Thermal Structures and Materials for High-Speed Flight [1 ed.] 9781600863936, 9781563470172

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Thermal Structures and Materials for High-Speed Flight

Edited by Earl A. Thornton University of Virginia Charlottesville, Virginia

Volume 140 PROGRESS IN ASTRONAUTICS AND AERONAUTICS A. Richard Seebass, Editor-in-Chief University of Colorado at Boulder Boulder, Colorado

Published by the American Institute of Aeronautics and Astronautics, Inc. 370 L'Enfant Promenade, SW, Washington, DC 20024-2518.

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Copyright © 1992 by the American Institute of Aeronautics and Astronautics, Inc. Printed in the United States of America. All rights reserved. Reproduction or translation of any part of this work beyond that permitted by Sections 107 and 108 of the U.S. Copyright Law without the permission of the copyright owner is unlawful. The code following this statement indicates the copyright owner's consent that copies of articles in this volume may be made for personal or internal use, on condition that the copier pay the per-copy fee ($2.00) plus the per-page fee ($0.50) through the Copyright Clearance Center, Inc., 21 Congress Street, Salem, Mass. 01970. This consent does not extend to other kinds of copying, for which permission requests should be addressed to the publisher. Users should employ the following code when reporting copying from this volume to the Copyright Clearance Center: 1-56347-017-9/92 $2.00+ .50 Data and information appearing in this book are for informational purposes only. AIAA is not responsible for any injury or damage resulting from use or reliance, nor does AIAA warrant that use or reliance will be free from privately owned rights. ISSN 0079-6050

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Progress in Astronautics and Aeronautics Editor-in-Chief A. Richard Seebass University of Colorado at Boulder

Editorial Board Richard G. Bradley General Dynamics

John L. Junkins Texas A&M University

John R. Casani California Institute of Technology Jet Propulsion Laboratory

John E. Keigler General Electric Company Astro-Space Division

Alien E. Fuhs Carmel, California

Daniel P. Raymer Lockheed Aeronautical Systems Company

George J. Gleghorn TRW Space and Technology Group

Joseph F. Shea Massachusetts Institute of Technology

Dale B. Henderson Los Alamos National Laboratory Carolyn L. Huntoon NASA Johnson Space Center Reid R. June Boeing Military Airplane Company

Martin Summerfield Princeton Combustion Research Laboratories, Inc. Charles E. Treanor Arvin/Calspan Advanced Technology Center

Jeanne Godette Series Managing Editor AIAA

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Preface The design of structures for winged flight vehicles that fly through the Earth's atmosphere—either to and from space or in sustained flight—poses severe challenges to structures engineers. Major components of the challenge are to design structures and select materials that are able to withstand the aerothermal loads of high-speed flight. Two current aeronautics initiatives, the National Aerospace Plane (NASP) and the High Speed Civil Transport (HSCT), provide strong impetus for current research and development in advanced thermal structures and materials. This volume presents a collection of papers originally presented at the First University of Virginia Thermal Structures Conference held in Charlottesville, Virginia on November 13-15, 1990. A fundamental goal of the conference was to expose participants to important problems and emerging technologies needed for the interdisciplinary design and development of thermal structures for high-speed flight. Aerothermal loads exerted on external surfaces of a flight vehicle consist of pressure, skin friction, and aerodynamic heating. Pressure and skin friction have important roles for aerodynamic forces and moments, but aerodynamic heating is the predominant structural load in high-speed flight. Aerodynamic heating is extremely important because induced elevated temperatures can affect structural behavior in several detrimental ways. Elevated temperatures degrade a material's ability to withstand loads because properties such as the elastic modulus and yield strength are reduced. Time-dependent inelastic behavior may come into play. Thermal stresses are introduced due to restrained local or global thermal expansions or contractions. Such stresses increase deformation, change buckling loads, and alter flutter behavior. In this volume, recent technological developments relevant to these issues are presented. Chapter 1 deals with Aerothermodynamics, one of the fundamental disciplines. Implementation of thermal-science principles in computational fluid dynamics (CFD) computer codes has been a major advance in characterizing aerothermal loads for new flight vehicles. In the first paper, Anderson presents an overview of aerothermodynamics with emphasis on high-temperature, chemically reacting flowfields around hypersonic bodies. Basic physical and chemical processes occurring in hypersonic flows are described. Computation results for inviscid and viscous flows illustrate important features of hypersonic flows for vehicles such as the Space Shuttle and the Galileo probe. In a closely related paper, key aerothermal issues for the structural design of high-speed vehicles are presented by Martellucci and Harris. CFD methods available for vehicle design and aerothermodynamic prediction methods are highlighted. Acoustic fatigue

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and thermal protection system design are described. The need for multidisciplinary design methods for high-speed vehicles is noted. The need for light weight, high-temperature materials is critical for future supersonic and hypersonic vehicles. Chapter 2 presents four papers that describe recent developments in Advanced Light Metals and Composites. The first three papers are concerned with new light metal alloys. In the first paper, prospects for developing superlight metals and laser surface alloying with ceramic phases is described by Hornbogen and Schemme. Alloying magnesium with lithium, magnesium matrix composites, and rapid solidification processing are discussed. Magnesium-lithium-based materials appear to be candidates to compete well in applications with highstrength aluminum alloys and polymer-based composites. In the next paper, characteristics of two new families of aluminum alloys are presented by Starke and Wilsdorf. One family, new aluminum-lithium alloys, has reduced density and greater stiffness. The second family, mechanically alloyed aluminum alloys, has improved strength at elevated temperatures. Although the fracture toughness of the high-temperature aluminum alloys needs improvement, the materials offer promise of replacing more expensive titanium alloys for supersonic aircraft structural applications. Then, a new series of rapidly solidified aluminum-iron-vanadium-silicon alloys for supersonic and hypersonic vehicles is presented by Oilman. Mechanical properties, fatigue, creep rupture properties, corrosion resistance, fasteners, welding and brazing, as well as finishing, are described. In the last paper in Chapter 2, Lisagor reviews material requirements for NASP and HSCT and identifies candidate emerging metallics and gives examples of properties. Critical needs and the current development status of candidate metallics are addressed. Another of the fundamental areas essential for the design of structures for high-speed flight is thermal-structural analysis and testing. Chapter 3 presents eight papers that reflect current research activities in Thermal Structures. Two papers study the buckling of panels at elevated temperatures under mechanical loads. Post buckling of multilayer, polymer matrix composite panels under uniform axial compression with a uniform temperature rise is examined by Noor and Peters. A finite element formulation of a first order shear deformation, von Karman-type nonlinear plate theory is employed. For quasi-isotropic panels, numerical results are presented showing the effects of the elastic modulus, coefficients of thermal expansion and fiber angles of different layers on the post-buckling response. The design, analysis, and testing of high-temperature, flight-representative, hat-stiffened panels is reported by Teare and Fields. Panels were tested under axial compression loads while subjected to a thermal environment controlled by a quartz lamp oven. Finite element analysis was used to provide pre-test strain and initial buckling load predictions. Agreement between test data and analysis suggest that post-buckling strength may potentially be used to reduce vehicle weight. A recent innovation in finite element analysis is the use of adaptive mesh refinement to resolve local response details accurately. An adaptive unstructured remeshing technique is applied to transient thermal-structural problems by Dechaumphai and Morgan. The technique generates a new mesh

