2 v22 = p2 − p1 ρ1 2

p1 + ρ1

By rearranging and simplifying the terms, we obtain

(5.58) (5.59) (5.60)

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5 Wave Phenomena

 v22 =

p2 − p1 ρ2 − ρ1



ρ1 ρ2

(5.61)

ρ2 ρ1

(5.62)

and combining it with Eq. (5.57), provides  v12 =

p2 − p1 ρ2 − ρ1



Now, eliminating v12 and v22 in Eq. (4.35) by using Eqs. (5.61) and (5.62), respectively, we get h1 +

1 2



ρ2 ρ1



    p2 − p1 p2 − p1 1 ρ1 = h2 + ρ2 − ρ1 2 ρ2 ρ2 − ρ1

(5.63)

By definition, h= e + ρp ; where e is the internal energy of the gas. Thus,       p2 − p1 p2 − p1 p1 1 ρ2 p2 1 ρ1 = e2 + e1 + + + ρ1 2 ρ1 ρ2 − ρ1 ρ2 2 ρ2 ρ2 − ρ1       p2 − p1 1 ρ1 p2 − p1 p1 p2 1 ρ2 − e2 − e1 = − + ρ1 ρ2 2 ρ1 ρ2 − ρ1 2 ρ2 ρ2 − ρ1 

 ρ2 p1 p2 1 p2 − p1 ρ1 e2 − e1 = − + − ρ1 ρ2 2 ρ2 − ρ1 ρ1 ρ2

p1 p2 1 (p2 − p1 ) (ρ2 + ρ1 ) (ρ2 − ρ1 ) e2 − e1 = − + ρ1 ρ2 2 (ρ2 − ρ1 ) ρ1 ρ2    1 p1 p2 p2 − p1 1 e2 − e1 = − + + ρ1 ρ2 2 ρ1 ρ2    1 p2 + p1 1 e2 − e1 = − 2 ρ1 ρ2

(5.64) (5.65) (5.66) (5.67) (5.68) (5.69)

i.e., the change in internal energy of a gas is proportional to the average pressure multiplied by the change in specific volume of the gas. Equation (5.69) is the well-known Rankine-Hugoniot equation, named after the Scottish engineer W. J. M. Rankine and the French engineer P. H. Hugoniot. This equation describes the thermodynamic states that exist upstream and downstream of a normal shock wave. It is analogous to the first law of thermodynamics, which states that the change in internal energy of a gas during an adiabatic process equals the work performed. Example 5.3 Show that the Rankine-Hugoniot equation for a perfect gas passing through a normal shock wave can be expressed as, 

p2 + p1 2



1 1 − ρ1 ρ2

Using this relation, show that

 =−

1 (γ − 1)



p1 p2 − ρ1 ρ2



5.6 The Rankine-Hugoniot Equation for a Normal Shock Wave



p2 − p1 ρ2 − ρ1



 =γ

p2 + p1 ρ2 + ρ1

209



Solution The Rankine-Hugoniot equation for a normal shock is:  e2 − e1 =

p2 + p1 2



1 1 − ρ1 ρ2

 (5.69)

For a perfect gas e = c∀ T and p = ρRT; thus   1 1 p2 + p1 − c∀ (T2 − T2 ) = 2 ρ1 ρ2      p2 + p1 1 p2 1 p1 1 = − − ρ1 2 ρ1 ρ2 (γ − 1) ρ2 

Hence, 

p2 + p1 2



1 1 − ρ1 ρ2

 =−

1 (γ − 1)



p1 p2 − ρ1 ρ2



Now, rewriting this equation as: 

  1 p2 ρ1 − p1 ρ2 = ρ1 ρ2 (γ − 1) 2 (p2 ρ1 − p1 ρ2 ) (γ − 1) (p2 + p1 ) = (ρ2 − ρ1 ) 2 (p2 ρ1 − p1 ρ2 ) (p2 + p1 ) = (γ − 1) (ρ2 + ρ1 ) (ρ2 + ρ1 ) (ρ2 − ρ1 ) 2 (p2 ρ1 − p1 ρ2 ) (p2 + p1 ) (p2 + p1 ) γ = + (ρ2 + ρ1 ) (ρ2 + ρ1 ) (ρ2 − ρ1 ) (ρ2 + ρ1 ) 2 (p2 ρ1 − p1 ρ2 ) + (p2 + p1 ) (ρ2 − ρ1 ) (p2 + p1 ) γ = (ρ2 + ρ1 ) (ρ2 + ρ1 ) (ρ2 − ρ1 ) 2p2 ρ1 − 2p1 ρ2 + p1 ρ2 − p1 ρ1 + p2 ρ2 − p2 ρ1 (p2 + p1 ) γ = (ρ2 + ρ1 ) (ρ2 + ρ1 ) (ρ2 − ρ1 ) p2 (ρ2 + ρ1 ) − p1 (ρ2 + ρ1 ) (p2 + p1 ) γ = (ρ2 + ρ1 ) (ρ2 + ρ1 ) (ρ2 − ρ1 ) (p2 + p1 ) (p2 − p1 ) (ρ2 + ρ1 ) γ = (ρ2 + ρ1 ) (ρ2 + ρ1 ) (ρ2 − ρ1 )     p2 − p1 p2 + p1 =γ ρ2 − ρ1 ρ2 + ρ1 p2 + p1 2



ρ2 − ρ1 ρ1 ρ2



210

5 Wave Phenomena

Fig. 5.5 Subsonic and supersonic Pitot probes

5.7 Supersonic Pitot Probe A Pitot-static probe1 is an instrument that measures both total and static pressures in a subsonic flow and estimates the dynamic pressure. As shown in Fig. 5.5a, the front hole facing the flow measures the total pressure, while the equally spaced holes mounted on the probe’s periphery measure the static pressure. The dynamic pressure is the difference between these pressures as sensed by a manometer or a pressure transducer connected to the probe. The flow speed of an incompressible flow is determined by entering the dynamic pressure value into the Bernoulli’s equation. For a compressible flow, however, the flow speed can be calculated by using the isentropic relations given by Eqs. (4.55) and (3.14). A shock wave forms ahead of a blunt body when it is put in a supersonic flow, as seen in Fig. 5.6. In general, this shock wave is curved and hence called the bow-shock wave. Since the central region of a bow-shock is essentially normal to the flow, the normal shock relations can be used to relate the upstream and downstream conditions across the shock wave. Besides that, since the flow downstream of a normal shock is subsonic, the flow deceleration between points 2 and 3 is presumed to be an isentropic process. Thus, the stagnation point is described as point 3, where the flow velocity is brought isentropically to rest. Interestingly, the pressure at the stagnation point can be calculated using this flow model for any given upstream conditions. An instrument that uses this type of flow model to measure the stagnation pressure is called the Pitot probe. Figure 5.5b depicts a typical Pitot probe in supersonic flow with a bow-shock ahead of the probe-nose. In this case, the measured stagnation pressure does not correspond to the freestream stagnation pressure. The probe essentially measures the stagnation pressure downstream of the shock wave. Since the flow experiences a change in stagnation pressure across the shock, the isentropic relation cannot be 1

For more information on the Pitot-Static probe, the readers may refer to Theoretical and Experimental Aerodynamics by Mrinal Kaushik, 1st Edition, 2019, Springer Nature, Singapore.

5.7 Supersonic Pitot Probe

211

Fig. 5.6 A blunt body in the supersonic flow

used to find the freestream Mach number. Thus, we need to find a relationship that connects the stagnation pressure (Pitot pressure) downstream of the shock to the static pressure upstream of the shock. This relationship is derived in Sect. 5.7.1.

5.7.1 Rayleigh Supersonic Pitot Probe Formula As stated in Sect. 5.7, the small area of flow enclosed by the pressure port at the nose of the Pitot probe over which the shock is essentially normal, and flow downstream of this section is thus subsonic and experiences isentropic deceleration. These assumptions are illustrated in Fig. 5.5b. The flow can be analyzed as follows: 1. The static pressure ratio across the shock wave, pp21 , can be calculated using the normal shock relation given by Eq. (5.23). 2. The stagnation pressure at the probe nose, p02 , can be found by applying the isentropic relation (Eq. (4.55)) between the flow downstream and the stagnation point. Hence, we may write p02 = p1



p02 p2



p2 p1

 (5.70)

where, the subscripts 1 and 2 denote the flow conditions upstream and downstream of the shock wave, respectively. By using Eqs. (4.55) and (5.23), the above equation becomes

212

5 Wave Phenomena

 γ

 2γ p02 γ − 1 2 γ −1 γ −1 M2 M12 − = 1+ p1 2 γ +1 γ +1

(5.71)

Eliminating M2 in this equation by using Eq. (5.18), yield  p02 = p1

γ +1 2 M1 2

2γ M2 γ +1 1

−

 (γ γ−1)

γ −1 γ +1

1  (γ −1)

(5.72)

This is the well-known Rayleigh Supersonic Pitot Probe formula. If p02 and p1 are measured then Eq. (5.72) allows for the calculation of M1 . Since the static pressure holes on a Pitot probe are not designed, the static pressure p1 must be obtained by some other means. In supersonic wind tunnels, for example, p1 is determined by static ports mounted on the test-section walls. Example 5.4 The pressure and temperature of the airstream in the test-section of a wind tunnel, as measured by a Pitot Probe and a thermocouple, are 200 kPa and 1300 K, respectively. The static pressure measured by an alternative method is 150 kPa. Calculate the flow velocity in the test-section if the flow is: (i) subsonic and (ii) supersonic with a detached bow-shock at the probe-nose. Solution Given, p0 = 200 kPa,

p = 150 kPa,

From the isentropic relation, T0 = T



p0 p

 γ γ−1

  1.4−1 200 γ 1300 = T 150 T = 1197.42 K

The density of air is: p RT 150 × 103 = 287 × 1197.42 = 0.436 kg/m3 ρ=

T0 = 1300 K

5.7 Supersonic Pitot Probe

213

(i) For a subsonic flow, the Bernoulli’s equation (Eq. (4.145) provides 1 p0 = p + ρv2 2 1 200 × 103 = 150 × 103 + × 0.436 × v2 2 $ 2 (200 − 150) × 103 v= 0.436 = 478.91 m/s (ii) For a supersonic flow with a detached bow-shock at the probe-nose, the Rayleigh Pitot Probe formula (Eq. (5.72)) yields  p02 = p1

γ +1 2 M1 2

2γ M2 γ +1 1

−

 (γ γ−1)

γ −1 γ +1

1  (γ −1)

1.4 % 1.4+1 2 & (1.4−1) M1 200 × 103 2 =% 1 & (1.4−1) 150 × 103 2×1.4 2 M − 1.4−1 1.4+1 1 1.4+1 &3.5 % 1.2M12 1.33 = % &2.5 1.67M12 − 0.17

M12.8 − 1.45217M12 + 0.147826 = 0 The relation above can be represented on a graph, which is similar to that shown in Fig. 5.7. The solutions are the points where the curve meets the x-axis. Thus, M1 = 0.38, and M1 = 1.50 Since M1 must be greater than one, thus M1 = 1.50 Therefore,  v = M1 γ RT1 √ = 1.5 × 1.4 × 287 × 1197.42 = 1040.44 m/s

214

5 Wave Phenomena

Fig. 5.7 The graph of M12.8 − 1.45217M12 + 0.147826 = 0

5.8 Fanno and Rayleigh Lines The aim of all flow processes is to change the state of a fluid from one state to another. Several driving potentials, such as area change, heat transfer, friction, and body forces, are involved in these flow processes to cause a change in state. In several cases of practical interest, however, one of the driving potentials may predominate to the exclusion of others. The steady one-dimensional flows in which friction, heat transfer, and body forces are negligible, i.e., for adiabatic and frictionless flows (isentropic flows), changes in flow passage area are the only driving potential responsible for variations in flow properties. The isentropic flow of a compressible fluid in a variable area flow passage is later discussed in Chap. 6. Since friction is always present in actual flow passages, the effects of wall friction in many cases are so small in comparison to those caused by area change that the assumption of frictionless flow is perfectly adequate. However, there are several realistic flow situations where the flow area is constant and friction is a major driving force. Such a steady one-dimensional adiabatic flow in a constant area flow passage is called the Fanno flow that takes friction into account. Likewise friction, in some flow situations, the heat transfer is also a major driving potential. In a constant area flow passage where the addition of heat or rejection takes place, the steady one-dimensional frictionless flow is called the Rayleigh Flow. This section briefly discusses Fanno and Rayleigh line flows, but more in detail in Chaps. 7 and 8, respectively.

5.8 Fanno and Rayleigh Lines

215

Fig. 5.8 Fanno and Rayleigh lines

5.8.1 Fanno Line The Fanno line is the locus of states defined by the equation obtained by solving the continuity equation (Eq. (4.3)), the energy equation (Eq. (4.36)), and the equation of state together. Since the momentum equation (Eq. (4.7)) is not considered in the analysis, the Fanno line depicts the states of the same mass flow rate and same stagnation enthalpy but different values of impulse function. Frictional effects are thus necessary to cause the change of states along the Fanno line, as shown in Fig. 5.8. The figure shows that in the presence of friction, entropy increases in the flow direction for an adiabatic flow and becomes maximum at point A, where the Mach number becomes unity. The point A is also known as the choking point. Furthermore, friction increases the Mach number for subsonic flow and decreases it for supersonic flow in a constant area flow passage. As a result of friction, a subsonic flow on the upper arm of the Fanno line (above point A) accelerates and approaches the choking point, where M = 1. Similarly, in the presence of friction, the supersonic flow on the lower arm (below point A) decelerates and, in the limiting case, attains the sonic speed.

5.8.2 Rayleigh Line The Rayleigh line is the locus of states defined by the equation obtained by solving the momentum equation (Eq. (4.7)), the continuity equation (Eq. (4.3)), and the equation of state. Because the energy equation (Eq. (4.36)) is not considered in the analysis, the Rayleigh line describes the states of the same mass flow rate and same impulse function but different stagnation enthalpies. Heat transfer effects are thus needed to cause the change of states along the Rayleigh line, as shown in Fig. 5.8. Figure reveals that the flow attains sonic speed (M = 1) at the point of maximum entropy (point B). The flow states on the upper arm of the Rayleigh line (above point B) are subsonic, while the flow states on the lower arm (below point B) are supersonic. The heat addition to the flow in a constant area duct increases the Mach number for

216

5 Wave Phenomena

subsonic flow while decreasing it for supersonic flow. During heat addition, however, the Mach number approaches unity in both cases.

5.8.3 Normal Shock Wave It is seen that the flow properties across a shock wave can be described by equations based on continuity equation (Eq. (4.3)), the momentum equation (Eq. (4.7)), the energy equation (Eq. (4.36)), and the equation of state. Since the points of intersection of the Rayleigh and Fanno lines (points 1 and 2 in Fig. 5.8) satisfy all of the equations simultaneously, they represent the two limits in an adiabatic flow process, such as the upstream and downstream flow conditions across a shock wave. Since the flow through a shock wave is so fast, the process is essentially adiabatic, and according to the second law of thermodynamics, the entropy in an adiabatic process either increases or remains constant. Thus, point 1 represents the flow condition upstream of the shock, while point 2 represents the flow condition downstream of the shock. This implies that the flow can occur only from 1 to 2 and not from 2 to 1. From earlier discussions, we know that a sudden change in flow speed from supersonic to subsonic can only occur from a normal shock wave. Therefore, points 1 and 2 in Fig. 5.8 basically reflect the upstream and downstream states across a normal shock wave.

5.9 The Plane Oblique Shock Wave We looked at the thermodynamic and kinematic changes encountered by a supersonic flow passing through a normal shock wave in Sect. 5.3. A normal shock is essentially a pressure discontinuity (compression front) that is perpendicular to the flow direction. However, in a wide variety of physical situations, a compression front formed in a supersonic flow is inclined to the flow. Such a compression front is called an oblique shock wave. The shock angle β denotes the angle of inclination formed by a shock wave with respect to the flow direction. Normal shock waves, with shock angle β = 90◦ , are clearly a special case of oblique shock waves. In fact, all shock waves that occur naturally in external flows are oblique. Oblique shocks normally occur when a supersonic flow encounters a concave corner in the flow field such that the flow is turned into itself. This concave corner at which the flow turns into itself, resulting in an oblique shock is also referred as compression corner. Contrary to the previous case, when a supersonic flow meets a convex corner in the flow field and is turned away from itself, expansion waves form. The convex corner at which the flow turns away from itself, creating expansion waves is also called the expansion corner. Both shock and expansion waves play a dominant role in all flow fields involving supersonic speed. Figure 5.9 depicts typical flows with an oblique shock and expansion waves.

5.9 The Plane Oblique Shock Wave

217

Fig. 5.9 Supersonic flow passing through a concave corner and b convex corner

Figure 5.9a shows how the flow is deflected into itself by the oblique shock produced at the concave corner, causing it to become parallel to the solid wall downstream of the shock. At the shock, all streamlines are deflected to the same angle θ , resulting in uniform parallel flow downstream of the shock. The angle formed by the oblique shock with the incoming (upstream) streamlines is known as the wave angle β, and the angle formed by the flow turning at the concave corner is known as the flow turning angle or the wedge angle θ . Flow properties such as pressure, temperature, and density jump across the shock wave while the Mach number decreases. In contrast, in an expansion or convex corner, the flow is turned away from itself, resulting in expansion waves, as shown in Fig. 5.9b. In this case, also, all of the streamlines

218

5 Wave Phenomena

are deflected to the same angle θ , resulting in uniform parallel flow downstream of the expansion waves. The pressure, temperature, and density decrease across the expansion waves, while the Mach number increases.

5.9.1 Governing Equations for a General Fluid When a uniform supersonic flow approaches a concave corner, the abrupt change of flow direction produces an oblique shock wave. Consider an inviscid uniform flow moving through a plane (two-dimensional) oblique shock, as shown in Fig. 5.10. The governing equations that correlate flow properties on both sides of a shock wave and are expressed with respect to the coordinate system moving with the shock can be derived. Let subscripts 1 and 2 denote fluid properties immediately upstream and downstream of the shock wave, respectively. The continuity equation (Eq. (2.49)) for steady two-dimensional flow applied to the control volume shown in Fig. 5.10 is: "

− → → ρ− v . ds = ρ2 v2n − ρ1 v1n = 0

(5.73)

CS

However, because of the small thickness of the shock wave, the area occupied by the flow through the shock wave remains constant. Consequently, the above equation becomes m ˚ = ρ1 v1n = ρ2 v2n A

Fig. 5.10 The plane oblique shock wave in a supersonic flow

(5.74)

5.9 The Plane Oblique Shock Wave

219

where v1n and v2n are the velocity components normal to the shock wave. The fluid is assumed to be a perfect gas, and the flow through the shock wave is assumed to be steady, adiabatic, and independent of body forces. The momentum equation (Eq. (2.72)) then becomes "  '− '− '− − → → → → − → → v ρ− v . ds F sys = Fb+ Fs=

(5.75)

CS

It should be noted that the shear force components in the normal direction of the shock wave are all zero. As a result, Eq. (5.75) in the normal direction to the shock wave becomes p2 − p1 = Eliminating

m ˚ A

m ˚ (v2n − v1n ) A

(5.76)

in this equation by using Eq. (5.74) yields 2 2 − ρ1 v1n p2 − p1 = ρ2 v2n

p1 +

2 ρ1 v1n =p2

+

2 ρ2 v2n

(5.77) (5.78)

Shear stresses can be neglected due to the small thickness of the shock wave. Hence, by applying the momentum equation (Eq. (5.75)) in the tangential direction, we obtain 0=

m ˚ (v2t − v1t ) A v1t = v2t

(5.79) (5.80)

where v1t and v2t are the velocity components parallel to the shock wave. Equation (5.80) states that the tangential component of flow velocity across an oblique shock remains constant. Therefore, the changes in flow properties across a plane oblique shock are caused by the normal component of upstream velocity. In addition, the flow velocity across an oblique shock decreases solely because its normal component decreases. Consequently, the slope of the flow upstream of the shock wave is greater than the slope of the flow downstream and the flow turns towards the wave. Now, for a steady adiabatic flow in the absence of body forces, the energy equation (Eq. (2.162)) gives 2 v1n v2 =h2 + 2n 2 2 i.e., h01 = h02

h1 +

(5.81) (5.82)

Thus, the stagnation enthalpy through an oblique shock wave remains constant, resulting in a constant stagnation temperature in the case of a perfect gas. It’s worth

220

5 Wave Phenomena

noting that Eqs. (5.74), (5.78), and (5.81) are the same as the continuity, momentum, and energy equations for a normal shock wave derived earlier in Sect. 5.3.1. It can, therefore, be inferred that all the previously derived relationships for the normal shock wave apply to the normal components of the velocities v1 and v2 upstream and downstream of the oblique shock wave, respectively, as long as the normal component v1n is supersonic.

5.9.2 Working Formulae for a Perfect Gas The tangential component of flow velocities across an oblique shock is shown to have no effect on flow properties. Furthermore, the flow through an oblique shock wave can be viewed as the flow through a normal shock with upstream velocity v1n . As a result, the governing equations for an oblique shock wave can be obtained by replacing the flow velocities (i.e., Mach numbers) in the normal shock relations with their normal components. From the velocity triangles shown in Fig. 5.10, the normal component of the Mach number upstream of the oblique shock is Mn1 = M1 sinβ

(5.83)

and the normal component of the Mach number downstream of the shock is M2n =M2 sin (β − θ )

(5.84)

Now, by using Eq. (5.18), M2n can be expressed in terms of M1n as 2 M2n =

2 +2 (γ − 1) M1n 2 2γ M1n − (γ − 1)

(5.85)

Substituting M1n in this equation from Eq. (5.83), yields 2 = M2n

(γ − 1) M12 sin2 β + 2 2γ M12 sin2 β − (γ − 1)

(5.86)

Combining Eqs. (5.86) and (5.84), we obtain M22 sin2 (β − θ ) =

(γ − 1) M12 sin2 β + 2 2γ M12 sin2 β − (γ − 1)

(5.87)

Now, if in normal shock relations M1 is replaced by M1 sin β and M2 is replaced by M2 sin (β − θ ), the following relations for oblique shocks are obtained:

5.9 The Plane Oblique Shock Wave

5.9.2.1

221

Static Pressure Ratio

 p2 2γ  2 2 2γ γ −1 M1 sin β − 1 = =1+ M12 sin2 β − p1 γ +1 γ +1 γ +1 5.9.2.2

Static Temperature Ratio



T2 2 + (γ − 1) M12 sin2 β 2γ γ −1 2 2 = sin β − M T1 γ +1 1 γ +1 (γ + 1) M12 sin2 β 5.9.2.3

(5.90)

Static Density Ratio

ρ2 (γ + 1) M12 sin2 β = ρ1 2 + (γ − 1) M12 sin2 β 5.9.2.5

(5.89)

Stagnation Pressure Ratio

1

(γ γ−1) 2γ p02 γ − 1 − (γ −1) (γ + 1) M12 sin2 β 2 2 M sin β − = p01 γ +1 1 γ +1 2 + (γ − 1) M12 sin2 β 5.9.2.4

(5.88)

(5.91)

Entropy Change

s 1 (s − s1 ) ln = 2 = R R (γ − 1)



2γ γ −1 M2 sin2 β − γ +1 1 γ +1



(γ − 1) M12 sin2 β + 2 (γ + 1) M12 sin2 β

γ where 1 ≤ γ ≤ 1.67

(5.92)

5.9.3 Minimum and Maximum Wave Angles It is clear from the discussion of normal shock waves that a normal shock occurs only if the flow upstream of the shock is supersonic, i.e., M1 > 1. Consequently, for

222

5 Wave Phenomena

an oblique shock wave, the normal component of the upstream Mach number, M1 , must also be greater than one. M1 sinβ ≥ 1

(5.93)

The minimum value that sinβ can have is, therefore, 1 M1   1 = sin−1 M1

sinβmin = βmin

(5.94) (5.95)

i.e., the Mach angle is the minimum shock wave angle. Equation (5.88) reflects that p2 is equal to one when the shock has this angle, indicating that the shock wave is p1 the Mach wave. The maximum shock wave angle is, of course, 90◦ , at which point the wave becomes a normal shock wave. Thus, the shock wave angle β varies as sin

−1



1 M1



≤ β ≤ 90◦

(5.96)

It should also be remembered that, since the flow downstream of a normal shock is subsonic, it follows for an oblique shock wave that M2 sin (β − θ ) ≤ 1

(5.97)

Thus, for an oblique shock wave, M2 can be greater than or less than one, indicating that the flow downstream of an oblique shock can be either supersonic or subsonic. This is in contrast to a normal shock case, where the flow downstream of the shock is always subsonic.

5.9.4 Relationship Between θ and β The proprieties of the flow downstream of an oblique shock wave can be calculated at the given values of M1 and β for any defined gas from Eqs. (5.88)–(5.92). This requires the determination of the flow deflection angle θ . On the other hand, for given values of M1 and θ , the shock angle β must first be determined before these equations can be used to determine the flow properties downstream of the shock. This is why a connection between β and θ is necessary. In this section, this relation is obtained from the governing equations for an oblique shock. First, consider the upstream and downstream velocity triangles across an oblique shock wave, as sketched in Fig. 5.10. It follows that

5.9 The Plane Oblique Shock Wave

223

v1n v1t v2n tan (β − θ )= v2t tan β =

(5.98) (5.99)

By dividing Eq. (5.99) by Eq. (5.98) and noting that v1t = v2t , it is clear that tan (β − θ ) v2n = tan β v1n

(5.100)

Combining Eq. (5.100) and the continuity equation (Eq. (5.74)) yields tan (β − θ ) ρ1 = tan β ρ2 Now, substituting

ρ1 ρ2

(5.101)

in Eq. (5.101) from Eq. (5.91), we get

tan (β − θ ) (γ − 1) M12 sin2 β + 2 = (5.102) tan β (γ + 1) M12 sin2 β

1 tan β − tan θ (γ − 1) M12 sin2 β + 2 = (5.103) tan β 1 + tan β tan θ (γ + 1) M12 sin2 β   % & tan θ 1− (γ + 1) M12 sin2 β = (1 + tan β tan θ ) (γ − 1) M12 sin2 β + 2 tan β (5.104) tan θ (γ + 1) M12 sin2 β − (γ + 1) M12 sin2 β = (γ − 1) M12 sin2 β tan β (5.105) + (γ − 1) M12 sin2 β tan β tan θ + 2 + 2 tan β tan θ

2   2 2 2 sin β 2 2 2 M1 sin β − 1 = tan θ tan β 2 + (γ + 1) M1 2 + (γ − 1) M1 sin β tan β (5.106)   2 2 2 cot β M1 sin β − 1 (5.107) tan θ = 2 + (γ + 1) M12 cos2 β + (γ − 1) M12 sin2 β  2 2  2 cot β M1 sin β − 1 tan θ = (5.108) 2 + M12 (γ + cos 2β) Equation (5.108) is a well-known relationship between the flow deflection angle θ , the wave angle β, and the upstream Mach number M1 . It is also known as the θ − β − M relation. It defines θ as a discrete function of M1 and β, and thus Eq. (5.108) may be used to calculate θ when M1 and β are given. Note that the flow turning angle  θ is zero under two conditions: (i) when β = 0 and also (ii) 1 −1 , with these two limits being, of course, a normal shock and when β = sin M1

224

5 Wave Phenomena

Fig. 5.11 Graphical representation of the θ − β − M relation (oblique shock chart)

an infinitely weak Mach wave, respectively. As previously stated, an oblique shock, therefore, exists between a normal shock and a Mach wave. There is no turning of the flow in any of these two limiting situations. The turning angle θ assumes a maximum value between these two limits. It is important to note that, for given values of M1 and θ , the values of β cannot be obtained explicitly from Eq. (5.108). However, β can be determined graphically, as shown in Fig. 5.11. The relation between β and M1 is plotted for various values of θ on Fig. 5.11. Let us take a closer look at this graph. Take special notice of the following: • The figure shows that for a given shock upstream Mach number, M1 , there exists a maximum flow deflection angle, θmax . For any given θ < θmax , there are two possible values of β predicted by θ − β − M relation for a given M1 , as illustrated in Fig. 5.12. The larger value of β is referred to as the strong shock solution, while the smaller value of β is referred to as the weak shock solution. The flow downstream of a strong oblique shock must become subsonic, while the flow downstream of a weak oblique shock may either become subsonic or remain supersonic. In nature, the weak shock solution is preferred and is more likely to occur. Therefore, in oblique shock analysis, the smaller value of β corresponding to a weak shock should always be considered, unless it is explicitly stated to consider a strong shock. • If the physical geometry is such that θ > θmax , there is no solution in the form of a straight oblique shock attached to the body; instead, a curved and detached shock, also called the bow-shock, is formed ahead of the body, as depicted in Fig. 5.13.

5.9 The Plane Oblique Shock Wave

225

Fig. 5.12 Strong and weak shock waves

Fig. 5.13 Detached shock wave

• If the physical geometry is such that θ = 0, either β = π2 or β = μ. The first case, where θ = 0 and β = π2 , corresponds to a normal shock wave, while the second case, where θ = 0 and β = μ, corresponds to the Mach wave. • In the case of the weak shock solution, a decrease in the shock upstream Mach number results in an increase in the shock angle for a given flow deflection angle θ . As the Mach number decreases, there comes a point below which no solutions exist; at this Mach number, θ = θmax . If the Mach number is reduced below this value, the shock will become detached (Fig. 5.13).

226

5 Wave Phenomena

Fig. 5.14 An oblique shock at a concave corner in Mach 2 airstream

Example 5.5 In a uniform supersonic flow, the shock waves may not occur, but the Mach waves are always present. Why?. Solution The shock wave may not arise in a uniform supersonic flow if an obstruction is not present in the flow field, the flow does not experience any area-change, or friction or heat transfer is absent. On the other hand, Mach waves develop as a result of surface roughness, and if the surface is absolutely smooth, these waves will not form. Because a completely smooth surface is only a theoretical assumption, Mach waves are always present in every flow field. Example 5.6 Consider a uniform supersonic flow with M∞ = 2.0, p∞ = 1 atm, and T∞ = 288 K, as illustrated in Fig. 5.14. The flow is deflected through 20◦ at a concave corner. Determine M2 , p2 , T2 , p02 , and T02 downstream of the resulting shock wave. Solution Given, M∞ = 2.0 p∞ = 1 atm T∞ = 288 K θ = 20◦ For M∞ = 2.0 and θ = 20◦ , from the oblique shock chart, β = 53.4◦ Therefore, Mn1 = M1 sin β = 2 × sin 53.4 = 1.61 For M1n = 1.61, from the normal shock table, we have

5.9 The Plane Oblique Shock Wave

227

p2 = 2.82 p1

M2n = 0.6684

T2 = 1.381 T1

p02 = 0.8952 p01

Thus, 0.6684 M2n = = 1.21 sin (β − θ ) sin (53.4 − 20) p2 p2 = p1 = 2.82 × 1 = 2.82 atm p1 T2 T2 = T1 = 1.381 × 288 = 399.7 K T1

M2 =

For M1 = 2.0, the isentropic table gives p01 = 7.824 and p1

T01 = 1.8 T1

i.e., p01 p1 = 7.824 × 1 = 7.824 atm p1 T01 = T1 = 1.8 × 288 = 514 K T1

p01 = T01 which gives, p02 =

p02 p01 = 0.8952 × 7.824 = 7 atm p01 T02 = T01 = 514 K

5.10 Prandtl’s Velocity Equation for an Oblique Shock Wave We deduced the Prandtl’s velocity equation for a normal shock wave in Sect. 5.5 as follows: v2 v1 = a∗2

(5.56)

A similar relationship for the product of the normal velocity components, v1n v2n , for an oblique shock wave is derived in this section. Consider the steady inviscid flow through a plane oblique shock wave shown in Fig. 5.10 to derive this relationship. Upstream and downstream Mach numbers of an oblique shock wave are related as follows:

228

5 Wave Phenomena

M22 sin2 (β − θ ) =

(γ − 1) M12 sin2 β + 2 2γ M12 sin2 β − (γ − 1)

(5.56)

By replacing M1 and M2 with the corresponding non-dimensional velocities M1∗ and M2∗ from Eq. (4.103), we get ⎡ ⎡

∗2

2

M ⎣ γ −1 2 ⎦ sin2 (β − θ) = γ +1 − M2∗2 γ −1

M2∗2 sin2 (β − θ)

v22 sin2 (β − θ) a∗2 2 v2n



a∗2

4γ (γ − 1)2



M1∗2 sin2 β −

γ −1

⎡

⎤

1

(5.109)

2 ∗2 γ −1 M1 ⎦ 2γ ⎣ γ +1 sin2 β − (γ − 1) ∗2 −M 1 γ −1



γ +1 γ +1 γ +1 + M1∗2 = − M2∗2 − M1∗2 + M1∗2 sin2 β γ −1 γ −1 γ −1

v12 v12 4γ γ +1 sin2 β − + ∗2 γ −1 a (γ − 1)2 a∗2 2 v1n

⎤

v12

4γ γ +1 − + ∗2 γ −1 a (γ − 1)2 a∗2



 =



 =

v22 γ +1 − ∗2 γ −1 a v22

γ +1 − ∗2 γ −1 a





(5.110) v2 sin2 β v12 γ +1 + 1 ∗2 − ∗2 γ −1 a a v12

γ +1 − ∗2 + γ −1 a

v12 sin2 β

4γ

2 v1n





(5.111)

a∗2

    v12 v22 v12 cos2 β γ +1 γ +1 γ +1 − + − − = γ −1 γ −1 γ −1 a∗2 (γ − 1)2 a∗2 a∗2 a∗2 a∗2      2 2 2 2 v2n v1n v1 v2 vt2 4γ γ +1 γ +1 γ +1 − + − − = γ −1 γ −1 γ −1 a∗2 (γ − 1)2 a∗2 a∗2 a∗2 a∗2





4γ γ + 1 ∗2 γ + 1 ∗2 γ + 1 ∗2 2 v2n v2 − a + v12 = a − v22 a − vt2 γ −1 γ −1 (γ − 1)2 1n γ − 1 2



 4γ γ −1 2 2 v2 − γ − 1 v2 a ∗2 + γ − 1 2 = a ∗2 − γ − 1 v2 a∗2 − v1n v12 v2n vt 2n 2n 2 2 γ +1 γ +1 γ +1 γ +1 (γ + 1)    4γ γ − 1 2 ∗2 γ −1 2 2 2 2 2 2 v v − v1n + v1t v2n v a + γ +1 (γ + 1)2 1n 2n γ + 1 2n    γ −1  2 γ −1 2 2 2 = a∗4 − v2 vt v + vt2 a∗2 + γ +1 2 γ +1 2 2   4γ 2 v2 − γ − 1 v2 a ∗2 + γ − 1 2 v2 + γ − 1 2 v2 v1n v1n v2n t 2n 2n 2n 2 γ + 1 γ + 1 γ + 1 (γ + 1)  2     γ −1 2 γ −1 2 + v2 v2 = a∗4 − v2n v + 2vt2 a∗2 + t t γ + 1 2n γ +1  2 2    4γ 2 v2 − γ − 1 v2 a ∗2 + γ − 1 2 v2 + γ − 1 2 v2 v1n v1n v2n t 2n 2n 2n 2 γ +1 γ +1 γ +1 (γ + 1)   2 2     γ −1 2 a ∗2 − 2 γ − 1 v2 v2 + γ − 1 2 v2 + γ − 1 v2n = a∗4 − v2n vt4 t 2n t γ +1 γ +1 γ +1 γ +1      2 4γ (γ − 1)2 2 v2 = a ∗4 − 2 γ − 1 v2 v2 + γ − 1 v1n + vt4 t 2n 2n 2 2 γ +1 γ +1 (γ + 1) (γ + 1)   2 2 v2 = a ∗2 − γ − 1 v2 v1n t 2n γ +1   γ − 1 vt2 v1n v2n = a∗2 − γ +1 2 v2n





2 ∗2 γ −1 M1

⎦ sin2 β + 2 (γ − 1) ⎣ γ +1 −M∗2

⎤

(5.112) (5.113) (5.114) (5.115) (5.116)

(5.117)

(5.118)

(5.119) (5.120) (5.121) (5.122)

5.10 Prandtl’s Velocity Equation for an Oblique Shock Wave

229

Equation (5.122) is considered to be the general form of the Prandtl’s velocity equation derived in Eq. (5.56). The tangential component of velocity, vt = 0, for a normal shock wave, so Eq. (5.122) reduces to Eq. (5.56).

5.11 The Rankine-Hugoniot Equation for an Oblique Shock Wave In Sect. 5.6, the Rankine-Hugoniot equation for a normal shock wave was derived, which connects the density ratio ρρ21 across a normal shock with the corresponding pressure ratio pp21 . This relationship was derived by combining the equation of state for a perfect gas with the continuity, momentum, and energy equations for the steady one-dimensional flow. Now, for an oblique shock wave, if v1 and v2 in Eqs. (5.1)– (5.5) are replaced by the normal velocity components v1n and v2n , respectively, the equations obtained govern the flow through the oblique shock wave. These governing equations, derived in Sect. 5.9.1, are rewritten for convenience as follows: ρ1 v1n = ρ2 v2n

(5.74)

2 2 p1 + ρ1 v1n =p2 + ρ2 v2n

(5.78)

h1 +

2 v1n

2

=h2 +

2 v2n

(5.81)

2

By introducing v1n = v1 sinβ and v2n = v2 sin (β − θ ) into the above equations, we obtain ρ2 sinβ v1 = ρ1 v2 sin (β − θ ) p1 + ρ1 v12 sin2 β=p2 + ρ2 v22 sin2 (β − θ ) h1 +

v12 2 v2 sin β=h2 + 2 sin2 (β − θ ) 2 2

(5.123) (5.124) (5.125)

Eliminating v22 in Eq. (5.124) using Eq. (5.123) provides  p1 + ρ1 v12 sin2 β=p2 + ρ2  ρ1 v12 sin2 β

− ρ2

ρ1 ρ2

2

ρ1 ρ2

2 v12 sin2 β

(5.126)

v12 sin2 β = p2 − p1

(5.127)

ρ1 v12 sin2 β (ρ2 − ρ1 ) = (p2 − p1 ) ρ   2 p2 − p1 ρ2 1 2 v1 = sin2 β ρ2 − ρ1 ρ1

(5.128) (5.129)

230

5 Wave Phenomena

which, when combined with Eq. (5.123), gives v22

1 = sin2 (β − θ )



p2 − p1 ρ2 − ρ1



ρ1 ρ2

(5.130)

Now, introducing v12 and v22 into Eq. (5.125) results in       p2 − p1 p2 − p1 1 ρ2 1 ρ1 = h2 + h1 + 2 ρ1 ρ2 − ρ1 2 ρ2 ρ2 − ρ1       p2 − p1 p2 − p1 p1 1 ρ2 p2 1 ρ1 = e2 + e1 + + + ρ1 2 ρ1 ρ2 − ρ1 ρ2 2 ρ2 ρ2 − ρ1 

 ρ2 p1 p2 1 p2 − p1 ρ1 e2 − e1 = − + − ρ1 ρ2 2 ρ2 − ρ1 ρ1 ρ2

p1 p2 1 (p2 − p1 ) (ρ2 + ρ1 ) (ρ2 − ρ1 ) e2 − e1 = − + ρ1 ρ2 2 (ρ2 − ρ1 ) ρ1 ρ2    1 p1 p2 p2 − p1 1 e2 − e1 = − + + ρ1 ρ2 2 ρ1 ρ2    1 p2 + p1 1 e2 − e1 = − 2 ρ1 ρ2

(5.131) (5.132) (5.133) (5.134) (5.135) (5.136)

This is the Rankine-Hugoniot equation for an oblique shock wave. It should be noted that Eq. (5.136) is the same relationship between ρρ21 and pp21 as derived for a normal shock in Sect. 5.6. Therefore, the Rankine-Hugoniot equation for a normal shock wave, given by Eq. (5.69), also applies to oblique shock waves. Accordingly, the pressure ratio, density ratio, and temperature ratio across an oblique shock wave are all associated in the same way as they are for a normal shock wave.

5.12 Weak Oblique Shock Waves The compression of the flow through the Mach waves was found to be an isentropic process in Sect. 5.18. In this section, it will be demonstrated that a weak oblique shock, which occurs when the flow deflection angle θ is very small and the Mach number downstream of the shock, M2 , is greater than one, will compress the flow with an entropy increase close to zero. Though the flow downstream of a weak oblique shock is supersonic, the downstream Mach number, M2 , is less than the upstream Mach number, M1 . Despite this, the decrease in the Mach number across a weak shock, M1 − M2 , is generally small. A shock wave with small M1 − M2 and θ can be approximated as the Mach wave, and therefore the shock angle β is nearly equal to the Mach angle μ. For small values of flow deflection angles, the oblique shock relations can be reduced to very simple forms. In this case,

5.12 Weak Oblique Shock Waves

231

sin θ ≈ θ

(5.137)

cos (β − θ ) = cos β

(5.138)

Using these approximations, Eq. (11.30) is reduced to M12 sin2 β − 1≈

(γ + 1) 2 M1 tan β θ 2

(5.139)

Since β ≈ μ for a weak oblique shock; thus 1 M

(5.140)

1 tan β = √ M2 − 1

(5.141)

sin β ≈ sin μ =

Now, by replacing tan β in Eq. (5.139) with Eq. (5.141), we get ⎤ 2 M + 1) (γ 1 ⎦θ M12 sin2 β − 1≈ ⎣ 2 M2 − 1 ⎡

(5.142)

1

Equation (5.142) is regarded as the fundamental relation to obtaining all other suitable expressions for weak oblique shock waves. Furthermore, the shock strength is defined as the non-dimensional pressure increase through a shock wave: p (p2 − p1 ) = p1 p1

(5.143)

Combining Eqs. (5.88) and (5.143) yields γ M12 p θ, i.e., Strength of shock ∝ θ ≈ p1 M12 − 1

(5.144)

Equation (5.144) reveals that the strength of a shock is proportional to the flow turning angle θ . Correspondingly, density and temperature variations can also be seen to be proportional to θ : ρ ∝θ ρ1 T ∝θ T1

(5.145) (5.146)

Let us rewrite Eq. (5.92) for convenience in order to examine the effects of small flow deflection angles on the change in entropy. Thus,

232

5 Wave Phenomena

1 s2 − s1 == ln R (γ − 1)



2γ γ −1 M12 sin2 β − γ +1 γ +1



γ (γ − 1) M12 sin2 β + 2 (γ + 1) M12 sin2 β (5.92)

But, for a weak oblique shock with M1 − M2  1 and θ very small, we can approximate M12 sin2 β≈ M12 . Thus, Eq. (5.92) becomes    1  γ  (γ −1) γ − 1 (γ −1) 2γ − (γ γ−1) (5.147) 1+ 1+ t t (1 + t) γ +1 γ +1     s2 − s1 2γ γ γ −1 γ 1 ln 1 + ln (1 + t) + ln 1 + = t − t R γ +1 γ +1 (γ − 1) (γ − 1) (γ − 1) s2 − s1 = ln R

(5.148)   where t = M12 − 1 .

3 2 + ·········∞ , Now, using the logarithmic sequence, log (1 + ) =  − 2 3 expanding the terms in Eq. (5.148) as follows: 



2

3 2γ 1 1 2γ 2γ t− t + t ·········∞ γ +1 2 γ +1 3 γ +1  



2 3 t γ −1 γ t γ 1 γ −1 2 1 γ −1 3 t− · · · · · · · · · ∞} − + ·········∞ + t− t + t 2 3 2 γ +1 3 γ +1 (γ − 1) (γ − 1) γ + 1 s2 − s1 1 = R (γ − 1)

(5.149)  8γ 3 s2 − s1 1 γ (γ − 1)3 1 1 {0} t 2 + {0} t − −γ + = R 2 (γ − 1) 3 (γ − 1) (γ + 1)3 (γ − 1) (γ + 1)3  ⎫ ⎧ 2 ⎨ 2γ γ − 1 ⎬ s2 − s1 1 t3 · · · · · · · · · ∞ = R 3 (γ − 1) ⎩ (γ + 1)3 ⎭

t3 · · · · · · · · · ∞

3  M12 − 1 s2 − s1 2γ = ·········∞ R 3 (γ + 1)2

(5.150) (5.151) (5.152)

Retaining the terms up to first order, we get  2 3 M1 − 1 2γ s ≈ R 3 (γ + 1)2 This equation can be rewritten in terms of shock strength, as: s (γ + 1) ≈ R 12γ 2



p p1

(5.153) p , p1

using Eq. (5.144)

3 (5.154)

5.12 Weak Oblique Shock Waves

233

However, according to Eq. (5.144), the shock strength s ∝ θ 3

p ∝ θ , and therefore p1 (5.155)

i.e, the entropy increase across a weak oblique shock is found to be proportional to the third power of the deflection angle θ .

5.13 The Hodograph Shock Polar Diagram In this section, we will look at an alternative method of evaluating oblique shock relations using shock polar diagrams. These diagrams are the traditional approach in the field of gas dynamics and they have an excellent view point of the oblique shock wave family. Each shock polar is, in fact, a graphical representation of the solutions that correspond to an oblique shock wave under specified flow parameters. Indeed, it indicates the conditions under which an oblique shock remains connected to the body’s vertex for a given shock upstream Mach number M1 and flow deflection angle θ . A shock polar is plotted in the hodograph plane, a coordinate system in which the velocity components are the coordinates, i.e., independent variables. In the first place, this plane is very useful for displaying data or solutions and for graphical analysis (vector diagram). However, a far more profound reason why it’s important is that certain nonlinear problems in the physical plane become linear when reinterpreted in the hodograph plane, i.e., with velocity components as independent variables. The oblique shock solutions in the hodograph plane are shown in Fig. 5.15. The flow velocities upstream and downstream of a shock as well as the flow deflection angle and shock angle in the physical plane can be represented on the vx − vy plane (u − v plane), commonly known as the hodograph plane. That is, a point in the physical  plane is located in the hodograph plane by plotting its velocity components  vx , vy . Thus, a vector in the hodograph plane, drawn from the origin to the point in question, is the corresponding velocity vector. The physical and hodograph planes are depicted in Fig. 5.15a, b, respectively. In the hodograph plane (Fig. 5.15b), the line OA represents the velocity vector upstream of the oblique shock and the line OB represents the velocity vector downstream of the shock. The entire flow field upstream of the shock plots into a single point A, whereas the flow downstream plots into the point B. They are the only two velocities considered in this study. The mapping from the physical plane to the hodograph plane, or vice versa, is sometimes most singular, but, it is not unique. Many different flows in the physical plane can be produced from the oblique shock solution by choosing alternative boundary conditions that are compatible with the streamlines. All of them will be represented in the hodograph plane by two points, A and B. Consider an oblique shock wave in a supersonic flow, as shown in Fig. 5.10. The flow velocities upstream and downstream of the shock wave, denoted by v1 and v2 , are decomposed into two components: normal to and along (tangential direction)

234

Fig. 5.15 Illustration of oblique shock solutions in the hodograph plane

5 Wave Phenomena

5.13 The Hodograph Shock Polar Diagram

235

the wave (Fig. 5.15a). Equation (5.80) already shows that the tangential components of v1 and v2 are equal, i.e., v1t = v2t . In hodograph plane, the velocities v1 and v2 are resolved along the coordinate axes (x − y axes) as shown in Fig. 5.15b. Let the velocity components along x- and y-directions be denoted by vx and vy , respectively, and let the flow approaching the shock be in the x-direction, so that v1x = v1 , and v1y = 0. It is interesting to see that Fig. 5.15c may be obtained by simply superimposing the velocity triangles of Fig. 5.10. Furthermore, all the flow parameters required to draw vector v2 and the flow deflection angle θ for a shock of given flow deflection angle θ may be derived from the oblique shock relations, and therefore point B (see Fig. 5.15c) can be located in the hodograph plane. When represented in non-dimensional velocity coordinates, the locus of point B that is allowed for a plane oblique shock is referred to as the hodograph shock polar. The following procedure can be adapted to get the shock polar equation in vx − vy coordinates from the Prandtl’s velocity equation for an oblique shock, derived in Sect. 5.10. The velocity triangles of Fig. 5.15c show that

v2n

v1n = v1x sin β v1t = v1x cos β = v2t 2 2 = v1n − v2y + (v1x − v2x )2 = v1x sin β − v2y + (v1x − v2x )2

(5.156) (5.157) (5.158)

Substituting the equations above into the Prandtl’s velocity equation (Eq. (5.122)), we obtain   γ −1 2 2 2 ∗2 2 2 v cos2 β (5.159) v1x sin β − v1x sin β v2y + (v1x − v2x ) = a − γ + 1 1x Another useful relation can be obtained from Eq. (5.80), v1x cos β = v2 cos (β − θ ) = v2x cos β + v2y sin β

(5.160)

Using this relation, the wave angle β can be expressed in terms of the velocity components: tan β = i.e., sin β = and cos β =

v1x − v2x v2y (v1x − v2x ) 2 v2y + (v1x − v2x )2 v2y

2 + (v1x − v2x )2 v2y

(5.161) (5.162) (5.163)

236

5 Wave Phenomena

Eliminating β in Eq. (5.159) by using (5.162) and Eq. (5.163), we get   2 v2 v1x γ −1 (v1x − v2x )2 2y  − v1x (v1x − v2x ) = a∗2 −   2 2 γ + 1 2 v2y + (v1x − v2x ) v2y + (v1x − v2x )2

2  v1x

(5.164)

      2 (v − v )2 − v (v − v ) v2 + (v − v )2 = a ∗2 v2 + (v − v )2 − γ − 1 v2 v2 v1x 1x 2x 1x 1x 2x 1x 2x 1x 2x 2y 2y 1x 2y γ +1 2 (v − v )2 − v (v − v )3 − v v2 (v − v ) = a ∗2 v2 + a ∗2 (v − v )2 − v1x 1x 2x 1x 1x 2x 1x 2y 1x 2x 1x 2x 2y



2 (v − v )2 − v (v − v )3 − a ∗2 (v − v )2 = a ∗2 − v1x 1x 2x 1x 1x 2x 1x 2x



γ −1 γ +1



γ −1 γ +1



(5.165) 2 v2 v1x 2y

(5.166)



2 + v (v − v ) v2 v1x 1x 1x 2x 2y

(5.167) 2 v2x )2 − v1x (v1x − v2x )3 − a∗2 (v1x − v2x )2 2 = v1x (v1x −     v2y γ −1 2 + v (v − v ) ∗2 a − γ +1 v1x 1x 1x 2x

(5.168)

(v − v2x )2 v1x v2x − a∗2 2 =  1x  v2y 2 v2 − v v a∗2 + (γ +1) 1x 2x 1x

(5.169)





 v2y 2 a∗



 2  v1x v v1x v2x − a2x ∗ ∗2 − 1 a∗ a

=   v1x 2 v v 2 − 1x∗22x + 1 γ +1 a∗ a 

(5.170)

v

Equation (5.170) allows us to draw a shock polar diagram, i.e., a plot of a2y∗ versus   v2x for a given value of va1x∗ . The typical results of a single shock polar va1x∗ = constant a∗ are presented in Fig. 5.15d. The figure reveals the following characteristics: • The intersectionof the shock polar with any line drawn from the origin O yields  v2x v2y for any flow deflection angle θ . A line from O may be the coordinates a∗ , 2 a∗2 observed cutting the shock polar at three points: A, B, and C. The point C has no physical importance because it implies v2x > v1x , which suggests a drop in entropy across the shock. Therefore, the theoretically feasible solutions of the oblique shock wave are points A and B (i.e., the two possible flow conditions downstream of the oblique shock). The sonic circle cuts the shock polar between the locations A and B at the point S, where M2 = 1 (i.e., the point S reflects the sonic flow conditions downstream of the shock wave). The subsonic flow downstream of the oblique shock is designated by point A, which is associated with a strong shock wave, and the supersonic flow downstream of the shock is denoted by point B, which is associated with a weak shock wave. It is worth noting that the strong shock wave can be obtained by drawing a line through the origin O that is orthogonal  to the line passing between A and va1x∗ , 0 . Likewise, the weak shock wave may be obtained by drawing  a line through the origin O and normal to the line passing through B and va1x∗ , 0 . • The intersections of the shock polar and va2x∗ axis yield v1xa∗2v2x = 1 and va1x∗ = va2x∗ , which correspond to the normal shock and the Mach wave, respectively. • The line through the origin O that is tangent to the shock polar subtends an angle θmax with va2x∗ axis. This is the maximum flow deflection angle for a given va1x∗ beyond which the shock is detached from the vertex of the body.

5.13 The Hodograph Shock Polar Diagram

237

• The region inside the sonic circle marked M∗ = 1 corresponds to subsonic flow, while the region between the sonic circle and the limit circle marked M∗ = corresponds to supersonic flow.

The MATLAB program given below shows the variation of compressible fluid, as seen in Fig. 5.16.

v2y a∗

γ +1 γ −1

versus va2x∗ for a general

Listing 5.1 A MATLAB program for generating the shock polar plot. % V e r s i o n 1.1 C o p y r i g h t M r i n a l Kaushik , IIT Kharagpur , 2 9 / 0 7 / 2 0 2 1 clc c l e a r all close format short gamma =1.4; % s p e c i f i c heat ratio b i n n u m = 7; % N u m b e r of c u r v e s to be p l o t t e d b i n n u m 2 = 100; % N u m b e r of V2x / a * to be c a l c u l a t e d s t r t m 1 =2; % S m a l l e s t M1 m1x = zeros ( binnum ,1) ; m 2 x s t a r t = z e r o s ( binnum ,1) ; m 2 x e n d = z e r o s ( binnum ,1) ; dm = z e r o s ( binnum ,1) ; for i =1: b i n n u m if i ==1 m1x ( i ,1) = s t r t m 1 ; else m1x ( i ,1) = m1x ( i -1 ,1) + 0 . 5 ; % C a l c u l a t i o n of M1 end m 1 x s t a r ( i ,1) = abs ( ((( g a m m a +1) * m1x (i ,1) ^2) /(2+( m1x (i ,1) ^2) *( gamma -1) ) ) ^0.5 ) ; % C h a r a c t e r i s t i c Mach no ( M1x *) for j =1: b i n n u m 2 m 2 x s t a r t (i ,1) =1/ m 1 x s t a r ( i ,1) ; % m2x * w h e r e V1x * V2x = a *^2 m 2 x e n d ( i ,1) = m 1 x s t a r ( i ,1) ; % m2x * w h e r e V1x / a * = V2x / a * dm ( i , j ) =( m 2 x e n d ( i ,1) - m 2 x s t a r t (i ,1) ) * 0 . 0 1 ; if j ==1 m 2 x s t a r (i , j ) = m 2 x s t a r t (i ,1) ; % c a l c u l a t i o n of V2x / a * values else m 2 x s t a r (i , j ) = m 2 x s t a r t (i ,1) + j * dm ( i ,1) ; end t e r m 1 ( i , j ) =(( m 1 x s t a r ( i ,1) - m 2 x s t a r ( i , j ) ) ^2) *( m 1 x s t a r ( i ,1) * m 2 x s t a r ( i , j ) -1) ; t e r m 2 ( i , j ) = ( ( 2 / ( g a m m a +1) ) *( m 1 x s t a r ( i ,1) ) ^2) -( m 1 x s t a r ( i ,1) * m 2 x s t a r ( i , j ) ) +1; t e r m 3 ( i , j ) = abs ( t e r m 1 ( i , j ) / t e r m 2 ( i , j ) ) ; m 2 y s t a r ( i , j ) = sqrt ( term3 (i , j ) ) ; % c a l c u l a t i o n of V2y / a * v a l u e s for g i v e n V2x / a * and M1x end end % M a t r i x o p e r a t i o n for c a l c u l a t i n g n e g a t i v e v a l u e s of m2y * m2xstar2 = fliplr ( m2xstar ); X1 = m 2 x s t a r . '; X2 = m 2 x s t a r 2 . '; X =[ X1 ; X2 ]; m 2 y s t a r 1 = -1* m 2 y s t a r ; m2ystar2 = fliplr ( m2ystar1 ); Y1 = m 2 y s t a r . '; Y2 = m 2 y s t a r 2 . '; Y =[ Y1 ; Y2 ]; % P l o t t i n g m2 plot (X ,Y , ' L i n e W i d t h ' ,2) hold on

238

5 Wave Phenomena

% c i r c l e for M =1 th = l i n s p a c e ( pi /3 , - pi /3 , 100) ; R = 1; x = R * cos ( th ) ; y = R * sin ( th ) ; p = plot (x ,y , 'g ' , ' L i n e W i d t h ' ,2) ; axis equal ; text (0.4 ,.8 , ' S o n i c c i r c l e ( v = a ^ { * } ) ' , ' f o n t n a m e ' , ' t i m e s ' , ' F o n t S i z e ' ,14) % c i r c l e for M = inf hold on th = l i n s p a c e ( pi /6 , - pi /6 , 100) ; R = 2.4495; x = R * cos ( th ) ; y = R * sin ( th ) ; plot (x ,y , 'g ' , ' L i n e W i d t h ' ,2) ; axis equal ; text (2.2 ,1 , ' L i m i t c i r c l e ( M =\ i n f t y ) ' , ' f o n t n a m e ' , ' t i m e s ' , ' F o n t S i z e ' ,14) text (2.25 , -1 , ' M =\ i n f t y ' , ' f o n t n a m e ' , ' t i m e s ' , ' F o n t S i z e ' ,14) % Annotation creation P1 =[ 0 0]; P2 =[ 0.6 -0.8]; D = P2 - P1 ; q u i v e r ( P1 (1) , P1 (2) , D (1) , D (2) , 0 , ' black - - ' , ' L i n e W i d t h ' ,2 ) ; % A r r o w c r e a t i o n for M =1 P1 =[ 0 0]; P2 =[ 2.23 -1]; D = P2 - P1 ; q u i v e r ( P1 (1) , P1 (2) , D (1) , D (2) , 0 , ' b l a c k : ' , ' L i n e W i d t h ' ,2) ; % A r r o w c r e a t i o n for M = inf text (1.6 , -1 , '$ M ^{*}=\ frac {\ surd (\ gamma +1) }{\ surd (\ gamma +1) }$ ' , ' i n t e r p r e t e r ' , ' l a t e x ' , ' F o n t S i z e ' ,14) % Box text ( s u b s o n i c & S u p e r s o n i c ) dim = [.34 .4 .3 .3]; str = ' S u b s o n i c ' ; a n n o t a t i o n ( ' t e x t b o x ' , dim , ' S t r i n g ' , str , ' F i t B o x T o T e x t ' , ' on ' , ' f o n t n a m e ' , ' t i m e s ' , ' F o n t S i z e ' ,14) ; dim1 = [.5 .58 .3 .3]; str1 = ' S u p e r s o n i c ' ; a n n o t a t i o n ( ' t e x t b o x ' , dim1 , ' S t r i n g ' , str1 , ' F i t B o x T o T e x t ' , ' on ' , ' f o n t n a m e ' , ' t i m e s ' , ' F o n t S i z e ' ,14) ; % Axis Label x l a b e l ( ' V_ {2 x }/ a ^{*} ' , ' f o n t n a m e ' , ' t i m e s ' , ' F o n t S i z e ' ,16) y l a b e l ( ' V_ {2 y }/ a ^{*} ' , ' f o n t n a m e ' , ' t i m e s ' , ' F o n t S i z e ' ,16) set ( get ( gca , ' y l a b e l ' ) , ' r o t a t i o n ' ,0) % In - f i g u r e text g e n e r a t i o n text (0.33 , -.4 , ' M ^ { * } = 1 ' , ' f o n t n a m e ' , ' t i m e s ' , ' F o n t S i z e ' ,14) text (0.63 , -.8 , 'M = 1 ' , ' f o n t n a m e ' , ' t i m e s ' , ' F o n t S i z e ' ,14) text (1.6330 ,0 , ' $\ l e f t a r r o w \ frac { V_ {1 x }}{ a ^ { * } } = \ frac { V_ {2 x }}{ a ^ { * } } $ ' , ' i n t e r p r e t e r ' , ' l a t e x ' , ' F o n t S i z e ' ,16) text (1.62 ,.05 , '\ d o w n a r r o w ' , ' F o n t S i z e ' ,14) text (1.45 ,.12 , ' Mach wave ' , ' f o n t n a m e ' , ' t i m e s ' , ' F o n t S i z e ' ,14) text (0.6 ,.05 , '\ d o w n a r r o w ' , ' F o n t S i z e ' ,14) text (.4 ,.12 , ' N o r m a l s h o c k ' , ' f o n t n a m e ' , ' t i m e s ' , ' F o n t S i z e ' ,14) text (0.6124 ,0 , '\ l e f t a r r o w V_ {1 x } V_ {2 x }= a ^ { * 2 } ' , ' f o n t n a m e ' , ' t i m e s ' , ' F o n t S i z e ' ,14) text (1 ,.34 , ' M_ {1 x }= 2 ' , ' f o n t n a m e ' , ' t i m e s ' , ' F o n t S i z e ' ,14) text (1.2 ,.45 , ' M_ {1 x }= 2.5 ' , ' f o n t n a m e ' , ' t i m e s ' , ' F o n t S i z e ' ,14) text (1.4 ,.55 , ' M_ {1 x }= 3 ' , ' f o n t n a m e ' , ' t i m e s ' , ' F o n t S i z e ' ,14) text (1.5513 ,0.6429 , '\ l e f t a r r o w M_ {1 x }= 3.5 ' , ' f o n t n a m e ' , ' t i m e s ' , ' F o n t S i z e ' ,14) text (1.8708 , -.5145 , '\ l e f t a r r o w M_ {1 x }= 4 ' , ' f o n t n a m e ' , ' t i m e s ' , ' F o n t S i z e ' ,14) text (1.608 ,.7667 , '\ l e f t a r r o w M_ {1 x }= 4.5 ' , ' f o n t n a m e ' , ' t i m e s ' , ' F o n t S i z e ' ,14) text (2.1 ,.4 , ' M_ {1 x }= 5 ' , ' f o n t n a m e ' , ' t i m e s ' , ' F o n t S i z e ' ,14) hold off

5.14 The Pressure-Deflection Diagram

239

Fig. 5.16 Hodograph shock polar plot by the MATLAB program

5.14 The Pressure-Deflection Diagram The pressure-deflection (p − θ ) diagram is yet another approach to represent the oblique shock properties. It is primarily utilized to flow situations in which a shock interacts with another shock or with an expansion fan, resulting in the formation of a surface of discontinuity known as the slipstream or vortex-sheet. A slipstream is a streamline that separates a flow into two segments with identical pressures and flow directions but different flow velocities. The oblique shock equations presented in Sect. 5.9.2 allow us to derive a relationship2 between the static pressure ratio pp21 and the flow deflection angle θ . A family of pressure-deflection curves may be obtained by plotting the resulting equation at various upstream Mach numbers. Figure 5.17 depicts a typical p − θ diagram for an oblique shock wave at different upstream Mach numbers. The figure shows that for every flow deflection angle θ smaller than the maximum deflection angle, i.e., θ < θmax , two shock solutions are possible: one weak shock solution and one strong shock solution. The maximum deflection angle θmax corresponding to an upstream Mach number M1 is also indicated. At a particular M1 , no attached shock solutions are possible for θ > θmax .

2

Mateescu, D. (2010), Explicit Exact and Third-Order-Accurate Pressure-Deflection Solutions for Oblique Shock and Expansion Waves, The Open Aerospace Engineering Journal, vol. 3, pp. 1–8 (open access).

240

5 Wave Phenomena

Fig. 5.17 Illustration of the pressure-deflection shock polar in p − θ plane

It is worth noting that all of the curves in Fig. 5.17 start at the same position where θ = 0, β = μ and pp21 = 1. This point effectively reflects the weakest form of the oblique shock, namely the Mach wave. The strongest type of oblique shock is, of course, a normal shock with the highest static pressure ratio. The end points of individual curves in Fig.5.17  indicate the conditions across a normal shock with θ = 0, β = 90◦ and pp21 = pp21 (across a normal shock). max

The following MATLAB program plots the variation of the pressure ratio, pp21 , versus the flow turning angle, θ , for a general compressible fluid, as shown in Fig. 5.18.

Listing 5.2 A MATLAB program for generating the pressure-deflection shock polar plot. % V e r s i o n 1.1 C o p y r i g h t M r i n a l Kaushik , IIT Kharagpur , 2 9 / 0 7 / 2 0 2 1 clc c l e a r all close format short gamma =1.4; % S p e c i f i c heat ratio b i n n u m 1 =6; % N u m b e r of M1 to be c o n s i d e r e d binnum2 =5001; % N u m b e r of t h e t a v a l u e to be c o n s i d e r e d

5.14 The Pressure-Deflection Diagram

241

for i =1: b i n n u m 1 if i ==1 M ( i ,1) = 1 . 5 ; else M ( i ,1) = M ( i -1 ,1) + 0 . 5 ; % Mach n u m b e r c a l c u l a t i o n end for j =1: b i n n u m 2 t h e t a d e g ( j ,1) =( j -1) * 0 . 0 1 ; % Calculating tjeta values t h e t a ( i , j ) = t h e t a d e g ( j ,1) * pi / 1 8 0 ; % Radian conversion t e r m 1 ( i , j ) =1 -(1/ M ( i ,1) ^2) +( gamma *( sin ( theta (i , j ) ) ) ^2) ; b ( i , j ) = t e r m 1 ( i , j ) * 4 / ( 3 * ( g a m m a +1) ) ; %b value t e r m 2 ( i , j ) = ( 4 * ( sin ( t h e t a ( i , j ) ) ) ^2) /(3* b ( i , j ) ^2) ; term3 (i , j ) =1 -( (4/( gamma +1) ) * (1/ M ( i ,1) ^2) ) ; q ( i , j ) =1 - t e r m 2 ( i , j ) * t e r m 3 ( i , j ) ; %q value t e r m 4 ( i , j ) =8* sin ( t h e t a ( i , j ) ) ^ 2 / ( ( g a m m a +1) * b ( i , j ) ^3* M ( i ,1) ^2) ; r ( i , j ) = ( ( 3 * q (i , j ) -1) /2) - t e r m 4 ( i , j ) ; % r v a l u e t e r m 5 ( i , j ) = r (i , j ) /( q ( i , j ) ^ ( 3 / 2 ) ) ; a l p h a ( i , j ) = pi * a c o s d ( t e r m 5 ( i , j ) ) / 1 8 0 ; % a l p h a v a l u e t e r m 6 ( i , j ) =2* q (i , j ) ^ 0 . 5 * cos (( pi + a l p h a ( i , j ) ) /3) ; c p w e a k ( i , j ) = M (i ,1) ^2* b ( i , j ) *(1 - t e r m 6 ( i , j ) ) ; % Cp for weak oblique shock p 2 b y p 1 ( i , j ) = 1 + ( c p w e a k ( i , j ) * g a m m a /2) ; % p2 / p1 for weak oblique shock if i s r e a l ( p 2 b y p 1 ( i , j ) ) % f i n d i n g out the t h e t a max v a l u e p 2 b y p 1 r e a l (i , j ) = p 2 b y p 1 ( i , j ) ; t h e t a m a x ( i ,1) = t h e t a d e g ( j ,1) ; p 2 b y p 1 t h e t a m a x (i ,1) = p 2 b y p 1 r e a l (i , j ) ; else p 2 b y p 1 r e a l (i , j ) = nan ; % R e m o v i n g the c o m p l e x n u m b e r s for t h e t a g r e a t e r than theta max end c p s t r o n g ( i , j ) = b ( i , j ) * ( 1 + ( 2 * ( q (i , j ) ^ 0 . 5 ) * cos (( a l p h a ( i , j ) ) /3) )); % cp for s t r o n g o b l i q u e s h o c k p 2 b y p 1 s t r o n g (i , j ) =1+( c p s t r o n g (i , j ) * M ( i ,1) ^2* g a m m a /2) ; % p2 / p1 for s t r o n g o b l i q u e s h o c k if i s r e a l ( p 2 b y p 1 s t r o n g ( i , j ) ) % % R e m o v i n g the c o m p l e x n u m b e r s for t h e t a g r e a t e r than theta max p2byp1strongreal (i ,j)= p2byp1strong (i ,j); else p 2 b y p 1 s t r o n g r e a l ( i , j ) = nan ; end end end % M a t r i x o p e r a t i o n for p l o t t i n g p 2 b y p 1 w = p 2 b y p 1 r e a l . '; p 2 b y p 1 s = f l i p l r ( p 2 b y p 1 s t r o n g r e a l ) . '; p 2 b y p 1 w s =[ p 2 b y p 1 w ; p 2 b y p 1 s ]; t h e t a d =[ t h e t a d e g ; flip ( t h e t a d e g ) ]; plot ( thetad , p2byp1ws , ' L i n e W i d t h ' ,2) ; hold on % plotting plot ( thetamax , p 2 b y p 1 t h e t a m a x , 'o ' , ' M a r k e r S i z e ' ,8 , ' M a r k e r E d g e C o l o r ' , ' b l a c k ' , ' M a r k e r F a c e C o l o r ' ,[.18 .35 . 1 5 ] ) ; axis square ; grid on ; set ( gca , ' f o n t n a m e ' , ' t i m e s ' , ' F o n t S i z e ' ,16) ; set ( gca , ' X T i c k ' , [ 0 : 4 : 4 0 ] ) ; xt = get ( gca , ' x t i c k ') ; for k =1: n u m e l ( xt ) ; % d e s p l a y i n g d e g r e e s y m b o l in x - axis xt1 { k }= s p r i n t f ( '% d° ' , xt ( k ) ) ; end % Axis s e t t i n g s set ( gca , ' x t i c k l a b e l ' , xt1 ) ; set ( gca , ' Y T i c k ' , [ 0 : 2 : 2 0 ] ) ; set ( gca , ' xlim ' ,[0 40] , ' ylim ' ,[0 20]) ;

242

5 Wave Phenomena

% A n n o t a t i o n s and in - f i g u r e text text (2.5 ,3 , 'M = 1.5 ' , ' f o n t n a m e ' , ' t i m e s ' , ' F o n t S i z e ' ,16) ; text (2.5 ,5.1 , 'M = 2 ' , ' f o n t n a m e ' , ' t i m e s ' , ' F o n t S i z e ' ,16) ; text (2.5 ,7.7 , 'M = 2.5 ' , ' f o n t n a m e ' , ' t i m e s ' , ' F o n t S i z e ' ,16) ; text (2.5 ,11 , 'M = 3 ' , ' f o n t n a m e ' , ' t i m e s ' , ' F o n t S i z e ' ,16) ; text (2.5 ,14.9 , 'M = 3.5 ' , ' f o n t n a m e ' , ' t i m e s ' , ' F o n t S i z e ' ,16) ; text (2.5 ,19 , 'M = 4 ' , ' f o n t n a m e ' , ' t i m e s ' , ' F o n t S i z e ' ,16) ; x l a b e l ( ' \ t h e t a ' , ' f o n t n a m e ' , ' t i m e s ' , ' F o n t S i z e ' ,16) ; y l a b e l ( ' p_ {2}/ p_ {1} ' , ' f o n t n a m e ' , ' t i m e s ' , ' F o n t S i z e ' ,16) ; a n n o t a t i o n ( ' t e x t a r r o w ' ,[0.55 ,0.6] ,[0.8 ,0.84] , ' S t r i n g ' , ' S t r o n g o b l i q u e s h o c k ' , ' f o n t n a m e ' , ' t i m e s ' , ' F o n t S i z e ' ,16) ; a n n o t a t i o n ( ' t e x t a r r o w ' ,[0.55 ,0.6] ,[0.8 ,0.67]) ; a n n o t a t i o n ( ' t e x t a r r o w ' ,[0.55 ,0.5] ,[0.16 ,0.21] , ' S t r i n g ' , ' Weak o b l i q u e s h o c k ' , ' f o n t n a m e ' , ' t i m e s ' , ' F o n t S i z e ' ,16) ; a n n o t a t i o n ( ' t e x t a r r o w ' ,[0.55 ,0.6] ,[0.16 ,0.31]) ; a n n o t a t i o n ( ' t e x t a r r o w ' ,[0.66 ,0.62] ,[0.345 ,0.345] , ' S t r i n g ' , ' \ t h e t a _ { max } ' , ' f o n t n a m e ' , ' t i m e s ' , ' F o n t S i z e ' ,16) ; annotation ( ' textarrow ' ,[0.66 ,0.66] ,[0.345 ,0.44]) ; text (34 ,15.2 , '\ t h e t a _ { max } \ l e f t a r r o w ' , ' f o n t n a m e ' , ' t i m e s ' , ' F o n t S i z e ' ,16) ; hold off

Fig. 5.18 The pressure-deflection (p − θ) plot by the MATLAB program

5.15 Compression and Expansion Waves

243

5.15 Compression and Expansion Waves 5.15.1 Supersonic Flow Around a Concave Corner: Isentropic Compression by Turning We have stated in Sect. 5.9 that turning a wall through an inclination angle θ (i.e., a sharp concave corner deflecting the supersonic flow into itself) produces an oblique shock wave of sufficient strength to turn the flow through this deflection angle (Fig. 5.9a). The flow is compressed as it passes through this shock wave, resulting in a sudden increase in pressure, temperature, and density, but a drop in velocity. Furthermore, when the flow passes through a sufficiently strong shock wave, as depicted in Fig. 5.9a, significant losses occur. Consequently, the flow experiences a finite change in entropy across the shock wave. In order to reduce the losses and hence the entropy change, it can be envisaged that if the single corner of Fig. 5.19a is replaced with a succession of small corners as represented in Fig. 5.19b, so that θ = nθ , we observe that the flow is now compressed through successive weak oblique shocks. The flow between successive shocks is uniform. Thus, in a supersonic flow, if the flow deflection angle θ is divided into n corners, each corner causing the flow to turn by a small angle θ, the pressure and entropy rise across each wave are, respectively, ∂p ∼ θ ∂s ∼ (θ )

(5.171) 3

(5.172)

and for the total deflection of Fig. 5.19b they are p ∼ nθ s ∼ n (θ )3 =

(5.173) θ n2

3

(5.174)

Therefore, if a large number of weak shock waves induce the compression, the entropy rise is greatly reduced compared to a single shock wave of sufficient strength for the same deflection angle θ . The continuous reduction in θ results in a smooth turn with θ → 0 in the limit. As a result, the entropy increase is virtually zero, and the compression is nearly isentropic. This means that if the number of turns in Fig. 5.19b is infinitely large or if the segments are extremely small, we obtain a smooth curving surface in the limiting situation as illustrated by Fig. 5.19c. In this case, the shocks are weak and the surface produces an infinite family of Mach waves of sufficient strength that the required flow curvature conforms to the solid boundary. According to Eq. (5.174), in this situation, the overall entropy change is infinitesimal, and hence, on a smooth curved surface, the compression is isentropic. In a compressive flow, the Mach number decreases as successive Mach waves form increasingly large angles with the flow direction (Fig. 5.19d). Consequently, as

244

5 Wave Phenomena

Fig. 5.19 Supersonic flow around sharp and smooth concave corners

seen in Fig. 5.19d, the Mach waves tend to converge towards each other and generate an envelope. Because of the convergence of the Mach waves, the gradient of flow properties becomes considerable and the conditions are no longer isentropic, resulting in the formation of an oblique shock wave. It may, therefore, be noted that conditions near the wall region (owing to Mach waves) are isentropic, but once sufficiently far away from the wall, they are not isentropic (due to oblique shock wave).

5.15.2 Supersonic Flow Around a Convex Corner: Isentropic Expansion by Turning Figure 5.20 depicts a supersonic flow over a convex corner of a wall, indicating that the wall is deflected away from the oncoming flow. As seen in Fig. 5.20a, the deflection is induced by a single oblique wave (a). It can be seen that, v1n = v1t tan (β − θ )

(5.175)

v2n = v2t tan β

(5.176)

Since tan β > tan (β − θ ) and v1t = v2t ; hence v2n > v1n

(5.177)

That is, the normal velocity component upstream of the wave exceeds the normal velocity component downstream of the wave. Because the tangential component of

5.15 Compression and Expansion Waves

245

Fig. 5.20 Supersonic flow around sharp and smooth convex corners

velocities across the oblique wave is equal, this result is readily observed in Fig. 5.20a. This type of flow configuration, however, is not possible according to the second law of thermodynamics since it causes a decrease in entropy across the wave. The fact that a Mach wave family tends to converge and form envelopes has already been seen in Fig. 5.19d. Since the flow through successive Mach waves is isentropic, thermodynamic expansion in the reverse direction is feasible. That is why a succession of Mach waves emanate from the corners and cause declining flow gradients through these waves when supersonic flow turns around a convex corner. All of the Mach waves, in this case, are centered on the corner, generating an expansion fan as seen in Fig. 5.20b. Flow conditions such as velocity, Mach number, and pressure remain constant throughout each Mach wave, but vary with the angular position at the corner. The expansion fan is a continuous expansion zone in which a supersonic flow passes through a countless number of Mach waves, each of which has a unique Mach angle μ with respect to the oncoming flow direction.  The  expansion 1 −1 with respect fan is bounded upstream by a Mach wave at an angle of μ1 = sin M1 to the oncoming flow, and it is bounded downstream by another Mach wave at an   1 −1 with respect to the oncoming flow, as shown in Fig. 5.20b. angle of μ2 = sin M2 It is worth mentioning that the supersonic flow expansion through a centered expansion fan is isentropic everywhere except at the fan’s vertex.3 However, if the turning is gradual, the Mach waves will be separated, as seen in Fig. 5.20c. Consequently, in this case, the expansion becomes isentropic everywhere, including at the 3

A large number of Mach waves combine at the vertex of the expansion fan, resulting in a substantial change in flow properties.

