Introduction to Fluid Mechanics [2 ed.] 9780081024379, 0081024379

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Introduction to Fluid Mechanics Second Edition Yasuki Nakayama

Butterworth-Heinemann is an imprint of Elsevier The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States Copyright © 2018 Elsevier Ltd. All rights reserved. This book is translated from Ryutai-no-Rikigaku 2nd edition (in Japanese) published by YOKENDO CO. LTD, 5-30-15 Hongo, Bunkyo-ku, Tokyo 113-0033 Japan, copyright 2013 by Yasuki Nakayam. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-08-102437-9 For information on all Butterworth-Heinemann publications visit our website at https://www.elsevier.com/books-and-journals

Publisher: Mathew Deans Acquisition Editor: Brian Guerin Editorial Project Manager: Anna Valutkevich Production Project Manager: Surya Narayanan Jayachandran Designer: Mark Rogers Typeset by TNQ Books and Journals

About the Author

Yasuki Nakayama was an internationally influential and respected expert in both mechanical engineering and visualization. During his career he held various important posts in Japan, including as President of the Future Technology Research Institute, as Professor of fluid mechanics and visualization at Tokai University, and at the National Railway Research Institute. He was also a Visiting Professor at Southampton University, President of the Visualization Society of Japan, and Director of the Japan Society of Mechanical Engineering. He published more than 180 research papers and more than 10 books and was a cofounder of the Journal of Visualization. His research earned him many awards and distinctions including the Medal with Purple Ribbon from the Emperor of Japan. He sadly passed away in 2016 before this book was published.

xi

Advisory Editor Biographies

Kazuyasu Izawa is an independent consultant in the field of lubrication, providing technical services to industrial companies and lubrication management training sessions to lubrication engineers. He was granted and received his doctorate (Engineering) in the field of fluid power control under the guidance of Professor Nakayama. Makoto Oki has a background in fluids engineering. He is currently a Professor at Tokai University in Japan, where he teaches and researches in information technology. Katsumi Aoki engaged in research and education of fluid dynamics and mechanical engineering for 40years at the Faculty of Engineering, Graduate School of Tokai University. Currently, he is a professor emeritus at Tokai University and an advisor to technology companies.

REVIEWER BIOGRAPHY John Tippetts is an independent consultant in the fields of mechanical and industrial engineering, based in Sheffield, UK.

xiii

Preface

This book was written as a textbook or guidebook on fluid mechanics for students or junior engineers studying mechanical or civil engineering. The recent progress in the science of visualization and computational fluid dynamics is astounding. In this book, effort has been made to introduce students/engineers to fluid mechanics by making explanations easy to understand, including recent information and comparing the theories with actual phenomena. Fluid mechanics has hitherto been divided into “hydraulics”, dealing with the experimental side, and “hydrodynamics”, dealing with the theoretical side. In recent years, however, both have merged into an inseparable single science. A great deal was contributed by developments in the science of visualization and by the progress in computational fluid dynamics using advances in computers. This book is written from this point of view. The following features are included in the book: 1. Many illustrations, photographs, and items of interest are presented for easy reading. 2. Portrait sketches of 18 selected pioneers who contributed to the development of fluid mechanics are inserted, together with brief descriptions of their achievements in the field. 3. Related major books and papers are presented in footnotes to facilitate advanced study. 4. Exercises appear at the ends of chapters to test understanding of the chapter topic. 5. Special emphasis is placed on flow visualization and computational fluid dynamics by including 25 color images in the frontispiece section to assist understanding. The 14 images in the previous edition have been revised and the 25 images are selected for the revision. 6. Many computational fluid dynamics and flow visualization videos (eBook) and illustrations are replaced or added in the second edition of the book. xv

xvi

Preface

All chapters have been reviewed and many changes in most chapters have been made to better understand fluid mechanics. Among them, Chapter 15, Computational Fluid Dynamics, especially has the most significant modification as a result of tremendous development in the last few decades. Professor Yasuki Nakayama sadly passed away in 2016 at the age of 99 years. He had been preparing the second edition of this book just before his death. His strong will to publish this book made us fulfill his last wish. We wish to acknowledge Dr John Tippetts in Sheffield, UK, who proofread the book and provided many helpful suggestions for improving the manuscript. We express our special appreciation to Yokendo Co. Ltd for giving permission to use figures and illustrations present in the Japanese version of this book for the second edition of English book. We would like to express special appreciation to The Visualization Society of Japan for permission to use various proceedings, journals and other publications. Our appreciation also goes to ANSYS, Inc. for permission to use ANSYSÒ FluentÒ 17.0 for creating animation of fluid flow by computational fluid dynamics for eBook. Finally, we thank the editorial and publication staff of Elsevier for their great assistance and cooperation. Kazuyasu Izawa Makoto Oki Katsumi Aoki

CHAPTER 1

History of Fluid Mechanics 1.1

FLUID MECHANICS IN EVERYDAY LIFE

There is air around us, and there are rivers and seas near us. ‘The flow of a river never ceases. The river endures but the water forever changes. Bubbles floating on the stagnant water now vanish and then develop but never remain.’ So stated Kamo no Choumei, the famous, 13th century essayist of Japan, in the prologue of H oj oki, his collection of essays. Such a movement of gas or liquid (collectively called fluid) is called the ‘flow’ and the study of this field is called ‘fluid mechanics’. While the flows of air in the atmosphere and water in rivers and the sea are flows of our concern, so also are the flows of water and sewage, in gas pipes, irrigation canals, and the flow around rockets, airplanes, bullet trains, automobiles and ships. The resistance which acts on such bodies is a ubiquitous problem. The trajectories of balls in baseball and golf are all influenced by flow. Furthermore, the movement of people on the platform of a railway station or at the intersection of a street can be regarded as forms of flow. In a wider sense, the movement of social phenomena and information, and history could be regarded as flows, too. In these ways we have a close relationship with flow, so ‘fluid mechanics’ which studies flow, is really a natural topic for our attention.

1.2

THE BEGINNING OF ‘FLUID MECHANICS’

The science of flow has been classified into hydraulics, which developed from experimental studies, and hydrodynamics which developed through theoretical studies. In recent years, however, both have merged into the single discipline called fluid mechanics. Hydraulics developed as a purely empirical science with practical techniques beginning in prehistorical times. As our ancestors settled to engage in farming and their hamlets developed into villages, the continuous supply of a proper 1 Introduction to Fluid Mechanics. http://dx.doi.org/10.1016/B978-0-08-102437-9.00001-2 Copyright © 2018 Elsevier Ltd. All rights reserved.

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CHAPTER 1:

History of Fluid Mechanics

quantity of water and the transport of essential food and materials posed the most important problem. In this sense, it is believed that hydraulics was born in the utilization of water channels and ships. Prehistoric relics of irrigation canals were discovered in Egypt and Mesopotamia, and it has been confirmed that canals had been constructed more than 4000 years BC. Water in cities is said to have begun in Jerusalem, where a reservoir to store water and a masonry channel to carry the water were constructed. Water canals were also constructed in Greece and other places. Above all, however, it was the Romans who constructed aqueducts throughout the Roman Empire. Even today their remains are still visible in many places in Europe (Fig. 1.1). The city water system in those days conveyed relatively clear water from far away to fountains, baths and public buildings. Citizens then fetched the water from water supply stations at high street corners, etc. The quantity of water used per day by a citizen in those days is said to be approximately 180 L. Today, the amount of water used per capita per day in an average household is said to be approximately 240 L. Therefore, according to this measure even about 2000 years ago, an impressive degree of amenity was attained. As stated above, the history of the city water system is very old. But in the development process of city water systems, to transport water effectively, the shape and size of the water conduit had to be designed and its inclination

FIGURE 1.1 Restored arch of Roman aqueduct in Campania Plain, Italy.

1.2 The Beginning of ‘Fluid Mechanics’

or supply pressure had to be adjusted to overcome wall friction in the conduit. This gave rise to much invention and progress in overcoming hydraulic problems. On the other hand, the origin of the ship is not clear, but it is easy to imagine the course of progress from log to raft, from manual propulsion to sails and from river to ocean navigation. The Phoenicians and Egyptians built huge, excellent ships. The relief work shown in Fig. 1.2, which was made about 2700 BC, clearly depicts a ship which existed at that time. The Greeks also left various records of ships. One of them is a beautiful picture of a ship depicted on an old Grecian vase, as shown in Fig. 1.3. As these objects indicate,

FIGURE 1.2 Relief of ancient Egyptian ship.

FIGURE 1.3 Ancient Greek ship depicted on old vase.

3

4

CHAPTER 1:

History of Fluid Mechanics

it was by progress in ship building and also navigation techniques that allowed much fundamental hydraulic knowledge to be accumulated. Another noteworthy finding is the discovery of the world’s first vortex produced by the Jomon people. A Jomon pot, shown in Fig. 1.4, was made about 5000 years ago in Japan and the vortex pattern depicted on the side wall is the world’s first to show twin vortices and Kármán vortices separately.1,2 In addition, many pots decorated with simple depictions of Kármán vortices have been discovered in this region. Before proceeding to describe the development of hydraulics, the Renaissance period of Leonard da Vinci in particular should be recalled. Popularly he is well known as a great artist, but he was an excellent scientist, too. He was so well versed in the laws of natural science that he stated that ‘a body gives air the same force as the resistance which air gives the body’. These statements preceded Newton’s law of gravity and the third law of motion (law of action and reaction).

FIGURE 1.4 Earthenware with flame or water vortex ornamentation, 3000 years BC (unearthed location, Umataka, Niigata, Japan).1,2

1

The Niigata Prefectural Museum of History, Country of Earthen Pots with a Flame Design, Niigata Nippo Jigyousha Corporation, 2009, 88. 2 Y. Nakayama, et al., Japan Society for Archaeological Information 10 (1) (2004) 1.

1.2 The Beginning of ‘Fluid Mechanics’

Leonardo da Vinci (1452e1519). An all-round genius born in Italy. His unceasing zeal for the truth and incomparable power of imagination are apparent in numerous sketches and astonishing design charts of implements, precise human anatomical charts and flow diagrams of fluids. He drew streamlines and vortices on these flow charts, which almost satisfy our present needs. It can therefore be said that he ingeniously suggested modern flow visualization.

Particularly interesting in the history of hydraulics is Leonardo’s note where an extensive description is made of the water movements, including vortices, waves, falling water, buoyancy, outflow, pipe flow and water channel flow in hydraulic machinery. As examples, Fig. 1.5 is a sketch of the flow around an obstacle, and Fig. 1.6 shows the development of vortices in the separation region. Leonardo was the first to find the least resistive streamline shape. In addition, he made many discoveries and observations in the field of hydraulics. He forecast laws such as the drag and movement of a jet or falling water which only later scholars were to discover. Furthermore, he advocated the observation of internal flow by floating particles in water, that is, ‘visualization of the flow’. Indeed, Leonardo was a great pioneer who opened up the field of hydraulics. Excellent researchers followed in his footsteps, and hydraulics progressed greatly from the 17th to the 20th century. On the other hand, the advent of hydrodynamics, which tackles fluid movement both mathematically and theoretically, was considerably later than that

5

6

CHAPTER 1:

History of Fluid Mechanics

FIGURE 1.5 Sketches from Leonardo da Vinci’s notes (No. 1).

FIGURE 1.6 Sketches from Leonardo da Vinci’s notes (No. 2).

of hydraulics. Its foundations were laid in the 18th century. Complete theoretical equations for the flow of non-viscous (non-frictional) fluid were derived by Euler (see Section 5.2.1) and other researchers. Thereby, various flows were mathematically describable. Nevertheless, the computation according to these theories of the force acting on a body or the state of flow differed greatly from the experimentally observed result. In this way, hydrodynamics was thought to lack practical use. In the 19th century, however, it advanced sufficiently to compete fully with hydraulics. One example of such progress was the derivation of the equation for the motion of a viscous fluid by Navier and Stokes. Unfortunately, because this equation has convection terms among the terms expressing the inertia (the terms expressing the force which varies from place to place), which renders

1.2 The Beginning of ‘Fluid Mechanics’

the equation nonlinear, it was not easy to obtain the analytical solution for general flows. Only such special flows as laminar flow between parallel plates or in a circular pipe were solved. Meanwhile, however, in 1869 an important paper was published which connected hydraulics and hydrodynamics. This was the report in which Kirchhoff, a German physicist (1824e87), computed the coefficient of contraction for the jet from a two-dimensional orifice as 0.611. This value coincided very closely with the experimental value for the case of an actual orificedapproximately 0.60. As it was then possible to compute a value near the actual value, hydrodynamics was re-evaluated by hydraulics scholars. Furthermore, in the present age, with the progress in electronic computers and the development of various numerical techniques in hydrodynamics, it is now possible to obtain numerical solutions of the NaviereStokes equation. Thus, the barrier between hydraulics and hydrodynamics has now been completely removed, and the field is probably on the eve of a big leap into a new age.

7

CHAPTER 2

Characteristics of a Fluid 2.1

FLUID

Fluids are divided into liquids and gases. Liquid is hard to compress and as in the ancient saying ‘Water takes the shape of the vessel containing it’, it changes its shape according to the shape of its container and has an upper free surface. Gas on the other hand is easy to compress, and fully expands to fill its container. There is thus no free surface. Consequently, an important characteristic of a fluid from the viewpoint of fluid mechanics is its compressibility. Another characteristic is its viscosity. Whereas a solid shows its elasticity in tension, compression, or shearing stress, a fluid does so only for compression. In other words, a fluid increases its pressure against compression, trying to retain its original volume. This characteristic is called compressibility. Furthermore, a fluid shows resistance whenever two layers slide over each other. This characteristic is called viscosity. In general, liquids are called incompressible fluids and gases compressible fluids. Nevertheless, for liquids, compressibility must be taken into account whenever they are highly pressurized, and for gases, compressibility may be disregarded whenever the change in pressure is small. Although a fluid is an aggregate of molecules in constant motion, the mean free path of these molecules is 0.06 mm or so even for air of normal temperature and pressure, so a fluid is treated as a continuous isotropic substance. Meanwhile, a nonexistent, assumed fluid without either viscosity or compressibility is called an ideal fluid or perfect fluid. A virtual fluid without compressibility or viscosity is called an ideal fluid or perfect fluid. Furthermore, a gas subject to Boyle’seCharles’ law is called a perfect gas or ideal gas.

2.2

UNITS AND DIMENSIONS

All physical quantities are given in a few fundamental quantities or their combinations. The units of such fundamental quantities are called base units, combination of them being called derived units. The system in which length, 9 Introduction to Fluid Mechanics. http://dx.doi.org/10.1016/B978-0-08-102437-9.00002-4 Copyright © 2018 Elsevier Ltd. All rights reserved.

10

CHAPTER 2:

Characteristics of a Fluid

mass, and time are adopted as the basic quantities, and from which the units of other quantities are derived, is called the absolute system of units.

2.2.1

Absolute System of Units

MKS System of Units This is the system of units where the metre (m) is used for the unit of length, kilogram (kg) for the unit of mass and second (s) for the unit of time as the base units.

CGS System of Units This is the system of units where the centimetre (cm) is used for length, gram (g) for mass, and second (s) for time as the base units.

International System of Units SI, the abbreviation of La Systeme International d’Unites, is the system developed from the MKS system of units. It is a consistent and reasonable system of units which makes it a rule to adopt only one unit for each of the various quantities used in such fields as science, education and industry. There are seven fundamental SI units, namely, metre (m) for length, kilogram (kg) for mass, second (s) for time, ampere (A) for electric current, Kelvin (K) for thermodynamic temperature, mole (mol) for mass quantity and candela (cd) for intensity of light (Table 2.1). Derived units consist of these units (Table 2.2).

SI Prefixes It is often useful to use prefixes with the SI units for very large or very small numerical values. The SI prefixes used to form decimal multiples and submultiples of SI units are specified as shown in Table 2.3.

Table 2.1 SI Basic and Supplementary Units Quantity

Name

Units

Length Mass Time Electric current Thermodynamic temperature Mass quantity Luminous intensity Plane angle

metre kilogram second ampere kelvin mole candela radian

m kg s A K mol cd rada

a

Supplementary unit.

2.2 Units and Dimensions

Table 2.2 SI Derived Units Quantity

Name

Unit

Velocity Acceleration Pressure Stress Viscosity Kinematic viscosity Force Moment of torque Energy Power Angular acceleration Rotational speed Frequency

Metre per second Metre per second squared Pascal Pascal Pascal second Metre squared per second Newton Newton metre Joule Watt Radian per second Rotation per second Hertz

m/s m/s2 Pa (¼N/m2) Pa (¼N/m2) Pa s m2/s N Nm J (N m) W (J/s) rad/s s 1 Hz

Table 2.3 SI Prefixes Factor

Name

Symbol

Factor

Name

Symbol

Factor

Name

Symbol

1024 1021 1018 1015 1012 109 106

Yotta Zetta Exa Peta Tera Giga Mega

Y Z E P T G M

103 102 101 10e1 10e2 10e3 10e6

Kilo Hecto Deka Deci Centi Milli Micro

k h da d c m m

10e9 10e12 10e15 10e18 10e21 10e24

Nano Pico Femto Atto Zepto Yocto

n p f a z y

2.2.2

Dimension

All physical quantities are expressed in combinations of base units. The index number of the combination of base units expressing a certain physical quantity is called the dimension, as follows. In the absolute system of units, the length, mass and time are, respectively, expressed by L, M and T. Put Q as a certain physical quantity and c as proportional constant, and assume that they are expressed as follows: Q ¼ cLa Mb T g

(2.1)

where a, b and g are, respectively, called the dimensions of Q for L, M and T. Table 2.4 shows the dimensions of various quantities.

11

12

CHAPTER 2:

Characteristics of a Fluid

Table 2.4 Dimensions and Units Absolute System of Units

2.3

Quantity

a

b

g

Units

Length Mass Time Velocity Acceleration Density Force Pressure, stress Energy, work Viscosity Kinematic viscosity

1 0 0 1 1

0 1 0 0 0 1 1 1 1 1 0

0 0 1

m kg s m/s m/s2 kg/m3 N Pa J Pa s m2/s

3 1 1 2 1 2

1 2 0 2 2 2 1 1

DENSITY, SPECIFIC GRAVITY AND SPECIFIC VOLUME

The mass per unit volume of material is called the density, which is generally expressed by the symbol r. The density of a gas changes according to the pressure, but that of a liquid may be considered unchangeable in general. The units of density are kg/m3 (SI). The density of water at 4 C and l atm (101 325 Pa, standard atmospheric pressure; see Section 3.1.1) is 1000 kg/m3. The ratio of the density of a material r to the density of water rw is called the specific gravity, which is expressed by the symbol s: density of a material r is called the specific gravity, which is expressed by the symbol s. s¼

r rw

(2.2)

The reciprocal of density, i.e., the volume per unit mass, is called the specific volume which is generally expressed by the sign n. n¼

1  3
 m kg r

(2.3)

Values for density r of water and air and specific weight g under the standard atmospheric pressure are shown on Table 2.5.

2.4 Viscosity

Table 2.5 Density of Water and Air (Standard Atmospheric Pressure) Temperature ( C) r (kg/m3)

2.4

Water Air

0

10

15

20

40

60

80

100

999.8 1.293

999.7 1.247

999.1 1.226

998.2 1.205

992.2 1.128

983.2 1.060

971.8 1.000

958.4 0.9464

VISCOSITY

As shown in Fig. 2.1, suppose that liquid fills the space between two parallel plates of area A each and gap h, the lower plate is fixed and force F is needed to move the upper plate in parallel at velocity U. Whenever Uh/n < 1500 (n ¼ m/r: kinematic viscosity), laminar flow (see Section 4.4) is maintained, and a linear velocity distribution, as shown in the figure, is obtained. Such a parallel flow of uniform velocity gradient is called Couette flow. In this case, the force per unit area necessary for moving the plate, i.e., the shearing stress (Pa) is proportional to U and inversely proportional to h. Using a proportional constant m, it can be expressed as follows: s¼

F U ¼m A h

(2.4)

The proportional constant m is called the viscosity, the coefficient of viscosity or the dynamic viscosity.

F

U A u h

y x O

FIGURE 2.1 Couette flow.

τ

τ

13

14

CHAPTER 2:

Characteristics of a Fluid

Isaac Newton (1642e1727). English mathematician, physicist and astronomer; studied at the University of Cambridge. His three big discoveries of the spectral analysis of light, universal gravitation and differential and integral calculus are only too well known. There are so many scientific terms named after Newton (Newton’s rings and Newton’s law of motion/viscosity/resistance) that he can be regarded as the greatest contributor to the establishment of modern natural science.

Newton’s statue at Grantham near Woolsthorpe, his birthplace.

Such a flow where the velocity u in the x direction changes in the y direction is called shear flow. Fig. 2.1 shows the case where the fluid in the gap is not flowing. However, the velocity distribution in the case where the fluid is flowing is

2.4 Viscosity

U

In the case of U = 0

dy

y

y x

du

O

FIGURE 2.2 Flow between parallel plates.

as shown in Fig. 2.2. Extending Eq. (2.4) to such a flow, the shear stress s on the section ddfsdsfsdgy, distance y from the solid wall, is given by the following equation: s¼m

du dy

(2.5)

This relation was found by Newton through experiment and is called Newton’s law of viscosity. In the case of gases, increased temperature makes the molecular movement more vigorous and increases molecular mixing so that the viscosity increases. In the case of a liquid, as its temperature increases molecules separate from each other, decreasing the attraction between them, and thus the velocity decreases. The relation between the temperature and the viscosity is thus reversed for gas and for liquid. Fig. 2.3 shows the change with temperature of the viscosity of air and of water. The units of viscosity are Pa s (Pascal second) in SI and g/(cm s) in CGS absolute system of units; 1 g/(cm s) in the absolute system of units is called 1 P (Poise) (because Poiseuille’s law, stated in Section 6.3.2, is utilized for measuring the velocity, the unit is named after him), while its 1/100th part is 1 cP (centipoise). Thus, 1 P ¼ 100 cP ¼ 0:1 Pa s Value n obtained by dividing viscosity m by density r is called the kinematic viscosity or the coefficient of kinematic viscosity. m n¼ (2.6) r Because the effect of viscosity on the movement of fluid is expressed by n, the name kinematic viscosity is given. This unit is m2/s regardless of the system of

15

Characteristics of a Fluid

×10–5 2.2

2.1

140

2.0

r

180

1.9

100

Viscosity of air, µ (Pa s)

×10–5 220

Ai

CHAPTER 2:

Viscosity of water, µ (Pa s)

16

1.8

60 wa

ter

20 0

20

40 60 80 Temperature (°C)

1.7 100

FIGURE 2.3 Change in viscosity of air and of water under 1 atm.

units. In the CGS system of units, l cm/s is called 1 St (Stokes) (because Stokes’ equation, to be stated in Section 9.3.3, is utilized for measuring viscosity, it is named after him), while its 1/100th part is 1 cSt (centistokes). Thus,
m2 s
1 cSt ¼ 1  10 6 m2 s 1 St ¼ 1  10

4

The viscosity m and the kinematic viscosity n of water and air under standard atmospheric pressure are given in Table 2.6. The kinematic viscosity n of oil is approximately 30e100 cSt. Viscosity sensitivity to temperature is expressed by the viscosity index VI,1 a nondimensional number. A large VI of 100 is assigned to the least temperature sensitive oil and 0 to the most sensitive. Although oil is used under high pressure in many cases, the viscosity of oil is apt to increase somewhat as the pressure increases.

1

ISO 2909.

2.5 Newtonian Fluid and Non-Newtonian Fluid

Table 2.6 Viscosity and Kinematic Viscosity of Water and Air at Standard Atmospheric Pressure Water

Air

Temp. ( C)

Viscosity, m (Pa s 3 10L5)

Kinematic Viscosity, n (m2/s 3 10L6)

Viscosity, m (Pa s 3 10L5)

Kinematic Viscosity, n (m2/s 3 10L6)

0 10 20 30 40

179.2 130.7 100.2 79.7 65.3

1.792 1.307 1.004 0.801 0.658

1.724 1.773 1.822 1.869 1.915

13.33 14.21 15.12 16.04 16.98

2.5

NEWTONIAN FLUID AND NON-NEWTONIAN FLUID

Viscous fluid is classified in terms of whether fluid flow follows Newton’s law of viscosity or not. For water, oil, or air, the shearing stress s is proportional to the velocity gradient du/dy. In other words, fluid with constant viscosity or coefficient of viscosity is Newtonian fluid and fluid with non-constant viscosity is non-Newtonian fluid. Fig. 2.4 shows the relationship between force applied and viscosity of various fluids. Typical examples of non-Newtonian fluids are Bingham fluid (plastic fluid), pseudoplastic fluid, dilatant fluid, etc.

2.5.1

Bingham Fluid (Plastic Fluid)

An example is solid butter. It does not flow until a critical force is applied. Once the critical force is exceeded, it flows like a Newtonian fluid with constant viscosity.

Viscosity

pseudoplastic fluid

Newtonian fluid

Bingham fluid

dilatant fluid

Force applied

FIGURE 2.4 Rheological diagram.

17

18

CHAPTER 2:

Characteristics of a Fluid

2.5.2

Pseudoplastic Fluid

In contrast to a Bingham fluid, a pseudoplastic fluid is a fluid that increases viscosity as force is applied. A typical example is a suspension of cornstarch in water with a concentration of one to one. This cornstarch behaves like water when no force is applied; however, it is solidified as force is applied. In addition to the above, there is fluid called time-dependent fluid. Fluid whose viscosity decreases over time is called Thixotropy fluid and fluid whose viscosity increases with time is called Rheopectic fluid. Non-Newtonian fluid with various properties exists. Their mechanical behaviour is minutely treated by rheology, the science relating to the deformation and flow of a substance.

2.6

SURFACE TENSION

The surface of liquid is apt to shrink, and its free surface is in such a state where each section pulls another as if an elastic film is stretched. The tensile strength per unit length of an assumed section of the free surface is called the surface tension. Surface tensions of various kinds of liquid are given in Table 2.7. As shown in Fig. 2.5, a dewdrop appearing on a plant leaf is spherical in shape. This is also because of the tendency to shrink because of surface tension. Consequently, its internal pressure is higher than its peripheral pressure. Putting d as the diameter of the liquid drop, T as the surface tension and p as the increase in internal pressure, the following equation is obtained owing to the balance of force as shown in Fig. 2.6. pdT ¼

pd2 Dp 4

or Dp ¼

4T d

(2.7)

The same applies to the case of small bubbles in liquid.

Table 2.7 Surface Tension of Liquid (20 C) Liquid

Surface Liquid

N/m

Water Mercury Mercury Methyl alcohol

Air Air Water Air

0.0728 0.476 0.373 0.023

2.6 Surface Tension

FIGURE 2.5 A dewdrop on a taro leaf.

π 2 d ∆p 4

d

T

∆p

FIGURE 2.6 Balance between the pressure increases within a liquid drop and the surface tension.

Whenever a fine tube is pushed through the free surface of liquid, the liquid rises up or falls in the tube as shown in Fig. 2.7 owing to the relation between the surface tension and the adhesive force between the liquid and the solid. This phenomenon is called capillarity. As shown in Fig. 2.8, d is the diameter of the tube, q the contact angle of the liquid to the wall, r the density of liquid and h the mean height of liquid surface. The following equations are obtained owing to the balance between the adhesive force of liquid stuck to the wall, trying to pull up the liquid up the tube by the surface tension, and the weight of liquid in the tube.

19

20

CHAPTER 2:

Characteristics of a Fluid

FIGURE 2.7 Change of liquid surface because of capillarity: (A) water (B) mercury. T

θ

h

ρ

d

FIGURE 2.8 Capillarity.

pdT cos q ¼

pd2 rgh 4

or h¼

4T cos q rgd

(2.8)

Whenever water or alcohol is in direct contact with a glass tube in air under normal temperature, q z 0. In the case of mercury, q ¼ 130e150 degrees. In the case where a glass tube is placed in liquid,

2.7 Compressibility

for water

h ¼ 30=d

for alcohol for mercury

h ¼ 11:6=d h¼

(2.9)

10=d

where the units are mm. Whenever pressure is measured using a liquid column, it is necessary to pay attention to the capillarity correction.

2.7

COMPRESSIBILITY

As shown in Fig. 2.9, assume that fluid of volume V at pressure p decreased its volume by DV because of the further increase in pressure by Dp. In this case, because the cubical dilatation of the fluid is DV=V, the bulk modulus K is expressed by the following equation: K¼

Dp ¼ DV=V

V

dp dV

(2.10)

Its reciprocal b b¼

1 K

(2.11)

is called the compressibility, whose value directly indicates how compressible the fluid is. For water of normal temperature/pressure, K ¼ 2.2  109 Pa and W

p+∆p

∆V V

FIGURE 2.9 Measuring of bulk modulus of fluid.