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based on the solution obtained from a previous mesh. Two applications illustrate the effectiveness of the technique. Severe aerothermal loads can introduce inelastic, rate-dependent thermalstructural behavior. Two papers describe finite element analyses using unified viscoplastic constitutive models to represent such nonlinear material behavior. Quasistatic responses of thin structures due to thermal loads are studied by Pandey, Dechaumphai, and Thornton. The finite element analysis uses a time-marching technique to solve the rate-dependent equations. Two applications illustrate the viscoplastic analysis. Inelastic stress behavior and regions of plastic deformations are determined. Dynamic effects in thermoviscoplastic structures are investigated by Byrom et al. Thermally induced vibrations are studied for an elevated temperature range where material behavior is viscoplastic. Significant damping caused by inelastic strain is demonstrated. Carbon-carbon composite material is used in flight structures at extreme elevated temperatures. Two papers describe thermal-structural analyses involving carbon-carbon structures. The design and flight certification of the reinforced carbon-carbon chin panel for the Space Shuttle Orbiter is described by Bohlmann, Curry, and Johnson. A discussion of the thermoelastic design process illustrates the role of analysis techniques in development of flight hardware. Preliminary thermal-structural analyses of a carbon-carbon/refractory-metal heat-pipe cooled wing loading edge concept for hypersonic vehicles is presented by Glass and Camarda. The concept utilizes high-temperature heat pipes to transfer energy from the stagnation region of the leading edge to upper and lower wing surfaces where it is radiated away. Thermal and structural analyses establish feasibility of the concept. The preceding papers illustrate that computational techniques such as the finite element method are widely used for the analysis and design of thermal structures. Equally important is thermal-structural testing, which has an essential role in validating computational techniques. Hot structures test technology developed and applied by the NASA Dryden Flight Research Facility are presented by DeAngelis and Fields. Key elements of hot-structures testing are identified and described. The need for performing combined thermal-mechanical tests of structures containing hydrogen is noted. Over the past several years, the design criteria for thermal structures has both changed and increased in complexity. The papers presented in Chapter 4 describe four different aspects of Material Performance and Failure Criteria for elevated temperature applications. The first paper by Van Stone and Kim describes methods for predicting crack growth in aircraft engine components. An overview of prediction of damage tolerance in monolithic and composite materials is given. Research on methods necessary for the prediction of elevated temperature crack growth in advanced structures is reviewed. A crucial requirement for hot structures for the NASP is the material capability to withstand elevated temperatures while exposed to hydrogen. The effects of hydrogen-containing environments on the mechanical behavior of advanced titanium-based materials is reported by Nelson. The general background is described, and potential problems are addressed. Recent data strongly suggest that titanium-based alloys should

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not be used in thermal structures exposed to hydrogen. Next, the paper by Chamis and Shiao describes a probabilistic-based structural analysis procedure being developed for hot structures. The paper provides a brief description of the fundamental aspects of the procedure and demonstrates the methodology for a turbopump blade of the Space Shuttle main engine. To conclude the chapter, a method for studying the influence of thermal effects on the inelastic response of metal matrix composites is reviewed by Herakovich. The method of cells is used to study yielding and inelastic response of the composite. Results are presented for yield surfaces and nonlinear stress-strain curves for two metal matrix composites including the effects of thermal stresses and imperfect bonding. To meet the need for new, high-temperature materials for hypersonic vehicles and propulsion systems, researchers are attempting to develop new advanced composite materials in parallel with emerging airframe and engine designs. The objective of the concurrent approach to materials development is to shorten the time between development and commercialization of new materials. The approach integrates process understanding with advanced sensing and controls for the production of advanced composite materials with tailored properties. Chapter 5, Intelligent Processing of Materials (IPM), contains five papers describing recent research in this new field. In the first paper, Russell describes a study of thermal and mechanical effects that occur in the manufacture of metal matrix composites monotapes. Analytical models are used to study the temperature and thermalstresses occurring in the composite during manufacture. The models are used to evaluate process conditions necessary to improve product quality. Next, recent progress in the development of IPM strategies for processing of intermetallic composites is reported by Wadley et al. Process models, advanced sensors, and control methodologies for understanding and controlling the consolidation processing of the composites are described. The approach offers promise for optimizing material performance and shortening the time from development to production. Then, application of intelligent processing to chemical vapor deposition is presented by Strife et al. Chemical vapor deposition is useful for coating refractory materials and composites to improve surface properties. Development and application of process models is described. Strategies for control architectures and advanced sensing needs are discussed. Research at the National Institute of Standards on process modeling and control of inert gas atomization is described by Ridder et al. Gas atomization is used in the production of metal powders. Studies described include gas and liquid flow imaging, gas flow modeling, particle size measurement, and process control. Consolidation of metal powders is achieved by hot isostatic pressing (HIP) in which powder in a can is exposed to high temperature and pressure for several hours. Schaefer describes a program for development of an intelligent HIP system. Sensors, process models, and controllers for the system are described. The intelligent HIP system has potential for closed-loop control of product properties and acceleration of the process design. The volume documents recent progress in the multidisciplinary aspects of thermal structures and materials for high-speed flight. It is hoped that the volume will be a useful reference for individuals working in these fields who

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wish to review recent developments and be helpful to others who in the future will face new design challenges. The editor gratefully acknowledges the contributions of the organizers of the technical sessions of the First Thermal Structures Conference at which these papers were presented: James C. Moss, Allan R. Wieting, Donald B. Rummler, and Charles E. Harris of NASA Langley; Richard D. Neumann and Donald B. Paul of Wright Research and Development Center; and Edgar A. Starke Jr., Carl T. Herakovich, Richard P. Gangloff, Haydn N. G. Wadley, and Thomas H. Courtney of the University of Virginia. The editor is also appreciative of the contributions of Jeanne Godette, Managing Editor, and A. Richard Seebass, Editor-in-Chief of the AIAA Progress in Astronautics and Aeronautics series. Finally, the contributors to this volume are thanked for their patience, cooperation, and care in the preparation of their papers. Earl A. Thornton March 1992

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Table of Contents Preface Chapter 1.

Aero thermodynamics

Aero thermodynamics: A Tutorial Discussion . . . . . . . . . . . . . . . . . . . . . . . . 3 John D. Anderson Jr., University of Maryland, College Park, Maryland

Assessment of Key Aerothermal Issues for the Structural Design of High Speed Vehicles..................................59 Anthony Martellucci and Thomas B. Harris, Science Applications International Corporation, Ft. Washington, Pennsylvania

Chapter 2.

Advanced Light Metals and Composites

Prospects of Superlight Metals and Their Laser Surface Alloying with Ceramic P h a s e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 5 Erhard Hornbogen and Knut Schemme, Ruhr—Universitat Bochum, Bochum, Germany

New Low Density and High-Temperature Aluminum Alloys .........113 E. A. Starke Jr. and H. G. F. Wilsdorf, University of Virginia, Charlottesville, Virginia

Light High-Temperature Aluminum Alloys for Supersonic and Hypersonic Vehicles ......................................... 141 Paul S. Oilman, Allied-Signal, Inc., Morristown, New Jersey

Advanced Metallics for High-Temperature Airframe Structures ...... 161 W. Barry Lisagor, NASA Langley Research Center, Hampton, Virginia

Chapter 3.

Thermal Structures

Postbuckling of Multilayered Composite Plates Subjected to Combined Axial and Thermal Loads ..................................... 183 Ahmed K. Noor and Jeanne M. Peters, NASA Langley Research Center, Hampton, Virginia

Transient Thermal-Structural Analysis Using Adaptive Unstructured Remeshing and Mesh M o v e m e n t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 0 5 Pramote Dechaumphai, NASA Langley Research Center, Hampton, Virginia, and Kenneth Morgan, University of Wales, Swansea, United Kingdom

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Finite Element Thermoviscoplastic Analysis of Aerospace Structures...................................................229 Ajay K. Pandey, Lockheed Engineering and Sciences Company, Hampton, Virginia, Pramote Dechaumphai, NASA Langley Research Center, Hampton, Virginia, and Earl A. Thornton, University of Virginia, Charlottesville, Virginia

Techniques for Hot Structures T e s t i n g . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 5 5 V. Michael DeAngelis and Roger A. Fields, NASA Dryden Flight Research Facility, Edwards, California

Thermal Structural Analysis of the Reinforced Carbon-Carbon Chin Panel.......................................................279 Frank J. Bohlmann, LTV Missiles and Electronics Group—Missiles Division, Dallas, Texas, Donald M. Curry, NASA Lyndon B. Johnson Space Center, Houston, Texas, and David W. Johnson, L TV Missiles and Electronics Group—Missiles Division, Dallas, Texas

Preliminary Thermal/Structural Analysis of a Carbon-Carbon/ Refractory-Metal Heat-Pipe-Cooled Wing Leading E d g ^ T . . . . . . . . . . 3 0 1 David E. Glass, Analytical Services & Materials, Inc., Hampton, Virginia, and Charles J. Camarda, NASA Langley Research Center, Hampton, Virginia

Dynamic Effects in Thermoviscoplastic S t r u c t u r e s . . . . . . . . . . . . . . . . . . 3 2 3 Ted G. Byrom and David H. Alien, Texas A & M University, College Station, Texas, and Earl A. Thornton and J. D. Kolenski, University of Virginia, Charlottesville, Virginia

Buckling Analysis and Test Correlation of High Temperature Structural P a n e l s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 7 Wendy P. Teare, McDonnell Douglas Corporation, St. Louis, Missouri, and Roger A. Fields, NASA Dryden Flight Research Facility, Edwards, California Chapter 4.