246

5 Wave Phenomena

wall. When it comes to changes in flow properties, however, it makes no difference whether the turn is sharp or smooth, until the maximum turning angle increases. Because the final properties largely depend on the overall deflection angle, a gradual turn simply indicates that the expansion takes place over a greater distance.

5.16 The Prandtl-Meyer Function We saw in Sect. 5.9 that if a supersonic flow turns into itself (around a concave corner of the wall), an oblique shock wave is formed (Fig. 5.9a); but, if the flow turns away from itself (around a convex corner of the wall), another type of waves termed an expansion fan is formed (Fig. 5.9b). The expansion fan formed at a sharp convex corner, as seen in Fig. 5.9b, is referred to as a centered expansion fan. Ludwig Prandtl, a German fluid dynamicist and physicist, and his student Theodor Meyer were the first to develop the theory of the centered expansion fan, therefore these waves are also known as Prandtl-Meyer expansion fans. The following are some of the unique characteristics of an expansion fan: • The streamlines that pass through an expansion fan are smooth curved lines. • An expansion fan is a continuous expansion zone made up of several Mach waves. Each Mach wave has a unique angle to the incoming flow direction, which is called the Mach angle μ. • There is no entropy change across a Mach wave. • As the Mach number increases, flow parameters such as static pressure, static temperature, and static density decrease across an expansion fan. The working formulae for calculating the change in flow properties of a perfect gas through an oblique shock wave were derived in Sect. 5.9.2. In this section, we will devise a method for calculating the changes in flow properties across an expansion fan. Consider a single Mach wave, as shown in Fig. 5.21. We have already mentioned that the flow conditions along any Mach line are constant and the flow is isentropic; thus, for a given set of initial conditions, we may develop a unique relationship between θ and any of the parameters v, M, or p. We may write the continuity equation and the momentum equation from Fig. 5.21 for the direction parallel to the wave, respectively, as follows: ρvn = (ρ + δρ) (vn + δvn ) ρvn vt = (ρ + δρ) (vn + δvn ) (vt + δvt )

(5.178) (5.179)

Combining Eqs. (5.178) and (5.179) results in δvt = 0

(5.180)

Equation (5.180) demonstrates that the velocity component in the direction parallel to the wave remains constant. Velocity triangles in Fig. 5.21 provide

5.16 The Prandtl-Meyer Function

247

Fig. 5.21 Prandtl-Meyer flow (a centered expansion fan at the convex corner of the wall)

vt = v cos μ = (v + δv) cos (μ − δθ )

(5.181)

v cos μ = (v + δv) (cos μ cos δθ + sin μ sin δθ )

(5.182)

Since δθ is very small, we have sin δθ ≈ δθ

(5.183)

cos δθ ≈ 1

(5.184)

v cos μ = (v + dv) (cos μ + dθ sin μ)

(5.185)

Then Eq. (5.182) becomes

v cos μ = v cos μ + vdθ sin μ + dv cos μ + dvdθ sin μ

(5.186)

When higher order terms are dropped, the above relationship is reduced to dv dθ = − cot μ v  dv dθ = − M2 − 1 v

(5.187) (5.188)

This is the governing differential equation for the Prandtl-Meyer flow. If the relationship between v and M is known, the integral in Eq. (5.188) can be readily evaluated. This relationship can now be determined by examining the expressions, respectively, v= aM

a02 =1+ a2



 γ −1 M2 2

(3.50)

(4.60)

248

5 Wave Phenomena

Taking log on both sides of Eq. (3.50) and differentiating, we have dv da dM = + v a M

(5.189)

Differentiating Eq. (4.60) with respect to M gives    −2 γ −1 γ −1 2 Ma0 1 + M2 dM ada= − 2 2 

(5.190)

By dividing the above relationship by Eq. (4.60) and simplifying the terms, we get 

γ −1 2



M2 dM da   =− a M2 M 1 + γ −1 2

(5.191)

Combining Eqs. (5.189) and (5.191) yields dv 1 dM   = γ −1 v 1 + 2 M2 M

(5.192)

Substituting Eq. (5.192) into Eq. (5.188) and integrating, we get ˆ −θ + Constant =

√ M2 − 1 dM   = ν (M) M2 M 1 + γ −1 2

(5.193)

where the integral ν (M) is evaluated as follows: √ √ ˆ M2 − 1 dM M2 − 1 dM2 1     ν (M) = = 2 2 M2 M M2 M 1 + γ −1 1 + γ −1 2 2   ⎤ ⎡ γ +1 ˆ 2 1 1 1 ⎣   ν (M) = − 2 ⎦ dM2 √ 2 M M2 − 1 1 + γ −1 M2 ˆ

(5.194)

(5.195)

2

By substituting t2 = M2 − 1, the equation above becomes ˆ ν (M) =

⎡

⎤ tdt tdt ⎣     −  ⎦  γ −1 t 1 + t2 2 + t t 1 + γ −1 2 2

(5.196)

5.16 The Prandtl-Meyer Function

249

After the basic integration, we obtain 2

t (γ −1) ν (M) = tan−1 − tan−1 t 2 2 1 + 1 + (γ −1) (γ −1) # #   γ +1 γ −1 2 tan−1 M − 1 − tan−1 M2 − 1 + constant = γ −1 γ +1

(5.197)

(5.198)

The integration constant is selected arbitrarily such that ν (1) = 0. Thus, Eq. (5.198) finally becomes # #   γ +1 γ −1 2 tan−1 M − 1 − tan−1 M2 − 1 (5.199) ν (M) = γ −1 γ +1 Equation (5.199) expresses the Prandtl-Meyer function ν in terms of the Mach number M. It is a significant result of supersonic flow, which reveals that there is a unique ν for a given M. Physically, ν can be defined as the angle through which a flow turns isentropically from sonic velocity (M1 = 1) to a Mach number greater than 1. Because calculating ν is tedious, the values of M and the corresponding ν for air calculated from Eq. (5.199) are tabulated in the isentropic table given in Appendix B. The table shows that when M varies from 1 to ∞, ν takes values from 0 to νmax , i.e., ν achieves a maximum when M → ∞, and this maximum is ) (# γ +1 π −1 (5.200) νmax = 2 γ −1 = 130.5◦ for γ = 1.4

(5.201)

Theoretically, the flow starts at M1 = 1 and expands to a vacuum, i.e., M → ∞ (Fig. 5.22). In other words, when a flow at Mach 1 is deflected by an angle 130.5◦ , the static pressure drops to zero and an infinite Mach number is generated. Thus, there is a maximum angle that the flow can negotiate for a given upstream Mach number, M1 . According to current theory, if the expansion occurs at an angle greater than νmax , a vacuum will form near the wall, as shown in Fig. 5.22. Of course, in reality, the continuum hypothesis and the perfect gas assumptions are rendered invalid well before this situation arises. The flow conditions for a uniform supersonic flow expanded around a sharp corner, as in Fig. 5.20b, can be determined using Eq. (5.199). If the original flow corresponds to M1 = 1, Eq. (5.199) produces the local Mach number M2 = M after expansion, which corresponds to the angle of turn θ . However, if the original flow corresponds to M1 > 1, Eq. (5.193) must be used to calculate the appropriate constant. Although Eq. (5.199) can be applied to any two arbitrary points within an expansion fan, it is most commonly used to relate the flow conditions upstream and downstream of the wave. If the wall inclination at the convex (expansion) corner is θ , the downstream Prandtl-Meyer function ν2 is obtained by adding θ to the upstream Prandtl-Meyer function ν1 , i.e.

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5 Wave Phenomena

Fig. 5.22 A flow begins at M = 1 and expands into a vacuum

Fig. 5.23 Prandtl-Meyer flow (non-centered expansion fan)

ν2 = ν1 + θ

(5.202)

Using this ν2 , the corresponding Mach number M2 can be found from the isentropic table given in Appendix B. Because the flow through the expansion fan is isentropic, the flow properties may be calculated using M2 from the isentropic relations. Furthermore, Eq. (5.199) can be used to calculate the flow conditions of a supersonic flow expanded continuously over a curved surface, as seen in Fig. 5.23. For an initial Mach number M1 , the local Mach number at any point P on the wall in a flow field dominated by an expansion fan can be measured by determining the angle of turn between the original flow direction and the tangent at the point P.

5.16.1 Important Remarks The Prandtl-Meyer flow theory finds its application in the areas of supersonic airfoils, convergent-divergent nozzles, ballistic missiles, etc. Supersonic airfoil is one of the most important applications in aerodynamics. An airfoil experiences the

5.16 The Prandtl-Meyer Function

251

Prandtl-Meyer expansion fan when it is at an angle of attack to a supersonic flow. A convergent-divergent nozzle experiences expansion fan when it operates under off-design conditions. The expansion fan on a ballistic missile is typically formed at the rear shoulder of the missile. For computation purposes, Eq. (5.202) can be rewritten with a positive angle +θ for a compression turn (concave corner) and a negative angle −θ for an expansion turn (convex corner). The resulting equations after determining the required integration constants are as follows: ν2 = ν1 + |θ − θ1 | = ν1 + θ ν2 = ν1 − |θ − θ1 | = ν1 − θ

at convex corner

(5.203)

at concave corner

(5.204)

Example 5.7 Using the Prandtl-Meyer flow theory, determine the Mach number downstream of an oblique wave in a uniform airstream at Mach 2.5 if the latter passes through (i) a concave corner with a deflection angle of 20◦ and (ii) a convex corner with a deflection angle of 20◦ . Solution For M1 = 2.5, from the isentropic table, ν1 = 39◦ (i) At the concave (compression) corner, Eq. (5.204) gives ν2 = ν1 − θ = 39 − 20 = 19◦ Therefore, M2 = 1.74 (ii) At the convex (expansion) corner, Eq. (5.203) provides ν2 = ν1 + θ = 39 + 20 = 59◦ Hence, M2 = 3.53

252

5 Wave Phenomena

5.17 Simple and Non-simple Regions The Prandtl-Meyer function for isentropic compression and expansion waves, as presented in Sect. 5.16, can be summarized as follows: ν2 = ν1 ± θ

(5.205)

where the positive sign (+) implies that the flow is turned away from itself, indicating an expansion, while the negative sign (−) implies that the flow is turned into itself, indicating a compression. Each of these waves (compression and expansion) is referred to as a simple wave. Straight Mach lines with constant flow properties characterize a simple wave, as does the simple relation (Eq. (5.205)) between the flow deflection angle, θ , and the Prandtl-Meyer function, ν. A simple wave is associated with one of two families, (+ or −), based on whether the wall that forms it is to the left or right of the flow, as shown in Fig. 5.24. These simple waves are also called the characteristic lines, are simply characteristics. Thus, a characteristic with a +sign is referred to as a left-running characteristic, and a characteristic with a −sign is referred to as a right-running characteristic. In the literature, these left- and right-running characteristics are also called η− and ξ −characteristics, respectively. The angle subtended by a characteristic with respect to the flow at any point is equal to the Mach angle, μ = sin−1 (1/M), where M is the local Mach number. The flow is non-simple in regions where two simple waves of opposite families interact, i.e. the relationship between ν and θ given by Eq. (5.205) is not applicable

Fig. 5.24 Simple and non-simple regions in an isentropic supersonic flow

5.17 Simple and Non-simple Regions

253

to this region. Such a non-simple flow region is often analyzed with the method of characteristics.4 Remark It is worth noting that, like the aforementioned characteristic waves, Mach waves are classified as left- or right-running based on whether the wall from which they arise is on their left or right in the flow field. Thus, the Mach waves of opposing families cross each other. Even after passing through a number of similar waves, a Mach wave, being the weakest degeneration of waves, would propagate as a linear wave. In other words, even after interacting with each another, the Mach wave remains a simple wave. Consequently, a region encompassed by the Mach waves is known as a simple region.

5.18 Supersonic Flow Past Wedges and Conical Bodies The previously described oblique shock characteristics apply to an oblique shock wave that forms in a two-dimensional compression corner. These concepts are extended in this section to a stationary two-dimensional (planar) wedge and a stationary three-dimensional conical body (cone) submerged in uniform supersonic flow. Because the shock wave in the latter also has a circular cone shape, the flow configuration frequently resembles a two-dimensional axisymmetric flow. In a supersonic wind tunnel, wedges and cones are frequently employed to measure the flow speed in the test section.

5.18.1 Flow over a Planar Wedge: The Two-Dimensional Case It was noticed that a Mach wave, which is formed by a tiny disturbance in fluid flows, is indeed the covering of many weaker wavelets, which are emitted by the disturbance traveling through the fluid to a velocity higher than the speed of sound, and the flow neither changes its direction through the wave nor changes its property. When the point disturbance is replaced with a finite object, such as a two-dimensional (planar) wedge as shown in Fig. 5.25a, an oblique shock is created and attached to the vertex of the wedge, causing the flow to abruptly change its direction and undergo a finite change in properties. Furthermore, as illustrated in Fig. 5.25a, the flow downstream of the shock becomes parallel to the surface of the wedge. The parallel streamlines upstream of the shock wave remain parallel downstream as well, but the distance between any two streamlines reduces. As a result, the density or velocity of the flow behind the shock wave must rise in order for the mass flow to remain constant 4

Readers interested in this topic should consult a more specialized literature, such as “Shapiro, A.H., Dynamics and Thermodynamics of Compressible Fluid Flow, 2 Vols., Ronald Press, New York, 1953.”

254

5 Wave Phenomena

Fig. 5.25 A symmetrical wedge in the supersonic flow

  m ˚ =ρAv . It should be remembered that in a low-speed subsonic flow, the density remains basically constant, hence the velocity must increase as the spacing between the streamlines decreases. The pressure distribution on the surface is, therefore, determined by the incompressible Bernoulli’s equation. However, the conditions are substantially different in supersonic flows. The velocities upstream and downstream of an oblique shock can be resolved into tangential and normal components for the flow through an oblique shock, as shown in Fig. 5.25b. It is demonstrated in Sect. 5.9.1 that v1t = v2t

(5.80)

This means that the tangential velocity components across an oblique shock are constant. Hence, the velocity change through the shock wave is caused only by a change in the normal component.

5.18 Supersonic Flow Past Wedges and Conical Bodies

255

Furthermore, due to the abrupt change in flow direction and the conditions behind the shock, the shock angle β is larger than the freestream Mach angle μ1 , indicating that β > μ1 . Eventually, the normal component of freestream velocity, v1n , exceeds the speed of sound in the freestream, i.e., v1n > a1 or M1n > 1. As previously stated, an oblique shock wave can be regarded as a normal shock with respect to M1n , and supersonic flow through the latter becomes subsonic. As a result, the normal component of the Mach number behind the oblique shock, M2n , will be smaller than one left (M2n < 1). Now, because v2n < a2 , the shock angle β is less than the Mach angle behind the shock, i.e., β < μ2 . The shock angle, therefore, lies in the range μ1 < β < μ2

(5.206)

The analysis above implies that the airflow passes through an oblique shock, in such a way that the acute angle between the streamline and the shock wave is decreased, that is to say, it deflects from the normal direction to the wave. Figure 5.25c depicts the relationship between the shock angle β and the Mach angles μ1 and μ2 . Because the normal component of velocities governs pressure and density changes across the shock, a more general conclusion may be drawn that pressure, temperature, and density increase while velocity and Mach number decrease across an oblique shock wave. These conditions, which were previously illustrated in Sect. 5.9.2, describe the exact solution for flow over a wedge or a two-dimensional compression corner. Indeed, there is a similarity between flow over a wedge and flow over a corner, which has a turning angle equal to the semi-vertex angle of the wedge; it is for this reason that flow over a wedge is commonly referred to as flow “in a corner.” It is worth noting that the shock angle β and pressure ratio across the wave p2/p1 are solely determined by the initial Mach number M1 and flow deflection angle θ . The variation of β with M1 for different values of θ is shown in Fig. 5.25d. It is likewise important to note that the flow downstream of an oblique shock remains supersonic except for a very small range of Mach numbers and flow deflection angles (semi-vertex angle of a wedge) in which the oblique shock is essentially strong. Example 5.8 A two-dimensional symmetrical wedge of the total included angle of 40◦ is kept in an airstream at Mach 2.5 at sea level conditions. If the shock angle is 38◦ , find the tangential component of the flow Mach number. Solution Given, 2 = 40◦

M∞ = 2.5

β = 38◦

Because the tangential component of the flow velocity across an oblique shock remains constant, so does the tangential component of the Mach number, i.e., M1t = M2t . From the velocity triangle depicted in Fig. 5.25b upstream of the shock, we have

256

5 Wave Phenomena

v1t a1 v1 cos β = a1 = M1 cos β M1t =

= 2.5 × cos (38◦ ) = 1.97 Example 5.9 A stationary two-dimensional wedge with an included angle of 20◦ is kept in a Mach 1.8 airstream. (i) Determine the possible shock angles. (ii) Which one is most likely? (iii) What is the greatest angle at which the flow can turn without being separated from the wedge? Solution Given, M∞ =1.8

and

2 = 20◦

(i) For M1 = 1.8 and θ =  = 10◦ , from the oblique shock chart, β≈42◦

and

β≈83◦

(ii) It is worth noting that, for a given M∞ and θ , the shock strength increases as β increases, with a lower value corresponding to a weak shock. Therefore, in this situation, the shock angle is β≈42◦ (iii) For M1 = 1.8, from the oblique shock chart, the maximum deflection angle θmax is θmax = 21◦ Recall that, θmax is the maximum angle possible at which the flow can turn through an oblique shock with shock remain attached to the vertex of the wedge. In this particular case, θ < θmax , and therefore the shock will remain attached to the wedge.

5.18.2 Flow over a Conical Body: The Three-Dimensional Case The supersonic airstream passing over a conical body, as shown in Fig. 5.26a, has some similarities and some differences with the two-dimensional wedge discussed in Sect. 5.18.1. For the same reasons, a straight oblique shock wave is formed and

5.18 Supersonic Flow Past Wedges and Conical Bodies

257

Fig. 5.26 A conical body in the supersonic flow

attached to the apex of the body in this situation, and the air has no knowledge about the body until it reaches the wave. The properties of air after entering the zone of influence of the conical body, on the other hand, are completely different. Of course, oblique shock relations are still used to compute the properties downstream of the shock wave. To begin, it is important to understand that supersonic flow over a shock wave deflects in a plane normal to the initial flow direction, rather than in only one plane as in the case of two-dimensional flow over a wedge. Because the flow field between the shock wave and the surface of the body is essentially three-dimensional, it is no longer uniform as in the case of a wedge. In reality, the presence of a third dimension allows the flow to relax from the barrier caused by the body. This phenomenon is known as the “three-dimensional relieving effect,” and it

258

5 Wave Phenomena

is a feature of all three-dimensional flows in nature. Therefore, in three-dimensional flows, the shock angle β for the same flow turning angle θ and freestream Mach number M∞ is much less than in two-dimensional flows. Figure 5.26b depicts the variation of β with M∞ at various θ for flow over a conical body. It is worth noting that a decrease in β for the same θ and M∞ implies the formation of a “weaker-shock” over the conical body than a wedge with the same included angle. As a result, the rise in pressure, temperature, and density is less than in the two-dimensional situation. Furthermore, as previously stated, the conditions between the shock wave and the surface of the body are constant in a two-dimensional situation, and hence the Mach lines are straight. In the three-dimensional situation, however, the conditions between the shock wave and the surface of the conical body change, causing the Mach lines to be curved. These curved lines are known as characteristic curves, and they are depicted in Fig. 5.26a by dashed lines. Note that, the characteristic curves subtend the Mach angle μ with the streamlines corresponding to the local Mach number at any location in the flow field. Another difference in flow characteristics between the two-dimensional and threedimensional cases is that, due to the extra degree of freedom experienced by the flow, the streamlines no longer turn immediately behind the shock and travel parallel to the surface, but rather approach it in an asymptotic manner, as shown in Fig. 5.26a. The third important difference in flow characteristics between the two-dimensional and the three-dimensional situations is that the flow properties are not constant between the shock wave and the surface of the body. In reality, following an initial rise across the wave, the pressure, temperature, and density continue to rise further downstream, but the velocity and Mach number drop. The pressure along any conical line emanating from the apex of the conical body, however, remains constant (Fig. 5.26c).

5.19 Shock-Expansion Theory: The Supersonic Airfoils The shock-expansion theory is made up of the oblique shock relations and the PrandtlMeyer flow relations, which may be used to determine the flow conditions and forces on simple two-dimensional geometries made up of straight-line segments in supersonic flows. This section examines two basic geometries: a flat plate and a symmetrical diamond-shaped airfoil immersed in uniform supersonic flows.

5.19.1 A Flat Plate Supersonic Airfoil The flat plate is the most basic airfoil design. Consider a flat plate in an inviscid supersonic flow with an angle of attack alpha, as shown in Fig. 5.27a. In this case, the leading edge functions as a convex corner to the flow passing through the top surface, with a flow deflection angle of θ = α. As a result, the flow at the upper

5.19 Shock-Expansion Theory: The Supersonic Airfoils

259

Fig. 5.27 A flat plate airfoil in the supersonic flow

surface passes through the resultant Prandtl-Meyer expansion fan, increasing its Mach number from M1 = M∞ to M2 = Mu . We can use Eq. (5.202) to calculate Mu for a given M∞ . Once the downstream Mach number, Mu , is determined, the isentropic relation is used to calculate the corresponding static pressure at the upper surface, pu :  ⎤ γ γ−1 ⎡ γ −1 2 1 + M ∞ 2 pu ⎦ =⎣ γ −1 2 p∞ 1 + 2 Mu

(5.207)

The leading edge, on the other hand, serves as a concave corner to the lower surface flow. This causes an increase in static pressure across the resultant oblique shock wave. The Mach number downstream of the shock, Ml , and the corresponding static pressure, pl , may be computed from the oblique shock relations using M∞ and θ. As illustrated in Fig. 5.27b, the pressure difference between the upper and lower surfaces of the flat plate creates a resultant force R (per unit span) acting normal to the plate. The components of R in the directions; normal to and along the flow are, by definition, called the lift L (per unit span) and the drag D (per unit span), respectively. From Fig. 5.27b, we have R = (pl − pu ) c L = (pl − pu ) c cos α D = (pl − pu ) c sin α

(5.208) (5.209) (5.210)

Example 5.10 Consider a flat plate in a uniform airstream at Mach 3 with a 20◦ angle of attack to the freestream direction, as shown in Fig. 5.28. The freestream

260

5 Wave Phenomena

Fig. 5.28 A flat plate in a supersonic flow

pressure and temperature are 100 kPa and 298 K. Calculate the flow properties in the regions depicted in the figure using shock-expansion theory. What is the slipstream’s inclination to the freestream direction? Solution Given, p-\ theta M∞ = M1 = 3.0,

p∞ = p1 = 100 kPa,

T∞ = T1 = 298 K,

θ = α = 20◦ ,

γ = 1.4

The freestream velocity is  v1 = γ RT1 √ = 1.4 × 287 × 298 m = 346.03 s Assuming air to be a perfect gas, the density of air becomes: p1 RT1 100 × 103 = 287 × 298 kg = 1.17 3 m ρ1 =

(i) Consider the flow from region 1 to region 2 across the expansion wave. For M1 = 3.0, from the isentropic table:

5.19 Shock-Expansion Theory: The Supersonic Airfoils

p1 = 0.0272 p01 T1 = 0.3571 T01 ρ1 = 0.0762 ρ01 ν1 = 49.757◦ From the Prandtl-Meyer relation, ν2 = ν1 + θ = 49.757 + 20 = 69.757◦ For ν2 = 69.757◦ , from the isentropic table: M2 = 4.32 p2 = 0.0043 p02 T2 =0.2113 T02 ρ2 =0.0205 ρ02 Hence, p2 p02 × × p1 p02 p1 p2 p01 = × × p1 p02 p1 1 × 100 × 103 = 0.0043 × 0.0272 = 15.8 kPa p2 =

T2 T02 × × T1 T02 T1 T2 T01 = × × T1 T02 T1 1 × 298 = 0.2113 × 0.3571 = 176.32 K T2 =

261

262

5 Wave Phenomena

From the ideal gas law, p2 RT2 15.8 × 103 = 287 × 176.32 kg = 0.312 3 m ρ2 =

The flow velocity in region 2 is v2 =

=

 γ RT2

√ 1.4 × 287 × 176.32 m ≈ 266.17 s

(ii) Consider the flow from region 2 to region 3 across the oblique shock. For M2 = 4.32 and θ = 20◦ , from the oblique shock chart: β ≈ 31◦ Thus, M2n = M2 sin β = 4.32 × sin (31◦ ) ≈ 2.22 For M2n = 2.22, from the normal shock table: M3n = 0.5477 p3 = 5.5831 p2 T3 = 1.8746 T2 ρ3 = 2.9784 ρ2 Therefore, M3n sin (β − θ ) 0.5477 = sin (31◦ − 20◦ ) = 2.87 M3 =

5.19 Shock-Expansion Theory: The Supersonic Airfoils

263

p3 × p2 p2 = 5.5831 × 15.8 × 103 p3 =

= 88.21 kPa T3 × T2 T2 = 1.8746 × 176.32 T3 =

= 330.52 K ρ3 × ρ2 ρ2 = 2.9784 × 0.312 kg = 0.93 3 m ρ3 =

The flow velocity in region 3 is:

=

√

v3 =

 γ RT3

1.4 × 287 × 330.52 m ≈ 364.42 s

(iii) Consider the flow from region 1 to region 2 through the oblique shock wave at the bottom surface. For M1 = 3 and θ = 20◦ , from the oblique shock chart: β ≈ 38◦ The normal component of M1n is M1n = M1 sin β = 3 × sin (38◦ ) ≈ 1.85 For M1n = 1.85, from the normal shock table: M2 n = 0.6057 p2 = 3.8262 p1 T2 = 1.5693 T1

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5 Wave Phenomena

ρ2 = 2.4381 ρ1 p02 = 0.7902 p01 Hence, M2 n sin (β − θ ) 0.6057 = sin (38◦ − 20◦ ) = 1.96

M2 =

p2 × p1 p1 = 3.8262 × 100 × 103 = 382.62 kPa p2 =

T2 × T1 T1 = 1.5693 × 298 = 467.65 K T2 =

ρ2 × ρ1 ρ1 = 2.4381 × 1.17 kg = 2.85 3 m ρ2 =

p02 p01 × × p1 p01 p1 1 × 100 × 103 = 0.7902 × 0.0272 = 2905.15 kPa p02 =

5.19 Shock-Expansion Theory: The Supersonic Airfoils

T02 = T01 =

265

T01 × T1 T1

1 × 298 0.3571 = 834.5 K

=

The flow velocity in region 2 is:

=

√

v2 =

 γ RT2

1.4 × 287 × 467.65 m ≈ 433.48 s

(iv) Finally, consider the flow from region 2 to region 3 across the expansion fan. For M2 = 1.96, the isentropic table yields: ν2 = 25.27◦ From the Prandtl-Meyer relation, ν3 = ν2 + θ = 25.27 + 20 = 45.27◦ Again from the isentropic table, we have M3 ≈ 2.78 p3 = 0.038 p03 T3 =0.3928 T03 Therefore, p3 =

p3 × p03 p03

p3 × p02 p03 = 0.038 × 2905.15 × 103 ≈ 110.4 kPa =

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5 Wave Phenomena

T3 × T03 T03 T3 = × T02 T03 = 0.3928 × 834.5 T3 =

≈ 327.8 K p3 RT3 110.4 × 103 = 287 × 327.8 kg ≈ 1.17 3 m ρ3 =

and  v3 = γ RT3 √ = 1.4 × 287 × 327.8 m ≈ 362.92 s Remark The calculations above show that the velocity jump across the slipstream (i.e., between the regions 3 and 3 ) is v = v3 − v3 = 364.42 − 362.92 m = 1.5 s Because this value is very small, the slipstream inclination to the freestream direction is negligible.

5.19.2 A Diamond-Shaped Supersonic Airfoil Let us now consider a symmetrical diamond-shaped airfoil in uniform supersonic flow, as shown in Fig. 5.29. An attached oblique shock is created at the leading edge of the airfoil if the semi-vertex angle of the airfoil, , does not exceed the maximum flow deflection angle, θmax , for the specified freestream Mach number, M∞ . Thus, the leading edge acts as a concave corner, compressing the incoming freestream by means of an oblique shock. The static pressure rise across the shock can be calculated using the oblique shock relations and the inputs: M∞ , θ and the corresponding freestream

5.19 Shock-Expansion Theory: The Supersonic Airfoils

267

Fig. 5.29 A symmetrical diamond-shaped airfoil in the supersonic flow

static pressure p∞ . Because the pressure ratio across an oblique shock is proportional to the flow deflection angle, the static pressure downstream of the shock, p2 , remains constant from the leading edge to the airfoil shoulder. In this case, the shoulder acts as a convex corner, turning the flow through a centered expansion fan generated at the shoulder. The static pressure downstream of the expansion fan drops to a constant value, p3 , which is obtained implicitly from the Prandtl-Meyer function. Finally, the trailing edge acts as a concave corner, recompressing the flow to freestream pressure, p∞ . The wave pattern and pressure distributions over the symmetrical diamond-shaped airfoil are depicted in Fig. 5.29a, b, respectively. Because the airfoil has a zero angle of attack to the freestream direction, the pressure distributions on the upper and lower surfaces of the airfoil are symmetrical. Consequently, there is no net force acting normal to the freestream direction. In other words, the diamond-shaped airfoil produces zero lift in supersonic flow when α = 0. However, as shown in Fig. 5.29b, the higher pressure p2 at the forward portion of the airfoil and the lower pressure p3 at the rearward portion produce a net force along the freestream direction, i.e., a drag

268

5 Wave Phenomena

Fig. 5.30 A symmetrical diamond-shaped airfoil in a Mach 1.8 flow

(per unit span), D = (p2 − p3 ) t

(5.211)

where t is the airfoil thickness at the shoulder. Equation (5.211) describes the drag induced by waves and is thus known as the wave drag. Example 5.11 Consider a symmetrical diamond-shaped airfoil at zero angle of attack to the airstream, with a Mach number of 1.8 and a pressure of 100 kPa, as shown in Fig. 5.30. The airfoil’s sides AB, BC, AD, and DC are all the same length, l = 1.5 m, and the airfoil has a maximum thickness of 0.3 m. Calculate the drag coefficient using the shock-expansion theory. Assume a unit length of 1 m in the spanwise direction. Solution Given, l = 1.5 m,

t =0.3 m,

α = 0◦ ,

M∞ = 1.8,

p∞ =100 kPa

The semi-vertex angle at the leading edge of the airfoil is:   t/2  = sin−1 l   0.15 = sin−1 1.5 = 5.74◦ (i) The flow at M1 = 1.8 is deflected at an angle of θ =  = 5.74◦ as it passes through the oblique shock (from zone 1 to zone 2). From the oblique shock table, we have

5.19 Shock-Expansion Theory: The Supersonic Airfoils

269

β ≈ 39◦ Hence, the normal component of shock upstream Mach number M1 becomes M1n = M1 sin β = 1.8 sin (39◦ ) = 1.13 For M1n = 1.13, from the normal shock table, we have p2 = 1.323 p1

M2n = 0.889, Thus,

M2n sin (β − θ ) 0.889 = sin (39◦ − 5.74◦ ) = 1.62 M2 =

and p2 p1 p1 = 1.323 × 100 = 132.2 kPa p2 =

For M2 = 1.62, the isentropic table gives ν2 = 15.4518◦ ,

p2 = 0.2284 p02

Therefore, p02 =

p2 p2/p02

132.2 0.2284 = 578.8 kPa =

The flow from zone 2 to zone 3 is turned by the angle 2θ = 11.48◦ through a centered expansion fan at the shoulder B. The Prandtl-Meyer function ν3 downstream of the fan is

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5 Wave Phenomena

ν3 = ν2 + 2θ = 15.4518 + 11.48 = 26.9318◦ For ν3 = 26.9318◦ , again from isentropic table, we have p3 = 0.1239 p03

M3 = 2.02, which gives

p3 p03 p03 = 0.1239 × 578.8 (p03 = p02 ) p3 =

= 71.71 kPa The drag coefficient can now be written as follows: CD =

D 1 2 S γ p∞ M∞ 2

where, S is the wing area. Because S =chord × span = c × b, thus CD =

D 1 2 γ p∞ M∞ 2 D/b

=

=

(c × b)

1 2 c γ p∞ M∞ 2 

D

1 2 c γ p∞ M∞ 2

The equation above can be expressed using Eq. (5.211) as CD = = =

1 2

(p2 − p3 ) t 1 2 c γ p∞ M∞ 2

(p2 − p3 ) t 1 2 (2 × l × γ p∞ M∞ 2

cos )

(132.2 − 71.71) × 103 × 0.3 × 1.4 × 100 × 103 × 1.82 × 2 × 1.5 × cos (5.74◦ ) ≈ 0.027

Remark We can see from the preceding calculations that the drag produced by both the flat plate airfoil and the diamond-section airfoil to a supersonic flow is finite. So

5.19 Shock-Expansion Theory: The Supersonic Airfoils

271

the d’Alembert’s paradox does not apply to two-dimensional supersonic flows. The drag encountered by supersonic airfoils is caused by viscous dissipation within the oblique shock waves and is, therefore, referred to as wave drag. In supersonic flows, the wave drag is an additional component that must be added to the skin friction drag and the induced drag in three-dimensional flows.

5.20 Supersonic Thin Airfoil Theory In Sect. 5.19, we discussed the shock-expansion theory, which gives a simple and general method for calculating lift and drag on supersonic airfoils. However, the accuracy of these parameters is subject to the attachment of shock and expansion waves to the airfoil. Even though the shock-expansion theory can easily calculate the properties, the results cannot be stated in a compact analytical form. The sole restriction of this theory is that it can only provide numerical solutions. If, however, the airfoil is considered thin and all flow inclinations, including the angle of attack, are small, the shock-expansion theory outcomes can be approximated by utilizing the approximate relations for weak shock and expansion waves. The aforementioned technique is commonly referred to as the thin airfoil theory, which yields easy analytical expressions for lift and drag. The pressure change across a weak oblique shock, as derived in Sect. 5.12, is given by γ M12 p θ ≈ p1 M12 − 1

(5.144)

where θ denotes the inclination relative to the freestream direction, and subscript 1 denotes the flow conditions upstream of the shock. Now, the pressure coefficient in compressible flow is   p2 − p1 2 p 2 = (4.164) Cp = 2 p1 γ M1 γ M12 p1 Substituting the expression for

p p1

into the above equation yields

Cp =

2θ

(5.212)

M12 − 1

Equation (5.212) is the fundamental relation for thin airfoil theory, which indicates that the pressure coefficient at any point in the flow is proportional to the local inclination of the flow to the freestream direction. Using Eq. (5.212), we can compute the lift and drag coefficients for various airfoil sections at different angles of attack. However, we will restrict ourselves to the three situations listed below.

272

5 Wave Phenomena

Case I: A Thin Flat Plate at a Small Angle of Attack to the Freestream Direction Consider a thin flat plate of the chord length, c, at a small angle of attack, α, in a supersonic flow, as shown in Fig. 5.27. The pressure coefficients at the upper and lower surfaces, respectively, are Cpu =

2α

(5.213)

M12 − 1

Cp = −

2α

(5.214)

M12 − 1

Thus, the lift and drag coefficients are  (p − pu ) c cos α  = Cp − Cpu cos α q1 c  (p − pu ) c sin α  CD = = Cp − Cpu sin α q1 c

CL =

(5.215) (5.216)

where q1 denotes the dynamic pressure in the freestream. Substituting Eqs. (5.213) and (5.214) into Eqs. (5.215) and (5.216), and applying the small-angle approximations, i.e., sin α ≈ α and cos α ≈ 1 to the resulting expressions yield CL =

4α

(5.217)

M12 − 1

CD =

4α 2

(5.218)

M12 − 1

Equations (5.217) and (5.218) are the lift and drag coefficients, respectively, for an inclined flat plate in the compressible flow. Case II: A Diamond-Shaped Thin Airfoil at Zero Degree Angle of Attack to the Freestream Direction Consider a diamond-shaped thin airfoil with a total included angle, 2, at zero degrees angle of attack to the freestream direction in a supersonic flow, as shown in Fig. 5.29. In this case, since the normal components of pressures on the top and bottom surfaces cancel each other, the lift of the airfoil is zero. However, the unbalanced components of pressures acting on the forward and rearward faces in the flow direction produce a drag:

5.20 Supersonic Thin Airfoil Theory

273

D = (p2 − p3 ) t

(5.211)

where p2 and p3 denote the pressures acting on the forward and rear faces of the airfoil, respectively. The above equation can be rewritten in terms of drag coefficient to give p2 − p3 CD = q1

   t t = Cpf − Cpr c c

(5.219)

Now, by using Eq. (5.212), the pressure coefficients on the forward and rearward faces of the airfoil are   2 t 2 (5.220) = Cpf = c M12 − 1 M12 − 1   2 t 2 Cpr = − = − (5.221) c M12 − 1 M12 − 1 Then Eq. (5.219) becomes  2 t CD = M2 − 1 c 4

(5.222)

1

Remark In the preceding two cases, the thin airfoil theory was applied to specific geometries of the airfoil section to compute the lift and drag coefficients. However, we can obtain a general result that applies to any airfoil shape, as shown below. Case III: A Thin Arbitrary-Shaped Airfoil at a Small Angle of Attack to the Freestream Direction Consider a thin arbitrary-shaped airfoil with finite thickness and camber at a small angle of attack in a supersonic flow, as shown in Fig. 5.31. The figure shows that the angle of attack is superimposed linearly on the camber and the thickness of the airfoil to form a thin arbitrary-shaped airfoil. According to thin airfoil theory, the pressure coefficients at the upper and lower surfaces of the airfoil are   2 dy 2θu = (5.223) Cpu = M12 − 1 M12 − 1 dx u   2 dy 2θ Cp = = − (5.224) M12 − 1 M12 − 1 dx 

274

5 Wave Phenomena

Fig. 5.31 Linear superposition of angle of attack, camber, and thickness into a thin arbitrary-shaped airfoil

    dy dy where the slopes dx and dx are obtained from the respective surface profiles. u  The upper and the lower surface profiles of the airfoil can be represented as the linear combination of a symmetrical thickness distribution, t (x), with zero camber, and a camber line, yc , of zero thickness. Hence, 

 dy dt dyc + = −α − αc (x) + = dx dx dx  u dy dt dyc − = −α − αc (x) − = dx  dx dx

dt dx dt dx

(5.225) (5.226)

where α denotes the angle of attack, and αc (x) denotes the angle of attack of the camber line of zero thickness. The expressions for lift and drag of the airfoil are ˆc L = q1



 Cp − Cpu dx

(5.227)

0

ˆc

 −Cp

D = q1 0

dy dx



 

+ Cpu

dy dx

 dx

(5.228)

u

Substituting Eqs. (5.223) and (5.224) into the above equations yields ˆc 4q1 L= (5.229) [α + αc (x)] dx M12 − 1 0  2  ˆc  2  2  ˆc  2q1 dy 4q1 dy dt {α + αc (x)}2 + D= + dx = dx dx dx 2 2  u M1 − 1 0 M1 − 1 0

(5.230)

5.20 Supersonic Thin Airfoil Theory

275

In the above expressions, the integrals may be replaced with the average values of the respective quantities. For instance, the integral of a total angle of attack, α + αc (x), experienced by the camber line of zero thickness is ˆc [α + αc (x)] dx = α + αc (x) = α + αc (x)

(5.231)

0

However, by definition, αc (x) = 0; thus ˆc [α + αc (x)] dx = α + αc (x) = α

(5.232)

0

Likewise, [α + αc (x)]2 = α 2 + αc2 (x) + 2ααc (x) == α 2 + αc2 (x) + 2ααc (x)

(5.233)

Thus, [α + αc (x)]2 = α 2 + αc2 (x)

(5.234)

Substituting Eqs. (5.232) and (5.234) into Eqs. (5.229) and (5.230), we obtain CL =

CD =



4 M12

−1

4α M12 − 1

α 2 + αc2 (x) +

(5.235) 

dt dx

2  (5.236)

Equations (5.235) and (5.236) are the general expressions for the lift and drag of a thin arbitrary-shaped airfoil at a small angle of attack in a supersonic flow. It is worth mentioning that the wave drag of an arbitrary-shaped airfoil comprises three components: drag caused by lift on the airfoil, drag caused by the camber of the airfoil, and drag caused by the thickness of the airfoil.

5.21 Reflection and Interaction of Oblique Shock Waves As seen in the preceding sections, when a supersonic flow approaches a concave corner, an oblique shock wave is formed at the corner, causing the flow to turn

276

5 Wave Phenomena

towards the latter. The rules governing the reflection and interaction phenomena of these shock waves are discussed in this section.

5.21.1 Reflection from a Solid Boundary Under certain conditions, such as the models mounted in the supersonic wind tunnel, the oblique shock waves reach the test-section wall, where they must either reflect or be dissipated. The flow passing through the shock wave must be turned in the direction of the wall. As it flows in the direction of the wall, a second shock wave must be created to return it to its original direction. We must note two important points in this case: 1. Since the Mach number downstream of the first oblique shock is reduced, the reflected shock does not have the same angle as the shock that created it; rather, the shock angle is increased, i.e., M2 < M1 ; hence β2 > β1 . 2. Although the final direction is parallel to the original direction, the final conditions are not the same as they would be if the air had been deflected twice in the same direction by values equal to the two alterations in the case of the wall. Consider a supersonic flow parallel to a wall that suddenly encounters a turning angle, θ , at point A, as shown in Fig. 5.32. Assuming that, θ < θmax , the sudden change in flow direction, i.e., the flow turning into itself, creates an oblique shock wave with a shock angle β1 . Downstream of the shock wave, to satisfy the boundary condition near the lower wall, the streamlines in region 2 are inclined at an angle θ . The Mach number, M1 , and the flow turning angle, θ , uniquely define the flow properties and shock wave strength in region 2. The shock originated at point A further extends into the flow and impinges on a solid wall at point B. This impinging shock wave is often called the incident shock wave. The nature of the shock wave after impinging the wall depends on the physical boundary conditions at the wall, where the streamlines adjacent to the wall must be parallel to it. An oblique shock with

Fig. 5.32 Reflection of an oblique shock wave from a solid boundary

5.21 Reflection and Interaction of Oblique Shock Waves

277

shock angle β2 is created at the impingement point (point B) to satisfy the physical boundary conditions at the downstream region of point B, i.e., region 3. This shock wave is referred to as the reflected shock wave. The strength of the reflected shock and the flow properties in region 3 depends on the Mach number, M2 and θ . Across the oblique shocks, M1 > M2 > M3 and thus p3 > p2 > p1 . Consequently, the reflected shock is comparatively weaker than the incident shock. Also note, the angle α is not equal to β1 because the reflection of the shock wave is not specular. It should be noted that if the reflected shock wave BC also hits a wall, another shock reflection occurs. In reality, such reflections may continue to happen as long as the upstream Mach number and flow turning angle allow for the creation of an attached oblique shock wave. These kinds of reflections are known as regular reflections.

5.21.1.1

Mach Reflection

It is seen in Sect. 5.21.1 that if the turning angle, θ , of the portion of the wall AP to the portion PA is small, the incident shock AB gets reflected by the wall QBQ at the point B, and thus regular shock reflection occurs. However, for a given Mach number, if the maximum flow turning angle through the reflected shock is less than the wall turning angle, θ , an attached oblique shock cannot be created at the wall QBQ . Rather, the shock becomes normal to the wall and bends to become tangent to the incident shock wave, as depicted in Fig. 5.33. Such a reflection is called the Mach reflection. In this case, the reflected shock cannot turn the flow parallel to the freestream direction, and get detached from the wall. A strong shock, called the Mach shock wave or the Mach stem, is formed between the reflection point B and the wall QBQ . The flow becomes subsonic just downstream of the Mach stem. Because the point C connects three shock waves: the incident shock wave, the reflected shock wave, and the Mach shock wave, it is called the triple-point. Furthermore, due to entropy variation between the streamlines, above and below the triple-point, a streamline extending downstream from point C is a slipstream. It is worth noting that the Mach stem has a curvature near the triple point, and the slipstream is slightly tilted downwards, ensuring that the reflected shock wave needs not to turn the flow to make it parallel to the freestream direction. Although slipstream separates the flow downstream of the Mach stem, the pressure is the same everywhere.

5.21.2 Reflection from a Free Pressure Boundary At the fluid-fluid interface, a free boundary exists across which the pressures on both sides of the interface are equal. Consider an oblique shock wave impinging on the free boundary, such as a jet of gas, as shown in Fig. 5.34. The static pressure at the free boundary is equal to pB . The incident shock wave deflects the flow towards the boundary and increases the static pressure downstream of the shock wave top

278

5 Wave Phenomena

Fig. 5.33 Mach reflection of an oblique shock

Fig. 5.34 Reflection of an oblique shock wave from a free pressure boundary

p2 > pB . However, the physical boundary condition at the point of incidence, O, requires that the pressure remains equal to pB . Thus, to balance the sudden rise in pressure, the flow must undergo an expansion process. Consequently, an expansion wave is reflected from the point O into the jet, as illustrated in Fig. 5.34. These expansion waves reduce the static pressure from p2 > pB to p3 = pB . Also, note that the expansion waves deflect the flow away from itself, and therefore the jet boundary deflects outward as depicted in the figure. The reflection phenomena of the oblique shock and expansion waves from a free pressure boundary are usually observed in the jet issuing from a convergent-divergent nozzle. If the ambient pressure at which the jet is discharging is greater than the nozzle exit pressure, the jet is said to be overexpanded. In this case, oblique shock waves are formed at the edge of the nozzle exit. These oblique shock waves get reflected as expansion waves from the boundary of the jet. Figure 5.35a schematically shows the shock and expansion waves prevailing in an overexpanded jet. Due to these waves, a periodic shock cell structure is generated in the jet, and the length of these periodic structures increases with Mach number. For an overexpanded jet, nozzle exit pressure, pe , is lower than the ambient pressure, p∞ . This causes an oblique shock to form at the nozzle exit plane. In order to approach the equilibrium with the surrounding environment, the exhaust gas undergoes compression through the oblique shock waves formed at the exit plane. The flow that has passed through the shock waves will

5.21 Reflection and Interaction of Oblique Shock Waves

279

Fig. 5.35 Schematic views of overexpanded jets

be turned towards the centerline. At the same time, the oblique shock wave, directed towards the jet centerline, cannot penetrate the center plane as a linear wave-front because it encounters the wave from the opposite lip of the nozzle. On meeting with the shock wave of the opposite family, it gets deflected, as shown in Fig. 5.35a. The flow passing through these reflected shock waves is further compressed; however, the flow is now turned parallel to the centerline, leading to an increased exhaust gas pressure above the ambient pressure. The deflected shock wave now strikes the free jet boundary (or the boundary where the outer edge of the flow interacts with the freestream fluid), leading to the generation of the Prandtl-Meyer expansion waves that reduce the fluid pressure to attain equilibrium with the ambient pressure. These expansion waves turn the flow away from the jet centerline; however, the waves themselves get deflected towards the jet periphery. The flow passing through these deflected expansion waves is turned back to parallel the jet centerline but experiences a further decrease in pressure. The deflected expansion waves now meet the jet periphery and reflect the jet centerline as compression waves. This allows the flow to

280

5 Wave Phenomena

Fig. 5.36 Shadowgraphic views of overexpanded jets

pass through the compression waves, increasing its pressure to the ambient pressure, but passage through the compression waves turns the flow towards the centerline. The compression waves are now deflected from the jet centerline plane as compression waves, further increasing the pressure above the ambient but turn the flow parallel to the jet centerline. The flow condition is now the same as it had while passing through the reflected shock wave, i.e., the pressure is above the ambient pressure, and the flow is parallel to the jet centerline. The formation of compression and expansion waves continues till the pressure of the jet becomes equal to the ambient pressure and the flow becomes parallel to the jet centerline and gives rise to a diamond pattern called the shock cell. It is worth noting that if the nozzle exit pressure is very low compared to the ambient pressure, i.e., the jet is highly overexpanded, then depending on the flow conditions, the compression achieved by the shock waves may not be sufficient to attain the equilibrium with the surroundings. This inability leads to the formation of the Mach disk at the jet centerline, and the reflected shock originates at the triple point. The schematic diagram of a highly overexpanded jet is shown in Fig. 5.35b. The shadowgraph images of the overexpanded jet are shown in Fig. 5.36. This being an overexpanded jet, there would be an oblique shock cone prevailing at the nozzle exit. The figure reveals that the oblique shock cone at the nozzle exit terminates as a shock cross-over at the jet centerline. The shock waves cross each other and tend to form a Mach disk. After crossing each other, the shock waves get reflected from the inner boundaries of the jet as expansion waves, and the process continues. Three prominent shock cells are explicitly seen in Fig. 5.36. When the pressure is in the surrounding medium, p∞ , to which the jet is discharging is lower than the nozzle exit pressure, pe , the jet is said to be underexpanded.

5.21 Reflection and Interaction of Oblique Shock Waves

281

Fig. 5.37 Schematic views of an underexpanded jets

The schematic view of a typical underexpanded jet is shown in Fig. 5.37. The nozzle operating pressure ratio, pp∞0 , and exit pressure ratio, pp∞e , determine the nature of the resulting flow field in the jet. The near field region of the flow is dominated by compressible effects and is largely considered stable. The curves AB and A B denote the jet boundary, where the pressure remains equal to p∞ . Since the flow is underexpanded, the flow must undergo expansion at the immediate downstream of the nozzle exit to satisfy the boundary condition at AB. The required expansion is achieved through the expansion waves formed the points A and A at the nozzle exit. These expansion waves extend towards the jet centerline and meet at point C. The region enclosed by points A, C, and A is denoted as region 1, where the static pressure equals p1 . After crossing the jet centerline at point C, the expansion waves extend and meet the jet boundary at point B. After passing through the expansion waves (AC), the fluid enters into the region 2, where the static pressure p2 is lower than p1 but is equal to p∞ to satisfy the boundary condition at AB. The flow from region 2 expands further as it passes through expansion waves (CB) and enters the

282

5 Wave Phenomena

region 3, where the pressure p3 is lower than p∞ . However, at the jet boundary, the constant pressure boundary condition must be satisfied. Consequently, the expansion waves incident on the jet boundary must be reflected as compression waves. These reflected compression waves coalesce and form a so-called barrel shock. The strength of the barrel shock depends on the strength of the expansion waves that caused it. The barrel shock meets the centerline at point D and gets reflected as the reflected shock wave. The reflected shock wave travels towards the jet boundary, meets the latter at point A1 , and the same cycle repeats. The preceding continues until the viscous forces predominate sufficiently away from the nozzle exit plane. The shock structure that results from the process above is commonly known as the shock-diamond in jet literature. The underexpanded jets are further classified as moderately underexpanded jets and highly underexpanded jets. In moderately underexpanded jets, the meeting of shock waves one family with the shock waves of the opposite family at the jet centerline forms the cross-over point at the end of the first shock cell, as shown in Fig. 5.37a. On the other hand, in highly underexpanded jets, the shock waves do not meet at the jet centerline but are reflected through the edges of a normal shock disk called the Mach disk, as depicted in Fig. 5.37b. We can summarize the above discussion on underexpanded jets as follows. If the ambient pressure (back pressure) is less than the static pressure at the nozzle exit, the flow gets relaxed through wedge-shaped expansion waves created at the nozzle lip. These expansion waves cross each other and get reflected from the opposite boundaries of the jet as compression waves. The compression waves cross at the jet centerline and undergo reflection from the jet periphery to become expansion waves. This process continues till the jet diffuses completely with the ambient fluid. The shadowgraph images of the underexpanded jets are shown in Fig. 5.38. To better understand, we should examine these images more closely and compare them with the shadowgraph image of Fig. 5.36. When the jet is highly underexpanded, as seen in Fig. 5.38b, the expansion process is so intense that the regular reflection of the barrel shock may not achieve the required pressure rise. This inability leads to the formation of the Mach disc at the jet centerline. Depending on the flow condition, the barrel shock may get reflected at the triple point instead of reflecting from the jet centerline. Since the Mach disc resembles a normal shock or a near-normal shock of considerable strength, the flow region downstream of the Mach disc becomes subsonic. In the jet literature,5 it is widely documented that a Mach disc arises when the exit pressure at the nozzle exit is four times or greater than the back pressure, i.e., pe ≥ 4pb . The slipstream originated from the triple point entirely bounds this subsonic region.

5

Kaushik, M., Innovative Passive Control Techniques for Supersonic Jet Mixing, Lambert Academic Publishing, Germany, 2012.

5.21 Reflection and Interaction of Oblique Shock Waves

283

Fig. 5.38 Shadowgraphic views of underexpanded jets

5.21.3 Neutralization or Cancellation of an Oblique Shock Wave As the preceding section demonstrates, an oblique shock wave impinging on a wall generates a reflected shock of about equal strength. This reflected shock keeps the flow adjacent to the wall parallel to it. The shock reflection can be prevented if the direction of the wall at the point of incidence is altered by a value equal to the new flow direction. There is no reflected shock since there is no change in direction. Similarly, an expansion wave striking a wall can be prevented from reflecting by changing the flow direction after the final change in conditions. This significant feature can mitigate the negative impacts of shock waves and improve flow conditions in diffuser designs. Nevertheless, it is worth emphasizing that the preceding applies exclusively to inviscid flows. The presence of a boundary layer complicates identifying the precise location at which the direction of the wall should be changed in actual flows. Figure 5.39 shows that a shock wave DB impinges on the wall ABC at the point of incidence B. The incident shock wave turns the flow by a turning angle, θ , towards the wave and increases the static pressure of the fluid passing through it. Had the wall been entirely straight, the incident shock DB forms a reflected shock to make the flow parallel to the wall at the point of incidence B. However, if the portion of the wall, denoted by BC, is turned through an angle θ to the wall AB, the preceding physical condition is satisfied. Consequently, the turning of the wall eliminates the creation of a reflected shock wave. This phenomenon is frequently referred to as wave neutralization or wave cancellation.

284

5 Wave Phenomena

Fig. 5.39 Cancellation of an oblique shock wave

5.21.4 Interactions of Shock-Shock and Shock-Expansion Waves of the Same Family Figure 5.40a illustrates the interaction of two oblique shock waves, AO and BO of the same family, formed at two successive corners of the same wall. In this case, the shock cannot pass through each other; instead, they merge to form a stronger shock OE. The regions above and below the merging location, O, have variations in entropy due to variations in the shock strengths. Consequently, a slipstream or shear-layer OD is produced. Moreover, the wave OE of the opposite family is also produced to balance the unequal pressures across the slipstream. The wave OE is compression or an expansion wave that depends on the strength of the shock BO. If the shock BO is weak, the wave OE is usually a compression wave. Furthermore, Fig. 5.40b shows the interaction of an oblique shock, and an expansion wave of the same family produced at the successive corners, one concave and the other convex, of the same wall. Similar to the previous case, the expansion waves also augment the shock strength; however, they get reflected partially along the Mach lines of the opposite family. Of course, these Mach lines are much weaker than the

5.21 Reflection and Interaction of Oblique Shock Waves

285

Fig. 5.40 Interactions of the same family of waves

expansion waves. Also, note that no slipstreams are produced in this case; rather, the entire region downstream of the interaction is full of vorticity.

5.21.5 Interaction of the Oblique Shock Waves of Opposite Family The interaction of the two oblique shock waves, AO and AO , of opposite families, i.e., produced by two opposite walls, is illustrated in Fig. 5.41. The shocks AO and AO are of different strengths. In turn, the streamlines passing through the intersection point O divide the flow region downstream of the interaction into two domains. Although these zones have the same pressures, they experience a change in entropy. Consequently, a slipstream OC is produced, as shown in the figure. Since the flow velocities across a slipstream are different, the latter is also called the shear layer. Note that the direction of the slipstream OC is not the same as those of freestream.

286

5 Wave Phenomena

Fig. 5.41 Interaction of the opposite family of shock waves

The slipstream is at an angle δ to the freestream direction. Had the shocks been of equal strengths, the slipstream direction would be parallel to the freestream. The conditions of the equal pressures across the slipstream and the same flow direction, i.e., along the slipstream, determine the angle δ. All other properties are then determined, although they do not have the same values on both sides of the slipstream.

5.22 Deviation from the Theoretical Pressure Distribution The preceding sections gave a qualitative description of the flow conditions caused by shock and expansion waves created by bodies traveling at supersonic speeds and the consequences of these waves on the pressure distribution over these bodies. Aside from the potential inaccuracies between theoretical and actual pressure produced due to the assumptions made in theoretical analysis, several other factors are involved in real flows that modify the pressure distribution and must, thus, be considered when comparing experimental data with those obtained by theoretical analysis. The formation of the boundary layer over the surface of the body is one such factor. When the thickness of the boundary layer grows along the surface of a body, its effective shape is modified to match the boundary layer contour, altering the pressure distribution along the surface. Furthermore, in supersonic, hypersonic, or transonic flows, the shock waves are always produced, owing to a change in the slope of the body over which the flow passes or to an imbalance between back pressure and nozzle exit pressure. In general, shock waves occur on a surface that already has a boundary layer, and so shock wave interaction with the boundary layer is a high possibility. This interaction is often called

5.22 Deviation from the Theoretical Pressure Distribution

287

Fig. 5.42 Shock Wave Boundary Layer Interaction (SBLI) over a flat surface with a weak impinging shock

the Shock Wave Boundary Layer Interaction (SBLI). The boundary layer thickens at the shock impingement location when pressure changes abruptly across the shock wave. Following that, the pressure signal transmission from the downstream side through the subsonic portion of the boundary layer results in retardation or thickening and/or separation of the boundary layer upstream of the shock location. As a result of the boundary layer thickening and/or separation, the flow is deflected towards the bulk flow (the main flow outside the boundary layer), resulting in a series of compression waves. These waves combine to form the separation shock outside the boundary layer. Similarly, reattachment shock will be produced when the separated flow reunites with the body. Furthermore, the impinging shock is reflected as expansion waves from the sonic line of the boundary layer. SBLIs cause complete pressure loss, flow separation, increased drag, unsteady shock oscillations, and other problems that can lead to engine failure. It also increases turbulent intensity, resulting in significant viscous dissipation. Figure 5.42a depicts the interaction pattern of a shock wave with the boundary layer in a two-dimensional domain when the impinging shock is relatively weak. The

288

5 Wave Phenomena

Fig. 5.43 The multi-layer structure of the boundary layer

shock gradually bends inside the boundary layer as the local flow speed drops towards the inner part of the boundary layer. Furthermore, the strength of the impinging shock rapidly decreases as it reaches the boundary layer and suddenly stops at the sonic line. The shock-induced pressure rise can be felt in the subsonic boundary layer’s upstream and downstream directions, as shown in Fig. 5.42b. The illustration depicts a smooth pressure rise in both the upstream and downstream directions. Thus, the subsonic portion of the boundary layer thickens over a finite region. Additionally, the deflection of the outer supersonic flow produces a series of compression waves. These compression waves combine outside the boundary to generate a reflected shock. Because the physical phenomena involved in this contact process are similar to those in an inviscid flow condition, it is called the weak viscous-inviscid interaction. The upstream interaction length, denoted by Li in Fig. 5.42b, is strongly dependent on the incoming flow conditions. It is worth noting, in this regard, that the boundary layer is essentially made up of three layers (Fig. 5.43): the outermost irrotational-inviscid layer (outer layer), the rotational-inviscid layer (middle layer), and the rotational-viscous layer (inner layer). The mutual interactions of these layers and the characteristics of the incoming flow have a considerable impact on Li . In a different flow situation, where the impinging shock is significant, the unfavorable pressure gradient is strong enough to cause flow reversal in the subsonic part of the boundary layer. The incident shock impinges on the subsonic portion of the boundary layer, causing it to separate, as shown in Fig. 5.44a. The compression waves are created at the commencement of the distorted thick boundary layer, the foot of which fall on the sonic line. When these compression waves merge outside the boundary layer, a separation shock is created. Furthermore, while interacting with the separation shock, the impinging shock slightly changes its direction and is reflected from the separated layer as expansion waves. This causes the separated layer to deflect, which then attaches after a certain downstream distance due to intense mixing between the fluid inside and outside the separated shear layer. This reattachment process is accomplished by generating a sequence of compression waves that merge above the boundary layer to generate the reattachment shock. Because the strong viscous effects dominate the interaction process, this type is called the strong viscous-inviscid interaction. The flow phenomenon, in this case, differs significantly

5.22 Deviation from the Theoretical Pressure Distribution

289

Fig. 5.44 Shock Wave Boundary Layer Interaction (SBLI) over a flat surface with a strong impinging shock

from that of pure inviscid flow. The increase in wall static pressure caused by the strong SBLIs is depicted in Fig. 5.44b. In contrast to the weak interaction process, the static pressure at the wall increases in two stages, separated by a pressure plateau in the strong SBLIs. The detachment and reattachment processes are principally responsible for the first and second pressure peaks. The pressure rise at the separation point is governed by the upstream flow characteristics and the viscous-inviscid interaction process. It is important to emphasize that the SBLI has several unintended implications that may cause the entire system to fail. An SBLI can cause boundary layer separation, stagnation pressure loss, flow field unsteadiness, increased thermal load,

290

5 Wave Phenomena

and other undesirable effects. The SBLIs that occur on the helicopter rotor blades generate much noise. Furthermore, SBLIs that occur above turbomachinery blades diminish overall efficiency. Additionally, the high viscous dissipation causes a significant increase in the flow field temperature near the body, resulting in significant aerodynamic heating. These drawbacks are exacerbated in hypersonic flows, owing primarily to viscous and vorticity interaction, as well as greater shock-layer temperatures. Therefore, suppressing Shock Wave Boundary Layer Interaction (SBLI) phenomena is essential. The MATLAB program below plots the variation of shock angle β against upstream Mach numbers and flow deflection angles θ for air (γ = 1.4), as illustrated in Fig. 5.45.

Listing 5.3 A MATLAB program for generating a graph of the shock angle versus the deflection angle. % V e r s i o n 1.1 C o p y r i g h t M r i n a l Kaushik , IIT Kharagpur , 2 9 / 0 7 / 2 0 2 1 c l o s e all ; c l e a r all ; clc g = 1.4; % S p e c i f i c heat ratio for air beta = 0 : 0 . 0 1 : 9 0 ; % Range for shock wave angle m = 0; % theta (= flow d e f l e c t i o n angle ) for M1 = 1 : 0 . 0 5 : 1 . 5 % U p s t r e a m Mach N u m b e r m = m +1; % theta - beta - M r e l a t i o n Nr = (( M1 ^2) *(( sind ( beta ) ) .^2) ) -1; Dr = (( g +( cosd (2* beta ) ) ) * M1 ^2) +2; theta = atan (2* cotd ( beta ) .* Nr ./ Dr ) *(180/ pi ) ; % max . t h e t a for a M1 a ( m ) = max ( t h e t a ) ; % max theta for the Mach No . b ( m ) = beta ( find ( theta == a ( m ) ) ) ; % find the beta for max . theta plot ( theta , beta , ' - b l a c k ') hold on end for M1 = 1 . 5 : 0 . 1 : 2 % U p s t r e a m Mach N u m b e r m = m +1; % theta - beta - M r e l a t i o n Nr = (( M1 ^2) *(( sind ( beta ) ) .^2) ) -1; Dr = (( g +( cosd (2* beta ) ) ) * M1 ^2) +2; theta = atan (2* cotd ( beta ) .* Nr ./ Dr ) *(180/ pi ) ; % max . t h e t a for a M1 a ( m ) = max ( t h e t a ) ; % max theta for the Mach No . b ( m ) = beta ( find ( theta == a ( m ) ) ) ; % find the beta for max . theta plot ( theta , beta , ' - b l a c k ') hold on end for M1 = 2 : 0 . 2 : 4 % U p s t r e a m Mach N u m b e r m = m +1; % theta - beta - M r e l a t i o n Nr = (( M1 ^2) *(( sind ( beta ) ) .^2) ) -1; Dr = (( g +( cosd (2* beta ) ) ) * M1 ^2) +2; theta = atan (2* cotd ( beta ) .* Nr ./ Dr ) *(180/ pi ) ; % max . t h e t a for a M1 a ( m ) = max ( t h e t a ) ; % max theta for the Mach No .

5.22 Deviation from the Theoretical Pressure Distribution

b ( m ) = beta ( find ( theta == a ( m ) ) ) ; theta plot ( theta , beta , ' - b l a c k ') hold on

291

% find the beta for max .

end for M1 = 4 : 0 . 5 : 5 % U p s t r e a m Mach N u m b e r m = m +1; % theta - beta - M r e l a t i o n Nr = (( M1 ^2) *(( sind ( beta ) ) .^2) ) -1; Dr = (( g +( cosd (2* beta ) ) ) * M1 ^2) +2; theta = atan (2* cotd ( beta ) .* Nr ./ Dr ) *(180/ pi ) ; % max . t h e t a for a M1 a ( m ) = max ( t h e t a ) ; % max theta for the Mach No . b ( m ) = beta ( find ( theta == a ( m ) ) ) ; % find the beta for max . theta plot ( theta , beta , ' - b l a c k ') hold on end for M1 = 5 : 1 : 8 % U p s t r e a m Mach N u m b e r m = m +1; % theta - beta - M r e l a t i o n Nr = (( M1 ^2) *(( sind ( beta ) ) .^2) ) -1; Dr = (( g +( cosd (2* beta ) ) ) * M1 ^2) +2; theta = atan (2* cotd ( beta ) .* Nr ./ Dr ) *(180/ pi ) ; % max . t h e t a for a M1 a ( m ) = max ( t h e t a ) ; % max theta for the Mach No . b ( m ) = beta ( find ( theta == a ( m ) ) ) ; % find the beta for max . theta plot ( theta , beta , ' - b l a c k ') hold on end for M1 = 8 : 2 : 1 0 % U p s t r e a m Mach N u m b e r m = m +1; % theta - beta - M r e l a t i o n Nr = (( M1 ^2) *(( sind ( beta ) ) .^2) ) -1; Dr = (( g +( cosd (2* beta ) ) ) * M1 ^2) +2; theta = atan (2* cotd ( beta ) .* Nr ./ Dr ) *(180/ pi ) ; % max . t h e t a for a M1 a ( m ) = max ( t h e t a ) ; % max theta for the Mach No . b ( m ) = beta ( find ( theta == a ( m ) ) ) ; % find the beta for max . theta plot ( theta , beta , ' - b l a c k ') hold on end for M1 = 1 0 : 1 0 : 2 0 % Upstream Mach Number m = m +1; % theta - beta - M r e l a t i o n Nr = (( M1 ^2) *(( sind ( beta ) ) .^2) ) -1; Dr = (( g +( cosd (2* beta ) ) ) * M1 ^2) +2; theta = atan (2* cotd ( beta ) .* Nr ./ Dr ) *(180/ pi ) ; % max . t h e t a for a M1 a ( m ) = max ( t h e t a ) ; % max theta for the Mach No . b ( m ) = beta ( find ( theta == a ( m ) ) ) ; % find the beta for max . theta plot ( theta , beta , ' - b l a c k ') hold on end for M1 = 1 0 0 0 0 : 1 : 1 0 0 0 2 % U p s t r e a m Mach N u m b e r ( c o s i d e r as i n f i n i t y ) m = m; % theta - beta - M r e l a t i o n Nr = (( M1 ^2) *(( sind ( beta ) ) .^2) ) -1; Dr = (( g +( cosd (2* beta ) ) ) * M1 ^2) +2; theta = atan (2* cotd ( beta ) .* Nr ./ Dr ) *(180/ pi ) ;

292

5 Wave Phenomena

Fig. 5.45 θ − β − M plot by the MATLAB program

% max . t h e t a for a M1 a ( m ) = max ( t h e t a ) ; No . b ( m ) = beta ( find ( theta == a ( m ) ) ) ; theta plot ( theta , beta , ' - b l a c k ') hold on

% max theta for the Mach % find the beta for max .

end plot (a ,b , ' -r ' , ' L i n e w i d t h ' ,2) x l a b e l ( ' D e f l e c t i o n a n g l e \ theta , D e g r e e s ') y l a b e l ( ' Shock - wave angle \ beta D e g r e e s ') axis ([0 50 0 90]) grid minor set ( gca , ' Y T i c k ' , [ 0 : 5 : 9 0 ] ) set ( gca , ' Y M i n o r G r i d ' , ' on ') set ( gca , ' X T i c k ' , [ 0 : 4 : 5 0 ] ) set ( gca , ' X M i n o r G r i d ' , ' on ')

Concluding Remarks In this chapter, the phenomena of shock and expansion waves in supersonic flows are discussed at length. It is shown that the normal shock wave is the strongest possible compression wave that can appear in a supersonic flow. It is shown that the flow properties across a shock wave jump. Also, from entropy considerations, it is shown that the Mach number upstream of a normal shock must be supersonic and always becomes subsonic downstream of the shock. A shock wave that forms at an angle to the incoming flow direction is called an oblique shock wave. We have shown that an oblique shock wave is created if a flow turns into itself at a concave corner. The well-known relationship between the flow turning angle, the shock angle, and

Concluding Remarks

293

the Mach number is derived. Since the shock waves are non-isentropic, the flow passing through these shocks experiences a loss of stagnation pressure. Instead of a concave corner, if a supersonic flow approaches a convex corner, a different set of waves, called the expansion waves, develops at the corner. It is shown that the expansion waves are isentropic, and hence the flow while passing through them does not experience a change in entropy. Consequently, the stagnation pressure across the expansion waves remains constant. It is, however, shown that the static properties across an expansion wave reduce. The reflection and interaction phenomena of oblique shocks and expansion waves are also discussed. Finally, we have discussed the shock wave and boundary layer interaction that causes a deviation in pressure distribution over the bodies traveling at supersonic speeds.

Exercise Problems J Exercise 5.1 The change in entropy across a normal shock wave is 300 (kg.K) . Calculate (i) the Mach number upstream of the shock, and (ii) the shock strength.

Exercise 5.2 Consider a stationary normal shock in a supersonic airflow with the following conditions ahead of shock: v1 = 1000 m/s, T1 = 298 K, and p1 = 1 atm. Determine v2 , T2 and p2 downstream of the shock. Exercise 5.3 Using the Rankine-Hugoniot equation, show that the density ratio for a perfect gas passing through a normal shock can be expressed as, p2 1 + γγ +1 ρ2 −1 p = γ +1 p21 ρ1 +p γ −1 1

Exercise 5.4 A Pitot Probe in the test section of a supersonic wind tunnel measures a pressure of 100 kPa. The static pressure ahead of the bow-shock is 25 kPa. Find the Mach number in the test-section. Exercise 5.5 The temperature and flow speed upstream of an oblique shock are 250 K and 800 m/s, respectively. Determine the flow deflection angle and the Mach number upstream of the shock. What is the shock strength? Exercise 5.6 An oblique shock wave turns the flow by 8◦ in a uniform supersonic airflow with a velocity of 550 m/s. The shock makes a 39◦ angle to the freestream direction, and the temperature downstream of the shock is 200 K. Calculate the freestream Mach number. Exercise 5.7 If the freestream Mach number is 2.0, construct a hodograph shock polar diagram. Calculate the shock angle, β, and the Mach number, M2 , downstream of the oblique shock waves. Also, determine the maximum flow deflection angle, θmax . Compare your results to those from oblique shock relations.

294

5 Wave Phenomena

Exercise 5.8 Using the isentropic table for the Prandtl-Meyer function, calculate the Mach number downstream of an oblique shock if a uniform supersonic flow at Mach 2.5 is deflected through (i) a concave corner of 15◦ , and (ii) a convex corner of 20◦ . Exercise 5.9 Wedges and cones are frequently employed in supersonic wind tunnels to determine the Mach number in the test section. Determine the corresponding value of the maximum Mach number, if any, for a wedge with a total vertex angle of 30◦ . Is the maximum Mach number affected if a cone is used instead of a wedge? Compare the Mach number values. Determine the drag coefficient. Assume that the base pressure is equal to the freestream pressure. Exercise 5.10 A wedge with an included angle of 30◦ is in the airstream at Mach 3. Determine the drag coefficient. Assume the base pressure to be equal to the freestream pressure. Exercise 5.11 Consider a stationary two-dimensional wedge in a supersonic airflow at Mach 2.0. (i) What are the different shock angles? (ii) What is the maximum angle at which the flow can turn without becoming detached? Exercise 5.12 Consider a cone with an apex angle of 30◦ in a uniform airstream at Mach 3.5, with the cone-axis parallel to the freestream direction. At the vertex, an attached oblique shock with a shock angle of 30◦ is created. Determine whether this shock is strong or weak? Exercise 5.13 A flat plate is put in a uniform airstream at Mach 2.5 with a 15◦ angle of attack to the freestream direction, as shown in Fig. 5.46. Determine the Mach numbers in the regions illustrated in figure, if p∞ = 100 kPa and T∞ = 288 K. Assume the slipstream inclination to the freestream direction is negligible.

Fig. 5.46 A flat plate in a Mach 2.5 airstream at α = 15◦

Exercise Problems

295

Fig. 5.47 A symmetrical diamond-shaped airfoil in a Mach 2.5 airstream at α = 15◦

Exercise 5.14 Consider a symmetrical diamond-shaped airfoil with a total vertex angle of 20◦ , as shown in Fig. 5.47, that operates at an angle of attack of 15◦ to the airstream when the flight Mach number and pressure are 2.5 and 100 kPa, respectively. The airfoil has a maximum thickness of 0.2 m. Determine the lift and drag on the airfoil. Assume a unit length of 1 m in the spanwise direction. Exercise 5.15 For the air passing through a normal shock, show that the density ratio across the shock cannot be greater than 6. Also, show that the Mach number downstream of the shock cannot be less than 0.38.

Chapter 6

Steady One-Dimensional Isentropic Flow in a Variable-Area Duct

Abstract This chapter will examine some of the interesting implications of fluid compressibility resulting in isentropic flows through variable area ducts. The typical application of this subject in gas dynamics is the design of nozzles and diffusers. A nozzle is a passage that accelerates the flow by expanding it from high to low pressure. Typical examples of a nozzle are the exit duct of a jet aircraft or a spacecraft. The diffuser acts oppositely to that of the nozzle; a diffuser is a passage that decelerates the flow by compressing it from low to high pressure. For instance, before entering the aircraft engine, the air passes through the diffuser, which raises the temperature and pressure of the air. For incompressible flows, the nozzle boundary converges in the direction of the flow to increase the flow velocity, while the diffuser boundary diverges to raise the static pressure level. We will see that this convergence and divergence are reversed for supersonic flow through the nozzles and diffusers.

6.1 Introduction In Chap. 4, the term one-dimensional flow was used to describe a flow in which the conditions are sensibly constant over a cross-sectional normal to the flow direction. Generally, such a condition is found in the fluid flow through long nozzles and diffusers. Since the flow in these passages is constrained in all other directions normal to the flow, it is known to be one-dimensional even though it may expand or contract in these directions. Simply put, the flow conditions in these passages are allowed to change only along the streamline direction and are usually constant in the plane normal to it. Thus, the changes in flow properties are only functions of the coordinate along the flow direction.

© Springer Nature Singapore Pte Ltd. 2022 M. Kaushik, Fundamentals of Gas Dynamics, https://doi.org/10.1007/978-981-16-9085-3_6

297

298

6 Steady One-Dimensional Isentropic Flow in a Variable-Area Duct

6.2 Steady One-Dimensional Isentropic Flow Through Ducts Since we are interested in the physical forms that can induce the one-dimensional changes in Sect. 6.1, the relationship between the nozzle and diffuser geometries to this type of flow will be deduced in this section.