21

22

CHAPTER 2:

Characteristics of a Fluid

for air K ¼ 1.4  105 Pa assuming adiabatic change. In the case of water, b ¼ 4.85  10 10 Pa 1 and shrinks only by approximately 0.005% even if the atmospheric pressure is increased by 1 atm. Putting r as the fluid density and M as its mass, because rV ¼ M ¼ constant, assume an increase in density Dr whenever the volume has decreased by DV, and K¼r

Dp dp ¼r Dr dr

(2.12)

The bulk modulus K is closely related to the velocity a of a pressure wave propagating in a liquid, which is given by the following equation (see Section 13.2). sffiffiffiffiffiffi sffiffiffiffi dp K a¼ (2.13) ¼ dr r

2.8

CHARACTERISTICS OF PERFECT GAS

Let p be the pressure of a gas, v the specific volume, T the absolute temperature and R the gas constant. Then the following equation results from Boyle’se Charles’ law: pv ¼ RT

(2.14)

This equation is called the equation of state of the gas, and v ¼ 1/r (SI) as shown in Eq. (2.3). The value and unit of R varies as given in Table 2.8. A gas subject to Eq. (2.14) is called a perfect gas or an ideal gas. Strictly speaking, all real gases are not perfect gases. However, any gas at a considerably higher temperature than its liquefied temperature may be regarded as approximating to a perfect gas.

Table 2.8 Gas Constant R and Ratio of Specific Heat k Density (kg/m3)

R (SI) m2/(s2 K)

k [ cp/cv

2078.1 287.1 296.9 4124.8 259.8 189.0 518.7

1.66 1.402 1.400 1.409 1.399 1.301 1.319

Gas

Symbol

(0 C,

Helium Air Carbon monoxide Hydrogen Oxygen Carbon dioxide Methane

He e CO H2 O2 CO2 CH4

0.1785 1.293 1.250 0.0899 1.429 1.977 0.717

760 mm Hg)

2.9 Problems

Adiabatic

Pressure, p

Polytropic

Isothermal Isobaric n=0

n=1 Isochoric 1 < n h2, p > 0. Consequently, it is possible to have the upper plane supported above the lower plane. This pressure distribution is illustrated in Fig. 6.23. By integrating this pressure, and the supporting load P per unit width of bearing is obtained.   Z l 6mUl2 h1 h1 h2 P¼ pdx ¼ log 2 (6.71) h2 h1 þ h2 ðh1 h2 Þ2 0 From Eq. (6.71), the force P due to the pressure reaches a maximum when h1/ h2 ¼ 2.2. At this condition, P is as follows: Pmax ¼ 0:16

mUl2 h22

(6.72)

This slide bearing is mostly used as a thrust bearing. The theory of lubrication above was first analysed by Reynolds.

6.6 Theory of Lubrication

The principle of the journal bearing is almost the same as the above case. However, since oil-film thickness h is not expressed by the linear equation of x as shown by Eq. (6.67), the computation is a little more complicated. This analysis was performed by Sommerfeld and others.

HOMER SOMETIMES NODS This is an example in which even such a great figure as Prandtl made a wrong assumption. On one occasion, under the guidance of Prandtl, Hiementz set up a tub to make an experiment for observing a separation point computed by the boundary layer theory. Contrary to his expectation, the flow observed in the tub showed violent vibrations. Hearing of the above vibration, Prandtl responded, ‘It was most likely caused by the imperfect circularity of the cylinder section shape.’ Nevertheless, however carefully the cylinder was reshaped, the vibrations never ceased. Karman, then an assistant to Prandtl, assumed there was some essential natural phenomenon behind it. He tried to compute the stability of vortex alignment. Summarizing the computation over the weekend, he showed the summary to Prandtl on Monday for his criticism. Then, Prandtl said ‘Not a bad job. Make it up into a paper as quickly as possible. I will submit it for you to the Academy.’

A BIRD STALLS Karman hit upon the idea of making a bird stall by utilizing his knowledge in aerodynamics. When he was standing on the bank of Lake Constance with a piece of bread in his hand, a gull approached him to snatch the bread. Then he slowly withdrew his hand, and the gull tried to slow down its speed for snatching. To do this, it had to increase the lift of its wings by increasing their angles of attack. In the course of this, the angles of attack probably exceeded their effective limits. Thus, the gull sometimes lost its speed and fell (see ‘stall’, Section 9.4.2).

131

132

CHAPTER 6:

Flow of Viscous Fluid

Benarl and Karman Karman’s train of vortices (‘vortex street’) has been known for so long that it is said to appear on a painting inside an ancient church in Italy. Even before Karman, however, Professor Henry Benarl (1874e1939) of a French university observed and photographed this train of vortices. Therefore, Benarl insisted on his priority in observing this phenomenon at a meeting on International Applied Dynamics. Karman responded at the occasion ‘I am agreeable to calling Henry Benarl Street in Paris what is called Karman Street in Berlin and London’. With this joke, the two became good friends.

6.7

PROBLEMS

1. Show that the continuity equation in the flow of a two-dimensional compressible fluid is as follows: vr vðruÞ vðrvÞ þ þ ¼0 vt vx vy 2. If the flow of an incompressible fluid is axially symmetric, develop the continuity equation using cylindrical coordinates. 3. If flow is laminar between parallel plates, derive equations expressing (1) the velocity distribution, (2) the mean and maximum velocity, (3) the flow quantity and (4) pressure loss. 4. If flow is laminar in a circular tube, derive equations expressing (1) the velocity distribution, (2) the mean and maximum velocity, (3) the flow quantity and (4) pressure loss. 5. If flow is turbulent in a circular tube, assuming a velocity 1=7 distribution u ¼ umaxðy=r0 Þ , obtain (1) the relationship between the

φ 122

φ 125

6.7 Problems

160

FIGURE 6.24 Friction force acting on the piston in a cylinder.

mean velocity and the maximum velocity and (2) the radius of the fluid flowing at mean velocity. 6. Water is flowing at a mean velocity of 4 cm/s in a circular tube of 1=7

diameter 50 cm. Assume the velocity distribution u ¼ umaxðy=r0 Þ . If the

7.

8.

9. 10.

shearing stress at a location 5 cm from the wall is 5.3  10 3N/m2, compute the turbulent kinematic viscosity and the mixing length. Assume that the water temperature is 20 C and the mean velocity is 0.8 times the maximum velocity. Consider viscous fluid flowing in a laminar state through the annular gap between concentric tubes. Derive an equation which expresses the amount of flow in this case. Assume that the inner diameter is d, the gap is h, and h  d. Oil of 0.09 Pa s (0.9 P) fills a slide bearing with a flat upper face of length 60 cm. A load of 5  102N per 1 cm of width is desired to be supported on the upper surface. What is the maximum oil-film thickness when the lower surface moves at a velocity of 5 m/s? ffi pffiffiffiffiffiffiffiffiffi Show that the friction velocity s0 =r (s0: shearing stress of the wall; r: fluid density) has the dimension of velocity. The piston shown in Fig. 6.24 is moving from left to right in a cylinder at a velocity of 6 m/s. Assuming that lubricating oil fills the gap between the piston and the cylinder to produce an oil-film, what is the friction force acting on the moving piston? Assume that kinematic viscosity of oil n ¼ 50 mm2/s, specific gravity ¼ 0.9, diameter of cylinder d1 ¼ 122 mm, diameter of piston d2 ¼ 125 mm, piston length l ¼ 160 mm and that the pressure on the left side of the piston is higher than that on the right side by 10 kPa.

133

CHAPTER 7

Flow in Pipes Consider the flow of an incompressible viscous fluid that flows in a full pipe. In the preceding chapter, efforts were made analytically to find the relationship between the velocity, pressure, etc. for this case. In this chapter, however, from a more practical and materialistic standpoint, a method of expressing the loss using an average flow velocity is stated. By extending this approach, studies will be made on how to express losses caused by a change in the cross-sectional area of a pipe, a pipe bend and a valve in addition to the frictional loss of a pipe. Sending water by pipe has a long history. Since the time of the Roman Empire (about 1 BC), lead pipes and clay pipes have been used for the water supply system in cities as shown in Fig. 7.1.

City water pipe

FIGURE 7.1 Lead city water pipe (Roman remains, Bath, England). Introduction to Fluid Mechanics. http://dx.doi.org/10.1016/B978-0-08-102437-9.00007-3 Copyright © 2018 Elsevier Ltd. All rights reserved.

135

136

CHAPTER 7:

Flow in Pipes

7.1

FLOW IN INLET REGION

Consider a case where fluid flows from a tank into a pipe whose entrance section is fully rounded. At the entrance, the velocity distribution is roughly uniform while the static pressure head falls by v2/2g (v is the average flow velocity). Because the velocity of a viscous fluid is zero on the wall, the fluid near the wall decelerates. Deceleration continues downstream, until eventually the boundary layers merge at the centre of the pipe. The zone shown in Fig. 7.2, between the pipe entry and the point where the velocity profile is fully developed (boundary

(A) υ2 2g

λ

l υ2 d 2g H

υ2 ζ 2g ξ υ 2 2g

Laminar boundary layer

Laminar flow

d L

(B)

Inlet region

Turbulent Laminar Transition boundary boundary layer flow layer

L

Turbulent flow

Inlet region

(C)

FIGURE 7.2 Flow in a circular pipe: (A) laminar flow; (B) turbulent flow; (C) laminar flow (flow visualisation using hydrogen bubble method).

7.1 Flow in Inlet Region

layers merge), is called the inlet or entry region having a length denoted by L. Equations for L are1: Laminar flow: 8 < computation by Boussinesq

L ¼ 0:065Re$d L ¼ 0:06Re$d

:

experiment by Nikuradse

computation by Asao; Iwanami and Mori1

Turbulent flow: L ¼ 0:693Re1=4 d L ¼ ð25e40Þd

Computation by Latzko Experiment by Nikuradze

Downstream of the inlet region, the static pressure in the pipe as measured by the liquid columns stemming from the pipe, shown in Fig. 7.2, drops by H below the water level in the tank, where H¼l

l v2 v2 þx d 2g 2g

(7.1)

The length l from the inlet, l(l/d) (v2/2g) expresses the frictional loss of head (the lost energy of fluid per unit weight). x(v2/2g) expresses the pressure loss equivalent to the sum of the kinetic energy stored when the velocity distribution is fully developed plus the additional frictional energy loss in excess of that in fully developed flow consumed during the change in velocity distribution. The kinetic energy of the fluid which has attained the fully developed velocity distribution when x ¼ L is Z d=2 ru2 E¼ dr (7.2) 2pru 2 0 E is calculated by substituting the equations of the velocity distribution for laminar flow (Eq. 6.33) into u of this equation. The kinetic energy for the same flow at the average velocity is E0 ¼

pd2 rv2 v 4 2

Putting E/E0 ¼ z gives z ¼ 2. For the case of turbulent flow, it is found by experiment to be 1.09. z is known as the kinetic energy correction factor.

1

Asao, et al., Bulletin of JSME 18 (66) (1952), p. 172.

137

138

CHAPTER 7:

Flow in Pipes

The dynamic head equivalent to this energy is E ðpd2 =4Þvrg

¼z

v2 2g

(7.3)

This means that, to compensate for this increase in dynamic head when the entrance length reaches L, the dynamic head must decrease by the same amount. Furthermore, with the extra energy loss caused by the changing velocity distribution included, the value of x turns out to be much larger than z. x(v2/2g) expresses how much further the pressure would fall than for frictional loss in the inlet region of the pipe if a constant velocity distribution existed. With respect to the value of x, for laminar flow values of x ¼ 2.24 (computation by Boussinesq), 2.16 (computation by Schiller), 2.7(experiment by Hagen) and 2.36 (experiment by Nakayama and Endo)2 were reported, while for turbulent flow x ¼ 1.4 (experiment by Hagen on a trumpet-like tube without entrance).

7.2

LOSS BY PIPE FRICTION

Let us study the flow in the region where the velocity distribution is fully developed after passing through the inlet region as shown in Fig. 7.3. If a fluid is flowing in the round pipe of diameter d at the average flow velocity v, let the pressures at two points distance l apart be p1 and p2, respectively. The relationship between the velocity v and the head loss h ¼ (p1 p2)/(rg) is illustrated in Fig. 7.4, where, for the laminar flow, the head loss h is proportional to the flow velocity v as can clearly be seen from Eq. (6.38). For the turbulent flow, it turns out to be proportional to v1.75～2. The head loss is expressed by the following equation as shown in Eq. (7.1): h¼l

l v2 d 2g

(7.4)

This equation is called the DarcyeWeisbach3 equation, where the coefficient l is called the friction coefficient of the pipe.

7.2.1

Laminar Flow

In this case, from Eqs (6.38) and (7.4), l ¼ 64

2

m 64 ¼ rvd Re

(7.5)

Nakayama, Endo, Bulletin of JSME 24 (145) (1958), p. 658. In place of l, many British texts use 4f in this equation. Because friction factor f ¼ l/4, it is essential to check the definition to which a value of friction factor refers. The symbol used is not a reliable guide.

3

7.2 Loss by Pipe Friction

h p1 ρg p2 ρg

υ p2

p1 l

FIGURE 7.3 Pipe frictional loss.

(A)

(B) h h ∝ υ 1.75 - 2

log h

Nearly 60° b'

b' a'

a' a

b

Transitional region

h ∝υ

b

a

Turbulent flow 45° O Laminar flow

υ

O

FIGURE 7.4 Relationship between flow velocity and loss head. (A) General display and (B) logarithmic display.

No effect of wall roughness is seen. The reason is probably that the flow turbulence caused by the wall face coarseness is limited to a region near the wall face because the velocity and therefore inertia are small, whereas viscous effects are large in such a laminar region.

7.2.2

Turbulent Flow

l generally varies according to Reynolds number and the pipe wall roughness.

log υ

139

140

CHAPTER 7:

Flow in Pipes

Smooth Circular Pipe The roughness is inside the viscous sublayer if the height ε of wall surface roughness is ε  5n=u ðfluid dynamically smoothÞ

(7.6)

From Eq. (6.47) and Fig. 6.15, no effect of roughness is seen and l varies according to Reynolds number only; thus, the pipe can be regarded as a smooth pipe. In the case of a smooth pipe, the following equations have been developed: Equation of Blasius   l ¼ 0:3164Re 1=4 Re ¼ 3 103 w1 105

(7.7)

Equation of Nikuradse l ¼ 0:0032 þ 0:221Re

0:237



Re ¼ 105 w3 106



Equation of KarmaneNikuradse .  pffiffiffi  2   l ¼ 1 2 log10 Re l Re ¼ 3  103 w3  106 0:8

(7.8)

(7.9)

Equation of Itaya4 l¼

0:314 0:7

1:65 log10 Re þ ðlog10 ReÞ2

(7.10)

By combining Eq. (7.4) with Eq. (7.7), the relationship h ¼ cv1.75 (here c is a constant) arises, giving the relationship for turbulent flow in Fig. 7.4.

Rough Circular Pipe From Eq. (6.52) and Fig. 6.15, where ε  70n=u ðfully roughÞ

(7.11)

the wall surface roughness extends into the turbulent flow region. This defines the rough pipe case where l is determined by the roughness only and is not related to Reynolds number value.

4

M. Itaya, Journal of JSME 48, 332e333 (1945-2 to 12), p. 84.

7.3 Frictional Loss on Pipes Other Than Circular Pipes

0.12 0.10 0.09 0.08 0.07

ε /d ×10–3 33.33

0.06

16.35

0.04

λ=

λ

0.05

Re 64/

0.03

λ=

8.33 0.

31

6/ Re 0

3.97 .2

1.98

5

0.02

0.986

0.015 5

103

2

5

104

2 υd Re = v

5

105

2

5

106

FIGURE 7.5 Friction coefficient of circular pipe roughened with sand grains.5

To simulate uniform roughness, Nikuradse performed an experiment in 1933 by lacquer-pasting screened sand grains of uniform diameter onto the inner wall of a tube and obtained the result shown in Fig. 7.5.5 According to this result, whenever Re > 900/(ε/d), it turns out that l¼

1 ½1:74

(7.12)

2 log10 ð2ε=dފ2

The velocity distribution for this case is expressed by the following equation: u=u ¼ 8:48 þ 5:75 log10 ðy=εÞ

(7.13) 6

For a pipe of irregular roughness found in practice, the Moody diagram shown in Fig. 7.6 is applicable. For a new commercial pipe, l can be easily obtained from the intersection of a ε/d curve and an Re value of the Moody diagram in Fig. 7.6 using ε/d in Fig. 7.7.

7.3

FRICTIONAL LOSS ON PIPES OTHER THAN CIRCULAR PIPES

In the case of a pipe other than a circular one (e.g., oblong or oval), how can the pressure loss be found?

5 6

J. Nikuradse, V.D.I.Forshungsheft 361 (1933). L.F. Moody, N.J. Princeton, Transactions of the ASME 66 (8) (1944), p. 671.

141

CHAPTER 7:

Flow in Pipes

e = d) ε/ —λ ( √

0.05

ε /d 0.05 0.04 0.03

R

0.06

Rough pipe (Eq. (7.12))

flow inar e Lam 64/R λ=

0.1 0.09 0.08 0.07

0.02 0.015 0.01 0.008 0.006 0.004

20

0.04

0

λ

0.03 0.025

0.002 0.001 0.0008 0.0006 0.0004 0.0002

0.02 0.015

0.01 0.009 0.008

Transitional region ε/d 2.51 1 — — = – 2 log 3.71 + Re √λ √λ Equation of Colebrook 103 2

4 6 8 104 2

Sm

oo

th

flo

4 6 8 105 2

w

(E

q.

(7.

9))

0.000005 0.000001

4 6 8 106 2

4 6 8 107 2

FIGURE 7.6 Moody diagram.6 0.05 4 2

Steel pipe (commercial) ε = 0.0

0.01 8 6 4

Rivet-joined steel pipe ε = 9.15mm

2 0.001 8 6 4

Seamless pipe 2 ε = 0.0015 0.0001 8 6 4 2 0.00001 8 6 101 2

FIGURE 7.7 Roughness of commercial pipe.

H 4 6 8 102 2 4 6 8 103 2 Pipe diameter, d (mm)

0.0001 0.00005

0.00001 4 6 8 108

Re

ε /d

142

ε = 3.05 ε = 0.92 Concrete pipe ε = 0.31 Cast-iron pipe ε = 0.26 Wooden pipe ε = 0.92 Asphalt-coated cast iron pipe ε = 0.1 ε = 0.1 Zinc-galvanized iron pipe ε = 0.1

7.3 Frictional Loss on Pipes Other Than Circular Pipes

h

Flow direction

p2

p1 l

Section area A

FIGURE 7.8 Flow in oblong pipe.

Where fluid flows in an oblong pipe as shown in Fig. 7.8, let the pressure drop over length l be h; the sides of the pipe be a and b, respectively; the crosssection area of the pipe be A and the wall perimeter in contact with the fluid on the section, called the length of the wetted perimeter, be s, where the shearing stress is s0; the shearing force acting on the pipe wall of length l is ls0s; and the balancing pressure force is rghA. Then, rghA ¼ s0 sl

(7.14)

This equation shows that for a given pressure loss, s0 is determined by A/s (the ratio of the flow section area to the wetted perimeter). A/s ¼ m is called the hydraulic mean depth (see Section 8.1). In the case of a filled circular section pipe, because A ¼ (p/4)d2, s ¼ pd, the relationship m ¼ d/4 is obtained. So, for pipes other than a circular pipe, calculation is made using the following equation and substituting 4m (which is called the hydraulic diameter) as the representative size in place of d in Eq. (7.4): h¼l

l v2 ; l ¼ f ðRe; ε=4mÞ 4m 2g

(7.15)

Here, assuming Re ¼ 4mv/n, ε/d ¼ ε/4m may be found from the Moody diagram for a circular pipe. Meanwhile, 4m is described by following equations, respectively, for an oblong section of a by b and for coaxial pipes of inner diameter d1 and outer diameter d2:   ðp=4Þ d2 2 d1 2 ab 2ab 4 ¼ ; 4 ¼ d2 2ð a þ b Þ a þ b pð d 1 þ d 2 Þ

d1

(7.16)

143

144

CHAPTER 7:

Flow in Pipes

7.4

VARIOUS LOSSES IN PIPE LINES7

In a pipe line, in addition to frictional loss, head loss is caused by additional turbulence arising when fluid flows through such components as change of area, change of direction, branching, junction, bend and valve. The head loss for such cases is generally expressed by the following equation: hs ¼ z

v2 2g

(7.17)

v in the above equation is the mean flow velocity on a section not affected by the section where the head loss is produced. Where the mean flow velocity changes upstream or downstream of the loss-producing section, the larger of the flow velocities is generally used.

7.4.1

Loss With Sudden Change of Area

Flow Expansion The flow expansion loss hs for a suddenly widening pipe becomes the following, as already shown by Eq. (5.44):   ðv1 v2 Þ2 A1 2 v1 2 ¼ 1 (7.18) hs ¼ 2g A2 2g In practice, however, it becomes hs ¼ x

ðv1

v2 Þ2 2g

(7.19)

or as follows: v1 2 2g   A1 2 z¼x 1 A2 hs ¼ z

(7.20) (7.21)

Here, x is a value near 1. At the outlet of the pipe as shown in Fig. 7.9, because v2 z 0, Eq. (7.19) becomes hs ¼ x

7

v1 2 2g

Technical literature, Fluid resistance in pipe lines and fluid conduit, The Japanese Society of Mechanical Engineers (1979).

(7.22)

7.4 Various Losses in Pipe Lines

υ1

FIGURE 7.9 Outlet of pipeline.

Flow Contraction Because of inertia, section 1 (section area A1) of the fluid shrinks to section 2 (section area Ac) and then widens to section 3 (section area A2). The loss when the flow is accelerated is extremely small, followed by a head loss similar to that in the case of sudden expansion as shown in Fig. 7.10. Like Eq. (7.18), it is expressed by  2 2  2 2 ðvc v2 Þ2 A2 v2 1 v2 1 1 hs ¼ ¼ ¼ (7.23) Cc 2g Ac 2g 2g

Contraction

v1

vc

v2

Separation region Section 1 A1

FIGURE 7.10 Sudden contraction pipe.

Section 2 Ac

Section 3 A2

145

146

CHAPTER 7:

Flow in Pipes

(A)

(B)

(C)

(D)

(E)

(F)

θ

FIGURE 7.11 Inlet shape and loss factor (A) z ¼ 0.50; (B) z ¼ 0.25; (C) z ¼ 0.06e0.005; (D) z ¼ 0.56; (E) z ¼ 3.0e1.3 and (F) z ¼ 0.5 þ 0.3 cos q þ 0.2 cos2 q.

Here Cc ¼ Ac/A2 is a contraction coefficient. For example, when A2/A1 ¼ 0.1, Cc ¼ 0.61.8

Inlet of Pipe Line As shown in Fig. 7.11, the loss of head in the case where fluid enters from a large vessel is expressed by the following equation: hs ¼ z

v2 2g

(7.24)

In this case, however, z is the inlet loss factor and v is the mean flow velocity in the pipe. The value of z will be the value as show in Fig. 7.11.9

Throttle A device which decreases the flow area, bringing about the extra resistance in a pipe, is generally called a throttle. There are three kinds of throttle, i.e., choke, orifice and nozzle. If the length of the narrow section is long compared with its diameter, the throttle is called a choke. Because the orifice is explained in Sections 5.2.2 and 11.2.2, and a nozzle is dealt with in Section 11.2.2, only the choke will be explained here.

8 9

H. Richter, Rohrhydraulik, third Aufl., Springer, (1985), p. 172. J. Weisbach, Ingenieur-und Machienen-Mechanik, 1, ( 1896), p. 1003.

7.4 Various Losses in Pipe Lines

d

l

FIGURE 7.12 Choke.

The coefficient of discharge C in Fig. 7.12 can be expressed as follows, as Eq. (5.25), where the difference between the pressure upstream and downstream of the throttle is Dp: sffiffiffiffiffiffiffiffiffi pd2 2Dp Q¼C (7.25) r 4 and C is expressed as a function of the choke number s ¼ Q/(nl), C is as shown in Fig. 7.13, and is expressed by the following equations10 if the entrance is not rounded: C¼

1 1:16 þ 6:25s

0:61

(7.26)

and if the entrance is rounded: C¼

1 pffiffiffi 1 þ 5:3= s

7.4.2

(7.27)

Loss With Gradual Change of Area

Divergent Pipe or Diffuser The head loss for a divergent pipe as shown in Fig. 7.14 is expressed in the same manner as Eq. (7.19) for a suddenly widening pipe: hs ¼ x

10

ðv1

v2 Þ2 2g

Hibi, et al. Journal of the Japan Hydraulics & Pneumatics Society 2, (1971), p. 72.

(7.28)

147

CHAPTER 7:

(A)

Flow in Pipes

1.0 0.9 0.8 0.7 0.6 0.5

0.4

C

0.3

C= 0.2

0.1 100

(B)

1 1.16+6.25σ –0.61

101

Experimantal values Iwanami (l/d = 0.964–98.9) Nakayama (l/d = 2.011–16.52)

102

σ

103

1.0 0.8 C=

0.6

1 — 1+5.3/√σ

C

148

0.4

Hibi, Ichikawa, Miyagawa (l/d = 1.75–9.5)

0.2 0

Nakayama (l/d = 2.00)

101

102

103

σ

104

FIGURE 7.13 Coefficient of discharge for cylindrical chokes: (A) entrance not rounded; (B) entrance rounded.10

hs

υ 22 2g

υ 21 2g

p2 ρg

p1 ρg

θ d1

υ1

υ2 d2

Section 1 A1

FIGURE 7.14 Divergent flow.

Section 2 A2

7.4 Various Losses in Pipe Lines

1.2 1.0

ξ

0.8 0.6

+ d1 = 2.0in d1 = 0.5in

A2 : A1 = 9

:1

0.4

× d1 = 1.5in + d1 = 1.0in

A2 : A1 = 4

:1

A2 : A1 = 9

:1

A2 : A1 = 2.25 : 1

0.2

0

20°

40°

60°

80°

100° 120° 140° 160° 180° θ

FIGURE 7.15 Loss factor for divergent pipes.11,12

The value of x for circular divergent pipes is shown in Fig. 7.15.11,12 The value of x varies according to q. For a circular section, x ¼ 0.135 (minimum) when q z 5 300 ; for the rectangular section, x ¼ 0.145 (minimum) when q z 6 degrees; and x z 1 (almost constant) whenever q ¼ 50e60 degrees or more. For a two-dimensional duct, if q is small the fluid flow attaches to one of the side walls as a result of a wall attachment phenomenon (the wall effect).13 In the case of a circular pipe, when q becomes larger than the angle which gives the minimum value of x, the flow separates midway as shown in Fig. 7.16. Because of the turbulence accompanying such a separation of flow, the loss of head suddenly increases. This phenomenon is visualised in Fig. 7.17. A divergent pipe is also used as a diffuser to convert kinetic energy into pressure energy. In the case of Fig. 7.14, the following equation is obtained by applying Bernoulli’s principle: p1 v1 2 p2 v2 2 þ ¼ þ þ hs rg 2g rg 2g

11

A.H. Gibson, Hydraulics, Constable, London, (1952), p. 91. T. Uematsu, Bulletin of JSME 2.7, (1936), p. 254. 13 Because such a phenomenon where fluid flows attached to the wall face was found in 1932 by H. Coanda, it is called the Coanda effect after him. The effect is the basic principle of the technology of fluidics. 12

149

150

CHAPTER 7:

Flow in Pipes

Separation point

FIGURE 7.16 Velocity distribution in a divergent pipe.

FIGURE 7.17 Separation occurring in a divergent pipe (hydrogen bubble method), in water; inlet velocity ¼ 6 cm/s, Re (inlet port) ¼ 900, divergent angle ¼ 20 degrees.

Therefore p2

p1 rg

¼

v1 2

v2 2

(7.29)

hs

2g

Putting p2th for p2 for the case where there is no loss, r

p2th p1 v1 2 v2 2 ¼ rg 2g

(7.30)

The pressure recovery efficiency h for a diffuser is therefore h¼

p2 p1 ¼1 p2th p1

ðv1 2

hs v2 2 Þ 2g

(7.31)

Substituting in Eq. (7.28), the above equation becomes h¼1

x

v1 v2 ¼1 v1 þ v2

x

1 A1 =A2 1 þ A1 =A2

(7.32)

7.4 Various Losses in Pipe Lines

υ

A

B

θ

R

A l

d B

FIGURE 7.18 Bend.

Convergent Pipe In the case where a pipe section gradually becomes smaller, because the pressure decreases in the direction of flow, the flow runs freely without extra turbulence. Therefore, losses other than the pipe friction are normally negligible.

7.4.3

Loss Whenever the Flow Direction Changes

Bend The gently curving part of a pipe shown in Fig. 7.18 is referred to as a pipe bend. In a bend, in addition to the head loss caused by pipe friction, a loss caused by the change in flow direction is also produced. The total head loss hb is expressed by the following equation:   v2 l v2 hb ¼ zb ¼ zþ l (7.33) d 2g 2g Here, zb is the total loss factor and z is the loss factor caused by the bend effect. The values of z are shown in Table 7.1.14,15 In a bend, secondary flow is produced as shown in the figure because of the introduction of the centrifugal force, and the loss increases. If guide blades are fixed in the bend section, the head loss can be very small.