Materials Performance and Failure Criteria

Methods for Predicting Crack Growth in Advanced Structures.......355 R. H. Van Stone and K. S. Kim, GE Aircraft Engines, Cincinnati, Ohio Hydrogen Environment Effects on Advanced Alloys and Composites in Aerospace Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 8 3 Howard G. Nelson, NASA Ames Research Center, Moffett Field, California

Probabilistic Assessment of Thermal Structures . . . . . . . . . . . . . . . . . . . . 4 0 1 Christos C. Chamis, NASA Lewis Research Center, Cleveland, Ohio, and Michael Shiao, Sverdrup Technology, Inc. Lewis Research Center Group, Brookpark, Ohio

Microlevel Thermal Effects in Metal Matrix Composites . . . . . . . . . . . . 4 1 7 Carl T. Herakovich, University of Virginia, Charlottesville, Virginia

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Chapter 5.

Intelligent Processing of Materials

Thermomechanical Effects in Plasma-Spray Manufacture of MMC Monotapes..................................................437 Edward S. Russell, GE Aircraft Engines, Lynn, Massachusetts

Intelligent Processing of Intermetallic Composite Consolidation..... .457 H. N. G. Wadley, D. M. Elzey, L. M. Hsiung, Y. Lu, J. M Duva, S. Parthasarathi, K. P. Dharmasena, J. M. Kunze, and D. G. Meyer, University of Virginia, Charlottesville, Virginia

Application of Intelligent Processing to Chemical Vapor Deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 8 5 J. R. Strife, W. Y. Lee, R. D. Veltri, and C. F. Sack, United Technologies Research Center, East Hartford Connecticut, R.J. Kee, G. H. Evans, R. S. Larson, and D. S. Dandy, Sandia National Laboratories, Livermore, California, and M. E. Coltrin, P. Ho, and R. J. Buss, Sandia National Laboratories, Albuquerque, New Mexico

Process Modeling and Control of Inert Gas A t o m i z a t i o n . . . . . . . . . . . . 4 9 9 S. D. Ridder, S. A. Osella, P. I. Espina, and F. S. Biancaniello, National Institute of Standards and Technology, Gaithersburg, Maryland

Intelligent Processing of Hot Isostatic Pressing . . . . . . . . . . . . . . . . . . . . 5 2 3 R. J. Schaefer, National Institute of Standards and Technology, Gaithersburg, Maryland

Author Index for Volume 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3 8 List of Series V o l u m e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3 9

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Aerothermodynamics: A Tutorial Discussion John D. Anderson Jr.* University of Maryland, College Park, Maryland Abstract Aerothermodynamics: What is it? What are the important physical phenomena associated with this term? What are the consequences? These and other questions are addressed in this paper in a purely tutorial fashion. The high-temperature chemically reacting flowfields around hypersonic bodies are the primary focus. Aspects of flows in which the transport mechanisms of viscosity, thermal conduction, and mass diffusion are neglected (inviscid flows), as well as those in which these transport phenomena are included (viscous flows), are discussed. Finally, the distinction between equilibrium and nonequilibrium chemically reacting flows is highlighted. Introduction On July 24, 1969, Apollo 11 successfully entered the atmosphere of the Earth, returning from the historic first manned flight to the moon. During its return to Earth, the Apollo vehicle acquired a velocity essentially equal to escape velocity from the Earth, approximately 11.2 km/s. At this entry velocity, the shock-layer temperature becomes very large. How large? Let us make an estimate based on the results of classical compressible flow.l The temperature ratio across a normal shock wave can easily be calculated from an analytic formula or simply found in a standard table. Let us assume that the temperature in the nose region of the Apollo lunar return vehicle is approximately that behind a normal shock wave. Considering a given point on the entry trajectory, at an altitude of 53 km, the vehicle's Mach number is 32.5. At this altitude, the freestream temperature is TOO = 283 K. From the classical results, this yields a shock-layer temperature behind the shock of 58,128 K, ungodly high, but also totally incorrect. It is totally Copyright© 1992 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved *Professor, Department of Aerospace Engineering.

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J. D. ANDERSON JR.

incorrect because the classical results are based on the assumption that the gas has constant specific heats. In our calculation, we have used y = 1.4 for the ratio of specific heats. In reality, at such high temperatures, the gas becomes chemically reacting, and y no longer equals 1.4 nor is it constant. A more realistic calculation, assuming the flow to be in local chemical equilibrium, yields a shock-layer temperature of 11,600 K, also a very high temperature but considerably lower than the 58,128 K originally predicted. (Note: The surface temperature of the sun is about 6000 K; the Apollo shock-layer temperature was about twice that of the sun's surface.) The major points here are: 1) The temperature in the shock layer of a high-speed entry vehicle can be very high. 2) If this temperature is not calculated properly, huge errors will result. The assumption of constant y = 1.4 does not even come close.

The above is a graphic example of aerothermodynamics in action. The "older" aerothermodynamics was used to obtain the classical result that predicted a shock-layer temperature of about 58,000 K, and the "newer" aerothermodynamics yields the more appropriate result of over 11,000 K. The above example is important to aerothermodynamicists and materials/structures people alike. The aerothermodynamicist's job is to predict the characteristics of the flow over a high-speed body, and the job of the materials/structures person is to comprehend and cope with the consequences of this prediction. Therefore, if the aerothermodynamicist makes a misjudgment, everybody suffers. Moreover, whether or not there is such a misjudgment, it is important for the materials/structures person to understand the analytical, computational, and experimental difficulties encountered by the aerothermodynamicist and to provide some sympathy in the process. The purpose of this paper is to promote such understanding and sympathy. This author leaves it to other papers in this conference to promote the understanding in reverse, i.e., to educate the aerothermodynamicist about the difficulties and problems encountered by the materials/structures person in designing lightweight thermal protection systems and structures for highenthalpy shock-layer environments. Aerothermodynamics: What does the word mean? The answer is simple. The "aero" stands for the vast field of aerodynamics that has been developing over the past two centuries since the seminal work of George Cayley in 1809 (see Chapter 1 of Ref. 2). Until the advent of high-speed flight in the 1940s, virtually all aerodynamic investigations assumed a low-speed, incompressible flow—a flow governed by purely mechanical laws. In contrast, a high-speed flow is a

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AEROTHERMODYNAMICS

5

high-energy flow, and changes in this energy become a dominant aspect of such flows. The science of energy is thermodynamics. Presto! The science of aerodynamics must be wed to the science of thermodynamics for the study of high-speed flows. This is the essence of the "thermo" in the word "aerothermodynamics". Indeed, the word was coined by General G. Arturo Crocco in 1931 (see Ref. 3), a leading Italian air force general and aeronautical engineer. The word was introduced and propagated in the United States by Theodore von Karman, beginning about 1940 (see Ref. 4). For the next two decades, aerothermodynamics concentrated on the analysis of high-speed flows germane to the period, flows at supersonic Mach numbers, where the temperature changes were still modest enough to assume a calorically perfect gas (constant specific heats) with a constant value of the ratio of specific heat at constant pressure Cp to that at constant volume cv - This ratio for air is y=Cp/c v =1.4. This is the "older" aerothermodynamics mentioned earlier; it is the essence of most standard compressible flow courses taught at the junior/senior level in universities to this day. However, with the advent of hypersonic flow and flight Mach numbers as high as 36 (for the Apollo), the flowfield temperature became high enough to cause the excitation of the vibrational energy mode in molecules and to promote chemical reactions in the flow, conditions under which y is definitely not constant. This is the "newer" aerothermodynamics mentioned earlier. Finally, for both the "older" and "newer" aerothermodynamics, the elevated temperatures in the shock layer over high-speed vehicles had a definite connotation associated with heat transfer to the surface. Hence, the problem of aerodynamic heating was born and was quickly absorbed under the more general heading of aerothermodynamics. This is perhaps the aspect of aerothermodynamics most important to materials/structures people. In the context of the present conference, the purpose of this paper is primarily tutorial. Here, we will describe some of the basic physical aspects of the "newer" aerothermodynamics, concentrating on hightemperature effects in shock layers around hypersonic vehicles, with the attendant effects on aerodynamic heating. A more in-depth discussion of the state of the art will be given in the papers that follow this one; the present paper is intended simply to set the stage. Finally, many aspects of this paper are abstracted from Ref. 5, which should be examined by those readers interested in obtaining more depth in the subject. Qualitative Aspects of High-Temperature, Hypersonic Shock Layers The high-temperature regions in the flowfield around a bluntnosed entry body are sketched in Figs. 1-3. The massive amount of flow

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J. D. ANDERSON JR.

kinetic energy in a hypersonic freestream is converted to internal energy of the gas across the strong bow shock wave, hence creating very high temperatures in the shock layer near the nose. In addition, downstream of the nose region, where the shock-layer gas has expanded and cooled around the body, we have a boundary layer wherein the effects of extreme viscous dissipation result in high temperatures. How does this high-temperature chemically reacting flow differ from the flow of a gas with constant Y? The answer is as follows:

1) The thermodynamic properties (e, h, p, T, p, s, etc.) are completely different. 2) The transport properties (|i and k) are completely different. Moreover, the additional transport mechanism of diffusion becomes important, with the associated diffusion coefficients Di,j. 3) High heat-transfer rates are usually a dominant aspect of any high-temperature application. 4) The ratio of specific heats, y= Cp/cv, is a variable. In fact, for the analysis of high-temperature flow, y loses the importance it has for the classical constant g flows. 5) In view of all this, virtually all analyses of high-temperature gas flows require some type of numerical, rather than closedform, solutions. 6) If the temperature is high enough to cause ionization, the gas becomes a partially ionized plasma, which has a finite Strong bow — shock wave // x

High-temperature boundary layer

M,

High-temperature shock layer region

Fig. 1.