6.2.1 Area-Velocity Relationship for the Flow Through a Duct Let us examine how the area of a duct needs to be varied to expand or compress the flow at subsonic or supersonic speeds. When analyzing the flow through a duct, the following assumptions are made: 1. The flow is one-dimensional and steady. 2. The flow is adiabatic (δQ = 0). 3. There is no work transfer (δW = 0), no body forces (gdz = 0), and no obstructions within the flow (δD = 0). 4. The flow process is isentropic (δs = 0). 5. The duct is frictionless (δFf = 0) but has a variable area (dA = 0). The assumptions above show that the only driving potential causing a state change in an isentropic flow is the area change. This form of flow under frictionless conditions may approximate the actual operation of many technological devices. The relationship between the change in the flow area dA and the corresponding change in the flow velocity and the pressure, represented by dv and dp, respectively, is obtained by combining the differential forms of the relevant governing equations. For the steady flow of a gas through a duct with gradually varying area, the continuity equation is m ˚ = ρvA = constant (4.3) Note that this equation applies to the flow through ducts, assuming that the curvature of the duct is very small and that the variable cross-sectional area is sufficiently small for the velocity, pressure, density, and temperature to be considered constant over each cross section. Taking the logarithmic differentiation of Eq. (4.3), we obtain dv dA dρ + + =0 ρ v A For incompressible flows,

dρ ρ

(6.1)

= 0; thus dA dv =− v A

(6.2)

6.2 Steady One-Dimensional Isentropic Flow Through Ducts

299

Equation (6.2) shows that decreasing the cross section of the duct in the direction of flow increases the flow velocity. The equation of motion for an inviscid compressible flow in the absence of body forces is dp + vdv = 0 (4.26) ρ If the flow through the duct is adiabatic and frictionless, the equation above can be modified as follows by using Eq. (3.9): vdv = −

dρ dp = a2 ρ ρ

(6.3)

Equation (6.3) requires that an increase in speed in the direction of flow, dv > 0, be followed by a decrease in pressure, dp < 0. Equation (6.3) is written in terms of the Mach number as dρ dv = −M2 ρ v

(6.4)

Therefore, for very low Mach numbers, the fractional change in density is significantly smaller than the fractional change in velocity. For incompressible flows, the density changes in the continuity equation (Eq. (6.1)) can thus be neglected, as confirmed by Eq. (6.2). from Eq. (6.1) using Eq. (6.4), we obtain Now, by eliminating dρ ρ  dv dA  2 = M −1 A v

(6.5)

This is the well-known area-velocity relationship for compressible flows. The following major conclusions can be drawn from Eq. (6.5) for different flow regimes based on the Mach number.   is negative and we 1. At subsonic Mach numbers (M < 1) the term M2 − 1 dv v come to the conclusion that a decrease in area induces an increase in flow speed. The subsonic nozzle must therefore have a converging profile (Fig. 6.1a), and the subsonic diffuser must have a diverging profile (Fig. 6.1b).   is positive and leads 2. At supersonic Mach numbers (M > 1) the term M2 − 1 dv v to the conclusion that an increase in the area leads to an increase in the fluid speed. This finding is consistent with Eq. (6.4), which shows that for M > 1 the density decreases faster than the speed increases, so the area must increase in the flow direction in order to have ρvA constant. For this reason, the supersonic portion of the nozzle has a diverging profile (Fig. 6.1c) and the supersonic portion of the diffuser has a converging profile (Fig. 6.1d). Assume a nozzle is used to generate a supersonic flow, beginning with a low-speed and high-pressure gas at the inlet, as shown in Fig. 6.2. Thus, the Mach number must

300

6 Steady One-Dimensional Isentropic Flow in a Variable-Area Duct

M2 < M 1

M1 < 1 M2 > M 1

M1 < 1 dp > 0 dv < 0

dp < 0 dv > 0

(a) Subsonic nozzle

M2 > M 1

(b) Subsonic diffuser

M1 > 1 M2 < M 1

M1 > 1 dp < 0 dv > 0

(c) Supersonic nozzle

dp > 0 dv < 0

(d) Supersonic diffuser

Fig. 6.1 Nozzle and diffuser profiles in subsonic and supersonic flows

gradually increase from M = 0 near the inlet to M > 1 at the exit. According to the preceding discussion, the nozzle must have a converging boundary in the subsonic portion and a divergence in the supersonic portion. This type of nozzle is known as a convergent-divergent nozzle1 or de Laval nozzle. The Mach number must be unity at the throat (minimum area location), where the area is neither increasing nor  → 0 . This is compatible with Eq. (6.5), which states that du can decreasing dA A be non-zero at the throat only if M = 1. Consequently, sonic speed can be obtained only at the throat of a nozzle or diffuser and nowhere else. It is not necessary, however, for M to be unity at the throat. According to Eq. (6.5), it is possible to have a situation where M = 1 at the throat if dv = 0 there. For example, the flow in a convergent-diverging tube may be subsonic everywhere, with M increasing in the converging portion and decreasing in the divergent portion, with M = 1 at the throat. In this case, the nozzle is commonly referred to as a venturi tube. For fully subsonic flow, the first half of the tube serves as a nozzle and the other as a diffuser. Alternatively, we may have a convergent-diverging tube with supersonic flow throughout, with M decreasing in the converging portion and increasing in the divergent portion, and M = 1 at the throat. 1

A convergent-divergent nozzle with straight boundaries creates a uniform flow at the exit, but this flow may not be unidirectional. Therefore, to achieve both uniform and unidirectional flow at the exit, the nozzle boundary is properly contoured.

6.2 Steady One-Dimensional Isentropic Flow Through Ducts

301

A exit

A entry

At M1

Throat

Convergent section

Divergent section

Fig. 6.2 A convergent-divergent nozzle

Example 6.1 Let v, A, ρ and p, respectively, denote the flow speed, cross-sectional area, density, and pressure at a point in an inviscid flow. Show that a2 dρ =0 ρ dv   dρ A v2 1 − = − 2 dv v a

1. v + 2.

Solution. In the absence of body forces Bernoulli’s equation becomes dp + vdv = 0 ρ 1 dp +v=0 ρ dv   1 dp dρ +v=0 ρ dρ dv Assume the flow to be reversible and adiabatic, i.e., isentropic. Hence,     dρ 1 dp +v=0 ρ dρ s dv  But,

dp dρ

 s

= a2 , where a is the speed of sound. Therefore,   a2 dρ +v=0 ρ dv

(1)

302

6 Steady One-Dimensional Isentropic Flow in a Variable-Area Duct

From the one-dimensional continuity equation for a given mass flow m ˙ = ρAv = constant Logarithmic differentiation of m ˙ gives dA dv dρ + + =0 ρ A v Dividing by dv throughout, we get 1 dρ 1 dA 1 + + =0 ρ dv A dv v   A v dρ dA =− 1+ dv v ρ dv in the above expression using the results obtained in Example We can replace dρ dv 6.1(1). Thus, substituting in dρ = −ρ av2 into dA expression yields dv dv   dA A v2 =− 1− 2 dv v a

6.2.2 Flow Through a Convergent Nozzle A nozzle is a flow passage used to accelerate a fluid stream that can be found in a wide range of applications, from household appliances to rocket science. Nozzles are used in wind tunnels, jet aircraft, spacecraft, rocket motors, fire hose pipes, and other applications. To accelerate the flow of fluid, a pressure drop is created across the nozzle. In this section, we will investigate the flow characteristics of a convergent nozzle by altering the back pressure pb at the nozzle exit while keeping the reservoir pressure (supply pressure at the nozzle intake) constant p0 (stagnation pressure). In this case, the back pressure pb is the pressure in the surrounding region where the flow is discharged. In the following discussions, it is assumed that if the flow at the exit plane is subsonic, pb equals pe (pressure at the exit of the nozzle), but not if it is supersonic. This is obvious because subsonic flow permits downstream pressure signals to pass upstream to the exit plane, whereas supersonic flow has steep pressure gradients that prohibit downstream pressure signals from being communicated upstream.

6.2 Steady One-Dimensional Isentropic Flow Through Ducts Fig. 6.3 A convergent nozzle

303

Convergent nozzle

pexit

p0

pb

p/p0

M p∗ (curve c), otherwise it decelerates to subsonic speeds, if pb7 < p∗ (curve g). As pb is reduced further, the flow upstream of the throat does not respond, and the nozzle is said to be choked in the sense that it has reached the maximum mass flow rate m ˚ max through the nozzle for a given reservoir conditions and throat area. There is a range of back pressures, shown by curves c and d, in which the flow initially becomes supersonic in the divergent portion, but then adjusts to pb by means of a normal shock standing inside the nozzle. The downstream of the normal shock

6.2 Steady One-Dimensional Isentropic Flow Through Ducts Convergent section

305

Me

Divergent section

Throat A entry

x

1.0 p pb

p

0

A exit

At

p

pb1 pb2

a b

m < m* t1

m < m*

t2

Isentropic diffusion p*

(b) m = m* (a) m = m*

d Normal shock

pb4 pb5

e

2

3

f pb6 g pb7

Isentropic expansion to supersonic speed

1

pb3

c

pb8

4

Overexpanded Mach 1.73 nozzle 5

h

Underexpanded Mach 1.73 nozzle

Fig. 6.4 The operation of a convergent-divergent nozzle at various back pressures

is, of course, subsonic. In this range, the normal shock moves towards exit as pb is reduced, and for curve d the normal shock stands right at the exit plane. At this stage, the flow throughout the nozzle up to the exit plane is now supersonic and remains so on further reduction of pb . When the back pressure is further lowered to pb6 , there is no normal shock anywhere inside the nozzle, and the jet pressure adjusts to pb by means of oblique shock waves downstream of the nozzle exit plane, as shown by curve f. Under this condition, the nozzle operation is said to be overexpanded. These oblique shocks vanish when pb = pb7 (curve g). On further reducing the back pressure corresponding to pb8 , the adjustment of pb with the jet pressure takes place downstream of the exit plane by means of expansion waves. At this stage, the nozzle operation is referred as underexpanded. It is worth noting that when a nozzle discharges the maximum mass flow rate and the gas is completely expanded to back pressure, the divergent portion of the nozzle experiences fully supersonic and isentropic flow. The further drop of back pressure has no influence on the flow conditions inside the nozzle; nevertheless, supersonic flow expands downstream of the nozzle exit plane in the same manner as the jet emanating from a convergent nozzle operating at supercritical pressure ratios. Because the gas is not completely expanded inside the nozzle, the kinetic energy of the flow at the exit is lower than in the fully isentropic case. Consequently, the thrust produced by an underexpanded nozzle will be less than that produced by a correctly expanded nozzle. Similarly, when a nozzle is operated with an overexpanded state, the thrust is reduced as compared to a correctly expanded nozzle. Because the static pressure decreases below the back pressure in the divergent portion of the overexpanded nozzle, the pressure is adjusted by compressing the gas through a succession of oblique shock waves (Fig. 6.5). Consider a convergent-divergent nozzle of constant-area ratio that functions under correctexpansion (pb = pb8 ) and provides fully supersonic and isentropic flow in the nozzle divergence. However, if pb is suddenly increased above pb8 , the change in back

306

6 Steady One-Dimensional Isentropic Flow in a Variable-Area Duct

Throat

Separated flow Oblique shock waves Separated flow

Convergent section

Divergent section

Fig. 6.5 An overexpanded convergent-divergent nozzle with flow separation

pressure cannot be conveyed upstream through the main flow since the flow leaves the nozzle exit at supersonic speeds. Pressure signals, on the other hand, can go upstream through the boundary layer on the wall. Because the fluid velocities within the boundary layer range from supersonic at the interface with the outer main fluid to zero at the wall, there are subsonic zones. Through these subsonic regions, acoustic signals in the form of pressure waves can travel upstream in the boundary layer. When pb is marginally greater than pb8 , oblique shocks generate right at the exit corner and the wall static pressure rises abruptly from pw to pb . If pb is increased to a level that is significantly more than pb8 , the flow splits from the wall and compresses through oblique shock waves (Fig. 6.5). The separation pressure, psep , is the static pressure at the location where the separation occurs. It is essential to alleviate or limit flow separation, particularly in rocket engines that fly at extremely high altitudes.

6.2.4 Performance of a Real Nozzle In the course of our discussion in Chap. 4, we have derived equations to calculate the throat area of a nozzle that allows to pass a specified mass flow rate to the ambiance isentropically under a given stagnation conditions. However, in a real nozzle, the flow always encounters a certain amount of friction due to boundary layer formation on the inner wall of the nozzle which eventually prevents the nozzle to operate in a prescribed manner even if the design conditions are imposed on the nozzle. Fortunately, in most of the cases, the deviation from ideal behavior is very small and hence requires only a mild correction in the isentropic analysis. To examine the effects of friction in a nozzle operating under non-ideal conditions, let us apply steady one-dimensional adiabatic flow energy equation to an arbitrary control volume bounded by sections 1 and 2, as shown in Fig. 6.6a. This provides 1 1 h1 + v12 = h2 + v22 2 2

(4.35)

6.2 Steady One-Dimensional Isentropic Flow Through Ducts Fig. 6.6 The enthalpy-entropy diagram of a real nozzle

307

p

1

1

p

2

2

h 2s s

The friction in an adiabatic flow tends to increase the temperature above that of an isentropic flow, and thus the static enthalpy increases as well. According to Eq. (4.35) a higher h2 corresponds to a lower v2 . Because the aim of a nozzle is to enhance kinetic energy, it is obvious that friction reduces nozzle effectiveness. This is represented in the conventional enthalpy-entropy diagram, as seen in Fig. 6.6b. If the flow in a nozzle expands isentropically from some initial pressure p1 to some pressure p2 , the process is denoted by the vertical line 1 to 2s. However, with an irreversible adiabatic flow process, entropy increases, which is consistent with the second law of thermodynamics. Thus, the final state must be on the constant pressure line p2 , to the right of 2s, as shown in Fig. 6.6b. Figure also shows that h2 > h2s and thus v2 < v2s . The nozzle efficiency is typically defined as the ratio of actual kinetic energy per unit mass flow to the theoretical kinetic energy per unit mass flow derived by an isentropic process as a measure of the frictional effects in the nozzle. Thus, the nozzle efficiency, denoted by η, is expressed as  2

 2

v2 2

v2 2

η =  2  act =  2  v2 2

isen

v1 2

isen

act

+ (h1 − h2s )

(6.7)

In most of the cases, the flow velocity at the nozzle inlet is very small, the kinetic v2 energy term 21 will consequently be smaller as compared to (h1 − h2s ). Thus, by dropping

v12 2

in Eq. (6.7), we obtain  2 η=

v2 2

act

h1 − h2s

For a perfect gas, h = cp T; Eq. (6.8) becomes

(6.8)

308

6 Steady One-Dimensional Isentropic Flow in a Variable-Area Duct

 2 η=

v2 2

act

(6.9)

cp (T1 − T2s )

Equation (6.9) is utilized to account for the frictional effects encountered during nozzle operation. Because the frictional effects are mostly limited to the divergent section of the nozzle, η is employed to change the exit area of the nozzle. It should be emphasized that Eqs. (6.8) and (6.9) are the results of a simplified one-dimensional analysis. Depending on the area of cross section, a two or threedimensional flow in the nozzle must be considered for more precise results. Therefore, these calculations are only applicable when the nozzle is operating at near to design conditions. The results of isentropic flow analysis modified by friction corrections may not be valid for more off-design nozzle operations. Example 6.2 Show that the minimum operating pressure ratio required to establish the supersonic flow up to the exit of a convergent-divergent nozzle is 

p0 pb

 min

 γ   2γ γ − 1 2 γ −1 γ − 1 −1 Me Me2 − = 1+ 2 γ +1 γ +1

where p0 is the reservoir pressure at the inlet of the nozzle, pb is the back pressure and Me is the Mach number at the exit of the nozzle. Solution. Assume the convergent-divergent nozzle is discharged into the ambiance at the back pressure pb . For a specific value of the operating pressure ratio pp0b , the normal shock willbe standing just at the nozzle exit. Indeed, this is the minimum at which the supersonic flow prevails up to the nozzle exit. pressure ratio pp0b min

The pressure ratio

p0 pb

can be expressed in terms of the nozzle exit pressure pe as p0 = pb



p0 pe



 ×

pe pb



From the isentropic relationship γ  p0 (γ − 1) 2 γ −1 Me = 1+ pe 2 and from the normal shock relationship   pb 2γ  2 Me − 1 = 1+ pe γ +1 Therefore,

6.2 Steady One-Dimensional Isentropic Flow Through Ducts





p0 pb

 min

p0 pb





1+

γ −1 2 Me 2

309

 γ γ−1

=

  1 + γ2γ Me2 − 1 +1  γ  2γ (γ − 1) 2 γ −1 (γ − 1) −1 = 1+ Me Me2 − 2 γ +1 (γ + 1) min

Example 6.3 Consider an adiabatic and non-isentropic flow through a convergentdivergent nozzle. If the efficiency of the nozzle is η, show that the static to stagnation pressure ratio at any section is given by  γ γ−1   p 1 (γ − 1) M2 = 1− p0 η 2 + (γ − 1) M2 Solution. By definition, the efficiency of the nozzle is η= For a perfect gas cp = η=

γR γ −1

v 2/2

cp (T0 − T)

and a2 = γ RT. Substituting in η expression yields v 2/2

 γR 1− T 0 γ −1

T T0

=

(γ − 1) v2   2γ RT TT0 1 −

T T0



From isentropic relationship T = T0



p p0

 γ γ−1

Therefore, η=

(γ − 1) M2 =   T0   2 T 1 − TT0 2 1+

(γ − 1) M2    γ γ−1  (γ −1) 2 1 − pp0 M 2

By rearranging the terms, we obtain p 1 (γ − 1) M2  =1−  p0 η 2 + (γ − 1) M2

310

6 Steady One-Dimensional Isentropic Flow in a Variable-Area Duct

6.2.5 Flow Through a Convergent-Divergent Diffuser: The Supersonic Wind Tunnel Diffuser A convergent-divergent diffuser is defined as the reverse operation of a convergentdivergent nozzle; thus, it is also referred to as the reverse-nozzle diffuser. The convergent-divergent diffuser immediately follows the test-section in a supersonic wind tunnel. The basic characteristics of the flow through a fixed-geometry convergent-divergent diffuser are investigated in this section. We shall assume that the flow is one-dimensional and steady and that it is isentropic everywhere except across the shock. The flow is discharged into the surrounding environment, where the back pressure is pb . The stagnation conditions p01 and T01 in the settling chamber are kept constant.

6.2.5.1

Normal Shock Recovery: The Shock Swallowing Phenomena

Consider Fig. 6.7a open-circuit supersonic wind tunnel with the diverging diffuser. We saw in Sect. 6.2.3 that for a specific value of back pressure, pb = pb8 , the normal shock stands right at the nozzle exit. In this case, the overall pressure ratio pp0201 across the standing normal shock equals the pressure ratio σ1 required to run the tunnel. Thus, − γ γ−1  1   γ −1 p01 2γ  2 (γ + 1) M12 M1 − 1 σ1 = = 1+ p02 γ +1 (γ − 1) M12 + 2

(6.10)

As pb is gradually reduced beyond pb8 , the normal shock moves in the test-section. If the flow leaving the test-section is immediately exposed to the environment, huge losses will occur due to the higher kinetic energy of the flow. However, these losses can be decreased by connecting a diverging duct to the test-section, as shown in Fig. 6.7b. Further decreases in pb8 will force the shock wave to move inside the diffuser, eventually resulting in a shock-free test-section. This is known as normal shock recovery or normal shock swallowing. The subsonic flow downstream of the normal shock is isentropically decelerated to a different stagnation pressure, p02 (> p02 ). Consequently, the tunnel can be operated at a lower pressure ratio, σ2 , rather than σ1 (Eq. (6.10)). − γ γ−1  1   γ −1 p01 2γ  2 (γ + 1) M12 σ2 =  = 1 + M1 − 1 p02 γ +1 (γ − 1) M12 + 2

(6.11)

It is therefore theoretically possible to reduce the operating pressure ratio of a wind tunnel by adding a diverging duct to the test-section.

6.2 Steady One-Dimensional Isentropic Flow Through Ducts

311

Divergent diffuser Convergent section

Divergent section

M=1

p 01

A*

A entry

Test−section p 02 ATS

A exit

Throat

(a) Normal shock at the nozzle exit Divergent diffuser Convergent section

Divergent section

Test−section

M=1 A entry

A*

p 01 A exit

p 02 ATS

Throat

(b) Normal shock in the divergent diffuser Fig. 6.7 The diverging duct in an open-circuit supersonic wind tunnel

6.2.5.2

Effects of Second Throat

As we saw in the preceding section, connecting a divergent duct to the test-section can reduce the operating pressure ratio of a supersonic wind tunnel. However, an alternative arrangement, which will be discussed in this section, can reduce the pressure ratio even further. It is possible to achieve a higher stagnation pressure at the diffuser exit if supersonic flow entering the test-section is isentropically compressed to sonic conditions at a downstream throat called the second throat and then decelerated in the diffuser to subsonic speeds. To produce these flow conditions, a fixed-geometry convergent-divergent diffuser is connected at the end of the test-section, as shown in Fig. 6.8. Assume that the flow passes through the convergent-divergent diffuser with constant stagnation conditions p01 and T01 in the settling chamber. The flow is discharged into the surroundings, where the back pressure is pb . A slight reduction in pb produces the flow inside the tunnel, which is initially subsonic throughout. However, successive reductions in pb result in a progressive increase in mass flow

312

6 Steady One-Dimensional Isentropic Flow in a Variable-Area Duct Convergent−divergent nozzle M=1 A entry

A*1

Test−section p 01

Convergent−divergent diffuser

p 02 A*2

ATS

A exit

Throat

(a) Normal shock in the test−section (startup condition) Convergent−divergent nozzle

Test−section

Convergent−divergent diffuser

M=1 A entry

A*1

A exit

ATS

p 01

p 02

A*2

Throat

(b) Normal shock at the second throat (running condition) Fig. 6.8 The convergent-divergent diffuser in an open-circuit supersonic wind tunnel

rate until the nozzle throat becomes choked. The mass flow rate through the nozzle reaches its maximum in this situation, and the nozzle is said to be operating at its first critical condition. In this situation, the pressure ratio pp01b is sufficient to choke the nozzle throat (first throat), resulting in a sonic condition. A normal shock appears immediately downstream of the throat when the value of pb is decreased further. The normal shock will migrate downstream in the nozzle divergence and swiftly pass through the test-section if pb is continuously decreased. With a normal shock in the test-section that happens during start-up, the tunnel is considered to be operating in its most unfavorable and off-design condition (Fig. 6.8a). When pp01b is reduced further, the supersonic diffuser swallows the normal shock and transports it to the divergent section. Thus, the test-section shock is eliminated, which results in an isentropic flow throughout. The power consumption of the running tunnel, on the other hand, remains higher due to pressure loss across the shock wave in the diffuser. A slight increase in pb sends the shock upstream to the diffuser throat (the position with the lowest shock strength), lowering power usage. Figure 6.8b illustrates the general running condition of a wind tunnel, commonly known as the most favorable condition. The mass flow rate through a convergent-divergent nozzle with a choked throat is expressed as   (γ +1) γ + 1 − 2(γ −1) γ m ˚ max = p (4.123) 0 A∗ RT0 2 For air (γ = 1.4), this reduces to

6.2 Steady One-Dimensional Isentropic Flow Through Ducts

313

m ˚ max 0.6847p0 = √ A∗ RT0

(4.124)

p01 A∗ m ˚ max ∝ √ 1 T01

(6.12)

where A1∗ denotes the throat area of the supersonic nozzle. It is worth noting that the mass flow rate given by Eq. (6.12) must also pass through the supersonic diffuser throat, so we can write p02 A∗ m ˚ max ∝ √ 2 T02

(6.13)

where A2∗ denotes the throat area of the supersonic diffuser; and p02 and T02 define the stagnation conditions of the surroundings to which the flow is directed. By combining Eqs. (6.12) and (6.13), we get p02 A∗ p01 A1∗ = √ 2 √ T01 T02

(6.14)

Because the flow across a shock wave is adiabatic (T01 = T02 ), Eq. (6.14) gives A∗ p01 = 2∗ p02 A1

(6.15)

Equation (6.15) is a key relationship in the operation of a supersonic wind tunnel. It indicates that the flow can only pass through the diffuser if A2∗ > A1∗ , i.e., when the diffuser throat area is greater than the nozzle throat area. In the limiting case, if ∗ , the shock will be positioned in the test-section causing the maximum A2∗ = Amin pressure loss. In this situation, Eq. (6.15) might be expressed as 

p01 p02

 = TS

∗ Amin A1∗

(6.16)

  where pp0201 denotes the ratio of stagnation pressures across the normal shock in TS the test-section. An example can aid in clarifying the preceding discussion. Consider a normal shock with a Mach number of 2.5 standing in the test-section, as depicted in Fig. 6.8a. Now, if we use Eq. (6.10) at M1 = 2.5, we get A1∗ = 0.7209 A2∗

314

6 Steady One-Dimensional Isentropic Flow in a Variable-Area Duct 01 A*1

A TS

Shock diamond A*2

Convergent−divergent nozzle

Weak normal shock

02

Test−section

Convergent−divergent diffuser with modified throat

Fig. 6.9 The modified diffuser geometry in an open-circuit supersonic wind tunnel

This reveals that the diffuser throat area must be enlarged by more than (1−0.7209) × 100 = 38.7% in order to swallow the shock. 0.7209 At the start of a supersonic wind tunnel with a fixed-geometry convergentdivergent diffuser, it is critical to keep A2∗ slightly higher than the theoretical minimum ∗ to account for inaccuracies in eliminating frictional losses due to fluid viscosity, Amin errors caused by deviation from the assumption of one-dimensional flow, and so on. Nevertheless, despite the fact that this is a simplified calculation with an isentropic flow assumption, the results apply to the actual flow through the diffuser where the boundary layer is sucked through the wall porosity. The interaction of a shock wave with the boundary layer on the diffuser wall affects the flow field structure, therefore even with the best-designed diffuser, 100 percent pressure recovery is never attained. Under design conditions, a fixed-geometry convergent-divergent diffuser is quite efficient, but suffers large losses due to the production of a normal shock at the diffuser inlet. Normal shock can be swallowed in two ways: by increasing the flow velocity above the design Mach number at the diffuser inlet, or by utilizing a variable throat area diffuser. With the exception of very low Mach numbers, however, the power required to accelerate the flow beyond the design Mach number is quite considerable, rendering this technique totally impractical. In addition, the use of a variable throat area diffuser causes a variety of technical problems in tunnel operation. Due to these factors, the convergent-divergent diffuser is rarely utilized in supersonic wind tunnels. It is evident from the preceding discussion that constructing a fully efficient diffuser is practically impossible. However, pressure losses can be decreased by altering the design of a supersonic diffuser, as demonstrated in Fig. 6.9. In this case, the incoming supersonic flow is compressed by a succession of weak oblique shock waves followed by a weak normal shock, resulting in better pressure recovery. Such a diffuser is called the oblique shock diffuser. In this case, the static pressure at

6.2 Steady One-Dimensional Isentropic Flow Through Ducts

315

the diffuser exit can be made to equal the back pressure. In actual diffusers, however, shock wave and boundary layer interactions limit the static pressure rise and cause additional stagnation pressure losses, limiting the benefits of an oblique shock diffuser. Example 6.4 A supersonic wind tunnel has the test-section size of 2.5 m × 2.5 m. If the Mach number in the test-section is 2.2, find the area of the nozzle throat. By neglecting all other losses except that occurred due to normal shock wave, calculate the area of the diffuser throat and the power required to start the tunnel. Assume the pressure and temperature at the tunnel inlet to be 100 kPa and 290 K. Solution. Given, ATS = 2.5 × 2.5 = 6.25 m2 MTS = 2.2 p01 = 100 kPa T01 = 290 K By isentropic table for M = 2.2, we have ATS = 2.01 A1∗ 6.25 = 3.11 m2 A1∗ = 2.01 At the starting of the wind tunnel, a normal shock appears in the test-section. By normal shock table for M = 2.2, we write p02 = 0.628 p01 By Eq. (6.15), we write A2∗ =

A1∗ 3.11 = 4.95 m2 = p02/p01 0.628

During the start-up of the wind tunnel, the shock is located in the test-section at MTS = 2.2 where pp0201 = 0.628. The power input to the compressor to start the tunnel is   T0e −1 ˚ p T0i P=m ˚ (h0e − h0i ) = mc T0i where 0i represents the stagnation conditions at the compressor entrance and 0e represents the stagnation conditions at the compressor exit.

316

6 Steady One-Dimensional Isentropic Flow in a Variable-Area Duct

If the compressor is isentropic, the temperature ratio can be related to pressure ratio by using the isentropic relation   γ γ−1   γ γ−1 T0e p0e p01 = = T0i p0i p02   γ γ−1  1.4−1  1.4 p01 1 T0e =T0i = 290 p02 0.628 = 331.23 K The power required to start the tunnel is  P = mc ˚ p T0i

T0e −1 T0i



For air (γ = 1.4), this can be written as   T0e 0.6847p01 A1∗ P= cp T0i −1 √ T0i RT01  331.23 0.6847 × 105 × 3.11 1.4 × 287 −1 × 290 × × = √ 290 (1.4 − 1) 287 × 290 = 30569.20 kW The power required to start the supersonic tunnel with a normal shock in the test-section is therefore 30569.2 kW.

6.3 Flow Through an Air-Breathing Intake In order to have an efficient combustion in a jet aircraft flying at supersonic Mach numbers, the incoming air must be decelerated and compressed to subsonic speeds before entering the combustor. This is accomplished by utilizing an air-breathing intake2 or inlet. The most basic way of compression is to utilize a normal shock ahead of a Pitot intake, but this incurs large stagnation pressure losses, making it unsuitable for Mach 2 or higher. A better technique would be to generate a series of oblique shock waves that can increase pressure and drop Mach number before changing the flow condition to subsonic through a terminating normal shock. A series of numerous shock waves produces less entropy and so suffers fewer losses than a single normal shock wave for a given incoming flow Mach number. Depending on whether the oblique shock waves are created outside the intake or within the inlet duct, these systems are referred to as external or internal compression intakes. In 2

An intake is the device which recovers pressure energy by reducing the kinetic energy of the flow.

6.3 Flow Through an Air-Breathing Intake

317

either instance, the shock wave interacts with the boundary layer that forms along the intake surface (see Sect. 5.22). Oblique shock waves with supersonic flow on both sides characterize the majority of these interactions. It is worth mentioning that the generation of a terminal normal shock in the intake is not necessary when the combustion occurs at supersonic Mach numbers (see Sect. 6.4). Nevertheless, in all other circumstances when combustion occurs at subsonic speeds, each intake design includes a final terminating, near-normal shock that converts the flow from supersonic to subsonic speeds. Although designing inlets with weaker terminating shock waves is better, overall system size restrictions often limit the compression that oblique shock waves can achieve. The strong interaction between the shock wave and the boundary layer reduces intake efficiency significantly. Strong normal or near-normal shock waves enhance entropy and produce stagnation pressure loss, both of which have a direct impact on system performance. Furthermore, because the boundary layer has already encountered a number of unfavorable pressure gradient regions in previous interactions with shock waves, it is more susceptible to flow separation when the final shock wave strikes. Clearly, flow separation has a negative impact on inlet performance. It adds significant non-uniformity to the flow entering the subsonic diffuser or combustor, as well as additional stagnation pressure losses. In addition, any flow separations are likely to bring significant unsteadiness into the flow, resulting in excessive dynamic loads on the engine. When the terminal (nearnormal) shock oscillation approaches the converging portion of the inlet geometry, it becomes unstable. At this point, it flows rapidly upstream, causing more of the flow inside the inlet to become subsonic until it is eventually expelled from the intake, resulting in unstart (or buzz, if this phenomenon is periodic). On transonic wings, this is analogous to shock stall or shock-induced buffet; such a severe event is extremely harmful to the engine. To prevent the problems associated with strong shock wave and boundary layer interactions in intakes, the researchers use flow control to keep the boundary layer attached even when the shock waves are strong. The most prevalent type of control is the boundary layer suction, often known as bleed. Such a technique effectively manipulates the boundary layer characteristics to achieve the desired results. Air-breathing engines have three basic components: an intake, a combustion chamber, and a nozzle. For every 1% drop in intake stagnation pressure, engine thrust decreases by 1 − 1.5% (Intake Aerodynamics by J. Seddon and E. Goldsmith, 1999). Therefore, the overall performance of the engine depends on the efficiency of its components. Because boundary layer separation is a result of an unfavorable pressure gradient, it is more critical to design engine components with high accuracy when static pressure is rising rather than falling. Of course, designing an intake is far more complex than designing a nozzle that experiences favorable pressure gradients. Intakes are widely employed, particularly in jet engines, and are categorized as subsonic, supersonic, or hypersonic based on flight Mach numbers.

318

6 Steady One-Dimensional Isentropic Flow in a Variable-Area Duct

6.3.1 Flow Through a Subsonic Intake The flow entering the compressor of a turbojet engine is known to have a Mach number in the range of 0.4 to 0.7, with the upper limit acceptable for transonic compressors or fans. When the engine is required to operate at Mach 0.85, the intake must induce a flow reduction from 0.85 to about 0.6. While passing through an intake, the flow encounters both external and internal decelerations. A well-designed intake should eliminate or minimize boundary layer separation even during the pitch and yaw motions of the aircraft (i.e., during aircraft maneuvers). An intake should have minimal stagnation pressure loss and produce a uniform flow to the compressor. A non-uniform flow to the compressor not only reduces its efficiency, but it can also trigger flow-induced vibrations, which can lead to blade failure. The intake design is more complicated since it must perform consistently at both subsonic and supersonic speeds. A subsonic intake typically suffers mostly from the three types of losses listed below. 1. Losses due to wall friction. 2. Losses due to shock waves (at high subsonic or transonic flight conditions). 3. Losses due to flow separation. All of the above factors contribute to the reduction of stagnation pressure as the flow passes through the intake. Thus, if the stagnation pressures at the intake lip and exit are p0i and p0e , respectively, the ratio of stagnation pressures, denoted by rd , may be expressed as rd =

p0e

0.8 on minimum nacelle size is rather small. Alternatively, partial internal v∞ deceleration is found to be particularly effective for reducing the maximum diameter of the intake since it allows for reductions in both Ai and AAmaxi . Despite the fact that the preceding analysis is based on a simplified flow field surrounding the intake, it demonstrated that the performance of an intake is dependent on the pressure gradients on both internal and external surfaces, as well as the area ratio AAmaxi . In a turbojet engine, the pressure rise on the outer surface is restricted by external compression and AAmax , but the internal pressure rise is determined by the i flow deceleration between the inlet and the compressor (or combustion chamber in

6.3 Flow Through an Air-Breathing Intake

325

Fig. 6.13 Variation of area ratio, AAmax , with velocity i vi ratio, vmax , for f = 0.5

3.0

0.4 0.5

Cp

0.6

max

Amax/ A i

2.0

1.0

0

0.2

0.4

0.6

0.8

1.0

8

vi / v

case of a ramjet). Compressibility effects must be considered for a more practical analysis.

6.3.2 Performance Criteria of a Subsonic Intake The performance of an intake is measured in terms of either isentropic efficiency or stagnation pressure ratio. Let’s take a closer look at them in the following sections to acquire a better understanding.

6.3.2.1

Isentropic Efficiency

The working performance of an intake or a diffuser is defined by a parameter known as diffuser efficiency. The enthalpy-entropy diagram depicts a typical diffusion process occurring inside a diffuser, as shown in Fig. 6.14. State 1 depicts the state of the flow entering the diffuser, while state 2 depicts the actual state of the flow leaving the diffuser. If the diffusion process had been carried out isentropically, the state 2s represents the state of flow leaving the diffuser exit. Assuming that the flow velocity at the diffuser exit is negligibly small (v2 ≈ 0), the diffuser efficiency can be calculated as follows:

ηD =

h2s − h∞ h2 − h∞

(6.39)

Equation (6.39) demonstrates that the diffuser efficiency is essentially a ratio of the isentropic change in enthalpy to the actual change in enthalpy. For a perfect gas, h = cp T, Eq. (6.39) becomes

6 Steady One-Dimensional Isentropic Flow in a Variable-Area Duct

p0 p 02

8

0

02

02s 2s

p2

1 v2 2 2

2

1 8

1 2 v 2 1

1

p1 p

8

8

T

2

8

Enthalpy (h)

T0

8

8

326

Inlet Entropy (s) Fig. 6.14 The enthalpy-entropy diagram of an actual subsonic intake

T2s − T∞ T2 − T∞

(6.40)

T2s − T∞ T2s/T∞ − 1 = T02 − T∞ T02/T∞ − 1

(6.41)

ηD = Because v2 ≈ 0, T2 ≈ T02 ; therefore ηD =

Using Eqs. (2.257) and (4.49), we can write  γ −1  γ −1   T2s p2s γ p2 γ = = T∞ p∞ p∞ T02 γ −1 2 M∞ =1+ T∞ 2

(6.42) (6.43)

Therefore, Eq. (6.41) becomes  ηD =

p2 p∞

 γ γ−1

−1

γ −1 2 M∞ 2

(6.44)

However, p2 ≈ p02 since v2 ≈ 0; therefore  ηD =

p02 p∞

 γ γ−1

−1

γ −1 2 M∞ 2

Using the isentropic relation and rearranging the terms, we finally obtain

(6.45)

6.3 Flow Through an Air-Breathing Intake

 ηD =

1+

327

γ −1 2 M∞ 2



p02 p0∞

 γ γ−1

−1 (6.46)

γ −1 2 M∞ 2

Equation (6.46) demonstrates that for a given Mach number, M∞ , the diffuser effi02 . The flow encounters ciency is solely determined by the stagnation pressure ratio, pp0∞ a decrease of stagnation pressure near the diffuser exit due to the production of shock 02 < 1. The diffuser efficiency, on the other hand, can waves inside the diffuser, i.e., pp0∞ be maximized if the Mach number upstream of the shock is just above one, indicating that the shock is just downstream of the diffuser throat.

6.3.2.2

Stagnation Pressure Ratio

The effectiveness of the diffuser is also quantified in terms of the stagnation pressure ratio or pressure recovery, which is expressed as rd =

p02 p0∞

(6.47)

Thus, Eq. (6.46) can be written as

ηD =

 1+

γ −1 2 M∞ 2



(rd )

γ −1 γ

−1 (6.48)

γ −1 2 M∞ 2

From Eq. (6.48), it is evident that for a given M∞ , the diffuser efficiency is simply dependent on rd across the shock wave. The variations of rd and ηd with flight Mach number for a typical subsonic diffuser are shown in Fig. 6.15.

Fig. 6.15 Performance curves for a typical subsonic intake

1.0 rd

0.95

η

0.90

d

0.85 0

0.2

0.4 Mach number

0.6

0.8

1.0

328

6 Steady One-Dimensional Isentropic Flow in a Variable-Area Duct

Example 6.5 The stagnation pressure at the inlet and the exit of a subsonic intake are 101 kPa and 97 kPa. Find the pressure recovery of the intake. Solution. The stagnation pressure ratio or pressure recovery of a subsonic intake is p02 p0∞ 97 = 101 ≈ 0.96

rd =

6.3.3 Flow Through a Supersonic Intake It is generally understood that reducing the area of a duct causes supersonic flow to decelerate. In supersonic wind tunnels, this concept is frequently used by incorporating a second throat of a convergent-divergent duct that immediately follows the test-section. Due to the formation of a normal shock, the flow decelerates as it passes through the duct. The shock is usually placed ahead of the second throat, where the Mach number is slightly greater than one, to minimize pressure losses. In principle, the operation of a supersonic diffuser is analogous to the reverse operation of a convergent-divergent nozzle. Because of this, the convergent-divergent diffuser is also known as the reverse nozzle diffuser. The primary distinction between the reverse-nozzle diffuser and the second throat is that the former has an aerodynamic streamtube effect ahead of it, while the latter does not have one when utilized in a supersonic wind tunnel. The usage of a supersonic diffuser poses a number of practical challenges. For example, it must be able to fly at a wide range of flight Mach numbers without experiencing severe nacelle drag. Losses caused by shock and boundary layer interaction on internal and external surfaces exacerbate the problem. Moreover, under some situations, the flow field becomes highly oscillatory. As the nozzle inlet pressure affects the exhaust velocity, the rise in static pressure caused by decelerating the flow is frequently crucial for jet engine operation. It has been found that a 1% decrease in inlet stagnation pressure results in a 1 − 1.5% decrease in engine gross thrust (Intake Aerodynamics by J. Seddon and E. Goldsmith, 1999). Therefore, the efficient design of a diffuser is critical for the overall engine operation. Because the maximum rise in static pressure is the isentropic stagnation pressure, a shock-free diffuser operation is highly desirable. In supersonic wind tunnels, the normal shock wave located in the test section should be transmitted downstream of the second throat to minimize pressure losses. This is accomplished by either increasing the operating stagnation pressure or temporarily enlarging the second throat area. In supersonic intakes, the position of the shock at the throat is achieved by either momentarily over-speeding the intake or by using a variable-area geometry diffuser. At this point, we must recognize that shock location is an independent phenomenon that is unaffected by the boundary layer at

6.3 Flow Through an Air-Breathing Intake

(a)

(b)

Ai

A th

M 0; density increases. ρ dM2 < 0; Mach number decreases. M2

√1 γ

> 0; Mach number increases.

 2. When heat is added δq > 0 at supersonic speeds (M > 1): h

•

It is worth noting that when heat is rejected from the system, i.e., when δq < 0, the h consequences opposite to those mentioned above occur at both subsonic and supersonic Mach numbers. The variation of the thermodynamic and the flow properties during heating and cooling in each flow regime is summarized in Table 8.1. Remark It should be noted that Eq. (8.39) assumes an indeterminate form when Mach number becomes one (see Sect. 8.3), and the gradient of all properties p, T, ρ, v, etc., become infinite. This means that the flow achieves the sonic state at an incredibly rapid rate.

384

8 Frictionless Flow Through Constant-Area Ducts with Heat Transfer

M= dh ds

1 γ

=0

M=1

Temperature (T)

w

lo

cf

i on

bs

pe

=

Su

dh ds

8

rso

nic

flo

w

Su

Entropy (s)

Fig. 8.3 Rayleigh line on T − s plane

The effect of heat addition is to accelerate a subsonic flow and to decelerate a supersonic flow. Nevertheless, a constant-area steady one-dimensional subsonic flow cannot become supersonic or vice versa. However, in both cases, the flow will always approach sonic state.

8.4 Relationships Among the Thermodynamic Properties for a Rayleigh Flow In this section, the thermodynamic properties of a perfect gas in a Rayleigh process are obtained in terms of the Mach number. Consider the constant area, steady onedimensional flow through the control surface of Fig. 8.2 to accomplish this. The equations governing the flow are formulated in such a way that the relationships between the ratio of corresponding properties at stations 1 and 2 separated by a finite distance are obtained. We can begin by calculating the static pressure ratio for a from Eqs. (8.25) and (8.30), we see that Rayleigh process. By eliminating δq h dp γ M2 dM2 =− p 1 + γ M2 M2

(8.41)

8.4 Relationships Among the Thermodynamic Properties for a Rayleigh Flow

385

Table 8.1 Effects of heat transfer on thermodynamic and flow properties of a perfect gas passing through a constant-area duct Property Heat addition Heat rejection Subsonic (M < 1)

Supersonic (M > 1)

Subsonic (M < 1)

Supersonic (M > 1)

dv v dp p

Increases

Decreases

Decreases

Increases

Decreases

Increases

Increases

Decreases

dρ ρ

Decreases

Increases

Increases

Decreases

dT T

Increases when M < √1γ

Increases always

Decreases when M < √1γ

Decreases always

Decreases when M > √1γ

Increases when M > √1γ

Increases

Decreases

Decreases

Increases

Increases

Increases

Decreases

Decreases

Decreases

Decreases

Increases

Increases

Increases when M < √1γ

Increases

Decreases when M < √1γ

Decreases

dM2 M2 dT0 T0 dp0 p0 ds cp

Decreases when M > √1γ

Increases when M > √1γ

This, when integrated across the two flow stations, yields

ln

1 + γ M12 p2 = ln p1 1 + γ M22

p2 1 + γ M12 = p1 1 + γ M22

(8.42) (8.43)

Recall that the relation between the stagnation pressure of a perfect gas in terms of static pressure and the Mach number is 

γ −1 2 p0 = p 1 + M 2

 γ γ−1 (4.55)

By applying this relation to stations 1 and 2 of Fig. 8.2, the ratio of stagnation pressures is p02 p2 = p01 p1



1+ 1+

γ −1 2 M2 2 γ −1 2 M1 2

γ γ−1 (8.44)

386

Substituting

8 Frictionless Flow Through Constant-Area Ducts with Heat Transfer p2 p1

in Eq. (8.44) with Eq. (8.43) yields p02 1 + γ M12 = p01 1 + γ M22

Similarly, by eliminating

δq h



1+ 1+

γ −1 2 M2 2 γ −1 2 M1 2

γ γ−1 (8.45)

from Eqs. (8.26) and (8.30), we get dT 1 − γ M2 dM2 = T 1 + γ M2 M2

(8.46)

Integrating Eq. (8.46) between state points 1 and 2 in the flow yields the following result:  2 M22 1 + γ M12 T2 = 2 (8.47) T1 M1 1 + γ M22 Now, using the isentropic relation for temperature stagnation,   γ −1 2 T0 = T 1 + M 2

(4.49)

the stagnation temperature ratio between flow states 1 and 2 can be represented as

γ −1 T02 T2 1 + 2 M22 = (8.48) T01 T1 1 + γ −1 M12 2 Substituting

T2 T1

in Eq. (8.48) with Eq. (8.47) yields

 2 2 1 + γ −1 M T02 M22 1 + γ M12 2 2 = 2 2 T01 M1 1 + γ M22 1 + γ −1 M 1 2

(8.49)

The density ratio is expressed using the equation of state for a perfect gas as ρ2 p2 T1 = ρ1 p1 T2 which when combined with Eqs. (8.43) and (8.47) results in   ρ2 M12 1 + γ M22 = 2 ρ1 M2 1 + γ M12 The velocity ratio is calculated using Eqs. (8.51) and (8.8). This gives   v2 M2 1 + γ M12 = 22 v1 M1 1 + γ M22

(8.50)

(8.51)

(8.52)

8.4 Relationships Among the Thermodynamic Properties for a Rayleigh Flow

387

According to the definition of impulse function   F = pA 1 + γ M2

(4.141)

and Eq. (8.43), it is interesting to see that F2 p2 = F1 p1 1 + γ M12 = 1 + γ M22

 

1 + γ M22 1 + γ M12 1 + γ M22 1 + γ M12

 (8.53) 

=1

(8.54) (8.55)

Finally, Eq. (8.32) may be used to calculate the entropy change during a Rayleigh process: ds 1 − M2 dM2 = cp 1 + γ M2 M2 This, when integrated between two flow state points, yields ⎡ ⎤ γ +1 2  2 γ M s2 − s1 1 + γ M 1 ⎦ = ln ⎣ 22 cp M1 1 + γ M22

(8.56)

(8.57)

Before we conclude, it is necessary to investigate the effect of heat addition on the change in stagnation temperature. By applying the energy equation (Eq. (8.17)) to the control volume of Fig. 8.2 and integrating between the conditions indicated at stations 1 and 2, we have q = cp (T2 − T1 ) +

v22 − v12 = cp (T02 − T01 ) 2

(8.58)

where T01 and T02 denote the stagnation temperatures at the reservoir conditions before station 1 and after station 2, respectively, and q denotes the heat transfer per unit mass. Equation (8.58) demonstrates that in a heat exchange process, the change in stagnation temperature is a direct measure of the amount of heat transfer to the system.

8.5 Working Formulae for a Perfect Gas For the same reasons as stated for Fanno flow in Sect. 7.6, the flow properties at any station where M = 1 are the convenient reference states to normalize the equations

388

8 Frictionless Flow Through Constant-Area Ducts with Heat Transfer

developed in the preceding sections for a Rayleigh process. Thus, by setting M1 = 1 and M2 = M in the Rayleigh line equations and writing the flow properties corresponding to station 1 with a superscript asterisk, we obtain the following normalized expressions: p γ +1 = ∗ p 1 + γ M2

(8.59)

 ⎤ γ γ−1 ⎡

γ −1 2 2 1 + M 2 p0 γ +1 ⎣ ⎦ = p∗0 1 + γ M2 γ +1 T (γ + 1)2 M2 =  2 T∗ 1 + γ M2

(8.60)

(8.61)

 γ −1 2 2 2 + 1) M 1 + M (γ 2 T0 =  2 2 T0∗ 1 + γM

(8.62)

  1 + γ M2 ρ = ρ∗ (γ + 1) M2

(8.63)

v (γ + 1) M2  =  ∗ v 1 + γ M2

(8.64)

s − s∗ = ln M2 cp



γ +1 1 + γ M2

 γ γ+1 (8.65)

Similarly to Fanno flow, the change in a flow property, say pressure, between two stations of a duct with Mach numbers of M1 and M2 may be computed in a Rayleigh flow by establishing

 p

p∗ M p2 =  2 p p1 p∗

 where

p p∗

M2

 and

p p∗

M1

are the values of

p p∗

(8.66)

M1

corresponding to M1 and M2 , respec-

tively. The following program computes the variation of the dimensional ratios given by Eqs. (8.59)–(8.64) with Mach number for air (γ = 1.4), and the results are plotted in Fig. 8.4 and tabulated in Appendix F.

8.5 Working Formulae for a Perfect Gas

389

Fig. 8.4 Variation in flow properties along the Rayleigh line for air (γ = 1.4)

Listing 8.1 A MATLAB program for calculating Rayleigh flow properties. The results are tabulated in Appendix F. % V e r s i o n 1.1 C o p y r i g h t M r i n a l Kaushik , IIT Kharagpur , 2 9 / 0 7 / 2 0 2 1 clc c l e a r all format short gamma =1.4; % s p e c i f i c heat ratio binnum = 251; % n u m b e r of rows M = z e r o s ( binnum ,1) ; for i = 1 : 2 5 0 M ( i +1 ,1) = M ( i ,1) + 0 . 0 2 ; % Mach n u m b e r end T 0 b y T 0 s t a r = z e r o s ( binnum ,1) ; % T0 / T0 * T b y T s t a r = z e r o s ( binnum ,1) ; %T/T* p b y p s t a r = z e r o s ( binnum ,1) ; %p/p* p 0 b y p 0 s t a r = z e r o s ( binnum ,1) ; % p0 / p0 * r h o s t a r b y r h o = z e r o s ( binnum ,1) ; % rho */ rho for i = 1 : 2 5 1 T 0 b y T 0 s t a r ( i ,1) = (( g a m m a +1) * M (i ,1) ^2) * ( 2 + ( gamma -1) * M ( i ,1) ^2) / (1+ g a m m a * M ( i ,1) ^2) ^2; T b y T s t a r ( i ,1) = M ( i ,1) ^2 * ( (1+ g a m m a ) /(1+ gamma * M (i ,1) ^2) ) ^2; p b y p s t a r ( i ,1) = (1+ g a m m a ) /(1+ gamma * M ( i ,1) * M ( i ,1) ) ; p 0 b y p 0 s t a r ( i ,1) = ( ( ( 2 + ( gamma -1) * M (i ,1) ^2) /( g a m m a +1) ) ^( g a m m a /( gamma -1) ) ) * (1+ g a m m a ) / (1+ g a m m a * M (i ,1) ^2) ; r h o s t a r b y r h o ( i ,1) = ((1/ M ( i ,1) ^2) *( (1+ gamma * M (i ,1) ^2) / (1+ g a m m a ) ) ) ^ -1; end T = t a b l e ( M , T 0 b y T 0 s t a r , TbyTstar , pbypstar , p 0 b y p 0 s t a r , r h o s t a r b y r h o ) ; T . P r o p e r t i e s . V a r i a b l e N a m e s = { 'M ' ' T 0 b y T 0 s t a r ' ' T b y T s t a r ' ' p b y p s t a r ' ' p 0 b y p 0 s t a r ' ' r h o s t a r b y r h o ' }; w r i t e t a b l e ( T , ' R a y l e i g h _ F l o w . xlsx ' , ' S h e e t ' ,1) ;

390

8 Frictionless Flow Through Constant-Area Ducts with Heat Transfer

Example 8.1 Air at 400 kPa and 340 K enters a constant-area duct with a velocity of heat is added to the duct, compute (i) the final stagnation pressure, 150 ms . If 180 KJ kg (ii) the final static pressure, (iii) the final static temperature, (iv) the final flow velocity and the Mach number. Solution Given, p1 = 400 kPa,

T1 = 340 K,

v1 = 150

m , s

q = 180

KJ kg

The initial Mach number is v1 γ RT1 150

M1 = √

=√ 1.4 × 287 × 340 ≈ 0.41 That is, the original flow is subsonic. From the isentropic flow table, for M1 = 0.41, p1 = 0.89071, p01 Thus,

T1 = 0.96747 T01

p01 = pp011 × p1 1 = 0.89071 × 400 × 103 = 449.07 kPa

and

T01 = TT011 × T1 1 = 0.96747 × 340 = 351.43 K

Using Eq. (8.58), we get q = cp (T02 − T01 ) 180 × 103 = 1004.5 × (T02 − 351.43) T02 = 530.6 K From the Rayleigh flow table, for M1 = 0.41, T01 = 0.54651, T0∗

T1 = 0.63448, T∗

p1 = 1.9428, p∗

p01 = 1.1523 p∗0

8.5 Working Formulae for a Perfect Gas

Now,

391

= TT01∗ × TT0201 0 530.6 = 0.54651 × 351.43 = 0.8251 T02 T0∗

From the Rayleigh flow table, for M2 = 0.61,

T02 T0∗

= 0.8251,

p02 = 1.0717, p∗0

p2 = 1.578, p∗

T2 = 0.92653 T∗

(i) The final stagnation pressure is p∗

p02 = pp02∗ × p010 × pp011 × p1 0 1 1 = 1.0717 × 1.1523 × 0.89071 × 449.07 × 103 = 468.9 kPa (ii) The final static pressure is ∗

p2 = pp∗2 × pp1 × p1 1 = 1.578 × 1.9428 × 400 × 103 = 324.89 kPa (iii) The final static temperature is ∗

T2 = TT2∗ × TT1 × T1 1 = 0.92653 × 0.63448 × 340 = 496.5 K (iv) The final Mach number is

M2 = 0.61

Thus, the final velocity is √ v2 = √ M2 γ RT2 = 0.61 × 1.4 × 287 × 496.5 = 272.45 ms Example 8.2 The stagnation temperature of air flowing through a frictionless constant-area duct is increased by heat addition from 310 K to 530 K. Calculate the final Mach number and percentage change in pressure if the initial Mach number is 0.3. Solution Given, M1 = 0.3,

T01 = 310 K,

T02 = 530 K

392

8 Frictionless Flow Through Constant-Area Ducts with Heat Transfer

From the Rayleigh table, for M1 = 0.3, T01 = 0.34686, T0∗

p01 = 1.1985 p∗0

Thus, the critical stagnation temperature is T0∗ × T01 T01 1 = × 310 0.34686 = 893.73 K T0∗ =

This gives T02 530 = T0∗ 893.73 = 0.593 (i) From the Rayleigh table, for

T02 T0∗

= 0.593,

M2 = 0.45,

p02 = 1.1351 p∗0

(ii) The percentage change in pressure is  p01 − p02 × 100 p01  ∗ p02 p = 1 − ∗ × 0 × 100 p p 01 0   1.1351 = 1− × 100 1.1985 = 5.29%

p0 % = p01



That means, the pressure in the flow drops by 5.29 percent. Example 8.3 Consider the steady, one-dimensional choked flow through a frictionless constant-area duct while heat is added. If the original Mach number of the air is 0.4 and the temperature is 330 K, calculate (i) the amount of heat added to the duct, (ii) the exit temperature, and (iii) the maximum temperature in the flow. Solution Given, M1 = 0.4,

T1 = 330 K

From the isentropic flow table, for M1 = 0.4,

8.5 Working Formulae for a Perfect Gas

393

T1 = 0.96899 T01 This gives T01 = =

T01 × T1 T1

1 × 330 0.96899 = 340.56 K

Also from the Rayleigh flow table, M1 = 0.4, T01 = 0.52903, T0∗

T1 = 0.61515 T∗

Thus, the critical stagnation temperature is T0∗ = =

T0∗ T01 × × T1 T01 T1

1 × 1 × 330 0.52903 0.96899 = 643.75 K

Because the flow through the duct is choked, it can be presumed to correspond to the end state of the flow. In other words, at the exit, the flow achieves sonic speed, i.e., M2 = 1. Thus, T02 = T0∗ = 643.75 K (i) The amount of heat added to the duct can be calculated by using Eq. (8.58): q = cp (T02 − T01 ) = 1004.5 × (643.75 − 340.56) KJ = 304.55 kg (ii) The static temperature at the exit is T2 = T∗ T∗ = × T1 T1 1 × 330 = 0.61515 = 536.45 K (iii) The critical stagnation temperature is indeed the maximum temperature in the flow. Thus, T0,max = T0∗ = 643.75 K

394

8 Frictionless Flow Through Constant-Area Ducts with Heat Transfer

Example 8.4 Consider a steady, one-dimensional flow of air entering a frictionless constant-area duct at 150 ms with a pressure of 200 kPa and a temperature of 350 K. If 250 KJ of heat is added to the flow, calculate (i) the mass flow rate per unit area, kg (ii) the flow properties at the exit, and (iii) the inlet Mach number and the mass flow of heat is transferred to the flow. rate per unit area, if 675 KJ kg Solution Given, v1 = 180

m , s

p1 = 200 kPa,

T1 = 350 K,

q = 250

KJ kg

The Mach number at the duct entrance is v1 γ RT1 150

M1 = √

=√ 1.4 × 287 × 350 = 0.4 That is, the flow is subsonic at the duct entrance. From the isentropic table, for M1 = 0.4, p1 = 0.89561, p01 Thus, we have T01 = =

Using Eq. (8.58),

T1 = 0.96899 T01 T01 × T1 T1

1 × 350 0.96899 = 361.2 K

q =cp (T02 − T01 ) q T02 = + T01 cp 3 250 × 10 + 361.2 = 1004.5 = 610.08 K

From the Rayleigh flow table, for M1 = 0.4, T01 = 0.52903, T0∗

T1 = 0.61515, T∗

p1 = 1.9608, p∗

p01 = 1.1566 p∗0

8.5 Working Formulae for a Perfect Gas

(i) The mass flow rate is

395

m ˚ =ρ1 Av1 m ˚ p1 v1 = A RT1 200 × 103 = × 150 287 × 350 kg = 298.65  2  m .s

(ii) The following ratio can be used to calculate the exit conditions: T02 T01 T1 T02 × ∗ = ∗ × T0 T0 T01 T1 610.08 = 0.52903 × 0.96899 × 350 = 0.893 From the Rayleigh flow table, for M2 = 0.68,

T02 T0∗

T2 = 0.98144, T∗

= 0.893, p2 = 1.4569, p∗

p02 = 1.0489 p∗0

Therefore, the flow properties at the exit are as follows: T2 T∗ × × T1 T∗ T1 1 × 350 = 0.98144 × 0.61515 = 558.4 K T2 =

p2 p∗ × × p1 p∗ p1 1 × 350 = 1.4569 × 1.9608 = 260.05 kPa p2 =

and

p02 p∗0 p01 × × p1 ∗ × p0 p01 p1 1 1 × × 200 × 103 = 1.0489 × 1.1566 0.89561 = 202.52 kPa p02 =

(iii) Given, q = 675

KJ kg

396

8 Frictionless Flow Through Constant-Area Ducts with Heat Transfer

The amount of heat required to cause thermal choking, q∗ , equals  ∗  q∗ = cp  T0 − T01  = T0∗ −1 = cp = T01 T01  1 −1 = 1004.5 × 361.2 × 0.52903 KJ = 323 kg Since q > q∗ , the mass flow rate will be lowered and the initial conditions will be altered to allow for the extra heat released. Thus, the flow process will follow a Rayleigh line with smaller G. The stagnation conditions at the inlet, however, will remain unchanged. As no information regarding the exit condition is supplied in the problem, assume the flow is choked at the exit. Therefore, we may write   q∗ = q=cp T’∗0 − T01 or

q cp + T01 675 × 103 + 361.2 = 1004.5 = 1033.17 K T ∗0 =

The stagnation temperature ratio under changed inlet conditions is T01 T ∗0

From the Rayleigh table, for conditions is

T01 T ∗0

361.2 1033.17 = 0.3496 =

= 0.3496, the Mach number under changed inlet M1 = 0.3

From the isentropic flow table, for M1 = 0.3, p1 = 0.93947, p01 Thus,

T1 = 0.98232 T01

p1 p01 × × p1 p01 p1 1 × 200 × 103 = 0.93947 × 0.89561 = 209.79 kPa p1 =

8.5 Working Formulae for a Perfect Gas

397

T1 × T01 T01 = 0.98232 × 361.2 = 354.81 K T1 =

The mass flow rate under changed inlet condition becomes m ˚  = ρ1 v1 A √ p1 m ˚ = M1 γ RT1 A RT1 √ p1 γ m ˚ =√ M1 A  RT1√ 209.79 × 103 × 1.4 = × 0.3 √ 287 × 354.81 kg = 233.36  2  m .s The percentage change in mass flow rate is ⎛ % Change in  =

m ˚ ⎜ =⎝ A

m ˚ A

−

m ˚⎞ A ⎟ × 100 ⎠

m ˚ A 298.65 − 233.36 × 100 298.65 = 21.86%

The preceding analysis shows that when q > q∗ , the mass flow rate is lowered by 21.86%.

Concluding Remarks This chapter examines a steady, one-dimensional flow in a constant-area duct with heat transfer without friction, body forces, and obstruction to the flow. Such a flow is called a simple diabatic flow, which exhibits distinct characteristics depending on whether the initial flow is subsonic or supersonic. If the flow is initially subsonic (M < 1), the heat addition increases the Mach number and decreases the corresponding static pressure. The opposite trends occur when heat is removed from the flow. The static enthalpy increases with heating and achieves a maximum when M = √1γ , decreasing as the Mach number approaches M = 1, even though the heat is added continuously. For a given initial condition, the flow state with M = 1 is called thermal choking. If the original flow is supersonic (M > 1), the heat addition reduces

398

8 Frictionless Flow Through Constant-Area Ducts with Heat Transfer

the Mach number and increases the static pressure. Once again, opposite effects are seen on cooling the flow. When the Mach number is reduced to M = 1, the flow is thermally choked.

Exercise Problems Exercise 8.1 Consider the steady, one-dimensional frictionless airflow through a constant-area duct with heat addition. Show that (i) the Mach number, at the point of maximum temperature, is equal to √1γ , where γ is the ratio of specific heats, and (ii) the Mach number, at the point of maximum entropy, is equal to one. Exercise 8.2 Under heat addition, air at 101 kPa and 298 K accelerates from Mach 0.3 to Mach 0.9 through a frictionless constant-area duct. Determine the amount of heat added to the duct as well as the percentage change in pressure. Exercise 8.3 Air at 300 kPa and 290 K enters a duct with a diameter of 10 cm. Calculate the amount of heat required to choke the flow at the duct exit. Determine the pressure, temperature, and flow velocity at the exit and at the section with a Mach number of 0.75. Exercise 8.4 Consider a one-dimensional steady flow through a frictionless constant-area duct. At a section with a Mach number of 4, the stagnation temperature is 290 K and the static pressure is 80 kPa. Find the stagnation temperature, the stagnation pressure, the static pressure, and the amount of heat transfer to cause the flow to decelerate in another section where the Mach number is 2. In this case, would heat be added or rejected? Exercise 8.5 Air at 101 kPa and 298 K is decelerated from Mach 2 to Mach 1 through a duct with a diameter of 30 mm. Calculate the change in static temperature as well as the amount of heat transfer that caused this deceleration.

Chapter 9

Steady, Inviscid, and Adiabatic Multi-dimensional Compressible Flows

Abstract The one-dimensional theory provided in the preceding chapters considerably increased our understanding of gas dynamics. This concept was also extended to two dimensions in the supersonic case. However, to analyze subsonic and transonic flows and general three-dimensional flows, general equations of motion must be derived. They are developed in this chapter.

9.1 Introduction The one-dimensional analysis of fluid flows presented in the previous chapters is well suited to flows where the properties vary only along the flow direction (axial direction) and have negligible variations in the other two transverse directions. For example, the flow through a constant-area duct is conveniently analyzed from the one-dimensional point of view. The properties are averaged over each cross section of the duct and vary only along the flow direction. In actual flows, however, the change in properties in the directions, along with and normal to flow, are of comparable magnitudes, making the flow three dimensional. Thus, the properties in a real flow must be computed at each point of the flow domain. For example, the flow through the turbomachinery, the flow past the wings and fuselage of an aircraft, the flow through a rapidly changing-area flow passage are some cases where the flow is three dimensional and hence cannot be analyzed from a one-dimensional perspective. A general three-dimensional flow occurs in the presence of many irreversibilities that include friction, heat transfer, shock and expansion waves, heat source or sink, etc. When all of these factors are considered, the resulting mathematical equations become too complected to solve even by modern computational techniques. Therefore, to obtain the closed-form solutions, it becomes necessary to employ suitable approximations that transform the governing equations in simple analytical form. One such assumption is to consider the large expanse of fluid outside the boundary layer, often termed the potential flow region, inviscid and adiabatic. This assumption is justified because, according to Prandtl’s boundary layer theory, the viscous and heat transfer phenomena are confined only to a thin layer adjacent to a surface, called the viscous layer or the boundary layer. To a first-order approximation, the potential © Springer Nature Singapore Pte Ltd. 2022 M. Kaushik, Fundamentals of Gas Dynamics, https://doi.org/10.1007/978-981-16-9085-3_9

399

400

9 Steady, Inviscid, and Adiabatic Multi-dimensional Compressible Flows

flow is almost independent of the flow in the boundary layer. Such an approximation is valid even in those cases where the shock waves are present in the flow field. It is assumed that the entire flow domain upstream and downstream of the shock is non-viscous and non-heat conducting. The shock is the only discontinuity present that divides the flow into two continuous domains. Another vital approximation, frequently employed in three-dimensional flows, is the irrotationality condition, which simplifies the mathematical equations to a great extent. Moreover, the assumption of fluid to be a perfect gas allows us to use the ideal gas law, which further simplifies the governing equations.

9.2 Governing Differential Equations This section derives the differential equations of motion for an irrotational, inviscid, adiabatic flow of a perfect gas in the absence of body forces under a shock-free environment.

9.2.1 Continuity Equation The first governing equation is the equation of continuity, which is based on the conservation of mass. Let u, v, w be the velocity components in the x-, y- and zdirections, and ρ be the density of the fluid, at an arbitrary point in Cartesian space, P (x, y, z), at a specified time, t. Consider a small control volume with sides dx, dy, and dz and a vertex P, as shown in Fig. 9.1. The mass leaving the control volume in the x-direction is   ∂ρu ∂ρu dx dydz − ρudydz = dxdydz (9.1) ρu + ∂x ∂x Likewise, the mass exiting the control volume in y- and z-directions are   ∂ρv ∂ρv ρv + dy dxdz − ρudxdz = dxdydz ∂y ∂y   ∂ρw ∂ρw ρw + dz dxdy − ρudxdy = dxdydz ∂z ∂z

(9.2) (9.3)

Thus, the total mass leaving the control volume is 

 ∂ρu ∂ρv ∂ρw dxdydz + + ∂x ∂y ∂z

(9.4)

9.2 Governing Differential Equations

401 y ρv +

δρv dy δy ρw

ρu +

dy

δ ρu dx δx

P + δ P dx δx dx ρu P

x

P (x, y, z) dz Infinitesimal control volume

z

ρw +

δ ρw dz δz

ρv

Fig. 9.1 Infinitesimal control volume in a flow

According to the conservation of mass principle, Eq. (9.4) must be equal to the time rate of mass decrease in the control volume. Thus,   ∂ρ ∂ρu ∂ρv ∂ρw + + dxdydz = − dxdydz (9.5) ∂x ∂y ∂z ∂t ∂ρ ∂ρu ∂ρv ∂ρw + + + =0 (9.6) ∂t ∂x ∂y ∂z In vector notation, Eq. (9.6) can be written as  → ∂ρ + ∇. ρ − v =0 ∂t

(9.7)

This is the general equation of continuity or the equation of conservation of → mass, where − v = uˆi + vˆj + wkˆ denotes the velocity vector. Under steady conditions, Eq. (9.7) yields  → ∂ρu ∂ρv ∂ρw ∇. ρ − v = + + =0 ∂x ∂y ∂z

(9.8)

which, in the case of a steady, two-dimensional compressible flow, becomes  → ∂ρu ∂ρv ∇. ρ − v = + =0 ∂x ∂y

(9.9)

402

9 Steady, Inviscid, and Adiabatic Multi-dimensional Compressible Flows

Example 9.1 Which of the following velocity vectors could describe the probable flow of an incompressible fluid? → v = (x + y) ˆi + (x − y + z) ˆj + (x + y + 3) kˆ (i) −   − → (ii) v = xyˆi + yzˆj + yz + z2 kˆ  → v = x (y + z) ˆi + y (x + z) ˆj + − (x + y) z − z2 kˆ (iii) − Solution The continuity equation is a necessary and sufficient condition for a velocity field to describe a conceivable flow. The continuity equation for an incompressible flow under steady conditions is → ∇.− v =0 (i) → ∇.− v =



∂ ˆ ∂ ˆ i + ∂∂y ˆj + ∂z k ∂x

=

∂ ∂x



. (x + y) ˆi + (x − y + z) ˆj + (x + y + 3) kˆ

(x + y) +

∂ ∂y

(x − y + z) + =1−1+0 =0

∂ ∂z

(x + y + 3)

Thus, the velocity field given represents the possible flow of an incompressible fluid. (ii)

   → ∂ ˆ ∂ ˆ ∂ ˆ ∇.− v = ∂x i + ∂v j + ∂z k . xyˆi + yzˆj + yz + z2   ∂ ∂ ∂ yz + z2 = ∂x xy + ∂y yz + ∂z = 2y + 3z = 0 So, the velocity field shown above does not represent the possible flow of an incompressible fluid. (iii) → ∇.− v =



  . x (y + z) ˆi + y (x + z) ˆj + − (x + y) z − z2 kˆ  ∂ ∂ − (x + y) z − z2 + ∂y y (x + z) + ∂z = y + z + x + z − x − y − 2z =0

∂ ˆ ∂ ˆ ∂ ˆ i + ∂y j + ∂z k ∂x ∂ = ∂x x (y + z)

which implies that the velocity field represents the possible flow of an incompressible fluid.

9.2.2 Momentum Equation Consider an inviscid, adiabatic flow of a compressible fluid through an infinitesimal control volume in the absence of body forces, as shown in Fig. 9.1. The flow

9.2 Governing Differential Equations

403

occurs only in the presence of inertial and pressure forces under the aforementioned conditions. Thus, by using Newton’s second law of motion in the x-direction, we get ρ

∂p Du dxdydz = − dxdydz Dt ∂x Du ∂p ρ =− Dt ∂x

(9.10) (9.11)

where D() is the substantial or material derivative defined by Eq. (2.11). Likewise, Dt applying Newton’s second law of motion in the y- and z-directions yields ∂p Dv =− Dt ∂y ∂p Dw =− ρ Dt ∂z ρ

(9.12) (9.13)

The vector addition of Eqs. (9.11)–(9.13) yields ρ

 −  → →  − D− v ∂→ v =ρ + − v .∇ → v = −∇p Dt ∂t

(9.14)

Using the vector identity,  ∇

v2 2

 =

 →  −   1 − → → → ∇ → v .− v = − v .∇ → v +− v × ∇ ×− v 2

(9.15)

Equation (9.14) becomes → ∂− v +∇ ∂t



v2 2



  1 → → −− v × ∇ ×− v = − ∇p ρ

(9.16)

9.2.3 Energy Equation The conservation of energy principle, or the first law of thermodynamics, yields for an inviscid and adiabatic flow:   dp 1 = dh − (9.17) dq = du + pd ρ ρ and the second law of thermodynamics states for a reversible process that dq = Tds

(9.18)

404

9 Steady, Inviscid, and Adiabatic Multi-dimensional Compressible Flows

By combining Eqs. (9.17) and (9.18), we have   1 1 = dh − dp Tds = du + pd ρ ρ

(9.19)

This, when written in vector notation, gives 1 ∇p ρ

T∇s = ∇h −

(9.20)

By definition, the stagnation enthalpy, h0 , is the sum of the static enthalpy, h, and 2 the kinetic energy per unit fluid mass, v2 , i.e., h0 = h +

v2 2

(9.21)

Differentiating the above equation with respect to time t results in → Dh0 Dh − D− v = +→ v. Dt Dt Dt

(9.22)

Equation (9.19) may be written in the form T Substituting

Dh Dt

Dh 1 Dp Ds = − Dt Dt ρ Dt

(9.23)

in Eq. (9.23) using Eq. (9.22), we obtain

→ Ds 1 Dp − D− v Dh0 =T + +→ v. Dt Dt ρ Dt Dt   →  Dh0 Ds 1 ∂p − D− v → =T + + → v .∇ p + − v. Dt Dt ρ ∂t Dt  −  → Dh0 Ds 1 ∂p − Dv 1 =T + +→ v. + ∇p Dt Dt ρ ∂t Dt ρ

(9.24) (9.25) (9.26)

Using Eq. (9.14), the equation above reduces to Ds 1 ∂p Dq 1 ∂p Dh0 =T + = + Dt Dt ρ ∂t Dt ρ ∂t

(9.27)

For an adiabatic flow, q = 0; thus  ∂h0 − Dh0 1 ∂p = + → v .∇ h0 = Dt ∂t ρ ∂t

(9.28)

9.2 Governing Differential Equations

405

and  ∂s − Ds = + → v .∇ s = 0 Dt ∂t

(9.29)

Equations (9.28) and (9.29) are the energy equations for the three-dimensional adiabatic flow of a compressible fluid. For the steady flow, these equations give →  Dh0 = − v .∇ h0 = 0 Dt  Ds − = → v .∇ s = 0 Dt

(9.30) (9.31)

These relationships show that for steady adiabatic flow, both stagnation enthalpy and entropy are constant along a streamline. Remark According to chemical kinetics, if the flowing fluid is in chemical equilibrium or its chemical composition is frozen (reaction rates are zero), then all of the flow processes are reversible. Under steady conditions, the above governing equations reduces to  → ∇. ρ − v =0 − → 1 Dv + ∇p = 0 Dt ρ   Dh0 D 1 2 = h+ v =0 Dt Dt 2 Ds =0 Dt

(9.32) (9.33) (9.34) (9.35)

→ where under steady conditions, D() =− v .∇ (). It is worth noting that Eqs. (9.32)– Dt (9.35) are derived for a small control volume (fluid particle). For example, the isentropic result of Eq. (9.35) only applies to the fluid particle; it does not guarantee that the entropy is constant throughout the flow. This result, however, can be extended to steady flow by recognizing that the particle paths (pathlines) coincide with streamlines, implying that the entropy along streamlines must be constant. It may, however, differ on different streamlines; the conditions that cause this will be examined in Sect. 9.5. Similarly, for the steady, adiabatic flow of a compressible fluid in the 2 absence of body forces, Eq. (9.34) for a fluid particle yields h + v2 = constant = h0 . Again, under steady conditions, since the pathlines and streamlines are the same, the stagnation enthalpy, h0 , would be constant for each streamline.

406

9 Steady, Inviscid, and Adiabatic Multi-dimensional Compressible Flows

9.2.4 Equation of State In order to describe the flow field of a compressible fluid, the equations of state for the fluid under consideration must be used in addition to the aforementioned set of equations. The equation of state is the relationship between the pressure p, temperature T, and density ρ of the fluid. The general form of this equation is as follows: p = p (ρ, T)

(9.36)

For a perfect gas, the equation of state is p = ρRT

(2.208)

where R is the specific gas constant. It is simple to demonstrate that the equation of state for an ideal gas following a non-isentropic process is represented as   γ  ρ s − s0 p (9.37) = exp p0 ρ0 c∀ Equation (9.37) usually applies to rotational flows. It is later proved that for irrotational flows, the entropy has a constant value throughout the flow domain. Thus, Eq. (9.37) reduces to  γ p ρ = (9.38) p0 ρ0 where the subscript 0 represents a reference state.

9.3 Circulation, Rotation, and Vorticity Circulation, rotation, and vorticity are three parameters that can accurately reflect the numerous features of multi-dimensional flows. This section describes these parameters and provides examples of how they might be utilized.