14 15

A. Hoffman, Mitt. Hydr. Inst. T. H. Munchen, 3, (1929), 45. R. Wasielewski, Mitt. Hydr. Inst. T. H. Munchen, 5, (1932), 66.

151

152

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Table 7.1 Loss Factor z for Bends (Smooth Wall Re ¼ 225,000, Coarse Wall Face Re ¼ 146,000)14,15 Wall Face

q (degrees)

R/d ¼ 1

2

3

4

5

Smooth

15 22.5 45 60 90 90

0.03 0.045 0.14 0.19 0.21 0.51

0.03 0.045 0.14 0.12 0.135 0.51

0.03 0.045 0.08 0.095 0.10 0.23

0.03 0.045 0.08 0.085 0.085 0.18

0.03 0.045 0.07 0.07 0.105 0.20

Coarse

Elbow As shown in Fig. 7.19, the section where the pipe curves sharply is called an elbow. The head loss hb is given in the same form as Eq. (7.33). Because the flow separates from the wall in the curving part, the loss is larger than in the case of a bend. Table 7.2 shows values of z for elbows.16

7.4.4

Branch Pipe and Junction Pipe

Pipe Branch As shown in Fig. 7.20, a pipe dividing into separate pipes is called a pipe branch. Putting hs1 as the head loss produced when the flow runs from pipe

d

θ

FIGURE 7.19 Elbow.

16

H. Kirchbach, W. Schubart, Mitt. Hydr. Inst. T. H. Munchen, 2, (1928), 72; 3 (1929), p. 121.

7.4 Various Losses in Pipe Lines

Table 7.2 Loss Factor z for Elbows16

z

q degrees

5 degrees

10 degrees

15 degrees

22.5 degrees

30 degrees

45 degrees

60 degrees

90 degrees

Smooth Coarse

0.016 0.024

0.034 0.044

0.042 0.062

0.066 0.154

0.130 0.165

0.236 0.320

0.471 0.687

1.129 1.265

Q1

υ1

d1

θ

Q3

d3

υ3

1

3 d2

Q2 2

υ2

FIGURE 7.20 Pipe branch.

① to pipe ③, and hs2 as the head loss produced when the flow runs from pipe ① to pipe ②, these are, respectively, expressed as follows: hs1 ¼ z1

v1 2 v1 2 ; hs2 ¼ z2 2g 2g

(7.34)

Because the loss factors z1, z2 vary according to the branch angle q, diameter ratio d1/d2 or d1/d3 and the discharge ratio Q1/Q2 or Q1/Q3, experiments were performed for various combinations. Such results were summarised.17,18

Pipe Junction As shown in Fig. 7.21, two pipe branches converging into one is called a pipe junction. Putting hs1 as the head loss when the flow runs from pipe ① to pipe ③ and hs2 as the head loss when the flow runs from pipe ② to pipe ③, these are expressed as follows: hs1 ¼ z1

v3 2 v3 2 ; hs2 ¼ z2 2g 2g

Values of z1 and z1 are similar to the case of pipe branch.17,18 17 18

G. Vogel, Mitt. Hydr. Inst. T. H. Munchen, 1 (1926), 75; 2, (1928), p. 61. F. Petermann, Mitt. Hydr. Inst. T. H. Munchen, 3 (1929), p. 98.

(7.35)

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Q1

υ1

d1

θ

1

d3

Q3 υ3 3

d2 Q2 υ2 2

FIGURE 7.21 Pipe junction.

7.4.5

Valve and Cock

Head loss in valves is brought about by changes in their section areas and is expressed by Eq. (7.17) provided that v indicates the mean flow velocity at the point not affected by the valve.

Gate Valve The valve as shown in Fig. 7.22 is called a gate valve. Putting d as the diameter and d0 as the valve opening, z varies according to d0 /d. Table 7.3 shows values of z for a l inch (2.54 cm) nominal diameter valve.19

Globe Valve Table 7.4 shows values of z for the globe valve shown in Fig. 7.23, at various openings.20

Butterfly Valve Table 7.5 shows values of z for a butterfly valve21 shown in Fig. 7.24. As the inclination angle q of the valve plate increases, the section area immediately downstream of the valve suddenly increases, bringing about an increased value of z. For a circular butterfly valve, when q ¼ 0 degree, the value of z is zz

t d

(7.36)

where t and d are thickness and diameter of the valve plate, respectively.

19

Corp, C.I., Bulletin of the University of Wisconsin, Engineering Series, vol. 9-1, (1922), p. 1. I. Oki, Suirikigaku (Hydraulics), Iwanami, Tokyo, (1942), p. 344. 21 J. Weisbach, Ingenieur-und Meschienen-Mechanik, 1, (1896), p. 1050. 20

7.4 Various Losses in Pipe Lines

d d'

FIGURE 7.22 Gate valve.

Table 7.3 Values for z for 1-inch Gate Valves (d ¼ 25.5 mm)19 d’/d

1/8

1/4

3/8

1/2

3/4

1

z

211

40.3

10.15

3.54

0.882

0.233

Table 7.4 Values of z for 1-inch Screw-in Globe Valves (d ¼ 25.5 mm)20 l/d

1/4

1/2

3/4

1

z

16.3

10.3

7.68

6.09

Cock Table 7.6 shows values21 of z for a cock shown in Fig. 7.25. For cocks, too, as angle q increases, large changes in section area of flow are brought about, increasing the value of z.

Other Valves Values22 of z for various valves are shown on Table 7.7.

22

F.D. Yeaple, Hydraulic and Pneumatic Power Control, McGraw-Hill, New York, (1966), p. 89.

155

156

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Flow in Pipes

l

d

FIGURE 7.23 Globe valve.

Table 7.5 Values of z for Circular Butterfly Valves21 q degrees

10 degrees

20 degrees

30 degrees

50 degrees

70 degrees

z

0.52

1.54

3.91

32.6

751

υ θ

FIGURE 7.24 Butterfly valve.

Table 7.6 Values of z for Cocks21 q degrees

10 degrees

30 degrees

50 degrees

60 degrees

z

0.29

5.47

52.6

206

7.5 Pumping to Higher Levels

υ

θ

FIGURE 7.25 Cock.

7.4.6

Total Loss Along a Pipe Line

For a pipe with flow velocity v, inner diameter d and length l, the total loss from pipe entrance to exit is   2 l X v h¼ l þ z (7.37) d 2g

The first term on the right expresses the total loss by friction, while Sz(v2/2g) represents the sum of the loss heads at such sections as the entrance, bends and valves. Whenever a pipe line consists of pipes of different diameters, it is necessary to use the appropriate valve for the flow velocity for each pipe.

When two tanks with a water-level differential h are connected by a pipe line, the exit velocity energy is generally lost. Therefore,  2  l X v h¼ l þ zþ1 (7.38) d 2g

However, when the pipe line is long such that l/d > 2000 and it has no valves of small opening etc., losses other than frictional loss may be neglected. Conversely, if h is known, the flow velocity could be obtained from Eq (7.37) or (7.38).

In general, for urban water pipes, v ¼ 1.0e1.5 m/s is typical for long pipe runs, while up to approximately 2.5 m/s is typical for short pipe runs. For the headrace of a hydraulic power plant, 2e5 m/s is the usual range.

7.5

PUMPING TO HIGHER LEVELS

A pump can deliver to high levels because it gives energy to the water as shown in Fig. 7.26. The head H across the pump is called the total head. The differential height Ha between two water levels is called the actual head and H ¼ Ha þ h where h is the sum of hs and hd expressing the total loss.

(7.39)

157

158

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Flow in Pipes

Table 7.7 Loss Factor for Various Valves22 Valve

Loss Coefficients, z

Relief valve

h/d z

d

d

0.05 3.35

0.1 2.85

0.15 2.4

0.2 2.4

0.25 1.7

0.3 1.35

d d h

h

Throttle area a [ pdx Section area of valve seat hole A ¼ pd2 4 When x [ d/4, a ¼ A Loss coefficient z ¼ 1.3 þ 0.2(A/a)2

Disc valve x d

Needle valve

 a ¼ p dx tan ðq=2Þ θ

d

d

Ax ¼ 0 when x ¼ 0 z ¼ 0.5 þ 0.15(A/a)2

x 2 tan2 ðq=2Þ



x

Ball valve

az0:75pdx z ¼ 0.5 þ 0.15(A/a)2

2.5 d d x

Spool valve

At full open position z ¼ 3e5.5

The volume of water that passes through a pump in a unit time is called the pump discharge. Because the energy that a pump gives water in a unit time is H per unit weight, the energy Lw given to water per unit time is Lw ¼ rgQH

(7.40)

The energy Lw as shown above is sometimes known as the water horsepower.

7.5 Pumping to Higher Levels

hd

H

Discharge water level

Ha Had

Has Suction water level hs

FIGURE 7.26 Storage pump: H, total head; Ha, actual head; Has, suction head; Had, discharge head; hs, losses on suction; hd, losses on discharge side.

The power Ls needed by a pump is called the shaft horsepower. Lw ¼h Ls

(7.41)

where h is the efficiency of the pump. Because the energy supplied to a pump is not all transmitted to the water as a result of the energy loss within the pump, it turns out that h < 1. As shown in Fig. 7.27, the curve that expresses the relationship between the pump discharge Q and the head H is called the characteristic curve or head curve. In general, the head loss h is proportional to the square of the mean flow velocity in the pipe, and therefore to the square of the pump discharge and is called the resistance curve. It must be summed with Ha to give the pump load curve. The pump discharge is given, as shown in Fig. 7.27, by the intersecting point of the head curve and this load curve.

159

160

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Flow in Pipes

H a b

Ha

a: Valve opening small b: Valve opening large

O

Qa

Qb

FIGURE 7.27 Total head and load curve of pump.

7.6

PROBLEMS

1. Verify that the kinetic energy for laminar flow in a circular pipe with a fully developed velocity distribution is twice that with uniform velocity. 2. What is the relationship between the flow velocity and the pressure loss in a circular pipe? 3. For laminar flow in a circular pipe, verify that the pipe frictional coefficient can be expressed by the following equation: l¼

64 Re

4. For turbulent flow in a circular pipe, show that, assuming the pipe frictional coefficient is subject to l ¼ 0.3164 Re 1/4, the pressure loss is proportional to a power of 1.75 of the mean flow velocity. 5. For flow in a circular pipe, with constant pipe friction coefficient, show that the frictional head loss is inversely proportional to the fifth power of the pipe diameter. Also, if the diameter is measured with a% error, what would be the percentage error in head loss? 6. How much head loss will be produced by sending 0.5 m3/min of water a distance of 2000 m using commercial steel pipes of diameter 50 mm? Also, what would be the head loss if the diameter is 100 mm? The water temperature is assumed to be 20 C.

100 m

7.6 Problems

20 m 45°

90° elbow Gate valve

10 m

45° elbow

FIGURE 7.28 System for Exercise 7.

7. What is the necessary shaft horsepower to send 1 m3/min of water through a conduit 100 mm in diameter as shown in Fig. 7.28? Assume pump efficiency h ¼ 80%, loss coefficient of sluice valve zv ¼ 0.175, of 90 degrees elbow z90 ¼ 1.265, of 45 degrees elbow z45 ¼ 0.320, and pipe frictional coefficient l¼ 0.026. 8. A flow of 0.6 m3/s of air discharges through a square duct of sides 20 cm. What is the pressure loss if the duct length is 50 m? Assume an air temperature of 20 C, standard atmospheric pressure, and smooth walls for the duct. 9. Water flows through a sudden expansion where a circular pipe of 40 mm diameter is directly connected to one of 80 mm. If the discharge is 0.08 mm3/min, find the expansion loss. 10. There is a pipeline connecting a circular tube with a diameter of 40 mm to a circular tube with a diameter of 80 mm via a diffuser spreading angle of 10 degree. If a flow rate of 0.3 m3/min is delivered to this pipe, calculate the loss head hs and the pressure recovery rate h of the spread tube.

161

CHAPTER 8

Flow in an Open Water Channel A man-made structure where water flows is called an open water channel. Roman waterworks were completed in 312 BC with a water channel as long as 16.5 km. It is said that in AD 305, 14 aqueducts were built with their water channels extending to 578 km in total. Thus, water channels have a long history. Fig. 8.1 shows a photograph of the remains of restored arch a Roman aqueduct. Open water channels have such large hydraulic mean depths that the Reynolds numbers are large too. Consequently, the flow is turbulent. Furthermore, at such large Reynolds number, the friction coefficient becomes constant and is determined by the roughness of the wall.

FIGURE 8.1 Remains of Roman aqueduct.

163 Introduction to Fluid Mechanics. http://dx.doi.org/10.1016/B978-0-08-102437-9.00008-5 Copyright © 2018 Elsevier Ltd. All rights reserved.

164

CHAPTER 8:

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8.1

FLOW IN AN OPEN WATER CHANNEL WITH CONSTANT SECTION AND FLOW VELOCITY

In an open water channel, the flowing water has a free surface and flows by the action of gravity. As shown in Fig. 8.2, assume that water flows with constant velocity v in an open channel having a constant cross-section and with a base inclined at an angle q. Now examine the balance of forces on water between two sections a distance l apart. Because the water depth is uniform, the forces F1 and F2 acting on the sections because of hydrostatic pressure balance each other. Therefore, the only force acting in the direction of the flow is that component of water weight. Because the flow is not accelerating, this force must equal the frictional force caused by the wall. If the cross-sectional area of the open channel is A, the length of wetted perimeter s and the mean value of wall shearing stress s0, then, rgAl sin q ¼ s0 sl Because q is very small, inclination i ¼ tan qxsin q Then, A s0 ¼ rg i ¼ rgmi s

(8.1)

Here, A/s ¼ m is the hydraulic mean depth.

Expressing s0 as s0 ¼ frv2/2 using the frictional coefficient f, then sffiffiffiffiffiffiffiffiffiffiffiffi 2g mi v¼ f

υ A

F1

τ0 θ

F2

τ0 s

l

ρ gAl

FIGURE 8.2 Open channel.

(8.2)

8.1 Flow in an Open Water Channel With Constant Section and Flow Velocity

Chezy pffiffiffiffiffiffiexpressed the velocity by the following equation as it was proportional to mi: pffiffiffiffiffiffi v ¼ c mi (8.3) This equation is called Chezy’s formula, with c the flow velocity coefficient. The value of c can be obtained using the GanguilleteKutter equation: c¼

23 þ ð1=nÞ þ ð0:00155=iÞ pffiffiffiffi 1 þ ½23þ ð0:00155=iފðn= mÞ

(8.4)

It is also obtainable from the Bazin equation: c¼

87 pffiffiffiffi 1 þ a= m

(8.5)

More recently, the Manning equation has often been used: 1 v ¼ m2=3 i1=2 n

(8.6)

n in Eqs (8.4) and (8.6) and a in Eq. (8.5) are coefficients varying according to the wall condition. Their values are shown in Table 8.1. In general, the flow velocity is 0.5e3 m/s. These equations and the values appearing in Table 8.1 are for the case of SI units (units m, s). The discharge of a water channel can be computed by the following equation: pffiffiffiffiffiffi 1 Q ¼ Av ¼ Ac mi ¼ Am2=3 i1=2 n

(8.7)

Table 8.1 Values of n in the GanuilleteKutter, the Manning and a in the Bazin Equations Wall Surface Condition

n

a

Smoothly shaved wooden board, smooth cement coated Rough wooden board, relatively smooth concrete Brick, coated with mortar or like, ashlar masonry Non-finished concrete Concrete with exposed gravel Rough masonry Both sides stone paved but bottom face irregular earth Deep, sand-bed river whose cross-sections are uniform Gravel-bed river whose cross-sections are uniform and whose banks are covered with wild grass Bending river with large stones and wild grass

0.010e0.013 0.012e0.018 0.013e0.017 0.015e0.018 0.016e0.020 0.017e0.030 0.028e0.035 0.025e0.033 0.030e0.040

0.06

1.30

0.035e0.050

2.0

0.46

165

166

CHAPTER 8:

Flow in an Open Water Channel

The flow velocities at various points of the cross-section are not uniform. The largest flow velocity is found to be 10%e40% of the depth below the water surface, while the mean flow velocity v is at 50%e70% depth.

8.2

BEST SECTION SHAPE OF AN OPEN WATER CHANNEL

If the section area A of the flow in an open channel is constant, and given that c and i in Eq. (8.3) are also constant, if the section shape is properly selected so that the wetted perimeter is minimized, both the mean flow velocity v and the discharge Q become maximum. Of all geometrical shapes, if fully charged, a circle has the shortest length of wetted perimeter for the given area. Consequently, a round water channel is important.

8.2.1

Circular Water Channel

Consider the relationship between water level, flow velocity and discharge for a round water channel of inner radius r as shown in Fig. 8.3. From Eqs (8.6) and (8.7)   1 A 2=3 1=2 1 A5=3 1=2 v¼ i ; Q¼ i n s n s2=3       q q r 2 ðq sin qÞ 2 q 2 A¼r r cos sin ¼ 2 2 2 2 s ¼ rq

r

FIGURE 8.3 Circular water channel.

θ

8.2 Best Section Shape of an Open Water Channel

  r sin q 1 2 q    1 1=2 r sin q 2=3 v¼ i 1 n 2 q   1 1=2 qr 8=3 sin q 5=3 1 Q¼ i n q 25=3

m¼

(8.8)

(8.9)

Consider vfull and Qfull, respectively, as the flow velocity and the discharge whenever the maximum capacity of channel is flowing.   v sin q 2=3 ¼ 1 (8.10) vfull q   Q q sin q 5=3 ¼ (8.11) 1 Qfull 2p q The relationship among q, v and Q is shown in Fig. 8.4.

1.2

1.0

υ υ full

υ υ full Q Qfull

Q Qfull

0.5

0

FIGURE 8.4 Relationship among q, v and Q.

π θ

2π

167

168

CHAPTER 8:

Flow in an Open Water Channel

B

A

H

FIGURE 8.5 Rectangular water channel.

8.2.2

Rectangular Water Channel

For the case of Fig. 8.5, obtain the section shape where s is a minimum. s ¼ B þ 2H ¼ ds ¼ dH

A þ 2H H

A þ2¼0 H2

A ¼ 2H2 Therefore, H 1 ¼ B 2 In other words, when c, A and i are constant, to maximize v and Q, the depth of the water channel should be one-half of the width.

8.3

SPECIFIC ENERGY

Many open water channel problems can be solved using the equation of energy. If the pressure is p at a point A in the open water channel in Fig. 8.6, the total head of fluid at point A is total head ¼

p v2 þ z þ z0 þ rg 2g

If the depth of open water channel is h, then h¼

p þz rg

8.3 Specific Energy

Energy line

2

υ 2g p ρg

υ

h

A z

z0 Datum level (horizontal)

FIGURE 8.6 Open water channel.

Consequently, the total head may be described as follows: total head ¼ h þ

v2 þ z0 2g

(8.12)

However, the total head relative to the channel bottom is called the specific energy E, which expresses the energy per unit weight, and if the cross-sectional area of the open water channel is A and the discharge Q, then E¼hþ

Q2 2gA2

(8.13)

This relationship is very important for analysing the flow in an open channel. There are three variables E, h and q. Keeping one of them constant gives the relation between the other two.

8.3.1

Constant Discharge

For constant discharge Q, the relation between the specific energy and the water depth is as shown in Fig. 8.7. The critical point of minimum energy occurs where dE/dh ¼ 0. dE ¼1 dh Then dA gA3 ¼ 2 dh Q

Q2 dA ¼0 gA3 dh

169

CHAPTER 8:

Flow in an Open Water Channel

Q2

h

2gA2

hc

υ c2 2g

h

170

Tranquil flow hc Rapid flow

O

Ec E

FIGURE 8.7 Curve for constant discharge.

When the channel width at the free surface is B, dA ¼ Bdh. Thus, the critical area Ac and the critical velocity vc are given as follows: 9  2 1=3 > BQ > > Ac ¼ > > = g (8.14)   > > Q gAc 1=2 > > > ; vc ¼ ¼ Ac B

Taking the rectangular open water channel as an example, when the discharge per unit width is q, Q ¼ qB. As the sectional area A ¼ hB, the water depth hc, Eq. (8.15), which makes the specific energy minimum, is obtained from Eq. (8.14). !1=3 q2 hc ¼ (8.15) g This water depth hc is called the critical water depth.

At the critical water depth hc, the specific energy (total head) in the critical situation Ec becomes: Ec ¼

q2 þ hc 2ghc 2

8.3 Specific Energy

From Eq. (8.15), q2 ¼ ghc 3 Ec ¼

hc þ hc ¼ 1:5hc 2

(8.16)

The specific energy (total head) in the critical situation Ec is thus 1.5 times the critical water depth hc. The corresponding critical velocity vc becomes, as follows from Eq. (8.15), pffiffiffiffiffiffiffi q (8.17) vc ¼ ¼ ghc hc In the critical condition, the flow velocity coincides with the travelling velocity of a wave in a water channel of small depth, a so-called long wave.

If the flow depth is deeper or shallower than hc, the flow behavior is different. When the water is deeper than hc, the velocity is smaller than the travelling velocity of the long wave and the flow is called tranquil (or subcritical) flow. When the water is shallower than hc, the velocity is larger than the travelling velocity of the long wave and the flow is called rapid (or supercritical) flow.

8.3.2

Constant Specific Energy

For the case of the rectangular open water channel, from Eq. (8.13),   q2 ¼ 2g h2 E h3 dq g  ¼ 2Eh dh q

Ec ¼ 1:5hc

 3h2 ¼ 0

(8.18)

For constant specific energy E, the relation between the discharge per unit width q and the depth of open water channel h is as shown in Fig. 8.8. Comparing Fig 8.7 with Fig 8.8, both the situation where the discharge is constant while the specific energy is minimum and that where the specific energy is constant while the discharge is maximum are found to be the same.

8.3.3

Constant Water Depth

For the case of the rectangular open water channel, from Eq. (8.13), E q2 ¼1þ h 2gh3

(8.19)

pffiffiffiffiffiffiffiffi gh3 and E/h is plotted in Fig. 8.9. In words, the The relationship between q specific energy increases parabolically from 1 with q and, when the water depth is critical, i.e., q2 ¼ gh3 , E/h ¼ 1.5.

171

172

CHAPTER 8:

Flow in an Open Water Channel

h=E

Curve, E constant

Tranquil flow hc Rapid flow

2 hc = — E 3

O q

qmax

FIGURE 8.8 Curve for constant specific energy. E/h 3.0

2.0 1.5 1.0

0

1.0

2.0

q gh3

FIGURE 8.9 Curve for constant water depth.

8.4

HYDRAULIC JUMP

Rapid flow is unstable, and if decelerated, it suddenly shifts to tranquil flow. This phenomenon is called a hydraulic jump. For example, as shown in Fig. 8.10A, when the inclination of the downstream face of a dam is steep, the flow is rapid. When the inclination becomes more gradual, the flow is unable to maintain rapid flow and suddenly shifts to tranquil flow. Fig. 8.11 shows an experimental result of this phenomenon.

8.4 Hydraulic Jump

(A)

(B) h Energy line

hl

υ 12

E2

υ 22 2g

2g

∆E

hl

h2 h2 h1

h1

O

E1

FIGURE 8.10 Hydraulic jump. (A) Flow for the bottom slope of the weir. (B) Water depth for energy change.

FIGURE 8.11 Rapid flow and hydraulic jump on a dam.

The p travelling velocity a of a long wave in an open water channel of small depth ffiffiffiffiffiffi h is gh. The ratio of the flow velocity to the wave velocity is called the Froude number. The Froude number of a tranquil flow is less than 1, i.e., the flow velocity is smaller than the wave velocity. On the other hand, the Froude number of a rapid flow is larger than 1, in other words, the flow velocity is larger than the wave velocity. Thus, tranquil flow and rapid flow in a water channel correspond to subsonic and supersonic flow, respectively, of a compressible gas. For the flow of a gas in a convergentedivergent nozzle (see Section 13.5.3), the supersonic flow which has gone through the nozzle stays supersonic if the back pressure is low. If the back pressure is high, however, the flow suddenly shifts to subsonic flow with a shock wave. In other words, there is an analogy between the hydraulic jump and the shock wave. When a hydraulic jump is brought about, energy is dissipated by it (Fig. 8.10B). Thus, erosion of the channel bottom further downstream can be prevented.

E

173

174

CHAPTER 8:

Flow in an Open Water Channel

William Froude (1810e79). He was born in England and engaged in shipbuilding. In his sixties, he started the study of ship resistance, building a boat-testing pool (approximately 75 m long) near his home. After his death, this study was continued by his son, Robert Edmund Froude (1846e1924). For similarity, under conditions of inertial and gravitational forces, the nondimensional number used carries his name.

8.5

PROBLEMS

1. It is desired to obtain 0.5 m3/s water discharge using a wooden open channel with rectangular section as shown in Fig. 8.12. Find the necessary inclination i using the Manning equation with n ¼ 0.01. 2. For a concrete-coated open water channel with the cross-section shown in Fig. 8.13, compare the discharge when the channel inclination is 0.002 obtained by the Chezy and Manning equation. Assume n ¼ 0.015.

40 cm

60 cm

FIGURE 8.12 Rectangular open channel.

8.5 Problems

2.5 m 1m

45° 3m

FIGURE 8.13 Concrete winding open channel.

r

θ

h

FIGURE 8.14 Circular open channel.

3. Find the discharge Q in a smooth, cement-coated, rectangular channel 5 m wide, water depth 2 m and inclination 1/2000 using Bazin equation. 4. Water is sent along the circular conduit in Fig. 8.14. What is the angle q and depth h which maximize the flow velocity and the discharge if the radius r ¼ 1.5 m? 5. In an open water channel with a rectangular section 5 m wide, 15 m3/s of water is flowing at 1.2 m depth. Is the flow rapid or tranquil and what is the specific energy E? 6. Find the critical water depth hc and the critical velocity vc when 12 m3/s of water is flowing in an open water channel with a rectangular section 4 m wide. 7. What is the maximum discharge Wmax for 2 m specific energy in an open water channel with a rectangular section 3 m wide? 8. Water is flowing at 20 m3/s in a rectangular channel 5 m wide. Find the downstream water depth hc necessary to cause this flow to jump to tranquil flow. 9. In what circumstances do the phenomena of rapid flow and hydraulic jump occur?

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Drag and Lift In Chapters 7 and 8 our study concerned ‘internal flow’ enclosed by solid walls. Now, how shall we consider such cases as the flight of a baseball or golf ball, the movement of an automobile or a train or when an aircraft flies in the air or where a submarine moves under the water? In this section, flows outside such solid walls, i.e., ‘external flows’, are discussed.

9.1

FLOWS AROUND A BODY

Generally speaking, flow around a body immersed in a uniform flow develops a thin layer along the body surface i.e., the boundary layer, across which is a large velocity gradient due to the viscosity of the fluid. Furthermore, the flow separates behind the body, discharging a wake with eddies. Fig. 9.1 shows the flows around a cylinder and a flat plate. The flow from an upstream point a is stopped at point b on the body surface with its velocity decreasing to zero; b is called a stagnation point. The flow divides into the upper and lower flows at point b. For a cylinder, the flow separates at point c producing a wake with eddies. Let the pressure upstream at a, which is not affected by the body be pN, the flow velocity be U and the pressure at the stagnation point be p0, Then p0 ¼ pN þ

rU2 2

(9.1)

(A) p∞ a

(B) U

U b p0

c

p∞ a

b p0

FIGURE 9.1 Flow around a body (A) cylinder; (B) plate.

177 Introduction to Fluid Mechanics. https://doi.org/10.1016/B978-0-08-102437-9.00009-7 Copyright © 2018 Elsevier Ltd. All rights reserved.

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9.2

FORCES ACTING ON A BODY

Whenever a body is placed in a flow, the body is subject to the force from the surrounding fluid. When a flat plate is aligned with the flow direction, it is only subject to a force in the downstream direction. A wing, however, is subject to the force R inclined to the flow as shown in Fig. 9.2. In general, the force R acting on a body is resolved into a component D in the flow direction U and the component L in a direction normal to U. The former is called drag and the latter lift. Drag and lift develop in the following manner. In Fig. 9.3, let the pressure of fluid acting on a given elemental area dA on the body surface be p, and the friction force per unit area be s. The force pdA due to the pressure p acts normal to dA, while the force due to the friction stress s acts tangentially. The drag Dp, which is the integration over the whole body surface of the component in the direction of the flow velocity U of this force pdA, is called form drag or pressure drag. The drag Df is the similar integration of sdA and is called the friction drag. Dp and Df are shown as follows in the form of equations: Z (9.2) Dp ¼ pdA cos q A

L

R

U D

FIGURE 9.2 Drag and lift. τ dA

U

dA

A

FIGURE 9.3 Force acting on body.