Schematic of the high-temperature regions in an entrybody flowfield.

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AEROTHERMODYNAMICS

Fig. 2.

Schematic of the plasma sheath

around an entry body.

Fig. 3. Schematic of the nonadiabatic radiating flowfield around a body.

electrical conductivity. In turn, if the flow is in the presence of an exterior electric or magnetic field, then electromagnetic body forces act on the fluid elements. This is the purview of an area called magnetohydrodynamics (MHD) (see Fig. 2). 7) If the gas temperature is high enough, there will be nonadiabatic effects resulting from radiation to or from the gas (see Fig. 3).

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For these reasons, a study of high-temperature flow is quite different from the more classical compressible flow of a calorically perfect gas. At what temperatures do chemically reacting effects become important in air? An answer is given in Fig. 4, which illustrates the ranges of dissociation and ionization in air at a pressure of 1 atm. Let us go through the following thought experiment. Imagine that we take the air in the room around us and progressively increase the temperature, holding the pressure constant at 1 atm. At a temperature of about 800 K, the vibrational energy of the molecules becomes significant (as noted on the right of Fig. 4). This is not a chemical reaction, but it does have some impact on the properties of the gas. When the temperature reaches about 2000 K, the dissociation of C>2 begins. At 4000 K, the O2 dissociation is essentially complete; most of the oxygen is in the form of atomic oxygen, O. Moreover, by an interesting quirk of nature, 4000 K is the temperature at which N2 begins to dissociate, as shown in Fig. 4. When the temperature reaches 9000 K, most of the N2 has dissociated. Coincidentally, this is the temperature at which both oxygen and nitrogen ionization occurs and, above 9000 K, we have a partially ionized plasma consisting mainly of O, O+, N, N"1", and electrons. Not shown in Fig. 4 (because it would

N - N ' + *•9000 K

__

N 2 almost completely dissociated; ionization begins

N , -*2N

4000 K

2500 K————;

N 2 begins to dissociate; O2 is almost completely dissociated 2-20 — O 2 begins to dissociate

No reactions

-Vibrational excitation

0 K———— I

Fig. 4.

Ranges of vibrational excitation, dissociation, and ionization for air at 1-atm pressure.

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AEROTHERMODYNAMICS Lifting reentry from orbit

10% 90° 0

90° n 10° 0

90 ° 0

I0° 0

km/s

Fig. 5.

Velocity-amplitude map with superimposed regions of vibrational excitation, dissociation, and ionization.

become too cluttered) is a region of mild ionization that occurs around 4000-6000 K; here small amounts of NO are formed, some of which ionize to form NO"1" and free electrons. In terms of the overall chemical composition of the gas, these are small concentrations; however, the electron number density due to NO ionization can be sufficient to cause a communications blackout. Reflecting on Fig. 4, it is very useful to fix in your mind the "onset" temperatures: 800 K for vibrational excitation, 2500 K for O2 dissociation, 4000 K for N2 dissociation, and 9000 K for ionization. With the exception of vibrational excitation, which is not affected by pressure, if the air pressure is lowered, these onset temperatures decrease; conversely, if the air pressure is increased, these onset temperatures are raised. The information on Fig. 4 leads directly to the velocity-altitude map shown in Fig. 5. Here, we show the flight paths of lifting entry vehicles with different values of the lift parameters, m / C^S. Superimposed on this velocity-altitude map are the flight regions associated with various chemical effects in air. The 10 and 90% labels at the top of Fig. 5 denote the effective beginning and end of various regions where these effects are important. Imagine that we start in the lower left corner and mentally "ride up" the flight path in reverse. As the velocity becomes larger, vibrational excitation is first encountered in the flowfield, at about V = \ km/s. At the higher velocity of about

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2.5 km/s, the vibrational mode is essentially fully excited, and oxygen dissociation begins. This effect covers the shaded region labeled "oxygen dissociation." The O2 dissociation is essentially complete at about 5 km/s, at which N2 dissociation commences. This effect covers the shaded region labeled "nitrogen dissociation." Finally, above 10 km/s, the N2 dissociation is complete, and ionization begins. It is most interesting that regions of various dissociations and ionization are so separate on the velocity altitude map, with very little overlap. This is, of course, consistent with the physical data shown in Fig. 4. In a sense, this is a situation in which nature is helping to simplify things for us. Finally, we can make the following general observation from Fig. 5. The entry flight paths slash across major sections of the velocityaltitude map where chemical reactions and vibrational excitation are important. Indeed, the vast majority of any given flight path is in such regions. From this, we can clearly understand why high-temperature effects are so important to entry-body flows. Some Basic Physics and Chemistry

We will distinguish between a real gas and a perfect gas. These are defined as follows. Consider the air around you as made up of molecules that are in random motion, frequently colliding with neighboring molecules. Imagine that you pluck one of these molecules out of the air around you. Examine it closely. You will find that a force field surrounds this molecule, as a result of the electromagnetic action of the electrons and nuclei of the molecule. In general, this force field will reach out from the given molecule and will be felt by neighboring molecules, and vice versa. Thus, the force field is called an intermolecular force. A schematic of a typical intermolecular force field due to a single particle is shown in Fig. 6. Here, the intermolecular force is sketched as a function of distance away from the particle. Note that, at small distances, the force is strongly repulsive, tending to push the two molecules away from each other. However, as we move further away from the molecule, the intermolecular force rapidly decreases and becomes a weak attractive force, tending to attract molecules together. At distances approximately 10 molecular diameters away from the molecule, the magnitude of the intermolecular force is negligible. Since the molecules are in constant motion, and this motion is what generates the macroscopic thermodynamic properties of the system, then the intermolecular force should affect these macroscopic properties. This leads to the following definition: 1) Real gas: a gas in which intermolecular forcesare important and must be accounted for.

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Repulsive force

Force field is usually negligible, about 10 molecular diameters away from the molecule Fig. 6.,

Sketch of the intermolecular force variation.

On the other hand, if the molecules are spaced, on the average, more than 10 molecular diameters apart, the magnitude of the intermolecular force is very small (see Fig. 6) and can be neglected. This, for example, is the case for air at standard conditions. This leads to the next definition: 2) Perfect

gas: a gas in which intermolecular forces are negligible.

For most problems in aerodynamics, the assumption of a perfect gas is very reasonable. Conditions that require the assumption of a real gas are very high pressures (p ~ 1000 atm) and/or low temperatures (T « 30 K). Under these conditions, the molecules in the system will be packed closely together and will be moving slowly with consequent low inertia. Thus, the intermolecular force has every opportunity to act on the molecules in the system and, in turn, to modify the macroscopic properties of the system. In contrast, at lower pressures (p ~ 10 atm, for example) and higher temperatures (T « 300 K, for example), the molecules are spaced widely apart and are moving more rapidly, with consequent higher inertia. Thus, on the average, the intermolecular force has little effect on the particle motion and, therefore, on the macroscopic properties of the system. Repeating again, we can assume such a gas to be a perfect gas, where the intermolecular force can be

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ignored. Deviations from perfect-gas behavior tend to be proportional

to p /T^, which makes qualitative sense based on the preceding discussion. Unless otherwise stated, we always deal with a perfect gas as defined herein; this is compatible with about 99% of all practical aerodynamic problems. Unfortunately, for the past three decades, the aerodynamics community has been misusing the term "real gas" in light of its classical definition from physical chemistry. We will return to this matter after the next few paragraphs. However, let us first examine a more precise distinction between different models of a gas.