9.3.1 Circulation Frederick Lanchester, Wilhelm Kutta, and Nikolai Joukowski were the first to use the term circulation independently. The circulation is defined as the line integral of the velocity vector around a closed spatial fluid curve in a flow field and is denoted

9.3 Circulation, Rotation, and Vorticity

407

Fluid curve ’C’

v n +Γ P

P

dA

(b) dA

(a)

Fig. 9.2 A closed fluid curve to evaluate the circulation

by . Consider a basic fluid curve C in a flow field around a point P, as shown in Fig. 9.2. Each arc segment on the curve C can be thought of as an infinitesimal length − → vector dl with the magnitude d and orienting in the tangent direction on the curve C. Thus, the circulation around the closed curve C is given by ˛ − → → = − v . dl (9.39) c

Circulation is a kinematic property that is solely determined by the velocity field and the choice of the curve C. When the line integral is performed in the counter− → → clockwise direction on C, the scalar product − v . dl is deemed positive. If we express − → → the velocity vector − v and the length vector dl in Cartesian coordinates as − → v = uˆi + vˆj + wkˆ

− → dl = (dx) ˆi + (dy) ˆj + (dz) kˆ

(9.40) (9.41)

Equation (9.39) becomes ˛ udx + vdy + wdz

= c

Alternatively, using the Stokes theorem, the circulation can be defined over the surface A bounded by the fluid curve C. " ‹

‹ ˛ − →− → − → − → B . dl = ∇ × B .d A (9.42) C

A

408

9 Steady, Inviscid, and Adiabatic Multi-dimensional Compressible Flows

− → where A is the vector area bounded by the fluid curve C. Thus, Eq. (9.39) yields ˛ "   − → − → → → = − ∇ ×− v .dA (9.43) v . dl = A

c



 ˆ Equation k. dz   → (9.43) demonstrates that the circulation is zero if the curl of velocity ∇ × − v over

→ → where ∇ × − v = curl of − v =

dw dy

−

dv dz



ˆi +

 du

−

dw dx



ˆj +



dv dx

−

du dy

any surface bounded by C is zero. If the curve C shrinks to the point that the circulation around it is given by d, Eq. (9.43) can be represented as   −  →  → → d = ∇ × − v .dA = ∇ × − v .ˆndA   d → ∇ ×− v .ˆn = dA

(9.44) (9.45)

where dA denotes the infinitesimal area bounded by the fluid curve C. Thus, when → the circulation is taken around the boundary of dA, the component of ∇ × − v normal to dA is equal to the circulation per unit area at any point P in the flow, as shown in Fig. 9.2b.

9.3.2 Fluid Rotation By definition, fluid rotation at a point is the mean or average rotation of two orthogonal lines passing through that point. In fact, this notion of fluid rotation is a generalization of solid body rotation. Because a point, say, P, always maintains a fixed relative position with some adjoining point, Q, the rotation defined by the perpendicular lines passing through the latter point is identical to that at the former. Consequently, specifying the rotation about one line on the solid body fully specifies the rotation about all other lines. In the case of fluids or deformable bodies, however, the situation is different. Fluid lines in any flow are allowed to change their relative positions with regard to one another, and such displacement is possible. Therefore, in the case of a fluid, the rotation of a fluid particle at any point is characterized in terms of the mean or average rotation about that point, which is known as the rotation of the fluid. This section will look at how a fluid element changes shape and orientation as it moves along a streamline. Consider a rectangular-shaped element at the beginning of the motion. If the flow velocity varies considerably across the span of the fluid element, the corners may not move in unison, causing the element to rotate and become deformed. In fact, the edges of the element are tilted and stretched in some way. However, because of its higher aerodynamic consequences, we shall solely address the titling motion in this discussion. Pure rotation occurs when the neighboring sides of the fluid element rotate equally and in the same direction; whereas, pure shearing motion occurs when the adjacent sides rotate equally but in opposite directions.

9.3 Circulation, Rotation, and Vorticity

409

The absence of rotational motion simplifies the fluid motion governing equations to a great extent. Furthermore, shearing stresses are produced by the shearing action between neighboring elements and the fluid viscosity, which are responsible for the drag and separation phenomena. The velocity field affects the quantum of distortion and rotation; our objective is to investigate this interdependence. Consider a fluid particle in a velocity field with velocity gradients in the x- and y-directions, as shown in Fig. 9.3a. At time t, the centroid of the fluid particle is positioned at point O, and its Cartesian velocity components along the x- and ydirections are u and v, respectively. The orthogonal fluid lines OA and OB, in Fig. 9.3, originate from the fluid particle at point O. As a result of the motion of the fluid particle and the fluid lines, the points O, A, and B have migrated higher and to the right at the later time t + dt, as depicted in Fig. 9.3b. The x-component of velocity at the bottom and top surfaces of the fluid element is calculated as u (y0 ) and u (y0 + dy), respectively. For small dy, the velocity component over the top surface of the fluid dy, using Taylor’s series element can be represented as u (y0 + dy) = u (y0 ) + du dy expansion. Similarly, the y-components of velocity on the left and right sides of the fluid element are v (x0 ) and v (x0 + dx), respectively. For small dx, the velocity on dx. The velocity gradients cause the right face of the fluid element is v (x0 ) + dv dx the fluid element to deform and rotate after the time dt. The amount of translation, deformation, and rotation that occurs during the time dt is expanded for clarity in Fig. 9.3b. In fact, the fluid element appears to change almost insignificantly during the short time dt. The deformation and rotation of the fluid element during the time dt travel with the element and become superimposed, as shown in Fig. 9.3c. The side OA of the fluid element rotates to the small angle dα (clockwise rotation) due to the gradient of the y-component of the velocity in the x-direction, whereas the side OB rotates to the small angle −dβ (counterclockwise rotation) due to the gradient of the x-component of the velocity in the y-direction. The distance moved by the point A relative to the point O in time dt equals the difference in velocity between the right and left sides multiplied by the time dt.   dv dx dt (9.46) = dx Similarly, the distance moved by point B relative to point O in time dt is equal to the difference in velocity between the top and bottom sides multiplied by the time dt.   du dy dt (9.47) = dy → Let − ω denote the angular velocity, and assuming counterclockwise rotation is positive, the angular velocity of the side OA is ωOA =

dα dt

(9.48)

410

9 Steady, Inviscid, and Adiabatic Multi-dimensional Compressible Flows y

y 0 + dy

y u(y0 + dy) = u(y0 ) + du dy dy (x +dx, y +dy) 0 0 B

B dβ

dy

y0

v(x 0 )

t + dt

t

v(x 0 + dx) = v(x 0) + dv dx dx

dx

A dα

A

Ο u(y0 ) x0

Ο x

x 0 + dx

x

(a)

(b) ( du dy dy ) dt B ω

dβ

+ dy

( dv dx dx ) dt

dα

O

dx

A

(c) Fig. 9.3 The rotational motion of a fluid element

and the angular velocity of side OB is ωOB = −

dβ dt

(9.49)

Because the time dt is very small, the angles dα and dβ are likewise very small. Therefore, we may write  dv

 dx dt dv tan (dα) ≈ dα = = dt dx

dx  du dy dt dy du tan (dβ) ≈ dβ = = dt dx dy dx

(9.50) (9.51)

9.3 Circulation, Rotation, and Vorticity

411

Combining Eqs. (9.48)–(9.51) yields dv dx du =− dy

ωOA = ωOB

(9.52) (9.53)

The component of fluid rotation or angular velocity, ω, about the z-axis is defined as the average of ωOA and ωOB . Thus, we may write ωz =

1 (ωOA + ωOB ) 2

Substituting Eqs. (9.52) and (9.53) into Eq. (9.54), we get   1 dv du − ωz = 2 dx dy

(9.54)

(9.55)

Correspondingly, the angular velocity components about x- and y-axes are   1 dw dv − 2 dy dz   1 du dw − ωy = 2 dz dx ωx =

(9.56) (9.57)

→ The vector addition of Eqs. (9.55)–(9.57) yields the angular velocity vector, − ω = ˆ given by ωx ˆi + ωyˆj + ωz k, 1 − → ω = 2



       dw dv ˆ du dw ˆ dv du ˆ 1 → − − − ∇ ×− v i+ j+ k = dy dz dz dx dx dy 2 (9.58)

Example 9.2 Show that the velocity field expressed by the vector − → v = −kyˆi + kxˆj has a uniform angular velocity around the origin of magnitude k. Also calculate the circulation around the circle, x2 + y2 = a2 , where a is the radius of the circle. Solution Given, − → v = −kyˆi + kxˆj When compared to the velocity vector in two dimensions,

412

9 Steady, Inviscid, and Adiabatic Multi-dimensional Compressible Flows

− → v = uˆi + vˆj yields the velocity components u = −ky and v = kx The angular velocity around the origin is



∂v 1 ∂(−ky) = − − ωz = 21 ∂u ∂y ∂x 2 ∂y =

1 2

(−k − k) = −k

∂(kx) ∂x



Thus, the magnitude of the angular velocity around the origin, |ωz |, is k. The circulation around the given circle, x2 + y2 = a2 , in the xy-plane may now be calculated using Eq. (9.43): ˜ dv du  dxdy − z = A dx ˜ dy = −2k A dxdy = −2k × (area of circle) = −2π ka2 That is, the magnitude of the circulation is 2π ka2 .

9.3.3 Vorticity and the Vortex Tubes The vorticity is another useful parameter to work with in fluid flows. It is defined as → the curl of the velocity vector, ∇ × − v , and is specified for each moment of time t and each location in a flow field. The vorticity vector, or vorticity in brief, is denoted − → by ζ , as shown below. − → → → ζ =∇ ×− v = 2− ω

(9.59)

which shows that the vorticity is twice the angular velocity. The vorticity is essentially a measure of the moment of momentum of a small circular fluid particle about its own center of mass. The dynamical theorems of Kelvin and Helmholtz (discussed later in this chapter) relate the vorticity of a fluid particle to the moments acting on it. − → In Cartesian space, the vorticity vector, ζ , can be expressed in terms of its components as

9.3 Circulation, Rotation, and Vorticity

− → ζ =



413

     dw dv ˆ du dw ˆ dv du ˆ − − − i+ j+ k dy dz dz dx dx dy

(9.60)

From Eqs. (9.45) and (9.60), it is seen that the vorticity for a two-dimensional flow is equal to the circulation per unit area. Thus, ζz =

d = 2ωz dA

(9.61)

A vortex line is an imaginary line whose points are all tangent to the vorticity vector (Fig. 9.4a), similar to the concept of a streamline in a flow, which stipulates that all of the points on a streamline are tangent to the velocity vector. Along a vortex → line the components of angular velocity vector, − ω , i.e., ωx , ωy , and ωz in the x-, yand z-directions, respectively, are related to the corresponding components dx, dy, and dz of the tangent vector to the vortex line as follows: dx dy dz = = ωx ωy ωz

(9.62)

Equation (9.62) is the equation of a vortex line in Cartesian coordinates. Figure 9.4b exhibits a flow field with a closed fluid curve C that permits vortex lines to pass through, resulting in the formation of a vortex tube. It is worth noting that a vortex tube with an infinitesimal cross section is referred to as a vortex filament. Remark Equation (9.61) leads to two significant conclusions. The flow is considered rotational if the curl of the velocity vector at each point in a flow field is not − → → zero, i.e., ∇ × − v = 0. This is a finite vorticity with a finite angular velocity situation. If, on the other hand, the curl of the velocity vector is zero at each point in a − → → flow field, i.e., ∇ × − v = 0, the flow is called irrotational. Because the flow field has zero vorticity everywhere, the fluid elements have zero angular velocity, i.e., − → → ζ = 2− ω = 0. Consequently, the fluid elements in an irrotational flow have only translational motion. By equating Eq. (9.58) to zero, we get

dw dv − =0 dy dz du dw − =0 dz dx dv du − =0 dx dy

(9.63) (9.64) (9.65)

Thus, in order to have an irrotational flow field, Eqs. (9.63)–(9.65) must be satisfied concurrently. It should be emphasized that in an irrotational flow, velocity gradients may exist, which are primarily responsible for fluid element deformations.

414

9 Steady, Inviscid, and Adiabatic Multi-dimensional Compressible Flows

Fig. 9.4 Illustrations of a vortex line and a vortex tube

y

Vorticity vector

Vortex line

o

x

z

(a) A vortex line in a flow

y Vortex tube

Fluid curve ’C’

Vortex lines passing through the curve ’C’

o

x

z

(b) A vortex tube in a flow

9.4 Kelvin’s Theorem In fluid flows, it is vital to investigate the effect on the circulation around a closed fluid curve C, such as that depicted in Fig. 9.5, as it moves with the flow. It is worth noticing that the curve C, which is closed at the beginning, will remain closed at later points in time due to the continuity of the flow. Despite this, the shape of the curve may change, but it will always be a closed curve. It should also be noted that the fluid particles in the curve C are the same at all times. Thus, the sole effect which we have to examine is the change in circulation, , around the curve C as a function of time t. According to the Lagrangian description (Sect. 2.2.3.1), the time derivative of a fluid element positioned on the fluid curve is, by definition, the substantial or material derivative. Thus, Eq. (9.39), the time derivative of the circulation, , around the curve C can be represented as j D D − → − → v . dl (9.66) = Dt Dt C

9.4 Kelvin’s Theorem

415

→ where − v is the velocity vector at any point on the curve C. When the scalar product on the right side of Eq. (9.66) is differentiated, the result is

− → j j → D dl D D− v − → → v. = . dl + − (9.67) Dt Dt C Dt C j = C

j → D− v − → → → . dl + − v .d− v Dt C

(9.68)

j → D− v − → 1 dv2 . dl + Dt 2 C

(9.69)

j = C

The second term in Eq. (9.69) vanishes because the line integral of an exact differential over a closed curve is zero. This results in j → D D− v − → = . dl (9.70) Dt C Dt By combining Eqs. (9.33) and (9.69), we get  j  1 1 D − → =− ∇p . dl Dt 2 C ρ Using the Stokes theorem (Eq. (9.42)), the equation above becomes  ‹ " ‹  ∇p − 1 D → ∇× =− .d A Dt 2 A ρ

(9.71)

(9.72)

− → where A is the vector area bounded by the fluid curve C. Take note of this:     ∇p 1 1 1 ∇× = ∇ × ∇p + ∇ × ∇p = ∇ × ∇p ρ ρ ρ ρ ∇p = ∇ × (H − T∇S) = −(∇T × ∇S) ∇× ρ Because the curl of a gradient identically vanishes, i.e., ∇ × ∇p = ∇ × ∇H = ∇ × ∇S = 0 Combining Eqs. (9.72)–(9.74), we obtain

(9.73) (9.74)

416

9 Steady, Inviscid, and Adiabatic Multi-dimensional Compressible Flows y

Fluid curve comprises of same particles at all times

ines

aml

Stre

C

Fluid curve at time t + dt

C

Fluid curve at time t o

x

z

Fig. 9.5 A fluid curve at different points in time

 ‹ ‹ " ‹    " ‹ 1 1 − → − → ∇ × ∇p .d A = (∇T × ∇S) .d A ρ 2 C C (9.75) This is an striking result that shows that the circulation around a closed curve is not a constant in general, even when the fluid under consideration is inviscid. If the will not be zero. stagnation enthalpy, H0 , and the entropy, S, are not constant, D Dt For a barotropic fluid, the pressure, p, is solely a function of the density, ρ, i.e., p = p (ρ). We may write   1 dp ∇p =∇× ∇ρ = ∇ × ∇ (F (ρ)) = 0 (9.76) ∇× ρ ρ dρ 1 D =− Dt 2

Therefore, we finally have D =0 Dt

(9.77)

which, when integrated, yields j =

− → − → v . dl = constant

(9.78)

C

which is the well-known Kelvin’s theorem According to which, for a barotropic fluid, the circulation around a closed fluid curve with a fixed number of particles is time

9.4 Kelvin’s Theorem

417

invariant. Thus, if the circulation is initially zero over the flow domain, it will remain zero at all times. Because a flow with  = 0 is referred to as an irrotational flow, Eq. (9.78) implies that the flow will remain irrotational forever. If, on the other hand, the flow is initially rotational, it will remain thus indefinitely. The results of Kelvin’s theorem may be extended to flows governed by conservative body forces, which can be described as the gradient of a scalar potential. One major application of Kelvin’s theorem is to initially uniform and thus irrotational flows. This initial condition causes the entire flow domain to be irrotational for all subsequent points in time, greatly simplifying the mathematical equations. It is worth noting that Kelvin’s theorem also applies to incompressible flows, = 0. In conwhere ρ = constant and ∇ × ρ1 ∇p = ρ1 (∇ × ∇p) = 0; therefore D Dt trast, Kelvin’s theorem is only valid for a compressible flow when the fluid is barotropic, i.e., p = p (ρ). Because pressure is a function of both density and entropy in a compressible fluid, i.e., p = p (ρ, S), Kelvin’s theorem applies only when the entropy is constant across the flow field. Kelvin’s theorem is described in its most basic form in Sect. 9.4, where the fluid is considered to be barotropic and the flow occurs under conservative body forces with negligible viscous forces. However, it is intriguing to investigate the consequences of Kelvin’s theory when all three forces, namely, the pressure force, the body force, and the viscous force, act in a flow. In the aforementioned situation, Kelvin’s theorem can be expressed as D Dt  Change of circulation

=

j dp − ρ   

j

term due to pressure forces

j

− →− → G .dr   

+

+

term due to body forces

μ 2− − → ∇ → v .dr ρ   

(9.79)

term due to viscous forces

Equation (9.79) show that the time rate of change of circulation around a given identity fluid curve is regulated by the torque produced by all the forces acting in a fluid, including pressure forces, body forces, and viscous forces.

Fluid particle

Torque

Center of gravity

Viscous force

Viscous force

Fig. 9.6 Torques exerted on a fluid particle as a result of pressure forces

418

9 Steady, Inviscid, and Adiabatic Multi-dimensional Compressible Flows

Let us first consider the torque caused by viscous forces acting on a fluid particle of fixed identity, as seen in Fig. 9.6. The force diagram of the fluid particle shows that viscous forces can induce torque around the center of gravity or center of mass. This torque has the potential to alter the vorticity of the fluid particle and hence the circulation. G

Circular fluid particle Center of gravity

Fig. 9.7 Torques exerted on a fluid particle as a result of body forces

− → Let us now consider the body forces, G , as the circulation altering component. If the body forces are irrotational, that is, conservative, the expression − →− → G .dr equals zero. However, for non-conservative forces, this term is rarely equal to zero. As seen in Fig. 9.7, the line of action of a body force acting on a fluid particle goes through the latter’s center of gravity. As is evident, this force produces no torque and hence has no effect on the circulation or vorticity. Gravity is an example of a centrally directed force. They are either irrotational or conservative in nature. The electrostatic forces generated by the electric charges ı− →− → on the particles are another form of conservative force. We have G .dr = 0 for irrotational flows. Furthermore, two key rotational forces affect circulation and vorticity: the Coriolis forces in a rotating reference frame and the electromagnetic forces caused by a current flowing at an angle to the magnetic field. In either case, the line of action of the body force does not have to pass through the center of gravity of the fluid particle. Because of such rotational forces, the ocean and atmosphere are filled with vorticity. The torque due to pressure forces is the third circulation altering factor. = 0; consequently, Incidentally, when the fluid is incompressible, the term dp ρ torque owing to pressure forces cannot modify the circulation. Furthermore, if the line of action of the pressure forces passes through the center of gravity, they create no torque and hence no vorticity or circulation. If, on the other hand, the line of action of the pressure forces does not pass through the center of gravity, it is capable of producing torque and so causes vorticity and circulation. From the preceding, it is clear that if the fluid is incompressible, the body forces are irrotational, and the viscous forces are small, the circulation and hence the vorticity will always be zero. Consequently, the flow will continue to be irrotational indefinitely.

9.5 Crocco’s Theorem and Shock-Induced Vorticity

419

9.5 Crocco’s Theorem and Shock-Induced Vorticity There are several theorems that relate vorticity to fluid flow dynamics. One of these is Crocco’s theorem, which is essentially a special form of the dynamical equation of motion. We saw in Sect. 9.2 that the stagnation enthalpy, h0 , and the entropy, s, are constant along a streamline if the flow is steady, inviscid, and adiabatic. In this section, we will look at how h0 and s vary in the direction normal to streamline, i.e., from one streamline to the next. For convenience, the analysis is performed in the streamline coordinates, commonly known as the natural coordinates (, n), where the direction along the streamline is denoted by  and the direction normal to it is denoted by n. Furthermore, as demonstrated in Fig. 9.8, we will limit our analysis to a two-dimensional flow; however, for a general three-dimensional case, an additional direction perpendicular to both  and n should also be accounted in the flow analysis. In terms of natural coordinates, the velocity vector in two-dimensions may be → represented as − v (v, θ ), where v is the magnitude and θ is the direction. Thus, the flow field, as shown in Fig. 9.8, becomes a function of the coordinates  and n: − → v (v, θ ) = f (, n)

(9.80)

It should be noted that the natural coordinates (, n) are curvilinear, therefore the governing equations derived in orthogonal space are not directly applicable. However, because these equations are developed through one-dimensional analysis, they may still be applied to a one-dimensional streamtube formed by two streamlines at n distance apart, as shown in Fig. 9.8. The main difference is that in this case, the momentum equation must be considered in both normal and streamline directions. Continuity Equation The streamlines are n distance apart, and if unit depth is considered orthogonal to the n−plane, Eq. (4.3) yields m ˚ = ρv n = constant

(9.81)

−Momentum Equation Using Eq. (4.26), one can obtain the momentum equation along the streamline direction, i.e., the −momentum equation as v

1 ∂p ∂v =− ∂ ρ ∂

(9.82)

n−Momentum Equation The component of momentum equation in normal to streamline direction, i.e., n−momentum equation, can be determined by taking into account the fact that a

420

9 Steady, Inviscid, and Adiabatic Multi-dimensional Compressible Flows

δ u Δn u + δn δΔl Δn + δn Δl

n l to rma e No amlin stre

θ+

δθ δn δn

+ Δn

δΔn Δl δl

l Δn

ine

aml

Stre

s of e adiu n r = r treamli s

u

Δl

θ

Fig. 9.8 Illustration of a two-dimensional flow in a streamline coordinate system

fluid element traveling along a curved path encounters centrifugal force that is balanced by the pressure differential across it. ρv2 ∂p ∂p =− = ρv2 of r ∂n ∂

(9.83)

Energy Equation The energy equation is written as follows: h+

v2 = h0 2

(9.84)

where the stagnation enthalpy, h0 , remains constant along a streamline. Also, by Gibbs equation, dp ρ

(2.203)

dh + vdv = dh0

(9.85)

Tds = dh − Differentiating Eq. (9.84) yields

9.5 Crocco’s Theorem and Shock-Induced Vorticity

By substituting Eq. (9.85) into the Gibbs equation, we get   dp + dh0 Tds = − vdv + ρ

421

(9.86)

Equation (9.86) can be expressed in two directions, normal to the streamline and along the streamline, as follows:

T

  ∂s ∂v 1 ∂p ∂h0 =− v + since = 0 along a streamline ∂ ∂ ρ ∂ ∂   ∂v 1 ∂p ∂h0 ∂s =− v + + T ∂n ∂n ρ ∂n ∂n

(9.87) (9.88)

Introducing Eq. (9.82) into Eq. (9.87) yields

T

∂s =0 ∂

(9.89)

Integrating, s = constant

along a streamline

(9.90)

By combining Eqs. (9.83) and (9.88), we obtain

T

  ∂s ∂v v ∂h0 = −v − + ∂n ∂n r ∂n

(9.91)

Equation (9.91) demonstrates the variation of the entropy normal to streamline. This is the well-known Crocco’s theorem, which is alternatively stated as

T

∂h0 ∂s = + vζ ∂n ∂n

(9.92)

where ζ is the vorticity as defined by

ζ =

v ∂v ∂θ ∂v − =v − r ∂n ∂ ∂n

(9.93)

It is evident from Eq. (9.92) that vorticity (or angular velocity) is governed by the gradients of stagnation enthalpy and entropy normal to streamlines. Thus, if the stagnation enthalpy, h0 , is uniform throughout the flow domain and the vorticity,

422

9 Steady, Inviscid, and Adiabatic Multi-dimensional Compressible Flows

ζ , is zero, the inviscid compressible flow under discussion will remain isentropic1 throughout. Furthermore, if the vorticity in a flow is zero, the inviscid property of the fluid has a direct correspondence with the isentropic behavior of the flow. Simply put, an inviscid flow with zero vorticity, i.e., irrotational flow, is always isentropic, and conversely, an isentropic flow is always inviscid and irrotational (zero vorticity) flow, provided h0 is uniform throughout in both circumstances. For a perfect gas, h0 = constant 2 indicates T0 = constant, consequently the entropy has a direct relationship with the stagnation pressure in a flow with ζ = 0 (Eq. (4.64)). In this situation, the vorticity can be used to calculate the change in stagnation pressure between the streamlines. By combining Eq. (4.64) with Eq. (9.92) and noting h0 = constant, we get

ζ =T

RT dp0 ∂s =− ∂n vp0 p0

(9.94)

It is possible that all streamlines have the same stagnation enthalpy, but the entropy varies from one to the next. Because the entropy change depends on the shock angle, this would be the case in the region downstream of a bow-shock. Under these conditions, the flow downstream of a shock would be rotational even though the flow upstream is irrotational. This suggests that the bow-shock causes rotational motion and vorticity. Consequently, there would be an entropy layer downstream of a curved shock, and this phenomenon occurs regardless of whether the flow upstream is inviscid. The fluid in a uniform parallel flow experiences adiabatic and reversible changes as it flows along the streamlines. Because reversible changes in a flow are identical to the statement that there is no friction involved, Eq. (9.92) implies that a steady, adiabatic, and frictionless3 flow is everywhere irrotational. The Prandtl theory of boundary layer asserts that the static pressure is uniform throughout the boundary layer, but the velocity and thus the stagnation pressure are variable. The change in stagnation pressure confirms the presence of vorticity within the boundary layer. Furthermore, a high-velocity gradient at the wall results in a high stagnation pressure gradient at the wall. Crocco’s theorem states that when the vorticity is zero, the stagnation pressure must be the same everywhere. In vector notation, Crocco’s theorem for an unsteady flow is often expressed as →   ∂− v → → T∇s + − v × ∇ ×− v = ∇h0 + ∂t

1

Also known as the homentropic. A constant stagnation enthalpy flow is also called a homenergetic or isoenergetic flow. 3 A reversible and adiabatic flow is called the isentropic flow. 2

(9.95)

9.5 Crocco’s Theorem and Shock-Induced Vorticity

423

Crocco’s theorem is discussed in Sect. 9.5 for a steady, inviscid flow of a barotropic compressible fluid. However, if the fluid is incompressible, what are the implications of Crocco’s theorem? In this discussion, we will attempt to provide a qualitative response to this question. Consider a two-dimensional incompressible and inviscid flow in the xy plane with conservative body forces, as depicted in Fig. 9.9. Crocco’s theorem asserts that for steady-flow conditions, the vector product of the velocity vector and the vorticity vector equals the gradient of the stagnation pressure divided by the fluid density, i.e., 

      1 Gradient of Velocity Vorticity × = stagnation pressure vector vector Density 1 − → − → v × ζ = ∇p0 ρ

(9.96) (9.97)

where p0 denotes the stagnation pressure, also known as Bernoulli’s number. It is defined using the incompressible Bernoulli’s equation as p0 =

p  Static pressure

+

1 2 ρv 2  

+

Dynamic pressure

ρU 

(9.98)

Pressure associated with body forces

y es

lin

eam

Str

v

ζ

p

0

O

z

Fig. 9.9 An incompressible flow in the two-dimensional (xy) plane

x

424

9 Steady, Inviscid, and Adiabatic Multi-dimensional Compressible Flows

− → As shown in Fig. 9.9, the vorticity vector, ζ , is normal to the xy plane (pointing outward in the z-direction) at a point in the flow, and the velocity vector, − → v , is tangent to the streamline at the point under consideration. According to − → → v and ζ , Eq. (9.97), the gradient of stagnation pressure, ∇p0 , is normal to both − and thus it lies in the xy plane. Consequently, the stagnation pressure is constant along each streamline and varies only when vorticity exists. However, if the vorticity in a flow field is zero, Crocco’s theorem implies that the stagnation pressure is the same everywhere in the flow. To better understand this, consider the boundary layer on a solid surface. According to Prandtl’s theory, the static pressure at each point in the boundary layer is constant. However, the velocity inside the boundary layer changes, resulting in large velocity gradients in the wall region. These significant velocity gradients create a strong stagnation pressure gradient, revealing the presence of vorticity near the wall. Example 9.3 Show that the variation of stagnation pressure normal to streamline direction for a perfect gas is given by   γ −1 2 γ − 1 2 dT0 1 dp0 = cp M + 1+ M vζ − ρ0 dn 2 dn 2 where symbols have their usual meaning. Hence, for an incompressible flow, the total pressure gradient is related to vorticity by dp0 = −ρ0 vζ dn Solution Using Crocco’s theorem, T

dh0 ds = + vζ dn dn

(9.92)

For a perfect gas, h0 = cp T0 and cp = constant. The preceding equation becomes T

dT0 ds = cp + vζ dn dn

By multiplying this equation by T0

T0 , T

we obtain

T0 ds T0 dT0 = cp + vζ dn T dn T

(A)

Using the isentropic relation, T0 γ −1 2 =1+ M T 2

(4.49)

9.5 Crocco’s Theorem and Shock-Induced Vorticity

425

Equation (A) becomes     γ − 1 2 dT0 ds γ −1 2 T0 = cp 1 + M + 1+ M vζ dn 2 dn 2

(B)

By using the Gibbs equation, Tds = dh −

1 dp ρ

(2.203)

which, in terms of stagnation properties, becomes T0 ds = dh0 − ρ10 dp0 ds 0 0 T0 dn = cp dT − ρ10 dp dn dn Combining Eqs. (B) and (C) yields

 

γ −1 2 dT0 γ −1 2 1 dp0 0 1 + vζ − = c M + 1 + M cp dT p dn ρ0 dn 2 dn 2 

0 0 0 0 − ρ10 dp = cp γ −1 M2 dT +cp dT − cp dT + 1 + γ −1 M2 vζ dn 2 dn dn  2

dn 0 0 − ρ10 dp = cp γ −1 M2 dT + 1 + γ −1 M2 vζ dn 2 dn 2

(C)

(D)

For an incompressible flow, M ≈ 0; thus Eq. (D) reduces to −

1 dp0 = vζ ρ0 dn

or dp0 = −ρ0 vζ dn

9.6 Irrotational Flows and the Velocity Potential Function (φ) For an irrotational flow, the vorticity, ζ , is zero everywhere in the flow. Thus, Eq. (9.59) requires that − → → ζ =∇ ×− v =0 (9.99) − → → where ζ is the vorticity vector, and − v is the velocity vector. As indicated earlier, if the flow field is irrotational, i.e., conservative, the velocity vector can be expressed as the gradient of a scalar function φ, such that − → v = ∇φ

(9.100)

426

9 Steady, Inviscid, and Adiabatic Multi-dimensional Compressible Flows

where the scalar function φ is usually called the velocity potential function or velocity potential in brief. Because, − → → ζ =∇ ×− v =∇ × ∇φ =0 the condition of irrotationality is satisfied. The velocity potential is in general a function of spatial coordinates (x, y, z) and time t, i.e., φ = φ (x, y, z, t). The relations between the velocity potential and the velocity components (u, v, w) are u=

∂φ ∂φ ∂φ , v= , w= ∂x ∂y ∂z

(9.101)

Example 9.4 Show that a flow field defined by the velocity vector     − → v = 3xy2 − 2x + 1 i + 3x2 y + y ˆj ı is irrotational. Also, compute the line integral c (udx + vdy) around the closed fluid curve C bounded by x = 0, y = 0, and 2x + 4y = 8. Solution Given,

    − → v = 3xy2 − 2x + 1 i + 3x2 y + y ˆj

When compared to the velocity vector in two dimensions, − → v = uˆi + vˆj yields the velocity components u = 3xy2 − 2x + 1 and v = 3x2 y + y The vorticity in the xy-plane for a two-dimensional flow is given by

=

d dx

− du ζz = dv dx  dy    d 2 3xy − 2x + 1 − dy 3x2 y + y = 6xy − 6xy =0

Because the vorticity is zero, the flow field is irrotational, i.e., conservative. Figure 9.10 illustrates that the fluid curve described by ı the coordinates x = 0, y = 0, and 2x + 4y = 8 is closed, and that the line integral c (udx + vdy) around the closed curve is always zero provided that the flow field is conservative. Consequently, we have j j     3xy2 − 2x + 1 dx + 3x2 y + y dy = 0 udx + vdy = c

c

9.6 Irrotational Flows and the Velocity Potential Function (φ) Fig. 9.10 Closed fluid curve to compute the line integral

427

y

(0, 4)

B

A O

(0, 0)

x

(4, 0)

9.6.1 The General Governing Equation in Terms of φ The governing differential equations derived in Sect. 9.2 can be represented in terms → of the velocity potential, φ, rather than the velocity vector, − v , by plugging Eq. (9.100) into Eq. (9.16). This results in  2 ∂∇φ v 1 → +∇ −− v × (∇ × ∇φ) + ∇p = 0 (9.102)    ρ ∂t 2 =0

Since ∇ operator has only spatial derivatives, Eq. (9.102) can be written as 

∇

 2 ˆ  ∂φ v dp +∇ + =0 ∂t 2 ρ ˆ ∂φ v2 dp + + = F (t) ∂t 2 ρ

(9.103) (9.104)

In general, F (t) is a constant that is unaffected by time. A unique time-dependent condition may occur in a closed wind tunnel if the pressure across the system is forced to fluctuate due to an overall change in the tunnel volume. Such changes may occur, but they are usually inconsequential unless there is an aim to adjust the tunnel volume for a specific reason, such as flow control. Thus, we may write

428

9 Steady, Inviscid, and Adiabatic Multi-dimensional Compressible Flows

∂φ v2 + + ∂t 2

ˆ

dp = Bc = constant ρ

(9.105)

Equation (9.105) denotes the general form of Bernoulli’s equation, which was earlier derived in Sect. 2.6, where Bc is Bernoulli’s constant. Differentiating Eq. (9.105) with respect to time t yields ˆ  → ∂ ∂ 2φ − ∂− v dp → + =0 +v. ∂t 2 ∂t ∂t ρ ˆ  → ∂ 2φ − ∂ ∂− v dp ∂p → + =0 + v . 2 ∂t ∂t ∂p ρ ∂t → ∂ 2φ − 1 ∂ρ ∂− v → + v . + a2 =0 2 ∂t ∂t ρ ∂t where, a =

(9.106) (9.107) (9.108)

 

∂p , ∂ρ s

is the local speed of sound. Let us now rewrite the continuity

equation as follows: 1 ∂ρ 1→ + ∇2φ + − v .∇ρ = 0 ρ ∂t ρ 1 ∂ρ 1→ 1 v . ∇p = 0 + ∇2φ + 2 − ρ ∂t a ρ  −  → →  − 1 ∂ρ ∂ v 1 → v. − +∇ 2 φ + 2 − − − v .∇ → v =0 ρ ∂t a ∂t

(9.109) (9.110) (9.111)

Combining Eqs. (9.108) and (9.111), we have →  → 1 ∂ 2φ v 2 → ∂− 1 → − = ∇2φ − 2 − v. v. → v .∇ − v + 2− 2 2 a ∂t a ∂t a

(9.112)

By writing (9.112) in Cartesian coordinates, we obtain

1−

u2 a2



∂2φ + ∂x2



1−

v2 a2

=







2 ∂2φ + 1 − wa2 ∂y2 ∂2φ 1 ∂2φ + 2u + a2 ∂t2 a2 ∂x∂t

∂2φ − ∂z2 2v ∂ 2 φ a2 ∂y∂t

∂ φ ∂ φ ∂ φ 2 uv − 2 vw − 2 wu a2 ∂x∂y a2 ∂y∂z a2 ∂z∂x 2

+

2w ∂ 2 φ a2 ∂z∂t

2

2

(9.113) This is the governing partial differential equation for the velocity potential, φ, in a three dimensional, irrotational, inviscid, isentropic flow of a compressible fluid. (9.113) is a nonlinear partial differential equation that is difficult to solve. Consequently, unlike incompressible flows, superposition of solutions is not a feasible technique. The intricacy is increased further by the fact that the speed of sound, a,

9.6 Irrotational Flows and the Velocity Potential Function (φ)

429

becomes a variable. For a perfect gas, it is related to the flow velocity, v, by the relation: a2 = a02 −

 γ −1 2 γ −1 2 φx + φy2 v = a02 − 2 2

(4.95)

where a0 is the stagnation speed of sound in the fluid. The advantage of Eq. (9.113) is that a single relation in φ concurrently satisfies the conservation laws of mass, momentum, and energy. For the steady, three-dimensional incompressible flow, ∂p = ∞; so Eq. (9.113) becomes a2 = ∂ρ ∇2φ =

∂ 2φ ∂ 2φ ∂ 2φ + + = φxx + φyy + φzz = 0 ∂x2 ∂y2 ∂z2

(9.114)

This is the well-known Laplace’s equation, which only applies to a steady, irrotational flow of an incompressible fluid. For the steady, two-dimensional irrotational flow of a compressible fluid, Eq. (9.113) yields     uv ∂ 2 φ u2 ∂ 2 φ v2 ∂ 2 φ −2 2 =0 1− 2 + 1− 2 a ∂x2 a ∂x∂y a ∂y2     u2 uv v2 1 − 2 φxx − 2 2 φxy + 1 − 2 φyy = 0 a a a

(9.115) (9.116)

9.6.2 The General Characteristics of the Velocity Potential Equation Consider the two-dimensional velocity potential equation, given by Eq. (9.116), and compare it with the general second-order partial differential equation in two variables: Aφxx + Bφxy + Cφyy + Dφx + Eφy + Fφ = G

(9.117)

We get     uv u2 v2 A = 1 − 2 , B = −2 2 , C = 1 − 2 , D = 0, E = 0, and F = 0 a a a (9.118) The discriminant of Eq. (9.116) is

430

9 Steady, Inviscid, and Adiabatic Multi-dimensional Compressible Flows

= B2 − 4AC   

uv 2 v2 u2 = −2 2 − 4 1 − 2 1− 2 a a a  2  u + v2 =4 −1 a2   = 4 M2 − 1

(9.119) (9.120) (9.121) (9.122)

Consider the following cases: 1. < 0 for M < 1; the velocity potential equation is said to be elliptic.4 2. = 0 for M = 1; the velocity potential equation is said to be parabolic.5 3. < 0 for M > 1; the velocity potential equation is said to be hyperbolic.6 It is worth noting that the subsonic, sonic, and supersonic flows have different types of potential equations due to the physical differences in their flow fields. However, if the flow field is of mixed type, as in a transonic flow, the governing equations likewise are of more than one type, and thus the analysis becomes cumbersome.

9.7 The Stream Function (ψ) In the preceding article, it is demonstrated that the irrotationality condition is a necessary and sufficient condition for the existence of the velocity potential. In this section. it is shown that the continuity equation is a necessary and sufficient condition to define another point function, ψ, called the stream function. For a steady, two-dimensional flow, the continuity equation is ∂ρu ∂ρv + =0 ∂x ∂y

(9.9)

It is apparent from the above equation that a point function, ψ, may be defined in such a way that it satisfies the foregoing equation identically. Therefore, ρu =

∂ψ = ψy ∂y

and

ρv = −

∂ψ = −ψx ∂x

(9.123)

Substituting Eq. (9.123) into the continuity equation confirms that the former equations are solutions to the latter. Consequently, it is worth noting that obeying the steady, two-dimensional continuity equation is the only necessary and sufficient condition for the existence of a stream function.7 4

An elliptic velocity potential equation has no integral or characteristic curves. A parabolic velocity potential equation has one characteristic curve. 6 A hyperbolic velocity potential equation has two characteristic curves. 7 Between two lines of constant ψ, the difference in ψ is a measure of the mass flow rate. 5

9.7 The Stream Function (ψ)

431

9.7.1 The General Governing Equation in Terms of ψ This section develops a single differential equation that describes the steady, two dimensional, inviscid, irrotational, isentropic flow of a compressible fluid. The approach for obtaining the governing equation in terms of the stream function, ψ, is identical to that used for the potential function, φ. If the velocity components, given by Eq. (9.123), are substituted in the following irrotationality condition: ωz =

dv du − =0 dx dy

(9.65)

We obtain     d ψx d ψy − − =0 dx ρ dy ρ 1 1 1 1 ψxx − 2 ψx ρx + ψyy − 2 ψy ρy = 0 ρ ρ ρ ρ     ρ ψxx + ψyy − ψx ρx + ψy ρy =0

(9.124) (9.125) (9.126)

For a steady, two dimensional, inviscid irrotational flow, the momentum equation (Eq. (9.16)) becomes  2 dp v +d =0 (9.127) ρ 2 dp ρ d =− dx 2 dx



ψx2 + ψy2



ρ2

 ⎤   2 2 2 ρ ψ − ψ ψ + ψ ψ + ψ x xx y yx x y ρρx dp ⎦ = −ρ ⎣ dx ρ4

(9.128)

⎡

1 dp = ρ dx

 ψx2 + ψy2 ρx ρ3

  ψx ψxx + ψy ψyx − ρ2

Since the flow is isentropic, the speed of sound in the gas is given by  dp a= dρ Combining Eqs. (9.130) and (9.131) yields

(9.129)

(9.130)

(9.131)

432

9 Steady, Inviscid, and Adiabatic Multi-dimensional Compressible Flows

   ψx2 + ψy2 ρx ψx ψxx + ψy ψyx a2 ρx = − ρ ρ3 ρ2 

  ρ 2 a2 ρx = ψx2 + ψy2 ρx − ρ ψx ψxx + ψy ψyx   ρ ψx ψxx + ψy ψyx  ρx =

ψx2 + ψy2 − ρ 2 a2

(9.132) (9.133) (9.134)

Similarly, we can obtain   ρ ψx ψxy + ψy ψyy  ρy =

ψx2 + ψy2 − ρ 2 a2

(9.135)

Substituting the expressions for ρx and ρy into Eq. (9.126), we get ⎧ ⎡  ⎡  ⎤  ⎤⎫ ⎨ ρ ψx ψxx + ψy ψyx ρ ψx ψxy + ψy ψyy ⎬ ⎦ + ψy ⎣

⎦ =0   ρ ψxx + ψyy − ψx ⎣

⎩ ψ 2 + ψ 2 − ρ 2 a2 ψ 2 + ψ 2 − ρ 2 a2 ⎭ 



x



ψxx + ψyy

y

x



 2     ψx + ψy2 − ρ 2 a2 − ψx ψx ψxx + ψy ψyx − ψy ψx ψxy + ψy ψyy = 0 

ψy2 1− 2 2 ρ a 



  2 ψ2 ψxx + 2 2 ψx ψy ψxy + 1 − 2 x 2 ψyy = 0 ρ a ρ a

u2 1− 2 a



 2uv ψxx + 2 ψxy + 1 − a

v2 a2

(9.136)

y

(9.137) (9.138)

 ψyy = 0

(9.139)

This is the governing partial differential equation for the stream function, ψ, in a two dimensional, irrotational, inviscid, isentropic flow of a compressible fluid. For a perfect gas, the speed of sound, a, is related to the flow velocity, v, by the relation:   γ −1 2 γ − 1 ψx2 + ψy2 2 2 2 v = a0 − a = a0 − (4.95) 2 2 ρ2 Equation (9.139) is a differential equation with the same form as the analogous differential equation for velocity potential (Eq. (9.113)). For an incompressible flow, the speed of sound becomes infinite, and Eq. (9.139) is reduced to ∇ 2 ψ = ψxx + ψyy = 0 This is a two-dimensional form of Laplace’s equation.

(9.140)

9.7 The Stream Function (ψ)

433

9.7.2 Relationship Between φ and ψ For a two-dimensional flow along a streamline, we know that φ (x, y) = constant, and so dφ = 0. Because φ is an exact differential, we can write dφ =

∂φ ∂φ dx + dy = φx dx + φy dy = 0 ∂x ∂y

(9.141)

Introducing φx and φy from Eq. (9.101) into Eq. (9.141), we get dφ= udx + vdy = 0

(9.142)

or 

dy dx

 φ=constant

=

u v

(9.143)

For a two-dimensional flow along a potential line, ψ (x, y) = constant, implying dψ = 0; and since ψ is an exact differential, we may write dψ=

∂ψ ∂ψ dx + dy = ψx dx + ψy dy = 0 ∂x ∂y

(9.144)

Substituting Eq. (9.123) into Eq. (9.144) yields dψ= −vdx + udy = 0 or



dy dx

 ψ=constant

=

v u

It is clear from the comparison of Eqs. (9.143) and (9.146) that     dy dy × = −1 dx φ=constant dx ψ=constant

(9.145)

(9.146)

(9.147)

i.e., lines of constant φ (equipotential lines) and constant ψ (streamlines) are orthogonal. Thus the potential lines and streamlines form an orthogonal set of lines that completely describes the flow in a two-dimensional irrotational flow field. Such a net is called the flow net, as shown in Fig. 9.11. Example 9.5 The velocity potential function in a steady, two-dimensional incompressible flow is φ = x + x2 − y2 . Find the stream function.

9 Steady, Inviscid, and Adiabatic Multi-dimensional Compressible Flows

Fig. 9.11 A two-dimensional flow with orthogonal potential lines and streamlines forms the flow net

ψ1

ψ2 90

o

ψ3

Streamlines

434

ψ4

ψ5 φ1

φ3

φ2

φ4

φ5

Potential lines

Solution Using Eq. (9.101), the velocity components are   ∂ u = ∂φ x + x2 − y2 = 2x + 1 = ∂x ∂x  ∂ x + x2 − y2 = −2y v = ∂φ = ∂y ∂y By applying Eq. (9.123), we get u = ∂ψ = 2x + 1 ∂y = −2y v = − ∂ψ ∂x Integrating, ψ = 2xy + y + f (x)

(A)

ψ = 2xy + f (y)

(B)

and

Because Eqs. (A) and (B) lack independent functions of either x or y, both f (x) and f (y) can be regarded zero. Hence, the stream function is ψ = 2xy + y

Concluding Remarks

435

Concluding Remarks In this chapter, the differential forms of the governing equations of the conservation of mass, momentum, and energy are derived for steady, multi-dimensional, irrotational, and isentropic flow. From these equations, the corresponding equations for steady, multi-dimensional, adiabatic, inviscid, and irrotational flow of a compressible fluid are derived in the Cartesian coordinate system.

Exercise Problems Exercise 9.1 Let C denotes the unit circle. Show that the velocity field represented by the vector y ˆ x ˆ − → v =− 2 i+ 2 j x + y2 x + y2 does not obey the Stokes theorem. Exercise 9.2 The components characterizing the velocity field for a free vortex flow are x y and v = 2 u=− 2 2 x +y x + y2 where the circulation  is a constant. Derive the expression for the velocity potential function for the flow field shown above. Exercise 9.3 Examine the following velocity vectors to determine whether they could represent the possible flow fields for an incompressible fluid. → v = −yˆi + x ˆj (i) − → v = xex ˆi + yeyˆj (ii) − 2    → v = (xyzt) ˆi + −xyzt 2 ˆj + z2 xt 2 − yt kˆ (iii) − Exercise 9.4 The stream function in a steady, two-dimensional incompressible flow is ψ = 5x − 6y + 11xy + 15. Determine the velocity potential. Exercise 9.5 The velocity potential function in a two-dimensional incompressible flow is φ = ln r. Find the stream function. (Hint: use polar coordinates)

Chapter 10

Linearized Potential Flow: The Small-Perturbation Theory

Abstract In the preceding chapter, the governing partial differential equations in terms of the velocity potential function and the stream function were derived. These equations are highly nonlinear and cannot be solved analytically in closed form. However, in some situations, they can be rendered linear by employing appropriate approximations. One example is the flow past a thin airfoil, where the assumptions of small perturbations in the flow properties allow for a linearized equation and, consequently, a closed-form solution. The present chapter discusses the analysis of such flows, commonly referred to as the small-perturbation theory.

10.1 Introduction So far in our discussion, we have examined one-dimensional and quasi-onedimensional flows, assuming that the flow parameters are uniform over specified cross-sections of the flow passage (duct). In these cases, we just looked at changes at the interfaces and did not solve the entire flow field. The control volume technique was the dominant approach to these investigations. We explored what happened across the control volume boundaries but did not discuss what happened within the flow field. However, we may require knowledge of the latter aspects in actual flow situations, necessitating our flow analysis approach change. One technique is, of course, to computationally solve the entire set of Navier-Stokes equations for the flow of a general compressible fluid. Alternatively, for the steady, adiabatic, inviscid, and irrotational flow of a compressible perfect gas, we can solve the governing partial differential equations expressed in terms of the velocity potential function computationally. Nevertheless, due to their nonlinearity, we cannot solve the equations mentioned above in closed form. However, we can simplify them in some situations to allow for an approximate solution. One typical strategy is to keep the relatively slender bodies in the flow to minimize the differences between the flow properties near the body and the corresponding properties in the freestream. In other words, the presence of slender bodies causes small perturbations to the mean flow, so this approach is called the method of small perturbations, or frequently, the small-perturbation theory. © Springer Nature Singapore Pte Ltd. 2022 M. Kaushik, Fundamentals of Gas Dynamics, https://doi.org/10.1007/978-981-16-9085-3_10

437

438

10 Linearized Potential Flow: The Small-Perturbation Theory

10.2 The Linearized Velocity Potential Equation for Steady, Two-Dimensional, Irrotational Flow with Small Perturbations The present section derives a single linearized velocity potential equation for steady, irrotational, and isentropic flow. This equation applies to both subsonic and supersonic flows. A separate equation for transonic flows is also derived. Consider a fluid flowing in the x-direction at a uniform velocity, U∞ , over a slender body, such as a thin airfoil, as shown in Fig. 10.1. Small changes in the fluid properties occur when it flows around the body. The properties in the perturbed flow can be approximated as if perturbation property components are superimposed on the properties of the mean velocity. For example, as the fluid flows around the body, the flow velocity changes slightly and is approximated as the superposition of the mean velocities and the corresponding perturbation velocity components. Let the local flow → → ˆ where u, v, and w velocity in the perturbed flow be − v , where − v = uˆi + vˆj + wk, denote the velocity components in the x−, y−, and z−direction, respectively. Since the direction of the mean flow is along the x−axis only, we can express u = U∞ + u

(10.1)



v=v w = w

(10.2) (10.3)

y

Shape of the airfoil, y = f(x)

U

8

8

U + u’ Thin airfoil

θ v

v’

x

Fig. 10.1 A slender body in the Cartesian space

10.2 The Linearized Velocity Potential Equation for Steady …

439

where u , v , and w denote the perturbation velocities in the x−, y−, and z−direction, → ˆ the respectively. If the perturbation velocity is represented as − q = uˆi + vˆj + w k, local flow velocity may be expressed as   → − → v = uˆi + vˆj + wkˆ = − q + U∞ˆi = U∞ + u ˆi + vˆj + w kˆ

(10.4)

If the total velocity potential and the perturbation-velocity potential are denoted by Φ and φ, respectively, we may express Eq. (10.4) as   ∂φ ˆ ∂φ ˆ ∂φ ˆ − → v = ∇Φ = U∞ + i+ j+ k ∂x ∂y ∂z = ∇φ + U∞ˆi

(10.5) (10.6)

where ∂φ = u = u−U∞ ∂x ∂φ = v = v ∂y ∂φ = w = w ∂z

(10.7) (10.8) (10.9)

Integrating Eq. (10.5) yields  = φ + U∞ x

(10.10)

Equation (10.10) shows that the total velocity potential in a perturbed flow is the sum of the perturbation-velocity potential and the velocity potential for the uniform freestream. Also, ∂ 2Φ = Φxx = ∂x2 ∂ 2Φ = Φyy = ∂y2 ∂ 2Φ = Φzz = ∂z2

∂ 2φ = φxx ∂x2 ∂ 2φ = φyy ∂y2 ∂ 2φ = φzz ∂z2

(10.11) (10.12) (10.13)

Under steady conditions, the velocity potential equation, derived in Sect. 9.6.1, gives

440 

10 Linearized Potential Flow: The Small-Perturbation Theory

u2 1− 2 a



 ∂2φ + 1− ∂x2

v2 a2



  ∂2φ vw ∂ 2 φ wu ∂ 2 φ w2 ∂ 2 φ uv ∂ 2 φ −2 2 −2 2 =0 + 1 − −2 2 2 2 2 ∂y a ∂z a ∂x∂y a ∂y∂z a ∂z∂x

(10.14) or 

  a2 − u2 φxx + a2 − v2 φyy + a2 − w2 φzz − 2φx φx φxy − 2φx φz φxz − 2φy φz φyz = 0

(10.15)  Substituting  = (U∞ x + φ), u = U∞ + Eq. (10.15) yields

∂φ ∂x

,

∂φ ∂y

= v, and

∂φ ∂z

= w into



2 a − (U∞ + φx )2 φxx + a2 − φy2 φyy + a2 − φy2 φzz − 2 (U∞ + φx ) φy φxy −2 (U∞ + φx ) φz φxz − 2φy φz φyz = 0

(10.16)

This relationship is known as the perturbation-velocity potential equation or frequently, the complete gas dynamic equation. In terms of velocities, Eq. (10.16) can be rewritten as 2 ∂u 2  ∂v 2  ∂w  ∂u   a2 − U∞ + u + a − v2 + a − w2 − 2 U∞ + u v ∂x ∂y ∂z ∂y     ∂u ∂v − 2v w =0 (10.17) −2 U∞ + u w ∂z ∂z Because the flow is isoenergetic (h0 = constant) throughout, the energy equation yields 2  2 U∞ + u + v2 + w2 v2 U∞ =h+ =h+ (10.18) h∞ + 2 2 2 √ For a perfect gas, h = cp T and a = γ RT; thus Eq. (10.18) becomes 2  2 2 U∞ + u + v2 + w2 U∞ a2 a∞ + = + γ −1 2 γ −1 2    2 2 2 2 2 2u U∞ + u + v2 + w2 a∞ U∞ a U∞ + = + + γ −1 2 γ −1 2 2   γ − 1 2 a2 = a∞ 2u U∞ + u2 + v2 + w2 − 2

(10.19) (10.20) (10.21)

By substituting Eq. (10.21) into Eq. (10.17) and performing algebraic rearrangement, we obtain

10.2 The Linearized Velocity Potential Equation for Steady …

2 +M∞



v U∞

441

  2 1 − M∞ u

∂u ∂v ∂w ∂x + ∂y + ∂z   2 (γ +1) u (γ −1) v2 +w2 ∂u 2 + 2 2 2  U∞ ∂x U∞ (γ +1) v2 (γ −1) w2 +u2 ∂v + 2 2 2 2 U∞  U∞ ∂y (γ +1) w2 (γ −1) u2 +v2 ∂w 2 + 2 2 2 ∂z U∞ U∞

2 (γ + 1) = M∞ U∞ +  2 +M∞ (γ − 1) Uu∞ + 2 (γ − 1) u + +M∞ U∞     u ∂u ∂v w 1 + U∞ ∂y + ∂x + U∞ 1 +

u U∞



∂u ∂z

+

∂w ∂x



+

v  w 2 U∞



1+

u U∞



∂w ∂y

+

∂v ∂z



(10.22) Equation (10.22) is the expanded version of the perturbation-velocity potential equation (Eq. (10.16)). For an irrotational isentropic flow, this equation continues to be exact. Also, notice that the left side of Eq. (10.22) consists of linear terms, while the right side consists of nonlinear terms. We can now employ the assumptions of small perturbations  in order to linearize  Eq. (10.22). Assume that the perturbation velocities u , v , w are small compared to the uniform flow velocity, U∞ , i.e.

so that

and

u v w  1,  1, 1 U∞ U∞ U∞

(10.23)

u2 v2 w2 ≪ 1, ≪ 1, ≪1 2 2 2 U∞ U∞ U∞

(10.24)

u v v  w w u  ≪ 1, ≪ 1, ≪1 2 2 2 U∞ U∞ U∞

(10.25)

Using these approximations, all of the squared and the products of perturbation velocities terms appearing in Eq. (10.22) may be set equal to zero. This gives

2 (γ − 1) +M∞

u

∂v

U∞ ∂y

  2 1 − M∞

∂u ∂x

+

2 (γ − 1) + M∞

  ∂v ∂w 2 + 1) Uu∞ ∂u ∂y + ∂z = M∞ (γ   ∂x     u ∂w ∂u ∂v 2 v 2 w U∞ ∂z + M∞ U∞ ∂y + ∂x + M∞ U∞



∂u ∂z

+

∂w ∂x



(10.26)  ∂u      2 2 + ∂v + ∂w = M∞ 1 − M∞ (γ + 1) Uu∞ ∂u ∂x ∂y ∂z  ∂x       ∂u ∂u 2 2 v 2 w + M∞ + M∞ M∞ + ∂w + ∂v + (γ − 1) Uu∞ ∂v ∂y ∂z U∞ ∂y ∂x U∞ ∂z

∂w ∂x



(10.27) Equation (10.27) can be observed to be significantly easier than Eq. (10.22), albeit it is still quite nonlinear. We need to simplify it even further before we try to solve it. Let us assume that all of the products of the perturbation velocities and their derivatives are insignificant. Therefore, all of the terms on the right side of Eq. (10.27) disappear, yielding the following linear equation:

442

10 Linearized Potential Flow: The Small-Perturbation Theory

  ∂u ∂v ∂w 2 1 − M∞ + + =0 ∂x ∂y ∂z

(10.28)

Alternatively, in terms of perturbation-velocity potential,  ∂ 2φ  ∂ 2φ ∂ 2φ 2 + + =0 1 − M∞ ∂x2 ∂y2 ∂z2   2 1 − M∞ φxx + φyy + φzz = 0

(10.29) (10.30)

Equations (10.28) and (10.30) are the linearized perturbation-velocity potential equations, valid for both subsonic and supersonic flows. Although they are linear, their application is somewhat restricted due to underlying assumptions made while deriving them, compared to Eq. (10.22). 2 , on the left At transonic speeds (0.8 < M∞ < 1.2), since the coefficient, 1 − M∞ side of Eq. (10.27) becomes very small, the first term on the right side has the same order of magnitude as the first term on the left side and thus the former term must be retained. However, all terms after the first term on the right side of Eq. (10.27) are negligible in transonic flows also and hence must be ignored. Thus, at transonic Mach numbers, the linearized perturbation-velocity potential equation is  ∂u  u ∂u ∂v ∂w 2 2 1 − M∞ + + = M∞ (γ + 1) ∂x ∂y ∂z U∞ ∂x

(10.31)

Because the Mach number becomes very large in hypersonic flows, the approximations mentioned above do not hold, and we require more sophisticated techniques to linearize Eq. (10.27). For steady, two-dimensional perturbed flows, Eq. (10.30) reduces to   2 φxx + φyy = 0 1 − M∞

(10.32)

Likewise Eq. (10.30), this equation is valid for both subsonic and supersonic flows.

10.2.1 The General Behavior of the Perturbation Equations In Sect. 9.6.2, we have classified the velocity potential equation as elliptic, parabolic, and hyperbolic based on a parameter called the discriminant of the partial differential equations. In this section, we can classify the linearized perturbation-velocity potential equation following the similar procedure. By comparing Eq. (10.32) with Eq. (9.117), we get the coefficients as   2 , B = 0, C = 1, D = 0, E = 0, and F = 0 A = 1 − M∞

(10.33)

10.2 The Linearized Velocity Potential Equation for Steady …

443

Hence, the discriminant, , is  2  −1  = B2 − 4AC = 4 M∞

(10.34)

Thus, if 1. the flow is subsonic (M∞ < 1), and the linearized perturbation equation is elliptic since  < 0 ; 2. the flow is sonic (M∞ = 1), and the linearized perturbation equation is parabolic since  = 0 ; 3. the flow is supersonic (M∞ > 1), and the linearized perturbation equation is hyperbolic since  > 0 . Remark The main distinction between elliptic and hyperbolic perturbation equations is that the solution in the former case is constantly reliant on the boundary conditions since the perturbation effects are perceived throughout the flow field. In contrast, the solution in the latter case does not depend continuously on the boundary conditions because the perturbation effects are not perceived everywhere in the flow; rather, there are zones of influence and domains of dependency.

10.2.2 Linearized Pressure Coefficient The pressure coefficient for the steady flow of a perfect gas, derived in Sect. 4.10, is repeated and renumbered here for convenience.     γ γ−1 γ −1 2 v2 2 1+ M∞ 1 − 2 −1 (10.35) Cp = 2 γ M∞ 2 U∞ Alternatively, Eq. (10.35) can be represented in terms of perturbation velocities as Cp =

2 2 γ M∞

⎫ ⎧⎡ ⎞⎤ γ γ−1  ⎛  ⎪ ⎪  2 2 2 ⎬ ⎨ U + u + v + w ∞ 2 ⎝ ⎣1 + γ − 1 M∞ ⎠ ⎦ − 1 1− 2 ⎪ ⎪ 2 U∞ ⎭ ⎩ (10.36)

⎫   " γ ⎬ γ −1 u2 v2 w2 2u γ −1 2 M Cp = + + + − 1 1 − 1 + 1 + ∞ 2 ⎩ 2 2 2 ⎭ 2 U∞ γ M∞ U∞ U∞ U∞ 2

⎧! ⎨

(10.37)

2 Cp = 2 γ M∞



γ −1 2 1− M∞ 2



2u u2 v2 w2 + 2 + 2 + 2 U∞ U∞ U∞ U∞

 γ γ−1

 −1

(10.38)

444

10 Linearized Potential Flow: The Small-Perturbation Theory

Since u2 v2 w2 ≪ 1, ≪ 1, ≪1 2 2 2 U∞ U∞ U∞

(10.24)

Thus, Eq. (10.38) reduces to 2 Cp = 2 γ M∞



γ −1 2 1− M∞ 2



2u U∞

 γ γ−1

 −1

(10.39)

We then use the following binomial theorem to expand the equation encompassed by the square bracket in Eq. (10.39): (1 − x)n = 1 − nx +

n (n − 1) 2 n (n − 1) (n − 2) 3 x − x + ... 2! 3!

(10.40)

This gives  ⎧⎡ γ γ     2 ⎨ − 1  γ − 1 2 2u 2u γ − 1 γ γ −1 γ −1 2 4 ⎣1 − + M Cp = M ∞ ∞ 2 ⎩ γ −1 2 U∞ 2! 2 U∞ γ M∞   ⎫ ⎤ γ γ γ   3   ⎬ 2u γ −1 3 6 γ −1 γ −1 − 1 γ −1 − 2 − M∞ + ...⎦ − 1 (10.41) ⎭ 3! 2 U∞ 2

In Eq. (10.41), discarding the square and higher powers of the perturbation velocities yields   ∂φ 2u 2 (10.42) =− Cp = − U∞ U∞ ∂x This is the linearized pressure coefficient for two-dimensional planar steady flows. It is clear that the x−component of the perturbation velocity solely determines the pressure coefficient.

10.3 Steady, Two-Dimensional, Irrotational, and Isentropic Flow Past an Infinite Wave-Shaped Wall This section will investigate a classical problem of fluid flows that will serve as an example showing the application of small perturbations. Consider a steady, adiabatic, inviscid, irrotational, uniform flow of a compressible fluid at velocity, U∞ , and Mach number, M∞ , over an infinite wave-shaped wall, as shown in Fig. 10.2. The following relation gives the ordinate of the wall:

10.3 Steady, Two-Dimensional, Irrotational, and Isentropic Flow Past …

445

y yw = h cos 2π x λ

8

8

U ,M

v

v v θ U + u’

θ dx

8

v

v’

θ=

v’ U + u’

tanθ

8

tanθ

dy

θ=(

dy ) dx w x

O

v h

λ

Fig. 10.2 A steady, two-dimensional compressible flow past an infinite wave-shaped wall

2π x λ

yw = h cos

(10.43)

where h denotes the amplitude and λ denotes the wavelength of the wavy wall. We assume that h is very small as compared to λ, i.e., h  λ, and 2πh  1, so that the λ first-order perturbation equation holds for this problem. At the wall, the geometric and velocity considerations require that the flow inclination must be equal to the slope of the surface and thus we have  tan θ ≈ θ =

dy dx



 = w

v U∞ + u



 = w

v U∞

 (10.44) y=0

In this case, our objective is to derive an expression for the velocity potential function and the pressure coefficient at the wall surface. Since the flow is irrotational, we can apply the two-dimensional perturbation-velocity potential equation derived in Sect. 10.2, which applies to both subsonic and supersonic flows. Let us rewrite and repeat this equation here for convenience.   2 φxx + φyy = 0 1 − M∞

(10.45)

where the perturbation-velocity potential, φ, is defined by Eq. (10.10). The accompanying boundary conditions are 

v U∞



 =

y=0

dy dx

 = w

1 U∞



∂φ ∂y

 =− y=0

φ (x, ∞) = ∞

2π x 2π h sin λ λ

(10.46) (10.47)

446

10 Linearized Potential Flow: The Small-Perturbation Theory

It is important to mention that Eq. (10.47) is the boundary condition that stipulates a finite velocity potential throughout the flow domain, i.e., for all cases of x and y. Consequently, even when y → ∞, the perturbation-velocity potential is finite. The boundary condition at y = ∞ depends on whether the uniform freestream is subsonic (M∞ < 1) or supersonic (M∞ > 1). These two cases are now discussed separately.

10.3.1 Uniform Subsonic Flow 2 Let us substitute 1 − M∞ = β 2 into Eq. (10.45). This gives

β 2 φxx + φyy = 0

(10.48)

It is seen in Sect. 10.2.1 that for subsonic flow, M∞ < 1, Eq. (10.45) is elliptic. We can use the method of separation of variables to solve Eq. (10.48) by writing φ (x, y) = F (x) .G (y)

(10.49)

where F (x) is a function of only the variable x, and G (y) is only the variable y. Thus, from Eq. (10.49), we have ∂F ∂φ =G , ∂x ∂x ∂G ∂φ =F , ∂y ∂y

∂ 2φ ∂ 2F = G ∂x2 ∂x2 ∂ 2φ ∂ 2G =F 2 ∂y2 ∂y

(10.50) (10.51)

Substituting Eqs. (10.50) and (10.51) into Eq. (10.48) yields ∂ 2F ∂ 2G + F =0 ∂x2 ∂y2 1 ∂ 2G ∂ 2F G 2 = − 2F 2 ∂x β ∂y

β2G

(10.52) (10.53)

Dividing Eq. (10.53) by F.G throughout, we get 1 1 ∂ 2G 1 ∂ 2F = − F ∂x2 β 2 G ∂y2

(10.54)

Notice that the left side of Eq. (10.54) is a function of x only, while the right side is of x only. So they will be equal to each other only if they are equal to a constant − 2 (say), i.e. 1 ∂ 2F 1 1 ∂ 2G = − = − 2 (10.55) F ∂x2 β 2 G ∂y2

10.3 Steady, Two-Dimensional, Irrotational, and Isentropic Flow Past …

447

The value of is determined later from the boundary conditions. This results in the following two relations: ∂ 2F + 2 F = 0 ∂x2 ∂ 2G − 2 β 2 G = 0 ∂y2

(10.56) (10.57)

The general solutions of Eqs. (10.56) and (10.57) can be expressed as F (x) = B1 sin x + B2 cos x G (y) = A1 e

− βy

+ A2 e

βy

(10.58) (10.59)

Therefore, Eq. (10.49) becomes   φ (x, y) = A1 e− βy + A2 e βy (B1 sin x + B2 cos x)

(10.60)

where A1 , A2 , B1 , and B2 are arbitrary constants to be determined from the boundary conditions. Equation (10.47) shows when y → ∞, φ = 0; thus the first term enclosed in the first bracket on the right side of Eq. (10.60) becomes zero, whereas the second term becomes infinitely large. Consequently, we must set A2 = 0 in Eq. (10.60). This gives (10.61) φ (x, y) = A1 e− βy (B1 sin x + B2 cos x) Differentiating Eq. (10.61) with respect to y gives ∂φ = − βA1 e− βy (B1 sin x + B2 cos x) ∂y

(10.62)

According to small-perturbation theory, the wall amplitude, h, is very small and so the ordinate at the wall. Thus, we may write 

dy dx

 = w

1 U∞



∂φ ∂y

 (10.63) y=0

Consequently, Eq. (10.62) becomes 

dy dx

 = w

1 U∞



∂φ ∂y

 =− y=0

β A1 (B1 sin x + B2 cos x) U∞

(10.64)

By equating Eqs. (10.46) and (10.64), we have −

2π x 2π h β sin A1 (B1 sin x + B2 cos x) = − U∞ λ λ

(10.65)

448

10 Linearized Potential Flow: The Small-Perturbation Theory

Note that the left side of Eq. (10.65) comprises both the sine and the cosine functions, whereas the right side has only the sine function. Thus, we must set B2 = 0 in Eq. (10.65). This gives −

2π x 2π h β sin A1 B1 sin x = − U∞ λ λ

(10.66)

In addition, we find from Eq. (10.66) that =

2π λ

(10.67)

Therefore, Eq. (10.66) becomes −

2π h 2π x 2π β 2π x =− sin A1 B1 sin λ U∞ λ λ λ hU∞ hU∞ A1 B1 = =# 2 β 1 − M∞

Now, by substituting B2 = 0, = 2π , and β = λ combining the result with Eq. (10.69), we obtain φ (x, y) = #

hU∞ 1−

2 M∞

e

(10.68) (10.69)

# 2 into Eq. (10.61) and 1 − M∞

 √ 2 y − 2π 1−M∞ λ

sin

2π x λ

(10.70)

Using Eq. (10.70), the total velocity potential in a two-dimensional perturbed flow may be written as 

hU∞ −  (x, y) = U∞ x + # e 2 1 − M∞

2π λ

√

2 y 1−M∞

sin

2π x λ

(10.71)

This is the total velocity potential for the two-dimensional, irrotational, and isentropic steady flow past the infinite wave-shaped wall. Using this solution, we may calculate all other physical properties, such as  √ 2 y ∂φ 2π hU∞ 2π x − 2π 1−M∞ # = U∞ + e λ cos 2 ∂x λ 1 − M∞ λ  √ 2π 2 y 2π hU∞ − λ 1−M∞ ∂φ 2π x v = =− e sin ∂y λ λ

u =

(10.72) (10.73)

and by combining Eqs. (10.42) and (10.72): 

Cp = −

h 2u 4π − # =− e 2 U∞ λ 1 − M∞

2π λ

√

2 y 1−M∞

cos

2π x λ

(10.74)

10.3 Steady, Two-Dimensional, Irrotational, and Isentropic Flow Past …

449

y Parallel streamline

8

At y =

Stre

Decreasing waviness in the streamlines

Streamline

aml

−

−

−

−

− +

−

−

−

−

− +

+

+

h cos 2πx C p = − 4π w λ λ 1 − M2

+

+

+

+

x

+

Symmetric distribution

8

w

−

+ +

Cp

−

ine

x

O

o

Surface pressure coefficient has 180 phase lag with respect to the wall shape

Fig. 10.3 Pressure coefficient variation at the infinite wave-shaped wall in a subsonic flow

Equation (10.74) is the pressure coefficient for the subsonic compressible flow over the infinite wave-shaped wall. The pressure coefficient at the wall, Cpw , may be obtained by substituting y = 0 into Eq. (10.74). This results in Cpw = −

h 4π 2π x # cos 2 λ 1 − M∞ λ

(10.75)

Equation (10.75) shows that the surface pressure coefficient distribution is a cosinusoidal curve which is schematically represented in Fig. 10.3. It is worth noting that Cpw has 180o phase-lag to the wall shape, and since the surface pressure has a symmetric distribution, the net pressure force acting on the wall in the x−direction is zero. An interesting result can be stated as follows: an infinite wave-shaped wall in a steady, adiabatic, irrotational, and inviscid subsonic flow of a compressible fluid does not encounter drag. It is, of course, an extension of the well-known d’Alembert’s paradox of inviscid incompressible flow to the present case of inviscid subsonic compressible flow. Figure 10.3 shows that for a fixed M∞ , the perturbations introduced by the wall into the flow die down at larger distances from the wall, and consequently, the waviness in the streamlines become lesser and lesser. In fact, at theoretical ∞, the streamlines

450

10 Linearized Potential Flow: The Small-Perturbation Theory

become parallel to the wall. Nevertheless, the spreading distance of perturbations increases with increasing M∞ .

10.3.2 Uniform Supersonic Flow Consider a two-dimensional, irrotational, and isentropic flow over an infinite waveshaped wall again. In this case, though, we shall examine a uniform supersonic flow. As shown in Sect. 10.3, the linearized perturbation-velocity potential equation for a supersonic flow is again Eq. (10.45). However, when the flow is supersonic 2 is negative, and thus Eq. (10.45) is represented in (M∞ > 1), the value of 1 − M∞ the form (10.76) m2 φxx − φyy = 0 2 − 1. Equation (10.76) is a hyperbolic partial differential equation.1 where m2 = M∞ Let us rewrite Eq. (10.76) as 1 ∂ 2φ ∂ 2φ = (10.77) ∂x2 m2 ∂y2

which is essentially a classical wave equation2 whose general solution is     $ $ 2 2 φ (x, y) = f x − M∞ − 1y + g x + M∞ − 1y = f (x − my) + g (x + my)

(10.78) (10.79)

Now, from the boundary condition (Eq. (10.46)),     2π x dy ∂φ 1 2π h sin = =− dx w U∞ ∂y y=0 λ λ We have

1 2

In literature, Eq. (10.76) is also referred to as the Prandtl-Glauert equation. The standard wave equation in one spatial coordinate is expressed as

1 ∂ 2 u (x, t) ∂ 2 u (x, t) = 2 (A) 2 ∂t a ∂x2 where u denotes the flow velocity, and a denotes the speed of sound in the fluid under consideration. The general solution to equation (A) is given by u (x, t) = F (x − at) + G (x + at)

(B)

Equation (B) shows that the solution to the one-dimensional wave equation is the sum of a lefttraveling function, F (x − at), and a right-traveling function, G (x + at).

10.3 Steady, Two-Dimensional, Irrotational, and Isentropic Flow Past …

f  (x) =



∂ (x − my) ∂f ∂ (x − my) ∂x

 = y=0

2π x 2π hU∞ sin λ m λ

451

(10.80)

Integrating, f (x) = −

2π x hU∞ cos + constant m λ

(10.81)

Because f (x) is defined throughout and not just at the wall, where y = 0, thus Eq. (10.81) can be rewritten, by replacing the argument x with x − my, as f (x − my) = −

2π hU∞ cos (x − my) + constant m λ

(10.82)

Hence, the perturbation-velocity potential, φ, in a supersonic flow is φ (x, y) = f (x − my) = − or

2π hU∞ cos (x − my) + constant m λ

2π cos φ (x, y) = − # 2 λ M∞ − 1 hU∞

  $ 2 x − M∞ − 1y + constant

(10.83)

(10.84)

where the value of the constant identifies a particular potential line. In addition, the perturbation-velocity components are   $ hU∞ 2π 2π ∂φ 2 =# sin x − M∞ − 1y u = 2 −1 λ ∂x λ M∞   $ 2π 2π ∂φ 2 − 1y = −hU∞ sin x − M∞ v = ∂y λ λ 

(10.85) (10.86)

In addition, from Eqs. (10.42) and (10.85), we get Cp = −

h 2u 4π 2π # =− sin 2 U∞ λ M∞ − 1 λ

  $ 2 − 1y x − M∞

(10.87)

Equation (10.87) is the linearized pressure coefficient for steady, irrotational, and isentropic supersonic flow past an infinite wave-shaped wall. At the wall (y = 0), Eq. (10.87) becomes h 4π 2π x # (10.88) Cpw = − sin 2 λ M∞ − 1 λ Equation (10.88) indicates that the surface pressure coefficient is a sinusoidal curve and is schematically shown in Fig. 10.4. The figure reveals that Cpw attains a maximum at those locations where the wall shape curve passes through the x−axis. It is worth noting that in contrast to a subsonic perturbed flow, the perturbationvelocity potential in a supersonic flow does not have any exponential term. This

452

10 Linearized Potential Flow: The Small-Perturbation Theory Left−running characteristics (Mach lines) 8

(x − y M2 − 1 = constant) y

Stre

dy θ=μ

aml

Stre

aml

p p + O

−

−

−

+ p

−

+

− −

+

ine

Velo φ = city pot c e thes onstant ntial, e ch arac along teris tics

dx

+

+

−

−

ine

p

−

+

−

x −

−

p +

Cp

w

x

O "Finite wave drag"

Fig. 10.4 Pressure coefficient variation at the infinite wave-shaped wall in a supersonic flow

means that the magnitude of perturbations does not diminish as we go away from the wall. The # when the argument of φ, # magnitude of perturbations remains constant 2 − 1, is a constant. The equation x − y M2 − 1 = constant reprei.e., x − y M∞ ∞ sents the equation of a straight line, also called the characteristic line (Fig. 10.4). Thus, we can state that the perturbations travel along a characteristic line to ∞ with # undiminished strength. Moreover, as discussed in Sect. 5.17, the lines 2 − 1 = constant are, by definition, the family of left-running characterisx − y M∞ # 2 −1= tics or Mach lines. Alternatively, if f = 0 in Eq. (10.78), the lines x − y M∞ constant would be referred to as the family of right-running Mach lines. The characteristic lines have a slope tan θ =

1 dy =# = tan μ 2 dx M∞ − 1

(10.89)

and are therefore identical to Mach lines with an angle, μ, to the freestream direction. At the wall (y = 0), by combining Eqs. (10.44) and (10.89) and considering the small

10.3 Steady, Two-Dimensional, Irrotational, and Isentropic Flow Past …

453

perturbations, we get  tan θ ≈ θ = This gives

dy dx



 =

w

v U∞ + u



 =

w

v U∞

 y=0

1 =# 2 M∞ − 1

  v y=0 ≈ U∞ θ

(10.90)

(10.91)

Again from the perturbation-velocity potential, φ (x, y), the perturbation-velocity components at the wall are   ∂φ = f  (x) u w= ∂x $   ∂φ 2 − 1f  (x) v w= = − M∞ ∂y and so

    v w # u w=− = U∞ θ 2 −1 M∞

(10.92) (10.93)

(10.94)

Substituting Eq. (10.94) into Eq. (10.42), the pressure coefficient at the wall becomes   dy 2θ 2 (10.95) Cpw = # =# 2 2 M∞ − 1 M∞ − 1 dx w This is a significant result that must be thoroughly examined. According to Eq. (10.95), the pressure coefficient is positive where the wall slope is positive, and the pressure coefficient is negative where the slope is negative. Consequently, between a crest and the next trough of the wall, since the slope is negative, the pressure coefficient is negative. On the other hand, between a trough and the next crest, the slope is positive, and hence the pressure coefficient is positive. These results are schematically shown in Fig. 10.4. Furthermore, in contrast to subsonic flow, Eq. (10.84) produces unsymmetrical streamline distributions about the normal drawn through the crest and trough of the wall. Instead, as seen in Fig. 10.4, they are geometrically similar between two characteristics. Hence, in the x−direction, the surface pressure distributions do not cancel each other, and a net force applies on the wall along the flow direction. Consequently, the wall experiences a finite drag known as wave drag. Remark In a subsonic flow, the amplitude of disturbances reduces rapidly with increasing distance from the wall, as seen in Fig. 10.3. However, as the Mach number increases, so does the rate at which the magnitude of the disturbances decreases. When the flow achieves sonic conditions, the disturbances propagate laterally without diminishing in amplitude. However, once the flow becomes supersonic, the perturbations propagate without distortion along the characteristic lines (Mach waves).