θ

pdA

9.3 The Drag of a Body

Df ¼

Z

sdA sin q

(9.3)

A

The drag D on a body is the sum of the pressure drag Dp and friction drag Df, whose proportions vary with the shape of the body. Table 9.1 shows the contribution of Dp and Df for various shapes. By integrating the component of pdA and sdA normal to U, the lift L is obtained.

9.3 9.3.1

THE DRAG OF A BODY Drag Coefficient

The drag D of a body placed in the uniform flow velocity U can be obtained from Eqs (9.2) and (9.3). This theoretical computation, however, is generally difficult except for bodies of simple shape and for a limited range of velocity. Therefore, there is no other way but to rely on experiments. In general, drag D is expressed as follows: D ¼ CD A

rU2 2

(9.4)

where A is the projected area of the body on the plane vertical to the direction of the uniform flow and CD is a non-dimensional number called the drag coefficient. Values of CD for bodies of various shapes are given in Table 9.2.

9.3.2

Drag for a Cylinder

Ideal Fluid Let us theoretically study (neglecting the viscosity of fluid) a cylinder placed in a flow. The flow around a cylinder placed at right angles to the flow of velocity U of an ideal fluid is as shown in Fig. 9.4. The velocity vq at a given point on the cylinder surface is as follows (see Section 12.5.2): vq ¼ 2U sin q

(9.5)

Table 9.1 Contributions of Dp and Df for Various Shapes Shape

Pressure Drag

Friction Drag

Dp (%)

Df (%)

U

0

100

U

z10

z90

U

z90

z10

U

100

0

179

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Table 9.2 Drag Coefficients for Various Bodies Dimensional Ratio

Body

l/d ¼ 1 2 4

Cylinder (flow direction) d U

l

7 l/d ¼ 1 2 5 10 40 N a/b ¼ 1 2 4 10 18 N

Cylinder (right angles to flow) l

U d

Oblong board (right angles to flow) b a

U

Hemisphere (bottomless) d U

Datum Area, A

pd 2 4

dl

ab

Drag Coefficient, CD 0.91 0.85 0.87 0.99 0.63 0.68 0.74 0.82 0.98 1.20 1.12 1.15 1.19 1.29 1.40 2.01 0.34 1.33

I II

pd 2 4

a ¼ 60 degrees a ¼ 30 degrees

pd 2 4

0.51 0.34

pd 2 4

1.2

I II

Cone α

U

d

Disc U

d

Ordinary passenger car

Front projection area

0.28e0.32

A U

Putting the pressure of the parallel flow as pN, and the pressure at a given point on the cylinder surface as p, Bernoulli’s equation produces the following result: pN þ

rv2 rU2 ¼pþ q 2 2

9.3 The Drag of a Body

y Vθ p

θ

U p∞

r x

O

FIGURE 9.4 Flow around a cylinder.

   r U 2 vq2 rU2  ¼ p pN ¼ 1 4 sin2 q 2 2 p pN ¼ 1 4 sin2 q ðCp : pressure coefficientÞ Cp ¼ rU2 =2

(9.6)

This pressure distribution is illustrated by the solid line for ideal fluid in Fig. 9.5, where there is left and right symmetry about the centre line at right angles to the flow. Consequently, the pressure resistance obtained by integrating

1 Turbulent separation point

0

ρ U2 /2

p — p∞

B Laminar separation point C

–1 A

–2 Ideal fluid

Reattachment point Laminar separation point

–3 0

20° 40° 60° 80° 100° 120° 140° 160° 180° θ

FIGURE 9.5 Pressure distribution around cylinder: A, Re ¼ 1.1  105; B, Re ¼ 6.7  105; C, Re ¼ 8.4  106.

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this pressure distribution turns out to be zero, i.e., no force at all acts on the cylinder. Since this phenomenon is contrary to actual experience, it is called d’Alembert’s paradox, after the French physicist (1717e83).

Viscous Fluid For a viscous flow, behind the cylinder, for very low values of Re < 5 (Re ¼ Ud/n, Re: Reynolds number; d: cylinder diameter), the streamlines come together symmetrically as at the front of the cylinder, as indicated in Fig. 9.4. If Re is increased to the range 5e40, the boundary layer separates symmetrically at the position a and a0 as shown in Fig. 9.6A and two eddies are formed rotating in opposite directions.1 Behind the eddies, the main streamlines come together. These eddies are called twin vortices. As Re increases, the eddies elongate and a periodic oscillation of the wake is observed. When Re is over 90, eddies are continuously shed alternately from the two sides of the cylinder as shown in Fig. 9.6B. When Re is up to 200, the wake is laminar eddies; where 200 < Re < 300, the wake is turbulent eddies after the transition zone. However, the boundary layer is laminar where 300 < Re < 3.0  105 and separation occurs near 80 degrees from the from stagnation point as shown in Fig. 9.6C. This arrangement of vortices is called a Kármán vortex street. When Re exceeds 3.0  105, the boundary layer starts to become turbulent and at 3.8  105, it completely becomes turbulent. The separation position moved further downstream to near 130 degrees as shown in Fig. 9.6D.2 This Reynolds number is called the critical Reynolds number Rec. For a viscous fluid, as shown in Fig. 9.6, the flow lines along the cylinder surface separate from the cylinder to develop eddies behind it. This is visualized in Fig. 9.7. For the rear half of the cylinder, just like the case of a divergent pipe, the flow gradually decelerates with the velocity gradient reaching zero. This point is now the separation point, downstream of which flow reversals occur, developing eddies (see Section 7.4.2). This separation point shifts downstream as shown in Fig. 9.6D with increased Re. The reason is that increased Re results in a turbulent boundary layer. Therefore, the fluid particles in and around the boundary layer mix with each other by the mixing action of the turbulent flow to make separation harder to occur. Fig. 9.8 shows a flow visualization of the development process from twin vortices to a Kármán vortex street.

1

V.L. Streeter, Handbook of Fluid Dynamics, McGraw-Hill, New York, (1961). B. Mutlusumer, Hydrodynamics around Cylindrical Structures, Advanced Series on Ocean Engineering, vol. 12, World Science, (1997), p. 2.

2

9.3 The Drag of a Body

(A)

(B) a a

(C)

(D)

Laminar boundary layer

l

Turbulent Laminar boundary layer boundary layer

h

h = 0.281 (Kármán’s calculation) l

Separation point

Separation point

FIGURE 9.6 Flow around a cylinder2 (A) 5 < Re < 40; (B) 40 < Re < 200; (C) 300 < Re < 3.0  105; (D) Rec < Re.

Wake

80° Separation point

FIGURE 9.7 Separation and Kármán vortex sheet (hydrogen bubble method) in water, velocity 2.4 cm/s, Re ¼ 195.

The pressure distribution on the cylinder surface in this case is like curves A, B and C in Fig. 9.5 with a reduced pressure behind the cylinder acting to produce a force in the downstream direction. Fig. 9.9 shows, for a cylinder of diameter d placed with its axis normal to a uniform flow velocity U, changes in drag coefficient CD with Re.3 When Re ¼ 1  103e2  105, CD ¼ 1e1.2 or roughly constant value. This region of

3

S.F. Hoerner, Fluid Dynamic Drag, Hoerner, Midland Park, NJ, (1965).

183

184

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Re = 1.06

Re = 32.1

Re = 212

Re = 275

FIGURE 9.8 Flow around a cylinder (Surface floating tracer method) (see Videos 1.1 and 1.2).

Re is called the subcritical region. When Re ¼ 3.8  105 or so, CD suddenly decreases to 0.3. This region of Re is called the critical region. To explain this phenomenon, it is surmised that the location of the separation point suddenly changes as it reaches this Re, as shown in Fig. 9.6D. Furthermore, CD approaches 0.8 as Re further increases. This region is called the supercritical region. G.I. Taylor (1886e1975, scholar of fluid dynamics at Cambridge University) calculated the number of vortices separating from the body every second, i.e., developing frequency f for 250 < Re < 2  105, by the following equation:   U 19:7 f ¼ 0:198 1 (9.7) d Re

9.3 The Drag of a Body

60 40 20

CD

10 8 6 4 2 1 0.8 0.6 0.4 0.2 0.1 0.08 0.06 0.04 10-1

Vertical plate

Cylinder l/d = ∞ Cylinder l/d = 5 l: Length d: Diameter

1:3 Ellipse

Streamline shape

1

10

102

103

104

105

106

Re

FIGURE 9.9 Drag coefficients for cylinders and other column-shaped bodies.3

fd/U is a dimensionless parameter called Strouhal number St [named after V. Strouhal (1850e1922), Czech physicist; in 1878, he first investigated the ‘singing’ of wires], which can be used to indicate the degree of regularity in a cyclically fluctuating flow. When the Kármán vortices develop, the body is acted on by a cyclic force, and as a result, it sometimes vibrates to produce sounds. The phenomenon where a power line ‘sings’ in the wind is an example of this. In general, most drag is produced because a stream separates behind a body, develops vortices and lowers its pressure. Therefore, to reduce the drag, it suffices to make the body into a shape from which the flow does not separate. This is the so-called streamlined shape.

9.3.3

Drag of a Sphere

The drag coefficient of a sphere changes as shown in Fig. 9.10.4 Within the range where Re is fairly high, Re ¼ 1  103e2  105, the resistance is proportional to the square of velocity and CD is approximately 0.44 (subcritical region). As Re reaches 3  105 or so, like the case of cylinder, the boundary layer changes from the laminar flow separation to the turbulent flow separation. Therefore, CD decreases to 0.1 or less (critical region). On reaching higher Re, CD gradually approaches 0.2 (supercritical region).

4

See footnote 1.

107

185

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103 102

CD

186

1

Stokes equation CD = 24 Re

Disc 1:0.75 Elliptical body

1:1.8 Elliptical body

10–1

Airship shape

10

–2

10–1

1

10

102

103

104

105

106

107

Re

FIGURE 9.10 Drag coefficients of a sphere and other three-dimensional bodies.4

Slow flow around a sphere is known as Stokes flow. From the continuity equation and the NaviereStokes equation, the drag D is as follows with velocity U and sphere diameter d: D ¼ 3pmUd CD ¼

24 Re

(9.8)

This is known as Stokes’s equation.5 This corresponds well with experiments within the range of Re < 1.

9.3.4

Drag of a Flat Plate

As shown in Fig. 9.11, as a uniform flow of velocity U flows parallel to a flat plate of length l, the boundary layer steadily develops owing to viscosity. Now, set the thickness of the boundary layer at a distance x from the leading edge of the flat plate to d. Consider the mass flow rate of the fluid rudy flowing in the layer dy within the boundary layer at the given point x. From the difference in momentum of this flow quantity rudy before and after passing over this plate, the drag D due to the friction on the plate is as follows: Z d D¼ ruðU uÞdy (9.9) 0

5

H. Lamb, Hydrodynamics, sixth ed., Cambridge University Press, (1932).

9.3 The Drag of a Body

y

U

U

u

dy

δ

y O

x l

dx

FIGURE 9.11 Flow around a flat plate.

Now, putting the wall surface friction stress as s0, and since dD ¼ s0dx, then from above Z dD d d ¼r uðU uÞdy (9.10) s0 ¼ dx dx 0

Laminar Boundary Layer Now, treating the distribution of u as a parabolic velocity distribution like the laminar flow in circular pipe, y u ¼ 2h h¼ ; d U

h2

Substituting the above into Eq. (9.10), Z dd 1 u  u dd s0 ¼ rU2 1 dh ¼ 0:133rU2 dx 0 U U dx

On the other hand,   du mU s0 ¼ m  ¼2 dy y¼0 d

(9.11)

(9.12)

(9.13)

Therefore, from Eqs (9.12) and (9.13), ddd ¼ 15:04

m dx rU

d2 n ¼ 15:04 x þ c U 2 From x ¼ 0 and d ¼ 0, c ¼ 0. Therefore, rffiffiffiffiffi nx 5:48 d ¼ 5:48 ¼ pffiffiffiffiffi x U Rx

(9.14)

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188

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However, since Rx ¼ Ux =n, substitute Eq. (9.14) into Eq. (9.13), rffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffi mrU3 rU2 n s0 ¼ 0:365 ¼ 0:730 Ux x 2

(9.15)

As shown in Fig. 9.12, the boundary layer thickness d increases in proportion to pffiffiffi pffiffiffi x, while the surface frictional stress reduces in inverse proportion to x.

The friction resistance for width b of the whole (but one face only) of that plate is expressed as follows by integrating Eq. (9.15): Z l pffiffiffiffiffiffiffiffiffiffiffiffiffi D¼ s0 dx ¼ 0:73 mrU3 l (9.16) 0

D ¼ Cf l

rU 2 2

(9.17)

Defining the friction drag coefficient as Cf, this becomes 1:46 Cf ¼ pffiffiffiffiffi Rl

(9.18)

where, Rl ¼ Ul/n. The above equations roughly coincide with experimental values within the range of Rl < 5  105.

Turbulent Boundary Layer Whenever Rl is large, the length of laminar boundary layer is so short that the layer can be regarded as a turbulent boundary layer over the full length of a flat plate. Now, assume the distribution of u to be given by, u g1=7 ¼ ¼ h1=7 (9.19) U d like turbulent flow in a circular pipe, and the following equations are obtained6:

d¼

0:37x

(9.20)

1=5

Rx

s0 ¼ 0:029rU2 D¼

0:036rU 2 l 1=5

(9.21) (9.22)

Rl

Cf ¼ 0:072Rl 6

 n 1=5 Ux

1=5

V.L. Streeter, E.B. Wylie, Fluid Mechanics, sixth ed., McGraw Hill New York, (1975), p. 272.

(9.23)

9.3 The Drag of a Body

y U

τ0

δ

O

x

FIGURE 9.12 Changes in boundary layer thickness and friction stress along a flat plate.

The above equations coincide well with experimental values within the range of 5  105 < Rl < 107. From experimental data, Cf ¼ 0:074Rl

1=5

(9.24)

gives better agreement. In the case where there is a significant length of laminar boundary layer at the front end of a flat plate, but later developing into a turbulent boundary layer, Eq. (9.12) is amended as follows: Cf ¼

0:074 1=5 Rl

1700 Rl

(9.25)

The relationship of Cf with Rl is shown in Fig. 9.13.

9.3.5

Friction Torque Acting on a Revolving Disc

If a disc revolves in a fluid at angular velocity u, a boundary layer develops around the disc owing to the fluid viscosity. Now, as shown in Fig. 9.14, let the radius of the disc be r0, the thickness be b, and the resistance acting on the elementary ring area 2prdr at a given radius r be dF. Assuming that dF is proportional to the square of the circular velocity ru of that section and the friction coefficient is f, the torque T1 due to this surface friction is as follows: dF ¼ f T1 ¼

Z

rðruÞ2 2prdr 2 r0 r¼0

rdF ¼

pf 2 5 ru r0 5

(9.26)

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CHAPTER 9:

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10–2 8 6 5 Cf

190

Cf = 0.074

Wieselsberger Gebers

Rl–1/5

Froude Kempf Schoenherr

4 3 Cf = 0.074 Rl–1/5 –(1700/Rl)

2 Cf = 1.46 Rl–1/2

10–3

2 3 4 5 6 8106

2 3 4 5 6 8107

2 3 4 5 6 8108

Rl

FIGURE 9.13 Friction drag coefficients of a flat plate.

dr

dr r0 r

r

b

FIGURE 9.14 A revolving disc.

2 3 4 5 6 8109

9.4 The Lift of a Body

Now, putting the friction coefficient at the cylindrical part of the disc as f 0 and the resistance acting on it as F 0 , F0 ¼ f 0

rðr0 uÞ2 2pr0 b 2

Torque T2 due to this surface friction is as follows: T2 ¼ pf 0 ru2 r04 b

(9.27)

Assuming f ¼ f 0 , the torque T needed for rotating this disc is   2 T ¼ 2T1 þ T2 ¼ pf ru2 r04 r0 þ b 5

(9.28)

and the power L needed in that case is   2 L ¼ Tu ¼ pf ru3 r04 r0 þ b 5

(9.29)

These relationships are used for such cases as computing the power loss due to the friction of the impeller of a centrifugal pump or water turbine.

9.4 9.4.1

THE LIFT OF A BODY Development of Lift

+

u

Consider a case where, as shown in Fig. 9.15, a cylinder placed in a uniform flow U rotates at angular velocity u but without flow separation. Since the fluid on the cylinder surface moves at circular velocity u ¼ r0u, sticking to the cylinder owing to the viscosity of the fluid, the flow velocity at a given point on the cylinder surface (angle q) is the tangential velocity vq caused by the uniform flow U plus u. In other words, 2Usin q þ r0u.

p

υθ

dθ

L

θ U p∞

FIGURE 9.15 Lift acting on a rotating cylinder.

r0

ω

x

191

192

CHAPTER 9:

Drag and Lift

Putting the pressure of the uniform flow as pN and the pressure at a given point on the cylinder surface as p, while neglecting the energy loss because it is too small, then from Bernoulli’s equation, r r pN þ U2 ¼ p þ ð2U sin qþ r0 uÞ2 2 2 Therefore,   2U sin q þ r0 u 2 U

p pN ¼1 rU 2 =2

(9.30)

Consequently, for unit width of the cylinder surface, integrate the component in the y direction of the force due to the pressure p pN acting on an elemental area r0dq, and the lift L acting on the unit width of cylinder is obtained: Z p=2 L¼2 ðp pN Þr0 sin qdq p=2

¼

r0 rU 2

¼

2

r0 rU

"

Z

p=2

Z

p=2

p=2

p=2

1



1

 r u 2 0 U

2U sin q þ r0 u U

2 #

4r0 u sin q U

sin qdq  4 sin q sin qdq 2

¼ 2pr02 urU ¼ 2pr0 urU

(9.31)

The circulation around the cylinder surface when a cylinder placed in a uniform flow U has circular velocity u is G ¼ 2pr0 u Substitute the above into Eq. (9.31), L ¼ rUG

(9.32)

This lift is the reason why a baseball, tennis or golf ball curves or slices if spinning7and is called the Magnus effect.8 The equation is called the Kuttae Joukowski equation.

7

The reason why the golf ball surface has many dimple-like hollows is to reduce the air resistance by producing turbulence around the ball and to produce an effective lift while keeping a stable flight by making the air circulation larger [see Plate 10 (see colour plate section)]. The number of rotations (called spin) per second of a golf ball can be 100 or more. 8 Magnus effect is a phenomenon in which a tangential force acts on a spinning cylinder or sphere in a uniform flow.

9.4 The Lift of a Body

In general, whenever circulation G develops owing to the shape of a body placed in the uniform flow U (e.g., aircraft wings or yacht sails) (see Section 9.4.2), lift L as in Eq. (9.32) is likewise produced for the unit width of its section.

9.4.2

Wing

Of the forces acting on a body placed in a flow, if the body is so manufactured as to make the lift larger than the drag, it is called a wing, aerofoil or blade. The shape of a wing section is called an aerofoil section, an example of which is shown in Fig. 9.16. The line connecting the leading edge with the trailing edge is called the chord, and its length is called the chord length. The line connecting the mid-points of the upper and lower faces of the aerofoil section is called the camber line. The height of the camber line from the chord is called the camber, which mostly means its maximum value in particular. The thickness of a wing as measured normal to the camber or chord is called its thickness, whose maximum value is called the maximum thickness. Furthermore, the angle a between the chord and the flow direction U is called the angle of attack. Putting the wing width as b, and the maximum projected area of the wing as A, b2/A is called the aspect ratio. Assuming the length of the chord is l, since A ¼ bl for an oblong wing, the aspect ratio becomes b2/A ¼ b/l. Of the studies on the characteristic of wing shape, those most well known have been performed by the U.S. National Advisory Committee for Aeronautics (renamed in 1959 as National Aeronautics and Space Administration)s, by the U.K. Royal Aircraft Establishment and by Germany’s Göttingen University. The particular wing shapes are named after them.

L Camber line

Thickness Leading edge

D

Trailing edge

α U

Camber

Chord l

FIGURE 9.16 An aerofoil section.

193

Drag and Lift

The lift L, drag D and moment M (moment about the wing’s leading edge or the point on the chord l/4 from the leading edge) acting on the wing are expressed, respectively, for unit width by the following equations: L ¼ CL l

rU2 2

D ¼ CD l

rU 2 2

M ¼ CM l2

(9.33)

rU2 2

CL, CD and CM are called, respectively, the lift coefficient, drag coefficient and moment coefficient to be determined by the aerofoil section, Mach number and Reynolds number. The wing characteristic is indicated by the values of CL, CD and CM for the angle of attack a, or by plotting CD and CM on the abscissa and CD on the ordinate. These plots are called the characteristic curves. Some examples of them are shown in Figs 9.17e9.19. The lift coefficient CL reaches zero at a certain angle of attack a, called the zero lift angle. As the angle of attack increases from the zero lift angle, the lift coefficient CL increases in a straight line. As it further increases, however, the

2.0

0.20 0.16

CL

1.2

0.12

0.8

0.08

0.4

CD

0.04 0

0

–0.04

–0.4

CMl/4

–0.08 –0.12

–8°

FIGURE 9.17 Characteristic curves of a wing.

CD

1.6

–4°

0°

4°

α

8°

12°

16°

20°

CMl/4

CHAPTER 9:

CL

194

9.4 The Lift of a Body

CL

α

O

α0

FIGURE 9.18 Aerofoil section and characteristic. CLmax

α = 15° 12° 9°

1.0 6° CL

3°

0.5 0°

 CL   CD  max CD min

–3°

α0 0 –6°

0.1 –9°

0.2 CD

FIGURE 9.19 Characteristic curve of a wing (liftedrag curve).

increase in CL gradually slows down, reaches a maximum value at a certain point, and thereafter suddenly decreases. This is due to the fact that, as shown in Fig. 9.20, the flow separates on the upper surface of the wing because the angle of attack has increased too much. This phenomenon is completely analogous to the separation occurring on a divergent pipe or flow behind a body and is called stall. The angle a at which CL reaches a maximum is the stalling angle and the maximum value of CL is the maximum lift coefficient. Fig. 9.18 shows the characteristic with changing wing section.

195

196

CHAPTER 9:

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FIGURE 9.20 Flow around a stalled wing (see Video 1.3).

Fig. 9.19 shows a wing characteristic by putting CD on the abscissa and CL on the ordinate and is called the liftedrag polar, from which the attack angle maximizing the liftedrag ratio CL/CD can easily be found. The reason why a wing produces lift is because a circulatory flow is produced just like for a rotating cylinder. In the case of a wing section, the circulatory flow is produced because the trailing edge is sharpened. A wing moves from a stationary state initially as shown in Fig. 9.21A. Owing to its behaviour as potential flow, a rear stagnation point develops at point A. Consequently, the flow develops into a flow running round the trailing edge B. Since the trailing edge is sharp, however, the flow is unable to run round the wing surface but separates from it producing a vortex as shown in (B) of the same figure. This vortex moves backwards being driven by the main flow. The flow on the upper surface of the wing is drawn towards the trailing edge, which itself develops into a stagnation point, and thus the flow is now as shown in (C) of the same figure. As one vortex is produced, another vortex of equal strength is also produced since the flow system as a whole should be in a net non-rotary movement. Therefore, a circulation is produced against the start-up vortex as if another vortex of equal strength in counterrotation had developed around the wing section. The former vortex is called a starting-up vortex because it is left at the starting point; the latter assumed vortex is a wing-bound vortex. The situation where the flow runs off the sharp trailing edge of a wing as stated above is called the Kutta condition or Joukowski’s hypothesis. Fig. 9.22 shows the visualized picture of a starting vortex. The blades of a blower, compressor, water wheel, steam turbine or gas turbine of the axial flow type are distributed radially in planes around the shaft and the blade sections of the same shape are found arranged at a certain spacing as shown in Fig. 9.23. This is called a cascade.

9.4 The Lift of a Body

(A) A

B

(B)

Γ=0

(C) –Γ

Γ

B

FIGURE 9.21 Development of circulation around aerofoil section. (A) Stagnation point occurs at point A, (B) vortex generation at the trailing edge, and (C) circulating flow occurs around the wing.

FIGURE 9.22 Starting vortex. Courtesy of the National Physical Laboratory.

197

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Drag and Lift

The action of a cascade is to change the flow direction with small loss by using the necessary stagger angle. The lift acting on a blade is expressed by rvNG from Eq. (9.32), where vN represents the mean flow velocity of v1 and v2. The magnitude of the circulation around a blade in a cascade is affected by the other blades giving less lift compared with a solitary blade. For the same blade section, setting the lifts of a solitary blade and a cascade blade to L0 and L, respectively, k¼

L L0

(9.34)

k is called the interference coefficient. It is a function of l/t and b and is near 1 whenever l/t is 0.5 or less.

9.5

CAVITATION

According to Bernoulli’s principle, as the velocity increases, the pressure decreases correspondingly. In the forward part on the upper surface of a wing section placed in a uniform flow as shown in Fig. 9.24, for example, the flow velocity increases while the pressure decreases. If a section of a body placed in liquid increases its velocity so much that the pressure there is less than the saturation pressure of the liquid, the liquid instantaneously boils, producing bubbles with cavities. This phenomenon is

α2

υ2

l

198

t

υ1 β α1

FIGURE 9.23 Cascade: l, chord length; l/t, solidity; t, space between blades; v1, v2, velocities at infinity in front of and behind the cascade; a1, inlet angle (angle of velocity v1 to axial direction); a2, exit angle (angle of velocity v2 to axial direction); b, stagger angle; q ¼ a1 a2, turning angle of flow.

9.5 Cavitation

υ p p∞

U

FIGURE 9.24 An aerofoil section inside the flow.

called cavitation. In addition, since gas dissolves in liquid in proportion to the pressure (Henry’s law), as the liquid pressure decreases, the dissolved gas separates from the liquid into bubbles even before the saturation pressure is reached. When these bubbles are conveyed downstream where the pressure is higher, they are suddenly squeezed and abnormally high pressure develops.9 At this point, noise and vibration occur eroding the neighbouring surface and leaving on it holes small in diameter but relatively deep, as if made by a slender drill in most cases. These phenomena as a whole are also referred to as cavitation in a wider sense. The blades of a pump or water wheel, or the propeller of a boat, are sometimes destroyed by such phenomena. They can develop on liquid-carrying pipe lines or on hydraulic devices and cause failures. The saturation pressures at various temperatures are shown in Table 9.3, while the volume ratios of air soluble in water at 1 atm are shown in Table 9.4. When an aerofoil section is placed in a flow of liquid, the pressure distribution on its surface is as shown in Fig. 9.25. As the cavity grows, the upper pressure characteristic curve lowers while vibration, etc. grow. When the liquid pressure is low and the flow velocity is large, the cavity grows further. When it grows beyond twice the chord length, the flow stabilizes, with noise and vibration reducing. This situation is called supercavitation and is applied to the wings of a hydrofoil boat. Let the upstream pressure not affected by the wing be pN, the flow velocity U and the saturation pressure pv. When the pressure at a point on the wing surface

9 According to actual measurement, a pressure of 100e200 atm, or sometimes as high as 500 atm, is brought about.

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Table 9.3 Saturation Pressure for Water Temp. ( C)

Temp. ( C)

Pa

0 10 20 30 40

608 1226 2334 4236 7375

Pa

50 60 70 80 100

12,330 19,920 31,160 47,360 101,320

Table 9.4 Solubility of Air in Water Temp. ( C)

0

20

40

60

80

100

Air

0.0288

0.0187

0.0142

0.0122

0.0113

0.0111

Cavitation development

U

B

A

Bottom

Top

C D

p∞

a

d Pressure on bottom Pressure on top Vapour pressure b

c

FIGURE 9.25 Development of cavitation on an aerofoil section.

or nearby has reached pv, cavitation develops. The ratio of pN namic pressure is expressed by the following equation: kd ¼

pN pv rU 2 =2

pv to the dy-

(9.35)

kd in this equation is called the cavitation number. When kd is small, cavitation is likely to develop.

Supplementary Data

9.6

PROBLEMS

1. Obtain the terminal velocity U of a spherical sand particle of diameter d dropping freely in water. 2. A wind of velocity 40 m/s is blowing against an electricity pole 50 cm in diameter and 5 m high. Obtain the drag and the maximum bending moment Mmax acting on the pole. Assume that the drag coefficient is 0.6 and the air density is 1.205kg/m3. 3. A smooth spherical body of diameter 12 cm is travelling at velocity of 30 m/s in windless open air under the conditions of 20 C temperature and standard atmospheric pressure. Obtain the drag D of the sphere. 4. If air at standard atmospheric pressure is flowing at velocity of 4 km/ h along a flat plate of length 2.5 m, what is the maximum value of the boundary layer thickness dmax ? What is it when the wind velocity is 120 km/h? 5. What are the torque T and the power L necessary to turn a rotor as shown in Fig. 9.14 at 600 rpm in oil of specific gravity 0.9? Assume that the friction coefficient f ¼ 0.047, r0 ¼ 30 cm and b ¼ 5 cm. 6. When walking on a country road in a cold wintry wind, whistling sounds can be heard from power lines blown by the wind. Explain the phenomenon by which such sounds develop. 7. In a baseball game, when the pitcher throws a drop or a curve, the ball significantly and suddenly goes down or curves. Find out why. 8. An oblong barge of length 10 m, width 2.5 m and draft 0.25 m is going up a river at the relative velocity of 1.5 m/s to the water flow. What are the friction resistance Df suffered by the barge and the power L necessary for navigation, assuming a water temperature of 20 C? 9. If a cylinder of radius r ¼ 3 cm and length l ¼ 50 cm is rotating at n ¼ 1000 rpm in air where a wind velocity U ¼ 10 m/s, how much lift L is produced on the cylinder? Assume that r ¼ 1.205 kg/m3 and that air on the cylinder surface does not separate. 10. A car, frontal projection area 2 m2, is running at 60 km/h in the calm air of temperature 20 C and standard atmospheric pressure. What is the drag D on the car? Assume that the resistance coefficient is 0.4.