Calorically Perfect Gas By definition, a calorically perfect gas is one with constant specific heats cp and Cy In turn, the ratio of specific heats, 7= cp/Cv, is constant. For this gas, the enthalpy and internal energy are functions of temperature, given explicitly by and e

The perfect-gas equation of state holds, for example, pv= RT

where R is a constant. In the introductory study of compressible flow, the assumption of a calorically perfect gas is almost always made; hence, the thermodynamics of a calorically perfect gas is probably quite familiar to you.

Thermally Perfect Gas By definition, a thermally perfect gas is one in which Cp and cv are variables and, specifically, are functions of temperature only.

Differential changes in the h and e are related to differential changes in T via

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Hence, h and e are functions of T only, i.e.,

h =h(T) e =e(T) The perfect-gas equation of state holds, for example, pv=RT

where R is a constant. The temperature variation of specific heats, hence the whole nature of a thermally perfect gas, is due to the excitation of vibrational energy within the molecules of the gas and to the electronic energy associated with electron motion within the atoms and molecules, as will be explained later. Chemically Reacting Mixture of Perfect Gases Here we are dealing with a multispecies, chemically reacting gas in which intermolecular forces are neglected; hence, each individual species obeys the perfect-gas equation of state in the familiar form.

pv = RT However, here R is a variable because, in a chemically reacting gas, the molecular weight of the mixture fl^is a variable, and R = 3t/M. For the special case of an equilibrium gas, the chemical composition is a unique function of p and T. Hence, for chemical equilibrium,

In the foregoing, it is frequently convenient to think of e and Cv as functions of T and V rather than T and p. It does not make any difference, however, because, for a thermodynamic system in equilibrium (including an equilibrium chemically reacting system), the state of the system is uniquely defined by any two state variables. The choice of T and p, or Tand V is somewhat arbitrary in this sense.

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Real Gas Here, we take into account the effect of intermolecular forces. In practice, a gas behaves as a real gas under conditions of very high pressure and low temperature, conditions that accentuate the influence of intermolecular forces on the gas. For these conditions, the gas is rarely chemically reacting. Therefore, for simplicity, we will consider a nonreacting gas here. Recall that for both the cases of a calorically perfect gas and thermally perfect gas, h and e were functions of T only. For a real gas, with intermolecular forces, h and e depend on p (or v ) as well:

e = e (T, V ) cp=fl (T,p

Moreover, the perfect-gas equation of state is no longer valid here. Instead, we must use a "real-gas" equation of state, of which there are many versions. Perhaps the most familiar is the Van der Waals equation, given by

where a and b are constants that depend on the type of gas. In summary, the preceding discussion has presented four different categories of gases. Any existing analyses of thermodynamic and gasdynamic problems will fall into one of these categories; they are presented here so that you can establish an inventory of such gases in your mind. It is extremely helpful to keep these categories in mind while performing any study of gasdynamics.

Also, to equate these different categories to a practical situation, let us once again take the case of air. Imagine that you take the air in the room around you and begin to increase its temperature. At room temperature, air is essentially a calorically perfect gas, and it continues to act as a calorically perfect gas until the temperature reaches approximately 800 K. Then, as the temperature increases further, we see from Fig. 4 that vibrational excitation becomes important. When this happens, air acts as a thermally perfect gas. Finally, above 2500 K, chemical reactions occur, and air becomes a chemically reacting mixture of perfect gases. If we were to go in the opposite direction, that is, to reduce the air temperature considerably

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below room temperature and/or to increase the pressure to a very high value, say 1000 atm, then the air would behave as a real gas. Finally, it is important to note a matter of nomenclature. We have followed classical physical chemistry in defining a gas where intermolecular forces are important as a real gas. Unfortunately, an ambiguous term has evolved in the aerodynamic literature that means something quite different. In the 1950s, aerodynamicists were suddenly confronted with hypersonic entry vehicles at velocities as high as 26,000 ft/s (8 km/s). As discussed earlier, the shock layers around such vehicles were hot enough to cause vibrational excitation, dissociation, and even ionization (see Fig. 5). These were "real" effects that happened in air in "real life." Hence, it became fashionable in the aerodynamic literature to denote such conditions as "real-gas effects." For example, the categories itemized earlier as a thermally perfect gas and as a chemically reacting mixture of perfect gases would come under the classification of "real-gas effects" in some of the aerodynamic literature. But, in light of classical physical chemistry, this is truly a misnomer. A real gas is truly one in which intermolecular forces are important, and this has nothing to do with vibrational excitation or chemical reactions. Therefore, in this paper we will talk about hightemperature effects and will discourage the use of the incorrect term "real-gas effects."

Microscopic Description of Cases A simple concept of a diatomic molecule (two atoms) is the "dumbbell" model sketched in Fig. 7a. The molecule has several modes (forms) of energy, as follows: 1) It is moving through space and, hence, it has translational energy Etrans/ as sketched in Fig. 7b. The source of this energy is the translational kinetic energy of the center of mass of the molecule.

2) It is rotating about the three orthogonal axes in space and, hence, it has rotational energy £rot, as sketched in Figs. 7c and 7d. The source of this energy is the rotational kinetic energy associated with the molecule's rotational velocity and its moment of inertia. 3) The atoms of the molecule are vibrating with respect to an equilibrium location within the molecule. For a diatomic molecule, this vibration is modeled by a spring connecting the two atoms, as illustrated in Fig. 7e. Hence, the molecule has vibrational energy Eyib- There are two sources of this vibrational energy: the kinetic energy of the linear motion of

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the atoms as they vibrate back and forth, and the potential energy associated with the intramolecular force (symbolized by the spring). 4) The electrons are in motion about the nucleus of each atom constituting the molecule, as sketched in Fig. 7f. Hence, the molecule has electronic energy Eel- There are two sources of electronic energy associated with each electron: kinetic energy

(a) Diatomic molecule

Source Translational kinetic energy of the center of mass (thermal degrees of freedom —3)

(/)) Translalional energy ev\b ' ^s a resu^ °^ molecular collisions, the excited particles will exchange this "excess" vibrational energy with

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the translational and rotational energy of the gas and, after a period of time, £ v ib will decrease and approach its equilibrium value. The relaxation time t in Eq.(5) is a function of both local pressure and temperature. For most diatomic gases, the variation of t is given by the form

where Ci and C2 are usually obtained from experimental measurements. For chemical nonequilibrium, we are interested in the time rate of change of the chemical composition. For example, let us consider the variation of atomic oxygen due to the dissociation reaction O2 + M 20 + M

(6)

where M is a collision partner (M can be any chemical species in the gas mixture, including O2 itself). Define the symbol [O] as the concentration of atomic oxygen (moles per unit volume), and k as a reaction rate constant. Consider Eq.(6) as it goes from left to right; this is called the forward rate, and k is really the forward rate constant kf. k

f

O2 + M -^2O + M

Hence, Eq.(6) is more precisely written as

forward rate :

The reaction in Eq.(6) that would proceed from right to left is called the reverse reaction, or backward reaction, O2 + M «- 2O + M

with an associated reverse or backward rate constant kb and a reverse or backward rate rate given by Reverse rate:

d[0]_^,.r0]2[M]

(g)

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Note that, in both Eqs.(7) and (8), the right-hand side is the product of the concentrations of those particular colliding molecules that produce the chemical change, raised to the power equal to their stoichiometric mole number in the chemical equation. Equation (7) gives the time rate of increase of O atoms due to the forward rate, and Eq.(8) gives the time rate of decrease of O atoms due to the reverse rate. However, what we would actually observe in the laboratory is the net time rate of change of O atoms due to the combined forward and reverse reactions M 2O + M

and the net reaction is given by Net rate: (9)

d/

The preceding example has been a special application of the more general case of a reacting mixture of n different species. Consider the general chemical reaction (but it must be an elementary reaction, i.e., one that can actually occur by means of a direct molecular collision process).

kf

n

I V/X/ I V/'Xi '' =1

(10)

**'' = 1

where vi and vi represent the stoichiometric mole numbers of the reactants and products, respectively. (Note that, in our earlier example for oxygen, where the chemical reaction was O2 + M 20 + M, v02 = 1, v,, = 0, V0'2 = 0, VM = 1, VM = l / a nd v0" = 2\ For the preceding general reaction, Eq.(lO), we write

Forward rate:

Reverse rate: dt

l

i

Net rate:

-LA = - (v.". v/) lkfIJ[Xi]vi Q/

\

I

- kb V[Xi]vi I

l

(13)