454

10 Linearized Potential Flow: The Small-Perturbation Theory

Real flow situation Subsonic flow (M < 1) 8

Supersonic flow (M > 1) 8

C p max

0

1

2 M

Fig. 10.5 Variation of maximum pressure coefficients with freestream Mach number

These results demonstrate that subsonic flow differs from the supersonic flow. For , while for M∞ > 1, M∞ < 1, the pressure coefficient, Cp , is proportional to cos 2πx λ it is proportional to sin 2πx . Notice that, for M = 1, the value of Cp becomes ∞, ∞ λ which shows that linearized theory is not valid for sonic flows (Fig. 10.5). In reality, the pressure coefficient remains finite at M∞ = 1. Similarly, for M∞ = ∞, the value of Cp is zero, which again emphasizes that the linearized theory does not hold for hypersonic flows.

10.4 Similarity Laws for the Steady, Two-Dimensional, Irrotational, and Isentropic Flows In Chap. 9, we have derived the velocity-potential equations for the subsonic and the supersonic flows. For subsonic flows, the governing equation is elliptic, whereas for supersonic flows, it is hyperbolic. In both cases, however, the equations are highly nonlinear, and thus, the solutions to them cannot be put in analytical form. This contrasts to incompressible flows where the simple analytical solutions to the governing equation (Laplace’s equation) are possible. Nevertheless, with suitable transformations, the nonlinear governing equations for compressible flows can be transformed into the forms applicable to incompressible flows lending the analytical solutions possible. Certain restrictions are enforced during the transformation process, and

10.4 Similarity Laws for the Steady, Two-Dimensional, Irrotational …

455

η

y

Velocity potential function, Φ (ξ, η)

Velocity potential function, φ(x, y)

η = g (ξ)

y = f (x)

Transformations ξ= x η = λy

M 1, passing over a thin airfoil at a small angle of attack, as shown in Fig. 10.7. Because the flow is compressed due to the deflection by the airfoil, shock waves form at the body. The linearized velocity-potential equation (Eq. (10.45)) is still the governing equation for a two-dimensional supersonic flow over a thin airfoil at a small angle of attack, but with M∞ > 1. As previously stated in Sect. 10.3.2, Eq. (10.45) is a simple wave equation with a general solution expressed as φ (x, y) = h (x − my) + g (x + my)

(10.132)

# 2 − 1, and h and g are arbitrary functions. Let x − my = ξ and where m = M∞ x + my = η, the above equation becomes φ (x, y) = h (ξ ) + g (η)

(10.133)

It is worth noting that the straight lines ξ = constant and η = constant are referred to as flow characteristics (Fig. 10.4). Let us limit ourselves to the flow over the airfoil’s upper surface. It is vital to remember that, unlike subsonic flows, supersonic flow perturbations cannot penetrate the flow zone above the airfoil and vice versa. Their mutual interaction is restricted to the region downstream of the trailing edge between two trailing edge characteristics. This means that the flow fields above and below the airfoil are independent and maybe analyzed separately. These perturbations in the flow field are only possible if another object is present above or below the airfoil. Therefore, while working with an isolated airfoil, we may set g (η) = 0 in

10.4 Similarity Laws for the Steady, Two-Dimensional, Irrotational …

461

Eq. (10.133). This gives φ (x, y) = h (ξ ) = h (x − my)

(10.134)

If y = fu (x) denotes the profile of the upper surface profile of airfoil, the boundary condition (Eq. (10.44)) to a flow with small perturbations at the wall yields 

   dy 1 ∂φ (x, y) = dx w U∞ ∂y y=0   ∂φ (x, y) dfu = U∞ ∂y dx y=0

(10.135) (10.136)

Substituting Eq. (10.134) into Eq. (10.136) provides −m

dh (x) dfu = U∞ dx dx

(10.137)

which, when integrated, gives h (x) = −

U∞ U∞ fu (x) fu (x) = − # 2 −1 m M∞

(10.138)

Thus, Eq. (10.134) becomes φ (x, y) = − #

U∞ 2 −1 M∞

  $ 2 fu x − M∞ − 1y

(10.139)

This is the perturbation-velocity potential function in supersonic flow over the upper surface of the airfoil. The pressure coefficient on the upper surface may be obtained from   ∂φ 2u 2 (10.42) =− Cpw = − U∞ U∞ ∂x Substituting Eq. (10.139) into the above equation leads to Cpw = #

2 2 M∞

−1

fu (x)

(10.140)

u (x) is the slope of the airfoil’s upper surface. Similarly, the pressure where fu (x) = dfdx coefficient on the lower surface of the airfoil is

Cpw = #

2 2 M∞

−1

f (x)

(10.141)

462

10 Linearized Potential Flow: The Small-Perturbation Theory

Equations (10.140) and (10.141) are the well-known Ackeret’s formula for supersonic flows. According to this formula, the pressure coefficient in a supersonic flow depends on the local slope of the airfoil surface. It provides a simple way to calculate the lift and drag on a thin airfoil in supersonic flows.

10.5 Supersonic Thin Airfoil Theory Consider a thin airfoil in the steady, irrotational, and isentropic supersonic flow, as shown in Fig. 10.8. The general solution of the linearized perturbation-velocity potential equation for supersonic flows, i.e.,     $ $ 2 2 (10.78) φ (x, y) = f x − M∞ − 1y + g x + M∞ − 1y may be applied in this case. We have seen in Sect. 10.3.2 that the perturbations propagate only along the left- and right-running characteristics in a two-dimensional supersonic flow past a wavy wall. Therefore, to examine the flow over a thin airfoil, we only require to consider the function f at the upper surface and the function g at the lower surface, i.e., ⎧  # ⎨ f x − M 2 − 1y y > 0 ∞  (10.142) φ (x, y) = # ⎩g x + M 2 − 1y y1

x

O

Function "g"

8

(x + y M2 − 1 = constant) Family of right−running Mach lines Fig. 10.8 A thin airfoil in the supersonic flow

Similarly, from the boundary condition on the lower surface of the airfoil,  U∞

dy dx

 

= v (x, 0) =



∂φ ∂y

 = y=0−

which yields U∞ g (x) = # 2 −1 M∞



$ 2 − 1g (x) M∞

dy dx

(10.147)

 (10.148) 

Hence, the pressure coefficient is Cp = −

2u 2 =− U∞ U∞



∂φ ∂x

 

=−

2  f (x) U∞

(10.149)

Substituting Eq. (10.144) into Eq. (10.149) yields 2 Cp = # 2 −1 M∞



dy dx

 (10.150) 

464

10 Linearized Potential Flow: The Small-Perturbation Theory

It is interesting to note that Eqs. (10.146) and (10.150) are the same results obtained in Sect. 5.20 by computing the pressures on the airfoil using the shock-expansion theory. Indeed, to calculate the velocity and pressure perturbations on a solid surface, the linearized theory discussed above may be referred to as the weak-wave theory. All other steps for calculating the lift and drag of the thin airfoil are, of course, the same as those stipulated in Sect. 5.20.

Concluding Remarks This chapter shows that the small-perturbation theory applies in cases where the flow velocity deviates only slightly from a uniform flow. The simpler mathematical procedure may then be employed to obtain the analytical solutions to the resulting linearized equations in terms of velocity-potential or the stream function. The linearized perturbation-velocity potential equation for steady, two-dimensional flow is derived. A single equation is deduced, which is valid for subsonic flow and supersonic flow. Besides that, a separate equation for transonic flow is obtained. It is observed that the equation governing the subsonic and supersonic flows is linear, while the equation for transonic flow is nonlinear. Moreover, a perturbation-velocity potential, φ, and a total-velocity potential, , are defined, and then using the method of small perturbations, the expression for the pressure coefficient is derived. The linearized governing equation is elliptic for subsonic flow, parabolic for sonic flow, and hyperbolic for supersonic flow. Applying the linearized perturbation equations to the classical flow problem past an infinite wave-shaped wall is demonstrated. The analysis is carried out for both subsonic and supersonic flows. In the subsonic case, it is shown that the strength of perturbations diminishes as we go away from the wall and completely disappear at ∞. In supersonic cases, however, the perturbations keep traveling with undiminished strength along the Mach lines, also called the characteristic lines.

Exercise Problems Exercise 10.1 Using Ackeret’s formula, derive the expressions for the lift and drag coefficients for a thin airfoil in a supersonic flow, as shown in Fig. 10.9. Assume that the flow is two-dimensional and steady. Exercise 10.2 Find the lift and drag coefficients for a thin airfoil in a Mach 2.0 flow at a small angle of attack, as shown in Fig. 10.10. Assume the flow to be steady, two-dimensional, irrotational, and isentropic. Exercise 10.3 A jet of uniform velocity, U∞ , is discharged into a quiescent environment. If the fluid density at the nozzle exit is only marginally different from that

Exercise Problems

465 y

8

M >1

Angle of attack, α , is small O

x

α y = f (x)

8

U

Thin air

foil

Fig. 10.9 A thin airfoil at a small angle of attack, α, in a supersonic flow 8

M = 2.0

10

o

10

o

8

p = 80 kPa

Fig. 10.10 A thin airfoil in a Mach 2.0 flow at a small angle of attack

of the surrounding medium, derive the linearized equation for the density in the jet. Assume the flow to be two-dimensional and steady.

Chapter 11

Elementary Concepts of Inviscid Hypersonic Flows

Abstract As demonstrated in the open literature, Mach 5 appears to be the lower bounds of the hypersonic flow regime. Still, a curious fellow might wonder why this limit is only Mach 5 but not Mach 4 or Mach 6? Put differently, what is so unusual about Mach 5 that it is considered to be the basic criteria for hypersonic flow? One might also wonder whether the expansion fans and shocks in the hypersonic flow field are different from those in the supersonic flow field? Undoubtedly, to answer these questions, one must have a profound understanding of the characteristics of hypersonic flow. This chapter explores the physical characteristics of fluids flowing at hypersonic speeds (at Mach numbers greater than 5), which will sooner or later become the “formalistic sketch” of hypersonic flow.

11.1 Introduction When a fluid element moves at a hypersonic speed, the kinetic energy is much more than its thermal energy. As a matter of fact, at higher Mach numbers, the ratio of kinetic energy to thermal energy is directly proportional to the Mach number squared. For a perfect gas, this ratio of kinetic energy to thermal energy is approximately equal to 21 γ (γ − 1) M2 . Thus, for a flow at Mach 10, this ratio is approximately 30, implying that the kinetic energy is 30 times greater than its thermal energy. On the other hand, hypersonic flow deceleration converts kinetic energy into thermal energy so that the flow field frequently experiences high temperatures, generating significant changes in thermodynamic properties. The strong shock waves in the flow field cause a substantial increase in temperature, contributing to the change in the thermodynamic properties. We know that the air is primarily composed of oxygen and nitrogen gases. At high temperatures, the vibrational excitation and dissociation of nitrogen and oxygen molecules result in an imperfect gas. Because molecular dissociation is an endothermic process, the imperfect gas has a greater reduction in flow temperature than the perfect gas. The changes in surface pressure strongly affect the dissociation of nitrogen and oxygen molecules.

© Springer Nature Singapore Pte Ltd. 2022 M. Kaushik, Fundamentals of Gas Dynamics, https://doi.org/10.1007/978-981-16-9085-3_11

467

468

11 Elementary Concepts of Inviscid Hypersonic Flows

Fig. 11.1 Thin shock layer in a hypersonic flow

θ = 15 deg β = 18 deg Μ a = 28

β

θ

11.1.1 Thin Shock Layer In our discussion on oblique shocks, it is observed that the rise in Mach number, M, causes a decrease in wave angle, β, for a fixed flow deflection angle, θ . That is, shocks tend to move closer to the surface of the body with an increase in Mach number. The flow area, bounded by a shock wave on one side and body surfaces on the other, is a shock layer. The structure of a typical shock layer in a hypersonic flow is shown in Fig. 11.1. It is found that the flow with a substantially high value of Mach number leads to a very thin layer of shock. Interestingly, the thin shock layer moves close to the body that the probability of its collision with the boundary layer on the body surface is greatly increased. This phenomenon is commonly referred to in the literature as shock-wave/boundary-layer interaction (SBLI), more pronounced at low Reynolds number where the boundary layer is relatively thicker. However, even at a high Reynolds number where the flow is almost inviscid, a thin shock layer can prove advantageous in analyzing the flow field. Indeed, the Newtonian flow model can be used to conveniently analyze the thin shock layer flow field and is commonly used in hypersonic aerodynamics for approximate calculations.

11.1.2 Entropy Layer In order to prevent a thin shock layer close to the body, a blunt-nosed aerodynamic body configuration is used for hypersonic flows. As is clear, the blunt-nosed configuration is usually the case where θ > θmax and, as a result, there is a detached bow-shock standing at some distance from the nose of the body (Fig. 11.2). A bowshock can be thought to be a combination of a strong normal shock at the middle and several oblique shocks at the periphery, which gradually degenerate into Mach waves at infinity. Furthermore, we know that the entropy of the flow increases across the shock wave; the higher the shock strength larger the entropy increase. Thus, the increase of the entropy along the streamline passing through the normal shock at the centerline of the flow is higher than that of the neighboring streamlines passing through the peripheral, weak oblique shock waves. As a result, there is a high entropy gradient in the nose region of the body, resulting in an entropy layer downstream that remains all along the body. From Crocco’s theorem, this layer of entropy can be seen as full of vorticity. As the boundary layer on the body surface grows inside the entropy layer, interactions of vorticity-boundary are inevitable.

11.1 Introduction

469

Fig. 11.2 Entropy layer in a hypersonic flow past a blunt-nosed body

Shock layer Entropy layer Boundary layer 8

M >5 Blunt nose

11.1.3 Viscous Interactions As the flow moves around a solid surface, a thin viscous layer forms over the surface. At high Mach numbers, the hypersonic flow has a considerable amount of kinetic energy, slowed by viscous motion inside the boundary layer. A portion of the lost kinetic energy is used to increase the internal energy of the gas, which in effect increases the temperature of the boundary layer. This process is known as viscous dissipation. The increase in temperature has a profound effect on the characteristics of the boundary layer. Since the coefficient of viscosity of the gas increases with the temperature, the boundary layer will become thicker by itself. In addition, due to constant static pressure within the boundary layer, increasing temperature induces a decrease in density following the ideal gas law. Thus, to maintain the same mass flow rate at reduced density, the boundary layer thickness should be increased. These factors are responsible for the rapid growth of the boundary layer at hypersonic speeds than at lower Mach numbers. Furthermore, the boundary layer theory for incompressible flows assumes that the viscous effects are confined to a thin region near the wall; and the outer inviscid flow is almost unaffected by the boundary layer. By comparison, due to the large boundary layer thickness at hypersonic Mach numbers, the outer inviscid flow is slightly or strongly influenced by the thick boundary layer. This is due to the greater displacement effects induced by a hypersonic boundary layer that dramatically alters the characteristics of the outer inviscid flow; changes in the outer flow contribute to further changes in the growth of the boundary layer. In this way, a feedback loop is created between the boundary layer and the external inviscid flow. This relationship is generally referred to as viscous-inviscid flow interactions. These interactions may be of two types. In the first, an exceptionally thick boundary layer grows over the surface, called the pressure interaction. In the second, a shock wave impinges on the boundary layer, due to which the boundary layer is subjected to an adverse pressure gradient present across the shock. As a result, the boundary layer is retarded or separated from the surface depending upon the boundary layer thickness and the wave strength. The viscous-inviscid flow interactions alter the surface

470

11 Elementary Concepts of Inviscid Hypersonic Flows

Fig. 11.3 Hypersonic boundary layer over a flat plate

Inviscid flow is weakly affected Shock wave

Inviscid flow is strongly affected

Outer edge of boundary layer 8

U

δ

y x Strong interaction

Weak interaction

pressure distributions around the flying object, affecting the lift and drag produced on the object. These interactions also increase heat transfer and skin friction. Consider a boundary layer on a flat plate in a hypersonic flow, as shown in Fig. 11.3. The thickness of the boundary layer (δ) at a distance of x from the leading edge is specified by x (11.1) δ∝√ Rex   where Rex = ρvx is the local Reynolds number based on the streamwise distance. μ The local Reynolds number, based on the wall temperature Tw , can also be described as ρw ue x (11.2) Rex = μw where ρw is the density of the gas, μw is the coefficient of viscosity, and ue is the velocity of the boundary layer fluid, at the given Tw . Combining Eqs. (11.1) and (11.2) yields  μw x δ∝√ ue x ρw     ρe μw ρe μw x x =√ δ∝ ρe ue x ρw μe Re∞ ρw μe

(11.3) (11.4)

μe

  where ρe is the density of the gas, μe is the coefficient of viscosity, and Re∞ = ρeμuee x is the Reynolds number based on the freestream conditions. In Eq. (11.4), the density ratio, ρρwe , can be expressed, using the perfect gas equation, as ρe pe Tw = ρw pw Te

(11.5)

11.1 Introduction

471

Recall that the static pressure inside an incompressible boundary layer remains constant and is equal to the freestream pressure, i.e., pw = pe . Applying the same approximation to the hypersonic boundary layer yields ρe Tw = ρw Te

(11.6)

Assuming a linear dependence between the temperature and viscosity, we have μw Tw = μe Te

(11.7)

Invoking Eqs. (11.6) and (11.7) into Eq. (11.4) yields δ∝√

x Re∞



Tw Te

 (11.8)

For an adiabatic wall obeying the no-slip condition, the wall temperature, Tw , may be approximated as the freestream stagnation temperature. Using isentropic relation, Tw can be expressed as Te γ −1 2 Tw M∞ =1+ (11.9) Te 2 Equation (11.8) becomes

x δ∝√ M2 Re∞ ∞

(11.10)

Equation (11.10) shows that the boundary layer thickness, δ, is directly proportional to the freestream Mach number, M∞ , squared. Consequently, the growth of the boundary layer at hypersonic Mach numbers is excessive.

11.1.4 High-Temperature Effects Another characteristic of hypersonic flow is a rapid temperature rise. The kinetic energy of a high-speed flow passing over a surface is dissipated as heat due to friction within the boundary layer. Extreme viscous dissipation within a hypersonic boundary layer will generate extremely high temperatures, which will excite the vibrational energy within the molecule, causing dissociation and even ionization of the gas. A chemically reacting boundary layer would therefore wet the surface of a hypersonic aircraft. Consider a space flight in which a re-entry spacecraft traveling at Mach 36 at an altitude of 59 km enters the atmosphere. The flow condition is depicted in Fig. 11.4. As previously mentioned, the center region of the bow shock can be approximated as a normal shock. Thus, the temperature ratio across a normal

472

11 Elementary Concepts of Inviscid Hypersonic Flows

Plasma 8

M = 36

8

M

Blunt body

Blunt body Central region is assumed to be a normal shock

Normal shock

Fig. 11.4 Atmospheric re-entry of a spacecraft

shock wave at Mach 36 is T2 = 253 T1

(11.11)

Since the temperature at 59 km altitude is T1 = −15 ◦ C + 273 = 258 K

(11.12)

The temperature downstream of the shock becomes T2 = 253 × 258 = 65274 K

(11.13)

which is an astonishingly high temperature; however, it is wrong. In reality, the air molecules will dissociate and ionize much before this temperature is attained. The shock layer essentially transforms into a partly ionized plasma, with the specific heats of the gas becoming a strong function of both pressure and temperature. Thus, it can be seen that the assumption of a calorically perfect gas in the preceding calculations is incorrect. Nonetheless, when proper calculations on this chemically reacting flow are performed, the temperature behind the shock wave is about 11000 K, which is still a high value at which the perfect gas assumption is invalid. At temperatures such as these, several high-temperature gas effects become apparent. For instance, the cp and c∀ values and their ratio γ are no more constants and become a function of temperature. Indeed, changes in γ may have to be considered at temperatures about 800 K. Two main physical properties allow high-temperature air to deviate from calorical perfectness. At room temperature, the air is often made up of diatomic oxygen and

11.1 Introduction

473

nitrogen gases. When the temperature of a diatomic gas is raised above its normal operating temperature, the vibrational mode of molecular motion becomes excited, consuming some of the energy that would otherwise go into the translational and rotational modes of motion. The excitation of the vibrational mode of molecular motion allows the specific heat to become temperature-dependent. If the temperature rises, these diatomic gases begin to dissociate into their monoatomic form, and with further rise in temperatures, the monoatomic gases ionize. Dissociation and ionization of the air molecules occur as follows. For 2000 K < T < 4000 K O2 → 2O

(11.14)

This means that between 2000 and 4000 K the diatomic oxygen molecules are broken down into their monoatomic form. For 4000 K < T < 9000 K N2 → 2N

(11.15)

i.e., between 4000 and 9000 K, the diatomic nitrogen molecules break down into their monoatomic form. Note that molecular dissociation is an endothermic mechanism that absorbs energy. The range of temperatures reveals that the gas is not all dissociated when a particular temperature is reached. In reality, the air is composed of a combination of diatomic and monoatomic molecules across the temperature range. The fraction of monoatomic gases, however, increases with the temperature rise. Likewise, For T > 9000 K O → O+ + e− N → N + + e−

(11.16) (11.17)

Again, this is an energy absorption endothermic process. Air consists of ionized and unionized atoms above 9000 K; the fraction of ionized atoms increases as the temperature increases. Besides these, additional chemical changes at higher temperatures can also occur. For example, the reaction between nitrogen and oxygen atoms may form nitrous oxide. From the discussion above, it can be understood that the temperature increase behind the shock wave can be large enough at high Mach numbers, causing changes in cp and c∀ dissociation of gas molecules, and at extremely high Mach numbers, ionization. The air behind the shock wave is essentially in the condition of plasma. The conventional shock wave relationships are therefore not applicable to air.

474

11 Elementary Concepts of Inviscid Hypersonic Flows

ρ p

1 1

M 1 = v1 /a 1 T1

ρ v 1n

v 1t

p

v2 θ

v1

v 2n

v 2t

2 2

M 2 = v2 /a 2 T2

β

Fig. 11.5 The plane oblique shock wave in a hypersonic flow

11.2 Hypersonic Oblique Shock Wave Relations We know that whenever the supersonic stream turns into itself, a shock wave is produced. Shock is an extremely thin region with the thickness of the order of 10−5 cm, where the viscosity and thermal conductivity are important mechanisms that make the shock process irreversible. Because of this irreversibility associated with the shock wave, stagnation pressure across the shock decreases with an increase of the Mach number, while the static pressure, static density, and static temperature rise. The hypersonic shock wave remains stationary if the static pressure downstream of the shock is sufficiently high. The relations obtained for the straight oblique shock waves at supersonic speeds are valid at moderately high hypersonic Mach numbers. Nevertheless, for very high Mach numbers (M → ∞), these exact relations eventually take some interesting and approximate forms. We will examine these results now. Consider the flow through an oblique shock wave, as illustrated in Fig. 11.5. Let subscripts 1 and 2 denote the upstream and downstream conditions, respectively. Moreover, only the fluid dynamics effects of Mach numbers are considered in the following analysis, while the viscous and real gas effects are ignored. Now, for a perfect gas, rewriting the static density ratio across an oblique shock wave as follows: ρ2 (γ + 1) M12 sin2 β = ρ1 (γ − 1) M12 sin2 β + 2

(5.91)

which, in the limit of high Mach numbers, i.e., M  1, becomes ρ2 γ +1 = ρ1 γ −1

(11.18)

The exact relation for the static pressure ratio across an oblique shock is p2 2γ  2 2 M1 sin β − 1 =1+ p1 γ +1

(5.88)

11.2 Hypersonic Oblique Shock Wave Relations

475

Now, including the high Mach number and low shock angle approximations, we can write: M2 sin2 β − 1 ≈ M2 sin2 β

(11.19)

M sin β  1

(11.20)

2

2

Hence the static pressure ratio across an oblique shock in a hypersonic flow becomes p2 2γ = (11.21) M2 sin2 β p1 γ +1 1 Similarly, the exact relation for the static temperature ratio across an oblique shock is ⎧ ⎨ 1+

T2 = T1 ⎩

γ −1 2 M1 2

sin2 β



(γ +1)2 M2 2(γ −1) 1

2γ M2 γ −1 1

⎫ sin2 β − 1 ⎬ ⎭

sin2 β

(5.89)

which, when combined with a high Mach number and small wave angle approximations, results in T2 = T1

2γ M2 sin2 γ +1 1 γ +1 γ −1

β

(11.22)

T2 2γ (γ − 1) 2 2 = M1 sin β T1 (γ + 1)2

(11.23)

Furthermore, let us rewrite here the θ − β − M relation for an oblique shock, derived in section 5.9, for convenience.  2 cot β M12 sin2 β − 1 tan θ = (5.108) 2 + M12 (γ + cos 2β) This relationship is schematically presented in Fig. 11.6. The figure indicates that the change in wave angle for small flow turning angles becomes smaller and smaller with the increase of Mach number. Thus, Eq. (5.108) can be simplified using the high Mach number and low shock angle approximations. Now, rearrange the terms in Eq. (5.108) as follows: tan θ 2 cot β tan θ tan β M12 sin2 β − 1 = 2 M12 sin2 β − 1 =

 

2 + M12 (γ + cos 2β) 2 + M12 (γ + cos 2β)

 

(11.24) (11.25)

476

11 Elementary Concepts of Inviscid Hypersonic Flows Μ = 1.5

Fig. 11.6 Wave angle variation at different Mach numbers

Μ=2

Μ=3 M = 10 M= θ = 10

  M2  sin β sin θ 1 + 1 γ + 1 − 2 sin2 β cos β cos θ 2    γ + 1 sin β sin θ M12 − M12 sin2 β − 1 M12 sin2 β − 1 = cos β cos θ 2      2 2 γ + 1 2 sin β sin θ sin β sin θ = M1 M1 sin β − 1 1 + cos β cos θ 2 cos β cos θ      2 2 cos (β − θ ) γ + 1 2 sin β sin θ = M1 M1 sin β − 1 cos β cos θ 2 cos β cos θ   sin β sin θ γ + 1 M12 M12 sin2 β − 1 = 2 cos (β − θ ) M12 sin2 β − 1 =

ο

(11.26) (11.27) (11.28) (11.29) (11.30)

In the limiting case of very small flow turning angles, we have sin θ ≈ θ

(11.31)

cos (β − θ ) ≈ cos β tan θ ≈ θ 1 1 cot θ = ≈ tan θ θ

(11.32) (11.33) (11.34)

Under the aforementioned approximations, Eq. (11.30) reduces to  M12 sin2 β − 1≈

 γ +1 2 M1 tan β θ 2

(11.35)

Furthermore, since a very high Mach number upstream of an oblique shock, i.e., M1  1, corresponds to a very low value of shock angle, it follows that

11.2 Hypersonic Oblique Shock Wave Relations

477

sin β ≈ β

(11.36)

cos 2β ≈ 1 tan β ≈ β

(11.37) (11.38)

Thus, Eq. (11.35) further reduces to β=

γ +1 θ 2

(11.39)

For air (γ = 1.4), β = 1.2θ

(11.40)

Thus, the wave angle is just 20% greater than the flow deflection angle at hypersonic speeds for very small flow turning angles. It should be noted that Eq. (11.39) is an excellent approximation for smaller turning angles and moderate flow turning angles. In fluid flows, the pressure distribution is typically described as a pressure coefficient rather than the pressure itself. The pressure coefficient in an incompressible flow is defined as Cp =

p − p∞ q∞

(11.41)

where p∞ and q∞ are the freestream static and dynamic pressures, respectively. It is important to remember that Eq. (11.41) is a concept that applies to all flow regimes, from incompressible to hypersonic. Nonetheless, the dynamic pressure in a compressible flow is expressed in terms of the Mach number. Consequently, q∞ =

1 2 γ 2 ρv∞ = p∞ M∞ 2 2

(11.42)

Therefore, the equation for Cp for a compressible flow is Cp =

2 2 γ M∞



 p −1 p∞

(11.43)

When we apply Eq. (11.43) to a supersonic flow past a body with an attached oblique shock wave, we get the pressure coefficient as Cp =

2 γ M12



 p2 −1 p1

(11.44)

where p1 and p2 are the static pressures upstream and downstream of the shock wave, respectively. By substituting pp21 in Eq. (11.44) using Eq. (5.88), we obtain

478

11 Elementary Concepts of Inviscid Hypersonic Flows

Cp =

2 γ M12



 2γ  2 2 M1 sin β − 1 γ +1

(11.45)

Again, for M  1, Cp =

4 sin2 β γ +1

(11.46)

11.3 Mach Number Independence Principle In Sect. 11.1, we have discussed the physical characteristics that make a hypersonic flow different than a supersonic flow. In this section, we will examine this difference between these flow regimes from a mathematical viewpoint. Consider an oblique shock again in the hypersonic freestream, as sketched in Fig. 11.5. Note that the pressure ratio, pp21 , across the shock wave (Eq. (11.21)) increases with Mach number, and in limiting case, i.e., if M → ∞ then pp21 → ∞. Similar is the case with the temperature ratio, TT21 , given by Eq. (11.23), where the temperature ratio increases with the increase of Mach number. Also, TT21 → ∞ when M → ∞. In contrast, the other relationships, such as the density ratio, ρρ21 , and the coefficient of pressure, Cp , given by Eqs. (11.18) and (11.46), respectively, are independent of the Mach number at very high hypersonic speeds (M → ∞). If, however, we make the shock downstream static pressure, p2 , non-dimensional by dividing it with the upstream dynamic pressure, q1 = 21 ρv12 = γ2 p1 M12 , i.e. p∗2 =

p2 p2 2 = q1 p1 γ M12

(11.47)

Introducing Eq. (11.21) into the above equation, we get p∗2 =

2 γ M12



 2γ M12 sin2 β γ +1 4 sin2 β p∗2 = γ +1

(11.48) (11.49)

Equation (11.49) is independent of Mach number. This certainly demonstrates the Mach number independence principle; at high Mach numbers, some aerodynamic parameters, for example, pressure coefficient (and henceforth the lift and drag coefficients), the density ratio, and the shock pattern end up noticeably independent of Mach number.

11.4 Hypersonic Expansion Wave Relations

479

11.4 Hypersonic Expansion Wave Relations Consider the centered Prandtl-Meyer expansion fan-created at the convex corner, as depicted in Fig. 11.7. We know that an expansion fan comprises many waves produced at the expansion corner and spread downstream. Let the Mach numbers upstream and downstream of the waves be M1 and M2 , respectively. The Prandtl-Meyer relation for the expansion waves is θ = ν (M2 ) − ν (M1 )

(5.202)

where ν is the Prandtl-Meyer function, defined as  ν (M) =

    γ +1 γ −1 2 −1 tan M − 1 − tan−1 M2 − 1 γ −1 γ +1

(5.199)

√ At high Mach numbers, we may write M2 − 1 ≈ M. Thus, the above equation becomes   γ +1 γ −1 −1 tan M − tan−1 M (11.50) ν (M) = γ −1 γ +1 Using trigonometric relation1 : tan−1 M =

1 π − tan−1 2 M

(11.51)

and the series expansion for inverse tan function2 : tan−1

1 1 1 1 1 1 = − + − + + ..... M M 3M3 5M5 7M7 9M9

(11.52)

Equation (11.50) becomes  ν (M) =

γ +1 γ −1



π − 2



   π 1 γ +1 1 + ... − − + ... γ −1M 2 M

Using the high Mach number approximation, i.e., M → ∞, the terms so on, in Eq. (11.53) can be neglected. This gives

  tan−1 y + tan−1 y1 = π2 .   2 arctan 1 = 1 − 1 + 1 − y y 3y3 5y5 1

1 7y7

+

1 9y9

+ ......

(11.53) 1 , 1 , and M3 M5

480

11 Elementary Concepts of Inviscid Hypersonic Flows

Fig. 11.7 Hypersonic flow turning at a convex corner

M1

μ

1

μ2

M2 θ

     1 γ +1 π γ +1 1 π − − ν (M) = − γ −1 2 γ −1M 2 M    γ +1 1 π 1 γ +1π − − + ν (M) = γ −1 2 γ −1 M 2 M 

(11.54)

(11.55)

Thus, we have  θ=

γ +1π − γ −1 2



γ +1 γ −1



 θ=

π 1 1 − + − M2 2 M2



γ +1π + γ −1 2



γ +1 γ −1

   1 1 1 1 + − − M1 M2 M2 M1   1 2 1 θ= − M2 (γ − 1) M1

γ +1 γ −1





π 1 1 + − M1 2 M1

(11.56) (11.57) (11.58)

This is the relation for Prandtl-Meyer expansion waves in hypersonic flow. It shows that the flow deflection angle θ is positive as M2 > M1 across an expansion fan. Alternatively, Eq. (11.58) can be expressed in another useful form: M1 γ −1 M1 θ =1− M2 2

(11.59)

Since the flow through an expansion fan is isentropic, i.e., p02 = p01 , thus we may rewrite Eq. (4.55) as p2/p02 p1/p01

 1+ p2 = = p1 1+

γ −1 2 M1 2 γ −1 2 M2 2

 γ γ−1  γ γ−1

(11.60)

11.4 Hypersonic Expansion Wave Relations

481

 1+ p2 = p1 1+

γ −1 2 M1 2 γ −1 2 M2 2

 γ γ−1  γ γ−1

(11.61)

At high Mach numbers (M  1), Eq. (11.61) becomes p2 = p1



M1 M2

 γ2γ−1 (11.62)

Substituting Eq. (11.59) into Eq. (11.62), we get   γ γ−1 p2 γ −1 M1 θ = 1− p1 2

(11.63)

It is worth noting that, Eq. (11.63) is equivalent to the hypersonic shock wave relation given by Eq. (11.21). The pressure coefficient is then Cp = 2 Cp = γ M12

 1−

2 γ M12

γ −1 M1 θ 2



 p2 −1 p1   γ γ −1

−1

(11.64) (11.65)

It is worth noting that Eq. (11.65) is analogous to Eq. (11.45) obtained for the hypersonic shock waves.

11.5 Hypersonic Similarity In fluid flows, similarity identifies the parameters or the set of parameters that result in identical flow features at different conditions. This enables one set of parameters obtained under specific experimental conditions to predict the results at another set of conditions. This section will not derive the criteria for hypersonic similarity mathematically; instead, we will focus our discussion on justifying the use of similarity. Consider a slender body in the hypersonic freestream with velocity U∞ at a low angle of attack, as shown in Fig. 11.8. If u and v are the perturbation-velocity components in the x- and y-direction, then at some distance downstream of the shock, we may write u = U∞ + u v = v

(11.66) (11.67)

482

11 Elementary Concepts of Inviscid Hypersonic Flows

Fig. 11.8 A slender body in the hypersonic freestream

Shock wave

U

8

θ Slender body

For the flow over a slender body u v, and thus the flow will be deflected by a small flow deflection angle, θ . Hence, from Fig. 11.8, we have  sin

v U∞ + u

 = sin θ

(11.68)

From small-perturbation theory, u U∞ . Thus, 

v sin U∞

 ≈ sin θ

(11.69)

For small flow turning angle, sin θ ≈ θ . Hence, v ≈θ U∞

v ≈ M∞ θ a∞

(11.70) (11.71)

where a∞ and M∞ are the speed of sound and the freestream Mach number, respectively. Eq. (11.71) is essentially a measure of the perturbation to the freestream velocity. In hypersonic flows, this disturbance indicator is defined as the hypersonic similarity parameter K, i.e., K = Mθ

(11.72)

Thus, if two different flow problems have the same values of K, then they are similar flows and will have like solutions.

11.6 Estimation of Aerodynamic Forces: The Newtonian Theory

483

crete w of dis o l f m r Unifo les id partic u l f f o t se 2θ

The fluid flow model is given by Sir Issac Newton (1687). The fluid particles lose their component of momentum normal to the solid boundary and thereby, moves further along the boundary.

Fig. 11.9 Fluid flow model proposed by Sir Issac Newton

11.6 Estimation of Aerodynamic Forces: The Newtonian Theory In 1687, Sir Issac Newton proposed a fluid flow modeling method where the flow was assumed as a stream of non-interacting particles. Essentially, this concept hypothesized fluid to be inviscid, giving inferior results when applied to predict the ship-hull drag. However, this model gives surprisingly good results when applied to hypersonic problems. The basic assumption in this theory was that each streamline of particles approaching a body would be deflected to parallel to the surface. Consequently, there will be a complete loss of the component of momentum normal to the body surface, while the component of momentum parallel to the surface will remain constant. According to Newton’s second law of motion, the time rate of momentum change is equal to the force exerted on the surface. Therefore, the total loss of the normal momentum component eventually results in the force normal to the surface (Fig. 11.9). Consider a stream with velocity U∞ impinging upon a surface of area A inclined at the angle θ to the incoming hypersonic freestream. From the geometry as shown in Fig. 11.10, we write The change in velocity normal to surface is = U∞ sin θ − 0 = U∞ sin θ

(11.73)

The mass flow rate incident upon the surface of area A is = ρ∞ AU∞ sin θ

(11.74)

Thus, the time rate of change of momentum of this fluid mass will be 2 sin2 θ = ρ∞ AU∞ sin θ (U∞ sin θ ) = ρ∞ AU∞

(11.75)

484

11 Elementary Concepts of Inviscid Hypersonic Flows

Fig. 11.10 Impingement of the non-interacting particles over a flat plate

Q

8

U

A sin θ

P

θ

But, from Newton’s second law of motion, the time rate of change of momentum is equal to the force F exerted on a surface. Thus, 2 sin2 θ F = ρ∞ AU∞ F 2 = ρ∞ U∞ sin2 θ A

(11.76) (11.77)

In reality, Newton has assumed a stream of non-interacting particles in rectilinear motion impinging upon the surface. This means that he has ignored the random motion of particles. However, from the kinetic theory of gases, we know that the static pressure experienced by a surface is solely due to the random motion of particles. Thus, the force per unit area exerted on the flat surface, given by Eq. (11.77), can be taken as the pressure difference above the freestream static pressure, i.e., 2 sin2 θ p − p∞ = ρ∞ U∞

(11.78)

where p∞ is the freestream static pressure acting on the surface and p is pressure experienced by the surface. Equation (11.78) can be rewritten as p − p∞ 1 ρ U2 2 ∞ ∞

= 2 sin2 θ

(11.79)

The term on the left side of Eq. (11.79) is the pressure coefficient, Cp . Thus, Eq. (11.79) becomes Cp = 2 sin2 θ

(11.80)

This is the well-known Newtonian sine-squared law for pressure coefficient. Equation (11.80) shows that the pressure coefficient is proportional to the sine squared of the flow deflection angle. At large Mach numbers and moderately small turning angles, the shock wave angle is almost equal to deflection. The fluid particles eventually hit the surface without any prior warning or deflection. This condition matches well with Eq. (11.80), and thus it is found to be useful in analyzing the flow at hypersonic Mach numbers.

11.6 Estimation of Aerodynamic Forces: The Newtonian Theory

485

Flat plate of area ‘A’

A Sin α

α 8

U

Fig. 11.11 An inclined flat plate in a hypersonic flow

11.6.1 Lift and Drag Coefficients for an Inclined Flat Plate Let us examine the applicability of Eq. (11.80) on a flat plate with chord length, c, at an angle of attack, α, to the freestream of velocity U∞ , as shown in Fig. 11.11. In the absence of friction and since the surface pressure acts normal to the plate, the resultant aerodynamic force R also acts perpendicular to the plate. Moreover, N can be resolved into its components L and D, called the lift and drag, respectively. From Eq. (11.80), the pressure coefficient on the lower surface of the plate is given by Cp (lower) = 2 sin2 α

(11.81)

But, on the upper surface, there is no flow, and hence the pressure coefficient will be Cp (upper) = 0

(11.82)

We have seen that the normal force coefficient is given by CN =

N q∞ Sr

(11.83)

where Sr = c (1). Since the normal force is produced due to the difference of pressures in lower and upper surfaces, thus CN =

1 c



c



Cp (lower) − Cp (upper) dx

(11.84)

0

where x is the distance measured from the leading edge. Introducing Eqs. (11.81) and (11.82) into the above equation, we get

486

11 Elementary Concepts of Inviscid Hypersonic Flows

CN = 2 sin2 α

(11.85)

The lift and drag coefficients can be defined as L q∞ Sr D CD = q∞ Sr CL =

(11.86) (11.87)

We can derive from the geometry, as illustrated in Fig. 11.11, that CL = CN cos α

(11.88)

CD = CN sin α

(11.89)

Substituting Eq. (11.85) into the above equations, we get CL = 2 sin2 α cos α

(11.90)

CD = 2 sin α

(11.91)

3

Therefore, the lift to drag ratio becomes CL L = = cot α D CD 11.6.1.1

(11.92)

The Maximum Lift Coefficient

Let us differentiate Eq. (11.90) with respect to α to obtain the value of the maximum lift coefficient, CL,max . dCL = 2 sin2 α (− sin α) + 4 sin α cos2 α = 0 dα 2 sin2 α = 3 α ≈ 55◦ Thus, Eq. (11.90) gives CL,max = 0.77

(11.93)

The MATLAB program below computes and presents the Newtonian results for a flat plate at various angles of attack in a hypersonic flow, as shown in Fig. 11.12.

11.6 Estimation of Aerodynamic Forces: The Newtonian Theory

487

Listing 11.1 A MATLAB programme for calculating Newtonian results in a hypersonic flow over a flat plate. % V e r s i o n 1.1 C o p y r i g h t M r i n a l Kaushik , IIT Kharagpur , 2 9 / 0 7 / 2 0 2 1 clc c l e a r all format short bin =91; adeg = zeros ( bin ,1) ; arad = zeros ( bin ,1) ; cl = zeros ( bin ,1) ; cd = zeros ( bin ,1) ; lbyd = zeros ( bin ,1) ; for i =1: bin adeg (i ,1) =i -1; % a l p h a in d e g r e e s arad (i ,1) = adeg (i ,1) * pi /180; % a l p h a in r a d i a n s cl (i ,1) =2*(( sin ( arad (i ,1) ) ) ^2) * cos ( arad (i ,1) ) ; % cl c a l c u l a t i o n cd (i ,1) =2*(( sin ( arad (i ,1) ) ) ^3) ; % cd c a l c u l a t i o n lbyd (i ,1) = cl ( i ,1) / cd ( i ,1) ; %L/D Calculation end % Plotting x = adeg ; y1 = cl ; y2 = lbyd ; y3 = cd ; f i g u r e (1) [ haxes , hline1 , h l i n e 2 ] = p l o t y y ( x , y1 , x , y2 ) ; % P l o t t i n g with m u l t i p l e y axis % Plot line p r o p e r t i e s s p e c i f i c a t i o n set ( hline1 , ' L i n e W i d t h ' ,2 , ' c o l o r ' , ' blue ') ; set ( hline2 , ' L i n e W i d t h ' ,2 , ' c o l o r ' , ' g r e e n ') ; set ( h a x e s (1) , ' Box ' , ' off ') ; set ( haxes , ' f o n t n a m e ' , ' t i m e s ' , ' F o n t S i z e ' ,16) ; % Y l i m i t and X l i m i t s p e c i f i c a t i o n set ( h a x e s (1) , ' ylim ' ,[0 2]) ; set ( h a x e s (2) , ' ylim ' ,[0 80]) ; set ( h a x e s (1) , ' Y T i c k ' , [ 0 : 0 . 2 : 2 ] ) set ( h a x e s (2) , ' Y T i c k ' , [ 0 : 1 0 : 8 0 ] ) ; set ( h a x e s (1) , ' x T i c k ' , [ 0 : 1 0 : 9 0 ] ) set ( h a x e s (2) , ' x T i c k ' , [ 0 : 1 0 : 9 0 ] ) ; set ( h a x e s (1) , ' X C o l o r ' , 'k ' , ' Y C o l o r ' , 'k ') ; set ( h a x e s (2) , ' X C o l o r ' , 'k ' , ' Y C o l o r ' , 'k ') hold on plot (x , y3 , ' L i n e W i d t h ' ,2 , ' c o l o r ' , ' red ') ; % P l o t t i n g Cd c u r v e l e g e n d ( ' C_ { L } ' , ' C_ { D } ' , 'L / D ' , ' L o c a t i o n ' , ' n o r t h w e s t ' ) ; % Legend creation x l a b e l ( ' \ a l p h a ' , ' f o n t n a m e ' , ' t i m e s ' , ' F o n t S i z e ' ,18) ; %X - axis title y l a b e l ( h a x e s (1) , ' C_ { L } , C_ { D } ' , ' f o n t n a m e ' , ' t i m e s ' , ' F o n t S i z e ' ,18) ; %Y - axis title y l a b e l ( h a x e s (2) , 'L / D ' , ' f o n t n a m e ' , ' t i m e s ' , ' F o n t S i z e ' ,18) ; %Y - axis title grid on

11.6.2 Some Observations on Newtonian Sine-Squared Law The results predicted using the Newtonian theory apply to inviscid hypersonic flow over a flat plate. The vital observations can be summarized as follows: • In hypersonic flows, the pressure coefficient, Cp , is predicted only by a local deflection angle θ , and by the surfaces directly in frontal flow conditions.

488

11 Elementary Concepts of Inviscid Hypersonic Flows

n

Fig. 11.12 Newtonian results plotted by the MATLAB program

pa

=8

Ex

8

M

ns

io

n

fa

1

A

Flat

plat

e

ck

2

l

e iqu

Ob

o

α = 15

sho

3

B 2’

3’

Fig. 11.13 A thin flat plate in a hypersonic flow

• The lift to drag ratio increases by decreasing the angle of attack and becomes infinitely large when α is almost equal to zero. This result is logical because there will not be any shear or viscous drag on the surface for an inviscid flow. • The lift coefficient increases with α and becomes maximum at around 55◦ , which is almost a practical condition. • At hypersonic speeds, the nonlinear variation of CL is exhibited for small angles of attack, i.e., α < 15◦ , in contrast to subsonic and supersonic flows, where linear variations are observed for this range of α.

11.6 Estimation of Aerodynamic Forces: The Newtonian Theory

489

Example 11.1 A thin flat plate at an angle of attack of 15◦ is in a Mach 8 airstream, as illustrated in Fig. 11.13. Find the lift and drag coefficients and lift-to-drag ratio using (i) the exact shock-expansion theory, and (ii) the Newtonian theory. Solution. Given, M1 = 8.0,

α = 15◦

(i) Using the exact shock-expansion theory: At the upper surface, for the flow through expansion waves from region 1 to region 2, from the isentropic table: For M1 = 8.0, ν1 = 95.62◦ ,

p01 = 9.763 × 103 p1

Thus, ν2 = ν1 + θ = 95.62 + 15 = 110.62◦ From isentropic table, for ν2 = 110.62◦ using interpolation, M2 = 14.32,

p02 = 4.808 × 105 p2

Since the expansion waves are isentropic, p02 = p01 The static pressure ratio across the expansion waves is p2 p02 p2 p01 p2 = × = × p1 p02 p1 p02 p1 9.763 × 103 = 4.808 × 105 = 0.0203 Now, the pressure coefficient on the upper surface is

(5.202)

490

11 Elementary Concepts of Inviscid Hypersonic Flows

Cp2 = =

2 :

2 γ M12



 p2 −1 p1

2 (0.0203 − 1) 1.4 × 82 = −0.0219

At the bottom surface, for the flow through shock waves from region 1 to region Using θ − β − M relation, for M1 = 8.0 and θ = 15◦ , β = 21◦ Thus, we get Mn1 = M1 sin β = 8 sin (21◦ ) = 2.87 From the normal shock table, for Mn1 = 2.87, p2 = 9.443 p1 The pressure coefficient at the bottom surface is   p2 2 Cp2 = −1 γ M12 p1 2 = (9.443 − 1) 1.4 × 82 = 0.1885

Note that the net axial force on the plate is zero as the pressure acts only normal to the plate. The pressure coefficient normal to plate is written as CN =

1 c

c

 Cp − Cpu dx = Cp2 −Cp2

= 0.1885 − (−0.0219) = 0.2104

11.6 Estimation of Aerodynamic Forces: The Newtonian Theory

491

The pressure coefficient normal to the freestream direction, i.e., lift coefficient, is CL = CN cos α = 0.2104 × cos (15◦ ) = 0.2032 Similarly, the pressure coefficient along the freestream direction, i.e., drag coefficient, is CD = CN sin α = 0.2104 × sin (15◦ ) = 0.0545 Therefore, the lift-to-drag ratio becomes CL L 0.2032 = = 3.73 = D CD 0.0545 (ii) From the Newtonian theory, the pressure coefficient at the bottom surface is Cp2 = 2 sin2 α = 2 × sin2 (15◦ ) = 0.134 Since the upper surface of the flat plate receives no direct impact of the freestream particles and remains in the shadow of the flow, the pressure coefficient at the upper surface is thus Cp2 = 0 The lift coefficient is  CL = Cp2 − Cp2 cos α = 0.134 × cos (15◦ ) = 0.1294 and the drag coefficient is  CD = Cp2 − Cp2 sin α = 0.134 × sin (15◦ ) = 0.03468 Therefore, CL 0.1294 L = = 3.73 = D CD 0.03468

492

11 Elementary Concepts of Inviscid Hypersonic Flows

11.7 Modified Newtonian Theory For blunt bodies, Lees (1955) has put forward the following modified form of Eq. (11.80): Cp = Cp,max sin2 θ

(11.94)

where Cp,max = Cp02 , i.e., the pressure coefficient at the stagnation point behind the shock wave. By definition, Cp02 =

p02 − p∞ 2 = 1 2 2 γ M ρ U ∞ 2 ∞ ∞



 p02 −1 p∞

(11.95)

Thus, Cp,max =

2 2 γ M∞



 p02 −1 p∞

(11.96)

But, from the Rayleigh-Pitot formula,  γ    1 γ −1 2γ  2 γ + 1 2 γ −1 p02 1+ M∞ M∞ − 1 = p∞ 2 γ +1

(11.97)

Substituting the above equation into Eq. (11.96) and rearranging the terms, we have    γ γ−1  2 2  1 − γ + 2γ M∞ 2 (γ + 1)2 M∞ −1 Cp,max = Cp02 = 2 2 − 2 (γ − 1) γ M∞ 4γ M∞ γ +1 (11.98) Note that Cp,max is a function of the specific heat ratio, γ , and the Mach number, M. Including high Mach number approximation, i.e., M  1, we get Cp,max as Cp,max

  γ 4 (γ + 1)2 γ −1 = γ +1 4γ

(11.99)

Therefore, for very high Mach numbers, the pressure coefficient according to the modified Newtonian theory is   γ 4 (γ + 1)2 γ −1 sin2 θ Cp = γ +1 4γ

(11.100)

11.7 Modified Newtonian Theory

493

For air (γ = 1.4), the coefficient of sin2 θ is equal to 1.839, whereas for γ = 1 the value of the coefficient is 2. For high Mach numbers and high geometric altitudes, e.g., M > 40 and hG > 90 km, the value of γ approaches one, which eventually leads to the Newtonian sine-squared law. Furthermore, the results obtained by the modified Newtonian theory are suitable for blunt-body configuration because this theory calculates the exact pressure at the stagnation point. In contrast, the direct Newtonian method is more suitable for slender bodies such as wedged or cones.

Concluding Remarks This chapter explores some of the basic characteristics of hypersonic flows, namely, think shock-layer, entropy-layer, viscous interactions, and high-temperature effects. We have derived the limiting form of the oblique shock relations, commonly referred to as hypersonic shock wave relations, by employing the high Mach number approximation, i.e., when M → ∞. These hypersonic shock wave relations demonstrate the existence of the Mach number independence principle. Similarly, for the hypersonic Prandtl-Meyer expansion waves at a convex corner, we have derived the relation:   1 1 2 (11.58) − θ= M2 (γ − 1) M1 We also examined the well-known Newtonian flow theory and demonstrated that it produces accurate results when applied to hypersonic flows. We have deduced the Newtonian sine-squared law as follows: Cp = 2 sin2 θ

(11.80)

Using this result for a thin flat plate at an angle of attack, α, in a hypersonic flow, the expressions for the lift coefficient, CL , the drag coefficient, CD , and their ratio, CL , are obtained. Finally, employing the correction coefficient as proposed by Lees CD (1955), we have derived the modified Newtonian sine-squared law as follows:   γ 4 (γ + 1)2 γ −1 sin2 θ Cp = γ +1 4γ

(11.100)

Exercise Problems Exercise 11.1 Consider a two-dimensional flow past a body as shown in Fig. 11.14. The freestream pressure and the Mach number are 8 kPa and 10, respectively. By using Newtonian flow model, find the pressure distribution on the body.

494

11 Elementary Concepts of Inviscid Hypersonic Flows

Fig. 11.14 A two-dimensional hypersonic flow over a body

D M = 10

C

p = 8 kPa

B

30

30

o

o

A Fig. 11.15 A two-dimensional Mach 8 flow over a body

10 p = 5 kPa

D o 30

C

M=8 o

B 30 A

Fig. 11.16 A two-dimensional hypersonic flow over a spherical-nosed cylindrical body

o

Base area = Sb

M = 10 R

L

2R

Exercise 11.2 A two-dimensional hypersonic airstream is flowing over a body, shown in Fig. 11.15. The freestream pressure and the Mach number are 8 and 5 kPa, respectively. By using Newtonian theory, find the pressure distribution on the surfaces AB, BC and CD. Exercise 11.3 A very thin flat plate with a 20◦ angle of attack is in a Mach 10 flow. Determine the pressure coefficients on the upper and lower surfaces, the lift and drag coefficients, and the lift-to-drag ratio using the Newtonian flow model. ◦ Exercise 11.4 A cone in a hypersonic flow has a half-angle of θc and is at α = 0 .  2 Newtonian sine-squared law Cp = 2 sin θc gives the pressure coefficient on the cone. Find the expression for the cone’s wave drag coefficient if the base region of the cone is Sb .

Exercise 11.5 Consider a spherical-nosed cylindrical body at zero angle of attack in a Mach 10 flow, as shown in Fig. 11.16. Find the drag coefficient of the body by using the modified Newtonian flow model.

Appendix A

The Standard Atmosphere

The properties of International Standard Atmosphere (ISA) are tabulated in SI units. • • • • •

The geo-potential altitude h and geometric altitude hG are measured in meters, m. The temperature T values are given in Kelvin, K. The speed of sound a is given in m/s. The pressure p is given in Pascals, Pa. The density ρ is shown in kg/m3 . h (m) 49825 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 525 550

hG (m) 50217.73 0 25 50 75 100 125 150 175 200.01 225.01 250.01 275.01 300.01 325.02 350.02 375.02 400.03 425.03 450.03 475.04 500.04 525.04 550.05

T (K) 270.65 288.15 287.99 287.83 287.66 287.5 287.34 287.18 287.01 286.85 286.69 286.53 286.36 286.2 286.04 285.88 285.71 285.55 285.39 285.23 285.06 284.9 284.74 284.58

© Springer Nature Singapore Pte Ltd. 2022 M. Kaushik, Fundamentals of Gas Dynamics, https://doi.org/10.1007/978-981-16-9085-3

a (m/s) 329.8 340.29 340.2 340.1 340.01 339.91 339.81 339.72 339.62 339.53 339.43 339.33 339.24 339.14 339.04 338.95 338.85 338.76 338.66 338.56 338.47 338.37 338.27 338.18

p (Pa) 77.64 101325 101025.03 100725.78 100427.25 100129.44 99832.34 99535.96 99240.29 98945.33 98651.08 98357.54 98064.7 97772.58 97481.16 97190.44 96900.42 96611.11 96322.5 96034.58 95747.36 95460.84 95175.01 94889.88

ρ (kg/m3 ) 0.001 1.225 1.222 1.219 1.216 1.213 1.21 1.207 1.205 1.202 1.199 1.196 1.193 1.19 1.187 1.184 1.182 1.179 1.176 1.173 1.17 1.167 1.164 1.162

495

496

Appendix A: The Standard Atmosphere h (m) 49825 575 600 625 650 675 700 725 750 775 800 825 850 875 900 925 950 975 1000 1025 1050 1075 1100 1125 1150 1175 1200 1225 1250 1275 1300 1325 1350 1375 1400 1425 1450 1475 1500 1525 1550 1575 1600 1625 1650 1675 1700 1725 1750

hG (m) 50217.73 575.05 600.06 625.06 650.07 675.07 700.08 725.08 750.09 775.09 800.1 825.11 850.11 875.12 900.13 925.13 950.14 975.15 1000.16 1025.16 1050.17 1075.18 1100.19 1125.2 1150.21 1175.22 1200.23 1225.24 1250.25 1275.26 1300.27 1325.28 1350.29 1375.3 1400.31 1425.32 1450.33 1475.34 1500.35 1525.37 1550.38 1575.39 1600.4 1625.41 1650.43 1675.44 1700.45 1725.47 1750.48

T (K) 270.65 284.41 284.25 284.09 283.93 283.76 283.6 283.44 283.28 283.11 282.95 282.79 282.63 282.46 282.3 282.14 281.98 281.81 281.65 281.49 281.33 281.16 281 280.84 280.68 280.51 280.35 280.19 280.03 279.86 279.7 279.54 279.38 279.21 279.05 278.89 278.73 278.56 278.4 278.24 278.08 277.91 277.75 277.59 277.43 277.26 277.1 276.94 276.78

a (m/s) 329.8 338.08 337.98 337.89 337.79 337.69 337.6 337.5 337.4 337.31 337.21 337.11 337.02 336.92 336.82 336.73 336.63 336.53 336.43 336.34 336.24 336.14 336.05 335.95 335.85 335.75 335.66 335.56 335.46 335.36 335.27 335.17 335.07 334.98 334.88 334.78 334.68 334.58 334.49 334.39 334.29 334.19 334.1 334 333.9 333.8 333.71 333.61 333.51

p (Pa) 77.64 94605.44 94321.68 94038.62 93756.25 93474.56 93193.56 92913.24 92633.61 92354.66 92076.39 91798.8 91521.88 91245.65 90970.09 90695.2 90420.98 90147.44 89874.57 89602.37 89330.83 89059.97 88789.76 88520.22 88251.35 87983.13 87715.58 87448.69 87182.45 86916.87 86651.95 86387.68 86124.06 85861.1 85598.78 85337.12 85076.1 84815.73 84556 84296.92 84038.49 83780.69 83523.54 83267.02 83011.15 82755.91 82501.3 82247.33 81994

ρ (kg/m3 ) 0.001 1.159 1.156 1.153 1.15 1.148 1.145 1.142 1.139 1.136 1.134 1.131 1.128 1.125 1.123 1.12 1.117 1.114 1.112 1.109 1.106 1.103 1.101 1.098 1.095 1.093 1.09 1.087 1.085 1.082 1.079 1.077 1.074 1.071 1.069 1.066 1.063 1.061 1.058 1.055 1.053 1.05 1.048 1.045 1.042 1.04 1.037 1.035 1.032

Appendix A: The Standard Atmosphere h (m) 49825 1775 1800 1825 1850 1875 1900 1925 1950 1975 2000 2025 2050 2075 2100 2125 2150 2175 2200 2225 2250 2275 2300 2325 2350 2375 2400 2425 2450 2475 2500 2525 2550 2575 2600 2625 2650 2675 2700 2725 2750 2775 2800 2825 2850 2875 2900 2925 2950

hG (m) 50217.73 1775.49 1800.51 1825.52 1850.54 1875.55 1900.57 1925.58 1950.6 1975.61 2000.63 2025.64 2050.66 2075.68 2100.69 2125.71 2150.73 2175.74 2200.76 2225.78 2250.79 2275.81 2300.83 2325.85 2350.87 2375.89 2400.9 2425.92 2450.94 2475.96 2500.98 2526 2551.02 2576.04 2601.06 2626.08 2651.1 2676.12 2701.14 2726.17 2751.19 2776.21 2801.23 2826.25 2851.28 2876.3 2901.32 2926.34 2951.37

T (K) 270.65 276.61 276.45 276.29 276.13 275.96 275.8 275.64 275.48 275.31 275.15 274.99 274.83 274.66 274.5 274.34 274.18 274.01 273.85 273.69 273.53 273.36 273.2 273.04 272.88 272.71 272.55 272.39 272.23 272.06 271.9 271.74 271.58 271.41 271.25 271.09 270.93 270.76 270.6 270.44 270.28 270.11 269.95 269.79 269.63 269.46 269.3 269.14 268.98

497 a (m/s) 329.8 333.41 333.31 333.22 333.12 333.02 332.92 332.82 332.73 332.63 332.53 332.43 332.33 332.23 332.14 332.04 331.94 331.84 331.74 331.64 331.55 331.45 331.35 331.25 331.15 331.05 330.95 330.86 330.76 330.66 330.56 330.46 330.36 330.26 330.16 330.07 329.97 329.87 329.77 329.67 329.57 329.47 329.37 329.27 329.17 329.07 328.98 328.88 328.78

p (Pa) 77.64 81741.3 81489.22 81237.78 80986.97 80736.78 80487.22 80238.28 79989.97 79742.28 79495.22 79248.77 79002.94 78757.73 78513.14 78269.16 78025.79 77783.04 77540.9 77299.37 77058.46 76818.14 76578.44 76339.34 76100.85 75862.96 75625.68 75388.99 75152.91 74917.42 74682.53 74448.24 74214.55 73981.45 73748.94 73517.02 73285.7 73054.96 72824.81 72595.25 72366.28 72137.89 71910.09 71682.87 71456.23 71230.17 71004.69 70779.79 70555.47

ρ (kg/m3 ) 0.001 1.029 1.027 1.024 1.022 1.019 1.017 1.014 1.012 1.009 1.006 1.004 1.001 0.999 0.996 0.994 0.991 0.989 0.986 0.984 0.981 0.979 0.976 0.974 0.972 0.969 0.967 0.964 0.962 0.959 0.957 0.954 0.952 0.95 0.947 0.945 0.942 0.94 0.938 0.935 0.933 0.93 0.928 0.926 0.923 0.921 0.919 0.916 0.914

498

Appendix A: The Standard Atmosphere h (m) 49825 2975 3000 3025 3050 3075 3100 3125 3150 3175 3200 3225 3250 3275 3300 3325 3350 3375 3400 3425 3450 3475 3500 3525 3550 3575 3600 3625 3650 3675 3700 3725 3750 3775 3800 3825 3850 3875 3900 3925 3950 3975 4000 4025 4050 4075 4100 4125 4150

hG (m) 50217.73 2976.39 3001.41 3026.44 3051.46 3076.48 3101.51 3126.53 3151.56 3176.58 3201.61 3226.63 3251.66 3276.68 3301.71 3326.74 3351.76 3376.79 3401.82 3426.84 3451.87 3476.9 3501.92 3526.95 3551.98 3577.01 3602.04 3627.06 3652.09 3677.12 3702.15 3727.18 3752.21 3777.24 3802.27 3827.3 3852.33 3877.36 3902.39 3927.42 3952.45 3977.48 4002.51 4027.54 4052.58 4077.61 4102.64 4127.67 4152.71

T (K) 270.65 268.81 268.65 268.49 268.33 268.16 268 267.84 267.68 267.51 267.35 267.19 267.03 266.86 266.7 266.54 266.38 266.21 266.05 265.89 265.73 265.56 265.4 265.24 265.08 264.91 264.75 264.59 264.43 264.26 264.1 263.94 263.78 263.61 263.45 263.29 263.13 262.96 262.8 262.64 262.48 262.31 262.15 261.99 261.83 261.66 261.5 261.34 261.18

a (m/s) 329.8 328.68 328.58 328.48 328.38 328.28 328.18 328.08 327.98 327.88 327.78 327.68 327.58 327.48 327.38 327.28 327.18 327.08 326.98 326.88 326.78 326.68 326.58 326.48 326.38 326.28 326.18 326.08 325.98 325.88 325.78 325.68 325.58 325.48 325.38 325.28 325.18 325.08 324.98 324.88 324.78 324.68 324.58 324.48 324.38 324.28 324.18 324.08 323.97

p (Pa) 77.64 70331.72 70108.54 69885.95 69663.92 69442.46 69221.58 69001.26 68781.52 68562.34 68343.72 68125.67 67908.19 67691.26 67474.9 67259.1 67043.86 66829.17 66615.05 66401.47 66188.46 65975.99 65764.08 65552.73 65341.92 65131.66 64921.95 64712.78 64504.16 64296.09 64088.56 63881.58 63675.13 63469.23 63263.86 63059.04 62854.75 62650.99 62447.78 62245.09 62042.94 61841.32 61640.24 61439.68 61239.65 61040.15 60841.17 60642.72 60444.8

ρ (kg/m3 ) 0.001 0.911 0.909 0.907 0.904 0.902 0.9 0.897 0.895 0.893 0.891 0.888 0.886 0.884 0.881 0.879 0.877 0.875 0.872 0.87 0.868 0.865 0.863 0.861 0.859 0.857 0.854 0.852 0.85 0.848 0.845 0.843 0.841 0.839 0.837 0.834 0.832 0.83 0.828 0.826 0.823 0.821 0.819 0.817 0.815 0.813 0.811 0.808 0.806

Appendix A: The Standard Atmosphere h (m) 49825 4175 4200 4225 4250 4275 4300 4325 4350 4375 4400 4425 4450 4475 4500 4525 4550 4575 4600 4625 4650 4675 4700 4725 4750 4775 4800 4825 4850 4875 4900 4925 4950 4975 5000 5025 5050 5075 5100 5125 5150 5175 5200 5225 5250 5275 5300 5325 5350

hG (m) 50217.73 4177.74 4202.77 4227.8 4252.84 4277.87 4302.9 4327.94 4352.97 4378.01 4403.04 4428.08 4453.11 4478.15 4503.18 4528.22 4553.25 4578.29 4603.32 4628.36 4653.4 4678.43 4703.47 4728.51 4753.54 4778.58 4803.62 4828.66 4853.69 4878.73 4903.77 4928.81 4953.85 4978.89 5003.93 5028.97 5054.01 5079.05 5104.09 5129.13 5154.17 5179.21 5204.25 5229.29 5254.33 5279.37 5304.41 5329.45 5354.5

T (K) 270.65 261.01 260.85 260.69 260.53 260.36 260.2 260.04 259.88 259.71 259.55 259.39 259.23 259.06 258.9 258.74 258.58 258.41 258.25 258.09 257.93 257.76 257.6 257.44 257.28 257.11 256.95 256.79 256.63 256.46 256.3 256.14 255.98 255.81 255.65 255.49 255.33 255.16 255 254.84 254.68 254.51 254.35 254.19 254.03 253.86 253.7 253.54 253.38

499 a (m/s) 329.8 323.87 323.77 323.67 323.57 323.47 323.37 323.27 323.17 323.07 322.97 322.86 322.76 322.66 322.56 322.46 322.36 322.26 322.16 322.05 321.95 321.85 321.75 321.65 321.55 321.45 321.34 321.24 321.14 321.04 320.94 320.83 320.73 320.63 320.53 320.43 320.33 320.22 320.12 320.02 319.92 319.82 319.71 319.61 319.51 319.41 319.3 319.2 319.1

p (Pa) 77.64 60247.39 60050.52 59854.16 59658.32 59463 59268.2 59073.92 58880.15 58686.9 58494.16 58301.93 58110.22 57919.02 57728.32 57538.14 57348.46 57159.29 56970.63 56782.47 56594.81 56407.66 56221 56034.85 55849.2 55664.04 55479.39 55295.23 55111.56 54928.39 54745.71 54563.53 54381.83 54200.63 54019.91 53839.69 53659.95 53480.69 53301.92 53123.64 52945.84 52768.52 52591.68 52415.33 52239.45 52064.05 51889.13 51714.68 51540.71

ρ (kg/m3 ) 0.001 0.804 0.802 0.8 0.798 0.796 0.794 0.791 0.789 0.787 0.785 0.783 0.781 0.779 0.777 0.775 0.773 0.771 0.769 0.766 0.764 0.762 0.76 0.758 0.756 0.754 0.752 0.75 0.748 0.746 0.744 0.742 0.74 0.738 0.736 0.734 0.732 0.73 0.728 0.726 0.724 0.722 0.72 0.718 0.716 0.714 0.713 0.711 0.709

500

Appendix A: The Standard Atmosphere h (m) 49825 5375 5400 5425 5450 5475 5500 5525 5550 5575 5600 5625 5650 5675 5700 5725 5750 5775 5800 5825 5850 5875 5900 5925 5950 5975 6000 6025 6050 6075 6100 6125 6150 6175 6200 6225 6250 6275 6300 6325 6350 6375 6400 6425 6450 6475 6500 6525 6550

hG (m) 50217.73 5379.54 5404.58 5429.62 5454.67 5479.71 5504.75 5529.8 5554.84 5579.88 5604.93 5629.97 5655.02 5680.06 5705.1 5730.15 5755.19 5780.24 5805.28 5830.33 5855.38 5880.42 5905.47 5930.52 5955.56 5980.61 6005.66 6030.7 6055.75 6080.8 6105.85 6130.89 6155.94 6180.99 6206.04 6231.09 6256.14 6281.19 6306.24 6331.29 6356.34 6381.39 6406.44 6431.49 6456.54 6481.59 6506.64 6531.69 6556.74

T (K) 270.65 253.21 253.05 252.89 252.73 252.56 252.4 252.24 252.08 251.91 251.75 251.59 251.43 251.26 251.1 250.94 250.78 250.61 250.45 250.29 250.13 249.96 249.8 249.64 249.48 249.31 249.15 248.99 248.83 248.66 248.5 248.34 248.18 248.01 247.85 247.69 247.53 247.36 247.2 247.04 246.88 246.71 246.55 246.39 246.23 246.06 245.9 245.74 245.58

a (m/s) 329.8 319 318.9 318.79 318.69 318.59 318.49 318.38 318.28 318.18 318.08 317.97 317.87 317.77 317.66 317.56 317.46 317.36 317.25 317.15 317.05 316.94 316.84 316.74 316.63 316.53 316.43 316.33 316.22 316.12 316.02 315.91 315.81 315.71 315.6 315.5 315.39 315.29 315.19 315.08 314.98 314.88 314.77 314.67 314.57 314.46 314.36 314.25 314.15

p (Pa) 77.64 51367.21 51194.19 51021.63 50849.55 50677.94 50506.8 50336.13 50165.92 49996.19 49826.91 49658.1 49489.76 49321.87 49154.45 48987.49 48820.99 48654.94 48489.36 48324.23 48159.56 47995.34 47831.57 47668.26 47505.4 47342.99 47181.03 47019.52 46858.45 46697.84 46537.67 46377.94 46218.66 46059.82 45901.43 45743.48 45585.96 45428.89 45272.25 45116.06 44960.29 44804.97 44650.08 44495.62 44341.6 44188.01 44034.85 43882.12 43729.81

ρ (kg/m3 ) 0.001 0.707 0.705 0.703 0.701 0.699 0.697 0.695 0.693 0.691 0.689 0.688 0.686 0.684 0.682 0.68 0.678 0.676 0.674 0.673 0.671 0.669 0.667 0.665 0.663 0.662 0.66 0.658 0.656 0.654 0.652 0.651 0.649 0.647 0.645 0.643 0.642 0.64 0.638 0.636 0.634 0.633 0.631 0.629 0.627 0.626 0.624 0.622 0.62

Appendix A: The Standard Atmosphere h (m) 49825 6575 6600 6625 6650 6675 6700 6725 6750 6775 6800 6825 6850 6875 6900 6925 6950 6975 7000 7025 7050 7075 7100 7125 7150 7175 7200 7225 7250 7275 7300 7325 7350 7375 7400 7425 7450 7475 7500 7525 7550 7575 7600 7625 7650 7675 7700 7725 7750

hG (m) 50217.73 6581.79 6606.84 6631.9 6656.95 6682 6707.05 6732.11 6757.16 6782.21 6807.27 6832.32 6857.37 6882.43 6907.48 6932.54 6957.59 6982.64 7007.7 7032.75 7057.81 7082.87 7107.92 7132.98 7158.03 7183.09 7208.15 7233.2 7258.26 7283.32 7308.37 7333.43 7358.49 7383.55 7408.61 7433.66 7458.72 7483.78 7508.84 7533.9 7558.96 7584.02 7609.08 7634.14 7659.2 7684.26 7709.32 7734.38 7759.44

T (K) 270.65 245.41 245.25 245.09 244.93 244.76 244.6 244.44 244.28 244.11 243.95 243.79 243.63 243.46 243.3 243.14 242.98 242.81 242.65 242.49 242.33 242.16 242 241.84 241.68 241.51 241.35 241.19 241.03 240.86 240.7 240.54 240.38 240.21 240.05 239.89 239.73 239.56 239.4 239.24 239.08 238.91 238.75 238.59 238.43 238.26 238.1 237.94 237.78

501 a (m/s) 329.8 314.05 313.94 313.84 313.73 313.63 313.53 313.42 313.32 313.21 313.11 313 312.9 312.8 312.69 312.59 312.48 312.38 312.27 312.17 312.06 311.96 311.86 311.75 311.65 311.54 311.44 311.33 311.23 311.12 311.02 310.91 310.81 310.7 310.6 310.49 310.39 310.28 310.18 310.07 309.96 309.86 309.75 309.65 309.54 309.44 309.33 309.23 309.12

p (Pa) 77.64 43577.94 43426.5 43275.48 43124.88 42974.71 42824.97 42675.65 42526.75 42378.27 42230.21 42082.57 41935.34 41788.54 41642.15 41496.18 41350.62 41205.47 41060.74 40916.42 40772.51 40629.02 40485.93 40343.25 40200.97 40059.1 39917.64 39776.59 39635.93 39495.68 39355.84 39216.39 39077.34 38938.7 38800.45 38662.6 38525.14 38388.09 38251.42 38115.16 37979.28 37843.8 37708.71 37574.01 37439.7 37305.78 37172.24 37039.1 36906.34

ρ (kg/m3 ) 0.001 0.619 0.617 0.615 0.613 0.612 0.61 0.608 0.606 0.605 0.603 0.601 0.6 0.598 0.596 0.595 0.593 0.591 0.59 0.588 0.586 0.584 0.583 0.581 0.579 0.578 0.576 0.575 0.573 0.571 0.57 0.568 0.566 0.565 0.563 0.561 0.56 0.558 0.557 0.555 0.553 0.552 0.55 0.549 0.547 0.545 0.544 0.542 0.541

502

Appendix A: The Standard Atmosphere h (m) 49825 7775 7800 7825 7850 7875 7900 7925 7950 7975 8000 8025 8050 8075 8100 8125 8150 8175 8200 8225 8250 8275 8300 8325 8350 8375 8400 8425 8450 8475 8500 8525 8550 8575 8600 8625 8650 8675 8700 8725 8750 8775 8800 8825 8850 8875 8900 8925 8950

hG (m) 50217.73 7784.5 7809.56 7834.62 7859.68 7884.75 7909.81 7934.87 7959.93 7985 8010.06 8035.12 8060.18 8085.25 8110.31 8135.38 8160.44 8185.5 8210.57 8235.63 8260.7 8285.76 8310.83 8335.89 8360.96 8386.02 8411.09 8436.16 8461.22 8486.29 8511.36 8536.42 8561.49 8586.56 8611.62 8636.69 8661.76 8686.83 8711.9 8736.97 8762.03 8787.1 8812.17 8837.24 8862.31 8887.38 8912.45 8937.52 8962.59