SUPPLEMENTARY DATA Supplementary data related to this article can be found online at doi:10.1016/ B978-0-08-102437-9.00009-7.

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Dimensional Analysis and Law of Similarity Dimensional analysis is used in every field of engineering, especially in such fields as fluid dynamics and thermodynamics where problems with many variables are handled. This method derives from the condition that each term summed in an equation depicting a physical relationship must have the same dimension. By constructing non-dimensional quantities expressing the relationship among the variables, it is possible to summarize the experimental results and to determine their functional relationship. Next, to determine the characteristics of a full-scale device through model tests, besides geometrical similarity, similarity of dynamical conditions between the two is also necessary. When the above dimensional analysis is employed, if the appropriate non-dimensional quantities such as Reynolds number and Froude number are the same for both devices, the results of the model device tests are applicable to the full-scale device.

10.1

DIMENSIONAL ANALYSIS

When the dimensions of all terms of an equation are equal, the equation is dimensionally correct. In this case, whatever unit system is used, this equation holds its physical meaning. If the dimensions of all terms of an equation are not equal, dimensions must be hidden in coefficients, so only the designated units can be used. Such an equation would be void of physical interpretation. Utilizing this principle that the terms of physically meaningful equations have equal dimensions, the method of obtaining dimensionless groups of which the physical phenomenon is a function is called dimensional analysis. If a phenomenon is too complicated to derive a formula describing it, dimensional analysis can be employed to identify groups of variables which would appear in such a formula. By supplementing this knowledge with experimental data, an analytic relationship between the groups can be constructed allowing numerical calculations to be conducted.

203 Introduction to Fluid Mechanics. http://dx.doi.org/10.1016/B978-0-08-102437-9.00010-3 Copyright © 2018 Elsevier Ltd. All rights reserved.

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10.2

BUCKINGHAM’S p THEOREM

To perform the dimensional analysis, it is convenient to use the p theorem. Consider a physical phenomenon having n physical variables v1, v2, v3, ., vn and k basic dimensions.1 (L, M, T or L, F, T or such) used to describe them. The phenomenon can be expressed by the relationship among n k ¼ m non-dimensional groups p1, p2, p3, ., pm. In other words, the equation expressing the phenomenon as a function f of the physical variables f ðv1 ; v2 ; v3 ; .; vn Þ ¼ 0

(10.1)

can be substituted by the following equation expressing it as a function f of a smaller number of non-dimensional groups fðp1 ; p2 ; p3 ; .; pm Þ ¼ 0

(10.2)

This is called Buckingham’s p theorem. To produce p1, p2, p3, ., pm, k core physical variables are selected which do not form a p themselves. Each p group will be a power product of these with each one of the m remaining variables. The powers of the physical variables in each p group are determined algebraically by the condition that the powers of each basic dimension must sum to zero. By this means, the non-dimensional quantities are found among which there is the functional relationship expressed by Eq. (10.2). If the experimental results are arranged in these non-dimensional groups, this functional relationship can clearly be appreciated.

10.3 10.3.1

APPLICATION EXAMPLES OF DIMENSIONAL ANALYSIS Flow Resistance of a Sphere

Let us study the resistance of a sphere placed in a uniform flow as shown in Fig. 10.1. In this case the effect of gravitational and buoyancy forces will be neglected. First of all, as the physical quantities influencing the drag D of a sphere, sphere diameter d, flow velocity U, fluid density r and fluid viscosity m are candidates. In this case n ¼ 5, k ¼ 3, and m ¼ 5 3 ¼ 2; so the number of necessary non-dimensional groups is two. Select r, U and d as the k core physical quantities, and the first non-dimensional group p1, formed with D, is

1

In general, the basic dimensions in dynamics are threedlength [L], mass [M] and time [T], dbut as the areas of study, e.g., heat and electricity, expand, the number of basic dimensions increases.

d

10.3 Application Examples of Dimensional Analysis

U

FIGURE 10.1 Sphere in uniform flow.


p1 ¼ Drx U y dz ¼ LMT ¼ L1

3xþyþz

M1þx T

2




L

3

x
M LT

 1 y

½LŠz

(10.3)

2 y

3x þ y þ z ¼ 0

L: 1

M: 1 þ x ¼ 0 T:

2

y¼0

Solving the above simultaneously gives x¼

1; y ¼

2; z ¼

2

Substituting these values into Eq. (10.3), then p1 ¼

D rU2 d2

(10.4)

Next, m, r, y and d are selected as another set of physical quantities, and x


y p2 ¼ mrx Uy d z ¼ L 1 MT 1 L 3 M LT 1 ½LŠz (10.5) ¼ L 1 3xþyþz M1þx T 1 y L:

3x þ y þ z ¼ 0

1

M: 1 þ x ¼ 0 T:

1

y¼0

Solving the above simultaneously gives x¼

1; y ¼

1; z ¼

1

205

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Substituting these values into Eq. (10.5), then p2 ¼

m rUd

(10.6)

Therefore, from the p theorem the following functional relationship is obtained: p1 ¼ f ðp2 Þ

(10.7)

Consequently,   D m ¼f rU 2 d2 rUd

(10.8)

In Eq. (10.8), because d2 is proportional to the projected area of sphere A ¼ (pd2/4), and rUd/m ¼ Ud/n ¼ Re (Reynolds number), the following general expression is obtained: D ¼ CD A

rU 2 2

(10.9)

where CD ¼ f(Re). Eq. (10.9) is just the same as Eq. (9.4). Because CD is found to be dependent on Re, it can be obtained through experiment and plotted against Re. The relationship is that shown in Fig. 9.9. Even through this result is obtained through an experiment using, say, water, it can be applied to other fluids such as air or oil and also used irrespective of the size of the sphere. Furthermore, the form of Eq. (10.9) is always applicable, not only to the case of the sphere but also where the resistance of any body is studied.

10.3.2

Pressure Loss Caused by Pipe Friction

As the quantities influencing pressure loss Dp/l per unit length caused by pipe friction, flow velocity v, pipe diameter d, fluid density r, fluid viscosity m and pipe wall roughness ε are candidates. In this case, n ¼ 6, k ¼ 3 and m ¼ 6 3 ¼ 3. Obtain p1, p2 and p3 by the same method as in the previous case, with r, v and d as core variables: Dp d l rv2 x
y m 4 FT 2 LT 1 ½LŠz ¼ rvd x
y
ε p3 ¼ εrx vy dz ¼ ½LŠ L 4 FT 2 LT 1 ½LŠz ¼ d

Dp x y z
3 
rvd ¼ L F L l

p2 ¼ mrx vy dz ¼ L 2 FT L p1 ¼

4

FT 2

x
LT

 1 y

½LŠz ¼

(10.10) (10.11) (10.12)

10.4 Law of Similarity

Therefore, from the p theorem, the following functional relationship is obtained: p1 ¼ f ðp2 ; p3 Þ

(10.13)

and   Dp d m ε ; ¼ f l rv2 rvd d That is,   l m ε Dp ¼ rv2 f ; d rvd d

(10.14)

The loss of head h is as follows:   Dp 1 ε l v2 l v2 ¼f ; ¼l ; h¼ rg Re d d 2g d 2g

(10.15)

where l ¼ f(Re, ε/d). Eq. (10.15) is just the same as Eq. (7.4) and l can be summarized against Re and ε/d as shown in Fig. 7.6.

10.4

LAW OF SIMILARITY

When the characteristics of a water wheel, pump, boat, or aircraft are obtained by means of a model, unless the flow condition are similar in addition to the shape, the characteristics of the prototype cannot be assumed from the model test result. To make the flow condition similar, the respective ratios of the corresponding forces acting on the prototype and the model should be equal. The forces acting on the flow element are caused by gravity FG, pressure FP, viscosity FV, surface tension FT (when the prototype model is on the boundary of water and air), inertia FI and elasticity FE. The forces can be expressed as shown below: gravity force pressure force viscous force surface tension force inertial force elasticity force

FG ¼ mg ¼ rL3 g FP ¼ ðDpÞA ¼ ðDpÞL2   v 2 FV ¼ m du dy A ¼ m L L ¼ mvL FT ¼ TL FI ¼ ma ¼ rL3 TL2 ¼ rL4 T FE ¼ KL2

2

¼ rv2 L2

Because it is not feasible to have the ratios of all such corresponding forces simultaneously equal, it will suffice to identify those forces that are closely related to the respective flows and to have them equal. In this way, the

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relationship which gives the conditions under which the flow is similar to the actual flow in the course of a model test is called the law of dynamical similarity. In the following section, the more common force ratios which ensure the flow similarity under appropriate conditions are developed.

10.4.1

Non-dimensional Groups Which Determine Flow Similarity

Reynolds Number Where the compressibility of the fluid may be neglected and in the absence of a free surface, e.g., where fluid is flowing in a pipe, an airship is flying in the air (Fig. 10.2) or a submarine is navigating under water, only the viscous force and inertia force are of importance: inertia force FI rv2 L2 vLr vL ¼ ¼ ¼ Re ¼ ¼ viscous force FV m n mvL which defines Reynolds number Re, Re ¼

Lv n

(10.16)

Consequently, when the Reynolds numbers of the prototype and the model are equal, the flow conditions are similar. This relationship is called Reynolds law of similarity and Eqs (10.16) and (4.5) are identical.

Froude Number When the resistance caused by the waves produced by motion of a boat (gravity wave) is studied, the ratio of inertia force to gravity force is important: inertia force FI rv2 L2 v2 ¼ ¼ 3 ¼ gravity force FG rL g gL In general, to change v2 above to v as in the case for Re, the square root of v2/(gL) is used. This square root is defined as the Froude number Fr. v Fr ¼ pffiffiffiffiffiffi gL

FIGURE 10.2 Airship.

(10.17)

10.4 Law of Similarity

FIGURE 10.3 Ship.

If a test is performed by making the Fr of the actual boat (Fig. 10.3) and of the model ship equal, the result is applicable to the actual boat so far as the wave resistance alone is concerned. This relationship is called Froude’s law of similarity. For the total resistance, the frictional resistance must be taken into account in addition to the wave resistance. Also included in the circumstances where gravity inertia forces are important are flow in an open ditch, the force of water acting on a bridge pier and flow running out of a water gate.

Weber Number When a moving liquid has its surface in contact with another fluid or a solid, the inertia and surface tension forces are important. inertia force FI rv2 L2 rv2 L ¼ ¼ ¼ surface tension FT T TL In this case, also, the square root is selected to be defined as the Weber number We. rffiffiffiffiffi rL We ¼ v (10.18) T This relationship is called Weber’s law of similarity and applicable to the development of surface tension waves and to a poured liquid.

Mach Number When a fluid flows at high velocity, or when an aircraft (Fig. 10.4), e.g., moves at high velocity in a fluid at rest, the compressibility of the fluid can dominate so that the ratio of the inertia force to the elasticity force is then important (Fig. 10.4). Inertia force FI rv2 L2 v2 v2 ¼ ¼ ¼ ¼ Elastic force FE KL2 K=r a2

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FIGURE 10.4 Boeing 777: full length, 63.7 m; full width, 60.9m; passenger capacity, 389 persons; engine thrust 34,900 kg  2 and cruising speed of 1028 km/h (M ¼ 0.82).

Again in this case, the square root is selected to be defined as the Mach number, M. M¼

v a

(10.19)

M < 1, M ¼ 1 and M > 1 are, respectively, called subsonic flow, sonic flow and supersonic flow. When M z 1 and M < 1 and M > 1 zones are coexistent, the flow is called transonic flow.

Ernst Mach (1838e1916). Austrian physicist and philosopher. After being professor at Graz and Prague Universities became professor at Vienna University. Studied high-velocity flow of air introducing the concept of Mach number. Criticized Newtonian dynamics and took initiatives on the theory of relativity. Also made significant achievements in thermodynamics and optical science.

10.4 Law of Similarity

10.4.2

Model Testing

From such external flows as over cars, trains, aircraft, boats, high-rise buildings and bridges to such internal flows as in tunnels and various machines like pumps, water wheels, etc., the prediction of characteristics through model testing is widely employed. Suppose that the drag D on a car is going to be measured on an 1:10 model (scale ratio S ¼ 10). Assume that the full length l of the car is 3 m and the running speed v is 60 km/h. In this case, the following three methods are conceivable. Subscript m refers to the model.

Test in a Wind Tunnel To make the Reynolds numbers equal, the velocity should be vm ¼ 167 m/s, but the Mach number is 0.49 including compressibility. Assuming that the maximum tolerable value M of incompressibility is 0.3, vm ¼ 102 m/s and Rem/Re ¼ vm/(Sv) ¼ 0.61. In this case, because the flows on both the car and model are turbulent, the difference in CD caused by the Reynolds numbers is modest. Assuming drag coefficients for both D/(rv2l2/2) are equal, the drag is obtainable from the following equation: D ¼ Dm



v l vm lm

2

¼ Dm



Sv vm

2

(10.20)

This method is widely used.

Test in a Circulating Flume or Towing Tank To make the Reynolds numbers for the car and the model equal, vm ¼ vSnm/ n ¼ 11.1 m/s. If water is made to flow at this velocity, or the model moved under calm water at this velocity, conditions of dynamical similarity can be realized. The conversion formula is D ¼ Dm

  r Sv 2 rn2 ¼ Dm rm vm rm n2m

(10.21)

Test in Variable Density Wind Tunnel If the density is increased, the Reynolds numbers can be equalized without increasing the air flow velocity. Assume that the test is made at the same velocity; it is then necessary to increase the wind tunnel pressure to 10 atm assuming the temperatures are equal. The conversion formula is D ¼ Dm

r 2 S rm

(10.22)

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TWO MYSTERIES SOLVED BY MACH [No. 1] The early Artillerymen knew that two bangs could be heard downrange from a gun when a high-speed projectile was fired, but only one from a low-speed projectile. But they did not know the reason and were mystified by these phenomena. Following Mach’s research, it was realized that in addition to the bang from the muzzle of the gun, an observer downrange would first hear the arrival of the bow shock which was generated from the head of the projectile when its speed exceeded the velocity of sound. By this reasoning, this mystery was solved.

Shock wave

Bang wave

[No. 2] This is a story of the Franco-Prussian war of 1870e71. It was found that the novel French Chassepot high-speed bullets caused large crater-shaped wounds. The French were suspected of using explosive projectiles and therefore violating the international Treaty of Petersburg prohibiting the use of explosive projectiles. Mach then gave the complete and correct explanation that the explosive-type wounds were caused by the highly pressurized air caused by the bullet’s bow wave and the bullet itself. So it was clear that the French did not use explosive projectiles and the mystery was solved.

10.5 Problems

Bow shock

High pressure Tail shock

REYNOLDS OF SIMILARITY USED FOR THE SHINKANSEN For the research on the aerodynamic characteristics of the body of the Shinkansen bullet train, a pair of rail tracks were constructed in a big water tank and running tests were conducted using a model of the Shinkansen. In this way the speed of the model of the Shinkansen could be reduced to 1/15 of the actual speed and the force acting on the model was increased 800-fold to obtain the same phenomenon as for the real Shinkansen body. The aerodynamic pressure when two car bodies pass each other, and, for example, the change in pressure for a car body entering a tunnel could be accurately measured. It was very easy to visualize the phenomenon.

10.5

PROBLEMS

1. Derive Torricelli’s principle by dimensional analysis. 2. Obtain the drag D on a sphere diameter d placed in a slow flow of velocity U. 3. Assuming that the travelling velocity a of a pressure wave in liquid depends upon the density r and the bulk modulus k of the liquid, derive the relationship for a by dimensional analysis. 4. Assuming that the wave resistance D of a boat is determined by the velocity v of the boat, the density r of fluid and the acceleration of gravity g, derive the relationship between them by dimensional analysis. 5. When fluid of viscosity m is flowing in a laminar state in a circular pipe of length l and diameter d with a pressure drop Dp, obtain by dimensional analysis a relationship between the discharge Q and d, Dp/l and m.

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6. Obtain by dimensional analysis the thickness d of the boundary layer distance x along a flat plate placed in a uniform flow of velocity U (density r, viscosity m). 7. Fluid of density r and viscosity m is flowing through an orifice of diameter d bringing about pressure difference Dp. For discharge Q, .h  pffiffiffiffiffiffiffiffiffiffiffiffiffiffi i pd2 4 2Dp=r , and discharge coefficient C ¼ Q pffiffiffiffiffiffiffiffiffiffiffi Re ¼ d 2rDp=m, show by dimensional analysis that there is a relationship C ¼ f(Re). 8. An aircraft wing, chord length 1.2 m, is moving through calm air at 20 C and 1 bar at a velocity of 200 km/h. If a model wing of scale 1: 3 is placed in a wind tunnel, assuming that the dynamical similarity conditions are satisfied by Re, then: a. If the temperature and the pressure in the wind tunnel are respectively equal to the above, what is the correct wind velocity in the tunnel? b. If the air temperature in the tunnel is the same but the pressure is increased by five times, what is the correct wind velocity? Assume that the viscosity m is constant. c. If the model is tested in a water tank of the same temperature, what is the correct velocity of the model? 9. Obtain the Froude number when a container ship of length 245 m is sailing at 28 knots. Also, when a model of scale 1:25 is tested under similarity conditions where the Froude numbers are equal, what is the proper towing velocity vm for the model in the water tank? Take 1 knot ¼ 0.514 m/s. 10. For a pump of head H, representative size l and discharge Q, assume that the following similarity rule is appropriate:  1=2   1=4 l Q H ¼ lm Qm Hm where, for the model, subscript m is used. If a pump of Q ¼ 0.1 m3/s and H ¼ 40 m is model tested using this relationship in the situation Qm ¼ 0.02 m3/s and Hm ¼ 50 m, what is the model scale necessary for dynamical similarity?

CHAPTER 11

Measurement of Flow Velocity and Flow Rate To clarify fluid phenomena, it is necessary to measure such quantities as pressure, flow velocity and flow rate. Since the measurement of pressure was covered in Section 3.1.5, in this chapter we cover the measurement of flow velocity and flow rate. Fluid includes gas and liquid. According to the type and condition of fluid, or if it flows in a pipe line or open channel, various methods of measurement were developed and are in practical use.

11.1 11.1.1

MEASUREMENT OF FLOW VELOCITY Pitot Tube

Fig. 11.1 shows the shape of a commonly used standard Pitot tube (also called a Pitot-static tube). The flow velocity is given by the following equation from total pressure pt and static pressure ps both to be measured as in the case of Eq. (5.20): pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v ¼ c 2ðpt ps Þ=r (11.1) where c is called the Pitot tube coefficient, which may be taken as having value 1 for a standard-type Pitot tube. However, when compressibility is to be taken into account, refer to Section 13.4.

A Pitot tube is also used to measure the flow in a large-diameter pipe. In this case, the cross-section of the pipe is divided into ring-like equal areas, and the flow velocity at the centre of the area of every ring is measured. The mean flow velocity is obtained from their mean value, and the total flow rate is obtained from the product of the mean velocity and the section area. Apart from the standard type, there are various other types of Pitot tube, as follows.

Cylinder-Type Pitot Tube This type of Pitot tube is used to measure simultaneously the direction and the flow velocity of a two-dimensional flow utilizing the pressure distribution on the cylinder surface wall, which is shown in Fig. 9.5. Fig. 11.2 shows the measuring principle. The body is rotated in a flow until Dh ¼ 0, and the centre-line direction is then the flow direction. The static pressure is obtained Introduction to Fluid Mechanics. http://dx.doi.org/10.1016/B978-0-08-102437-9.00011-5 Copyright © 2018 Elsevier Ltd. All rights reserved.

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Φ1mm

1 d — Total pressure hole 2 U

Static pressure hole

Φd

216

6d 1 R= — d 2

8d (Nose tube)

Supporting stem Static pressure pick-up port

FIGURE 11.1 Pitot-static tube.

Δh h

θ

U

θ U

FIGURE 11.2 Cylinder-type Pitot tube.

11.1 Measurement of Flow Velocity

Angle plate

Enlarged bulb section 135° 45°

Side view

Side view

Front view

Front view

(A)

(B)

U

FIGURE 11.3 Spherical Pitot tubes: (A) 5-hole spherical type; (B) 13-hole spherical type.

if q ¼ 33e35 degrees. Then, if one of the holes is made to face the flow direction by rotating the cylinder, it measures the total pressure. If a third measuring hole is provided on the centre line, the flow direction and both pressures can be measured at the same time. This is called a three-hole Pitot tube. A device which measures the flow direction and velocity in this way is called a yawmetre.

Five-Hole Spherical Pitot Tube This is constructed as shown in Fig. 11.3A and is capable of measuring the velocity and direction of a three-dimensional flow.

Thirteen-Hole Spherical Pitot Tube1 This is constructed as shown in Fig. 11.3B and is capable of measuring the velocity and direction in the range of 135 degrees half-cone angle from the probe axes, instead of the range of 45 degrees in case of a five-hole spherical Pitot tube.

1

I. Yamaguchi, et al., Measurement of Wake Flow Fields, Including Reverse Flow, of Scale Vehicle Models Using a New 13-Hole Pitot Tube, SAE Technical Paper 960676, 1996.

217

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Measurement of Flow Velocity and Flow Rate

(A)

(B)

Wall B

A

A–A

Pressure pickup hole

A

FIGURE 11.4 Pitot tubes for measuring the velocity of flow near the wall face: (A) total pressure tube; (B) surface Pitot tube.

Pitot Tube for Measuring the Flow Velocity Near the Wall Face For measuring the velocity of a flow very near the wall face, a total pressure tube from a flattened fine tube as shown in Fig. 11.4A is used. For measuring the velocity of a flow even nearer to the wall face, a surface Pitot tube as shown in Fig. 11.4B is used. By changing the width of opening B while moving the tube, the whole pressure distribution can be determined. In this case, the static pressure is measured by another hole on the wall face.

11.1.2

Hot-Wire Anemometre

If a heated fine wire is placed in a flow, the temperature of the hot wire changes according to the velocity of the fluid thus changing its electrical resistance. A metre which measures the flow by utilizing this change in resistance is called a hot-wire anemometre. One method is shown in Fig. 11.5A. The flow velocity is obtained by reading the changing hot-wire temperature as a change of electrical resistance (using the galvanometre G) while keeping the voltage between C and D constant. This is called the constant-voltage anemometre. A second method is shown in Fig. 11.5B. The flow velocity is obtained by reading the voltmetre when the galvanometre (G) reading is zero after adjusting the variable electric resistance to maintain the hot-wire temperature, i.e., the electric resistance, constant as the velocity changes. This is called the constant-temperature anemometre (CTA). Because the CTA has a good frequency response characteristic because thermal inertia effects are minimized, almost all currently used metres are of this type. It is capable of giving a flat characteristic up to a frequency of 100 kHz.

11.1.3

Laser Doppler Anemometre

If we point laser light at a tracer particle carried by the flow, the scattered light from the particle develops a difference in frequency from the original incident

11.1 Measurement of Flow Velocity

(A)

(B)

(C)

B

R

B

R

R2

R1

R2

R1

G

C

D

G

C

R3

R3

D Voltmeter

U

υ Hot wire

υ Hot wire

Hot wire (Platinum wire 5–20 μm length 1 mm)

FIGURE 11.5 Hot-wire anemometre: (A) constant-voltage type; (B) constant-temperature type; (C) probe.

light (reference light). The difference is because of the Doppler effect and is proportional to the particle velocity. A device by which the flow velocity is obtained from the velocity of tracer particles by measuring the difference in frequency fD using a photocell or photodiode is called a laser Doppler anemometre. Laser Doppler anemometres include the three types shown in Fig. 11.6.

Reference Beam Type When a particle is moving in a fluid at velocity u as shown in Fig. 11.6A, by measuring the difference in frequency fD between the reference light and the scattered light observed in the direction of angle 2q, the flow velocity u can be obtained from the following equation: u¼

lfD 2 sin q

Pole

(11.2)

where l is the wavelength of the laser light.

Interference Fringe Type As shown in Fig. 11.6B, the flow velocity is obtained by using a photomultiplier to detect the alternating light intensity scattered when a particle passes the interference fringes. The velocity is again calculated using Eq. (11.2).

Single-Beam Type As shown in Fig. 11.6C, by using the interference of the scattered light in two directions from a single incident beam, the flow velocity can be obtained as for the interference type.

219

220

CHAPTER 11:

Measurement of Flow Velocity and Flow Rate

Lens

(A)

Mirror Flow u 2θ Mask Lightscattering particle

Laser Half-mirror

Pinhole Pho

Lens

tom

ultip

lier

(B) Lens

Lens

Mirror

Flow

Pinhole

u 2θ Laser Half-mirror

Photomultiplier

Lightscattering particle

Mask

(C)

Lens Flow Lens

u

2θ

Pinhole

Laser

Photomultiplier

Lightscattering particle

Mask

FIGURE 11.6 Laser Doppler anemometres: (A) reference beam type; (B) interference fringe type; (C) single-beam type.

11.2 11.2.1

MEASUREMENT OF FLOW DISCHARGE Method Using a Collecting Vessel

This method involves measuring the fluid discharge by collecting it in a vessel and measuring its weight or volume. In the case of a gas, the temperature and pressure of the gas in the vessel are measured allowing conversion to another volume under standard conditions of temperature and pressure or to mass.

11.2.2

Methods Using Flow Restrictions

Discharge measurement using flow restrictions is widely used in industry. Restrictions include the orifice, nozzle and Venturi tube. The flow rate is obtained by detecting the difference in pressures upstream and downstream of the device. Flow measurement methods are stipulated in British Standards BS1042 (1992).

11.2 Measurement of Flow Discharge

Flow

φD

φd

Orifice plate

FIGURE 11.7 Orifice plate with pressure tappings (corner and flange).

Orifice Plate2 The construction of an orifice plate is shown in Fig. 11.7. It is set inside a straight pipe. The flow rate is found by measuring the difference in pressures across it. The mass flow rate m is calculated as follows: p pffiffiffiffiffiffiffiffiffiffiffiffiffi 9 m ¼ aε d2 2Dpr1 > > > 4 =

C a ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 b4

(11.3)

> > > ;

where a is called the flow coefficient, ε is the expansion coefficient of gas, Dp is the pressure difference across the orifice plate, r1 is the upstream fluid density, C is the coefficient of discharge and b is the throttle diameter ratio. The volumetric flow rate is calculated as follows: Q¼

m r

(11.4)

where r is the fluid density of the fluid volume at a temperature and pressure.

2

British Standard BS1042, Measurement of Fluid in Closed Conduits, British Standards Institution.

221

222

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Measurement of Flow Velocity and Flow Rate

(A)

(B) Short pipe type

ϕ/2 Flow Flow

φd

φD

φd

φd

φD

φD

Long pipe type

ϕ/2

FIGURE 11.8 (A) ISA 1932 Nozzle; (B) nozzle-type Venturi tube.

φD

ϕ

φd 21° ± 1°

Flow

d

7° > =ϕ > = 15°

FIGURE 11.9 Geometry of cone-type Venturi tube.

Nozzle and Nozzle-Type Venturi Tube3 The designs of a nozzle and nozzle-type Venturi tube are shown in Fig. 11.8, and the measuring method and calculation formula are therefore the same as those for an orifice plate. For a nozzle, the pressure loss is smaller than that for an orifice and also the flow coefficient is larger.

Geometry of Cone-Type Venturi Tube4 The design of a cone-type Venturi tube is shown in Fig. 11.9, and the calculation of the discharge is the same as that for an orifice plate; therefore, Eqs (11.3) and (11.4) are used.

3 4

ISO 5167-2:2003. ISO 5167-4:2003.

11.2 Measurement of Flow Discharge

Tapered tube

Float

FIGURE 11.10 Float-type area flowmetre (Rotametre).