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The chemical rate constants in the preceding equations are usually obtained from experimental measurement. This brings to an end our short review of the basic physics and chemistry necessary for the analysis of high-temperature flows. For more description and details, see, for example, Ref. 5. Some Examples of Inviscid Chemically Reacting Shock Waves and Blunt-Body Flows

In this section, we will examine some high-temperature inviscid flows, i.e., flows in which the dissipative transport mechanism of viscosity, thermal conductivity, and diffusion are neglected. We will first treat these flows for the case of local chemical equilibrium; nonequilibrium flow will be discussed at the end of this section. Equilibrium Normal Shock-Wave Flows If points 1 and 2 represent conditions immediately in front of and behind a normal shock wave, respectively, the governing continuity, momentum, and energy equations are (see, for example, Ref. 1)

Continuity:

Pl"l=P2u2

Momentum:

Energy:

For a flow in local chemical equilibrium, these equations must be solved numerically in conjunction with the appropriate equilibrium thermodynamic properties, say in the form of T2 = T (H2y p2) and P2 = P (^2, P2) • Some results for equilibrium normal shock properties in air are given in Figs. 10 and 11, obtained from Ref. 7. In Figs. lOa and lOb, the temperature behind a normal shock wave is plotted as a function of velocity in front of the wave, with altitude as a parameter. The velocity range in Fig. lOa is below orbital velocity and, hence, the results are affected primarily by dissociation. In contrast, the velocities in Fig. lOb cover the superorbital range (above 26,000 ft/s) and therefore reflect the effects of substantial ionization. Note the effect of pressure in these results: at a given velocity, T2 increases with decreasing altitude (i.e., increasing pressure) because the amount of

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Gcopotcntial altitude, ft 35,900 59,80082,200 100,000 120,300 154,800 173,500

a) 7000

200,100

6000

230,400 259,700 294,800 322,900

5000

4000 3000

2000 1000i

0

I

10

16

14

12 MI,

b)

18

J

22 x 103

20

Geopotential altitude, ft 120,300 154.800 173,500

14.000

200,100 100.000/

13,000 12.000 11,000

294,800

10,000

322,900

9,000

8,000 7,000 6,000 5,000 4,000

16

18

20

22

24

26

28

30

32

34

36

38

40

42

44

46 x 103

M,,ft/S

Fig. 10.

Variation of normal shock temperature with velocity and altitude: a) velocity range below orbital velocity; b) velocity range near and above orbital velocity (from Ref. 7).

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a)

18

Gcopotentiul altitude, ft 322,9\

10

154,800 120,300 100,000

82,200 59,800 35,900 22 x 103

20

Geopotential altitude, ft 322,900

294,800 259,700 230,400

16

Fig. 11.

18

20

22

24

26

28

30

32

34

36

38

40

42

44

46 x 10J

Variation of normal shock density with velocity and altitude: a) velocity range below orbital velocity; b) velocity range near and above orbital velocity (from Ref. 7).

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dissociation and ionization in an equilibrium gas decrease at higher pressure. Also, note the general magnitude of the temperatures encountered. At W j = 10,000 ft/s (typical of a hypersonic cruise transport), T2 « 3000 K. At u2 = 26,000 ft/s (orbital velocity typical of a Space Shuttle or transatmospheric vehicle), T2 « 7000 K. For atmospheric entry at escape velocity, MJ = 36,000 ft/s (typical of Apollo-type vehicles and aeroassisted orbital transfer vehicles), T2 « 11,000 K. Moreover, Fig. lOa illustrates that chemically reacting effects begin to impact the normal shock properties at velocities above 6000 ft/s (approximately Mach 6). The density ratio across a normal shock wave P2/P1 *s shown in Figs, lla and lib, plotted vs. velocity with altitude as a parameter. For a calorically perfect gas, P2/P1 approaches the limiting value of (7+ !)/(/- 1) as M^ —» °o . For air with 7 = 1.4, this limiting ratio is 6. Note, from Figs, lla and lib, that P2 / Pi is strongly affected by chemical reactions and that its values range far above 6, reaching as high as 22. The value of P2/P1 has an important effect on the shock detachment distance in front of a hypersonic blunt body. An approximate expression for the shock detachment distance 5 on a sphere of radius £ is

SP1/P2 R 1 + Y2(pi/p 2 )

< 14)

In the limit of high velocities, Pi / P2 becomes small compared to unity, and Eq.(14) is approximated by

8 _ Pi *

=

P2

1 (P2/P1)

(15)

Therefore, the value of the density ratio across a normal shock wave has a major impact on shock detachment distance; the higher the density ratio P2/P1, the smaller is S. From Figs, lla and lib, we see that the effect of chemical reactions is to increase P2 / Pi which, in turn, decreases the shock detachment distance. Therefore, in comparison to the calorically perfect-gas blunt-body results, the shock wave for a chemically reacting gas (at the same velocity and altitude conditions) will lie closer to the body. This is emphasized in Fig. 12, where 5cp and >

III 1 6 SJ 14

Equilibrium values

-J

P/Pi

10

0.01

0.1

1

10

100

Distance behind shock, cm

Fig. 21.

Distributions of the temperature and density for the nonequilibrium flow through a normal shock wave in air. Af = 12.28, poo = 1.0-mm Hg, Too = 300 K.

shock waves, as discussed earlier. On a qualitative basis, the nonequilibrium flow over a blunt body behaves as sketched in Fig. 22. In the nose region, the chemical composition resembles that in the nonequilibrium region behind a normal shock wave, as already discussed. However, consider the streamline that goes through the stagnation point; this streamline is labeled abc in Fig. 22. Between a and b, the flow is compressed and slowed; it reaches zero velocity at the stagnation point b. In so doing, it can be shown that a fluid element takes an infinite time to traverse the distance ab. This means that local equilibrium conditions must exist at the stagnation point (b), with its attendant highly dissociated and ionized state. The flow then expands rapidly downstream of the stagnation point; indeed, the surface streamline be encounters very large pressure and temperature gradients in the region near the sonic point c, that is, dp/ds and dT/ds are large negative quantities. This is very similar to the nonequilibrium flow through a convergent-divergent nozzle, where sudden freezing of the flow can occur downstream of the throat. The same type of sudden freezing can be experienced near point c in Fig. 22. In turn, the surface of the body downstream of the sonic point can be bathed in a region of nearly frozen flow. Since the streamline started with a large amount of dissociation and ionization at point b, then this frozen flow is

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Equilibrium flow

Stagnation point (equilibrium)

Fig. 22.

Schematic of different regions in a high-temperature blunt-body flowfield.

characterized by a thin region of high dissociated and ionized gas that

flows downstream over the body.

To examine a typical case of nonequilibrium chemically reacting flow over a hypersonic blunt body, consider the work by Hall et al™ Results from this analysis are presented along two streamlines in the blunt-body flowfield, streamlines A and B shown in Fig. 23. This figure is drawn to scale, showing the assumed axisymmetric catenary shock in cylindrical coordinates, where z and r are coordinates parallel and perpendicular, respectively, to the freestream direction. The resulting body shape is nearly spherical and is shown in Fig. 23 for the case of Voo = 23,000 ft/s, altitude equal to 200,000 ft, and a given shock radius of curvature at the point of symmetry Rs = 0.0692 ft. In Figs. 24-26, results are given for the variation of flow properties along streamlines A and B for the velocity-altitude point just given. Figure 24 shows the

results for T, p, and p as a function of distance s along the streamlines,

nonequilibrium behavior similar to that behind a normal shock wave.

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The temperature along streamline A, TA, exhibits an initial rapid decrease behind the shock; this is due to the finite rate dissociation of both O2 and N2- The more gradual decrease in TA for s /Rs > 0.2 is due primarily to the gasdynainic expansion around the body. Similarly, the initially slight increase in p^ and the substantial increase in p^ are due to the nonequilibrium effects, and their subsequent decrease beyond s/R$ = 0.2 is indicative of the gasdynamic expansion around the body. In contrast, streamline B crosses a much weaker portion of the bow shock wave. The flow behind an oblique shock front experiences a much longer relaxation distance than a normal shock at the same upstream conditions although, at the same time, the actual quantitative degree of dissociation behind the oblique shock is less because of the lower temperature. These comparisons carry over to the blunt-body flow. In Fig. 24, the behavior of T^f pg, and pg along streamline B is a Axisymmetric catenary shock z/R, = cosh (r/KJ - 1

0.1

Fig. 23.

0.2 0.3 0.4 Axial coordinate, z/R>

0.5

Shock and body shapes, and calculated streamline for the nonequilibrium flow over a blunt body (from Ref. 10).