T (K) 270.65 237.61 237.45 237.29 237.13 236.96 236.8 236.64 236.48 236.31 236.15 235.99 235.83 235.66 235.5 235.34 235.18 235.01 234.85 234.69 234.53 234.36 234.2 234.04 233.88 233.71 233.55 233.39 233.23 233.06 232.9 232.74 232.58 232.41 232.25 232.09 231.93 231.76 231.6 231.44 231.28 231.11 230.95 230.79 230.63 230.46 230.3 230.14 229.98

a (m/s) 329.8 309.02 308.91 308.8 308.7 308.59 308.49 308.38 308.27 308.17 308.06 307.96 307.85 307.74 307.64 307.53 307.43 307.32 307.21 307.11 307 306.89 306.79 306.68 306.58 306.47 306.36 306.26 306.15 306.04 305.94 305.83 305.72 305.62 305.51 305.4 305.29 305.19 305.08 304.97 304.87 304.76 304.65 304.54 304.44 304.33 304.22 304.12 304.01

p (Pa) 77.64 36773.96 36641.98 36510.37 36379.15 36248.31 36117.85 35987.77 35858.07 35728.75 35599.81 35471.25 35343.06 35215.24 35087.81 34960.74 34834.05 34707.73 34581.78 34456.2 34330.99 34206.15 34081.68 33957.57 33833.83 33710.46 33587.45 33464.8 33342.52 33220.6 33099.04 32977.84 32857 32736.52 32616.4 32496.63 32377.22 32258.17 32139.47 32021.13 31903.13 31785.49 31668.21 31551.27 31434.68 31318.44 31202.55 31087.01 30971.81

ρ (kg/m3 ) 0.001 0.539 0.538 0.536 0.534 0.533 0.531 0.53 0.528 0.527 0.525 0.524 0.522 0.521 0.519 0.518 0.516 0.514 0.513 0.511 0.51 0.508 0.507 0.505 0.504 0.502 0.501 0.5 0.498 0.497 0.495 0.494 0.492 0.491 0.489 0.488 0.486 0.485 0.483 0.482 0.481 0.479 0.478 0.476 0.475 0.473 0.472 0.471 0.469

Appendix A: The Standard Atmosphere h (m) 49825 8975 9000 9025 9050 9075 9100 9125 9150 9175 9200 9225 9250 9275 9300 9325 9350 9375 9400 9425 9450 9475 9500 9525 9550 9575 9600 9625 9650 9675 9700 9725 9750 9775 9800 9825 9850 9875 9900 9925 9950 9975 10000 10025 10050 10075 10100

hG (m) 50217.73 8987.66 9012.73 9037.8 9062.87 9087.95 9113.02 9138.09 9163.16 9188.23 9213.3 9238.38 9263.45 9288.52 9313.6 9338.67 9363.74 9388.82 9413.89 9438.96 9464.04 9489.11 9514.19 9539.26 9564.34 9589.41 9614.49 9639.56 9664.64 9689.71 9714.79 9739.87 9764.94 9790.02 9815.1 9840.17 9865.25 9890.33 9915.41 9940.49 9965.56 9990.64 10015.72 10040.8 10065.88 10090.96 10116.04

T (K) 270.65 229.81 229.65 229.49 229.33 229.16 229 228.84 228.68 228.51 228.35 228.19 228.03 227.86 227.7 227.54 227.38 227.21 227.05 226.89 226.73 226.56 226.4 226.24 226.08 225.91 225.75 225.59 225.43 225.26 225.1 224.94 224.78 224.61 224.45 224.29 224.13 223.96 223.8 223.64 223.48 223.31 223.15 222.99 222.83 222.66 222.5

503 a (m/s) 329.8 303.9 303.79 303.69 303.58 303.47 303.36 303.26 303.15 303.04 302.93 302.82 302.72 302.61 302.5 302.39 302.28 302.18 302.07 301.96 301.85 301.74 301.64 301.53 301.42 301.31 301.2 301.09 300.99 300.88 300.77 300.66 300.55 300.44 300.33 300.23 300.12 300.01 299.9 299.79 299.68 299.57 299.46 299.35 299.25 299.14 299.03

p (Pa) 77.64 30856.96 30742.46 30628.3 30514.48 30401.01 30287.87 30175.08 30062.63 29950.52 29838.75 29727.31 29616.21 29505.45 29395.03 29284.94 29175.18 29065.76 28956.67 28847.91 28739.48 28631.38 28523.62 28416.18 28309.07 28202.28 28095.82 27989.69 27883.88 27778.4 27673.24 27568.4 27463.89 27359.69 27255.82 27152.26 27049.03 26946.11 26843.51 26741.23 26639.26 26537.61 26436.27 26335.24 26234.53 26134.13 26034.04

ρ (kg/m3 ) 0.001 0.468 0.466 0.465 0.464 0.462 0.461 0.459 0.458 0.457 0.455 0.454 0.452 0.451 0.45 0.448 0.447 0.446 0.444 0.443 0.442 0.44 0.439 0.438 0.436 0.435 0.434 0.432 0.431 0.43 0.428 0.427 0.426 0.424 0.423 0.422 0.42 0.419 0.418 0.417 0.415 0.414 0.413 0.411 0.41 0.409 0.408

504

Appendix A: The Standard Atmosphere h (m) 49825 10125 10150 10175 10200 10225 10250 10275 10300 10325 10350 10375 10400 10425 10450 10475 10500 10525 10550 10575 10600 10625 10650 10675 10700 10725 10750 10775 10800 10825 10850 10875 10900 10925 10950 10975 11000 11025 11050 11075 11100 11125 11150 11175 11200 11225 11250 11275 11300

hG (m) 50217.73 10141.12 10166.2 10191.28 10216.36 10241.44 10266.52 10291.6 10316.68 10341.76 10366.84 10391.92 10417 10442.09 10467.17 10492.25 10517.33 10542.42 10567.5 10592.58 10617.67 10642.75 10667.83 10692.92 10718 10743.09 10768.17 10793.25 10818.34 10843.42 10868.51 10893.59 10918.68 10943.77 10968.85 10993.94 11019.03 11044.11 11069.2 11094.29 11119.37 11144.46 11169.55 11194.64 11219.72 11244.81 11269.9 11294.99 11320.08

T (K) 270.65 222.34 222.18 222.01 221.85 221.69 221.53 221.36 221.2 221.04 220.88 220.71 220.55 220.39 220.23 220.06 219.9 219.74 219.58 219.41 219.25 219.09 218.93 218.76 218.6 218.44 218.28 218.11 217.95 217.79 217.63 217.46 217.3 217.14 216.98 216.81 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65

a (m/s) 329.8 298.92 298.81 298.7 298.59 298.48 298.37 298.26 298.15 298.04 297.93 297.82 297.71 297.6 297.49 297.38 297.27 297.16 297.05 296.94 296.83 296.72 296.61 296.5 296.39 296.28 296.17 296.06 295.95 295.84 295.73 295.62 295.51 295.4 295.29 295.18 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07

p (Pa) 77.64 25934.26 25834.8 25735.64 25636.79 25538.24 25440.01 25342.08 25244.45 25147.13 25050.12 24953.4 24857 24760.89 24665.08 24569.57 24474.37 24379.46 24284.85 24190.54 24096.52 24002.8 23909.38 23816.25 23723.42 23630.87 23538.62 23446.67 23355 23263.62 23172.54 23081.74 22991.23 22901.01 22811.08 22721.43 22632.06 22543.02 22454.32 22365.98 22277.98 22190.33 22103.02 22016.06 21929.44 21843.16 21757.22 21671.62 21586.35

ρ (kg/m3 ) 0.001 0.406 0.405 0.404 0.403 0.401 0.4 0.399 0.398 0.396 0.395 0.394 0.393 0.391 0.39 0.389 0.388 0.387 0.385 0.384 0.383 0.382 0.38 0.379 0.378 0.377 0.376 0.374 0.373 0.372 0.371 0.37 0.369 0.367 0.366 0.365 0.364 0.362 0.361 0.36 0.358 0.357 0.355 0.354 0.353 0.351 0.35 0.348 0.347

Appendix A: The Standard Atmosphere h (m) 49825 11325 11350 11375 11400 11425 11450 11475 11500 11525 11550 11575 11600 11625 11650 11675 11700 11725 11750 11775 11800 11825 11850 11875 11900 11925 11950 11975 12000 12025 12050 12075 12100 12125 12150 12175 12200 12225 12250 12275 12300 12325 12350 12375 12400 12425 12450 12475 12500

hG (m) 50217.73 11345.17 11370.26 11395.35 11420.44 11445.53 11470.62 11495.71 11520.8 11545.89 11570.98 11596.07 11621.16 11646.25 11671.34 11696.43 11721.53 11746.62 11771.71 11796.8 11821.9 11846.99 11872.08 11897.18 11922.27 11947.36 11972.46 11997.55 12022.65 12047.74 12072.83 12097.93 12123.02 12148.12 12173.22 12198.31 12223.41 12248.5 12273.6 12298.7 12323.79 12348.89 12373.99 12399.08 12424.18 12449.28 12474.38 12499.48 12524.57

T (K) 270.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65

505 a (m/s) 329.8 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07

p (Pa) 77.64 21501.42 21416.82 21332.56 21248.63 21165.03 21081.75 20998.81 20916.19 20833.9 20751.93 20670.28 20588.95 20507.95 20427.26 20346.89 20266.83 20187.1 20107.67 20028.56 19949.76 19871.27 19793.08 19715.21 19637.64 19560.38 19483.42 19406.76 19330.41 19254.35 19178.6 19103.14 19027.98 18953.11 18878.54 18804.27 18730.28 18656.59 18583.19 18510.07 18437.24 18364.7 18292.45 18220.48 18148.79 18077.38 18006.26 17935.42 17864.85

ρ (kg/m3 ) 0.001 0.346 0.344 0.343 0.342 0.34 0.339 0.338 0.336 0.335 0.334 0.332 0.331 0.33 0.328 0.327 0.326 0.325 0.323 0.322 0.321 0.32 0.318 0.317 0.316 0.315 0.313 0.312 0.311 0.31 0.308 0.307 0.306 0.305 0.304 0.302 0.301 0.3 0.299 0.298 0.296 0.295 0.294 0.293 0.292 0.291 0.277 0.288 0.287

506

Appendix A: The Standard Atmosphere h (m) 49825 12525 12550 12575 12600 12625 12650 12675 12700 12725 12750 12775 12800 12825 12850 12875 12900 12925 12950 12975 13000 13025 13050 13075 13100 13125 13150 13175 13200 13225 13250 13275 13300 13325 13350 13375 13400 13425 13450 13475 13500 13525 13550 13575 13600 13625 13650 13675 13700

hG (m) 50217.73 12549.67 12574.77 12599.87 12624.97 12650.07 12675.17 12700.27 12725.37 12750.47 12775.57 12800.67 12825.77 12850.87 12875.97 12901.07 12926.17 12951.27 12976.38 13001.48 13026.58 13051.68 13076.79 13101.89 13126.99 13152.09 13177.2 13202.3 13227.41 13252.51 13277.61 13302.72 13327.82 13352.93 13378.03 13403.14 13428.24 13453.35 13478.45 13503.56 13528.67 13553.77 13578.88 13603.99 13629.09 13654.2 13679.31 13704.42 13729.52

T (K) 270.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65

a (m/s) 329.8 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07

p (Pa) 77.64 17794.56 17724.55 17654.81 17585.35 17516.16 17447.25 17378.6 17310.22 17242.12 17174.28 17106.71 17039.4 16972.36 16905.59 16839.07 16772.82 16706.83 16641.1 16575.62 16510.41 16445.45 16380.74 16316.29 16252.1 16188.15 16124.46 16061.02 15997.83 15934.89 15872.19 15809.74 15747.54 15685.58 15623.87 15562.4 15501.17 15440.18 15379.43 15318.92 15258.65 15198.62 15138.82 15079.26 15019.93 14960.83 14901.97 14843.34 14784.94

ρ (kg/m3 ) 0.001 0.286 0.285 0.284 0.283 0.282 0.281 0.279 0.278 0.277 0.276 0.275 0.274 0.273 0.272 0.271 0.27 0.269 0.268 0.267 0.265 0.264 0.263 0.261 0.261 0.26 0.259 0.258 0.257 0.256 0.255 0.254 0.253 0.252 0.251 0.25 0.249 0.248 0.247 0.246 0.245 0.244 0.243 0.242 0.242 0.241 0.24 0.239 0.238

Appendix A: The Standard Atmosphere h (m) 49825 13725 13750 13775 13800 13825 13850 13875 13900 13925 13950 13975 14000 14025 14050 14075 14100 14125 14150 14175 14200 14225 14250 14275 14300 14325 14350 14375 14400 14425 14450 14475 14500 14525 14550 14575 14600 14625 14650 14675 14700 14725 14750 14775 14800 14825 14850 14875 14900

hG (m) 50217.73 13754.63 13779.74 13804.85 13829.96 13855.07 13880.17 13905.28 13930.39 13955.5 13980.61 14005.72 14030.83 14055.94 14081.05 14106.16 14131.27 14156.39 14181.5 14206.61 14231.72 14256.83 14281.94 14307.06 14332.17 14357.28 14382.39 14407.51 14432.62 14457.73 14482.85 14507.96 14533.08 14558.19 14583.31 14608.42 14633.53 14658.65 14683.77 14708.88 14734 14759.11 14784.23 14809.34 14834.46 14859.58 14884.69 14909.81 14934.93

T (K) 270.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65

507 a (m/s) 329.8 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07

p (Pa) 77.64 14726.77 14668.83 14611.11 14553.63 14496.36 14439.33 14382.52 14325.93 14269.57 14213.42 14157.5 14101.8 14046.32 13991.05 13936.01 13881.17 13826.56 13772.16 13717.97 13664 13610.24 13556.69 13503.35 13450.23 13397.31 13344.6 13292.09 13239.79 13187.7 13135.82 13084.13 13032.66 12981.38 12930.31 12879.43 12828.76 12778.28 12728.01 12677.93 12628.05 12578.37 12528.88 12479.58 12430.48 12381.57 12332.86 12284.34 12236.01

ρ (kg/m3 ) 0.001 0.237 0.236 0.235 0.234 0.233 0.232 0.231 0.23 0.229 0.229 0.228 0.227 0.226 0.225 0.224 0.223 0.222 0.221 0.221 0.22 0.219 0.218 0.217 0.216 0.215 0.215 0.214 0.213 0.212 0.211 0.21 0.21 0.209 0.208 0.207 0.206 0.205 0.205 0.204 0.203 0.202 0.201 0.201 0.2 0.199 0.198 0.198 0.197

508

Appendix A: The Standard Atmosphere h (m) 49825 14925 14950 14975 15000 15025 15050 15075 15100 15125 15150 15175 15200 15225 15250 15275 15300 15325 15350 15375 15400 15425 15450 15475 15500 15525 15550 15575 15600 15625 15650 15675 15700 15725 15750 15775 15800 15825 15850 15875 15900 15925 15950 15975 16000 16025 16050 16075 16100

hG (m) 50217.73 14960.05 14985.16 15010.28 15035.4 15060.52 15085.64 15110.75 15135.87 15160.99 15186.11 15211.23 15236.35 15261.47 15286.59 15311.71 15336.83 15361.95 15387.07 15412.19 15437.32 15462.44 15487.56 15512.68 15537.8 15562.92 15588.05 15613.17 15638.29 15663.41 15688.54 15713.66 15738.78 15763.91 15789.03 15814.16 15839.28 15864.41 15889.53 15914.66 15939.78 15964.91 15990.03 16015.16 16040.28 16065.41 16090.54 16115.66 16140.79

T (K) 270.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65

a (m/s) 329.8 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07

p (Pa) 77.64 12187.86 12139.91 12092.15 12044.57 11997.18 11949.98 11902.96 11856.13 11809.48 11763.02 11716.74 11670.64 11624.72 11578.99 11533.43 11488.05 11442.85 11397.83 11352.99 11308.32 11263.83 11219.51 11175.37 11131.4 11087.6 11043.98 11000.53 10957.25 10914.14 10871.19 10828.42 10785.82 10743.38 10701.11 10659.01 10617.07 10575.3 10533.69 10492.25 10450.97 10409.85 10368.89 10328.09 10287.46 10246.98 10206.67 10166.51 10126.51

ρ (kg/m3 ) 0.001 0.196 0.195 0.194 0.194 0.193 0.192 0.191 0.191 0.19 0.189 0.188 0.188 0.187 0.186 0.185 0.185 0.184 0.183 0.183 0.182 0.181 0.18 0.18 0.179 0.178 0.178 0.177 0.176 0.175 0.175 0.174 0.173 0.173 0.172 0.171 0.171 0.17 0.169 0.169 0.168 0.167 0.167 0.166 0.165 0.165 0.164 0.163 0.163

Appendix A: The Standard Atmosphere h (m) 49825 16125 16150 16175 16200 16225 16250 16275 16300 16325 16350 16375 16400 16425 16450 16475 16500 16525 16550 16575 16600 16625 16650 16675 16700 16725 16750 16775 16800 16825 16850 16875 16900 16925 16950 16975 17000 17025 17050 17075 17100 17125 17150 17175 17200 17225 17250 17275 17300

hG (m) 50217.73 16165.92 16191.04 16216.17 16241.3 16266.43 16291.55 16316.68 16341.81 16366.94 16392.07 16417.2 16442.33 16467.45 16492.58 16517.71 16542.84 16567.97 16593.1 16618.23 16643.37 16668.5 16693.63 16718.76 16743.89 16769.02 16794.15 16819.29 16844.42 16869.55 16894.68 16919.82 16944.95 16970.08 16995.22 17020.35 17045.48 17070.62 17095.75 17120.89 17146.02 17171.16 17196.29 17221.43 17246.56 17271.7 17296.83 17321.97 17347.1

T (K) 270.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65

509 a (m/s) 329.8 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07

p (Pa) 77.64 10086.67 10046.98 10007.45 9968.08 9928.86 9889.8 9850.88 9812.13 9773.52 9735.07 9696.77 9658.61 9620.61 9582.76 9545.06 9507.5 9470.1 9432.84 9395.72 9358.76 9321.94 9285.26 9248.73 9212.34 9176.09 9139.99 9104.03 9068.21 9032.53 8996.99 8961.6 8926.34 8891.22 8856.23 8821.39 8786.68 8752.11 8717.68 8683.38 8649.21 8615.18 8581.29 8547.52 8513.89 8480.4 8447.03 8413.8 8380.69

ρ (kg/m3 ) 0.001 0.162 0.162 0.161 0.16 0.16 0.159 0.158 0.158 0.157 0.157 0.156 0.155 0.155 0.154 0.153 0.153 0.152 0.152 0.151 0.15 0.15 0.149 0.149 0.148 0.148 0.147 0.146 0.146 0.145 0.145 0.144 0.144 0.143 0.142 0.142 0.141 0.141 0.14 0.14 0.139 0.139 0.138 0.137 0.137 0.136 0.136 0.135 0.135

510

Appendix A: The Standard Atmosphere h (m) 49825 17325 17350 17375 17400 17425 17450 17475 17500 17525 17550 17575 17600 17625 17650 17675 17700 17725 17750 17775 17800 17825 17850 17875 17900 17925 17950 17975 18000 18025 18050 18075 18100 18125 18150 18175 18200 18225 18250 18275 18300 18325 18350 18375 18400 18425 18450 18475 18500

hG (m) 50217.73 17372.24 17397.38 17422.51 17447.65 17472.79 17497.93 17523.06 17548.2 17573.34 17598.48 17623.62 17648.75 17673.89 17699.03 17724.17 17749.31 17774.45 17799.59 17824.73 17849.87 17875.01 17900.15 17925.29 17950.43 17975.57 18000.72 18025.86 18051 18076.14 18101.28 18126.43 18151.57 18176.71 18201.85 18227 18252.14 18277.28 18302.43 18327.57 18352.72 18377.86 18403.01 18428.15 18453.29 18478.44 18503.59 18528.73 18553.88

T (K) 270.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65

a (m/s) 329.8 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07

p (Pa) 77.64 8347.72 8314.88 8282.16 8249.58 8217.12 8184.79 8152.59 8120.51 8088.56 8056.74 8025.04 7993.46 7962.01 7930.69 7899.48 7868.4 7837.45 7806.61 7775.9 7745.3 7714.83 7684.47 7654.24 7624.13 7594.13 7564.25 7534.49 7504.84 7475.32 7445.91 7416.61 7387.43 7358.36 7329.41 7300.58 7271.85 7243.24 7214.74 7186.36 7158.08 7129.92 7101.87 7073.93 7046.09 7018.37 6990.76 6963.25 6935.86

ρ (kg/m3 ) 0.001 0.134 0.134 0.133 0.133 0.132 0.132 0.131 0.131 0.13 0.13 0.129 0.129 0.128 0.128 0.127 0.127 0.126 0.126 0.125 0.125 0.124 0.124 0.123 0.123 0.122 0.122 0.121 0.121 0.12 0.12 0.119 0.119 0.118 0.118 0.117 0.117 0.116 0.116 0.116 0.115 0.115 0.114 0.114 0.113 0.113 0.112 0.112 0.112

Appendix A: The Standard Atmosphere h (m) 49825 18525 18550 18575 18600 18625 18650 18675 18700 18725 18750 18775 18800 18825 18850 18875 18900 18925 18950 18975 19000 19025 19050 19075 19100 19125 19150 19175 19200 19225 19250 19275 19300 19325 19350 19375 19400 19425 19450 19475 19500 19525 19550 19575 19600 19625 19650 19675 19700

hG (m) 50217.73 18579.02 18604.17 18629.31 18654.46 18679.61 18704.75 18729.9 18755.05 18780.2 18805.34 18830.49 18855.64 18880.79 18905.94 18931.09 18956.23 18981.38 19006.53 19031.68 19056.83 19081.98 19107.13 19132.28 19157.43 19182.58 19207.73 19232.89 19258.04 19283.19 19308.34 19333.49 19358.64 19383.8 19408.95 19434.1 19459.25 19484.41 19509.56 19534.71 19559.87 19585.02 19610.18 19635.33 19660.48 19685.64 19710.79 19735.95 19761.1

T (K) 270.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65

511 a (m/s) 329.8 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07

p (Pa) 77.64 6908.57 6881.39 6854.31 6827.34 6800.48 6773.73 6747.08 6720.53 6694.09 6667.75 6641.52 6615.39 6589.36 6563.43 6537.61 6511.89 6486.27 6460.75 6435.33 6410.01 6384.79 6359.67 6334.64 6309.72 6284.9 6260.17 6235.54 6211 6186.57 6162.23 6137.98 6113.83 6089.78 6065.82 6041.95 6018.18 5994.5 5970.92 5947.43 5924.03 5900.72 5877.5 5854.38 5831.34 5808.4 5785.55 5762.78 5740.11

ρ (kg/m3 ) 0.001 0.111 0.111 0.11 0.11 0.109 0.109 0.108 0.108 0.108 0.107 0.107 0.106 0.106 0.106 0.105 0.105 0.104 0.104 0.103 0.103 0.103 0.102 0.102 0.101 0.101 0.101 0.1 0.1 0.099 0.099 0.099 0.098 0.098 0.098 0.097 0.097 0.096 0.096 0.096 0.095 0.095 0.095 0.094 0.094 0.093 0.093 0.093 0.092

512

Appendix A: The Standard Atmosphere h (m) 49825 19725 19750 19775 19800 19825 19850 19875 19900 19925 19950 19975 20000 20025 20050 20075 20100 20125 20150 20175 20200 20225 20250 20275 20300 20325 20350 20375 20400 20425 20450 20475 20500 20525 20550 20575 20600 20625 20650 20675 20700 20725 20750 20775 20800 20825 20850 20875 20900

hG (m) 50217.73 19786.26 19811.42 19836.57 19861.73 19886.88 19912.04 19937.2 19962.35 19987.51 20012.67 20037.82 20062.98 20088.14 20113.3 20138.46 20163.61 20188.77 20213.93 20239.09 20264.25 20289.41 20314.57 20339.73 20364.89 20390.05 20415.21 20440.37 20465.53 20490.69 20515.85 20541.01 20566.18 20591.34 20616.5 20641.66 20666.82 20691.99 20717.15 20742.31 20767.48 20792.64 20817.8 20842.97 20868.13 20893.29 20918.46 20943.62 20968.79

T (K) 270.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.65 216.68 216.7 216.73 216.75 216.78 216.8 216.83 216.85 216.88 216.9 216.93 216.95 216.98 217 217.03 217.05 217.08 217.1 217.13 217.15 217.18 217.2 217.23 217.25 217.28 217.3 217.33 217.35 217.38 217.4 217.43 217.45 217.48 217.5 217.53 217.55

a (m/s) 329.8 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.07 295.09 295.1 295.12 295.14 295.15 295.17 295.19 295.21 295.22 295.24 295.26 295.27 295.29 295.31 295.32 295.34 295.36 295.38 295.39 295.41 295.43 295.44 295.46 295.48 295.49 295.51 295.53 295.55 295.56 295.58 295.6 295.61 295.63 295.65 295.66 295.68

p (Pa) 77.64 5717.53 5695.03 5672.62 5650.31 5628.07 5605.93 5583.88 5561.91 5540.02 5518.23 5496.51 5474.89 5453.35 5431.9 5410.53 5389.25 5368.06 5346.95 5325.93 5304.99 5284.14 5263.38 5242.69 5222.09 5201.58 5181.14 5160.79 5140.52 5120.34 5100.23 5080.21 5060.26 5040.4 5020.62 5000.92 4981.29 4961.75 4942.29 4922.9 4903.59 4884.36 4865.21 4846.14 4827.14 4808.22 4789.37 4770.6 4751.91

ρ (kg/m3 ) 0.001 0.092 0.092 0.091 0.091 0.09 0.09 0.09 0.089 0.089 0.089 0.088 0.088 0.088 0.087 0.087 0.087 0.086 0.086 0.086 0.085 0.085 0.085 0.084 0.084 0.084 0.083 0.083 0.083 0.082 0.082 0.082 0.081 0.081 0.081 0.08 0.08 0.08 0.079 0.079 0.079 0.078 0.078 0.078 0.077 0.077 0.077 0.076 0.076

Appendix A: The Standard Atmosphere h (m) 49825 20925 20950 20975 21000 21025 21050 21075 21100 21125 21150 21175 21200 21225 21250 21275 21300 21325 21350 21375 21400 21425 21450 21475 21500 21525 21550 21575 21600 21625 21650 21675 21700 21725 21750 21775 21800 21825 21850 21875 21900 21925 21950 21975 22000 22025 22050 22075

hG (m) 50217.73 20993.95 21019.12 21044.28 21069.45 21094.61 21119.78 21144.95 21170.11 21195.28 21220.45 21245.61 21270.78 21295.95 21321.12 21346.28 21371.45 21396.62 21421.79 21446.96 21472.12 21497.29 21522.46 21547.63 21572.8 21597.97 21623.14 21648.31 21673.48 21698.65 21723.82 21748.99 21774.16 21799.34 21824.51 21849.68 21874.85 21900.02 21925.19 21950.37 21975.54 22000.71 22025.89 22051.06 22076.23 22101.41 22126.58 22151.75

T (K) 270.65 217.58 217.6 217.63 217.65 217.68 217.7 217.73 217.75 217.78 217.8 217.83 217.85 217.88 217.9 217.93 217.95 217.98 218 218.03 218.05 218.08 218.1 218.13 218.15 218.18 218.2 218.23 218.25 218.28 218.3 218.33 218.35 218.38 218.4 218.43 218.45 218.48 218.5 218.53 218.55 218.58 218.6 218.63 218.65 218.68 218.7 218.73

513 a (m/s) 329.8 295.7 295.72 295.73 295.75 295.77 295.78 295.8 295.82 295.83 295.85 295.87 295.89 295.9 295.92 295.94 295.95 295.97 295.99 296 296.02 296.04 296.06 296.07 296.09 296.11 296.12 296.14 296.16 296.17 296.19 296.21 296.23 296.24 296.26 296.28 296.29 296.31 296.33 296.34 296.36 296.38 296.39 296.41 296.43 296.45 296.46 296.48

p (Pa) 77.64 4733.29 4714.75 4696.28 4677.89 4659.57 4641.32 4623.15 4605.05 4587.02 4569.07 4551.19 4533.38 4515.64 4497.98 4480.38 4462.86 4445.41 4428.02 4410.71 4393.47 4376.29 4359.19 4342.15 4325.18 4308.28 4291.45 4274.69 4257.99 4241.36 4224.8 4208.3 4191.87 4175.51 4159.21 4142.98 4126.81 4110.71 4094.67 4078.7 4062.79 4046.95 4031.16 4015.45 3999.79 3984.2 3968.67 3953.2

ρ (kg/m3 ) 0.001 0.076 0.075 0.075 0.075 0.075 0.074 0.074 0.074 0.073 0.073 0.073 0.072 0.072 0.072 0.072 0.071 0.071 0.071 0.07 0.07 0.07 0.07 0.069 0.069 0.069 0.069 0.068 0.068 0.068 0.067 0.067 0.067 0.067 0.066 0.066 0.066 0.066 0.065 0.065 0.065 0.065 0.064 0.064 0.064 0.063 0.063 0.063

514

Appendix A: The Standard Atmosphere h (m) 49825 22100 22125 22150 22175 22200 22225 22250 22275 22300 22325 22350 22375 22400 22425 22450 22475 22500 22525 22550 22575 22600 22625 22650 22675 22700 22725 22750 22775 22800 22825 22850 22875 22900 22925 22950 22975 23000 23025 23050 23075 23100 23125 23150 23175 23200 23225 23250

hG (m) 50217.73 22176.93 22202.1 22227.28 22252.45 22277.63 22302.8 22327.98 22353.15 22378.33 22403.51 22428.68 22453.86 22479.03 22504.21 22529.39 22554.57 22579.74 22604.92 22630.1 22655.28 22680.45 22705.63 22730.81 22755.99 22781.17 22806.35 22831.53 22856.71 22881.89 22907.07 22932.25 22957.43 22982.61 23007.79 23032.97 23058.15 23083.33 23108.51 23133.7 23158.88 23184.06 23209.24 23234.43 23259.61 23284.79 23309.97 23335.16

T (K) 270.65 218.75 218.78 218.8 218.83 218.85 218.88 218.9 218.93 218.95 218.98 219 219.03 219.05 219.08 219.1 219.13 219.15 219.18 219.2 219.23 219.25 219.28 219.3 219.33 219.35 219.38 219.4 219.43 219.45 219.48 219.5 219.53 219.55 219.58 219.6 219.63 219.65 219.68 219.7 219.73 219.75 219.78 219.8 219.83 219.85 219.88 219.9

a (m/s) 329.8 296.5 296.51 296.53 296.55 296.56 296.58 296.6 296.61 296.63 296.65 296.67 296.68 296.7 296.72 296.73 296.75 296.77 296.78 296.8 296.82 296.83 296.85 296.87 296.89 296.9 296.92 296.94 296.95 296.97 296.99 297 297.02 297.04 297.05 297.07 297.09 297.11 297.12 297.14 297.16 297.17 297.19 297.21 297.22 297.24 297.26 297.27

p (Pa) 77.64 3937.79 3922.45 3907.17 3891.95 3876.79 3861.69 3846.65 3831.67 3816.75 3801.9 3787.1 3772.36 3757.68 3743.05 3728.49 3713.99 3699.54 3685.15 3670.82 3656.54 3642.33 3628.17 3614.06 3600.02 3586.03 3572.09 3558.21 3544.39 3530.62 3516.91 3503.25 3489.64 3476.09 3462.6 3449.16 3435.77 3422.43 3409.15 3395.93 3382.75 3369.63 3356.56 3343.54 3330.57 3317.66 3304.8 3291.98

ρ (kg/m3 ) 0.001 0.063 0.062 0.062 0.062 0.062 0.061 0.061 0.061 0.061 0.06 0.06 0.06 0.06 0.06 0.059 0.059 0.059 0.059 0.058 0.058 0.058 0.058 0.057 0.057 0.057 0.057 0.056 0.056 0.056 0.056 0.056 0.055 0.055 0.055 0.055 0.054 0.054 0.054 0.054 0.054 0.053 0.053 0.053 0.053 0.053 0.052 0.052

Appendix A: The Standard Atmosphere h (m) 49825 23275 23300 23325 23350 23375 23400 23425 23450 23475 23500 23525 23550 23575 23600 23625 23650 23675 23700 23725 23750 23775 23800 23825 23850 23875 23900 23925 23950 23975 24000 24025 24050 24075 24100 24125 24150 24175 24200 24225 24250 24275 24300 24325 24350 24375 24400 24425 24450

hG (m) 50217.73 23360.34 23385.53 23410.71 23435.89 23461.08 23486.26 23511.45 23536.63 23561.82 23587 23612.19 23637.37 23662.56 23687.75 23712.93 23738.12 23763.31 23788.49 23813.68 23838.87 23864.05 23889.24 23914.43 23939.62 23964.81 23990 24015.18 24040.37 24065.56 24090.75 24115.94 24141.13 24166.32 24191.51 24216.7 24241.89 24267.08 24292.27 24317.46 24342.66 24367.85 24393.04 24418.23 24443.42 24468.62 24493.81 24519 24544.19

T (K) 270.65 219.93 219.95 219.98 220 220.03 220.05 220.08 220.1 220.13 220.15 220.18 220.2 220.23 220.25 220.28 220.3 220.33 220.35 220.38 220.4 220.43 220.45 220.48 220.5 220.53 220.55 220.58 220.6 220.63 220.65 220.68 220.7 220.73 220.75 220.78 220.8 220.83 220.85 220.88 220.9 220.93 220.95 220.98 221 221.03 221.05 221.08 221.1

515 a (m/s) 329.8 297.29 297.31 297.33 297.34 297.36 297.38 297.39 297.41 297.43 297.44 297.46 297.48 297.49 297.51 297.53 297.54 297.56 297.58 297.6 297.61 297.63 297.65 297.66 297.68 297.7 297.71 297.73 297.75 297.76 297.78 297.8 297.81 297.83 297.85 297.87 297.88 297.9 297.92 297.93 297.95 297.97 297.98 298 298.02 298.03 298.05 298.07 298.08

p (Pa) 77.64 3279.22 3266.51 3253.86 3241.25 3228.69 3216.18 3203.72 3191.32 3178.96 3166.65 3154.39 3142.17 3130.01 3117.9 3105.83 3093.81 3081.84 3069.92 3058.04 3046.21 3034.43 3022.7 3011.01 2999.37 2987.78 2976.23 2964.73 2953.27 2941.86 2930.49 2919.17 2907.9 2896.67 2885.48 2874.34 2863.24 2852.19 2841.18 2830.21 2819.29 2808.41 2797.58 2786.78 2776.03 2765.33 2754.66 2744.04 2733.46

ρ (kg/m3 ) 0.001 0.052 0.052 0.052 0.051 0.051 0.051 0.051 0.051 0.05 0.05 0.05 0.05 0.05 0.049 0.049 0.049 0.049 0.049 0.048 0.048 0.048 0.048 0.048 0.047 0.047 0.047 0.047 0.047 0.046 0.046 0.046 0.046 0.046 0.046 0.045 0.045 0.045 0.045 0.045 0.044 0.044 0.044 0.044 0.044 0.044 0.043 0.043 0.043

516

Appendix A: The Standard Atmosphere h (m) 49825 24475 24500 24525 24550 24575 24600 24625 24650 24675 24700 24725 24750 24775 24800 24825 24850 24875 24900 24925 24950 24975 25000 25025 25050 25075 25100 25125 25150 25175 25200 25225 25250 25275 25300 25325 25350 25375 25400 25425 25450 25475 25500 25525 25550 25575 25600 25625

hG (m) 50217.73 24569.39 24594.58 24619.77 24644.97 24670.16 24695.35 24720.55 24745.74 24770.94 24796.13 24821.33 24846.52 24871.72 24896.91 24922.11 24947.31 24972.5 24997.7 25022.9 25048.09 25073.29 25098.49 25123.68 25148.88 25174.08 25199.28 25224.48 25249.67 25274.87 25300.07 25325.27 25350.47 25375.67 25400.87 25426.07 25451.27 25476.47 25501.67 25526.87 25552.07 25577.27 25602.47 25627.68 25652.88 25678.08 25703.28 25728.48

T (K) 270.65 221.13 221.15 221.18 221.2 221.23 221.25 221.28 221.3 221.33 221.35 221.38 221.4 221.43 221.45 221.48 221.5 221.53 221.55 221.58 221.6 221.63 221.65 221.68 221.7 221.73 221.75 221.78 221.8 221.83 221.85 221.88 221.9 221.93 221.95 221.98 222 222.03 222.05 222.08 222.1 222.13 222.15 222.18 222.2 222.23 222.25 222.28

a (m/s) 329.8 298.1 298.12 298.14 298.15 298.17 298.19 298.2 298.22 298.24 298.25 298.27 298.29 298.3 298.32 298.34 298.35 298.37 298.39 298.4 298.42 298.44 298.46 298.47 298.49 298.51 298.52 298.54 298.56 298.57 298.59 298.61 298.62 298.64 298.66 298.67 298.69 298.71 298.72 298.74 298.76 298.77 298.79 298.81 298.83 298.84 298.86 298.88

p (Pa) 77.64 2722.92 2712.42 2701.97 2691.56 2681.19 2670.85 2660.57 2650.32 2640.11 2629.94 2619.81 2609.73 2599.68 2589.67 2579.7 2569.77 2559.88 2550.03 2540.22 2530.45 2520.72 2511.02 2501.37 2491.75 2482.17 2472.63 2463.12 2453.65 2444.22 2434.83 2425.48 2416.16 2406.88 2397.63 2388.43 2379.25 2370.12 2361.02 2351.96 2342.93 2333.94 2324.98 2316.06 2307.17 2298.32 2289.51 2280.73

ρ (kg/m3 ) 0.001 0.043 0.043 0.043 0.042 0.042 0.042 0.042 0.042 0.042 0.041 0.041 0.041 0.041 0.041 0.041 0.04 0.04 0.04 0.04 0.04 0.04 0.039 0.039 0.039 0.039 0.039 0.039 0.039 0.038 0.038 0.038 0.038 0.038 0.038 0.037 0.037 0.037 0.037 0.037 0.037 0.037 0.036 0.036 0.036 0.036 0.036 0.036

Appendix A: The Standard Atmosphere h (m) 49825 25650 25675 25700 25725 25750 25775 25800 25825 25850 25875 25900 25925 25950 25975 26000 26025 26050 26075 26100 26125 26150 26175 26200 26225 26250 26275 26300 26325 26350 26375 26400 26425 26450 26475 26500 26525 26550 26575 26600 26625 26650 26675 26700 26725 26750 26775 26800

hG (m) 50217.73 25753.69 25778.89 25804.09 25829.29 25854.5 25879.7 25904.9 25930.11 25955.31 25980.52 26005.72 26030.93 26056.13 26081.34 26106.54 26131.75 26156.95 26182.16 26207.36 26232.57 26257.78 26282.98 26308.19 26333.4 26358.6 26383.81 26409.02 26434.23 26459.43 26484.64 26509.85 26535.06 26560.27 26585.48 26610.69 26635.9 26661.11 26686.32 26711.53 26736.74 26761.95 26787.16 26812.37 26837.58 26862.79 26888 26913.21

T (K) 270.65 222.3 222.33 222.35 222.38 222.4 222.43 222.45 222.48 222.5 222.53 222.55 222.58 222.6 222.63 222.65 222.68 222.7 222.73 222.75 222.78 222.8 222.83 222.85 222.88 222.9 222.93 222.95 222.98 223 223.03 223.05 223.08 223.1 223.13 223.15 223.18 223.2 223.23 223.25 223.28 223.3 223.33 223.35 223.38 223.4 223.43 223.45

517 a (m/s) 329.8 298.89 298.91 298.93 298.94 298.96 298.98 298.99 299.01 299.03 299.04 299.06 299.08 299.09 299.11 299.13 299.14 299.16 299.18 299.19 299.21 299.23 299.25 299.26 299.28 299.3 299.31 299.33 299.35 299.36 299.38 299.4 299.41 299.43 299.45 299.46 299.48 299.5 299.51 299.53 299.55 299.56 299.58 299.6 299.61 299.63 299.65 299.66

p (Pa) 77.64 2271.98 2263.27 2254.59 2245.95 2237.34 2228.76 2220.22 2211.71 2203.24 2194.8 2186.39 2178.02 2169.68 2161.37 2153.09 2144.85 2136.64 2128.46 2120.32 2112.2 2104.12 2096.07 2088.05 2080.07 2072.11 2064.19 2056.29 2048.43 2040.6 2032.8 2025.03 2017.29 2009.58 2001.91 1994.26 1986.64 1979.05 1971.49 1963.97 1956.47 1949 1941.56 1934.15 1926.77 1919.41 1912.09 1904.8

ρ (kg/m3 ) 0.001 0.036 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.034 0.034 0.034 0.034 0.034 0.034 0.034 0.034 0.033 0.033 0.033 0.033 0.033 0.033 0.033 0.033 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.031 0.031 0.031 0.031 0.031 0.031 0.031 0.031 0.03 0.03 0.03 0.03 0.03 0.03 0.03

518

Appendix A: The Standard Atmosphere h (m) 49825 26825 26850 26875 26900 26925 26950 26975 27000 27025 27050 27075 27100 27125 27150 27175 27200 27225 27250 27275 27300 27325 27350 27375 27400 27425 27450 27475 27500 27525 27550 27575 27600 27625 27650 27675 27700 27725 27750 27775 27800 27825 27850 27875 27900 27925 27950 27975 28000

hG (m) 50217.73 26938.42 26963.64 26988.85 27014.06 27039.27 27064.49 27089.7 27114.91 27140.13 27165.34 27190.55 27215.77 27240.98 27266.19 27291.41 27316.62 27341.84 27367.05 27392.27 27417.49 27442.7 27467.92 27493.13 27518.35 27543.57 27568.78 27594 27619.22 27644.43 27669.65 27694.87 27720.09 27745.31 27770.52 27795.74 27820.96 27846.18 27871.4 27896.62 27921.84 27947.06 27972.28 27997.5 28022.72 28047.94 28073.16 28098.38 28123.6

T (K) 270.65 223.48 223.5 223.53 223.55 223.58 223.6 223.63 223.65 223.68 223.7 223.73 223.75 223.78 223.8 223.83 223.85 223.88 223.9 223.93 223.95 223.98 224 224.03 224.05 224.08 224.1 224.13 224.15 224.18 224.2 224.23 224.25 224.28 224.3 224.33 224.35 224.38 224.4 224.43 224.45 224.48 224.5 224.53 224.55 224.58 224.6 224.63 224.65

a (m/s) 329.8 299.68 299.7 299.71 299.73 299.75 299.77 299.78 299.8 299.82 299.83 299.85 299.87 299.88 299.9 299.92 299.93 299.95 299.97 299.98 300 300.02 300.03 300.05 300.07 300.08 300.1 300.12 300.13 300.15 300.17 300.18 300.2 300.22 300.23 300.25 300.27 300.28 300.3 300.32 300.33 300.35 300.37 300.38 300.4 300.42 300.43 300.45 300.47

p (Pa) 77.64 1897.53 1890.29 1883.08 1875.9 1868.75 1861.62 1854.53 1847.46 1840.42 1833.4 1826.42 1819.46 1812.53 1805.62 1798.74 1791.89 1785.07 1778.27 1771.5 1764.76 1758.04 1751.35 1744.69 1738.05 1731.44 1724.85 1718.29 1711.75 1705.25 1698.76 1692.3 1685.87 1679.46 1673.08 1666.72 1660.39 1654.08 1647.79 1641.53 1635.3 1629.09 1622.9 1616.74 1610.6 1604.49 1598.4 1592.33 1586.29

ρ (kg/m3 ) 0.001 0.03 0.029 0.029 0.029 0.029 0.029 0.029 0.029 0.029 0.029 0.028 0.028 0.028 0.028 0.028 0.028 0.028 0.028 0.028 0.027 0.027 0.027 0.027 0.027 0.027 0.027 0.027 0.027 0.026 0.026 0.026 0.026 0.026 0.026 0.026 0.026 0.026 0.026 0.025 0.025 0.025 0.025 0.025 0.025 0.025 0.025 0.025 0.025

Appendix A: The Standard Atmosphere h (m) 49825 28025 28050 28075 28100 28125 28150 28175 28200 28225 28250 28275 28300 28325 28350 28375 28400 28425 28450 28475 28500 28525 28550 28575 28600 28625 28650 28675 28700 28725 28750 28775 28800 28825 28850 28875 28900 28925 28950 28975 29000 29025 29050 29075 29100 29125 29150 29175

hG (m) 50217.73 28148.82 28174.04 28199.27 28224.49 28249.71 28274.93 28300.15 28325.38 28350.6 28375.82 28401.05 28426.27 28451.49 28476.72 28501.94 28527.17 28552.39 28577.61 28602.84 28628.06 28653.29 28678.52 28703.74 28728.97 28754.19 28779.42 28804.65 28829.87 28855.1 28880.33 28905.55 28930.78 28956.01 28981.24 29006.46 29031.69 29056.92 29082.15 29107.38 29132.61 29157.84 29183.07 29208.3 29233.53 29258.76 29283.99 29309.22

T (K) 270.65 224.68 224.7 224.73 224.75 224.78 224.8 224.83 224.85 224.88 224.9 224.93 224.95 224.98 225 225.03 225.05 225.08 225.1 225.13 225.15 225.18 225.2 225.23 225.25 225.28 225.3 225.33 225.35 225.38 225.4 225.43 225.45 225.48 225.5 225.53 225.55 225.58 225.6 225.63 225.65 225.68 225.7 225.73 225.75 225.78 225.8 225.83

519 a (m/s) 329.8 300.48 300.5 300.52 300.53 300.55 300.57 300.59 300.6 300.62 300.64 300.65 300.67 300.69 300.7 300.72 300.74 300.75 300.77 300.79 300.8 300.82 300.84 300.85 300.87 300.89 300.9 300.92 300.94 300.95 300.97 300.99 301 301.02 301.04 301.05 301.07 301.09 301.1 301.12 301.14 301.15 301.17 301.19 301.2 301.22 301.24 301.25

p (Pa) 77.64 1580.27 1574.28 1568.3 1562.35 1556.43 1550.53 1544.65 1538.79 1532.96 1527.15 1521.36 1515.59 1509.85 1504.13 1498.43 1492.75 1487.1 1481.47 1475.86 1470.27 1464.7 1459.16 1453.64 1448.13 1442.65 1437.19 1431.76 1426.34 1420.95 1415.57 1410.22 1404.89 1399.57 1394.28 1389.01 1383.76 1378.53 1373.32 1368.13 1362.96 1357.82 1352.69 1347.58 1342.49 1337.42 1332.37 1327.34

ρ (kg/m3 ) 0.001 0.025 0.024 0.024 0.024 0.024 0.024 0.024 0.024 0.024 0.024 0.024 0.023 0.023 0.023 0.023 0.023 0.023 0.023 0.023 0.023 0.023 0.023 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.02

520

Appendix A: The Standard Atmosphere h (m) 49825 29200 29225 29250 29275 29300 29325 29350 29375 29400 29425 29450 29475 29500 29525 29550 29575 29600 29625 29650 29675 29700 29725 29750 29775 29800 29825 29850 29875 29900 29925 29950 29975 30000 30025 30050 30075 30100 30125 30150 30175 30200 30225 30250 30275 30300 30325 30350 30375

hG (m) 50217.73 29334.45 29359.68 29384.91 29410.14 29435.37 29460.6 29485.84 29511.07 29536.3 29561.53 29586.77 29612 29637.23 29662.46 29687.7 29712.93 29738.17 29763.4 29788.63 29813.87 29839.1 29864.34 29889.57 29914.81 29940.04 29965.28 29990.51 30015.75 30040.99 30066.22 30091.46 30116.7 30141.93 30167.17 30192.41 30217.65 30242.88 30268.12 30293.36 30318.6 30343.84 30369.08 30394.31 30419.55 30444.79 30470.03 30495.27 30520.51

T (K) 270.65 225.85 225.88 225.9 225.93 225.95 225.98 226 226.03 226.05 226.08 226.1 226.13 226.15 226.18 226.2 226.23 226.25 226.28 226.3 226.33 226.35 226.38 226.4 226.43 226.45 226.48 226.5 226.53 226.55 226.58 226.6 226.63 226.65 226.68 226.7 226.73 226.75 226.78 226.8 226.83 226.85 226.88 226.9 226.93 226.95 226.98 227 227.03

a (m/s) 329.8 301.27 301.29 301.3 301.32 301.34 301.35 301.37 301.39 301.4 301.42 301.44 301.45 301.47 301.49 301.5 301.52 301.54 301.55 301.57 301.59 301.6 301.62 301.64 301.65 301.67 301.69 301.7 301.72 301.74 301.75 301.77 301.79 301.8 301.82 301.84 301.85 301.87 301.89 301.9 301.92 301.94 301.95 301.97 301.99 302 302.02 302.04 302.05

p (Pa) 77.64 1322.33 1317.34 1312.37 1307.42 1302.48 1297.57 1292.68 1287.8 1282.94 1278.1 1273.29 1268.49 1263.7 1258.94 1254.2 1249.47 1244.76 1240.07 1235.4 1230.75 1226.11 1221.49 1216.89 1212.31 1207.75 1203.2 1198.67 1194.16 1189.67 1185.19 1180.73 1176.29 1171.87 1167.46 1163.07 1158.7 1154.34 1150 1145.68 1141.37 1137.08 1132.81 1128.55 1124.31 1120.09 1115.88 1111.69 1107.52

ρ (kg/m3 ) 0.001 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.019 0.019 0.019 0.019 0.019 0.019 0.019 0.019 0.019 0.019 0.019 0.019 0.019 0.019 0.018 0.018 0.018 0.018 0.018 0.018 0.018 0.018 0.018 0.018 0.018 0.018 0.018 0.018 0.017 0.017 0.017 0.017 0.017 0.017 0.017 0.017

Appendix A: The Standard Atmosphere h (m) 49825 30400 30425 30450 30475 30500 30525 30550 30575 30600 30625 30650 30675 30700 30725 30750 30775 30800 30825 30850 30875 30900 30925 30950 30975 31000 31025 31050 31075 31100 31125 31150 31175 31200 31225 31250 31275 31300 31325 31350 31375 31400 31425 31450 31475 31500 31525 31550 31575

hG (m) 50217.73 30545.75 30570.99 30596.23 30621.47 30646.72 30671.96 30697.2 30722.44 30747.68 30772.92 30798.17 30823.41 30848.65 30873.89 30899.14 30924.38 30949.62 30974.87 31000.11 31025.35 31050.6 31075.84 31101.09 31126.33 31151.58 31176.82 31202.07 31227.31 31252.56 31277.81 31303.05 31328.3 31353.54 31378.79 31404.04 31429.29 31454.53 31479.78 31505.03 31530.28 31555.52 31580.77 31606.02 31631.27 31656.52 31681.77 31707.02 31732.27

T (K) 270.65 227.05 227.08 227.1 227.13 227.15 227.18 227.2 227.23 227.25 227.28 227.3 227.33 227.35 227.38 227.4 227.43 227.45 227.48 227.5 227.53 227.55 227.58 227.6 227.63 227.65 227.68 227.7 227.73 227.75 227.78 227.8 227.83 227.85 227.88 227.9 227.93 227.95 227.98 228 228.03 228.05 228.08 228.1 228.13 228.15 228.18 228.2 228.23

521 a (m/s) 329.8 302.07 302.09 302.1 302.12 302.14 302.15 302.17 302.19 302.2 302.22 302.24 302.25 302.27 302.28 302.3 302.32 302.33 302.35 302.37 302.38 302.4 302.42 302.43 302.45 302.47 302.48 302.5 302.52 302.53 302.55 302.57 302.58 302.6 302.62 302.63 302.65 302.67 302.68 302.7 302.72 302.73 302.75 302.77 302.78 302.8 302.82 302.83 302.85

p (Pa) 77.64 1103.36 1099.22 1095.09 1090.98 1086.88 1082.81 1078.74 1074.7 1070.66 1066.65 1062.65 1058.66 1054.69 1050.74 1046.8 1042.87 1038.97 1035.07 1031.19 1027.33 1023.48 1019.65 1015.83 1012.02 1008.23 1004.46 1000.7 996.95 993.22 989.5 985.8 982.11 978.43 974.77 971.13 967.49 963.88 960.27 956.68 953.1 949.54 945.99 942.46 938.93 935.43 931.93 928.45 924.98

ρ (kg/m3 ) 0.001 0.017 0.017 0.017 0.017 0.017 0.017 0.017 0.016 0.016 0.016 0.016 0.016 0.016 0.016 0.016 0.016 0.016 0.016 0.016 0.016 0.016 0.016 0.016 0.015 0.015 0.015 0.015 0.015 0.015 0.015 0.015 0.015 0.015 0.015 0.015 0.015 0.015 0.015 0.015 0.015 0.015 0.014 0.014 0.014 0.014 0.014 0.014 0.014

522

Appendix A: The Standard Atmosphere h (m) 49825 31600 31625 31650 31675 31700 31725 31750 31775 31800 31825 31850 31875 31900 31925 31950 31975 32000 32025 32050 32075 32100 32125 32150 32175 32200 32225 32250 32275 32300 32325 32350 32375 32400 32425 32450 32475 32500 32525 32550 32575 32600 32625 32650 32675 32700 32725 32750

hG (m) 50217.73 31757.52 31782.77 31808.02 31833.27 31858.52 31883.77 31909.02 31934.27 31959.52 31984.77 32010.03 32035.28 32060.53 32085.78 32111.03 32136.29 32161.54 32186.79 32212.05 32237.3 32262.55 32287.81 32313.06 32338.32 32363.57 32388.83 32414.08 32439.34 32464.59 32489.85 32515.1 32540.36 32565.61 32590.87 32616.13 32641.38 32666.64 32691.9 32717.15 32742.41 32767.67 32792.93 32818.19 32843.44 32868.7 32893.96 32919.22

T (K) 270.65 228.25 228.28 228.3 228.33 228.35 228.38 228.4 228.43 228.45 228.48 228.5 228.53 228.55 228.58 228.6 228.63 228.65 228.72 228.79 228.86 228.93 229 229.07 229.14 229.21 229.28 229.35 229.42 229.49 229.56 229.63 229.7 229.77 229.84 229.91 229.98 230.05 230.12 230.19 230.26 230.33 230.4 230.47 230.54 230.61 230.68 230.75

a (m/s) 329.8 302.87 302.88 302.9 302.92 302.93 302.95 302.97 302.98 303 303.02 303.03 303.05 303.06 303.08 303.1 303.11 303.13 303.18 303.22 303.27 303.32 303.36 303.41 303.46 303.5 303.55 303.59 303.64 303.69 303.73 303.78 303.83 303.87 303.92 303.97 304.01 304.06 304.1 304.15 304.2 304.24 304.29 304.34 304.38 304.43 304.47 304.52

p (Pa) 77.64 921.53 918.08 914.66 911.24 907.84 904.45 901.07 897.71 894.36 891.02 887.7 884.39 881.09 877.8 874.53 871.27 868.02 864.78 861.56 858.35 855.15 851.97 848.8 845.64 842.49 839.36 836.24 833.13 830.04 826.96 823.88 820.83 817.78 814.75 811.73 808.72 805.72 802.73 799.76 796.8 793.85 790.91 787.99 785.07 782.17 779.28 776.4

ρ (kg/m3 ) 0.001 0.014 0.014 0.014 0.014 0.014 0.014 0.014 0.014 0.014 0.014 0.014 0.013 0.013 0.013 0.013 0.013 0.013 0.013 0.013 0.013 0.013 0.013 0.013 0.013 0.013 0.013 0.013 0.013 0.013 0.013 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012

Appendix A: The Standard Atmosphere h (m) 49825 32775 32800 32825 32850 32875 32900 32925 32950 32975 33000 33025 33050 33075 33100 33125 33150 33175 33200 33225 33250 33275 33300 33325 33350 33375 33400 33425 33450 33475 33500 33525 33550 33575 33600 33625 33650 33675 33700 33725 33750 33775 33800 33825 33850 33875 33900 33925 33950

hG (m) 50217.73 32944.48 32969.74 32995 33020.26 33045.52 33070.78 33096.04 33121.3 33146.56 33171.82 33197.08 33222.34 33247.6 33272.87 33298.13 33323.39 33348.65 33373.92 33399.18 33424.44 33449.7 33474.97 33500.23 33525.49 33550.76 33576.02 33601.29 33626.55 33651.82 33677.08 33702.35 33727.61 33752.88 33778.14 33803.41 33828.67 33853.94 33879.21 33904.47 33929.74 33955.01 33980.28 34005.54 34030.81 34056.08 34081.35 34106.61 34131.88

523 T (K) 270.65 230.82 230.89 230.96 231.03 231.1 231.17 231.24 231.31 231.38 231.45 231.52 231.59 231.66 231.73 231.8 231.87 231.94 232.01 232.08 232.15 232.22 232.29 232.36 232.43 232.5 232.57 232.64 232.71 232.78 232.85 232.92 232.99 233.06 233.13 233.2 233.27 233.34 233.41 233.48 233.55 233.62 233.69 233.76 233.83 233.9 233.97 234.04 234.11

a (m/s) 329.8 304.57 304.61 304.66 304.7 304.75 304.8 304.84 304.89 304.94 304.98 305.03 305.07 305.12 305.17 305.21 305.26 305.3 305.35 305.4 305.44 305.49 305.53 305.58 305.63 305.67 305.72 305.76 305.81 305.86 305.9 305.95 305.99 306.04 306.09 306.13 306.18 306.22 306.27 306.32 306.36 306.41 306.45 306.5 306.55 306.59 306.64 306.68 306.73

p (Pa) 77.64 773.53 770.67 767.83 764.99 762.17 759.36 756.56 753.77 750.99 748.23 745.47 742.73 739.99 737.27 734.56 731.86 729.17 726.49 723.82 721.16 718.51 715.88 713.25 710.63 708.03 705.43 702.85 700.27 697.7 695.15 692.61 690.07 687.55 685.03 682.53 680.03 677.55 675.07 672.61 670.15 667.71 665.27 662.84 660.43 658.02 655.62 653.23 650.85

ρ (kg/m3 ) 0.001 0.012 0.012 0.012 0.012 0.011 0.011 0.011 0.011 0.011 0.011 0.011 0.011 0.011 0.011 0.011 0.011 0.011 0.011 0.011 0.011 0.011 0.011 0.011 0.011 0.011 0.011 0.011 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01

524

Appendix A: The Standard Atmosphere h (m) 49825 33975 34000 34025 34050 34075 34100 34125 34150 34175 34200 34225 34250 34275 34300 34325 34350 34375 34400 34425 34450 34475 34500 34525 34550 34575 34600 34625 34650 34675 34700 34725 34750 34775 34800 34825 34850 34875 34900 34925 34950 34975 35000 35025 35050 35075 35100 35125 35150

hG (m) 50217.73 34157.15 34182.42 34207.69 34232.96 34258.23 34283.5 34308.77 34334.04 34359.31 34384.58 34409.85 34435.12 34460.39 34485.66 34510.93 34536.21 34561.48 34586.75 34612.02 34637.29 34662.57 34687.84 34713.11 34738.39 34763.66 34788.93 34814.21 34839.48 34864.76 34890.03 34915.31 34940.58 34965.86 34991.13 35016.41 35041.68 35066.96 35092.23 35117.51 35142.79 35168.06 35193.34 35218.62 35243.89 35269.17 35294.45 35319.73 35345.01

T (K) 270.65 234.18 234.25 234.32 234.39 234.46 234.53 234.6 234.67 234.74 234.81 234.88 234.95 235.02 235.09 235.16 235.23 235.3 235.37 235.44 235.51 235.58 235.65 235.72 235.79 235.86 235.93 236 236.07 236.14 236.21 236.28 236.35 236.42 236.49 236.56 236.63 236.7 236.77 236.84 236.91 236.98 237.05 237.12 237.19 237.26 237.33 237.4 237.47

a (m/s) 329.8 306.78 306.82 306.87 306.91 306.96 307 307.05 307.1 307.14 307.19 307.23 307.28 307.32 307.37 307.42 307.46 307.51 307.55 307.6 307.64 307.69 307.74 307.78 307.83 307.87 307.92 307.96 308.01 308.06 308.1 308.15 308.19 308.24 308.28 308.33 308.38 308.42 308.47 308.51 308.56 308.6 308.65 308.69 308.74 308.79 308.83 308.88 308.92

p (Pa) 77.64 648.48 646.12 643.77 641.43 639.1 636.77 634.46 632.15 629.86 627.57 625.29 623.02 620.76 618.51 616.27 614.03 611.81 609.59 607.39 605.19 603 600.81 598.64 596.48 594.32 592.17 590.03 587.9 585.78 583.66 581.56 579.46 577.37 575.29 573.21 571.15 569.09 567.04 565 562.97 560.94 558.92 556.91 554.91 552.92 550.93 548.95 546.98

ρ (kg/m3 ) 0.001 0.01 0.01 0.01 0.01 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008

Appendix A: The Standard Atmosphere h (m) 49825 35175 35200 35225 35250 35275 35300 35325 35350 35375 35400 35425 35450 35475 35500 35525 35550 35575 35600 35625 35650 35675 35700 35725 35750 35775 35800 35825 35850 35875 35900 35925 35950 35975 36000 36025 36050 36075 36100 36125 36150 36175 36200 36225 36250 36275 36300 36325

hG (m) 50217.73 35370.28 35395.56 35420.84 35446.12 35471.4 35496.68 35521.96 35547.24 35572.52 35597.8 35623.08 35648.36 35673.64 35698.92 35724.2 35749.48 35774.76 35800.04 35825.33 35850.61 35875.89 35901.17 35926.46 35951.74 35977.02 36002.3 36027.59 36052.87 36078.16 36103.44 36128.72 36154.01 36179.29 36204.58 36229.86 36255.15 36280.43 36305.72 36331 36356.29 36381.58 36406.86 36432.15 36457.44 36482.72 36508.01 36533.3

525 T (K) 270.65 237.54 237.61 237.68 237.75 237.82 237.89 237.96 238.03 238.1 238.17 238.24 238.31 238.38 238.45 238.52 238.59 238.66 238.73 238.8 238.87 238.94 239.01 239.08 239.15 239.22 239.29 239.36 239.43 239.5 239.57 239.64 239.71 239.78 239.85 239.92 239.99 240.06 240.13 240.2 240.27 240.34 240.41 240.48 240.55 240.62 240.69 240.76

a (m/s) 329.8 308.97 309.01 309.06 309.1 309.15 309.2 309.24 309.29 309.33 309.38 309.42 309.47 309.51 309.56 309.6 309.65 309.7 309.74 309.79 309.83 309.88 309.92 309.97 310.01 310.06 310.1 310.15 310.19 310.24 310.29 310.33 310.38 310.42 310.47 310.51 310.56 310.6 310.65 310.69 310.74 310.78 310.83 310.87 310.92 310.96 311.01 311.06

p (Pa) 77.64 545.02 543.06 541.11 539.17 537.24 535.31 533.4 531.49 529.58 527.69 525.8 523.92 522.04 520.18 518.32 516.46 514.62 512.78 510.95 509.13 507.31 505.5 503.7 501.9 500.11 498.33 496.55 494.78 493.02 491.27 489.52 487.78 486.04 484.32 482.6 480.88 479.17 477.47 475.78 474.09 472.41 470.73 469.06 467.4 465.74 464.09 462.45

ρ (kg/m3 ) 0.001 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007

526

Appendix A: The Standard Atmosphere h (m) 49825 36350 36375 36400 36425 36450 36475 36500 36525 36550 36575 36600 36625 36650 36675 36700 36725 36750 36775 36800 36825 36850 36875 36900 36925 36950 36975 37000 37025 37050 37075 37100 37125 37150 37175 37200 37225 37250 37275 37300 37325 37350 37375 37400 37425 37450 37475 37500

hG (m) 50217.73 36558.59 36583.87 36609.16 36634.45 36659.74 36685.03 36710.32 36735.61 36760.89 36786.18 36811.47 36836.76 36862.05 36887.34 36912.63 36937.93 36963.22 36988.51 37013.8 37039.09 37064.38 37089.67 37114.97 37140.26 37165.55 37190.84 37216.14 37241.43 37266.72 37292.01 37317.31 37342.6 37367.9 37393.19 37418.48 37443.78 37469.07 37494.37 37519.66 37544.96 37570.26 37595.55 37620.85 37646.14 37671.44 37696.74 37722.03

T (K) 270.65 240.83 240.9 240.97 241.04 241.11 241.18 241.25 241.32 241.39 241.46 241.53 241.6 241.67 241.74 241.81 241.88 241.95 242.02 242.09 242.16 242.23 242.3 242.37 242.44 242.51 242.58 242.65 242.72 242.79 242.86 242.93 243 243.07 243.14 243.21 243.28 243.35 243.42 243.49 243.56 243.63 243.7 243.77 243.84 243.91 243.98 244.05

a (m/s) 329.8 311.1 311.15 311.19 311.24 311.28 311.33 311.37 311.42 311.46 311.51 311.55 311.6 311.64 311.69 311.73 311.78 311.82 311.87 311.91 311.96 312 312.05 312.09 312.14 312.18 312.23 312.27 312.32 312.36 312.41 312.45 312.5 312.54 312.59 312.63 312.68 312.72 312.77 312.81 312.86 312.9 312.95 312.99 313.04 313.08 313.13 313.17

p (Pa) 77.64 460.81 459.18 457.56 455.94 454.32 452.72 451.12 449.52 447.94 446.35 444.78 443.21 441.64 440.09 438.53 436.99 435.45 433.91 432.39 430.86 429.35 427.84 426.33 424.83 423.34 421.85 420.37 418.89 417.42 415.95 414.49 413.04 411.59 410.15 408.71 407.28 405.85 404.43 403.01 401.6 400.19 398.79 397.4 396.01 394.62 393.25 391.87

ρ (kg/m3 ) 0.001 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006

Appendix A: The Standard Atmosphere h (m) 49825 37525 37550 37575 37600 37625 37650 37675 37700 37725 37750 37775 37800 37825 37850 37875 37900 37925 37950 37975 38000 38025 38050 38075 38100 38125 38150 38175 38200 38225 38250 38275 38300 38325 38350 38375 38400 38425 38450 38475 38500 38525 38550 38575 38600 38625 38650 38675

hG (m) 50217.73 37747.33 37772.63 37797.93 37823.22 37848.52 37873.82 37899.12 37924.42 37949.71 37975.01 38000.31 38025.61 38050.91 38076.21 38101.51 38126.81 38152.11 38177.41 38202.71 38228.01 38253.31 38278.61 38303.92 38329.22 38354.52 38379.82 38405.12 38430.43 38455.73 38481.03 38506.33 38531.64 38556.94 38582.24 38607.55 38632.85 38658.16 38683.46 38708.77 38734.07 38759.38 38784.68 38809.99 38835.29 38860.6 38885.9 38911.21

527 T (K) 270.65 244.12 244.19 244.26 244.33 244.4 244.47 244.54 244.61 244.68 244.75 244.82 244.89 244.96 245.03 245.1 245.17 245.24 245.31 245.38 245.45 245.52 245.59 245.66 245.73 245.8 245.87 245.94 246.01 246.08 246.15 246.22 246.29 246.36 246.43 246.5 246.57 246.64 246.71 246.78 246.85 246.92 246.99 247.06 247.13 247.2 247.27 247.34

a (m/s) 329.8 313.22 313.26 313.31 313.35 313.4 313.44 313.49 313.53 313.58 313.62 313.67 313.71 313.76 313.8 313.85 313.89 313.94 313.98 314.03 314.07 314.11 314.16 314.2 314.25 314.29 314.34 314.38 314.43 314.47 314.52 314.56 314.61 314.65 314.7 314.74 314.79 314.83 314.88 314.92 314.96 315.01 315.05 315.1 315.14 315.19 315.23 315.28

p (Pa) 77.64 390.5 389.14 387.78 386.43 385.08 383.74 382.4 381.06 379.74 378.41 377.1 375.78 374.47 373.17 371.87 370.58 369.29 368.01 366.73 365.45 364.19 362.92 361.66 360.41 359.16 357.91 356.67 355.43 354.2 352.97 351.75 350.53 349.32 348.11 346.91 345.71 344.51 343.32 342.14 340.95 339.78 338.6 337.43 336.27 335.11 333.95 332.8

ρ (kg/m3 ) 0.001 0.006 0.006 0.006 0.006 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005

528

Appendix A: The Standard Atmosphere h (m) 49825 38700 38725 38750 38775 38800 38825 38850 38875 38900 38925 38950 38975 39000 39025 39050 39075 39100 39125 39150 39175 39200 39225 39250 39275 39300 39325 39350 39375 39400 39425 39450 39475 39500 39525 39550 39575 39600 39625 39650 39675 39700 39725 39750 39775 39800 39825 39850 39875

hG (m) 50217.73 38936.52 38961.82 38987.13 39012.44 39037.74 39063.05 39088.36 39113.67 39138.97 39164.28 39189.59 39214.9 39240.21 39265.52 39290.83 39316.14 39341.45 39366.76 39392.07 39417.38 39442.69 39468 39493.31 39518.62 39543.93 39569.24 39594.55 39619.86 39645.18 39670.49 39695.8 39721.11 39746.43 39771.74 39797.05 39822.37 39847.68 39872.99 39898.31 39923.62 39948.94 39974.25 39999.57 40024.88 40050.2 40075.51 40100.83 40126.14

T (K) 270.65 247.41 247.48 247.55 247.62 247.69 247.76 247.83 247.9 247.97 248.04 248.11 248.18 248.25 248.32 248.39 248.46 248.53 248.6 248.67 248.74 248.81 248.88 248.95 249.02 249.09 249.16 249.23 249.3 249.37 249.44 249.51 249.58 249.65 249.72 249.79 249.86 249.93 250 250.07 250.14 250.21 250.28 250.35 250.42 250.49 250.56 250.63 250.7

a (m/s) 329.8 315.32 315.37 315.41 315.46 315.5 315.54 315.59 315.63 315.68 315.72 315.77 315.81 315.86 315.9 315.95 315.99 316.03 316.08 316.12 316.17 316.21 316.26 316.3 316.35 316.39 316.43 316.48 316.52 316.57 316.61 316.66 316.7 316.75 316.79 316.83 316.88 316.92 316.97 317.01 317.06 317.1 317.15 317.19 317.23 317.28 317.32 317.37 317.41

p (Pa) 77.64 331.66 330.51 329.38 328.24 327.11 325.98 324.86 323.75 322.63 321.52 320.42 319.32 318.22 317.13 316.04 314.95 313.87 312.8 311.72 310.66 309.59 308.53 307.47 306.42 305.37 304.33 303.29 302.25 301.21 300.18 299.16 298.14 297.12 296.1 295.09 294.09 293.08 292.08 291.09 290.09 289.11 288.12 287.14 286.16 285.19 284.22 283.25 282.29

ρ (kg/m3 ) 0.001 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004

Appendix A: The Standard Atmosphere h (m) 49825 39900 39925 39950 39975 40000 40025 40050 40075 40100 40125 40150 40175 40200 40225 40250 40275 40300 40325 40350 40375 40400 40425 40450 40475 40500 40525 40550 40575 40600 40625 40650 40675 40700 40725 40750 40775 40800 40825 40850 40875 40900 40925 40950 40975 41000 41025 41050

hG (m) 50217.73 40151.46 40176.77 40202.09 40227.41 40252.72 40278.04 40303.36 40328.68 40353.99 40379.31 40404.63 40429.95 40455.27 40480.58 40505.9 40531.22 40556.54 40581.86 40607.18 40632.5 40657.82 40683.14 40708.46 40733.78 40759.1 40784.42 40809.74 40835.07 40860.39 40885.71 40911.03 40936.35 40961.68 40987 41012.32 41037.64 41062.97 41088.29 41113.62 41138.94 41164.26 41189.59 41214.91 41240.24 41265.56 41290.89 41316.21

529 T (K) 270.65 250.77 250.84 250.91 250.98 251.05 251.12 251.19 251.26 251.33 251.4 251.47 251.54 251.61 251.68 251.75 251.82 251.89 251.96 252.03 252.1 252.17 252.24 252.31 252.38 252.45 252.52 252.59 252.66 252.73 252.8 252.87 252.94 253.01 253.08 253.15 253.22 253.29 253.36 253.43 253.5 253.57 253.64 253.71 253.78 253.85 253.92 253.99

a (m/s) 329.8 317.46 317.5 317.54 317.59 317.63 317.68 317.72 317.77 317.81 317.85 317.9 317.94 317.99 318.03 318.08 318.12 318.16 318.21 318.25 318.3 318.34 318.38 318.43 318.47 318.52 318.56 318.61 318.65 318.69 318.74 318.78 318.83 318.87 318.91 318.96 319 319.05 319.09 319.13 319.18 319.22 319.27 319.31 319.36 319.4 319.44 319.49

p (Pa) 77.64 281.33 280.37 279.42 278.47 277.52 276.58 275.64 274.7 273.77 272.84 271.92 271 270.08 269.16 268.25 267.34 266.44 265.54 264.64 263.74 262.85 261.96 261.08 260.19 259.32 258.44 257.57 256.7 255.83 254.97 254.11 253.25 252.4 251.55 250.7 249.86 249.02 248.18 247.34 246.51 245.68 244.86 244.03 243.21 242.4 241.58 240.77

ρ (kg/m3 ) 0.001 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003

530

Appendix A: The Standard Atmosphere h (m) 49825 41075 41100 41125 41150 41175 41200 41225 41250 41275 41300 41325 41350 41375 41400 41425 41450 41475 41500 41525 41550 41575 41600 41625 41650 41675 41700 41725 41750 41775 41800 41825 41850 41875 41900 41925 41950 41975 42000 42025 42050 42075 42100 42125 42150 42175 42200 42225 42250

hG (m) 50217.73 41341.54 41366.86 41392.19 41417.51 41442.84 41468.17 41493.49 41518.82 41544.15 41569.47 41594.8 41620.13 41645.46 41670.78 41696.11 41721.44 41746.77 41772.1 41797.43 41822.76 41848.09 41873.42 41898.75 41924.08 41949.41 41974.74 42000.07 42025.4 42050.73 42076.06 42101.39 42126.72 42152.05 42177.39 42202.72 42228.05 42253.38 42278.72 42304.05 42329.38 42354.72 42380.05 42405.38 42430.72 42456.05 42481.39 42506.72 42532.06

T (K) 270.65 254.06 254.13 254.2 254.27 254.34 254.41 254.48 254.55 254.62 254.69 254.76 254.83 254.9 254.97 255.04 255.11 255.18 255.25 255.32 255.39 255.46 255.53 255.6 255.67 255.74 255.81 255.88 255.95 256.02 256.09 256.16 256.23 256.3 256.37 256.44 256.51 256.58 256.65 256.72 256.79 256.86 256.93 257 257.07 257.14 257.21 257.28 257.35

a (m/s) 329.8 319.53 319.58 319.62 319.66 319.71 319.75 319.8 319.84 319.88 319.93 319.97 320.02 320.06 320.1 320.15 320.19 320.23 320.28 320.32 320.37 320.41 320.45 320.5 320.54 320.59 320.63 320.67 320.72 320.76 320.81 320.85 320.89 320.94 320.98 321.02 321.07 321.11 321.16 321.2 321.24 321.29 321.33 321.37 321.42 321.46 321.51 321.55 321.59

p (Pa) 77.64 239.96 239.16 238.35 237.56 236.76 235.97 235.17 234.39 233.6 232.82 232.04 231.26 230.49 229.72 228.95 228.19 227.42 226.66 225.91 225.15 224.4 223.65 222.91 222.16 221.42 220.68 219.95 219.21 218.48 217.76 217.03 216.31 215.59 214.87 214.16 213.45 212.74 212.03 211.33 210.62 209.92 209.23 208.53 207.84 207.15 206.47 205.78 205.1

ρ (kg/m3 ) 0.001 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003

Appendix A: The Standard Atmosphere h (m) 49825 42275 42300 42325 42350 42375 42400 42425 42450 42475 42500 42525 42550 42575 42600 42625 42650 42675 42700 42725 42750 42775 42800 42825 42850 42875 42900 42925 42950 42975 43000 43025 43050 43075 43100 43125 43150 43175 43200 43225 43250 43275 43300 43325 43350 43375 43400 43425 43450

hG (m) 50217.73 42557.39 42582.73 42608.06 42633.4 42658.73 42684.07 42709.41 42734.74 42760.08 42785.42 42810.75 42836.09 42861.43 42886.76 42912.1 42937.44 42962.78 42988.12 43013.46 43038.79 43064.13 43089.47 43114.81 43140.15 43165.49 43190.83 43216.17 43241.51 43266.85 43292.19 43317.53 43342.88 43368.22 43393.56 43418.9 43444.24 43469.58 43494.93 43520.27 43545.61 43570.96 43596.3 43621.64 43646.99 43672.33 43697.67 43723.02 43748.36