11.2.3

Area Flowmetre5

The flowmetres explained in the previous section indicate the flow from the pressure difference across the restriction. An area flowmetre, however, has a changing level of restriction such that the pressure difference remains constant, and the flow rate is induced by the flow area. Area metres include float and piston type. A float-type area flowmetre (rotametre) has, as shown in Fig. 11.10, a float which is suspended in a vertical tapered tube. The flow produces a pressure difference across the float. The float rests in a position where the combined forces of pressure drag, frictional drag and buoyancy balance its weight. In this case, ignoring friction, volumetric flow Q is expressed by the following equation: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2gVðrf rÞ Q ¼ Cax (11.5) ra0

5

British Standards BS7405.

223

224

CHAPTER 11:

Measurement of Flow Velocity and Flow Rate

where, r is the fluid density, C is the coefficient of discharge, ax is the area of the annulus through which the fluid passes outside the float, V is the float volume, rf is the float density and a0 is the maximum section area of the float. Because ax is a definite function of float position, if C is constant the equilibrium height of the float in the tube is proportional to the flow.

11.2.4

Positive Displacement Flowmetre

A positive displacement flowmetre with continuous flow relies on some form of measuring chamber of constant volume. It is then possible to obtain the integrated volume by counting the number of times the volume is filled and the flow rate by measuring the number of times this is done per second. As a typical example, Fig. 11.11 shows oval gear and Roots-type positive displacement metres. Because of the difference between the flow inlet pressure p1 and the flow outlet pressure p2 of fluid, the vertically set gear (Fig. 11.11A) turns in the direction of the arrow. Thus, every complete revolution sends out fluid of volume 4V. Therefore, volume flow rate can be calculated by counting the number of revolutions.

11.2.5

Turbine Flowmetre

If a turbine is placed in the course of a flow, the turbine rotates owing to the velocity energy of the fluid. Because they are almost proportional, the flow velocity is obtainable from the rotational velocity of the turbine, while the integrated volume can be calculated by counting the number of revolutions. The turbine flowmetre has long been used as a water metre. Fig. 11.12 shows a turbine flowmetre used industrially for flow rate measurement of various fluids. A pulse is induced every time the blade of the turbine passes the magnetic coil face and the pulse frequency is proportional to the volume flow rate.

(A)

(B) p

1

V

p2

FIGURE 11.11 Positive displacement flowmetres: (A) oval gear type; (B) roots type.

11.2 Measurement of Flow Discharge

Flow rectifier

Impeller of axial flow turbine Rotor bearing

Flow

Frequency pickup To frequency meter

FIGURE 11.12 Turbine flowmetre.

11.2.6

Vortex Shedding Flowmetre6

If a cylinder is placed in a flow, Karman vortices develop behind it. A vortex shedding flowmetre consists of the vortex shedder to generate Karman vortices, the vortex detector to detect them and the converter to transform a signal detected by the vortex detector as shown in Fig. 11.13. The frequency f of vortex shedding from the cylinder is proportional to the flow velocity U as shown in Eq. (9.7). The Strouhal number St changes with the Reynolds number Re, but it is almost constant at Reynolds numbers within a certain range. The Strouhal number is almost constant and St ¼ 0.25 at Reynolds numbers of 20,000 and greater when the triangular vortex shedder is used. In other words, the flow velocity U is expressed by the following equation: U ¼ fd=St ¼ fd=0:25

(11.6)

where D is the inner diameter of the tube and d is the width of the vortex shedder. As shown above, flow velocity in the tube can be measured by detecting vortex frequency and volumetric flow rate can be calculated by its flow velocity multiplied by the cross-section of the tube. The vortex detector is constructed with a piezoelectric device and is located inside the vortex shedder for nominal sizes of 40 mm or greater and downstream of the vortex shedder for nominal sizes of 25 mm or smaller.

11.2.7

Ultrasonic Flowmetre

As shown in Fig. 11.14, piezocrystals A and B are located a distance l apart on a line passing obliquely through the pipe centre line. Assume that an ultrasonic wave pulse sent from a transmitter at A is received by the detector at B t1

6

H. Yamazaki, et al., Journal of Instrumentation and Control 10 (1971) p. 173.

225

226

CHAPTER 11:

Measurement of Flow Velocity and Flow Rate

Measurement tube

Flow

Vortices Vortex shedder

FIGURE 11.13 Vortex flowmetre.

A

υ

θ

B

Send–receive switch

FIGURE 11.14 Ultrasonic flowmetre.

seconds later. Then, exchanging the functions of A and B by the sendereceive switch, an ultrasonic wave pulse sent from B is detected by A t2 seconds later. Thus, t1 ¼

1 ; t2 ¼ a þ v cos q a

1 v cos q

1 t1

1 a þ v cos q ¼ t2 l

v cos q 2v cos q ¼ l l

a

where a is the sonic velocity in the fluid. From this equation,   1 1 1 v¼ 2 cos q t1 t2

(11.7)

11.2 Measurement of Flow Discharge

Exiting coil Measuring tube

Output signal (Electromotive force E)

Power Terminal

Flow υ

Exiting coil Magnetic field lines (Flux density B)

FIGURE 11.15 Magnetic flowmetre.

This ultrasonic flowmetre has the same merits as an electromagnetic flowmetre and an additional benefit of usability in a non-conducting fluid. On the other hand, it has the disadvantages of complex construction and high price.

11.2.8

Magnetic Flowmetre7

As shown in Fig. 11.15, when a conducting fluid flows in a non-conducting section of a measuring tube to which a magnetic field of flux density B is applied normal to the flow direction, an electromotive force E proportional to the mean flow velocity v is induced in the liquid (Faraday’s law of electromagnetic induction) which, after amplification, permits computation of the volume flow rate Q. The electromotive force is detected by inserting two electrodes into the tube in contact the fluid and normal to both the flow and magnetic field directions. In other words, if the tube diameter is d, then E ¼ Bdv

(11.8)

and Q¼

pdE 4B

(11.9)

Because this magnetic flowmetre has no pressure loss, measurement can be made irrespective of the viscosity, specific gravity, pressure and Reynolds number of the fluid.

7

ISO 6817:1992.

227

228

CHAPTER 11:

Measurement of Flow Velocity and Flow Rate

Transmittor

Flow

Temperature sensor Flow tubes

Electromagnetic pickoff Electromagnetic oscillator

FIGURE 11.16 Coriolis mass flowmetre. Provide materials from OVAL Corp.

11.2.9

Coriolis Flowmetre8

A Coriolis flowmetre operates on the principle of Coriolis force which is an inertial force acting on objects that are in motion relative to a rotating reference frame. It is called Coriolis mass flowmetre as well. Fig. 11.16 shows a typical construction of the Coriolis flowmetre. Fluid flows into the inlet flange and is separated into a pair of the flow tubes in parallel. Fluid flows out from these flow tubes, merges at the outlet flange and flows out from the Coriolis flowmetre. Measurement of mass flow rate takes place by an electromagnetic pickoff on each flow tube, an electromagnetic oscillator and a temperature sensor located as shown in Fig. 11.16. A pair of electromagnetic pickoffs detect phase difference of a twist of these flow tubes because of Coriolis force and the electromagnetic oscillator causes the flow tubes to oscillate at resonant frequency. The temperature sensor is used to compensate modulus of elasticity of flow tubes for change in temperature. Oscillation of the flow tubes created by the electromagnetic oscillator corresponds to rotating motion and phase shift detected by a pair of the electromagnetic pickoffs is proportional to mass flow rate.

8

ISO 10790:1999.

11.2 Measurement of Flow Discharge

Bridge curcuit Temperature sensor

Flow rate signal Heater Temperature sensor

Flow sensor line Flow Flow element Main flow channel

FIGURE 11.17 Thermal mass flowmetre.

11.2.10

Thermal Mass Flowmetre

A thermal mass flowmetre consists of the main flow channel and flow sensor line as shown in Fig. 11.17. The flow sensor line is constructed with a capillary, a heater and temperature sensors upstream and downstream of the heater. The temperature sensors form a part of the bridge circuit. Fluid passing through the flow sensor line is heated to provide stable temperature difference between upstream and downstream of the heater. The temperature difference is converted to electric voltage in the bridge circuit and amplified to output a signal on mass flow rate. The flow element mounted in the main flow channel consists of a bundle of small metal tubes, a metal plate with many small holes or a sintered metal plate to make proportional flow division by a suitable flow resistance. Flow rate can be calculated by knowing sensor flow rate because flow rate in the main flow channel is proportional to flow rate in the flow sensor line.

11.2.11

Fluidic Flowmetre9,10

Fluidics is a mechanical element to control flow without any mechanical moving parts. As shown in Fig. 11.18, with an appropriate feedback mechanism a wall attachment amplifier can become a fluidic oscillator whose jet spontaneously oscillates at a frequency proportional to the volume flow rate of the main jet flow. The device can thus be used as a flowmetre.

9 10

R.F. Boucher, C. Mazharoglu, Int. Gas Research Conference (1989) p. 522. H. Yamazaki, et al., Proc. FLUCOME’85 vol. 2 (1985) p. 617.

229

230

CHAPTER 11:

Measurement of Flow Velocity and Flow Rate

(A)

(B)

FIGURE 11.18 Fluidic flowmetre. (A) Wall attachment type amplifier A9 and (B) wall attachment type amplifier B.10 B

(A)

(B)

B b

90°

h

h

D

D

B

(C)

3hmax–B h

50 or more D

50 or more

φ 10–30

FIGURE 11.19 Weirs11: (A) right-angled triangular; (B) rectangular; (C) full width.

11.2.12

Weir11

As shown in Fig. 11.19, the three principle weir configurations are classified by shape into right-angled triangular, rectangular and full-width weirs. Table 11.1 shows the flow computation formulae and the applicable scope for such wires.

11

British Standards BS3680.

Table 11.1 Flow Computation Formulae for Weirs in British Standard BS 3680 Kind of Weir Discharge computational formula

Right-Angled Triangular Weir    pffiffiffiffiffiffi 8 C 2gH5=2 m3 s Q ¼ 15

C ¼ 0:5785

H ¼ H þ 0:00085

H D H B

< 0:4 < 0:2 0:05 m < H < 0:38 m D > 0:45 m B > 1:0 m

Full-Width Weir    pffiffiffiffiffiffi Q ¼ C 23 2gbH3=2 m3 s C ¼ 0:596 þ 0:091 H D H ¼ H þ 0:001

b B H D

b B H D

 2 b #  2 0:003615 0:0030 b B C ¼ 0:578þ 0:037 þ H þ 0:0016 B " 2 #  4  b H  1þ 0:5 B HþD "

> 0:3 < 1:0 0:025 m < H < 0:80 m D > 0:30 m

¼ 1:0 < 2:5 H > 0:03 b > 0:20 D > 0:10

11.2 Measurement of Flow Discharge

Applicable range

Rectangular Weir    pffiffiffiffiffiffi Q ¼ C 23 2gbH3=2 m3 s

231

232

CHAPTER 11:

Measurement of Flow Velocity and Flow Rate

11.3

PROBLEMS

1. The velocity of water flowing in a pipe was measured with a Pitot tube, and the differential pressure read on the connected mercury manometre was 8 cm. Assuming that the velocity coefficient for the Pitot tube is 1, obtain the flow velocity. Assume that the water temperature is 20 C, and the specific gravity of mercury s ¼ 13.5. 2. Air flow was measured with the three-hole Pitot tube shown in Fig. 11.20, and it was found that the heights B and C of the water manometre were equal while A was 5 cm lower. What was the air flow velocity? Assume that the temperature was 20 C and the air density is 1.205 kg/m3. 3. An orifice of diameter 50 cm on a pipe of inner diameter 100 mm was used to measure air flow. The differential pressure read on a connected mercury manometre was 120 mm. Assuming that the discharge coefficient a ¼ 0.62 and the gas expansion coefficient ε ¼ 0.98, obtain the mass flow rate. Assume that the pressure and temperature upstream of the orifice are 196 kPa and 20 C, respectively. 4. A pipe line contains both an orifice and a nozzle. When Re ¼ 1  105, and with a throttle diameter ratio b ¼ 0.6 for both, the flow coefficient a is 0.65 for the orifice but 1.03 for the nozzle. Explain why. 5. Explain the principle of a hot-wire flow anemometer. Over what points should caution be especially exercised? 6. Explain the principles and features of the laser Doppler anemometre. 7. With a vortex-shedding flowmetre, cylinder diameter 2 cm, the shedding frequency was measured as 5 Hz. What was the flow velocity? 8. Obtain the flow formulae for a rectangular weir and a right-angled triangular weir. 9. Assuming a reading error of 2% for both rectangular and right-angled triangular weirs, what are the resulting percentage flow errors? A

B

C

B 5 cm

A

C

Three-hole Pitot tube Manometer

FIGURE 11.20 Three-hole Pitot tube and its static pressure characteristics.

CHAPTER 12

Flow of Ideal Fluid When the Reynolds number Re is large, because the diffusion of vorticity is now small (Eq. 6.19) and the boundary layer is very thin, the overwhelming majority of the flow is the main flow. Consequently, although the fluid itself is viscous, it can be treated as an ideal fluid subject to Euler’s equation of motion, thus disregarding the viscous term. In other words, the applicability of ideal flow is large. For an irrotational flow, the velocity potential f can be defined so this flow is called potential flow. Originally the definition of potential flow did not distinguish between viscous and non-viscous flows. However, now, as studied below, potential flow refers to an ideal fluid. In the case of two-dimensional flow, a stream function j can be defined from the continuity equation, establishing a relationship where the Cauchye Riemann equation is satisfied by both f and j. This fact allows theoretical analysis through the application of the theory of complex variables so that f and j can be obtained. Once f or j is obtained, velocities u and v in x and y directions, respectively, can be obtained, and the nature of the flow is revealed. In the case of three-dimensional flow, the theory of complex variables cannot be used. Rather, Laplace’s equation Df ¼ 0 for a velocity potential f ¼ 0 is solved. Using this approach, the flow around a sphere etc. can be determined. Here, however, only two-dimensional flows will be considered.

12.1

EULER’S EQUATION OF MOTION

Consider the force acting on the small element of fluid in Fig. 12.1. Because the fluid is an ideal fluid, no force caused by viscosity acts. Therefore, by Newton’s second law of motion, the sum of all forces acting on the element in any direction must balance the inertia force in the same direction. The pressure acting on

233 Introduction to Fluid Mechanics. http://dx.doi.org/10.1016/B978-0-08-102437-9.00012-7 Copyright © 2018 Elsevier Ltd. All rights reserved.

234

CHAPTER 12:

Flow of Ideal Fluid

y p+

dy

∂p dy ∂y

p

p+

∂p dx ∂x

p dx

x

O

FIGURE 12.1 Balance of pressures on fluid element.

the small element of fluid dxdy per unit thickness is, as shown in Fig. 12.1, similar to Fig. 6.3B. In addition, taking account of the body force and also assuming that the sum of these two forces is equal to the inertial force, the equation of motion for this case can be obtained. This is the case where the viscous term of Eq. (6.13) is omitted. Consequently, the following equations are obtainable:   9 vu vu vu vp > > þu þv r ¼ rX > vt vx vy vx > = (12.1)   > vv vv vv vp > > > ; r þu þv ¼ rY vt vx vy vy These equations are similar to Eq. (5.4) and are called Euler’s equations of motion for two-dimensional flow.

For a steady flow, if the body force term is neglected, then:   9 vu vu vp > > r u þv ¼ > vx vy vx > =   vv vv r u þv ¼ vx vy

(12.2)

> vp > > > ; vy

If u and v are known, the pressure is obtainable from Eqs (12.1) or (12.2). Generally speaking, to obtain the flow of an ideal fluid, the continuity equation (6.2) and Euler’s equation of motion (12.1) or Eq. (12.2) must be solved under the given initial conditions and boundary conditions. In the flow fluid, three

12.2 Velocity Potential

quantities are to be obtained, namely u, v and p, as functions of t and x, y. However, because the acceleration term, i.e., inertial term, is nonlinear, it is so difficult to obtain these terms analytically that a solution can only be obtained for a particular restricted case.

12.2

VELOCITY POTENTIAL

Velocity potential f as a function of x and y will be studied. Assume that1 u¼

vf vf ; v¼ vx vy

(12.3)

From vu v2 f v2 f vv ¼ ¼ ¼ vy ðvyvxÞ ðvxvyÞ vx the following relationship is obtained: vu vy

vv ¼0 vx

(12.4)

This is the condition for irrotational motion. Conversely, if a flow is irrotational, a function f as in the following equation must exist for u and v: df ¼ udx þ vdy

(12.5)

Using Eq. (12.3): df ¼

vf vf dx þ dy vx vy

(12.6)

Consequently, when the function f has been obtained, velocities u and v can also be obtained by differentiation, and thus, the flow pattern is found. This function f is called velocity potential, and such a flow is called potential or irrotational flow. In other words, the velocity potential is a function whose gradient is equal to the velocity vector. Eq. (12.3) turns out as follows if expressed in polar co-ordinates: vr ¼

vf vf ; vq ¼ vr rvq

(12.7)

1

In general, whenever u, v and w are, respectively, expressed as vf=vx; vf=vy; and vf=vz for vector V (x, y and z components are, respectively, u, v and w), vector V is written as grad f or Vf:

 vf vf vf ; ; vx vy vz Eq. (12.3) is the case of two-dimensional flow where w ¼ 0 and can be written as grad f or Vf. V ¼ grad f ¼ V f ¼



235

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CHAPTER 12:

Flow of Ideal Fluid

For the potential flow of an incompressible fluid, substitute Eq. (12.3) into continuity Eq. (6.2), and the following relationship is obtained2: v2 f v 2 f þ ¼0 vx2 vy2

(12.8)

Eq. (12.8), called Laplace’s equation, is thus satisfied by the velocity potential used in this manner to express the continuity equation. From any solution which satisfies Laplace’s equation and the particular boundary conditions, the velocity distribution can be determined. It is particularly noteworthy that the pattern of potential flow is determined solely by the continuity equation and the momentum equation serves only to determine the pressure. A line along which f has a constant value is called equipotential line, and on this line, because df ¼ 0 and the inner product of both vectors of velocity and the tangential line is zero, the direction of fluid velocity is at right angles to the equipotential line.

12.3

STREAM FUNCTION

For incompressible flow, from the continuity Eq. (6.2), vu vv þ ¼0 vx vy

(12.9)

This is Eq. (12.4) but with u and v replaced by v and u, respectively,. Consequently, corresponding to Eq. (12.5), it turns out that there exists a function j for x and y shown by the following equation: dj ¼

vdx þ udy

(12.10)

In general, because dj ¼

2

vj vj dx þ dy vx vy

(12.11)

That is,

div V ¼ div ½u; v; wŠ ¼ div ðgrad fÞ ¼ div Vf ¼ div

  vf vf vf vu vv vw ¼ þ þ ; ; vx vy vz vx vy vz

v2 f v2 f v2 f 2 þ 2 þ 2 vz   2 vx 2  vy 2 2 v vx þ v vy þ v2 vz2 is called the Laplace operator (Laplacian), abbreviated to D. Eq. (12.8) is for a two-dimensional flow where w ¼ 0, expressed as Df ¼ 0. ¼

12.3 Stream Function

u and v are, respectively, expressed as follows: u¼

vj vj ; v¼ vy vx

(12.12)

Consequently, once function j has been obtained, differentiating it by y and x gives velocity u and v, revealing the detail of the fluid motion. j is called the stream function. Expressing the above equation using polar coordinate gives vr ¼

vj ; vq ¼ rvq

vj vr

(12.13)

In general, for two-dimensional flow, the streamline is as follows, from Eq. (4.1): dx dy ¼ u v or vdx þ udy ¼ 0

(12.14)

From Eqs (12.12) and (12.14), the corresponding dj ¼ 0, i.e., j ¼ constant, defines a streamline. The product of the tangents of a streamline and an equipotential line at the crossing point of both lines is as follows from Eqs (12.3) and (12.12):           dy dy vj vj vf vf  ¼ 1 ¼ dx 4 dx j vx vy vx vy This relationship shows that the streamline intersects normal to the equipotential line at the crossing point of the two lines. As shown in Fig. 12.2, consider points A and B on two closely neighbouring streamlines, j and j þ dj. The volume flow rate dQ flowing in unit time and crossing line AB is as follows from the figure: dQ ¼ udy

vdx ¼

vj vj dy þ dx ¼ dj vy vx

The volume flow rate Q of fluid flowing between two streamlines j ¼ j1 and j ¼ j2 is thus given by the following equation: Q¼

Z

dQ ¼

Z

j2

j1

dj ¼ j2

j1

(12.15)

237

238

CHAPTER 12:

Flow of Ideal Fluid

y

ψ + dψ

ψ

dQ B u A

dy

v dx

x

O

FIGURE 12.2 Relationship between flow rate and stream function.

Substituting Eq. (12.12) into Eq. (4.8) for flow without vorticity, the following is obtained, clarifying that the stream function satisfies Laplace’s equation: v 2 j v2 j þ ¼0 vx2 vy2

12.4

(12.16)

COMPLEX POTENTIAL

For a two-dimensional incompressible potential flow, because the velocity potential f and stream function j exist, the following equations result from Eqs (12.3) and (12.12): vf vj ¼ ; vx vy

vf ¼ vy

vj vx

(12.17)

These equations are called the CauchyeRiemann equations in the theory of complex variables. In this case, they express the relationship between the velocity potential and stream function. The CauchyeRiemann equations clarify the fact that f and j both satisfy Laplace’s equation. They also clarify the fact that a combination of f and j satisfying the CauchyeRiemann conditions expresses a twodimensional incompressible potential flow.

12.5 Example of Potential Flow

Now, consider a regular function3 w(z) of complex variable z ¼ x þ iy and express it as follows by dividing it into real and imaginary parts: wðzÞ ¼ f þ ij z ¼ x þ iy ¼ rðcos qþ i sin qÞ ¼ reiq

(12.18)

f ¼ fðx; yÞ; j ¼ jðx; yÞ and f and j above satisfy Eq. (12.17) as a result of the nature of a regular function. Consequently, the real part f(x,y) and the imaginary part j(x,y) of the regular function w(z) of complex number z can, respectively, be regarded as the velocity potential and the stream function of the two-dimensional incompressible potential flow. In other words, there exists an irrotational motion whose equipotential line is f(x,y) ¼ constant and streamline j(x,y) ¼ constant. Such a regular function w(z) is called the complex potential. From Eq. (12.18) dw ¼

vw vw dx þ dy ¼ vx vy

¼ ðu



   vf vj vf vj þi þi dx þ dy vx vx vy vy

ivÞdx þ ðv þ iuÞdy ¼ ðu

ivÞðdx þ idyÞ ¼ ðu

ivÞdz

Therefore, dw ¼u dz

iv

(12.19)

Consequently, whenever w(z) is differentiated with respect to z, as shown in Fig. 12.3, its real part yields velocity u in the x direction, and the negative of its imaginary part yields velocity v in y direction. The actual velocity u þ iv is called the complex velocity while u iv in the above equation is the conjugate complex velocity.

12.5 12.5.1

EXAMPLE OF POTENTIAL FLOW Basic Example

Parallel Flow For the uniform flow U shown in Fig. 12.4, from Eq. (12.3) u¼

3

vf vf ¼ U; v ¼ ¼0 vx vy

The function whose differential at any point with respect to z is independent of direction in the z plane is called a regular function. A regular function satisfies the CauchyeRiemann equations.

239

240

CHAPTER 12:

Flow of Ideal Fluid

u + iv v

u

–v u – iv

FIGURE 12.3 Complex velocity. φ = constant

y

U

ψ = constant

x

O

FIGURE 12.4 Parallel flow.

Therefore, df ¼

vf vf dx þ dy ¼ Udx vx vy

f ¼ Ux From Eq. (12.12), u¼

vj ¼ U; v ¼ vy

vj ¼0 vx

12.5 Example of Potential Flow

Therefore, dj ¼

vj vj dx þ dy ¼ Udy vx vy

j ¼ Uy wðzÞ ¼ f þ ij ¼ Uðxþ iyÞ ¼ Uz

(12.20)

The complex potential of parallel flow U in x direction emerges as w(z) ¼ Uz. Furthermore, if the complex potential is given as w(z) ¼ Uz, the conjugate complex velocity is dw ¼U dz

(12.21)

clarifying again that it expresses a uniform flow in the direction of the x axis.

Source As shown in Fig. 12.5, consider a case where fluid discharges from the origin (point O) at quantity q per unit time. Putting velocity in the radial direction on a circle of radius r to vr, the discharge q per unit thickness is q ¼ 2prvr ¼ constant

(12.22)

From Eqs (12.7) and (12.22) vr ¼

vf q ¼ vr 2pr

y

ψ = constant

φ = constant

O

FIGURE 12.5 Source.

x

241

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Also from Eq. (12.7) vq ¼

vf ¼0 rvq

Integrating df in the above equation gives: f¼

q log r 2p

(12.23)

Then, from Eqs (12.13) and (12.22) vr ¼

vj q ¼ ; vq ¼ rvq 2pr

vj ¼0 vr

Therefore, j¼

q q 2p

(12.24)

Consequently, the complex potential is expressed by the following equation: w ¼ f þ ij ¼


q q q ðlog r þ iqÞ ¼ log reiq ¼ log z 2p 2p 2p

(12.25)

From Eqs (12.23) and (12.24) it is known that the equipotential lines are a set of circles centred at the origin while the streamlines are a set of radial lines radiating from the origin. Also, it is noted that the flow velocity vr is inversely proportional to the distance r from the origin. Whenever q > 0, fluid flows out evenly from the origin towards the periphery. Such a point is called a source. Conversely, whenever q < 0, fluid is absorbed evenly from the periphery. Such a point is called a sink. jqj is called the strength of the source or sink.

Free Vortex In Fig. 12.6, fluid rotates around the origin with tangential velocity vq at any given radius r. The circulation G is as follows from Eq. (4.9): Z 2p Z 2p G¼ vq ds ¼ vq r dq ¼ 2prvq q¼0

0

The velocity potential f is vq ¼

vf G vf ¼ ; vr ¼ ¼0 rvq 2pr vr

Therefore, f¼

G q 2p

(12.26)

It emerges that vq is inversely proportional to the distance from the centre.

12.5 Example of Potential Flow

y

υθ r

θ

x

FIGURE 12.6 Vortex.

The stream function j is vq ¼

vj G vj ¼ ; vr ¼ ¼0 vr 2pr rvq

Therefore, j¼

G log r 2p

(12.27)

Consequently, the complex potential is wðzÞ ¼ f þ ij ¼

G ðq 2p

i log r Þ ¼

i

G ðlog r þ iqÞ ¼ 2p

i

G log z 2p

(12.28)

For clockwise circulation, wðzÞ ¼ ½iG=ð2pފlog z. From Eqs (12.26) and (12.17), it is known that the equipotential lines are a group of radial straight lines passing through the origin while the flow lines are a group of concentric circles centred on the origin. This flow appears in Fig. 12.5 with broken lines representing streamlines and solid lines as equipotential lines. The circulation G is positive counterclockwise and negative clockwise. This flow consists of rotary motion in concentric circles around the origin with the velocity inversely proportional to the distance from the origin. Such a flow is called a free vortex while the origin point itself is a point vortex. The circulation is also called the strength of the vortex.

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12.5.2

Synthesizing of Flows

When there are two regular functions w1(z) and w2(z), the function obtained as their sum wðzÞ ¼ w1 ðzÞ þ w2 ðzÞ

(12.29)

is also a regular function. If w1 and w2 represent the complex potentials of two flows, another complex potential is obtained from their sum. By combining two two-dimensional incompressible potential flows in such a manner, another flow can be obtained.

Combining of Source and a Sink Assume that, as shown in Fig. 12.7, the source q is at point A(z ¼ sink q is at point B(z ¼ a).

a) and

The complex potential w1 at any point z due to the source whose strength is q at point A is w1 ¼

q logðzþ aÞ 2p

(12.30)

The complex potential w2 at any point z due to the sink whose strength is q is w2 ¼

q logðz 2p

aÞ

(12.31)

Because of the linearity of Laplace’s equation, the complex potential w of the flow which is the combination of these two flows is w¼

q ½logðzþ aÞ 2p

logðz

aފ

(12.32)

y

z

θ r1

r2

θ2 –a

θ1

+q A

FIGURE 12.7 Definition of variables for source A and sink B combination.

+a O

–q B

x

12.5 Example of Potential Flow

Now, from Fig. 12.7, because z þ a ¼ r1 eiq1 ; z

a ¼ r2 eiq2

from Eq. (12.32)  q r1 w¼ log þ iðq1 2p r2

q2 Þ



(12.33)

Therefore, f¼

q r1 log 2p r2

j¼

q ðq1 2p

(12.34) q2 Þ

(12.35)

Assuming f ¼ constant from the first equation, equipotential lines are obtainable which are Appolonius circles for points A and B (a group of circles whose ratios of distances from fixed points A and B are constant). Taking j ¼ constant, streamlines are obtainable which are found to be another set of circles whose vertical angles are the constant angle (q1 q2) for chord AB (Fig. 12.8). Consider the case where a / 0 in Fig. 12.8 under the condition of aq ¼ constant. Then from Eq. (12.32):     q 1 þ a=z q a 1 a3 1 a5 aq m log þ (12.36) w¼ þ þ/ z ¼ ¼ 2p 1 a=z p z 3 z 5 z pz z

y

ψ = constant

φ = constant

A

FIGURE 12.8 Flow as a result of the combination of source and sink.