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1000 K

Infinite-rate values T = 5500 K

p Pr

Fig. 24.

Temperature, pressure, and density variations along streamlines A and B in the nonequilibrium blunt-body flowfield.

combination of the nonequilibrium effects and the gasdynamic expansion around the body, a combination that persists over the complete length of streamline B shown in Fig 24. Also shown in Fig. 24 are the equilibrium (infinite rate) values of p^ and p$ just behind the shock front, at s = 0. The pressure is least affected by chemically reacting effects in a compression region. Also shown in Fig. 24 are the values of p^ and p% at s/Rs = 0.5 from a calculation of the blunt-body

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• denotes concentration for infinite-rate equilibrium

0.6

Fig. 25.

Atomic oxygen and nitrogen concentrations along streamlines A and B in the nonequilibrium blunt-body flowfield.

flowfield assuming local thermodynamic and chemical equilibrium. Note that these infinite rate values are considerably above the nonequilibrium results for density. As expected, the nonequilibrium effects are strongest on temperature. In Fig. 24, both T^ and Tg are far above the local equilibrium values shown. The variation of atomic oxygen and nitrogen is given in Fig. 25. Note that the amount of atomic oxygen denoted by (O)^ increases rapidly behind the shock front along streamline A) this is due to the nonequilibrium dissociation behind the strong shock front and is

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analogous to the normal shock results discussed earlier. However, for s/Rs > 0.1, the oxygen freezes as a result of the gasdynamic expansion and essentially plateaus at a value slightly less than the equilibrium value shown at s/Rs > 0.5. Along streamline B, the oxygen relaxation is slower, and (O)g freezes at a level even less than that for streamline A. (Note that the equilibrium values for both A and B shown at s/Rs = 0 and 0.5 are the same; this is because the temperatures along streamlines A and B are high enough such that, at local equilibrium conditions, the oxygen is fully dissociated.) Also shown in Fig. 25 is the variation of atomic nitrogen along streamlines A and B, denoted by (N)^ and (N)^ respectively. For N, the nonequilibrium relaxation distances are much longer than for O and, hence, (N)^ and (N)g exhibit strong nonequilibrium behavior. Note that (N)^ is frozen at about one-half its local equilibrium value when compared at s/Rs = 0.5 and that (N)g is about one-fourth its local equilibrium value at the same location. The variations of nitric oxide and electrons are shown in Fig. 26. Note that (NO)^ exhibits the same type of overshoot observed behind a normal shock, as discussed in Fig. 20, whereas (NO)g shows a monotonic increase. Also, note that both (NO)^ and (NO)# are behind a normal shock, as discussed in Fig. 20, whereas (NO)j£ shows a monotonic increase. Also, note that both (NO)^ and (NO)# are considerably above their local equilibrium values. Examining the electron concentrations shown in Fig. 26, we see that (^~}A freezes at a level above the local equilibrium value but that (^~)B is considerably below its equilibrium value. This brings to an end our discussion of inviscid, nonequilibrium, chemically reacting blunt-body flows. Let us now move on and examine the nature of hypersonic viscous flows. High Temperature Viscous Flow Mechanism of Diffusion In addition to the transport phenomena of viscosity and thermal conduction, in a chemically reacting viscous flow, the mechanism of mass diffusion becomes important. Let us first look at some physical aspects of diffusion. It is common knowledge that, if you are in a room and someone in the corner opens a bottle of ammonia, after a period of time, you will smell the ammonia. This is because, over a period of time, some of the ammonia molecules will work their way over to you, just by virtue of their random motion in the gas. To be a little more precise, in the immediate vicinity of the ammonia bottle after it is opened, there is a locally high concentration of ammonia, with a resulting concentration

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10"

o 13

(O,

10-

c '5b

•

L Jo 10-

2 x ID' 5 -

J____L

J____L

(NO),

(NO),

• denotes concentration for infinite-rate equilibrium

0

0.1

0.2

I 0.4

0.3

fB 0.5

I 0.6

0.7

s/R,

Fig. 26.

Nitric oxide and electron concentrations along streamlines A and B in the nonequilibrium blunt-body flowfield.

gradient. Under the influence of this gradient, the ammonia molecules will gradually diffuse away from the bottle. Imagine the ammonia molecules colored green: you see a "green cloud" form in the vicinity of the bottle, and this green cloud moves toward you at some mean velocity. This velocity is defined as the diffusion velocity of the ammonia. Let us now be more precise. Consider a stationary slab of gas mixture in which there exists a gradient in mass fraction of species i; the variation of c{ is sketched in Fig. 27a, and the resulting gradient Vc/ is shown at a given point in the stationary slab in Fig. 27b. Because of this

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J. D. ANDERSON JR.

gradient, at the same point, there is a mass motion of species i in the opposite direction; the velocity of this mass motion of species i is defined as the diffusion velocity of species i, denoted by U{. The corresponding mass flux of species i is pf U{, shown in Fig. 27b, which is given approximately by Pick's law as

Let us now imagine that the slab in Fig. 16.2b is set into motion with the velocity V , as sketched in Fig. 27c. The mass motion of species // relative to us standing in the laboratory, is now Vj where

V{ = 4, Mass motion of species i (relative to the lab, or just simply the

mass motion of species i)

V + 4, Mass motion of the mixture relative to the lab)

Ui 4, Diffusion velocity of species z (relative to the mass motion of the mixture)

In conjunction with a gasdynamic flow, V is the familiar flow velocity at a point in the flowfield; for a gas mixture, the flow velocity is in reality a mass average of all the V;, that is,

In addition to its property as a mass transport mechanism, diffusion is also an energy transport mechanism. This is easily seen by visualizing a chemical species i diffusing from location 1 to location 2, where at location 2 the species participates in a chemical reaction, thus exchanging some energy with the gas. That is, as species i diffuses through the gas, it carries with it the enthalpy of species i, h{r which is a form of energy transport. (Keep in mind that h{ contains the heat of formation of species i.) Hence, at a point in the gas, we can write I Energy flux due to 1 ,, , \ diffusion of species i: | ^ l l

In turn,

/Energy flux due to diffusion) .. , j of all species at the point { ~ q° = }pi Ui hi

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(b) Flux of i due to diffusion

Mass fraction gradient

I

Stationary slab

Moving slab

Fig. 27.

Illustration of diffusion velocity.

This energy flux due to diffusion appears explicitly as a term in the fluid flow general energy equation.

Catalytic Walls In a chemically reacting flow, the mass fraction of species i is one of the dependent variables. Therefore, we need boundary conditions for c{ as well as for u, v, and T, discussed earlier. At the wall, the boundary condition on c{ deserves some discussion because it involves, in general, a surface chemistry interaction with the gas at the wall. The wall may be made of a material that tends to catalyze (i.e., enhance) chemical reactions right at the surface. Such surfaces are called catalytic walls.

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This leads to the following definitions:

1) Fully catalytic wall: a wall at which chemical reactions are catalyzed at an infinite rate; i.e., the mass fractions at the wall are their local equilibrium values at the local pressure and temperature at the wall. 2) Partially catalytic wall: a wall at which chemical reactions are catalyzed at a finite rate. For a fully catalytic wall, the boundary condition is simply:

At y = 0,

c{ = (c/)equii

(fully catalytic wall)

For a partially catalytic wall, the boundary condition can be developed as follows. For a wall with an arbitrary degree of catalyticity, the chemical reactions occur at a finite rate. Let wc denote the catalytic rate at the surface. Then,

\WcK = mass of species i lost at the surface per unit area per unit time due to surface-catalyzed chemical reaction Right at the surface, the mechanism that feeds particles of species i from the gas to the surface is diffusion, as sketched in Fig. 28. The diffusion flux to the surface element of area dS is - (Pi Ui)w do. From Pick's law (assuming a binary gas),

For steady-state conditions, the amount of species i "gobbled up" at the surface because of the catalytic rate (wc)i must be exactly balanced by the rate at which species i is diffused to the surface. Hence, (wc)i dS = - (pi Ui^ dS = pD 12 fe) dS

or

Ww

Equation (17) is the boundary condition for a surface with finite catalyticity. It dictates the gradient of the mass fraction at the wall.

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A noncatalytic wall is one where no recombination occurs at the wall, that is, (wc)/ = 0. For this case, from Eq.(17),

Stagnation-Point Heat Transfer in a Dissociated Gas One of the most important aspects of a viscous flow solution is the calculation of heat transfer to the surface. In turn, one of the critical areas of high heating rates is a stagnation point on a body. The proper inclusion of a high-temperature dissociating gas in the calculation of stagnation-point heat transfer was first made by Fay and Riddell11 over 30 years ago-their results are still in use today. The Fay and Riddell formulas follow: Equilibrium boundary layer (spherical nose):

(18)

where ho = Zi C[t \Ahfy and U/*/); is the heat of formation of species i at absolute zero.