531 T (K) 270.65 257.42 257.49 257.56 257.63 257.7 257.77 257.84 257.91 257.98 258.05 258.12 258.19 258.26 258.33 258.4 258.47 258.54 258.61 258.68 258.75 258.82 258.89 258.96 259.03 259.1 259.17 259.24 259.31 259.38 259.45 259.52 259.59 259.66 259.73 259.8 259.87 259.94 260.01 260.08 260.15 260.22 260.29 260.36 260.43 260.5 260.57 260.64 260.71

a (m/s) 329.8 321.64 321.68 321.72 321.77 321.81 321.86 321.9 321.94 321.99 322.03 322.07 322.12 322.16 322.21 322.25 322.29 322.34 322.38 322.42 322.47 322.51 322.55 322.6 322.64 322.69 322.73 322.77 322.82 322.86 322.9 322.95 322.99 323.03 323.08 323.12 323.16 323.21 323.25 323.29 323.34 323.38 323.43 323.47 323.51 323.56 323.6 323.64 323.69

p (Pa) 77.64 204.42 203.74 203.07 202.4 201.73 201.06 200.39 199.73 199.07 198.41 197.76 197.1 196.45 195.81 195.16 194.52 193.87 193.23 192.6 191.96 191.33 190.7 190.07 189.45 188.82 188.2 187.58 186.97 186.35 185.74 185.13 184.52 183.91 183.31 182.71 182.11 181.51 180.92 180.32 179.73 179.14 178.56 177.97 177.39 176.81 176.23 175.65 175.08

ρ (kg/m3 ) 0.001 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002

532

Appendix A: The Standard Atmosphere h (m) 49825 43475 43500 43525 43550 43575 43600 43625 43650 43675 43700 43725 43750 43775 43800 43825 43850 43875 43900 43925 43950 43975 44000 44025 44050 44075 44100 44125 44150 44175 44200 44225 44250 44275 44300 44325 44350 44375 44400 44425 44450 44475 44500 44525 44550 44575 44600 44625

hG (m) 50217.73 43773.71 43799.05 43824.4 43849.74 43875.09 43900.43 43925.78 43951.12 43976.47 44001.82 44027.16 44052.51 44077.86 44103.21 44128.55 44153.9 44179.25 44204.6 44229.94 44255.29 44280.64 44305.99 44331.34 44356.69 44382.04 44407.39 44432.74 44458.09 44483.44 44508.79 44534.14 44559.49 44584.84 44610.19 44635.54 44660.89 44686.25 44711.6 44736.95 44762.3 44787.66 44813.01 44838.36 44863.72 44889.07 44914.42 44939.78

T (K) 270.65 260.78 260.85 260.92 260.99 261.06 261.13 261.2 261.27 261.34 261.41 261.48 261.55 261.62 261.69 261.76 261.83 261.9 261.97 262.04 262.11 262.18 262.25 262.32 262.39 262.46 262.53 262.6 262.67 262.74 262.81 262.88 262.95 263.02 263.09 263.16 263.23 263.3 263.37 263.44 263.51 263.58 263.65 263.72 263.79 263.86 263.93 264

a (m/s) 329.8 323.73 323.77 323.82 323.86 323.9 323.95 323.99 324.03 324.08 324.12 324.16 324.21 324.25 324.29 324.34 324.38 324.42 324.47 324.51 324.55 324.6 324.64 324.68 324.73 324.77 324.81 324.86 324.9 324.94 324.99 325.03 325.07 325.12 325.16 325.2 325.25 325.29 325.33 325.38 325.42 325.46 325.51 325.55 325.59 325.64 325.68 325.72

p (Pa) 77.64 174.5 173.93 173.37 172.8 172.23 171.67 171.11 170.55 170 169.44 168.89 168.34 167.79 167.24 166.7 166.16 165.61 165.08 164.54 164 163.47 162.94 162.41 161.88 161.35 160.83 160.31 159.79 159.27 158.75 158.24 157.72 157.21 156.7 156.19 155.69 155.18 154.68 154.18 153.68 153.18 152.69 152.2 151.7 151.21 150.72 150.24

ρ (kg/m3 ) 0.001 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002

Appendix A: The Standard Atmosphere h (m) 49825 44650 44675 44700 44725 44750 44775 44800 44825 44850 44875 44900 44925 44950 44975 45000 45025 45050 45075 45100 45125 45150 45175 45200 45225 45250 45275 45300 45325 45350 45375 45400 45425 45450 45475 45500 45525 45550 45575 45600 45625 45650 45675 45700 45725 45750 45775 45800

hG (m) 50217.73 44965.13 44990.48 45015.84 45041.19 45066.55 45091.9 45117.26 45142.61 45167.97 45193.33 45218.68 45244.04 45269.39 45294.75 45320.11 45345.46 45370.82 45396.18 45421.54 45446.89 45472.25 45497.61 45522.97 45548.33 45573.69 45599.05 45624.41 45649.76 45675.12 45700.48 45725.84 45751.2 45776.56 45801.93 45827.29 45852.65 45878.01 45903.37 45928.73 45954.09 45979.46 46004.82 46030.18 46055.54 46080.91 46106.27 46131.63

533 T (K) 270.65 264.07 264.14 264.21 264.28 264.35 264.42 264.49 264.56 264.63 264.7 264.77 264.84 264.91 264.98 265.05 265.12 265.19 265.26 265.33 265.4 265.47 265.54 265.61 265.68 265.75 265.82 265.89 265.96 266.03 266.1 266.17 266.24 266.31 266.38 266.45 266.52 266.59 266.66 266.73 266.8 266.87 266.94 267.01 267.08 267.15 267.22 267.29

a (m/s) 329.8 325.77 325.81 325.85 325.89 325.94 325.98 326.02 326.07 326.11 326.15 326.2 326.24 326.28 326.33 326.37 326.41 326.46 326.5 326.54 326.58 326.63 326.67 326.71 326.76 326.8 326.84 326.89 326.93 326.97 327.01 327.06 327.1 327.14 327.19 327.23 327.27 327.32 327.36 327.4 327.44 327.49 327.53 327.57 327.62 327.66 327.7 327.75

p (Pa) 77.64 149.75 149.27 148.79 148.31 147.83 147.35 146.88 146.4 145.93 145.46 144.99 144.53 144.06 143.6 143.13 142.67 142.22 141.76 141.3 140.85 140.4 139.95 139.5 139.05 138.6 138.16 137.71 137.27 136.83 136.39 135.96 135.52 135.09 134.65 134.22 133.79 133.37 132.94 132.51 132.09 131.67 131.25 130.83 130.41 130 129.58 129.17

ρ (kg/m3 ) 0.001 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002

534

Appendix A: The Standard Atmosphere h (m) 49825 45825 45850 45875 45900 45925 45950 45975 46000 46025 46050 46075 46100 46125 46150 46175 46200 46225 46250 46275 46300 46325 46350 46375 46400 46425 46450 46475 46500 46525 46550 46575 46600 46625 46650 46675 46700 46725 46750 46775 46800 46825 46850 46875 46900 46925 46950

hG (m) 50217.73 46157 46182.36 46207.72 46233.09 46258.45 46283.82 46309.18 46334.55 46359.91 46385.28 46410.64 46436.01 46461.37 46486.74 46512.11 46537.47 46562.84 46588.21 46613.57 46638.94 46664.31 46689.67 46715.04 46740.41 46765.78 46791.15 46816.52 46841.88 46867.25 46892.62 46917.99 46943.36 46968.73 46994.1 47019.47 47044.84 47070.21 47095.58 47120.96 47146.33 47171.7 47197.07 47222.44 47247.81 47273.19 47298.56

T (K) 270.65 267.36 267.43 267.5 267.57 267.64 267.71 267.78 267.85 267.92 267.99 268.06 268.13 268.2 268.27 268.34 268.41 268.48 268.55 268.62 268.69 268.76 268.83 268.9 268.97 269.04 269.11 269.18 269.25 269.32 269.39 269.46 269.53 269.6 269.67 269.74 269.81 269.88 269.95 270.02 270.09 270.16 270.23 270.3 270.37 270.44 270.51

a (m/s) 329.8 327.79 327.83 327.87 327.92 327.96 328 328.05 328.09 328.13 328.17 328.22 328.26 328.3 328.35 328.39 328.43 328.47 328.52 328.56 328.6 328.65 328.69 328.73 328.77 328.82 328.86 328.9 328.94 328.99 329.03 329.07 329.12 329.16 329.2 329.24 329.29 329.33 329.37 329.41 329.46 329.5 329.54 329.59 329.63 329.67 329.71

p (Pa) 77.64 128.75 128.34 127.94 127.53 127.12 126.72 126.31 125.91 125.51 125.11 124.71 124.32 123.92 123.53 123.13 122.74 122.35 121.96 121.58 121.19 120.81 120.42 120.04 119.66 119.28 118.9 118.53 118.15 117.78 117.4 117.03 116.66 116.29 115.93 115.56 115.19 114.83 114.47 114.11 113.74 113.39 113.03 112.67 112.32 111.96 111.61

ρ (kg/m3 ) 0.001 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001

Appendix A: The Standard Atmosphere h (m) 49825 46975 47000 47025 47050 47075 47100 47125 47150 47175 47200 47225 47250 47275 47300 47325 47350 47375 47400 47425 47450 47475 47500 47525 47550 47575 47600 47625 47650 47675 47700 47725 47750 47775 47800 47825 47850 47875 47900 47925 47950 47975 48000 48025 48050 48075 48100 48125

hG (m) 50217.73 47323.93 47349.3 47374.68 47400.05 47425.42 47450.8 47476.17 47501.55 47526.92 47552.29 47577.67 47603.04 47628.42 47653.79 47679.17 47704.55 47729.92 47755.3 47780.67 47806.05 47831.43 47856.8 47882.18 47907.56 47932.94 47958.31 47983.69 48009.07 48034.45 48059.83 48085.21 48110.58 48135.96 48161.34 48186.72 48212.1 48237.48 48262.86 48288.24 48313.62 48339 48364.38 48389.77 48415.15 48440.53 48465.91 48491.29

535 T (K) 270.65 270.58 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65

a (m/s) 329.8 329.76 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8

p (Pa) 77.64 111.26 110.91 110.56 110.21 109.86 109.52 109.17 108.83 108.48 108.14 107.8 107.46 107.12 106.79 106.45 106.11 105.78 105.45 105.11 104.78 104.45 104.12 103.79 103.47 103.14 102.82 102.49 102.17 101.85 101.53 101.21 100.89 100.57 100.25 99.94 99.62 99.31 99 98.68 98.37 98.06 97.75 97.45 97.14 96.83 96.53 96.22

ρ (kg/m3 ) 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001

536

Appendix A: The Standard Atmosphere h (m) 49825 48150 48175 48200 48225 48250 48275 48300 48325 48350 48375 48400 48425 48450 48475 48500 48525 48550 48575 48600 48625 48650 48675 48700 48725 48750 48775 48800 48825 48850 48875 48900 48925 48950 48975 49000 49025 49050 49075 49100 49125 49150 49175 49200 49225 49250 49275 49300 49325

hG (m) 50217.73 48516.67 48542.06 48567.44 48592.82 48618.2 48643.59 48668.97 48694.35 48719.74 48745.12 48770.51 48795.89 48821.27 48846.66 48872.04 48897.43 48922.81 48948.2 48973.59 48998.97 49024.36 49049.74 49075.13 49100.52 49125.9 49151.29 49176.68 49202.07 49227.45 49252.84 49278.23 49303.62 49329.01 49354.4 49379.78 49405.17 49430.56 49455.95 49481.34 49506.73 49532.12 49557.51 49582.9 49608.29 49633.69 49659.08 49684.47 49709.86

T (K) 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65

a (m/s) 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8

p (Pa) 77.64 95.92 95.62 95.32 95.02 94.72 94.42 94.12 93.83 93.53 93.24 92.94 92.65 92.36 92.07 91.78 91.49 91.2 90.91 90.62 90.34 90.05 89.77 89.49 89.21 88.92 88.64 88.37 88.09 87.81 87.53 87.26 86.98 86.71 86.43 86.16 85.89 85.62 85.35 85.08 84.81 84.55 84.28 84.01 83.75 83.49 83.22 82.96 82.7

ρ (kg/m3 ) 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001

Appendix A: The Standard Atmosphere h (m) 49825 49350 49375 49400 49425 49450 49475 49500 49525 49550 49575 49600 49625 49650 49675 49700 49725 49750 49775 49800 49850 49875 49900 49925 49950 49975 50000

hG (m) 50217.73 49735.25 49760.64 49786.04 49811.43 49836.82 49862.21 49887.61 49913 49938.39 49963.79 49989.18 50014.57 50039.97 50065.36 50090.76 50116.15 50141.55 50166.94 50192.34 50243.13 50268.52 50293.92 50319.32 50344.71 50370.11 50395.51

537 T (K) 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65 270.65

a (m/s) 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8 329.8

p (Pa) 77.64 82.44 82.18 81.92 81.66 81.4 81.15 80.89 80.64 80.38 80.13 79.88 79.63 79.38 79.13 78.88 78.63 78.38 78.13 77.89 77.4 77.15 76.91 76.67 76.43 76.18 75.94

ρ (kg/m3 ) 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001

Appendix B

Isentropic Table (γ = 1.4)

M

p p0

T T0

ρ ρ0

A A∗

a a0

M∗

μ

ν

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27

1.0000 0.9999 0.9997 0.9994 0.9989 0.9983 0.9975 0.9966 0.9955 0.9944 0.9930 0.9916 0.9900 0.9883 0.9864 0.9844 0.9823 0.9800 0.9776 0.9751 0.9725 0.9697 0.9668 0.9638 0.9607 0.9575 0.9541 0.9506

1.0000 1.0000 0.9999 0.9998 0.9997 0.9995 0.9993 0.9990 0.9987 0.9984 0.9980 0.9976 0.9971 0.9966 0.9961 0.9955 0.9949 0.9943 0.9936 0.9928 0.9921 0.9913 0.9904 0.9895 0.9886 0.9877 0.9867 0.9856

1.0000 0.9999 0.9998 0.9996 0.9992 0.9988 0.9982 0.9976 0.9968 0.9960 0.9950 0.9940 0.9928 0.9916 0.9903 0.9888 0.9873 0.9857 0.9840 0.9822 0.9803 0.9783 0.9762 0.9740 0.9718 0.9694 0.9670 0.9645

∞ 57.8738 28.9421 19.3005 14.4815 11.5914 9.6659 8.2915 7.2616 6.4613 5.8218 5.2992 4.8643 4.4969 4.1824 3.9103 3.6727 3.4635 3.2779 3.1123 2.9635 2.8293 2.7076 2.5968 2.4956 2.4027 2.3173 2.2385

1.0000 1.0000 1.0000 0.9999 0.9998 0.9998 0.9996 0.9995 0.9994 0.9992 0.9990 0.9988 0.9986 0.9983 0.9980 0.9978 0.9974 0.9971 0.9968 0.9964 0.9960 0.9956 0.9952 0.9948 0.9943 0.9938 0.9933 0.9928

0.0000 0.0110 0.0219 0.0329 0.0438 0.0548 0.0657 0.0766 0.0876 0.0985 0.1094 0.1204 0.1313 0.1422 0.1531 0.1639 0.1748 0.1857 0.1965 0.2074 0.2182 0.2290 0.2398 0.2506 0.2614 0.2722 0.2829 0.2936

-

-

© Springer Nature Singapore Pte Ltd. 2022 M. Kaushik, Fundamentals of Gas Dynamics, https://doi.org/10.1007/978-981-16-9085-3

539

Appendix B: Isentropic Table (γ = 1.4)

540 M

p p0

T T0

ρ ρ0

A A∗

a a0

M∗

μ

ν

0.28 0.29 0.30 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.40 0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.50 0.51 0.52 0.53 0.54 0.55 0.56 0.57 0.58 0.59 0.60 0.61 0.62 0.63 0.64 0.65 0.66 0.67 0.68 0.69 0.70 0.71 0.72 0.73 0.74 0.75

0.9470 0.9433 0.9395 0.9355 0.9315 0.9274 0.9231 0.9188 0.9143 0.9098 0.9052 0.9004 0.8956 0.8907 0.8857 0.8807 0.8755 0.8703 0.8650 0.8596 0.8541 0.8486 0.8430 0.8374 0.8317 0.8259 0.8201 0.8142 0.8082 0.8022 0.7962 0.7901 0.7840 0.7778 0.7716 0.7654 0.7591 0.7528 0.7465 0.7401 0.7338 0.7274 0.7209 0.7145 0.7080 0.7016 0.6951 0.6886

0.9846 0.9835 0.9823 0.9811 0.9799 0.9787 0.9774 0.9761 0.9747 0.9733 0.9719 0.9705 0.9690 0.9675 0.9659 0.9643 0.9627 0.9611 0.9594 0.9577 0.9559 0.9542 0.9524 0.9506 0.9487 0.9468 0.9449 0.9430 0.9410 0.9390 0.9370 0.9349 0.9328 0.9307 0.9286 0.9265 0.9243 0.9221 0.9199 0.9176 0.9153 0.9131 0.9107 0.9084 0.9061 0.9037 0.9013 0.8989

0.9619 0.9592 0.9564 0.9535 0.9506 0.9476 0.9445 0.9413 0.9380 0.9347 0.9313 0.9278 0.9243 0.9207 0.9170 0.9132 0.9094 0.9055 0.9016 0.8976 0.8935 0.8894 0.8852 0.8809 0.8766 0.8723 0.8679 0.8634 0.8589 0.8544 0.8498 0.8451 0.8405 0.8357 0.8310 0.8262 0.8213 0.8164 0.8115 0.8066 0.8016 0.7966 0.7916 0.7865 0.7814 0.7763 0.7712 0.7660

2.1656 2.0979 2.0351 1.9765 1.9219 1.8707 1.8229 1.7780 1.7358 1.6961 1.6587 1.6234 1.5901 1.5587 1.5289 1.5007 1.4740 1.4487 1.4246 1.4018 1.3801 1.3595 1.3398 1.3212 1.3034 1.2865 1.2703 1.2549 1.2403 1.2263 1.2130 1.2003 1.1882 1.1767 1.1656 1.1552 1.1451 1.1356 1.1265 1.1179 1.1097 1.1018 1.0944 1.0873 1.0806 1.0742 1.0681 1.0624

0.9923 0.9917 0.9911 0.9905 0.9899 0.9893 0.9886 0.9880 0.9873 0.9866 0.9859 0.9851 0.9844 0.9836 0.9828 0.9820 0.9812 0.9803 0.9795 0.9786 0.9777 0.9768 0.9759 0.9750 0.9740 0.9730 0.9721 0.9711 0.9700 0.9690 0.9680 0.9669 0.9658 0.9647 0.9636 0.9625 0.9614 0.9603 0.9591 0.9579 0.9567 0.9555 0.9543 0.9531 0.9519 0.9506 0.9494 0.9481

0.3043 0.3150 0.3257 0.3364 0.3470 0.3576 0.3682 0.3788 0.3893 0.3999 0.4104 0.4209 0.4313 0.4418 0.4522 0.4626 0.4729 0.4833 0.4936 0.5038 0.5141 0.5243 0.5345 0.5447 0.5548 0.5649 0.5750 0.5851 0.5951 0.6051 0.6150 0.6249 0.6348 0.6447 0.6545 0.6643 0.6740 0.6837 0.6934 0.7031 0.7127 0.7223 0.7318 0.7413 0.7508 0.7602 0.7696 0.7789

-

-

Appendix B: Isentropic Table (γ = 1.4)

541

M

p p0

T T0

ρ ρ0

A A∗

a a0

M∗

μ

ν

0.76 0.77 0.78 0.79 0.80 0.81 0.82 0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.90 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.20 1.21 1.22 1.23 1.24 1.25

0.6821 0.6756 0.6691 0.6625 0.6560 0.6495 0.6430 0.6365 0.6300 0.6235 0.6170 0.6106 0.6041 0.5977 0.5913 0.5849 0.5785 0.5721 0.5658 0.5595 0.5532 0.5469 0.5407 0.5345 0.5283 0.5221 0.5160 0.5099 0.5039 0.4979 0.4919 0.4860 0.4800 0.4742 0.4684 0.4626 0.4568 0.4511 0.4455 0.4398 0.4343 0.4287 0.4232 0.4178 0.4124 0.4070 0.4017 0.3964 0.3912 0.3861

0.8964 0.894 0.8915 0.8890 0.8865 0.8840 0.8815 0.8789 0.8763 0.8737 0.8711 0.8685 0.8659 0.8632 0.8606 0.8579 0.8552 0.8525 0.8498 0.8471 0.8444 0.8416 0.8389 0.8361 0.8333 0.8306 0.8278 0.8250 0.8222 0.8193 0.8165 0.8137 0.8108 0.8080 0.8052 0.8023 0.7994 0.7966 0.7937 0.7908 0.7879 0.7851 0.7822 0.7793 0.7764 0.7735 0.7706 0.7677 0.7648 0.7619

0.7609 0.7557 0.7505 0.7452 0.7400 0.7347 0.7295 0.7242 0.7189 0.7136 0.7083 0.7030 0.6977 0.6924 0.6870 0.6817 0.6764 0.6711 0.6658 0.6604 0.6551 0.6498 0.6445 0.6392 0.6339 0.6287 0.6234 0.6181 0.6129 0.6077 0.6024 0.5972 0.5920 0.5869 0.5817 0.5766 0.5714 0.5663 0.5612 0.5562 0.5511 0.5461 0.5411 0.5361 0.5311 0.5262 0.5213 0.5164 0.5115 0.5067

1.0570 1.0519 1.0471 1.0425 1.0382 1.0342 1.0305 1.0270 1.0237 1.0207 1.0179 1.0153 1.0129 1.0108 1.0089 1.0071 1.0056 1.0043 1.0031 1.0021 1.0014 1.0008 1.0003 1.0001 1.0000 1.0001 1.0003 1.0007 1.0013 1.0020 1.0029 1.0039 1.0051 1.0064 1.0079 1.0095 1.0113 1.0132 1.0153 1.0175 1.0198 1.0222 1.0248 1.0276 1.0304 1.0334 1.0366 1.0398 1.0432 1.0468

0.9468 0.9455 0.9442 0.9429 0.9416 0.9402 0.9389 0.9375 0.9361 0.9347 0.9333 0.9319 0.9305 0.9291 0.9277 0.9262 0.9248 0.9233 0.9219 0.9204 0.9189 0.9174 0.9159 0.9144 0.9129 0.9113 0.9098 0.9083 0.9067 0.9052 0.9036 0.9020 0.9005 0.8989 0.8973 0.8957 0.8941 0.8925 0.8909 0.8893 0.8877 0.886 0.8844 0.8828 0.8811 0.8795 0.8778 0.8762 0.8745 0.8729

0.7883 0.7975 0.8068 0.8160 0.8251 0.8343 0.8433 0.8524 0.8614 0.8704 0.8793 0.8882 0.8970 0.9058 0.9146 0.9233 0.9320 0.9406 0.9493 0.9578 0.9663 0.9748 0.9832 0.9916 1.0000 1.0083 1.0166 1.0248 1.0330 1.0411 1.0492 1.0573 1.0653 1.0733 1.0812 1.0891 1.097 1.1048 1.1126 1.1203 1.128 1.1356 1.1432 1.1508 1.1583 1.1658 1.1732 1.1806 1.1879 1.1952

90.0000 81.9307 78.6351 76.1376 74.0576 72.2472 70.6300 69.1603 67.8084 66.5534 65.3800 64.2767 63.2345 62.2461 61.3056 60.4082 59.5497 58.7267 57.9362 57.1756 56.4427 55.7354 55.0520 54.3909 53.7507 53.1301

0.0000 0.0447 0.1257 0.2294 0.3510 0.4874 0.6367 0.7973 0.9680 1.1479 1.3362 1.5321 1.735 1.9445 2.16 2.381 2.6073 2.8385 3.0742 3.3142 3.5582 3.8060 4.0572 4.3117 4.5694 4.8299

Appendix B: Isentropic Table (γ = 1.4)

542 M

p p0

T T0

ρ ρ0

A A∗

a a0

M∗

μ

ν

1.26 1.27 1.28 1.29 1.30 1.31 1.32 1.33 1.34 1.35 1.36 1.37 1.38 1.39 1.40 1.41 1.42 1.43 1.44 1.45 1.46 1.47 1.48 1.49 1.50 1.51 1.52 1.53 1.54 1.55 1.56 1.57 1.58 1.59 1.60 1.61 1.62 1.63 1.64 1.65 1.66 1.67 1.68 1.69 1.70 1.71 1.72 1.73 1.74 1.75

0.3809 0.3759 0.3708 0.3658 0.3609 0.3560 0.3512 0.3464 0.3417 0.3370 0.3323 0.3277 0.3232 0.3187 0.3142 0.3098 0.3055 0.3012 0.2969 0.2927 0.2886 0.2845 0.2804 0.2764 0.2724 0.2685 0.2646 0.2608 0.2570 0.2533 0.2496 0.2459 0.2423 0.2388 0.2353 0.2318 0.2284 0.2250 0.2217 0.2184 0.2152 0.2119 0.2088 0.2057 0.2026 0.1996 0.1966 0.1936 0.1907 0.1878

0.7590 0.7561 0.7532 0.7503 0.7474 0.7445 0.7416 0.7387 0.7358 0.7329 0.7300 0.7271 0.7242 0.7213 0.7184 0.7155 0.7126 0.7097 0.7069 0.7040 0.7011 0.6982 0.6954 0.6925 0.6897 0.6868 0.6840 0.6811 0.6783 0.6754 0.6726 0.6698 0.6670 0.6642 0.6614 0.6586 0.6558 0.6530 0.6502 0.6475 0.6447 0.6419 0.6392 0.6364 0.6337 0.6310 0.6283 0.6256 0.6229 0.6202

0.5019 0.4971 0.4923 0.4876 0.4829 0.4782 0.4736 0.4690 0.4644 0.4598 0.4553 0.4508 0.4463 0.4418 0.4374 0.4330 0.4287 0.4244 0.4201 0.4158 0.4116 0.4074 0.4032 0.3991 0.3950 0.3909 0.3869 0.3829 0.3789 0.3750 0.3710 0.3672 0.3633 0.3595 0.3557 0.3520 0.3483 0.3446 0.3409 0.3373 0.3337 0.3302 0.3266 0.3232 0.3197 0.3163 0.3129 0.3095 0.3062 0.3029

1.0504 1.0542 1.0581 1.0621 1.0663 1.0706 1.0750 1.0796 1.0842 1.0890 1.0940 1.0990 1.1042 1.1095 1.1149 1.1205 1.1262 1.1320 1.1379 1.1440 1.1501 1.1565 1.1629 1.1695 1.1762 1.1830 1.1899 1.1970 1.2042 1.2116 1.2190 1.2266 1.2344 1.2422 1.2502 1.2584 1.2666 1.2750 1.2836 1.2922 1.3010 1.3100 1.3190 1.3283 1.3376 1.3471 1.3567 1.3665 1.3764 1.3865

0.8712 0.8695 0.8679 0.8662 0.8645 0.8628 0.8611 0.8595 0.8578 0.8561 0.8544 0.8527 0.8510 0.8493 0.8476 0.8459 0.8442 0.8425 0.8407 0.8390 0.8373 0.8356 0.8339 0.8322 0.8305 0.8287 0.8270 0.8253 0.8236 0.8219 0.8201 0.8184 0.8167 0.8150 0.8133 0.8115 0.8098 0.8081 0.8064 0.8046 0.8029 0.8012 0.7995 0.7978 0.7961 0.7943 0.7926 0.7909 0.7892 0.7875

1.2025 1.2097 1.2169 1.2240 1.2311 1.2382 1.2452 1.2522 1.2591 1.2660 1.2729 1.2797 1.2864 1.2932 1.2999 1.3065 1.3131 1.3197 1.3262 1.3327 1.3392 1.3456 1.3520 1.3583 1.3646 1.3708 1.3770 1.3832 1.3894 1.3955 1.4015 1.4075 1.4135 1.4195 1.4254 1.4313 1.4371 1.4429 1.4487 1.4544 1.4601 1.4657 1.4713 1.4769 1.4825 1.4880 1.4935 1.4989 1.5043 1.5097

52.5280 51.9434 51.3752 50.8226 50.2849 49.7612 49.2510 48.7535 48.2682 47.7946 47.3321 46.8803 46.4387 46.0070 45.5847 45.1715 44.7670 44.3709 43.9830 43.6028 43.2302 42.8649 42.5067 42.1552 41.8103 41.4718 41.1395 40.8132 40.4927 40.1778 39.8684 39.5642 39.2653 38.9713 38.6822 38.3978 38.1181 37.8428 37.5719 37.3052 37.0427 36.7842 36.5296 36.2789 36.0319 35.7885 35.5488 35.3125 35.0795 34.8499

5.0931 5.3590 5.6272 5.8977 6.1703 6.4449 6.7213 6.9995 7.2794 7.5607 7.8435 8.1276 8.4130 8.6995 8.9870 9.2756 9.5650 9.8553 10.1463 10.4381 10.7305 11.0234 11.3169 11.6109 11.9052 12.1999 12.4949 12.7901 13.0856 13.3812 13.6769 13.9728 14.2686 14.5645 14.8603 15.1561 15.4518 15.7473 16.0427 16.3379 16.6328 16.9275 17.2220 17.5161 17.8099 18.1033 18.3964 18.6891 18.9813 19.2732

Appendix B: Isentropic Table (γ = 1.4)

543

M

p p0

T T0

ρ ρ0

A A∗

a a0

M∗

μ

ν

1.76 1.77 1.78 1.79 1.80 1.81 1.82 1.83 1.84 1.85 1.86 1.87 1.88 1.89 1.90 1.91 1.92 1.93 1.94 1.95 1.96 1.97 1.98 1.99 2.00 2.01 2.02 2.03 2.04 2.05 2.06 2.07 2.08 2.09 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 2.20 2.21 2.22 2.23 2.24

0.1850 0.1822 0.1794 0.1767 0.1740 0.1714 0.1688 0.1662 0.1637 0.1612 0.1587 0.1563 0.1539 0.1516 0.1492 0.1470 0.1447 0.1425 0.1403 0.1381 0.1360 0.1339 0.1318 0.1298 0.1278 0.1258 0.1239 0.1220 0.1201 0.1182 0.1164 0.1146 0.1128 0.1111 0.1094 0.1077 0.1060 0.1043 0.1027 0.1011 0.0996 0.0980 0.0965 0.0950 0.0935 0.0921 0.0906 0.0892 0.0878

0.6175 0.6148 0.6121 0.6095 0.6068 0.6041 0.6015 0.5989 0.5963 0.5936 0.5910 0.5885 0.5859 0.5833 0.5807 0.5782 0.5756 0.5731 0.5705 0.5680 0.5655 0.5630 0.5605 0.5580 0.5556 0.5531 0.5506 0.5482 0.5458 0.5433 0.5409 0.5385 0.5361 0.5337 0.5313 0.5290 0.5266 0.5243 0.5219 0.5196 0.5173 0.5150 0.5127 0.5104 0.5081 0.5059 0.5036 0.5014 0.4991

0.2996 0.2964 0.2931 0.2900 0.2868 0.2837 0.2806 0.2776 0.2745 0.2715 0.2686 0.2656 0.2627 0.2598 0.2570 0.2542 0.2514 0.2486 0.2459 0.2432 0.2405 0.2378 0.2352 0.2326 0.2300 0.2275 0.2250 0.2225 0.2200 0.2176 0.2152 0.2128 0.2104 0.2081 0.2058 0.2035 0.2013 0.1990 0.1968 0.1946 0.1925 0.1903 0.1882 0.1861 0.1841 0.1820 0.1800 0.1780 0.1760

1.3967 1.4070 1.4175 1.4282 1.4390 1.4499 1.4610 1.4723 1.4836 1.4952 1.5069 1.5187 1.5308 1.5429 1.5553 1.5677 1.5804 1.5932 1.6062 1.6193 1.6326 1.6461 1.6597 1.6735 1.6875 1.7016 1.7160 1.7305 1.7451 1.7600 1.7750 1.7902 1.8056 1.8212 1.8369 1.8529 1.8690 1.8853 1.9018 1.9185 1.9354 1.9525 1.9698 1.9873 2.0050 2.0229 2.0409 2.0592 2.0777

0.7858 0.7841 0.7824 0.7807 0.7790 0.7773 0.7756 0.7739 0.7722 0.7705 0.7688 0.7671 0.7654 0.7637 0.7621 0.7604 0.7587 0.7570 0.7553 0.7537 0.7520 0.7503 0.7487 0.7470 0.7454 0.7437 0.7420 0.7404 0.7388 0.7371 0.7355 0.7338 0.7322 0.7306 0.7289 0.7273 0.7257 0.7241 0.7225 0.7208 0.7192 0.7176 0.7160 0.7144 0.7128 0.7112 0.7097 0.7081 0.7065

1.5150 1.5203 1.5256 1.5308 1.5360 1.5411 1.5463 1.5514 1.5564 1.5614 1.5664 1.5714 1.5763 1.5812 1.5861 1.5909 1.5957 1.6005 1.6052 1.6099 1.6146 1.6192 1.6239 1.6284 1.6330 1.6375 1.6420 1.6465 1.6509 1.6553 1.6597 1.6640 1.6683 1.6726 1.6769 1.6811 1.6853 1.6895 1.6936 1.6977 1.7018 1.7059 1.7099 1.7139 1.7179 1.7219 1.7258 1.7297 1.7336

34.6236 34.4003 34.1802 33.9631 33.7490 33.5378 33.3293 33.1237 32.9207 32.7205 32.5228 32.3276 32.1349 31.9447 31.7569 31.5714 31.3882 31.2072 31.0285 30.8519 30.6774 30.5051 30.3347 30.1664 30.0000 29.8356 29.6730 29.5123 29.3535 29.1964 29.0411 28.8875 28.7357 28.5855 28.4369 28.2900 28.1446 28.0008 27.8585 27.7178 27.5785 27.4407 27.3043 27.1693 27.0357 26.9035 26.7726 26.6430 26.5148

19.5645 19.8554 20.1458 20.4357 20.7250 21.0138 21.3021 21.5898 21.8768 22.1633 22.4491 22.7344 23.0189 23.3029 23.5861 23.8687 24.1506 24.4318 24.7122 24.9920 25.2710 25.5493 25.8269 26.1037 26.3797 26.6550 26.9295 27.2032 27.4762 27.7483 28.0197 28.2903 28.5600 28.8289 29.0971 29.3644 29.6308 29.8965 30.1613 30.4252 30.6884 30.9507 31.2121 31.4727 31.7325 31.9914 32.2494 32.5066 32.7629

Appendix B: Isentropic Table (γ = 1.4)

544 M

p p0

T T0

ρ ρ0

A A∗

a a0

M∗

μ

ν

2.25 2.26 2.27 2.28 2.29 2.30 2.31 2.32 2.33 2.34 2.35 2.36 2.37 2.38 2.39 2.40 2.41 2.42 2.43 2.44 2.45 2.46 2.47 2.48 2.49 2.50 2.51 2.52 2.53 2.54 2.55 2.56 2.57 2.58 2.59 2.60 2.61 2.62 2.63 2.64 2.65 2.66 2.67 2.68 2.69 2.70 2.71 2.72 2.73 2.74

0.0865 0.0851 0.0838 0.0825 0.0812 0.0800 0.0787 0.0775 0.0763 0.0751 0.0740 0.0728 0.0717 0.0706 0.0695 0.0684 0.0673 0.0663 0.0653 0.0643 0.0633 0.0623 0.0613 0.0604 0.0594 0.0585 0.0576 0.0567 0.0559 0.0550 0.0542 0.0533 0.0525 0.0517 0.0509 0.0501 0.0493 0.0486 0.0478 0.0471 0.0464 0.0457 0.0450 0.0443 0.0436 0.0430 0.0423 0.0417 0.0410 0.0404

0.4969 0.4947 0.4925 0.4903 0.4881 0.4859 0.4837 0.4816 0.4794 0.4773 0.4752 0.4731 0.4709 0.4688 0.4668 0.4647 0.4626 0.4606 0.4585 0.4565 0.4544 0.4524 0.4504 0.4484 0.4464 0.4444 0.4425 0.4405 0.4386 0.4366 0.4347 0.4328 0.4309 0.4289 0.4271 0.4252 0.4233 0.4214 0.4196 0.4177 0.4159 0.4141 0.4122 0.4104 0.4086 0.4068 0.4051 0.4033 0.4015 0.3998

0.1740 0.1721 0.1702 0.1683 0.1664 0.1646 0.1628 0.1609 0.1592 0.1574 0.1556 0.1539 0.1522 0.1505 0.1488 0.1472 0.1456 0.1439 0.1424 0.1408 0.1392 0.1377 0.1362 0.1346 0.1332 0.1317 0.1302 0.1288 0.1274 0.1260 0.1246 0.1232 0.1218 0.1205 0.1192 0.1179 0.1166 0.1153 0.1140 0.1128 0.1115 0.1103 0.1091 0.1079 0.1067 0.1056 0.1044 0.1033 0.1022 0.1010

2.0964 2.1153 2.1345 2.1538 2.1734 2.1931 2.2131 2.2333 2.2537 2.2744 2.2953 2.3164 2.3377 2.3593 2.3811 2.4031 2.4254 2.4479 2.4706 2.4936 2.5168 2.5403 2.5640 2.5880 2.6122 2.6367 2.6615 2.6864 2.7117 2.7372 2.7630 2.7891 2.8154 2.8420 2.8688 2.8960 2.9234 2.9511 2.9791 3.0073 3.0359 3.0647 3.0938 3.1233 3.1530 3.1830 3.2133 3.2439 3.2749 3.3061

0.7049 0.7033 0.7018 0.7002 0.6986 0.6971 0.6955 0.6940 0.6924 0.6909 0.6893 0.6878 0.6863 0.6847 0.6832 0.6817 0.6802 0.6786 0.6771 0.6756 0.6741 0.6726 0.6711 0.6696 0.6682 0.6667 0.6652 0.6637 0.6622 0.6608 0.6593 0.6578 0.6564 0.6549 0.6535 0.6521 0.6506 0.6492 0.6477 0.6463 0.6449 0.6435 0.6421 0.6406 0.6392 0.6378 0.6364 0.6350 0.6337 0.6323

1.7374 1.7412 1.7450 1.7488 1.7526 1.7563 1.7600 1.7637 1.7673 1.7709 1.7745 1.7781 1.7817 1.7852 1.7887 1.7922 1.7956 1.7991 1.8025 1.8059 1.8092 1.8126 1.8159 1.8192 1.8225 1.8257 1.8290 1.8322 1.8354 1.8386 1.8417 1.8448 1.8479 1.8510 1.8541 1.8571 1.8602 1.8632 1.8662 1.8691 1.8721 1.8750 1.8779 1.8808 1.8837 1.8865 1.8894 1.8922 1.8950 1.8978

26.3878 26.2621 26.1376 26.0144 25.8923 25.7715 25.6518 25.5332 25.4158 25.2995 25.1844 25.0702 24.9572 24.8452 24.7343 24.6243 24.5154 24.4075 24.3005 24.1946 24.0895 23.9854 23.8823 23.7800 23.6787 23.5782 23.4786 23.3799 23.2820 23.1850 23.0888 22.9934 22.8988 22.8051 22.7121 22.6199 22.5284 22.4378 22.3478 22.2586 22.1702 22.0824 21.9954 21.9091 21.8234 21.7385 21.6542 21.5706 21.4876 21.4053

33.0184 33.2730 33.5267 33.7796 34.0316 34.2828 34.5330 34.7825 35.0310 35.2787 35.5255 35.7714 36.0165 36.2607 36.5040 36.7465 36.9881 37.2288 37.4687 37.7077 37.9458 38.1831 38.4195 38.6550 38.8897 39.1235 39.3565 39.5886 39.8198 40.0502 40.2798 40.5084 40.7363 40.9633 41.1894 41.4147 41.6391 41.8627 42.0855 42.3074 42.5285 42.7487 42.9682 43.1867 43.4045 43.6214 43.8376 44.0528 44.2673 44.4810

Appendix B: Isentropic Table (γ = 1.4)

545

M

p p0

T T0

ρ ρ0

A A∗

a a0

M∗

μ

ν

2.75 2.76 2.77 2.78 2.79 2.80 2.81 2.82 2.83 2.84 2.85 2.86 2.87 2.88 2.89 2.90 2.91 2.92 2.93 2.94 2.95 2.96 2.97 2.98 2.99 3.00 3.01 3.02 3.03 3.04 3.05 3.06 3.07 3.08 3.09 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.20 3.21 3.22 3.23 3.24

0.0398 0.0392 0.0386 0.0380 0.0374 0.0368 0.0363 0.0357 0.0352 0.0347 0.0341 0.0336 0.0331 0.0326 0.0321 0.0317 0.0312 0.0307 0.0302 0.0298 0.0293 0.0289 0.0285 0.0281 0.0276 0.0272 0.0268 0.0264 0.0260 0.0256 0.0253 0.0249 0.0245 0.0242 0.0238 0.0234 0.0231 0.0228 0.0224 0.0221 0.0218 0.0215 0.0211 0.0208 0.0205 0.0202 0.0199 0.0196 0.0194 0.0191

0.3980 0.3963 0.3945 0.3928 0.3911 0.3894 0.3877 0.3860 0.3844 0.3827 0.3810 0.3794 0.3777 0.3761 0.3745 0.3729 0.3712 0.3696 0.3681 0.3665 0.3649 0.3633 0.3618 0.3602 0.3587 0.3571 0.3556 0.3541 0.3526 0.3511 0.3496 0.3481 0.3466 0.3452 0.3437 0.3422 0.3408 0.3393 0.3379 0.3365 0.3351 0.3337 0.3323 0.3309 0.3295 0.3281 0.3267 0.3253 0.3240 0.3226

0.0999 0.0989 0.0978 0.0967 0.0957 0.0946 0.0936 0.0926 0.0916 0.0906 0.0896 0.0886 0.0877 0.0867 0.0858 0.0849 0.084 0.0831 0.0822 0.0813 0.0804 0.0796 0.0787 0.0779 0.0770 0.0762 0.0754 0.0746 0.0738 0.0730 0.0723 0.0715 0.0707 0.0700 0.0692 0.0685 0.0678 0.0671 0.0664 0.0657 0.0650 0.0643 0.0636 0.0630 0.0623 0.0617 0.0610 0.0604 0.0597 0.0591

3.3377 3.3695 3.4017 3.4342 3.4670 3.5001 3.5336 3.5674 3.6015 3.6359 3.6707 3.7058 3.7413 3.7771 3.8133 3.8498 3.8866 3.9238 3.9614 3.9993 4.0376 4.0762 4.1153 4.1547 4.1944 4.2346 4.2751 4.3160 4.3573 4.3989 4.4410 4.4835 4.5263 4.5696 4.6132 4.6573 4.7018 4.7467 4.7920 4.8377 4.8838 4.9304 4.9774 5.0248 5.0727 5.1209 5.1697 5.2189 5.2685 5.3186

0.6309 0.6295 0.6281 0.6268 0.6254 0.6240 0.6227 0.6213 0.6200 0.6186 0.6173 0.6159 0.6146 0.6133 0.6119 0.6106 0.6093 0.6080 0.6067 0.6054 0.6041 0.6028 0.6015 0.6002 0.5989 0.5976 0.5963 0.5951 0.5938 0.5925 0.5913 0.5900 0.5887 0.5875 0.5862 0.5850 0.5838 0.5825 0.5813 0.5801 0.5788 0.5776 0.5764 0.5752 0.5740 0.5728 0.5716 0.5704 0.5692 0.5680

1.9005 1.9033 1.9060 1.9087 1.9114 1.9140 1.9167 1.9193 1.9219 1.9246 1.9271 1.9297 1.9323 1.9348 1.9373 1.9398 1.9423 1.9448 1.9472 1.9497 1.9521 1.9545 1.9569 1.9593 1.9616 1.9640 1.9663 1.9686 1.9709 1.9732 1.9755 1.9777 1.9800 1.9822 1.9844 1.9866 1.9888 1.9910 1.9931 1.9953 1.9974 1.9995 2.0016 2.0037 2.0058 2.0079 2.0099 2.0119 2.0140 2.0160

21.3237 21.2427 21.1623 21.0825 21.0034 20.9248 20.8469 20.7696 20.6928 20.6166 20.5410 20.4659 20.3915 20.3175 20.2441 20.1713 20.0990 20.0272 19.9559 19.8852 19.8149 19.7452 19.6760 19.6072 19.5390 19.4712 19.4039 19.3371 19.2708 19.2049 19.1395 19.0745 19.0100 18.9459 18.8823 18.8191 18.7563 18.6940 18.6320 18.5705 18.5094 18.4487 18.3884 18.3286 18.2691 18.2100 18.1513 18.0929 18.0350 17.9774

44.6938 44.9058 45.1170 45.3275 45.5371 45.7459 45.9539 46.1611 46.3675 46.5731 46.7779 46.9819 47.1852 47.3877 47.5894 47.7903 47.9904 48.1898 48.3884 48.5862 48.7833 48.9796 49.1752 49.3700 49.5640 49.7573 49.9499 50.1417 50.3327 50.5231 50.7127 50.9015 51.0897 51.2771 51.4637 51.6497 51.8349 52.0195 52.2033 52.3864 52.5688 52.7505 52.9315 53.1118 53.2914 53.4703 53.6485 53.8260 54.0029 54.1791

Appendix B: Isentropic Table (γ = 1.4)

546 M

p p0

T T0

ρ ρ0

A A∗

a a0

M∗

μ

ν

3.25 3.26 3.27 3.28 3.29 3.30 3.31 3.32 3.33 3.34 3.35 3.36 3.37 3.38 3.39 3.40 3.41 3.42 3.43 3.44 3.45 3.46 3.47 3.48 3.49 3.5 3.51 3.52 3.53 3.54 3.55 3.56 3.57 3.58 3.59 3.60 3.61 3.62 3.63 3.64 3.65 3.66 3.67 3.68 3.69 3.70 3.71 3.72 3.73 3.74

0.0188 0.0185 0.0183 0.0180 0.0177 0.0175 0.0172 0.0170 0.0167 0.0165 0.0163 0.0160 0.0158 0.0156 0.0153 0.0151 0.0149 0.0147 0.0145 0.0143 0.0141 0.0139 0.0137 0.0135 0.0133 0.0131 0.0129 0.0127 0.0126 0.0124 0.0122 0.0120 0.0119 0.0117 0.0115 0.0114 0.0112 0.0111 0.0109 0.0108 0.0106 0.0105 0.0103 0.0102 0.0100 0.0099 0.0098 0.0096 0.0095 0.0094

0.3213 0.3199 0.3186 0.3173 0.3160 0.3147 0.3134 0.3121 0.3108 0.3095 0.3082 0.3069 0.3057 0.3044 0.3032 0.3019 0.3007 0.2995 0.2982 0.2970 0.2958 0.2946 0.2934 0.2922 0.2910 0.2899 0.2887 0.2875 0.2864 0.2852 0.2841 0.2829 0.2818 0.2806 0.2795 0.2784 0.2773 0.2762 0.2751 0.2740 0.2729 0.2718 0.2707 0.2697 0.2686 0.2675 0.2665 0.2654 0.2644 0.2633

0.0585 0.0579 0.0573 0.0567 0.0561 0.0555 0.0550 0.0544 0.0538 0.0533 0.0527 0.0522 0.0517 0.0511 0.0506 0.0501 0.0496 0.0491 0.0486 0.0481 0.0476 0.0471 0.0466 0.0462 0.0457 0.0452 0.0448 0.0443 0.0439 0.0434 0.0430 0.0426 0.0421 0.0417 0.0413 0.0409 0.0405 0.0401 0.0397 0.0393 0.0389 0.0385 0.0381 0.0378 0.0374 0.0370 0.0367 0.0363 0.0359 0.0356

5.3691 5.4201 5.4715 5.5234 5.5758 5.6286 5.6820 5.7357 5.7900 5.8448 5.9000 5.9558 6.0120 6.0687 6.1260 6.1837 6.2419 6.3007 6.3600 6.4197 6.4801 6.5409 6.6023 6.6642 6.7266 6.7896 6.8531 6.9172 6.9819 7.0470 7.1128 7.1791 7.2460 7.3134 7.3815 7.4501 7.5193 7.5891 7.6595 7.7304 7.8020 7.8742 7.9470 8.0204 8.0944 8.1690 8.2443 8.3202 8.3967 8.4739

0.5668 0.5656 0.5645 0.5633 0.5621 0.5609 0.5598 0.5586 0.5575 0.5563 0.5552 0.554 0.5529 0.5517 0.5506 0.5495 0.5484 0.5472 0.5461 0.5450 0.5439 0.5428 0.5417 0.5406 0.5395 0.5384 0.5373 0.5362 0.5351 0.5340 0.5330 0.5319 0.5308 0.5298 0.5287 0.5276 0.5266 0.5255 0.5245 0.5234 0.5224 0.5213 0.5203 0.5193 0.5183 0.5172 0.5162 0.5152 0.5142 0.5132

2.0180 2.0200 2.0220 2.0239 2.0259 2.0278 2.0297 2.0317 2.0336 2.0355 2.0373 2.0392 2.0411 2.0429 2.0447 2.0466 2.0484 2.0502 2.0520 2.0537 2.0555 2.0573 2.059 2.0607 2.0625 2.0642 2.0659 2.0676 2.0693 2.0709 2.0726 2.0743 2.0759 2.0775 2.0792 2.0808 2.0824 2.0840 2.0856 2.0871 2.0887 2.0903 2.0918 2.0933 2.0949 2.0964 2.0979 2.0994 2.1009 2.1024

17.9202 17.8634 17.8069 17.7508 17.6951 17.6397 17.5847 17.5300 17.4757 17.4217 17.3680 17.3147 17.2617 17.2090 17.1567 17.1046 17.0529 17.0016 16.9505 16.8997 16.8493 16.7991 16.7493 16.6998 16.6505 16.6016 16.5529 16.5045 16.4564 16.4086 16.3611 16.3139 16.2669 16.2202 16.1738 16.1276 16.0817 16.0361 15.9908 15.9457 15.9008 15.8562 15.8119 15.7678 15.7240 15.6804 15.6370 15.5939 15.5510 15.5084

54.3546 54.5294 54.7035 54.8770 55.0498 55.2219 55.3934 55.5642 55.7344 55.9039 56.0728 56.2410 56.4086 56.5755 56.7418 56.9075 57.0725 57.2369 57.4007 57.5638 57.7263 57.8882 58.0495 58.2102 58.3703 58.5297 58.6886 58.8468 59.0045 59.1615 59.318 59.4738 59.6291 59.7838 59.9379 60.0914 60.2444 60.3967 60.5485 60.6997 60.8504 61.0005 61.1500 61.2990 61.4474 61.5952 61.7425 61.8893 62.0355 62.1811

Appendix B: Isentropic Table (γ = 1.4)

547

M

p p0

T T0

ρ ρ0

A A∗

a a0

M∗

μ

ν

3.75 3.76 3.77 3.78 3.79 3.80 3.81 3.82 3.83 3.84 3.85 3.86 3.87 3.88 3.89 3.90 3.91 3.92 3.93 3.94 3.95 3.96 3.97 3.98 3.99 4.00 4.01 4.02 4.03 4.04 4.05 4.06 4.07 4.08 4.09 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18 4.19 4.20 4.21 4.22 4.23

0.0092 0.0091 0.0090 0.0089 0.0087 0.0086 0.0085 0.0084 0.0083 0.0082 0.0081 0.0080 0.0078 0.0077 0.0076 0.0075 0.0074 0.0073 0.0072 0.0071 0.0070 0.0069 0.0069 0.0068 0.0067 0.0066 0.0065 0.0064 0.0063 0.0062 0.0062 0.0061 0.0060 0.0059 0.0058 0.0058 0.0057 0.0056 0.0055 0.0055 0.0054 0.0053 0.0053 0.0052 0.0051 0.0051 0.0050 0.0049 0.0049

0.2623 0.2613 0.2602 0.2592 0.2582 0.2572 0.2562 0.2552 0.2542 0.2532 0.2522 0.2513 0.2503 0.2493 0.2484 0.2474 0.2465 0.2455 0.2446 0.2436 0.2427 0.2418 0.2408 0.2399 0.2390 0.2381 0.2372 0.2363 0.2354 0.2345 0.2336 0.2327 0.2319 0.2310 0.2301 0.2293 0.2284 0.2275 0.2267 0.2258 0.2250 0.2242 0.2233 0.2225 0.2217 0.2208 0.2200 0.2192 0.2184

0.0352 0.0349 0.0345 0.0342 0.0339 0.0335 0.0332 0.0329 0.0326 0.0323 0.0320 0.0316 0.0313 0.0310 0.0307 0.0304 0.0302 0.0299 0.0296 0.0293 0.0290 0.0287 0.0285 0.0282 0.0279 0.0277 0.0274 0.0271 0.0269 0.0266 0.0264 0.0261 0.0259 0.0256 0.0254 0.0252 0.0249 0.0247 0.0245 0.0242 0.0240 0.0238 0.0236 0.0234 0.0231 0.0229 0.0227 0.0225 0.0223

8.5517 8.6302 8.7093 8.7891 8.8695 8.9506 9.0323 9.1148 9.1979 9.2816 9.3661 9.4513 9.5371 9.6237 9.7110 9.7989 9.8876 9.9770 10.0672 10.1580 10.2496 10.3419 10.4350 10.5288 10.6234 10.7187 10.8148 10.9117 11.0093 11.1077 11.2069 11.3068 11.4076 11.5091 11.6115 11.7146 11.8186 11.9234 12.0290 12.1354 12.2427 12.3508 12.4597 12.5695 12.6802 12.7917 12.9040 13.0172 13.1313

0.5121 0.5111 0.5101 0.5091 0.5081 0.5072 0.5062 0.5052 0.5042 0.5032 0.5022 0.5013 0.5003 0.4993 0.4984 0.4974 0.4964 0.4955 0.4945 0.4936 0.4926 0.4917 0.4908 0.4898 0.4889 0.4880 0.4870 0.4861 0.4852 0.4843 0.4833 0.4824 0.4815 0.4806 0.4797 0.4788 0.4779 0.4770 0.4761 0.4752 0.4743 0.4735 0.4726 0.4717 0.4708 0.4699 0.4691 0.4682 0.4673

2.1039 2.1053 2.1068 2.1082 2.1097 2.1111 2.1125 2.1140 2.1154 2.1168 2.1182 2.1195 2.1209 2.1223 2.1236 2.1250 2.1263 2.1277 2.1290 2.1303 2.1316 2.1329 2.1342 2.1355 2.1368 2.1381 2.1394 2.1406 2.1419 2.1431 2.1444 2.1456 2.1468 2.1480 2.1493 2.1505 2.1517 2.1529 2.1540 2.1552 2.1564 2.1576 2.1587 2.1599 2.1610 2.1622 2.1633 2.1644 2.1655

15.4660 15.4239 15.3819 15.3402 15.2988 15.2575 15.2165 15.1757 15.1352 15.0948 15.0547 15.0147 14.9750 14.9355 14.8963 14.8572 14.8183 14.7796 14.7412 14.7029 14.6649 14.6270 14.5893 14.5519 14.5146 14.4775 14.4406 14.4039 14.3674 14.3311 14.2950 14.2590 14.2232 14.1876 14.1522 14.1170 14.0819 14.0470 14.0123 13.9778 13.9434 13.9092 13.8752 13.8414 13.8077 13.7741 13.7408 13.7076 13.6745

62.3263 62.4708 62.6149 62.7584 62.9013 63.0438 63.1857 63.3271 63.4679 63.6082 63.7481 63.8874 64.0262 64.1644 64.3022 64.4395 64.5762 64.7125 64.8483 64.9835 65.1183 65.2526 65.3863 65.5196 65.6525 65.7848 65.9166 66.0480 66.1789 66.3093 66.4393 66.5687 66.6978 66.8263 66.9544 67.0820 67.2092 67.3359 67.4621 67.5879 67.7132 67.8381 67.9626 68.0866 68.2101 68.3333 68.4559 68.5782 68.7000

Appendix B: Isentropic Table (γ = 1.4)

548 M

p p0

T T0

ρ ρ0

A A∗

a a0

M∗

μ

ν

4.24 4.25 4.26 4.27 4.28 4.29 4.30 4.31 4.32 4.33 4.34 4.35 4.36 4.37 4.38 4.39 4.40 4.41 4.42 4.43 4.44 4.45 4.46 4.47 4.48 4.49 4.50 4.51 4.52 4.53 4.54 4.55 4.56 4.57 4.58 4.59 4.60 4.61 4.62 4.63 4.64 4.65 4.66 4.67 4.68 4.69 4.70 4.71

0.0048 0.0047 0.0047 0.0046 0.0046 0.0045 0.0044 0.0044 0.0043 0.0043 0.0042 0.0042 0.0041 0.0041 0.0040 0.0040 0.0039 0.0039 0.0038 0.0038 0.0037 0.0037 0.0036 0.0036 0.0035 0.0035 0.0035 0.0034 0.0034 0.0033 0.0033 0.0032 0.0032 0.0032 0.0031 0.0031 0.0031 0.0030 0.0030 0.0029 0.0029 0.0029 0.0028 0.0028 0.0028 0.0027 0.0027 0.0027

0.2176 0.2168 0.2160 0.2152 0.2144 0.2136 0.2129 0.2121 0.2113 0.2105 0.2098 0.2090 0.2082 0.2075 0.2067 0.2060 0.2053 0.2045 0.2038 0.2030 0.2023 0.2016 0.2009 0.2002 0.1994 0.1987 0.1980 0.1973 0.1966 0.1959 0.1952 0.1945 0.1938 0.1932 0.1925 0.1918 0.1911 0.1905 0.1898 0.1891 0.1885 0.1878 0.1872 0.1865 0.1859 0.1852 0.1846 0.1839

0.0221 0.0219 0.0217 0.0215 0.0213 0.0211 0.0209 0.0207 0.0205 0.0203 0.0202 0.0200 0.0198 0.0196 0.0194 0.0193 0.0191 0.0189 0.0187 0.0186 0.0184 0.0182 0.0181 0.0179 0.0178 0.0176 0.0174 0.0173 0.0171 0.0170 0.0168 0.0167 0.0165 0.0164 0.0163 0.0161 0.0160 0.0158 0.0157 0.0156 0.0154 0.0153 0.0152 0.0150 0.0149 0.0148 0.0146 0.0145

13.2463 13.3622 13.4789 13.5966 13.7151 13.8346 13.9550 14.0762 14.1984 14.3216 14.4456 14.5706 14.6966 14.8235 14.9513 15.0802 15.2100 15.3407 15.4724 15.6052 15.7389 15.8736 16.0093 16.1460 16.2838 16.4225 16.5623 16.7031 16.8450 16.9879 17.1318 17.2768 17.4229 17.5701 17.7183 17.8676 18.0180 18.1694 18.3220 18.4757 18.6305 18.7864 18.9435 19.1017 19.2610 19.4214 19.583 19.7458

0.4665 0.4656 0.4648 0.4639 0.4631 0.4622 0.4614 0.4605 0.4597 0.4588 0.458 0.4572 0.4563 0.4555 0.4547 0.4539 0.4530 0.4522 0.4514 0.4506 0.4498 0.4490 0.4482 0.4474 0.4466 0.4458 0.4450 0.4442 0.4434 0.4426 0.4418 0.4411 0.4403 0.4395 0.4387 0.4380 0.4372 0.4364 0.4357 0.4349 0.4341 0.4334 0.4326 0.4319 0.4311 0.4304 0.4296 0.4289

2.1667 2.1678 2.1689 2.1700 2.1711 2.1721 2.1732 2.1743 2.1754 2.1764 2.1775 2.1785 2.1796 2.1806 2.1816 2.1827 2.1837 2.1847 2.1857 2.1867 2.1877 2.1887 2.1897 2.1907 2.1917 2.1926 2.1936 2.1946 2.1955 2.1965 2.1974 2.1984 2.1993 2.2002 2.2012 2.2021 2.2030 2.2039 2.2048 2.2057 2.2066 2.2075 2.2084 2.2093 2.2102 2.2110 2.2119 2.2128

13.6417 13.6090 13.5764 13.5440 13.5117 13.4797 13.4477 13.4159 13.3843 13.3528 13.3215 13.2903 13.2592 13.2284 13.1976 13.1670 13.1365 13.1062 13.076 13.0460 13.0161 12.9863 12.9567 12.9272 12.8979 12.8687 12.8396 12.8106 12.7818 12.7531 12.7245 12.6961 12.6678 12.6396 12.6116 12.5836 12.5558 12.5281 12.5006 12.4732 12.4458 12.4186 12.3916 12.3646 12.3378 12.3111 12.2845 12.2580

68.8214 68.9423 69.0628 69.1829 69.3026 69.4219 69.5407 69.6591 69.7771 69.8947 70.0118 70.1286 70.2449 70.3608 70.4764 70.5915 70.7062 70.8206 70.9345 71.0480 71.1612 71.2739 71.3863 71.4982 71.6098 71.7210 71.8318 71.9423 72.0523 72.1620 72.2713 72.3802 72.4888 72.5969 72.7048 72.8122 72.9193 73.0260 73.1323 73.2383 73.3439 73.4492 73.5541 73.6587 73.7629 73.8668 73.9703 74.0734

Appendix B: Isentropic Table (γ = 1.4)

549

M

p p0

T T0

ρ ρ0

A A∗

a a0

M∗

μ

ν

4.72 4.73 4.74 4.75 4.76 4.77 4.78 4.79 4.80 4.81 4.82 4.83 4.84 4.85 4.86 4.87 4.88 4.89 4.90 4.91 4.92 4.93 4.94 4.95 4.96 4.97 4.98 4.99 5.00

0.0026 0.0026 0.0026 0.0025 0.0025 0.0025 0.0025 0.0024 0.0024 0.0024 0.0023 0.0023 0.0023 0.0023 0.0022 0.0022 0.0022 0.0022 0.0021 0.0021 0.0021 0.0021 0.0020 0.0020 0.0020 0.0020 0.0019 0.0019 0.0019

0.1833 0.1827 0.1820 0.1814 0.1808 0.1802 0.1795 0.1789 0.1783 0.1777 0.1771 0.1765 0.1759 0.1753 0.1747 0.1741 0.1735 0.1729 0.1724 0.1718 0.1712 0.1706 0.1700 0.1695 0.1689 0.1683 0.1678 0.1672 0.1667

0.0144 0.0143 0.0141 0.0140 0.0139 0.0138 0.0137 0.0135 0.0134 0.0133 0.0132 0.0131 0.0130 0.0129 0.0128 0.0126 0.0125 0.0124 0.0123 0.0122 0.0121 0.0120 0.0119 0.0118 0.0117 0.0116 0.0115 0.0114 0.0113

19.9098 20.0749 20.2411 20.4086 20.5773 20.7471 20.9182 21.0905 21.264 21.4387 21.6147 21.7918 21.9703 22.1500 22.3309 22.5131 22.6966 22.8814 23.0675 23.2548 23.4435 23.6334 23.8247 24.0173 24.2112 24.4065 24.6031 24.8011 25.0004

0.4281 0.4274 0.4267 0.4259 0.4252 0.4245 0.4237 0.4230 0.4223 0.4216 0.4208 0.4201 0.4194 0.4187 0.4180 0.4173 0.4166 0.4159 0.4152 0.4145 0.4138 0.4131 0.4124 0.4117 0.4110 0.4103 0.4096 0.4089 0.4082

2.2136 2.2145 2.2154 2.2162 2.2171 2.2179 2.2187 2.2196 2.2204 2.2212 2.2220 2.2228 2.2236 2.2245 2.2253 2.2261 2.2268 2.2276 2.2284 2.2292 2.2300 2.2308 2.2315 2.2323 2.2331 2.2338 2.2346 2.2353 2.2361

12.2316 12.2053 12.1792 12.1532 12.1272 12.1014 12.0757 12.0501 12.0247 11.9993 11.9740 11.9489 11.9238 11.8989 11.8740 11.8493 11.8247 11.8001 11.7757 11.7514 11.7271 11.7030 11.6790 11.6551 11.6312 11.6075 11.5839 11.5604 11.5369

74.1762 74.2787 74.3808 74.4826 74.584 74.6851 74.7859 74.8863 74.9864 75.0862 75.1856 75.2847 75.3835 75.4819 75.5801 75.6779 75.7753 75.8725 75.9693 76.0658 76.1620 76.2579 76.3535 76.4487 76.5437 76.6383 76.7327 76.8267 76.9204

Appendix C

Normal Shock Table (γ = 1.4)

M1

M2

p2 p1

ρ2 ρ1

T2 T1

a2 a1

p02 p01

1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.20 1.21 1.22 1.23 1.24 1.25 1.26 1.27 1.28

0.9901 0.9805 0.9712 0.9620 0.9531 0.9444 0.9360 0.9277 0.9196 0.9118 0.9041 0.8966 0.8892 0.8820 0.8750 0.8682 0.8615 0.8549 0.8485 0.8422 0.8360 0.8300 0.8241 0.8183 0.8126 0.8071 0.8016 0.7963

1.0234 1.0471 1.0710 1.0952 1.1196 1.1442 1.1690 1.1941 1.2194 1.2450 1.2708 1.2968 1.3230 1.3495 1.3762 1.4032 1.4304 1.4578 1.4854 1.5133 1.5414 1.5698 1.5984 1.6272 1.6562 1.6855 1.7150 1.7448

1.0167 1.0334 1.0502 1.0671 1.0840 1.1009 1.1179 1.1349 1.1520 1.1691 1.1862 1.2034 1.2206 1.2378 1.2550 1.2723 1.2896 1.3069 1.3243 1.3416 1.3590 1.3764 1.3938 1.4112 1.4286 1.4460 1.4634 1.4808

1.0066 1.0132 1.0198 1.0263 1.0328 1.0393 1.0458 1.0522 1.0586 1.0649 1.0713 1.0776 1.0840 1.0903 1.0966 1.1029 1.1092 1.1154 1.1217 1.1280 1.1343 1.1405 1.1468 1.1531 1.1594 1.1657 1.1720 1.1783

1.0033 1.0066 1.0099 1.0131 1.0163 1.0195 1.0226 1.0258 1.0289 1.0320 1.0350 1.0381 1.0411 1.0442 1.0472 1.0502 1.0532 1.0561 1.0591 1.0621 1.0650 1.0680 1.0709 1.0738 1.0767 1.0797 1.0826 1.0855

1.0000 1.0000 1.0000 0.9999 0.9999 0.9998 0.9996 0.9994 0.9992 0.9989 0.9986 0.9982 0.9978 0.9973 0.9967 0.9961 0.9953 0.9946 0.9937 0.9928 0.9918 0.9907 0.9896 0.9884 0.9871 0.9857 0.9842 0.9827

© Springer Nature Singapore Pte Ltd. 2022 M. Kaushik, Fundamentals of Gas Dynamics, https://doi.org/10.1007/978-981-16-9085-3

551

Appendix C: Normal Shock Table (γ = 1.4)

552 M1

M2

p2 p1

ρ2 ρ1

T2 T1

a2 a1

p02 p01

1.29 1.30 1.31 1.32 1.33 1.34 1.35 1.36 1.37 1.38 1.39 1.40 1.41 1.42 1.43 1.44 1.45 1.46 1.47 1.48 1.49 1.50 1.51 1.52 1.53 1.54 1.55 1.56 1.57 1.58 1.59 1.60 1.61 1.62 1.63 1.64 1.65 1.66 1.67 1.68 1.69 1.70 1.71 1.72 1.73 1.74 1.75 1.76

0.7911 0.7860 0.7809 0.7760 0.7712 0.7664 0.7618 0.7572 0.7527 0.7483 0.7440 0.7397 0.7355 0.7314 0.7274 0.7235 0.7196 0.7157 0.7120 0.7083 0.7047 0.7011 0.6976 0.6941 0.6907 0.6874 0.6841 0.6809 0.6777 0.6746 0.6715 0.6684 0.6655 0.6625 0.6596 0.6568 0.6540 0.6512 0.6485 0.6458 0.6431 0.6405 0.6380 0.6355 0.6330 0.6305 0.6281 0.6257

1.7748 1.8050 1.8354 1.8661 1.8970 1.9282 1.9596 1.9912 2.0230 2.0551 2.0874 2.1200 2.1528 2.1858 2.2190 2.2525 2.2862 2.3202 2.3544 2.3888 2.4234 2.4583 2.4934 2.5288 2.5644 2.6002 2.6362 2.6725 2.7090 2.7458 2.7828 2.8200 2.8574 2.8951 2.9330 2.9712 3.0096 3.0482 3.0870 3.1261 3.1654 3.2050 3.2448 3.2848 3.3250 3.3655 3.4062 3.4472

1.4983 1.5157 1.5331 1.5505 1.5680 1.5854 1.6028 1.6202 1.6376 1.6549 1.6723 1.6897 1.7070 1.7243 1.7416 1.7589 1.7761 1.7934 1.8106 1.8278 1.8449 1.8621 1.8792 1.8963 1.9133 1.9303 1.9473 1.9643 1.9812 1.9981 2.0149 2.0317 2.0485 2.0653 2.0820 2.0986 2.1152 2.1318 2.1484 2.1649 2.1813 2.1977 2.2141 2.2304 2.2467 2.2629 2.2791 2.2952

1.1846 1.1909 1.1972 1.2035 1.2099 1.2162 1.2226 1.2290 1.2354 1.2418 1.2482 1.2547 1.2612 1.2676 1.2741 1.2807 1.2872 1.2938 1.3003 1.3069 1.3136 1.3202 1.3269 1.3336 1.3403 1.3470 1.3538 1.3606 1.3674 1.3742 1.3811 1.3880 1.3949 1.4018 1.4088 1.4158 1.4228 1.4299 1.4369 1.4440 1.4512 1.4583 1.4655 1.4727 1.4800 1.4873 1.4946 1.5019

1.0884 1.0913 1.0942 1.0971 1.0999 1.1028 1.1057 1.1086 1.1115 1.1144 1.1172 1.1201 1.1230 1.1259 1.1288 1.1317 1.1346 1.1374 1.1403 1.1432 1.1461 1.1490 1.1519 1.1548 1.1577 1.1606 1.1635 1.1664 1.1694 1.1723 1.1752 1.1781 1.1811 1.1840 1.1869 1.1899 1.1928 1.1958 1.1987 1.2017 1.2046 1.2076 1.2106 1.2136 1.2165 1.2195 1.2225 1.2255

0.9811 0.9794 0.9776 0.9758 0.9738 0.9718 0.9697 0.9676 0.9653 0.9630 0.9607 0.9582 0.9557 0.9531 0.9504 0.9476 0.9448 0.9420 0.9390 0.9360 0.9329 0.9298 0.9266 0.9233 0.9200 0.9166 0.9132 0.9097 0.9062 0.9026 0.8989 0.8952 0.8915 0.8877 0.8838 0.8799 0.8760 0.8720 0.8680 0.8639 0.8599 0.8557 0.8516 0.8474 0.8431 0.8389 0.8346 0.8302

Appendix C: Normal Shock Table (γ = 1.4)

553

M1

M2

p2 p1

ρ2 ρ1

T2 T1

a2 a1

p02 p01

1.77 1.78 1.79 1.80 1.81 1.82 1.83 1.84 1.85 1.86 1.87 1.88 1.89 1.90 1.91 1.92 1.93 1.94 1.95 1.96 1.97 1.98 1.99 2.00 2.01 2.02 2.03 2.04 2.05 2.06 2.07 2.08 2.09 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 2.20 2.22 2.23 2.24

0.6234 0.6210 0.6188 0.6165 0.6143 0.6121 0.6099 0.6078 0.6057 0.6036 0.6016 0.5996 0.5976 0.5956 0.5937 0.5918 0.5899 0.5880 0.5862 0.5844 0.5826 0.5808 0.5791 0.5774 0.5757 0.5740 0.5723 0.5707 0.5691 0.5675 0.5659 0.5643 0.5628 0.5613 0.5598 0.5583 0.5568 0.5554 0.5540 0.5525 0.5511 0.5498 0.5484 0.5471 0.5444 0.5431 0.5418

3.4884 3.5298 3.5714 3.6133 3.6554 3.6978 3.7404 3.7832 3.8262 3.8695 3.9130 3.9568 4.0008 4.0450 4.0894 4.1341 4.1790 4.2242 4.2696 4.3152 4.3610 4.4071 4.4534 4.5000 4.5468 4.5938 4.6410 4.6885 4.7362 4.7842 4.8324 4.8808 4.9294 4.9783 5.0274 5.0768 5.1264 5.1762 5.2262 5.2765 5.3270 5.3778 5.4288 5.4800 5.5831 5.6350 5.6872

2.3113 2.3273 2.3433 2.3592 2.3751 2.3909 2.4067 2.4228 2.4381 2.4537 2.4693 2.4848 2.5003 2.5157 2.5310 2.5463 2.5616 2.5767 2.5919 2.6069 2.6220 2.6369 2.6518 2.6667 2.6815 2.6962 2.7108 2.7255 2.7400 2.7545 2.7689 2.7833 2.7976 2.8119 2.8261 2.8402 2.8543 2.8683 2.8823 2.8962 2.9100 2.9238 2.9376 2.9512 2.9784 2.9918 3.0053

1.5093 1.5167 1.5241 1.5316 1.5391 1.5466 1.5541 1.5617 1.5693 1.5770 1.5847 1.5924 1.6001 1.6079 1.6157 1.6236 1.6314 1.6394 1.6473 1.6553 1.6633 1.6713 1.6794 1.6875 1.6956 1.7038 1.7120 1.7203 1.7285 1.7369 1.7452 1.7536 1.7620 1.7704 1.7789 1.7875 1.7960 1.8046 1.8132 1.8219 1.8306 1.8393 1.8481 1.8569 1.8746 1.8835 1.8924

1.2285 1.2315 1.2346 1.2376 1.2406 1.2436 1.2467 1.2497 1.2527 1.2558 1.2588 1.2619 1.2650 1.2680 1.2711 1.2742 1.2773 1.2804 1.2835 1.2866 1.2897 1.2928 1.2959 1.2990 1.3022 1.3053 1.3084 1.3116 1.3147 1.3179 1.3211 1.3242 1.3274 1.3306 1.3338 1.3370 1.3402 1.3434 1.3466 1.3498 1.3530 1.3562 1.3594 1.3627 1.3691 1.3724 1.3756

0.8259 0.8215 0.8171 0.8127 0.8082 0.8038 0.7993 0.7948 0.7902 0.7857 0.7811 0.7765 0.7720 0.7674 0.7627 0.7581 0.7535 0.7488 0.7442 0.7395 0.7349 0.7302 0.7255 0.7209 0.7162 0.7115 0.7069 0.7022 0.6975 0.6928 0.6882 0.6835 0.6789 0.6742 0.6696 0.6649 0.6603 0.6557 0.6511 0.6464 0.6419 0.6373 0.6327 0.6281 0.6191 0.6145 0.6100

Appendix C: Normal Shock Table (γ = 1.4)

554 M1

M2

p2 p1

ρ2 ρ1

T2 T1

a2 a1

p02 p01

2.25 2.26 2.27 2.28 2.29 2.30 2.31 2.32 2.33 2.34 2.35 2.36 2.37 2.38 2.39 2.40 2.41 2.42 2.43 2.44 2.45 2.46 2.47 2.48 2.49 2.50 2.51 2.52 2.53 2.54 2.55 2.56 2.57 2.58 2.59 2.60 2.61 2.62 2.63 2.64 2.65 2.66 2.67 2.68 2.69 2.70

0.5406 0.5393 0.5381 0.5368 0.5356 0.5344 0.5332 0.5321 0.5309 0.5297 0.5286 0.5275 0.5264 0.5253 0.5242 0.5231 0.5221 0.5210 0.5200 0.5189 0.5179 0.5169 0.5159 0.5149 0.5140 0.5130 0.5120 0.5111 0.5102 0.5092 0.5083 0.5074 0.5065 0.5056 0.5047 0.5039 0.5030 0.5022 0.5013 0.5005 0.4996 0.4988 0.4980 0.4972 0.4964 0.4956

5.7396 5.7922 5.8450 5.8981 5.9514 6.0050 6.0588 6.1128 6.1670 6.2215 6.2762 6.3312 6.3864 6.4418 6.4974 6.5533 6.6094 6.6658 6.7224 6.7792 6.8362 6.8935 6.9510 7.0088 7.0668 7.1250 7.1834 7.2421 7.3010 7.3602 7.4196 7.4792 7.5390 7.5991 7.6594 7.7200 7.7808 7.8418 7.9030 7.9645 8.0262 8.0882 8.1504 8.2128 8.2754 8.3383

3.0186 3.0319 3.0452 3.0584 3.0715 3.0845 3.0975 3.1105 3.1234 3.1362 3.1490 3.1617 3.1743 3.1869 3.1994 3.2119 3.2243 3.2367 3.2489 3.2612 3.2733 3.2855 3.2975 3.3095 3.3215 3.3333 3.3452 3.3569 3.3686 3.3803 3.3919 3.4034 3.4149 3.4263 3.4377 3.4490 3.4602 3.4714 3.4826 3.4936 3.5047 3.5156 3.5266 3.5374 3.5482 3.5590