O

B

x

245

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CHAPTER 12:

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y

ψ = constant

φ = constant

x

FIGURE 12.9 Doublet.

A flow given by the complex potential of Eq. (12.36) is called a doublet, while m ¼ aq/p is its strength. The concept of a doublet is the extremity of a source and a sink of equal strength approaching infinitesimally close to each other while increasing their strength. From Eq. (12.36), w¼

m x iy ¼m 2 x þ iy x þ y2

(12.37)

f¼

mx x2 þ y 2

(12.38)

j¼

my x2 þ y 2

(12.39)

From these equations, as shown in Fig. 12.9, an equipotential line is a circle whose centre is on the x axis while being tangential to the y axis, and a streamline is a circle whose centre is on the y axis while being tangential to the x axis.

Flow Around a Cylinder Consider a circle of radius r0 centred at the origin in uniform parallel flows. In general, by placing a number of sources and sinks in parallel flows, flows

12.5 Example of Potential Flow

around variously shaped bodies are obtainable. In this case, however, by superimposing parallel flows onto the same doublet shown in Fig. 12.9, flows around a circle are obtainable as follows. From Eqs (12.20) and (12.36) the complex potential when a doublet is in uniform flows U is   m m1 wðzÞ ¼ Uz þ ¼ U zþ z U z Now, put m/U ¼ r0 2, then   r0 2 wðzÞ ¼ U z þ z

(12.40)

Decompose the above using the relationship z ¼ r(cos q þ i sin q), then     r0 2 r0 2 cos q þ iU r sin q wðzÞ ¼ U r þ r r   r0 2 f¼U rþ cos q (12.41) r   r0 2 sin q (12.42) j¼U r r Also, the conjugate complex velocity is dw ¼U dz

Ur0 2 z2

(12.43)

with stagnation points at z ¼ r0 . The streamline passing the stagnation point j ¼ 0 is given by the following equation:   r0 2 r sin q ¼ 0 r This streamline consists of the real axis and the circle of radius r0 centring at the origin. By replacing this streamline with a solid surface, the flow around a cylinder is obtained as shown in Fig. 12.10. The tangential velocity of flow around a cylinder is, from Eq. (12.41)   1 vf r0 2 ¼ U 1 þ 2 sin q (12.44) vq ¼ r vq r Because r ¼ r0 on the cylinder surface, vq ¼

2U sin q

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y

U r0

x

O

FIGURE 12.10 Flow around a cylinder.

When the directions of q and vq are arranged as shown in Fig. 12.11, this becomes: vq ¼ 2U sin q

(12.45)

The complex potential when there is clockwise circulation G around the cylinder is, as follows from Eqs (12.28) and (12.40):   r0 2 G (12.46) w ð zÞ ¼ U z þ þ i log z 2p z

y

υθ

r0

θ O

FIGURE 12.11 Definitions of vq and q.

x

12.6 Conformal Mapping

y

υ′θ

θ

r0

U

O

x

FIGURE 12.12 Flow around a cylinder with circulation.

The flow in this case turns out as shown in Fig. 12.12. The tangential velocity vq0 on the cylinder surface is as follows: vq0 ¼ 2U sin q þ

12.6

G 2pr0

(12.47)

CONFORMAL MAPPING

A simple flow can be studied within the limitations of the z plane as in the preceding section. For a complex flow, however, there may be some established cases of useful mapping of a transformation to another plane. For examples, by transforming flow around a cylinder, etc. through mapping functions onto some other planes, such complex flows as the flow around a wing, and between the blades of a pump, blower or turbine, can be determined. Assume that there is the relationship z ¼ f ðzÞ

(12.48)

between two complex variables z ¼ x þ iy and z ¼ x þ ih and that z is the regular function of z. Consider a mesh composed of x ¼ constant and y ¼ constant on the z plane as shown in Fig. 12.13. That mesh transforms to another mesh composed of x ¼ constant and h ¼ constant on the z plane. In other words, the pattern on the z plane is different from the pattern on the z plane but they are related to each other. Further, assume that, as shown in Fig. 12.14, point z0 corresponds to point z0 and that the points corresponding to points z1 and z2 both minutely off z0 are z1 and z2. Then

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Flow of Ideal Fluid

η

y

ζ

z

x

ξ

FIGURE 12.13 Corresponding mesh on z and z planes. y

η

z

ζ

ζ2 z2

R2

θ2

r2 z1 θ 1

z0

ζ0

x

β2 ζ1 β1 R1

r1

ξ

x

FIGURE 12.14 Conformal mapping.

z1

z0 ¼ r1 eiq1

z1

z0 ¼ R1 eib1

z2

z0 ¼ r2 eiq2

z2

z0 ¼ R2 eib2

From Eq. (12.48),      z1 z0 dz z2 lim ¼ ¼ lim z1 /z0 z1 z0 dz z¼z0 z2 /z0 z2

z0 z0

or R1 eib1 R2 eib2 ¼ r1 eiq1 r2 eiq2 From the above, it turns out that r2 R2 ¼ ; r1 R1

q2

q1 ¼ b2

b1

and the minute triangles on the z plane are



12.6 Conformal Mapping

Dz0 z1 z2 wDz0 z1 z2

(12.49)

This shows that even though the pattern as a whole on the z plane may be very different from that on the z plane, their minute sections are similar and equiangularly mapped. Such a manner of pattern mapping is called conformal mapping, and f(z) is the mapping function. Now, consider the mapping function z¼zþ

a2 ða > 0Þ z

(12.50)

Substitute a circle of radius a on the z plane z ¼ aeiq into Eq. (12.50), 

z ¼ a eiq þ 1 eiq ¼ a eiq þ e iq ¼ 2a cos q (12.51)

At the time when q changes from 0 to 2p, z corresponds in 2a / 0 / 2a / 0 / 2a. In other words, as shown in Fig. 12.15A, the cylinder on the z plane is conformally mapped onto the flat plate on the z plane. The mapping function in Eq. (12.50) is renowned, and is called Joukowski’s transformation. If conformal mapping is made onto the z plane using Joukowski’s mapping function (Eq. 12.50) while changing the position and size of a cylinder on the z plane, the shape on the z plane changes variously as shown in Fig. 12.15. The flow around the asymmetrical wing appearing in Fig. 12.15D can be obtained by utilizing Joukowski’s conversion. Consider the flow in the case where a cylinder of eccentricity z0 and radius r0 is placed in a uniform flow U whose circulation strength is G. The complex potential of this flow can be obtained by substituting z z0 for z in Eq. (12.46),   r0 2 G logðz z0 Þ þi (12.52) w ¼ U ð z z0 Þ þ 2p z z0 Putting z ¼ z0 þ reiq , from w ¼ f þ ij,   r0 2 G q f¼U rþ cos q 2p r   r0 2 G log r j¼U r sin q þ 2p r

(12.53) (12.54)

On the circle r ¼ r0, j ¼ constant comprises a streamline. According to the Kutta condition4 (where the trailing edge must become a stagnation point).

4

If the trailing edge was not a stagnation point, the flow would go around the sharp edge at infinite velocity from the lower surface of the wing towards the upper surface. The Kutta condition avoids this physical impossibility.

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η

y

(A)

a x

O

–2a

2a

η

y

(B)

ξ

O

b a x

O

2a

η

y

(C)

ξ

O –2a

b a x d

O

η

y

(D)

z0

ξ

2a

–2a

r0 O β

z plane

a

x –2a

O

2a

ξ

ζ plane

FIGURE 12.15 Mapping of cylinders through Joukowski’s transformation: (A) flat plate; (B) elliptical section; (C) symmetrical wing; (D) asymmetrical wing.

12.6 Conformal Mapping

(A)

(B)

y

η

φ

φ

ψ ξ

ψ O

x

O

FIGURE 12.16 Mapping of flow around cylinder onto flow around wing (A) z plane and (B) z plane.

  df ¼ 2Ur0 sin b dq q ¼ b r ¼ r0

G ¼0 2p

(12.55)

Therefore, G ¼ 4pUr0 sin b

(12.56)

Equipotential lines and streamlines produced by substituting value of G satisfying Eq. (12.56) into Eqs (12.53) and (12.54) are shown in Fig. 12.16A. They can be conformally mapped onto the z plane by utilizing Joukowski’s conversion by eliminating z from Eqs (12.50) and (12.52) to obtain the complex potential on the z plane. The resulting flow pattern around a wing can be found as shown in Fig. 12.16B. In this way, by means of conformal mapping of simple flows, such as around a cylinder, flow around complex-shaped bodies can be found. Because the existence of analytical functions which shift z to the outside territory of given wing shapes is generally known, the behaviour of flow around these wings can be found from the flow around a cylinder through a process similar to the previous one. In addition, there are examples where it can be used for computing the contraction coefficient5 of flow out of an orifice in a large vessel and the drag6 caused by the flow behind a flat plate normal to the flow.

5 6

H. Lamb, Hydrodynamics, sixth ed., Cambridge University Press, (1932), p. 98. G. Kirchhoff, Grelles Journal 70 (1869), p.289.

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Flow of Ideal Fluid

12.7

PROBLEMS

1. Obtain the velocity potential and the flow function for a flow whose components of velocity in the x and y directions at a given point in the flow are u0 and v0, respectively. 2. Show the existence of the following relationship between flow function j and the velocity components vr, vq in a two-dimensional flow: vq ¼

vj vj ; vr ¼ vr rvq

3. What is the flow whose velocity potential is expressed as f ¼ Gq=ð2pÞ? 4. Obtain the velocity potential and the stream function for radial flow from the origin at quantity q per unit time. 5. Assuming j ¼ U(r r0 2/r)sin q expresses the stream function around a cylinder of radius r0 in a uniform flow of velocity U. Obtain the velocity distribution and the pressure distribution on the cylinder surface. 6. Obtain the pattern of flow whose complex potential is expressed as w ¼ z2. 7. What is the flow expressed by the following complex potential? wðzÞ ¼ f þ ij ¼ i

G log z 2p

8. Obtain the complex potential of a uniform flow at angle a to the x axis. 9. Obtain the streamline y ¼ k and the equipotential line x ¼ c of a flow parallel to the x axis on the z plane when mapped onto the z plane by mapping function z ¼ 1/z. 10. Obtain the flow in the case where parallel flow w ¼ Uz on the z plane is mapped onto the z plane by mapping function z ¼ z1/3.

CHAPTER 13

Flow of a Compressible Fluid Fluids have the capacity to change volume and density, i.e., they are subject to compressibility. Gas is much more compressible than liquid. Because liquid has a low compressibility, when its motion is studied its density is normally regarded as constant. However, where an extreme change in pressure occurs, such as in water hammer, compressibility is taken into account. Gas has a large compressibility but when its velocity is low compared with the sonic velocity the change in density is small and it is then treated as an incompressible fluid. Nevertheless, when studying the atmosphere with large altitude changes, highvelocity gas flow in a pipe with large pressure difference, the drag sustained by a body moving with significant velocity in a calm gas and the flow which accompanies combustion, etc., changes of density must be taken into account. As described later, the parameter expressing the degree of compressibility is the Mach number M. Supersonic flow, where M > 1, behaves very differently from subsonic flow where M < 1. In this chapter, thermodynamic characteristics will be explained first, followed by the effects of sectional change in isentropic flow, flow through a convergent nozzle and flow through a convergentedivergent nozzle. Then the adiabatic but irreversible shock wave will be explained, and finally adiabatic pipe flow with friction (Fanno flow) and pipe flow with heat transfer (Rayleigh flow).

13.1

THERMODYNAMIC CHARACTERISTICS

Now, with the specific volume v and density r, rv ¼ 1

(13.1)

A gas having the following relationship between absolute temperature T and pressure p pv ¼ RT

(13.2)

or p ¼ rRT

(13.3) 255

Introduction to Fluid Mechanics. http://dx.doi.org/10.1016/B978-0-08-102437-9.00013-9 Copyright © 2018 Elsevier Ltd. All rights reserved.

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Flow of a Compressible Fluid

is called a perfect gas. And Eqs (13.2) and (13.3) are called its equations of state. Here R is the gas constant, and R¼

R0 M

where R0 is the universal gas constant ðR0 ¼ 8:3144 J=ðmol KÞÞ and is constant despite any kind of gas and M is the molecular weight. For example, for air, assuming M ¼ 28.96 ðg=molÞ, the gas constant is R¼


  8:3144 J=ðmol KÞ ¼ 0:287 J=ðg KÞ ¼ 287 m2 s2 K 28:96ðg=molÞ

Then, assuming internal energy per unit mass is e, entropy per unit mass h is defined as: h ¼ e þ pv

(13.4)

Using internal energy e and entropy h, specific heat at constant volume cv and specific heat at constant pressure cp are expressed, respectively, as follow.   ve Specific heat at constant volume cv ¼ ; de ¼ cv dT (13.5) vT v   vh ; dh ¼ cp dT (13.6) Specific heat at constant pressure cp ¼ vT p According to the first law of thermodynamics, when a quantity of heat dq is supplied to a system, the internal energy of the system increases by de, and work pdv is done by the system. In other words, dq ¼ de þ pdv

(13.7)

From the equation of state (Eq. 13.2) pdv þ vdp ¼ RdT

(13.8)

From Eq. (13.6) dh ¼ de þ pdv þ vdp

(13.9)

Now, because dp ¼ 0 in the case of constant pressure change, Eqs (13.8) and (13.9) become pdv ¼ RdT

(13.10)

dh ¼ de þ pdv ¼ dq

(13.11)

Substitute Eqs (13.5), (13.6), (13.10) and (13.11) into Eq. (13.7), cp dT ¼ cv dT þ RdT

13.1 Thermodynamic Characteristics

which becomes cp

cv ¼ R

(13.12)

Now, cp/cv ¼ k [k: ratio of specific heats (isentropic index)], so cp ¼ cv ¼

k

1

k 1 k

1

R

(13.13)

R

(13.14)

Whenever heat energy dq is supplied to a substance of absolute temperature T, the change in entropy ds of the substance is defined by the following equation: ds ¼

dq T

(13.15)

As is clear from this equation, if a substance is heated the entropy increases, while if it is cooled the entropy decreases. Also, the higher the gas temperature, the greater the added quantity of heat for the increment in entropy. Rewrite Eq. (13.15) using Eqs (13.1), (13.2), (13.12) and (13.13), the following equation is obtained1: dq ¼ cv dðlog pvk Þ T

(13.16)

When changing from state ðp1 ; v1 Þ to state ðp2 ; v2 Þ, if reversible, the change in entropy is as follows from Eqs (13.15) and (13.16): s2

s1 ¼ cv log

p2 v2k p1 v1k

(13.17)

In addition, the relationships of Eqs (13.18)e(13.20) are also obtained.2

1

From rv ¼ 1; dr=r þ dv=v ¼ 0; from pv ¼ RT; dp=p þ dv=v ¼ dT=T   dq p dp dp dT dT dv dv dv T ¼ cv T þ T dv ¼ cv T þ R v ¼ cv p þ cp v ¼ cv p þ k v .

Therefore, 2

Eqs (13.18), (13.19) and (13.20) are, respectively, induced from the following equations:

ds ¼ dq ¼ cv dT T T

R dr ¼ cv dT r T

ðk 1Þcv dr r

ds ¼ dq ¼ cv dT þ R dvv ¼ ðcp RÞ dT þ R dvv ¼ kcv dT T T T T ds ¼ dq ¼ cv dpp þ cp dvv ¼ cv dpp T

cp dr ¼ cv dpp r

kcv dr . r

ðk 1Þcv dpp

257

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CHAPTER 13:

Flow of a Compressible Fluid

s2

s2

s2

"

  # T2 r1 k 1 s1 ¼ cv log T1 r2 "    T2 k p1 k s1 ¼ cv log T1 p2  k

p 2 r1 s1 ¼ cv log p 1 r2

(13.18) 1

#

(13.19)

(13.20)

for the reversible adiabatic (isentropic) change, ds ¼ 0. Putting the proportional constant equal to c, Eq. (13.17) gives Eq. (13.21), or Eq. (13.22) from Eq. (13.20). That is, pvk ¼ c

(13.21)

p ¼ crk

(13.22)

Eqs (13.18) and (13.19) give the following equation: T ¼ crk

1

¼ cpðk

1Þ=k

(13.23)

When a quantity of heat DQ transfers from a high-temperature gas at T1 to a low-temperature gas at T2, the changes in entropy of the respective gases are DQ/T1 and DQ/T2. Also, the value of their sum is never negative.3 Using entropy, the second law of thermodynamics could be expressed as “Although the grand total of entropies in a closed system does not change if a reversible change develops therein, it increases if any irreversible change develops.” This is expressed by the following equation: ds  0

(13.24)

Consequently, it can also be said that “entropy in nature increases.”

13.2

SONIC VELOCITY

It is well known that when a minute disturbance develops in a gas, the resulting change in pressure propagates in all directions as a compression wave (longitudinal wave, pressure wave), which we feel as a sound. Its propagation velocity is called the sonic velocity. Here, for the sake of simplicity, assume a plane wave in a stationary fluid in a tube of uniform cross-sectional area A as shown in Fig. 13.1. Assume that,

3

In a reversible change where an ideal case is assumed, the heat shifts between gases of equal temperature. Therefore, ds ¼ 0.

13.2 Sonic Velocity

u

a

p + dp p

l

FIGURE 13.1 Propagation of pressure wave.

because of a disturbance, the velocity, pressure and density increase by u, dp and dr, respectively. Between the wave front which has advanced at sonic velocity a and the starting plane is a section of length l where the pressure has increased. Because the wave travel time, during which the pressure increases in this section, is t ¼ l/a, the mass in this section increases by Aldr/t ¼ Aadr per unit time. To supplement it, gas of mass Au(r þ dr) z Aur flows in through the left plane. In other words, the continuity equation in this case is: Aadr ¼ Aur or adr ¼ ur

(13.25)

The fluid velocity in this section changes from 0 to u in t time. In other words, the velocity can be regarded as having uniform acceleration u/t ¼ ua/l. Taking its mass as Alr and neglecting dr in comparison with r, the equation of motion is: Alr

ua ¼ Adp l

or rau ¼ dp

(13.26)

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CHAPTER 13:

Flow of a Compressible Fluid

Eliminate u in Eqs (13.25) and (13.26), and pffiffiffiffiffiffiffiffiffiffiffiffiffi a ¼ dp=dr

(13.27)

is obtained.

Because a sudden change in pressure is regarded as adiabatic, the following equation is obtained from Eqs (13.3) and (13.23)4: pffiffiffiffiffiffiffiffiffi a ¼ kRT (13.28)

In other words, the sonic velocity is proportional to the square root of absolute temperature. For example, for k ¼ 1.4 and R ¼ 287 m2/(s2 K), pffiffiffi a z 20 T ða ¼ 340 m=s at 16 C ð289 KÞÞ (13.29) Next, if the bulk modulus of fluid is K, from Eqs (2.10) and (2.12)

dp ¼

K

dv dr ¼K v r

and dp K ¼ dr r Therefore, Eq. (13.27) can also be expressed as follows: pffiffiffiffiffiffiffiffiffi a ¼ K=r

13.3

(13.30)

MACH NUMBER

The ratio of flow velocity u to sonic velocity a, i.e., M ¼ u/a, is called the Mach number (See Section 10.4.1). Now, consider a body placed in a uniform flow of velocity u. At the stagnation point, the pressure increases by Dp ¼ ru2/2 in approximation of Eq. (9.1). This increased pressure brings about an increased density Dr z Dp/a2 from Eq. (13.27). Consequently, sffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffi u 1 2Dp 2Dr M¼ z z (13.31) a a r r In other words, the Mach number is a nondimensional number expressing the compressive effect on the fluid. From the above equation, the Mach number M corresponding to a density charge of 5% is approximately 0.3. For this reason,

4


From p ¼ crk ; dp dr ¼ ckrk

1


¼ kp r ¼ kRT.

13.3 Mach Number

steady flow can be treated as incompressible flow up to around a Mach number of 0.3. Now, consider the propagation of a sonic wave. This minute change in pressure, like a sound, propagates at sonic velocity a from the sonic source in all directions as shown in Fig. 13.2A. A succession of sonic waves is produced cyclically from a sonic source placed in a parallel flow of velocity u. When u is smaller than a, as shown in Fig. 13.2B, i.e., if M < 1, the wave fronts propagate at velocity a u upstream but at velocity a þ u downstream. Consequently, the interval between the wave fronts is dense upstream while being sparse downstream. When the upstream wave fronts therefore develop a higher frequency tone than those downstream, this produces the Doppler effect. When u ¼ a, i.e., M ¼ 1, the propagation velocity is just zero with the sound propagating downstream only. The wave front is now as shown in Fig. 13.2C, producing a Mach wave normal to the flow direction. When u > a, i.e., M > 1, the wave fronts cannot propagate upstream as in Fig. 13.2D, but propagate in sequence downstream. The envelope of these

(A)

(B) 3a 2a

M1

0 1 23

45

t

t

01 234 5

α

u 0

1

2

3

4 ut

5

t

Mach cone

FIGURE 13.2 Mach number and propagation range of a sonic wave: (A) calm; (B) subsonic (M < 1); (C) sonic (M ¼ 1); (D) supersonic (M > 1).

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Flow of a Compressible Fluid

wave fronts forms a Mach cone. The propagation of sound is limited to the inside of the cone only. If the included angle of this Mach cone is 2a, then5 sin a ¼ a=u ¼ 1=M

(13.32)

is called the Mach angle.

13.4

BASIC EQUATIONS FOR ONE-DIMENSIONAL COMPRESSIBLE FLOW

For a constant mass flow m of fluid density r flowing at velocity u through section area A, the continuity equation is m ¼ ruA ¼ constant

(13.33)

or by logarithmic differentiation dr du dA þ þ ¼0 r u A

(13.34)

Euler’s equation of motion in the steady state along a streamline is   1 dp d 1 2 þ u ¼0 r ds ds 2 or Z

dp 1 2 þ u ¼ constant r 2

(13.35)

Assuming adiabatic conditions from p ¼ crk Z Z dp k p ¼ ckrk 2 dr ¼ þ constant r k 1r Substituting into Eq. (13.35), k k

p 1 2 þ u ¼ constant 1r 2

(13.36)

or k k

1

1 RT þ u2 ¼ constant 2

(13.37)

Eqs (13.36) and (13.37) correspond to Bernoulli’s equation for an incompressible fluid.

5

Actually, the three-dimensional Mach line forms a cone, and the Mach angle is equal to its semiangle.

13.4 Basic Equations for One-Dimensional Compressible Flow

If fluid discharges from a very large vessel, u ¼ u0 z 0 (using subscript 0 for the state variable in the vessel), Eq. (13.37) gives, k k

1 k RT þ u2 ¼ RT0 1 2 k 1

or T0 1 k 1 u2 k 1 2 M ¼1þ ¼1þ RT k 2 2 T

(13.38)

2

In this equation, T0, T and R1 k k 1 u2 are, respectively, called the total temperature, the static temperature and the dynamic temperature. From Eqs (13.23) and (13.38)  k=ðk 1Þ   p0 T0 k 1 2 k=ðk M ¼ 1þ ¼ 2 p T

1Þ

(13.39)

This is applicable to a body placed in the flow, e.g., between the stagnation point of a Pitot tube and the main flow.

13.4.1

Correction to a Pitot Tube (See Section 11.1.1)

Putting pN as the pressure at a point not affected by a body and making a binominal expansion of Eq. (13.39), then (in the case where M < 1)6   k 2 k 4 kð2 kÞ 6 M þ/ p0 ¼ pN 1 þ M þ M þ 2 8 48 (13.40)   1 2 1 2 2 k 4 M þ/ ¼ pN þ ru 1 þ M þ 2 4 24 For an incompressible fluid, p0 ¼ pN þ 12 ru2 . Consequently, for the case when the compressibility of fluid is taken into account, the correction appearing in Table 13.1 is necessary.

Table 13.1 Pilot Tube Correction M ðp0

pN Þ 12 ru2 ¼ c

Relative pffiffiffierror of u ¼ ð c 1Þ  100% 6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1.000

1.003

1.010

1.023

1.041

1.064

1.093

1.129

1.170

0

0.15

0.50

1.14

2.03

3.15

4.55

6.25

8.17

ku ¼ pN u2 ¼ ru2 . pN kM2 ¼ pN k ua2 ¼ pN kRT RT 2

2

263

264

CHAPTER 13:

Flow of a Compressible Fluid

From Table 13.1, it is found that, when M ¼ 0.7, the true flow velocity is approximately 6% less than if the fluid was considered to be incompressible.

13.5

ISENTROPIC FLOW

13.5.1

Flow in a Pipe (Effect of Sectional Change)

Consider the flow in a pipe with a gradual sectional change, as shown in Fig. 13.3, having its properties constant across any section. For the fluid at sections 1 and 2 in Fig. 13.3, continuity equation:

dr du dA þ þ ¼0 r u A

equation of momentum conservation: isentropic relationship : p ¼ crk sonic velocity : a2 ¼

(13.41) dpA ¼ ð AruÞdu

(13.42) (13.43)

dp dr

(13.44)

From Eqs (13.41), (13.42) and (13.44) a2 dr ¼ rudu ¼ ru2 M2

du ¼ u

du u

dr du dA ¼ þ r u A

Therefore, 

M2

 du dA 1 ¼ u A

(13.45)

A

A + dA

p

p + dp

u

u + du

ρ

ρ + dρ

1 2

FIGURE 13.3 Flow in pipe with a gradual sectional change.

13.5 Isentropic Flow

or du 1 u ¼ 2 dA M 1A

(13.46)

Also, dr ¼ r

M2

du u

(13.47)

Therefore, dr du ¼ M2 r u

(13.48)

From Eq. (13.46), when M < 1, du/dA < 0, i.e., the flow velocity decreases with increased sectional area, but when M > 1, dr/r > du/u, i.e., for supersonic flow the density decreases at a faster rate than the velocity increase. Consequently, for mass continuity, the surprising fact emerges that to increase the flow velocity the section area should increase rather than decrease, by contrast with subsonic flow. From Eq. (13.47), the change in density has an inverse relationship to the velocity. Also from Eq. (13.23), the pressure and the temperature change in proportion to the density. The above results are summarized in Table 13.2.

13.5.2

Convergent Nozzle

Gas of pressure p0, density r0 and temperature T0 flows from a large vessel through a convergent nozzle into the open air of back pressure pb is entropically at velocity u, as shown in Fig. 13.4. Putting p as the outer plane pressure, from Eq. (13.36) u2 k p k p0 ¼ þ 2 k 1 r k 1 r0

Table 13.2 Subsonic Flow and Supersonic Flow in One-Dimensional Isentropic Flow Flow State Subsonic

Supersonic

Changing Item Changing area Changing velocity/Mach number Changing density/pressure/temperature

þ

þ þ

þ

þ þ

265

266

CHAPTER 13:

Flow of a Compressible Fluid

(A) Vessel p0 T0

p pb

u0 = 0

ρ0

(B) p/p0 1

(pb = p0)

p/p0 1

p = pb (pb > p*) 0.528

0.528

Choked flow p = p* (pb ≤ p*)

O

Position in nozzle

O

Mass flow

FIGURE 13.4 Flow passing through convergent nozzle. (A) Pressure distribution in nozzle. (B) Relationship between Mass flow and pressure distribution.

Using Eq. (13.23) with the above equation vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " u  ðk 1Þ=k # u k p0 p t u¼ 2 1 k 1 r0 p0

Therefore, the flow rate is vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u  2=k "  ðk 1Þ=k # u k p p m ¼ ruA ¼ At2 p0 r 1 k 1 0 p0 p0

Writing p=p0 ¼ x, then  k=ðk vm p 2 ¼ 0; x ¼ ¼ vx p0 kþ1

1Þ

(13.49)

(13.50)

(13.51)

When p/p0 has the value of Eq. (13.51), m is maximum. The corresponding pressure is called the critical pressure and is written as p*. For air, p =p0 ¼ 0:528

(13.52)

Using the relationship between m and p/p0 of Eq. (13.50), the maximum flow rate occurs when p=p0 ¼ 0:528 as shown in Fig. 13.4B. Thereafter, however

13.5 Isentropic Flow

much the pressure pb downstream is reduced, the pressure there cannot propagate towards the nozzle because it is discharging at sonic velocity. Therefore, the pressure of the air in the outlet plane remains p*, and the mass flow rate does not change. In this state the flow is called choked.

Substitute Eq. (13.51) into Eq. (13.49) and use the relationship p0 rk0 ¼ p rk to obtain rffiffiffiffiffiffi p (13.53) u ¼ k ¼ a r In other words, for M ¼ 1, under these conditions u is called the critical velocity and is written as u*. At the same time  1=ðk 1Þ r 2 ¼ ¼ 0:634 (13.54) kþ1 r0 T 2 ¼ 0:833 ¼ T0 k þ 1

(13.55)

The relationships of the above Eqs (13.52), (13.54) and (13.55) show that, at the critical outlet state M ¼ 1, the critical pressure falls to 52.8% of the pressure in the vessel, while the critical density and the critical temperature, respectively, decrease by 37% and 17% from those of the vessel.