Frozen boundary layer with a fully catalytic wall (spherical nose):

qw = 0.76 Pr-0-6 (Pe (19)

W/////////M^^^

Fig. 28.

Catalytic wall of some a r b i t r a r y degree

Model for catalytic wall effects.

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J. D. ANDERSON JR.

Frozen boundary layer with a noncatalytic wall (spherical nose):

aw = 0.76 TV'0-6 (pe ue)OA (pw /UW -7T, I1 ~ ^

(20)

In these equations, the stagnation-point velocity gradient is given by Newtonian theory as . 1 R

pe

Also, Le is the Lewis number, defined as Le -pD\2 cp Ik. Note from Eqs. (18) and (19) that the driving potential for heat transfer is the enthalpy difference (hoe - hw) in the case of equilibrium flow or frozen flow with a catalytic wall. However, in Eqs.(18-20), the enthalpies are absolute values, i.e., they contain the heats of formation, thus including the powerful chemical energy associated with the reacting gas. Also note that Eqs. (18) and (19) are essentially the same, varying only in the slightly different exponent on the Lewis number. This demonstrates that the surface heat transfer is essentially the same whether the flow is in local chemical equilibrium or frozen with a fully catalytic wall. In the former case (local chemical equilibrium), recombination occurs within the cooler regions of the boundary layer itself, releasing chemical energy throughout the interior of the boundary layer, most of which is transported by thermal conduction to the surface. In the latter case (frozen flow with a fully catalytic wall), the chemical energy release due to recombination is right at the wall itself. Equations (18) and (19) indicate that the new heat transfer to the surface is essentially the same whether the chemical energy is released within the boundary layer or right at the surface. This trend is graphically illustrated in Fig.29, which shows the heat-transfer coefficient Nu tfRe as a function of a recombination rate parameter C\. The Nusselt number Nu is defined as Nu = q^X fik (Taw - rw)] . On the abscissa, going from right to left in the direction of decreasing C\, the chemical state of the boundary layer changes from equilibrium (large Q) to frozen (small Ci). The results in Fig.29 are from a large number of different nonequilibrium boundary-layer cases, with different values of Ci calculated by Fay and Riddell. Shown on this graph are two categories of solutions, one with a fully catalytic wall (curve 1), and the other with a noncatalytic wall (curve 2). Curve 1 for the fully catalytic wall gives the highest heat transfer, and it is essentially constant for all values of Ci. For large values of Ci, this curve

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AEROTHERMODYNAMICS 0.5 0.4

Tolal heat transfer catalytic wall cj)

0.3

Nu 0.2

- 300 K Conductive part of heat transfer to a catalytic wall

0.1 0

J_

10-" 10" 5

_L 10 4

J_

10

3

10^2

_L

10'1

_L

1

10

102

104

Recombination rate parameter C,

Fig. 29.

Catalytic wall effect on stagnation-point transfer.

corresponds to an equilibrium boundary layer and, for small values of Ci7 it corresponds to a frozen boundary layer. Clearly, as long as the wall is fully catalytic to atom recombination, the heat transfer is essentially the same. In contrast, curve 2 in Fig. 29 is for a noncatalytic wall. Here, as we move from right to left along this curve (and as the boundary layer becomes progressively more nonequilibrium, approaching a frozen flow), we see that the heat transfer drops by more than a factor of 2. This is an important point: For nonequilibrium and frozen flows, there is a substantial decrease in heat transfer if the wall

is noncatalytic in comparison to a catalytic wall. Finally, curve 3 in

Fig. 29 goes along with curve 1 for a catalytic wall; curve 3 gives just the conductive part of the heat transfer to a catalytic wall. The difference between curves 1 and 3 represents the heat transfer due to diffusion. Hence, for equilibrium flows, q is essentially all conductive; however, as we examine flows that progressively become more nonequilibrium with a fully catalytic wall, diffusion progressively becomes a larger part of q. As a final note in this section, the work of Fay and Riddell^ 1 represents an excellent example of chemically reacting boundary-layer analysis, and their results convey virtually all the important physical trends to be observed in chemically reacting viscous flows. Although this analysis is old, carried out more than 40 years ago, it is classical, and just as viable today as it was then. This is why we have chosen to highlight it here. The reader is strongly encouraged to study Ref. 11 closely for more details and insight.

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Viscous Shock Layer Solutions In light of the modern techniques associated with computational fluid dynamics, the analysis of aerodynamic heating around a bluntnosed body (including the stagnation-point heat transfer) can be obtained by computing the entire shock layer, assuming a fully viscous flow. One of the more popular viscous shock-layer (VSL) methods is the technique pioneered by the late Tom Davis^, who first applied it with the assumption of a calorically perfect gas. However, the v/ork of Moss* ^ is one of the first detailed investigations of a chemically reacting viscous shock layer using the VSL technique and, in this sense, is a classic contribution to the field. Moss considered the cases of frozen, equilibrium, and nonequilibrium laminar flow. Five chemical species were included: O2, O, N2, N, and NO. Surface catalysis and mass injection were also treated. The viscous shock layer over hyperboloids with included angles of 20 and 45 deg was calculated. Sample results are shown in Figs. 30-36 for flow over a 45 deg hyperboloid with R =

o.30r

Nonequilibrium

Equilibrium Frozen 0.25 -

0.20

0.15

0.10

0.05

Fig. 30.

Shock-layer velocity profiles on a hyperboloid. VSL calculations by Moss 13.

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0.25

Nonequilihrium

0.20

-—

Equilibrium

—

Frozen

0.15

0.10

s* = 2.0

0.05 -

J 16 x I03

12 Temperature, K

Fig. 31.

Shock-layer temperature profiles on a hyperboloid. VSL calculations by Moss 13 .

0.4 0.3

Nonequilibrium and frozen

0.2

Equilibrium

0.1 0

Fig. 32.

Newtonian

0.5

_L

1.0

1.5

2.0

2.5

3.0

Pressure distributions along a hyperboloid.

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J. D. ANDERSON JR.

Noncatalylic wall E q u i l i b r i u m catalytic wall

10

5

0

Fig. 33.

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Shock-layer mass fraction on a hyperboloid.

2.54 cm, Voo = 6.10 km/s, Tw = 1500 K, and an altitude of 60.96 km. In Figs. 30 and 31, shock-layer profiles of velocity and temperature, respectively, are shown, as a function of n* at a streamwise location of s* = 2. Here, n* is the local normal coordinate to the body, and s* is measured along the surface, both in nondimensional form. Three cases are shown in each figure: frozen, equilibrium, and nonequilibrium flow. Comparing Figs. 30 and 31, note that the temperature is much more sensitive to chemically reacting flow than is the velocity, another indication that the thermodynamic properties rather than the more purely fluid dynamic variables (such as velocity and pressure) are more affected by chemical reactions. The tops of the curves in Figs. 30 and 31 correspond to the location of the bow shock wave and, hence, give n* at the shock. Once again, we see that the equilibrium shock layer is thinner than the frozen shock layer. Also, note that, for the flowfield conditions in these figures, the nonequilibrium flow is closer to frozen than to equilibrium. The surface pressure distribution is given in Fig. 32 and graphically demonstrates the insensitivity of pressure to the chemically reacting effects. Catalytic wall effects on the chemical species profiles are shown in Fig. 33. The nonequilibrium flow is calculated for two cases, a fully catalytic wall and a noncatalytic wall.

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The abscissa is the nondirnensional distance across the shock layer n/n$, where ns is the coordinate of the shock wave. The effect of the catalytic wall reaches across more than 70% of the shock layer and, of course, is strongest near the wall. The catalytic wall effect on the heattransfer distribution along the body surface is shown in Fig. 34. These results are consistent with our discussion surrounding Fig. 29 from Fay and Riddell. Note, in Fig. 34, that the nonequilibrium heat transfer is reduced by a noncatalytic wall in comparison to a fully catalytic wall. Also, note that the nonequilibrium, fully catalytic wall case yields essentially the same heat transfer as the local chemical equilibrium flow. Finally, note that the relative influence of wall catalyticity diminishes as a function of downstream distance. All of the foregoing results were obtained with no mass injection into the shock layer through the wall. Moss^ examined the case of wall mass injection; Fig. 35 gives results for heat transfer at the stagnation point with mass injection q divided by its value with no mass injection, (