1.9014 1.9104 1.9194 1.9285 1.9376 1.9468 1.9560 1.9652 1.9745 1.9838 1.9931 2.0025 2.0119 2.0213 2.0308 2.0403 2.0499 2.0595 2.0691 2.0788 2.0885 2.0982 2.1080 2.1178 2.1276 2.1375 2.1474 2.1574 2.1674 2.1774 2.1875 2.1976 2.2077 2.2179 2.2281 2.2383 2.2486 2.2590 2.2693 2.2797 2.2902 2.3006 2.3111 2.3217 2.3323 2.3429

1.3789 1.3822 1.3854 1.3887 1.3920 1.3953 1.3986 1.4019 1.4052 1.4085 1.4118 1.4151 1.4184 1.4217 1.4251 1.4284 1.4317 1.4351 1.4384 1.4418 1.4451 1.4485 1.4519 1.4553 1.4586 1.4620 1.4654 1.4688 1.4722 1.4756 1.4790 1.4824 1.4858 1.4893 1.4927 1.4961 1.4995 1.5030 1.5064 1.5099 1.5133 1.5168 1.5202 1.5237 1.5272 1.5307

0.6055 0.6011 0.5966 0.5921 0.5877 0.5833 0.5789 0.5745 0.5702 0.5658 0.5615 0.5572 0.5529 0.5486 0.5444 0.5401 0.5359 0.5317 0.5276 0.5234 0.5193 0.5152 0.5111 0.5071 0.5030 0.4990 0.4950 0.4911 0.4871 0.4832 0.4793 0.4754 0.4715 0.4677 0.4639 0.4601 0.4564 0.4526 0.4489 0.4452 0.4416 0.4379 0.4343 0.4307 0.4271 0.4236

Appendix C: Normal Shock Table (γ = 1.4)

555

M1

M2

p2 p1

ρ2 ρ1

T2 T1

a2 a1

p02 p01

2.71 2.72 2.73 2.74 2.75 2.76 2.77 2.78 2.79 2.80 2.81 2.82 2.83 2.84 2.85 2.86 2.87 2.88 2.89 2.90 2.91 2.92 2.93 2.94 2.95 2.96 2.97 2.98 2.99 3.00 3.01 3.02 3.03 3.04 3.05 3.06 3.07 3.08 3.09 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17

0.4949 0.4941 0.4933 0.4926 0.4918 0.4911 0.4903 0.4896 0.4889 0.4882 0.4875 0.4868 0.4861 0.4854 0.4847 0.4840 0.4833 0.4827 0.4820 0.4814 0.4807 0.4801 0.4795 0.4788 0.4782 0.4776 0.4770 0.4764 0.4758 0.4752 0.4746 0.4740 0.4734 0.4729 0.4723 0.4717 0.4712 0.4706 0.4701 0.4695 0.4690 0.4685 0.4679 0.4674 0.4669 0.4664 0.4659

8.4014 8.4648 8.5284 8.5922 8.6562 8.7205 8.7850 8.8498 8.9148 8.9800 9.0454 9.1111 9.1770 9.2432 9.3096 9.3762 9.4430 9.5101 9.5774 9.6450 9.7128 9.7808 9.8490 9.9175 9.9862 10.0552 10.1244 10.1938 10.2634 10.3333 10.4034 10.4738 10.5444 10.6152 10.6862 10.7575 10.8290 10.9008 10.9728 11.0450 11.1174 11.1901 11.2630 11.3362 11.4096 11.4832 11.5570

3.5697 3.5803 3.5909 3.6015 3.6119 3.6224 3.6327 3.6431 3.6533 3.6635 3.6737 3.6838 3.6939 3.7039 3.7138 3.7238 3.7336 3.7434 3.7532 3.7629 3.7725 3.7821 3.7917 3.8012 3.8106 3.8200 3.8294 3.8387 3.8479 3.8571 3.8663 3.8754 3.8845 3.8935 3.9025 3.9114 3.9203 3.9291 3.9379 3.9466 3.9553 3.9639 3.9725 3.9811 3.9896 3.9981 4.0065

2.3536 2.3642 2.3750 2.3858 2.3966 2.4074 2.4183 2.4292 2.4402 2.4512 2.4622 2.4733 2.4844 2.4955 2.5067 2.5179 2.5292 2.5405 2.5518 2.5632 2.5746 2.5861 2.5975 2.6091 2.6206 2.6322 2.6439 2.6555 2.6673 2.6790 2.6908 2.7026 2.7145 2.7264 2.7383 2.7503 2.7623 2.7744 2.7865 2.7986 2.8108 2.8230 2.8352 2.8475 2.8598 2.8722 2.8846

1.5341 1.5376 1.5411 1.5446 1.5481 1.5516 1.5551 1.5586 1.5621 1.5656 1.5691 1.5727 1.5762 1.5797 1.5833 1.5868 1.5903 1.5939 1.5974 1.6010 1.6046 1.6081 1.6117 1.6153 1.6188 1.6224 1.6260 1.6296 1.6332 1.6368 1.6404 1.6440 1.6476 1.6512 1.6548 1.6584 1.6620 1.6656 1.6693 1.6729 1.6765 1.6802 1.6838 1.6875 1.6911 1.6947 1.6984

0.4201 0.4166 0.4131 0.4097 0.4062 0.4028 0.3994 0.3961 0.3928 0.3895 0.3862 0.3829 0.3797 0.3765 0.3733 0.3701 0.3670 0.3639 0.3608 0.3577 0.3547 0.3517 0.3487 0.3457 0.3428 0.3398 0.3369 0.3340 0.3312 0.3283 0.3255 0.3227 0.3200 0.3172 0.3145 0.3118 0.3091 0.3065 0.3038 0.3012 0.2986 0.2960 0.2935 0.2910 0.2885 0.2860 0.2835

Appendix C: Normal Shock Table (γ = 1.4)

556 M1

M2

p2 p1

ρ2 ρ1

T2 T1

a2 a1

p02 p01

3.18 3.19 3.20 3.21 3.22 3.23 3.24 3.25 3.26 3.27 3.28 3.29 3.30 3.31 3.32 3.33 3.34 3.35 3.36 3.37 3.38 3.39 3.40 3.41 3.42 3.43 3.44 3.45 3.46 3.47 3.48 3.49 3.50 3.51 3.52 3.53 3.54 3.55 3.56 3.57 3.58 3.59 3.60 3.61 3.62 3.63 3.64

0.4654 0.4648 0.4643 0.4639 0.4634 0.4629 0.4624 0.4619 0.4614 0.4610 0.4605 0.4600 0.4596 0.4591 0.4587 0.4582 0.4578 0.4573 0.4569 0.4565 0.4560 0.4556 0.4552 0.4548 0.4544 0.4540 0.4535 0.4531 0.4527 0.4523 0.4519 0.4515 0.4512 0.4508 0.4504 0.4500 0.4496 0.4492 0.4489 0.4485 0.4481 0.4478 0.4474 0.4471 0.4467 0.4463 0.4460

11.6311 11.7054 11.7800 11.8548 11.9298 12.0050 12.0805 12.1562 12.2322 12.3084 12.3848 12.4614 12.5383 12.6154 12.6928 12.7704 12.8482 12.9262 13.0045 13.0830 13.1618 13.2408 13.3200 13.3994 13.4791 13.5590 13.6392 13.7196 13.8002 13.8810 13.9621 14.0434 14.1250 14.2068 14.2888 14.3710 14.4535 14.5362 14.6192 14.7024 14.7858 14.8694 14.9533 15.0374 15.1218 15.2064 15.2912

4.0149 4.0232 4.0315 4.0397 4.0479 4.0561 4.0642 4.0723 4.0803 4.0883 4.0963 4.1042 4.1120 4.1198 4.1276 4.1354 4.1431 4.1507 4.1583 4.1659 4.1734 4.1809 4.1884 4.1958 4.2032 4.2105 4.2178 4.2251 4.2323 4.2395 4.2467 4.2538 4.2609 4.2679 4.2749 4.2819 4.2888 4.2957 4.3026 4.3094 4.3162 4.3229 4.3296 4.3363 4.3429 4.3496 4.3561

2.8970 2.9095 2.9220 2.9345 2.9471 2.9597 2.9724 2.9851 2.9979 3.0106 3.0234 3.0363 3.0492 3.0621 3.0751 3.0881 3.1011 3.1142 3.1273 3.1405 3.1537 3.1669 3.1802 3.1935 3.2069 3.2203 3.2337 3.2471 3.2607 3.2742 3.2878 3.3014 3.3150 3.3287 3.3425 3.3562 3.3701 3.3839 3.3978 3.4117 3.4257 3.4397 3.4537 3.4678 3.4819 3.4961 3.5103

1.7021 1.7057 1.7094 1.7130 1.7167 1.7204 1.7241 1.7277 1.7314 1.7351 1.7388 1.7425 1.7462 1.7499 1.7536 1.7513 1.7610 1.7647 1.7684 1.7721 1.7759 1.7796 1.7833 1.7870 1.7908 1.7945 1.7982 1.8020 1.8057 1.8095 1.8132 1.8170 1.8207 1.8245 1.8282 1.8320 1.8358 1.8395 1.8433 1.8471 1.8509 1.8546 1.8584 1.8622 1.8660 1.8698 1.8736

0.2811 0.2786 0.2762 0.2738 0.2715 0.2691 0.2668 0.2645 0.2622 0.2600 0.2577 0.2555 0.2533 0.2511 0.2489 0.2468 0.2446 0.2425 0.2404 0.2383 0.2363 0.2342 0.2322 0.2302 0.2282 0.2263 0.2243 0.2224 0.2205 0.2186 0.2167 0.2148 0.2129 0.2111 0.2093 0.2075 0.2057 0.2039 0.2022 0.2004 0.1987 0.1970 0.1953 0.1936 0.1920 0.1903 0.1887

Appendix C: Normal Shock Table (γ = 1.4)

557

M1

M2

p2 p1

ρ2 ρ1

T2 T1

a2 a1

p02 p01

3.65 3.66 3.67 3.68 3.69 3.70 3.71 3.72 3.73 3.74 3.75 3.76 3.77 3.78 3.79 3.80 3.81 3.82 3.83 3.84 3.85 3.86 3.87 3.88 3.89 3.90 3.91 3.92 3.93 3.94 3.95 3.96 3.97 3.98 3.99 4.00 4.01 4.02 4.03 4.04 4.05 4.06 4.07 4.08 4.09 4.10

0.4456 0.4453 0.4450 0.4446 0.4443 0.4439 0.4436 0.4433 0.4430 0.4426 0.4423 0.4420 0.4417 0.4414 0.4410 0.4407 0.4404 0.4401 0.4398 0.4395 0.4392 0.4389 0.4386 0.4383 0.4380 0.4377 0.4375 0.4372 0.4369 0.4366 0.4363 0.4360 0.4358 0.4355 0.4352 0.4350 0.4347 0.4344 0.4342 0.4339 0.4336 0.4334 0.4331 0.4329 0.4326 0.4324

15.3762 15.4615 15.5470 15.6328 15.7188 15.8050 15.8914 15.9781 16.0650 16.1522 16.2396 16.3272 16.4150 16.5031 16.5914 16.6800 16.7688 16.8578 16.9470 17.0365 17.1262 17.2162 17.3063 17.3968 17.4874 17.5783 17.6694 17.7608 17.8524 17.9442 18.0362 18.1285 18.2210 18.3138 18.4068 18.5000 18.5934 18.6871 18.7810 18.8752 18.9696 19.0642 19.1590 19.2541 19.3494 19.4450

4.3627 4.3692 4.3756 4.3821 4.3885 4.3949 4.4012 4.4075 4.4138 4.4200 4.4262 4.4324 4.4385 4.4447 4.4507 4.4568 4.4628 4.4688 4.4747 4.4807 4.4866 4.4924 4.4983 4.5041 4.5098 4.5156 4.5213 4.5270 4.5326 4.5383 4.5439 4.5494 4.5550 4.5605 4.5660 4.5714 4.5769 4.5823 4.5876 4.5930 4.5983 4.6036 4.6089 4.6141 4.6193 4.6245

3.5245 3.5388 3.5531 3.5674 3.5818 3.5962 3.6107 3.6252 3.6397 3.6549 3.6689 3.6836 3.6983 3.7130 3.7278 3.7426 3.7574 3.7723 3.7873 3.8022 3.8172 3.8323 3.8473 3.8625 3.8776 3.8928 3.9080 3.9233 3.9386 3.9540 3.9694 3.9848 4.0002 4.0157 4.0313 4.0469 4.0625 4.0781 4.0938 4.1096 4.1253 4.1412 4.1570 4.1729 4.1888 4.2048

1.8774 1.8812 1.8850 1.8888 1.8926 1.8964 1.9002 1.9040 1.9078 1.9116 1.9154 1.9193 1.9231 1.9269 1.9307 1.9346 1.9384 1.9422 1.9461 1.9499 1.9538 1.9576 1.9615 1.9653 1.9692 1.9730 1.9769 1.9807 1.9846 1.9885 1.9923 1.9962 2.0001 2.0039 2.0078 2.0117 2.0156 2.0194 2.0233 2.0272 2.0311 2.0350 2.0389 2.0428 2.0467 2.0506

0.1871 0.1855 0.1839 0.1823 0.1807 0.1792 0.1777 0.1761 0.1746 0.1731 0.1717 0.1702 0.1687 0.1673 0.1659 0.1645 0.1631 0.1617 0.1603 0.1589 0.1576 0.1563 0.1549 0.1536 0.1523 0.1510 0.1497 0.1485 0.1472 0.1460 0.1448 0.1435 0.1423 0.1411 0.1399 0.1388 0.1376 0.1364 0.1353 0.1342 0.1330 0.1319 0.1308 0.1297 0.1286 0.1276

Appendix C: Normal Shock Table (γ = 1.4)

558 M1

M2

p2 p1

ρ2 ρ1

T2 T1

a2 a1

p02 p01

4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18 4.19 4.20 4.21 4.22 4.23 4.24 4.25 4.26 4.27 4.28 4.29 4.30 4.31 4.32 4.33 4.34 4.35 4.36 4.37 4.38 4.39 4.40 4.41 4.42 4.43 4.44 4.45 4.46 4.47 4.48 4.49 4.50 4.51 4.52 4.53 4.54 4.55

0.4321 0.4319 0.4316 0.4314 0.4311 0.4309 0.4306 0.4304 0.4302 0.4299 0.4297 0.4295 0.4292 0.4290 0.4288 0.4286 0.4283 0.4281 0.4279 0.4277 0.4275 0.4272 0.4270 0.4268 0.4266 0.4264 0.4262 0.426 0.4258 0.4255 0.4253 0.4251 0.4249 0.4247 0.4245 0.4243 0.4241 0.4239 0.4237 0.4236 0.4234 0.4232 0.423 0.4228 0.4226

19.5408 19.6368 19.7331 19.8295 19.9262 20.0232 20.1204 20.2178 20.3155 20.4133 20.5115 20.6098 20.7084 20.8072 20.9063 21.0056 21.1051 21.2048 21.3048 21.4050 21.5055 21.6062 21.7071 21.8082 21.9096 22.0113 22.1131 22.2152 22.3175 22.4201 22.5229 22.6259 22.7291 22.8326 22.9363 23.0403 23.1445 23.2489 23.3535 23.4584 23.5635 23.6689 23.7745 23.8803 23.9864

4.6296 4.6348 4.6399 4.6450 4.6500 4.6550 4.6601 4.6650 4.6700 4.6749 4.6798 4.6847 4.6896 4.6944 4.6992 4.704 4.7087 4.7135 4.7182 4.7229 4.7275 4.7322 4.7368 4.7414 4.746 4.7505 4.755 4.7595 4.764 4.7685 4.7729 4.7773 4.7817 4.7861 4.7904 4.7948 4.7991 4.8034 4.8076 4.8119 4.8161 4.8203 4.8245 4.8287 4.8328

4.2208 4.2368 4.2529 4.2690 4.2852 4.3014 4.3176 4.3339 4.3502 4.3666 4.383 4.3994 4.4159 4.4324 4.4489 4.4655 4.4821 4.4988 4.5155 4.5322 4.5490 4.5658 4.5827 4.5996 4.6165 4.6335 4.6505 4.6675 4.6846 4.7017 4.7189 4.7361 4.7533 4.7706 4.7879 4.8053 4.8227 4.8401 4.8576 4.8751 4.8927 4.9102 4.9279 4.9455 4.9632

2.0545 2.0584 2.0623 2.0662 2.0701 2.0740 2.0779 2.0818 2.0857 2.0896 2.0936 2.0975 2.1014 2.1053 2.1092 2.1132 2.1171 2.1210 2.1250 2.1289 2.1328 2.1368 2.1407 2.1447 2.1486 2.1525 2.1565 2.1604 2.1644 2.1683 2.1723 2.1763 2.1802 2.1842 2.1881 2.1921 2.1961 2.2000 2.2040 2.2080 2.2119 2.2159 2.2199 2.2239 2.2278

0.1265 0.1254 0.1244 0.1234 0.1223 0.1213 0.1203 0.1193 0.1183 0.1173 0.1164 0.1154 0.1144 0.1135 0.1126 0.1116 0.1107 0.1098 0.1089 0.1080 0.1071 0.1062 0.1054 0.1045 0.1036 0.1028 0.1020 0.1011 0.1003 0.0995 0.0987 0.0979 0.0971 0.0963 0.0955 0.0947 0.094 0.0932 0.0924 0.0917 0.0910 0.0902 0.0895 0.0888 0.0881

Appendix C: Normal Shock Table (γ = 1.4)

559

M1

M2

p2 p1

ρ2 ρ1

T2 T1

a2 a1

p02 p01

4.56 4.57 4.58 4.59 4.60 4.61 4.62 4.63 4.64 4.65 4.66 4.67 4.68 4.69 4.70 4.71 4.72 4.73 4.74 4.75 4.76 4.77 4.78 4.79 4.80 4.81 4.82 4.83 4.84 4.85 4.86 4.87 4.88 4.89 4.9 4.91 4.92 4.93 4.94 4.95 4.96 4.97 4.98 4.99 5.00

0.4224 0.4222 0.422 0.4219 0.4217 0.4215 0.4213 0.4211 0.421 0.4208 0.4206 0.4204 0.4203 0.4201 0.4199 0.4197 0.4196 0.4194 0.4192 0.4191 0.4189 0.4187 0.4186 0.4184 0.4183 0.4181 0.4179 0.4178 0.4176 0.4175 0.4173 0.4172 0.4170 0.4169 0.4167 0.4165 0.4164 0.4162 0.4161 0.4160 0.4158 0.4157 0.4155 0.4154 0.4152

24.0926 24.1992 24.3059 24.4129 24.5201 24.6276 24.7353 24.8432 24.9513 25.0597 25.1683 25.2772 25.3863 25.4956 25.6051 25.7149 25.8249 25.9352 26.0457 26.1564 26.2674 26.3785 26.4900 26.6016 26.7135 26.8256 26.9380 27.0506 27.1634 27.2764 27.3897 27.5032 27.6170 27.7310 27.8452 27.9597 28.0743 28.1893 28.3044 28.4198 28.5354 28.6513 28.7674 28.8837 29.0002

4.8369 4.841 4.8451 4.8492 4.8532 4.8572 4.8612 4.8652 4.8692 4.8731 4.8771 4.881 4.8849 4.8887 4.8926 4.8964 4.9002 4.9040 4.9078 4.9116 4.9153 4.9190 4.9227 4.9264 4.9301 4.9338 4.9374 4.9410 4.9446 4.9482 4.9518 4.9553 4.9589 4.9624 4.9659 4.9694 4.9728 4.9763 4.9797 4.9831 4.9865 4.9899 4.9933 4.9967 5.0000

4.981 4.9988 5.0166 5.0344 5.0523 5.0703 5.0883 5.1063 5.1243 5.1424 5.1605 5.1787 5.1969 5.2152 5.2335 5.2518 5.2701 5.2885 5.3070 5.3255 5.3440 5.3625 5.3811 5.3998 5.4184 5.4372 5.4559 5.4747 5.4935 5.5124 5.5313 5.5502 5.5692 5.5882 5.6073 5.6264 5.6455 5.6647 5.6839 5.7032 5.7225 5.7418 5.7612 5.7806 5.8000

2.2318 2.2358 2.2398 2.2438 2.2477 2.2517 2.2557 2.2597 2.2637 2.2677 2.2717 2.2757 2.2797 2.2837 2.2877 2.2917 2.2957 2.2997 2.3037 2.3077 2.3117 2.3157 2.3197 2.3237 2.3278 2.3318 2.3358 2.3398 2.3438 2.3478 2.3519 2.3559 2.3599 2.3639 2.3680 2.3720 2.3760 2.3801 2.3841 2.3881 2.3922 2.3962 2.4002 2.4043 2.4083

0.0874 0.0867 0.0860 0.0853 0.0846 0.0839 0.0832 0.0826 0.0819 0.0813 0.0806 0.0800 0.0793 0.0787 0.0781 0.0775 0.0769 0.0762 0.0756 0.075 0.0745 0.0739 0.0733 0.0727 0.0721 0.0716 0.0710 0.0705 0.0699 0.0694 0.0688 0.0683 0.0677 0.0672 0.0667 0.0662 0.0657 0.0652 0.0647 0.0642 0.0637 0.0632 0.0627 0.0622 0.0617

Appendix D

Oblique Shock Chart (γ = 1.4)

© Springer Nature Singapore Pte Ltd. 2022 M. Kaushik, Fundamentals of Gas Dynamics, https://doi.org/10.1007/978-981-16-9085-3

561

562

Appendix D: Oblique Shock Chart (γ = 1.4)

The θ − β − M diagram. (Source From NACA Report 1135, Ames Research Staff, Equations, Tables, and Carts for Compressible Flow, 1953)

Appendix E

One-Dimensional Flow with Friction (γ = 1.4)

M

T T∗

p p∗

p0 p∗0

ρ ρ∗

F F∗

¯ max 4fL D

0.02000 0.04000 0.06000 0.08000 0.10000 0.12000 0.14000 0.16000 0.18000 0.20000 0.22000 0.24000 0.26000 0.28000 0.30000 0.32000 0.34000 0.36000 0.38000 0.40000 0.42000 0.44000 0.46000 0.48000 0.50000 0.52000 0.54000

1.19990 1.19962 1.19914 1.19847 1.19760 1.19655 1.19531 1.19389 1.19227 1.19048 1.18850 1.18633 1.18399 1.18147 1.17878 1.17592 1.17288 1.16968 1.16632 1.16279 1.15911 1.15527 1.15128 1.14714 1.14286 1.13843 1.13387

54.77006 27.38175 18.25085 13.68431 10.94351 9.11559 7.80932 6.82907 6.06618 5.45545 4.95537 4.53829 4.18505 3.88199 3.61906 3.38874 3.18529 3.00422 2.84200 2.69582 2.56338 2.44280 2.33256 2.23135 2.13809 2.05187 1.97192

28.94213 14.48149 9.66591 7.26161 5.82183 4.86432 4.18240 3.67274 3.27793 2.96352 2.70760 2.49556 2.31729 2.16555 2.03507 1.92185 1.82288 1.73578 1.65870 1.59014 1.52890 1.47401 1.42463 1.38010 1.33984 1.30339 1.27032

0.02191 0.04381 0.06570 0.08758 0.10944 0.13126 0.15306 0.17482 0.19654 0.21822 0.23984 0.26141 0.28291 0.30435 0.32572 0.34701 0.36822 0.38935 0.41039 0.43133 0.45218 0.47293 0.49357 0.51410 0.53452 0.55483 0.57501

22.83364 11.43462 7.64285 5.75288 4.62363 3.87473 3.34317 2.94743 2.64223 2.40040 2.20464 2.04344 1.90880 1.79503 1.69794 1.61440 1.54200 1.47888 1.42356 1.37487 1.33185 1.29371 1.25981 1.22962 1.20268 1.17860 1.15705

1778.44988 440.35221 193.03108 106.71822 66.92156 45.40796 32.51131 24.19783 18.54265 14.53327 11.59605 9.38648 7.68757 6.35721 5.29925 4.44674 3.75195 3.18012 2.70545 2.30849 1.97437 1.69152 1.45091 1.24534 1.06906 0.91742 0.78663

© Springer Nature Singapore Pte Ltd. 2022 M. Kaushik, Fundamentals of Gas Dynamics, https://doi.org/10.1007/978-981-16-9085-3

563

Appendix E: One-Dimensional Flow with Friction (γ = 1.4)

564 M

T T∗

p p∗

p0 p∗0

ρ ρ∗

F F∗

¯ max 4fL D

0.56000 0.58000 0.60000 0.62000 0.64000 0.66000 0.68000 0.70000 0.72000 0.74000 0.76000 0.78000 0.80000 0.82000 0.84000 0.86000 0.88000 0.90000 0.92000 0.94000 0.96000 0.98000 1.00000 1.02000 1.04000 1.06000 1.08000 1.10000 1.12000 1.14000 1.16000 1.18000 1.20000 1.22000 1.24000 1.26000 1.28000 1.30000 1.32000 1.34000 1.36000 1.38000 1.40000 1.42000 1.44000 1.46000

1.12918 1.12435 1.11940 1.11433 1.10914 1.10383 1.09842 1.09290 1.08727 1.08155 1.07573 1.06982 1.06383 1.05775 1.05160 1.04537 1.03907 1.03270 1.02627 1.01978 1.01324 1.00664 1.00000 0.99331 0.98658 0.97982 0.97302 0.96618 0.95932 0.95244 0.94554 0.93861 0.93168 0.92473 0.91777 0.91080 0.90383 0.89686 0.88989 0.88292 0.87596 0.86901 0.86207 0.85514 0.84822 0.84133

1.89755 1.82820 1.76336 1.70261 1.64556 1.59187 1.54126 1.49345 1.44823 1.40537 1.36470 1.32605 1.28928 1.25423 1.22080 1.18888 1.15835 1.12913 1.10114 1.07430 1.04854 1.02379 1.00000 0.97711 0.95507 0.93383 0.91335 0.89359 0.87451 0.85608 0.83826 0.82103 0.80436 0.78822 0.77258 0.75743 0.74274 0.72848 0.71465 0.70122 0.68818 0.67551 0.66320 0.65122 0.63958 0.62825

1.24029 1.21301 1.18820 1.16565 1.14515 1.12654 1.10965 1.09437 1.08057 1.06814 1.05700 1.04705 1.03823 1.03046 1.02370 1.01787 1.01294 1.00886 1.00560 1.00311 1.00136 1.00034 1.00000 1.00033 1.00131 1.00291 1.00512 1.00793 1.01131 1.01527 1.01978 1.02484 1.03044 1.03657 1.04323 1.05041 1.05810 1.06630 1.07502 1.08424 1.09396 1.10419 1.11493 1.12616 1.13790 1.15015

0.59507 0.61501 0.63481 0.65448 0.67402 0.69342 0.71268 0.73179 0.75076 0.76958 0.78825 0.80677 0.82514 0.84335 0.86140 0.87929 0.89703 0.91460 0.93201 0.94925 0.96633 0.98325 1.00000 1.01658 1.03300 1.04925 1.06533 1.08124 1.09699 1.11256 1.12797 1.14321 1.15828 1.17319 1.18792 1.20249 1.21690 1.23114 1.24521 1.25912 1.27286 1.28645 1.29987 1.31313 1.32623 1.33917

1.13777 1.12050 1.10504 1.09120 1.07883 1.06777 1.05792 1.04915 1.04137 1.03449 1.02844 1.02314 1.01853 1.01455 1.01115 1.00829 1.00591 1.00399 1.00248 1.00136 1.00059 1.00014 1.00000 1.00014 1.00053 1.00116 1.00200 1.00305 1.00429 1.00569 1.00726 1.00897 1.01081 1.01278 1.01486 1.01705 1.01933 1.02170 1.02414 1.02666 1.02925 1.03189 1.03459 1.03733 1.04012 1.04295

0.67357 0.57568 0.49082 0.41720 0.35330 0.29785 0.24978 0.20814 0.17215 0.14112 0.11447 0.09167 0.07229 0.05593 0.04226 0.03097 0.02179 0.01451 0.00891 0.00482 0.00206 0.00049 0.00000 0.00046 0.00177 0.00384 0.00658 0.00994 0.01382 0.01819 0.02298 0.02814 0.03364 0.03943 0.04547 0.05174 0.05820 0.06483 0.07161 0.07850 0.08550 0.09259 0.09974 0.10694 0.11419 0.12146

Appendix E: One-Dimensional Flow with Friction (γ = 1.4)

565

M

T T∗

p p∗

p0 p∗0

ρ ρ∗

F F∗

¯ max 4fL D

1.48000 1.50000 1.52000 1.54000 1.56000 1.58000 1.60000 1.62000 1.64000 1.66000 1.68000 1.70000 1.72000 1.74000 1.76000 1.78000 1.80000 1.82000 1.84000 1.86000 1.88000 1.90000 1.92000 1.94000 1.96000 1.98000 2.00000 2.02000 2.04000 2.06000 2.08000 2.10000 2.12000 2.14000 2.16000 2.18000 2.20000 2.22000 2.24000 2.26000 2.28000 2.30000 2.32000 2.34000 2.36000 2.38000 2.40000 2.42000

0.83445 0.82759 0.82075 0.81393 0.80715 0.80038 0.79365 0.78695 0.78027 0.77363 0.76703 0.76046 0.75392 0.74742 0.74096 0.73454 0.72816 0.72181 0.71551 0.70925 0.70304 0.69686 0.69073 0.68465 0.67861 0.67262 0.66667 0.66076 0.65491 0.64910 0.64334 0.63762 0.63195 0.62633 0.62076 0.61523 0.60976 0.60433 0.59895 0.59361 0.58833 0.58309 0.57790 0.57276 0.56767 0.56262 0.55762 0.55267

0.61722 0.60648 0.59602 0.58583 0.57591 0.56623 0.55679 0.54759 0.53862 0.52986 0.52131 0.51297 0.50482 0.49686 0.48909 0.48149 0.47407 0.46681 0.45972 0.45278 0.44600 0.43936 0.43287 0.42651 0.42029 0.41421 0.40825 0.40241 0.39670 0.39110 0.38562 0.38024 0.37498 0.36982 0.36476 0.35980 0.35494 0.35017 0.34550 0.34091 0.33641 0.33200 0.32767 0.32342 0.31925 0.31516 0.31114 0.30720

1.16290 1.17617 1.18994 1.20423 1.21904 1.23438 1.25024 1.26663 1.28355 1.30102 1.31904 1.33761 1.35674 1.37643 1.39670 1.41755 1.43898 1.46101 1.48365 1.50689 1.53076 1.55526 1.58039 1.60617 1.63261 1.65972 1.68750 1.71597 1.74514 1.77502 1.80561 1.83694 1.86902 1.90184 1.93544 1.96981 2.00497 2.04094 2.07773 2.11535 2.15381 2.19313 2.23332 2.27440 2.31638 2.35928 2.40310 2.44787

1.35195 1.36458 1.37705 1.38936 1.40152 1.41353 1.42539 1.43710 1.44866 1.46008 1.47135 1.48247 1.49345 1.50429 1.51499 1.52555 1.53598 1.54626 1.55642 1.56644 1.57633 1.58609 1.59572 1.60523 1.61460 1.62386 1.63299 1.64201 1.65090 1.65967 1.66833 1.67687 1.68530 1.69362 1.70183 1.70992 1.71791 1.72579 1.73357 1.74125 1.74882 1.75629 1.76366 1.77093 1.77811 1.78519 1.79218 1.79907

1.04581 1.04870 1.05162 1.05456 1.05752 1.06049 1.06348 1.06647 1.06948 1.07249 1.07550 1.07851 1.08152 1.08453 1.08753 1.09053 1.09351 1.09649 1.09946 1.10242 1.10536 1.10829 1.11120 1.11410 1.11698 1.11984 1.12268 1.12551 1.12831 1.13110 1.13387 1.13661 1.13933 1.14204 1.14471 1.14737 1.15001 1.15262 1.15521 1.15777 1.16032 1.16284 1.16533 1.16780 1.17025 1.17268 1.17508 1.17746

0.12875 0.13605 0.14335 0.15063 0.15790 0.16514 0.17236 0.17954 0.18667 0.19377 0.20081 0.20780 0.21474 0.22162 0.22844 0.23519 0.24189 0.24851 0.25507 0.26156 0.26798 0.27433 0.28061 0.28681 0.29295 0.29901 0.30500 0.31091 0.31676 0.32253 0.32822 0.33385 0.33940 0.34489 0.35030 0.35564 0.36091 0.36611 0.37124 0.37631 0.38130 0.38623 0.39109 0.39589 0.40062 0.40529 0.40989 0.41443

Appendix E: One-Dimensional Flow with Friction (γ = 1.4)

566 M

T T∗

p p∗

p0 p∗0

ρ ρ∗

F F∗

¯ max 4fL D

2.44000 2.46000 2.48000 2.50000 2.52000 2.54000 2.56000 2.58000 2.60000 2.62000 2.64000 2.66000 2.68000 2.70000 2.72000 2.74000 2.76000 2.78000 2.80000 2.82000 2.84000 2.86000 2.88000 2.90000 2.92000 2.94000 2.96000 2.98000 3.00000 3.02000 3.04000 3.06000 3.08000 3.10000 3.12000 3.14000 3.16000 3.18000 3.20000 3.22000 3.24000 3.26000 3.28000 3.30000 3.32000 3.34000 3.36000

0.54777 0.54291 0.53810 0.53333 0.52862 0.52394 0.51932 0.51474 0.51020 0.50571 0.50127 0.49687 0.49251 0.48820 0.48393 0.47971 0.47553 0.47139 0.46729 0.46323 0.45922 0.45525 0.45132 0.44743 0.44358 0.43977 0.43600 0.43226 0.42857 0.42492 0.42130 0.41772 0.41418 0.41068 0.40721 0.40378 0.40038 0.39702 0.39370 0.39041 0.38716 0.38394 0.38075 0.37760 0.37448 0.37139 0.36833

0.30332 0.29952 0.29579 0.29212 0.28852 0.28498 0.28150 0.27808 0.27473 0.27143 0.26818 0.26500 0.26186 0.25878 0.25575 0.25278 0.24985 0.24697 0.24414 0.24135 0.23861 0.23592 0.23326 0.23066 0.22809 0.22556 0.22307 0.22063 0.21822 0.21585 0.21351 0.21121 0.20895 0.20672 0.20453 0.20237 0.20024 0.19814 0.19608 0.19405 0.19204 0.19007 0.18812 0.18621 0.18432 0.18246 0.18063

2.49360 2.54031 2.58801 2.63672 2.68645 2.73723 2.78906 2.84197 2.89598 2.95109 3.00733 3.06472 3.12327 3.18301 3.24395 3.30611 3.36952 3.43418 3.50012 3.56737 3.63593 3.70584 3.77711 3.84977 3.92383 3.99932 4.07625 4.15466 4.23457 4.31599 4.39895 4.48347 4.56959 4.65731 4.74667 4.83769 4.93039 5.02481 5.12096 5.21887 5.31857 5.42008 5.52343 5.62865 5.73576 5.84479 5.95577

1.80587 1.81258 1.81921 1.82574 1.83219 1.83855 1.84483 1.85103 1.85714 1.86318 1.86913 1.87501 1.88081 1.88653 1.89218 1.89775 1.90325 1.90868 1.91404 1.91933 1.92455 1.92970 1.93479 1.93981 1.94477 1.94966 1.95449 1.95925 1.96396 1.96861 1.97319 1.97772 1.98219 1.98661 1.99097 1.99527 1.99952 2.00372 2.00786 2.01195 2.01599 2.01998 2.02392 2.02781 2.03165 2.03545 2.03920

1.17981 1.18214 1.18445 1.18673 1.18899 1.19123 1.19344 1.19563 1.19780 1.19995 1.20207 1.20417 1.20625 1.20830 1.21033 1.21235 1.21433 1.21630 1.21825 1.22017 1.22208 1.22396 1.22582 1.22766 1.22948 1.23128 1.23307 1.23483 1.23657 1.23829 1.23999 1.24168 1.24334 1.24499 1.24662 1.24823 1.24982 1.25139 1.25295 1.25449 1.25601 1.25752 1.25901 1.26048 1.26193 1.26337 1.26479

0.41891 0.42332 0.42768 0.43198 0.43621 0.44039 0.44451 0.44858 0.45259 0.45654 0.46044 0.46429 0.46808 0.47182 0.47551 0.47915 0.48273 0.48627 0.48976 0.49321 0.49660 0.49995 0.50326 0.50652 0.50973 0.51290 0.51603 0.51912 0.52216 0.52516 0.52813 0.53105 0.53393 0.53678 0.53958 0.54235 0.54509 0.54778 0.55044 0.55307 0.55566 0.55822 0.56074 0.56323 0.56569 0.56812 0.57051

Appendix E: One-Dimensional Flow with Friction (γ = 1.4)

567

M

T T∗

p p∗

p0 p∗0

ρ ρ∗

F F∗

¯ max 4fL D

3.38000 3.40000 3.42000 3.44000 3.46000 3.48000 3.50000 3.52000 3.54000 3.56000 3.58000 3.60000 3.62000 3.64000 3.66000 3.68000 3.70000 3.72000 3.74000 3.76000 3.78000 3.80000 3.82000 3.84000 3.86000 3.88000 3.90000 3.92000 3.94000 3.96000 3.98000 4.00000 4.02000 4.04000 4.06000 4.08000 4.10000 4.12000 4.14000 4.16000 4.18000 4.20000 4.22000 4.24000 4.26000 4.28000

0.36531 0.36232 0.35936 0.35643 0.35353 0.35066 0.34783 0.34502 0.34224 0.33949 0.33677 0.33408 0.33141 0.32877 0.32616 0.32358 0.32103 0.31850 0.31600 0.31352 0.31107 0.30864 0.30624 0.30387 0.30151 0.29919 0.29688 0.29460 0.29235 0.29011 0.28790 0.28571 0.28355 0.28140 0.27928 0.27718 0.27510 0.27304 0.27101 0.26899 0.26699 0.26502 0.26306 0.26112 0.25921 0.25731

0.17882 0.17704 0.17528 0.17355 0.17185 0.17016 0.16851 0.16687 0.16526 0.16367 0.16210 0.16055 0.15903 0.15752 0.15604 0.15458 0.15313 0.15171 0.15030 0.14892 0.14755 0.14620 0.14487 0.14355 0.14225 0.14097 0.13971 0.13846 0.13723 0.13602 0.13482 0.13363 0.13246 0.13131 0.13017 0.12904 0.12793 0.12683 0.12574 0.12467 0.12362 0.12257 0.12154 0.12052 0.11951 0.11852

6.06873 6.18370 6.30070 6.41976 6.54092 6.66419 6.78962 6.91723 7.04705 7.17912 7.31346 7.45011 7.58910 7.73045 7.87421 8.02040 8.16907 8.32023 8.47393 8.63020 8.78907 8.95059 9.11477 9.28167 9.45131 9.62373 9.79897 9.97707 10.15806 10.34197 10.52886 10.71875 10.91168 11.10770 11.30684 11.50915 11.71465 11.92340 12.13543 12.35079 12.56951 12.79164 13.01722 13.24629 13.47890 13.71509

2.04290 2.04656 2.05017 2.05374 2.05727 2.06075 2.06419 2.06759 2.07094 2.07426 2.07754 2.08077 2.08397 2.08713 2.09026 2.09334 2.09639 2.09941 2.10238 2.10533 2.10824 2.11111 2.11395 2.11676 2.11954 2.12228 2.12499 2.12767 2.13032 2.13294 2.13553 2.13809 2.14062 2.14312 2.14560 2.14804 2.15046 2.15285 2.15522 2.15756 2.15987 2.16215 2.16442 2.16665 2.16886 2.17105

1.26620 1.26759 1.26897 1.27033 1.27167 1.27300 1.27432 1.27562 1.27691 1.27818 1.27944 1.28068 1.28191 1.28313 1.28433 1.28552 1.28670 1.28787 1.28902 1.29016 1.29128 1.29240 1.29350 1.29459 1.29567 1.29674 1.29779 1.29883 1.29987 1.30089 1.30190 1.30290 1.30389 1.30487 1.30583 1.30679 1.30774 1.30868 1.30960 1.31052 1.31143 1.31233 1.31322 1.31410 1.31497 1.31583

0.57287 0.57521 0.57751 0.57978 0.58203 0.58424 0.58643 0.58859 0.59072 0.59282 0.59490 0.59695 0.59898 0.60098 0.60296 0.60491 0.60684 0.60874 0.61062 0.61247 0.61431 0.61612 0.61791 0.61968 0.62142 0.62315 0.62485 0.62653 0.62819 0.62984 0.63146 0.63306 0.63465 0.63622 0.63776 0.63929 0.64080 0.64230 0.64377 0.64523 0.64668 0.64810 0.64951 0.65090 0.65228 0.65364

Appendix E: One-Dimensional Flow with Friction (γ = 1.4)

568 M

T T∗

p p∗

p0 p∗0

ρ ρ∗

F F∗

¯ max 4fL D

4.30000 4.32000 4.34000 4.36000 4.38000 4.40000 4.42000 4.44000 4.46000 4.48000 4.50000 4.52000 4.54000 4.56000 4.58000 4.60000 4.62000 4.64000 4.66000 4.68000 4.70000 4.72000 4.74000 4.76000 4.78000 4.80000 4.82000 4.84000 4.86000 4.88000 4.90000 4.92000 4.94000 4.96000 4.98000 5.00000

0.25543 0.25357 0.25172 0.24990 0.24809 0.24631 0.24453 0.24278 0.24105 0.23933 0.23762 0.23594 0.23427 0.23262 0.23098 0.22936 0.22775 0.22616 0.22459 0.22303 0.22148 0.21995 0.21844 0.21694 0.21545 0.21398 0.21252 0.21108 0.20965 0.20823 0.20683 0.20543 0.20406 0.20269 0.20134 0.20000

0.11753 0.11656 0.11560 0.11466 0.11372 0.11279 0.11188 0.11097 0.11008 0.10920 0.10833 0.10746 0.10661 0.10577 0.10494 0.10411 0.10330 0.10249 0.10170 0.10091 0.10013 0.09936 0.09860 0.09785 0.09711 0.09637 0.09564 0.09492 0.09421 0.09351 0.09281 0.09212 0.09144 0.09077 0.09010 0.08944

13.95490 14.19838 14.44557 14.69652 14.95127 15.20987 15.47236 15.73879 16.00921 16.28366 16.56219 16.84486 17.13170 17.42277 17.71812 18.01779 18.32185 18.63032 18.94328 19.26076 19.58283 19.90953 20.24091 20.57703 20.91795 21.26371 21.61437 21.96999 22.33061 22.69631 23.06712 23.44311 23.82434 24.21086 24.60272 25.00000

2.17321 2.17535 2.17747 2.17956 2.18163 2.18368 2.18571 2.18771 2.18970 2.19166 2.19360 2.19552 2.19742 2.19930 2.20116 2.20300 2.20482 2.20662 2.20841 2.21017 2.21192 2.21365 2.21536 2.21705 2.21872 2.22038 2.22202 2.22365 2.22526 2.22685 2.22842 2.22998 2.23153 2.23306 2.23457 2.23607

1.31668 1.31752 1.31836 1.31919 1.32000 1.32081 1.32161 1.32241 1.32319 1.32397 1.32474 1.32550 1.32625 1.32700 1.32773 1.32846 1.32919 1.32990 1.33061 1.33131 1.33201 1.33269 1.33338 1.33405 1.33472 1.33538 1.33603 1.33668 1.33732 1.33796 1.33859 1.33921 1.33983 1.34044 1.34104 1.34164

0.65499 0.65632 0.65763 0.65893 0.66022 0.66149 0.66275 0.66399 0.66522 0.66643 0.66763 0.66882 0.67000 0.67116 0.67231 0.67345 0.67457 0.67569 0.67679 0.67788 0.67895 0.68002 0.68107 0.68211 0.68315 0.68417 0.68518 0.68618 0.68717 0.68814 0.68911 0.69007 0.69102 0.69196 0.69288 0.69380

Appendix F

One-Dimensional Flow with Heat Transfer (γ = 1.4)

M

T0 T∗0

T T∗

p p∗

p0 p∗0

ρ ρ∗

0.00000 0.02000 0.04000 0.06000 0.08000 0.10000 0.12000 0.14000 0.16000 0.18000 0.20000 0.22000 0.24000 0.26000 0.28000 0.30000 0.32000 0.34000 0.36000 0.38000 0.40000 0.42000 0.44000 0.46000 0.48000 0.50000 0.52000 0.54000 0.56000

0.00000 0.00192 0.00765 0.01712 0.03022 0.04678 0.06661 0.08947 0.11511 0.14324 0.17355 0.20574 0.23948 0.27446 0.31035 0.34686 0.38369 0.42056 0.45723 0.49346 0.52903 0.56376 0.59748 0.63007 0.66139 0.69136 0.71990 0.74695 0.77249

0.00000 0.00230 0.00917 0.02053 0.03621 0.05602 0.07970 0.10695 0.13743 0.17078 0.20661 0.24452 0.28411 0.32496 0.36667 0.40887 0.45119 0.49327 0.53482 0.57553 0.61515 0.65346 0.69025 0.72538 0.75871 0.79012 0.81955 0.84695 0.87227

2.40000 2.39866 2.39464 2.38796 2.37869 2.36686 2.35257 2.33590 2.31696 2.29586 2.27273 2.24770 2.22091 2.19250 2.16263 2.13144 2.09908 2.06569 2.03142 1.99641 1.96078 1.92468 1.88822 1.85151 1.81466 1.77778 1.74095 1.70425 1.66778

1.26788 1.26752 1.26646 1.26470 1.26226 1.25915 1.25539 1.25103 1.24608 1.24059 1.23460 1.22814 1.22126 1.21400 1.20642 1.19855 1.19045 1.18215 1.17371 1.16517 1.15658 1.14796 1.13936 1.13082 1.12238 1.11405 1.10588 1.09789 1.09011

0.00000 0.00096 0.00383 0.00860 0.01522 0.02367 0.03388 0.04578 0.05931 0.07439 0.09091 0.10879 0.12792 0.14821 0.16955 0.19183 0.21495 0.23879 0.26327 0.28828 0.31373 0.33951 0.36556 0.39178 0.41810 0.44444 0.47075 0.49696 0.52302

© Springer Nature Singapore Pte Ltd. 2022 M. Kaushik, Fundamentals of Gas Dynamics, https://doi.org/10.1007/978-981-16-9085-3

569

Appendix F: One-Dimensional Flow with Heat Transfer (γ = 1.4)

570 M

T0 T∗0

T T∗

p p∗

p0 p∗0

ρ ρ∗

0.58000 0.60000 0.62000 0.64000 0.66000 0.68000 0.70000 0.72000 0.74000 0.76000 0.78000 0.80000 0.82000 0.84000 0.86000 0.88000 0.90000 0.92000 0.94000 0.96000 0.98000 1.00000 1.02000 1.04000 1.06000 1.08000 1.10000 1.12000 1.14000 1.16000 1.18000 1.20000 1.22000 1.24000 1.26000 1.28000 1.30000 1.32000 1.34000 1.36000 1.38000 1.40000 1.42000 1.44000 1.46000 1.48000 1.50000

0.79648 0.81892 0.83983 0.85920 0.87708 0.89350 0.90850 0.92212 0.93442 0.94546 0.95528 0.96395 0.97152 0.97807 0.98363 0.98828 0.99207 0.99506 0.99729 0.99883 0.99971 1.00000 0.99973 0.99895 0.99769 0.99601 0.99392 0.99148 0.98871 0.98564 0.98230 0.97872 0.97492 0.97092 0.96675 0.96243 0.95798 0.95341 0.94873 0.94398 0.93914 0.93425 0.92931 0.92434 0.91933 0.91431 0.90928

0.89552 0.91670 0.93584 0.95298 0.96816 0.98144 0.99290 1.00260 1.01062 1.01706 1.02198 1.02548 1.02763 1.02853 1.02826 1.02689 1.02452 1.02120 1.01702 1.01205 1.00636 1.00000 0.99304 0.98554 0.97755 0.96913 0.96031 0.95115 0.94169 0.93196 0.92200 0.91185 0.90153 0.89108 0.88052 0.86988 0.85917 0.84843 0.83766 0.82689 0.81613 0.80539 0.79469 0.78405 0.77346 0.76294 0.75250

1.63159 1.59574 1.56031 1.52532 1.49083 1.45688 1.42349 1.39069 1.35851 1.32696 1.29606 1.26582 1.23625 1.20734 1.17911 1.15154 1.12465 1.09842 1.07285 1.04793 1.02365 1.00000 0.97698 0.95456 0.93275 0.91152 0.89087 0.87078 0.85123 0.83222 0.81374 0.79576 0.77827 0.76127 0.74473 0.72865 0.71301 0.69780 0.68301 0.66863 0.65464 0.64103 0.62779 0.61491 0.60237 0.59018 0.57831

1.08256 1.07525 1.06822 1.06147 1.05503 1.04890 1.04310 1.03764 1.03253 1.02777 1.02337 1.01934 1.01569 1.01241 1.00951 1.00699 1.00486 1.00311 1.00175 1.00078 1.00019 1.00000 1.00019 1.00078 1.00175 1.00311 1.00486 1.00699 1.00952 1.01243 1.01573 1.01942 1.02349 1.02795 1.03280 1.03803 1.04366 1.04968 1.05608 1.06288 1.07007 1.07765 1.08563 1.09401 1.10278 1.11196 1.12155

0.54887 0.57447 0.59978 0.62477 0.64941 0.67366 0.69751 0.72093 0.74392 0.76645 0.78853 0.81013 0.83125 0.85190 0.87207 0.89175 0.91097 0.92970 0.94797 0.96577 0.98311 1.00000 1.01645 1.03246 1.04804 1.06320 1.07795 1.09230 1.10626 1.11984 1.13305 1.14589 1.15838 1.17052 1.18233 1.19382 1.20499 1.21585 1.22642 1.23669 1.24669 1.25641 1.26587 1.27507 1.28402 1.29273 1.30120

Appendix F: One-Dimensional Flow with Heat Transfer (γ = 1.4)

571

M

T0 T∗0

T T∗

p p∗

p0 p∗0

ρ ρ∗

1.52000 1.54000 1.56000 1.58000 1.60000 1.62000 1.64000 1.66000 1.68000 1.70000 1.72000 1.74000 1.76000 1.78000 1.80000 1.82000 1.84000 1.86000 1.88000 1.90000 1.92000 1.94000 1.96000 1.98000 2.00000 2.02000 2.04000 2.06000 2.08000 2.10000 2.12000 2.14000 2.16000 2.18000 2.20000 2.22000 2.24000 2.26000 2.28000 2.30000 2.32000 2.34000 2.36000 2.38000 2.40000 2.42000

0.90424 0.89920 0.89418 0.88917 0.88419 0.87922 0.87429 0.86939 0.86453 0.85971 0.85493 0.85019 0.84551 0.84087 0.83628 0.83174 0.82726 0.82283 0.81845 0.81414 0.80987 0.80567 0.80152 0.79742 0.79339 0.78941 0.78549 0.78162 0.77782 0.77406 0.77037 0.76673 0.76314 0.75961 0.75613 0.75271 0.74934 0.74602 0.74276 0.73954 0.73638 0.73326 0.73020 0.72718 0.72421 0.72129

0.74215 0.73189 0.72173 0.71168 0.70174 0.69190 0.68219 0.67259 0.66312 0.65377 0.64455 0.63545 0.62649 0.61765 0.60894 0.60036 0.59191 0.58359 0.57540 0.56734 0.55941 0.55160 0.54392 0.53636 0.52893 0.52161 0.51442 0.50735 0.50040 0.49356 0.48684 0.48023 0.47373 0.46734 0.46106 0.45488 0.44882 0.44285 0.43698 0.43122 0.42555 0.41998 0.41451 0.40913 0.40384 0.39864

0.56676 0.55552 0.54458 0.53393 0.52356 0.51346 0.50363 0.49405 0.48472 0.47562 0.46677 0.45813 0.44972 0.44152 0.43353 0.42573 0.41813 0.41072 0.40349 0.39643 0.38955 0.38283 0.37628 0.36988 0.36364 0.35754 0.35158 0.34577 0.34009 0.33454 0.32912 0.32382 0.31865 0.31359 0.30864 0.30381 0.29908 0.29446 0.28993 0.28551 0.28118 0.27695 0.27281 0.26875 0.26478 0.26090

1.13153 1.14193 1.15274 1.16397 1.17561 1.18768 1.20017 1.21309 1.22644 1.24024 1.25447 1.26915 1.28428 1.29987 1.31592 1.33244 1.34943 1.36690 1.38486 1.40330 1.42224 1.44168 1.46164 1.48210 1.50310 1.52462 1.54668 1.56928 1.59244 1.61616 1.64045 1.66531 1.69076 1.71680 1.74345 1.77070 1.79858 1.82708 1.85623 1.88602 1.91647 1.94759 1.97939 2.01187 2.04505 2.07895

1.30945 1.31748 1.32530 1.33291 1.34031 1.34753 1.35455 1.36140 1.36806 1.37455 1.38088 1.38705 1.39306 1.39891 1.40462 1.41019 1.41562 1.42092 1.42608 1.43112 1.43604 1.44083 1.44551 1.45008 1.45455 1.45890 1.46315 1.46731 1.47136 1.47533 1.47920 1.48298 1.48668 1.49029 1.49383 1.49728 1.50066 1.50396 1.50719 1.51035 1.51344 1.51646 1.51942 1.52232 1.52515 1.52793

Appendix F: One-Dimensional Flow with Heat Transfer (γ = 1.4)

572 M

T0 T∗0

T T∗

p p∗

p0 p∗0

ρ ρ∗

2.44000 2.46000 2.48000 2.50000 2.52000 2.54000 2.56000 2.58000 2.60000 2.62000 2.64000 2.66000 2.68000 2.70000 2.72000 2.74000 2.76000 2.78000 2.80000 2.82000 2.84000 2.86000 2.88000 2.90000 2.92000 2.94000 2.96000 2.98000 3.00000 3.02000 3.04000 3.06000 3.08000 3.10000 3.12000 3.14000 3.16000 3.18000 3.20000 3.22000 3.24000 3.26000 3.28000 3.30000 3.32000 3.34000 3.36000 3.38000

0.71842 0.71558 0.71280 0.71006 0.70736 0.70471 0.70210 0.69952 0.69700 0.69451 0.69206 0.68964 0.68727 0.68494 0.68264 0.68037 0.67815 0.67595 0.67380 0.67167 0.66958 0.66752 0.66550 0.66350 0.66154 0.65960 0.65770 0.65583 0.65398 0.65216 0.65037 0.64861 0.64687 0.64516 0.64348 0.64182 0.64018 0.63857 0.63699 0.63543 0.63389 0.63237 0.63088 0.62940 0.62795 0.62652 0.62512 0.62373

0.39352 0.38850 0.38356 0.37870 0.37392 0.36923 0.36461 0.36007 0.35561 0.35122 0.34691 0.34266 0.33849 0.33439 0.33035 0.32638 0.32248 0.31864 0.31486 0.31114 0.30749 0.30389 0.30035 0.29687 0.29344 0.29007 0.28675 0.28349 0.28028 0.27711 0.27400 0.27094 0.26792 0.26495 0.26203 0.25915 0.25632 0.25353 0.25078 0.24808 0.24541 0.24279 0.24021 0.23766 0.23515 0.23268 0.23025 0.22785

0.25710 0.25337 0.24973 0.24615 0.24266 0.23923 0.23587 0.23258 0.22936 0.22620 0.22310 0.22007 0.21709 0.21417 0.21131 0.20850 0.20575 0.20305 0.20040 0.19780 0.19525 0.19275 0.19029 0.18788 0.18551 0.18319 0.18091 0.17867 0.17647 0.17431 0.17219 0.17010 0.16806 0.16604 0.16407 0.16212 0.16022 0.15834 0.15649 0.15468 0.15290 0.15115 0.14942 0.14773 0.14606 0.14442 0.14281 0.14122

2.11356 2.14891 2.18499 2.22183 2.25944 2.29782 2.33699 2.37696 2.41774 2.45935 2.50179 2.54509 2.58925 2.63429 2.68021 2.72704 2.77478 2.82346 2.87308 2.92366 2.97521 3.02775 3.08129 3.13585 3.19145 3.24809 3.30579 3.36457 3.42445 3.48544 3.54756 3.61082 3.67524 3.74084 3.80764 3.87565 3.94488 4.01537 4.08712 4.16015 4.23449 4.31014 4.38714 4.46549 4.54522 4.62635 4.70889 4.79287

1.53065 1.53331 1.53591 1.53846 1.54096 1.54341 1.54581 1.54816 1.55046 1.55272 1.55493 1.55710 1.55922 1.56131 1.56335 1.56536 1.56732 1.56925 1.57114 1.57300 1.57482 1.57661 1.57836 1.58008 1.58178 1.58343 1.58506 1.58666 1.58824 1.58978 1.59129 1.59278 1.59425 1.59568 1.59709 1.59848 1.59985 1.60119 1.60250 1.60380 1.60507 1.60632 1.60755 1.60877 1.60996 1.61113 1.61228 1.61341

Appendix F: One-Dimensional Flow with Heat Transfer (γ = 1.4)

573

M

T0 T∗0

T T∗

p p∗

p0 p∗0

ρ ρ∗

3.40000 3.42000 3.44000 3.46000 3.48000 3.50000 3.52000 3.54000 3.56000 3.58000 3.60000 3.62000 3.64000 3.66000 3.68000 3.70000 3.72000 3.74000 3.76000 3.78000 3.80000 3.82000 3.84000 3.86000 3.88000 3.90000 3.92000 3.94000 3.96000 3.98000 4.00000 4.02000 4.04000 4.06000 4.08000 4.10000 4.12000 4.14000 4.16000 4.18000 4.20000 4.22000 4.24000 4.26000 4.28000 4.30000 4.32000 4.34000

0.62236 0.62101 0.61968 0.61837 0.61708 0.61580 0.61455 0.61331 0.61209 0.61089 0.60970 0.60853 0.60738 0.60624 0.60512 0.60401 0.60292 0.60184 0.60078 0.59973 0.59870 0.59768 0.59667 0.59568 0.59470 0.59373 0.59278 0.59184 0.59091 0.58999 0.58909 0.58819 0.58731 0.58644 0.58558 0.58473 0.58390 0.58307 0.58225 0.58145 0.58065 0.57987 0.57909 0.57832 0.57757 0.57682 0.57608 0.57535

0.22549 0.22317 0.22087 0.21861 0.21639 0.21419 0.21203 0.20990 0.20780 0.20573 0.20369 0.20167 0.19969 0.19773 0.19581 0.19390 0.19203 0.19018 0.18836 0.18656 0.18478 0.18303 0.18131 0.17961 0.17793 0.17627 0.17463 0.17302 0.17143 0.16986 0.16831 0.16678 0.16527 0.16378 0.16231 0.16086 0.15943 0.15802 0.15662 0.15524 0.15388 0.15254 0.15121 0.14990 0.14861 0.14734 0.14607 0.14483

0.13966 0.13813 0.13662 0.13513 0.13367 0.13223 0.13081 0.12942 0.12805 0.12670 0.12537 0.12406 0.12277 0.12150 0.12024 0.11901 0.11780 0.11660 0.11543 0.11427 0.11312 0.11200 0.11089 0.10979 0.10871 0.10765 0.10661 0.10557 0.10456 0.10355 0.10256 0.10159 0.10063 0.09968 0.09875 0.09782 0.09691 0.09602 0.09513 0.09426 0.09340 0.09255 0.09171 0.09089 0.09007 0.08927 0.08847 0.08769

4.87830 4.96521 5.05362 5.14355 5.23501 5.32804 5.42264 5.51885 5.61668 5.71615 5.81730 5.92013 6.02468 6.13097 6.23902 6.34884 6.46048 6.57394 6.68926 6.80646 6.92557 7.04660 7.16958 7.29454 7.42151 7.55050 7.68156 7.81469 7.94993 8.08731 8.22685 8.36858 8.51252 8.65872 8.80718 8.95794 9.11104 9.26649 9.42433 9.58459 9.74729 9.91247 10.08015 10.25037 10.42316 10.59854 10.77656 10.95723

1.61453 1.61562 1.61670 1.61776 1.61881 1.61983 1.62085 1.62184 1.62282 1.62379 1.62474 1.62567 1.62660 1.62750 1.62840 1.62928 1.63014 1.63100 1.63184 1.63267 1.63348 1.63429 1.63508 1.63586 1.63663 1.63739 1.63814 1.63888 1.63960 1.64032 1.64103 1.64172 1.64241 1.64309 1.64375 1.64441 1.64506 1.64570 1.64633 1.64696 1.64757 1.64818 1.64878 1.64937 1.64995 1.65052 1.65109 1.65165

Appendix F: One-Dimensional Flow with Heat Transfer (γ = 1.4)

574 M

T0 T∗0

T T∗

p p∗

p0 p∗0

ρ ρ∗

4.36000 4.38000 4.40000 4.42000 4.44000 4.46000 4.48000 4.50000 4.52000 4.54000 4.56000 4.58000 4.60000 4.62000 4.64000 4.66000 4.68000 4.70000 4.72000 4.74000 4.76000 4.78000 4.80000 4.82000 4.84000 4.86000 4.88000 4.90000 4.92000 4.94000 4.96000 4.98000 5.00000

0.57463 0.57392 0.57322 0.57252 0.57183 0.57116 0.57049 0.56982 0.56917 0.56852 0.56789 0.56726 0.56663 0.56602 0.56541 0.56480 0.56421 0.56362 0.56304 0.56246 0.56190 0.56133 0.56078 0.56023 0.55969 0.55915 0.55862 0.55809 0.55758 0.55706 0.55655 0.55605 0.55556

0.14360 0.14239 0.14119 0.14000 0.13883 0.13767 0.13653 0.13540 0.13429 0.13319 0.13210 0.13102 0.12996 0.12891 0.12787 0.12685 0.12583 0.12483 0.12384 0.12286 0.12190 0.12094 0.12000 0.11906 0.11814 0.11722 0.11632 0.11543 0.11455 0.11367 0.11281 0.11196 0.11111

0.08691 0.08615 0.08540 0.08465 0.08392 0.08319 0.08248 0.08177 0.08107 0.08039 0.07970 0.07903 0.07837 0.07771 0.07707 0.07643 0.07580 0.07517 0.07456 0.07395 0.07335 0.07275 0.07217 0.07159 0.07101 0.07045 0.06989 0.06934 0.06879 0.06825 0.06772 0.06719 0.06667

11.14060 11.32669 11.51554 11.70717 11.90163 12.09894 12.29914 12.50226 12.70834 12.91740 13.12949 13.34464 13.56288 13.78425 14.00879 14.23653 14.46750 14.70174 14.93930 15.18020 15.42449 15.67220 15.92337 16.17803 16.43624 16.69801 16.96341 17.23245 17.50519 17.78167 18.06192 18.34598 18.63390

1.65220 1.65275 1.65329 1.65382 1.65434 1.65486 1.65537 1.65588 1.65638 1.65687 1.65735 1.65783 1.65831 1.65878 1.65924 1.65969 1.66014 1.66059 1.66103 1.66146 1.66189 1.66232 1.66274 1.66315 1.66356 1.66397 1.66436 1.66476 1.66515 1.66554 1.66592 1.66629 1.66667

Appendix G

Letter of Admittance

A letter of admittance to the author’s book Essentials of Aircraft Armaments, published by Springer Nature, from the Honorable Defense Minister of India, Late Mr. Manohar Parrikar (2016).

© Springer Nature Singapore Pte Ltd. 2022 M. Kaushik, Fundamentals of Gas Dynamics, https://doi.org/10.1007/978-981-16-9085-3

575

References

1. Courant, R., Fredrichs, K.O.: Supersonic Flow and Shock Waves. Interscience Publishers, New York (1948) 2. Ferri, A.: Elements of Aerodynamics of Supersonic Flows. Macmillan, New York (1949) 3. Bonney, E.A.: Engineering Supersonic Aerodynamics. McGraw-Hill, New York (1950) 4. Miles, E.R.C.: Supersonic Aerodynamics. Dover Publications, New York (1950) 5. Shapiro, A.H.: Dynamics and Thermodynamics of Compressible Fluid Flow, 2 Vols. Ronald Press, New York (1953) 6. Cambel, A.B., Jennings, B.H.: Gas Dynamics. Dover Publications, New York (1958) 7. Hayes, W.D., Probstein, R.F.: Hypersonic Flow Theory. Academic Press, New York (1959) 8. Shames, H.: Mechanics of Fluids. McGraw-Hill, New York (1962) 9. Streeter, V.L.: Fluid Mechanics. McGraw-Hill, New York (1962) 10. Liepmann, H.W., Roshko, A.: Elements of Gas Dynamics. Wiley, New York (1963) 11. Vincenti, W.G., Kruger, C.H., Jr.: Introduction to Physical Gas Dynamics. Krieger Publishing, Florida (1965) 12. Benedict, R.P., Steltz, W.G.: Handbook of Generalized Gas Dynamics. Plenum Press, New York (1966) 13. Becker, E.: Gas Dynamics. Academic Press, New York (1968) 14. Curle, N., Davies, H.J.: Modern Fluid Dynamics. Van Nostrand Reinhold, London (1971) 15. Thompson, P.A.: Compressible Fluid Dynamics. McGraw-Hill, New York (1972) 16. Imrie, B.W.: Compressible Fluid Flow. Butterworths, London (1973) 17. Bird, G.A.: Molecular Gas Dynamics. Clarendon Press, London (1976) 18. Zucrow, M.J., Hoffman, J.D.: Gas Dynamics, 2 Vols. Wiley, New York (1976) 19. Daneshyar, H.: One-Dimensional Compressible Flow. Pergamon Press, New York (1976) 20. Anderson, J.D., Jr.: Modern Compressible Flow. McGraw-Hill, New York (1982) 21. Anderson, J.D., Jr.: Hypersonic and High Temperature Gas Dynamics. McGraw-Hill, New York (1989) 22. Kaushik, M.: Innovative Passive Control Techniques for Supersonic Jet Mixing. Lambert Academic Publishing, Germany (2012) 23. Kaushik, M.: Essentials of Aircraft Armaments. Springer Nature, Singapore (2016) 24. Kaushik, M.: Theoretical and Experimental Aerodynamics. Springer Nature, Singapore (2019)

© Springer Nature Singapore Pte Ltd. 2022 M. Kaushik, Fundamentals of Gas Dynamics, https://doi.org/10.1007/978-981-16-9085-3

577

Index

A Activity envelope, 123 Air-breathing propulsion systems, 333 Amplitude, 445 Area-Mach number relationship, 304 Area-velocity relationship, 299 Average or mean translational kinetic energy, 20 Average speed, 7 Avogadro’s number, 13, 18

B Back pressure, 302, 304, 310, 311 Barotropic fluid, 416 Bernoulli’s constant, 428 Bernoulli’s equation, 43, 74, 423 Bernoulli’s number, 423 Bow-shock, 210, 422 Bulk modulus of elasticity, 37 isentropic bulk modulus, 37 isothermal bulk modulus, 37

C Centrifugal force, 420 Characteristic length scale, 30 Characteristic line, 126, 452 Characteristic speeds, 158 Circulation, 406 Combustor, 337 Compressibility, 37 Compressibility factor, 114 Compressible, 1 Compressible Bernoulli’s equation, 184 Concave corner, 217, 251 Conical body, 256 © Springer Nature Singapore Pte Ltd. 2022 M. Kaushik, Fundamentals of Gas Dynamics, https://doi.org/10.1007/978-981-16-9085-3

Conservation of energy, 47, 77 conservation form, 83 control mass form, 79 integral form, 82 non-conservation form, 85 Conservation of momentum, 47, 67 conservation form, 70 integral form, 67 non-conservation form, 72 Continuity equation, 47, 61, 419 conservation form, 66 integral form, 63 non-conservation form, 66 Control volume, 50 Convergent-divergent diffuser, 310 Convergent-divergent nozzle, 304 Convergent nozzle, 302 Convex corner, 217, 251 Coriolis forces, 418 Correctly expanded nozzle, 305 Crest and trough, 453 Critical pressure, 115 Critical speed of sound, 160 Critical temperature, 115 Crocco’s theorem, 419

D D’Alembert’s paradox, 43, 271, 449 Deflection angle, 222 de Laval nozzle, 304 Density, 62 Diabatic flow, 375 Diamond-shaped airfoil, 266 Diffuser efficiency, 325 Dimensionless speed, 165 Dissociation, 337 579

580 Drag, 271 Dynamic pressure, 183 compressible flow, 184 correction coefficient, 186

E Elliptic, 443 Energy distribution function, 3 Energy equation, 420 Enthalpy, 79 Entropy, 47, 416, 421 Equation of state, 406 non-isentropic process, 406 perfect gas, 406 Equipartition of energy, 22 Equipotential lines, 433 Eulerian description, 53 Euler’s equation, 73 Expansion waves, 216 Extensive properties, 50

F Fanno flow, 214, 342 continuity equation, 342 D’Arcy-Weisbach equation, 344 energy equation, 342 Fanning equation, 345 friction parameter, 352 Fanno line, 215, 353 First law of thermodynamics, 77 First throat, 312 Flat plate airfoil, 258 Fluid rotation, 408 Frictional choking, 356 Fundamental postulates, 2

G Gas dynamics, 47 Gaussian integrals, 5 Gibbs equations, 91

H Highly overexpanded, 280 Highly underexpanded jets, 282 High-speed flows, 48 Hyperbolic, 443 Hypersonic flow, 135, 454 Hypervelocity flow, 135

Index I Ideal gas law, 48 Impulse function, 181 Incident shock wave, 276 Incompressible, 1 Incompressible flow, 133 Infinite wave-shaped wall, 444 subsonic flow, 446 pressure coefficient, 449 surface pressure coefficient, 449 supersonic flow, 450 linearized pressure coefficient, 451 surface pressure coefficient, 451 Intake, 316 hypersonic intake, 333 subsonic intake, 318 supersonic intake, 328 fixed-Geometry intake, 329 Kantrowitz-Donaldson inlet, 332 variable-Geometry intake, 331 Intake lip, 319 Intensive properties, 50 Intermolecular forces, 2 Irrotational flow, 413, 425 Isentropic compression, 243 Isentropic expansion, 245 Isentropic relation, 48 Isolator, 335

K Kelvin’s theorem, 416 Kinematic viscosity, 40 Knudsen number, 30

L Lagrangian description, 52, 414 Laplace’s equation, 454 Lift, 271 Lift to drag ratio, 486 Linearized perturbation-velocity potential equation, 442 Linearized pressure coefficient, 444, 462 Linearized velocity potential equation, 438 −momentum equation, 419 Low-speed flows, 48

M Mach angle, 123, 230 Mach cone, 123 Mach disk, 280, 282 Mach lines, 126, 284, 452

Index Mach number, 127 Mach reflection, 277 Mach shock wave, 277 Mach stem, 277 Mach waves, 2, 230 Mass, 62 Mass flow rate, 59, 141, 312 Material derivative, 54, 414 Maximum deflection angle, 224 Maximum isentropic discharge speed, 158 Maxwell-Boltzmann distribution law, 3 Maxwell’s velocity distribution, 3 Mayer’s relation, 95 Mean free path, 12 Mean molecular speed, 39 Method of separation of variables, 446 Moderately underexpanded jets, 282 Molar-specific heat at constant pressure, 26 Molar-specific heat at constant volume, 26 Molecular collision theory, 12 Molecular weight, 7 Most probable speed, 7

N Natural coordinates, 419 Newtonian fluid, 40 n−momentum equation, 419 Non-air-breathing propulsion systems, 333 Normal shock wave, 195, 216 change in entropy, 196 continuity equation, 195 energy equation, 196 momentum equation, 196 Prandtl’s velocity relation, 207 Rankine-Hugoniot equation, 208 stagnation pressure ratio, 199 static density ratio, 201 static pressure ratio, 198 static temperature ratio, 199 Nozzle, 302 Nozzle efficiency, 307 Number density, 14

O Oblique shock diffuser, 314 Oblique shock wave, 216 continuity equation, 218 energy equation, 219 entropy change, 221 momentum equation, 219 Prandtl’s velocity equation, 227

581 Rankine-Hugoniot equation, 229 stagnation pressure ratio, 221 static density ratio, 221 static pressure ratio, 221 static temperature ratio, 221 One-dimensional, 437 One-dimensional flow, 137 Bernoulli’s Equation, 144 continuity equation, 141 energy equation, 145 momentum equation, 141 Operating pressure ratio of a wind tunnel, 310 Opposite family, 284, 285 Overexpanded jet, 278 Overexpanded nozzle, 305 P Parabolic, 443 Perfect fluid, 38 Perfect gas, 38, 91 Boyle’s law, 92 calorical equation of state, 93 calorical properties, 91 Charles law, 92 equation of state, 92 thermal properties, 91 Perfect liquid, 38 Perfectly elastic, 3 Perfect solid, 37 Perfect substance, 35 Perturbation-velocity, 439 Perturbation velocity potential, 439 Perturbation-velocity potential equation, 441 Phase-lag, 449 Pitot intake, 316 Pitot probe, 210 Pitot-static probe, 210 Planar wave front, 123 Planar wedge, 253 Potential flow, 399 Prandtl–Glauert rule, 459 Prandtl–Meyer function, 249 Prandtl–Meyer expansion fans, 246 Prandtl–Meyer expansion waves, 279 Prandtl number, 130 Prandtl’s boundary layer theory, 399 Pressure, 182 Pressure coefficient, 187, 271, 453 Pressure-deflection diagram, 239 Pressure recovery, 327 Property field, 52

582 Q Quasi-one-dimensional, 437 R Ramjet, 333, 337 Ratio of specific heats, 131 Rayleigh effect, 378 Rayleigh flow, 214, 375 continuity equation, 376 momentum equation, 376 Rayleigh line, 215, 376 Rayleigh Supersonic Pitot Probe formula, 212 Reduced pressure, 115 Reduced temperature, 115 Reflected shock wave, 277 Regular reflections, 277 Reverse nozzle diffuser, 328 Reynolds number, 129 Reynolds transport theorem, 55 Right-running characteristics, 462 Root mean square speed, 7 S Saint Venant-Wantzel formula, 159 Scramjet, 333, 338 Second law of thermodynamics, 47, 86 conservation form, 89 entropy, 87 integral form, 89 non-conservation form, 90 Second throat, 311 Self-diffusion, 42 Separation bubble, 335 Shadowgraph, 280, 282 Shear-layer, 284 Shock angle, 216 Shock-cell, 280 Shock-diamond, 282 Shock-expansion theory, 258, 464 Shock polar, 233 Shock wave, 193 Shock-Wave Boundary-Layer Interaction, 287 strong interaction, 288 weak interaction, 288 Similarity laws, 455 Similarity parameters, 126 dynamic similarity, 127 geometric similarity, 126 kinematic similarity, 127 Slipstream, 277, 284, 285

Index Small-perturbation theory, 437 Specular, 277 Speed of sound, 109 Laplace’s correction, 112 pressure wave, 110 Stagnation condition, 150 change in entropy, 157 stagnation density, 155 stagnation enthalpy, 150 stagnation pressure, 153 stagnation speed of sound, 156 stagnation temperature, 152 Stagnation enthalpy, 404, 416, 421 Stagnation pressure, 183, 422 Static pressure, 183 Steady adiabatic flow ellipse, 167 Stokes theorem, 407 Stream function, 430 governing partial differential equation, 432 Laplace’s equation, 432 Streamline coordinates, 419 Streamlines, 433 Stream thrust, 182 Streamtube, 319 Strong normal shock, 204 Subsonic flows, 134, 442 Subsonic intake ratio of stagnation pressures, 318 ratio of stagnation temperatures, 318 Superposition of solutions, 428 Supersonic combustion, 337 Supersonic flow, 134, 442 Surface pressure coefficient, 449 co-sinusoidal curve, 449 Sutherland’s formula, 40 System, 49 T Thermal choking, 378 Thin airfoil theory, 271 Torque, 417 Total velocity potential, 439 Transonic flow, 134, 442 Triple-point, 277, 280 Turbofan, 333 Turbojet, 333 Turbo-prop, 333 Turning angle, 217 U Underexpanded nozzle, 305

Index Universal gas constant, 18, 21

V Velocity potential, 426 governing partial differential equation, 428 Laplace’s equation, 429 Velocity vector, 4 Von Karman’s rules for supersonic flows, 123 Vortex filament, 413 Vortex line, 413 Vortex tube, 413 Vorticity, 285, 412, 421

583 W Wave cancellation, 283 Wave drag, 268 Wavelength, 445 Wave neutralization, 283 Wavy wall, 445 Weak normal shock, 204 Weak oblique shock, 233 Weak-wave theory, 464 Wedge angle, 217 Wind tunnel, 427

Z Zero-point energy, 21