13.5.3

ConvergenteDivergent Nozzle

A convergentedivergent nozzle (also called de Laval nozzle) shown in Fig. 13.57 is a convergent nozzle followed by a divergent section. When back pressure pb outside the nozzle is reduced below p0, flow is established. So long as the fluid flows out through the throat section without reaching the critical pressure the general behaviour is the same as for incompressible fluid. When the back pressure decreases further, the pressure at the throat section reaches the critical pressure and M ¼ 1; thereafter, the flow in the divergent port is at least initially supersonic. However, unless the back pressure is low enough, supersonic velocity cannot be maintained. Instead, a shock wave develops, after which the flow becomes subsonic. As the back pressure is recovered, the shock moves further away from the diverging section to the exit plane and eventually disappears, giving a perfect expansion. The ratio A/A* between the outlet section and the throat giving this perfect expansion is called the area ratio and using Eqs (13.50) and (13.51),

7

H.W. Liepmann, A. Roshko, Elements of Gasdynamics, John Wiley & Sons, Inc., New York, (1975), p. 127.

267

268

CHAPTER 13:

Flow of a Compressible Fluid

Vessel p0

A* A

T0

p pb

u0 = 0

ρ0

1 p/p0

d′

a b c d

0.528 e s O

T s

M

d′

1 O

f g E f

Position on nozzle

d c b a

FIGURE 13.5 Compressible flow in a divergent nozzle.

A ¼ A



13.6

2 kþ1

1=ðk

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " 1=k ,u  ð1 kÞ=k # uk þ 1 p0 p0 t 1 k 1 p p

1Þ 

(13.56)

SHOCK WAVES

When air undergoes large and rapid compression (e.g., following an explosion, the release of engine gases into an exhaust pipe or where an aircraft or a bullet flies at supersonic velocity), a thin wave of large pressure change is produced as shown in Figs 13.6 and 13.7. Because the state of gas changes adiabatically, an increased temperature accompanies this increased pressure. As shown in Fig. 13.8A, the wave face at the rear of the compression wave, being at a higher temperature, propagates faster than the wave face at the front. The rear therefore gradually catches up with the front until finally, as shown in Fig. 13.8B, the wave faces combine into a thin wave increasing the pressure discontinuously. Such a pressure discontinuity is called a shock wave, which is only associated with an increase, rather than a reduction, in pressure in the flow direction.

13.6 Shock Waves

Shock wave

FIGURE 13.6 Jet plane flying at supersonic velocity.

Shock wave

Air, Mach 3

FIGURE 13.7 Cone in supersonic flow (Schlieren method).

Because a shock wave is essentially different from a sound wave because of the large change in pressure, the propagation velocity of the shock is larger, and the larger the pressure rise, the greater the propagation velocity. If a long cylinder is partitioned with Cellophane film or aluminium foil to give a pressure difference between the two sections, and then the partition is ruptured, a shock wave develops. The shock wave in this case is at right angles to the flow and is called a normal shock wave. The device itself is called a shock tube.

269

270

CHAPTER 13:

Flow of a Compressible Fluid

p

(A)

(B) Shock wave a2 a1

a1 < a2 Propagating direction

x

FIGURE 13.8 Propagation of a compression wave. (A) Time t1 and (B) time t2.

As shown in Fig. 13.9, the states upstream and downstream of the shock wave are, respectively, represented by subscripts 1 and 2. A shock wave Dx is so thin, approximately micrometres at thickest, that it is normally regarded as having no thickness. Now, assuming A1 z A2, the continuity equation is r1 u1 ¼ r2 u2

(13.57)

the equation of momentum conservation is p 1 þ r1 u 1 2 ¼ p 2 þ r2 u 2 2

(13.58)

and the equation of energy conservation is k k

p1 u1 2 k p 2 u2 2 ¼ þ þ 1 r1 k 1 r2 2 2

or u1

2

2k

2

u2 ¼

k



p2 1 r2

p1 r1



(13.59)

From Eqs (13.57) and (13.58) u1 2 ¼

p2 r2

p1 r2 r1 r1

(13.60)

u2 2 ¼

p2 r2

p1 r1 r1 r2

(13.61)

Substituting Eqs (13.60) and (13.61) into Eq. (13.59), r2 ½ðk þ 1Þ=ðk 1ފðp2 =p1 Þ þ 1 u1 ¼ ¼ ½ðk þ 1Þ=ðk 1ފ þ p2 =p1 r1 u2

(13.62)

13.6 Shock Waves

M1

u1

u2

M1

M2

a1

a2

p1

p2

ρ1

ρ2

A1

A2

M2

∆x

FIGURE 13.9 Normal shock wave.

Or, using Eq. (13.3), T2 ½ðk þ 1Þ=ðk ¼ T1 ½ðk þ 1Þ=ðk

1ފ þ p2 =p1 1ފ þ p1 =p2

(13.63)

Eqs (13.62) and (13.63), which are called the RankineeHugoniot equations, show the relationships between the pressure, density and temperature ahead of and behind a shock wave. From the change of entropy associated with these equations, it can be deduced that a shock wave develops only when the upstream flow is supersonic.8 It has already been explained that when a supersonic flow strikes a particle, a Mach line develops. On the other hand, when a supersonic flow flows along a plane wall, numerous parallel Mach lines develop as shown in Fig. 13.10A.

8

From Eqs (13.57) and (13.58),

p2 p1

2k ¼ 1 þ kþ1 ðM 1 2 1Þ

Likewise

p2 p1

Therefore,

 2k M 2 ¼ 1 þ kþ1 2

2 þ ðk 1ÞM1 2 . 2 ðk 1Þ 1

M2 2 ¼ 2kM

1



271

272

CHAPTER 13:

Flow of a Compressible Fluid

(A)

(B)

Mach line

M>1

M>1

µ=

sin–1

Expansion wave

M>1  1  M

M >

(D)

Shock wave

1

(C)

Envelope of Mach line group Shock wave M>1 M


dm (Fig. 13.12B), however, the shock wave detaches and stands off from nose A.

13.7

FANNO FLOW AND RAYLEIGH FLOW

Because an actual flow of compressible fluid in pipelines and similar conduits is always affected by the friction between the fixed wall and the fluid, it can be adiabatic but not isentropic. Such an adiabatic but irreversible (i.e., nonisentropic) flow is called Fanno flow. Alternatively, in a system of flow forming a heat exchanger or combustion process, friction may be neglected but transfer of heat must be taken into account. Such a flow without friction through a pipe with heat transmission is called Rayleigh flow.

273

Flow of a Compressible Fluid

Shock wave

(A)

(B) Shock wave

σ δ

δ

A

A

FIGURE 13.12 Flow pattern and shock wave around a body placed in supersonic flow: (A) shock wave attached to wedge; (B) detached shock wave.

Fig. 13.13 shows a diagram of both of those flows in a pipe with fixed sectional area. The lines appearing there are called the Fanno line and Rayleigh line, respectively. For both of them, points a or b of maximum entropy corresponds to the sonic state M ¼ 1. The curve above these points corresponds to subsonic velocity and that below to supersonic velocity. The states immediately ahead of and behind the normal shock wave are expressed by the intersection points 1 and 2 of these two curves. For the flow through the shock wave, only the direction of increased entropy, i.e., the discontinuous change, 1 / 2 is possible.

M < 1

M
1

1

g

M g

M>

an sh

ock w

a

1

ave

Co

Norm

CHAPTER 13:

Enthalpy h

274

lin oo

C

Rayleigh line Fanno line Entropy s

FIGURE 13.13 Fanno line and Rayleigh line.

n ati He

b

M=1

13.8 Problems

13.8

PROBLEMS

1. When air is regarded as a perfect gas, what is the density r in kg/m3 of air at 15 C and 760 mmHg? 2. Find the velocity a of sound propagating in hydrogen at 16 C. 3. When the velocity is 30 m/s, pressure 3.5  105 Pa and temperature 150 C at a point on a streamline in an isentropic air flow, obtain the pressure and temperature at the point on the same streamline of velocity 100 m/s. 4. Find the temperature T0, pressure p0 and density r0 at the front edge (stagnation point) of a wing of an aircraft flying at 900 km/h in calm air of pressure 4.5  104 Pa and temperature 26 C. 5. From a Schlieren photograph of a small bullet flying in air of 15 C and standard atmospheric pressure, it was noticed that the Mach angle was 50 degrees. Find the velocity u of this bullet. 6. When a Pitot tube was inserted into an air flow at high velocity, the pressure at the stagnation point was 1  105 Pa, the static pressure was 7  104 Pa and the air temperature was 10 C. Find the velocity u of this air flow. 7. Air of gauge pressure 6  104 Pa and temperature 20 C is stored in a large tank. When this air is released through a convergent nozzle into air of 760 mmHg, find the flow velocity u at the nozzle exit. 8. Air of gauge pressure 1.2  105 Pa and temperature 15 C is stored in a large tank. When this air is released through a convergent nozzle of exit area 3 cm2 into air of 760 mmHg, what is the mass flow m? 9. Find the divergence ratio necessary for perfectly expanding air under standard conditions down to 100 mmHg absolute pressure through a convergentedivergent nozzle. 10. The nozzle for propelling a rocket is a convergentedivergent nozzle of throat cross-sectional area of 500 cm2. Regard the combustion gas as a perfect gas of mean molecular weight 25.8 and k ¼ 1.25. To make the combustion gas of pressure 32  105 Pa and temperature 3300K expand perfectly from the combustion chamber into air of 1  105 Pa, what should be the cross-sectional area at the nozzle exit? 11. When the rocket in Problem 10 flies at an altitude where the pressure is 2  104 Pa, what is the available thrust from the rocket? 12. A supersonic flow of Mach 2, pressure 5  104 Pa and temperature 5 C develops a normal shock wave. What is the Mach number M, flow velocity u and pressure p behind the wave?

275

CHAPTER 14

Unsteady Flow Since olden times fluid has mostly been used mechanically for generating motive power, but in recent times it has been used for power transmission or automatic power control. High-pressure fluid has to be used in these systems for high speed and fast response. Consequently, the issue of unsteady flow has become very important. In this chapter, the oscillation of a liquid column with frictional resistance will be explained first, followed by the propagation of pressure in a pipeline, the time-dependent return to steady flow conditions after the sudden opening of a valve, the velocity of pressure waves in an elastic pipeline, water hammer in the case of instantaneous valve closure and a fluidic oscillator.

14.1

OSCILLATION OF A LIQUID COLUMN IN A U-TUBE

14.1.1

No Frictional Resistance

When the viscous frictional resistance is zero, by Newton’s second law, rgðz2 g ðz 2

z1 ÞA ¼ z1 Þ þ l

rAl

dv dt

dv ¼0 dt

(14.1) (14.2)

Moving the datum for height to the steady-state position as shown in Fig. 14.1, then, gðz2

z1 Þ ¼ 2gz

Also, dv d2 z ¼ dt dt 2 and from above, d2 z ¼ dt 2

2g z l

(14.3) 277

Introduction to Fluid Mechanics. https://doi.org/10.1016/B978-0-08-102437-9.00014-0 Copyright © 2018 Elsevier Ltd. All rights reserved.

278

CHAPTER 14:

Unsteady Flow

A z O

O –z

z2

d l z1

O

O

FIGURE 14.1 Oscillation of a liquid column in a U-tube.

Therefore, rffiffiffiffiffiffi ! rffiffiffiffiffiffi ! 2g 2g t þ C2 sin t z ¼ C1 cos l l

(14.4)

Assuming the initial conditions are t ¼ 0 and z ¼ z0, then dz/dt ¼ 0, C1 ¼ z0, C2 ¼ 0. Therefore, rffiffiffiffiffiffi ! 2g t (14.5) z ¼ z0 cos l This means that the liquid surface makes an oscillatory cycle with a period of pffiffiffiffiffiffiffiffiffi T ¼ 2p l=2g.

14.1.2

Laminar Frictional Resistance

In this case, with the viscous frictional resistance (Eq. 6.38) in Eq. (14.1), gðz2

z1 Þ þ l

dv 32nvl þ 2 ¼0 dt d

Substitute 2z ¼ (z2

(14.6)

z1) as above,

d2 z 32n dz 2g þ z¼0 þ 2 dt 2 d dt l d2 z dz þ 2zun þ u2n z ¼ 0 dt 2 dt

rffiffiffiffiffiffi 2g 16n 1 and z ¼ 2 where un ¼ l d un

(14.7)

14.2 Propagation of Pressure in a Pipeline

The general solution of Eq. (14.7) is as follows: 1. when z < 1

z¼e

zun t

  qffiffiffiffiffiffiffiffiffiffiffiffiffiffi   qffiffiffiffiffiffiffiffiffiffiffiffiffiffi  C1 sin un 1 z2 t þ C2 cos un 1 z2 t

(14.8)

Assume z ¼ z0 and dz/dt ¼ 0 when t ¼ 0. Then, "  qffiffiffiffiffiffiffiffiffiffiffiffiffiffi   qffiffiffiffiffiffiffiffiffiffiffiffiffiffi # z zun t 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin un 1 z t þ cos un 1 z2 t z ¼ z0 e 1 z2 z0 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi e 1 z2

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 sin un 1 z t þ f

(14.9)

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 1 z2 z

1

f ¼ tan

zun t

2. when z > 1

z ¼ z0 e

zun t

"

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi   qffiffiffiffiffiffiffiffiffiffiffiffiffiffi # z pffiffiffiffiffiffiffiffiffiffiffiffiffiffi sinh un z2 1 t þ cosh un z2 1 t z2 1

z0 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi e z2 1 f ¼ tanh

1

zun t

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 sinh un z 1 tþf

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! z2 1 z (14.10)

Eqs (14.9) and (14.10) can be plotted using the non-dimensional quantities of unt, z/z0, as shown in Fig. 14.2. With large frictional resistance, there is no oscillation but as it becomes smaller a damped oscillation occurs.

14.2

PROPAGATION OF PRESSURE IN A PIPELINE

In the system shown in Fig. 14.3, a tank (capacity V) is connected to a pipeline (diameter d, section area A and length l). If the inlet pressure is suddenly changed (from 0 to p1, say), it is desirable to know the response of the outlet

279

CHAPTER 14:

Unsteady Flow

1.0 0.8

ζ=2

0.6 0.4

ζ=1

0.2 z z0

280

0

–0.2

1

2

3

4

5

–0.4

6

7

8

9

10 11 12

ωn t

ζ = 0.2

–0.6 –0.8 –1.0

FIGURE 14.2 Motion of liquid column with frictional resistance.

p1

d p2 V

υ l

FIGURE 14.3 System comprising pipeline and tank.

pressure p2. Assuming a pressure loss Dp due to pipe friction, with instantaneous flow velocity v the equation of motion is rAl

dv ¼ Aðp1 dt

p2

DpÞ

(14.11)

If v is within the range of laminar flow, then Dp ¼

32ml v d2

(14.12)

14.3 Transient Flow in a Pipeline

The pipe is regarded as rigid so compressibility b is due to the fluid, dp2 ¼

1 Avdt b V

(14.13)

Substituting Eqs (14.12) and (14.13) into Eq. (14.11) gives d2 p2 32n dp2 A þ 2 ðp 2 þ d dt rlbV dt 2

p1 Þ ¼ 0

Now, writing sffiffiffiffiffiffiffiffiffiffi A 16n 1 un ¼ ; z¼ 2 ; p2 rlbV d un

p1 ¼ p

d2 p dp þ 2zun þ u2n p ¼ 0 dt 2 dt

(14.14)

Since Eq. (14.14) has the same form as Eq. (14.7), the solution also has the same form as Eqs. (14.9) and (14.10) with the response trend being similar to that shown in Fig. 14.2.

14.3

TRANSIENT FLOW IN A PIPELINE

When the valve at the end of a pipeline of length l as shown in Fig. 14.4 is instantaneously opened, there is a time lapse before the flow reaches steady state. When the valve first opens, the whole of head H is used for accelerating the flow. As the velocity increases, however, the head used for acceleration decreases owing to the fluid friction loss h1 and discharge energy h2. Consequently, the effective head available to accelerate the liquid in the pipe becomes rg(H h 1 h 2). Assuming the tank volume is large enough so that the effective

H

d

υ

l

FIGURE 14.4 Transient flow in a pipe.

281

282

CHAPTER 14:

Unsteady Flow

head H does not change, the equation of motion of the liquid in the pipe is as follows, putting A as the sectional area of the pipe, rgAðH

h2 Þ ¼

h1

rgAl dv g dt

(14.15)

Giving h1 ¼ l

l v2 v2 v2 ¼ k ; h2 ¼ d 2g 2g 2g

and ðk þ 1Þ

H

v2 l dv ¼ 2g g dt

(14.16)

Assume that velocity v became v0 (terminal velocity) in the steady state (dv/dt ¼ 0), and ðk þ 1Þv02 ¼ 2gH k¼

2gH v02

1

Substituting the above value of k into Eq. (14.16),   v2 l dv ¼ H 1 2 g dt v0 l v02 dv 2 gH v0 v2   lv0 v0 þ v t¼ log v0 v 2gH

dt ¼

(14.17)

Thus, time t for the flow to become steady is obtainable as shown in Fig. 14.5. Now, calculating the time from v/v0 ¼ 0 to v/v0 ¼ 0.99, the following equation can be obtained:   lv0 1:99 lv0 t¼ log (14.18) ¼ 2:646 0:01 2gH gH

14.4

VELOCITY OF PRESSURE WAVES IN A PIPE

The velocity of a pressure wave (sonic velocity) depends on the bulk modulus K (Eq. 13.30) as explained in Section 13.2. The bulk modulus expresses the relationship between change of pressure on a fluid and the corresponding change

14.4 Velocity of Pressure Waves in a Pipe

υ — υ0

1.0

0.5

t

FIGURE 14.5 Development of steady flow.

in its volume. When a small volume V of fluid in a short length of rigid pipe experiences a pressure wave, the resulting reduction in volume dV1 produces a reduction in length. If the pipe is elastic, however, it will experience radial expansion causing an increase in volume dV2. This produces a further reduction in the length of volume V. Therefore, to the wave, the fluid appears more compressible, i.e., to have a lower bulk modulus. A modified bulk modulus K0 is thus required which incorporates both effects. From the definition in Eq. (2.12), dp dV1 ¼ K V

(14.19)

where the minus sign was introduced solely for the convenience of having positive values of K. Similarly, for positive K0 , dp dV1 dV2 ¼ K0 V

(14.20)

where the negative dV2 indicates that, despite being a volume increase, it produces the equivalent effect of a volume reduction dV1. Thus, 1 1 dV2 ¼ þ K 0 K V dP

(14.21)

If the elastic modulus (Young’s modulus) of a pipe of inside diameter D and thickness b is E, the stress increase ds is ds ¼ E

dD D

283

284

CHAPTER 14:

Unsteady Flow

This hoop stress in the wall balances the internal pressure dp, ds ¼

D dp 2b

Therefore, dD Ddp ¼ D 2bE since V ¼ pD2/4, dV2 ¼ pDdD/2 per unit length, dV2 dD Ddp ¼ ¼2 D bE V

(14.22)

Substituting Eq. (14.22) into Eq. (14.21), then 1 1 D ¼ þ K 0 K bE or K0 ¼

K DK 1þ b E

(14.23)

The sonic velocity a in the fluid is, from Eq. (13.30) sffiffiffiffi K a¼ r Therefore, the propagation velocity a0 of a pressure wave in an elastic pipe is vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffi u u K vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 1 K0 u r 0 u (14.24) ¼ a a ¼ ¼u t t DK DK r 1þ 1þ b E b E

Since the values of D for steel, cast iron and concrete are respectively 206, 92.1 and 20.6 GPa, a0 is in the range 600e1200 m/s in an ordinary water pipeline.

14.5

WATER HAMMER

Water flows in a pipe as shown in Fig. 14.6. If the valve at the end of the pipe is suddenly closed, the velocity of the fluid will abruptly decrease causing a mechanical impulse to the pipe due to a sudden increase in pressure of the fluid. Such a phenomenon is called water hammer. This phenomenon poses a very important problem in cases where, for example, a valve is closed to reduce the water flow in a hydraulic power station when the load on the water turbine

14.5 Water Hammer

∆p g ρ—

aʹ

p0 g ρ— x

aʹ

υ

B

C

l

FIGURE 14.6 Water hammer.

is reduced. In general, water hammer is a phenomenon which is always possible whenever a valve is closed in a system where liquid is flowing.

14.5.1

Case of Instantaneous Valve Closure

When the valve at pipe end C in Fig. 14.6 is instantaneously closed, the flow velocity v of the fluid in the pipe, and therefore also its momentum, becomes zero. Therefore, the pressure increases by dp. Since the following portions of fluid are also stopped one after another, dp propagates upstream. The propagation velocity a0 of this pressure wave is expressed by Eq. (14.24). Given that an impulse is equal to the change of momentum dpA

l ¼ rAlv a0

or dp ¼ rva0

(14.25)

When this pressure wave reaches the pipe inlet, the pressurized pipe begins to discharge backwards into the tank at velocity v. The pressure reverts to the original tank pressure p0, and the pipe, too, begins to contract to its original state. The low pressure and pipe contraction proceed from the tank end towards the valve at velocity a0 with the fluid behind the wave flowing at velocity v. In time 2l/a0 from the valve closing, the wave reaches the valve. The pressure in the pipe has reverted to the original pressure, with the fluid in the pipe flowing at velocity v. Since the valve is closed, however, the velocity there must be zero. This

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(A)

2l — aʹ

∆p

— ρg t

(B) 2x — aʹ

∆p

— ρg t

FIGURE 14.7 Change in pressure due to water hammer (A) at point C and (B) at point B in Fig. 14.6.

requires a flow at velocity v to propagate from the valve. This outflow causes the pressure to fall by dp. This dp propagates upstream at velocity a0 . At time 3l/a0 , from the valve closing, the liquid in the pipe is at rest with a uniform low pressure of dp. Then, once again, the fluid flows into the low-pressure pipe from the tank at velocity v and pressure p. The wave propagates downstream at velocity a0 . When it reaches the valve, the pressure in the pipe has reverted to the original pressure and the velocity to its original value. In other words, at time 4l/a0 , the pipe reverts to the state when the valve was originally closed. The changes in pressure at points C and B in Fig. 14.6 are as shown in Fig. 14.7A and B, respectively. The pipe wall around the pressurized liquid also expands, so that the waves propagate at velocity a0 as shown in Eq. (14.24).

14.5.2

Case of Slow Valve Closure

When the valve closing time tc is less than time 2l/a0 for the wave round-trip of the pipeline, the maximum pressure increase when the valve is closed is equal to that in Eq. (14.25). When the valve closing time tc is longer than time 2l/a0 , it

14.6 Fluid Oscillation

is called slow closing, to which Allievi’s equation applies [named after L. Allievi (1856e1941), Italian hydraulics scholar]. That is, pffiffiffiffiffiffiffiffiffiffiffiffiffiffi  pmax 1 2 ¼1þ n þ n n2 þ 4 (14.26) 2 p0

Here, pmax is the highest pressure generated when the valve is closed, p0 is the pressure in the pipe when the valve is open, v is the flow velocity when the valve is open and n ¼ rlv/(p0tc). This equation does not account for pipe friction and the valve is assumed to be uniformly closed. In practice, however, there is pipe friction and valve leakage occurs. To obtain such changes in the flow velocity or pressure, either graphical analysis1 or method of characteristics (see Section 15.3.3) using the method of characteristics may be used.

14.6

FLUID OSCILLATION2,3

Consider water flow in a fluidic vortex chamber oscillator shown in Fig. 14.8 when water is supplied to its supply port. This supply flow passes through the main nozzle, crosses the vortex chamber towards the output port and forms a jet stream discharging from the output port. Because the width of the output Jet Stream

Output Port

Jet Stream

Vortex Chamber

Air Vent Main Nozzle Supply Port

FIGURE 14.8 Vortex chamber oscillation device.

1

J. Parmakian, Waterhammer Analysis, second ed., Dover, New York, 1963. Y. Nakayama, et al., The Transactions of JSME (B) 52 (474) (1986-2) 727. 3 Y. Nakayama, et al., Journal of the Visualization Society of Japan 14 (Suppl. 2) (1994) 137. 2

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port is slightly wider than the jet stream, the jet stream hits either one of the outlet ports and the jet stream adheres to the surface of the output port due to viscosity and then spreads out from the output port. Let the jet stream adhere to the left surface of the output port as shown in Fig. 14.8. A small portion of the jet stream flows back into the left side of the vortex chamber clinging to its surface and merges into the jet stream causing a vortex to form. In the right side of the vortex chamber, the jet stream entrains air from the right side of the vortex chamber thereby reducing the pressure in the right side of the vortex chamber. The pressure difference between the right and the left chambers causes the jet stream to bend towards the right. This behaviour of the jet stream repeats and becomes a stable oscillation. The oscillation frequency is obtained as follows: Let the differential pressure between the right and left sides of the chamber to bend the jet stream to one side be called the critical differential pressure pcr. From the balance between centrifugal force acting on jet stream and force caused by the differential pressure, J rh

pcr ¼

(14.27)

J ¼ rw U 2 A

(14.28)

where J is the momentum of the jet stream, r is the curvature of the jet stream, h is the height of the jet stream, rw is water density, U is the average velocity of the jet stream and A is the cross-section of the jet stream. Putting the diameter of air vent d, flow coefficient Ci, the mass flow rate mi through the air vent becomes, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mi ¼ 1:11Ci d2 ra ðp0 pÞ (14.29) where ra is the density of air, p0 is atmospheric pressure (absolute pressure) and p is the pressure (absolute pressure) in the vortex chamber. Letting the thickness of the boundary layer dD of air and flow coefficient C0, mass flow rate m0 of air flowing out together with the jet stream is, 1 C0 dD Ura h 3 p ra ¼ RT m0 ¼

(14.30) (14.31)

where R is gas constant and T is absolute temperature. The oscillation frequency of the jet stream can be calculated as: mi

m0 ¼

dðra V Þ dt

(14.32)

14.6 Fluid Oscillation

Substituting Eqs (14.29)e(14.31) into Eq. (14.32), pffiffiffiffiffiffi h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 1 dp h RT 1:11Ci d2 pðp0 pÞ þ C0 dD U p ¼ 0 dt 3 V V

(14.33)

Using the above equation, the time to reach the critical differential pressure pcr for a vortex oscillator with a vortex chamber diameter of 35 mm, a main nozzle area of 4  13 mm2, an air vent diameter of 3 mm and water supply pressure of 0.15 MPa becomes t ¼ 0.008 s; therefore, the oscillation frequency is 62 Hz.

FLUIDICS SUCCESSFULLY APPLIED TO THE ‘SHINKANSEN’ BULLET TRAIN A vortex chamber oscillation device has been developed and used as a water sprinkler. Because tons of snow fall in the northern part of Japan in winter, it was thought that snow on the railways could be melted by water sprinklers. Vortex chamber oscillators have been applied very successfully as water sprinklers for snow clearing. The photo shows the melting and successful clearance of snow on the Shinkansen lines by means of the fluidic vortex oscillators.

This vortex oscillator has also been successfully used more widely in washing equipment.

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z

l

FIGURE 14.9 U-tube.

14.7

PROBLEMS

1. As shown in Fig. 14.9, a liquid column of length 1.225 m in a U-shaped pipe is allowed to oscillate freely. Given that at t ¼ 0, z ¼ z0 ¼ 0.4 m and dz/dt ¼ 0, obtain a. the velocity of the liquid column when z ¼ 0.2 m, and b. the oscillation cycle time T. Ignore frictional resistance. 2. Oil of viscosity n ¼ 3  10 5 m2/s extends over a 3-m length of a tube of diameter 2.5 cm, as shown in Fig. 14.9. Air pressure in one arm of the U-tube, which produces 40 cm of liquid column difference, is suddenly released causing the liquid column to oscillate. What is the maximum velocity of the liquid column if laminar frictional resistance occurs? 3. Obtain the cycle time T for the oscillation of liquid in a U-shaped tube whose arms are both oblique as shown in Fig. 14.10. Ignore frictional resistance.

θ1 θ2

l

FIGURE 14.10 U-shaped tube whose arms are both oblique.

14.7 Problems

H l d

FIGURE 14.11 Water tank.

4. As shown in Fig. 14.11, a pipeline of diameter 2 m and length 4000 m is connected to a tank of head 18 m. Find the time t from the sudden opening of the valve for the exit velocity to reach 90% of the final velocity. Use friction coefficient for the pipe as 0.03. 5. Find the velocity a of a pressure wave propagating in a water-filled steel pipe of inside diameter 2 m and wall thickness 1 cm. If the bulk modulus K ¼ 2.1  109 Pa, density r ¼ 1000 kg/m3 and Young’s modulus for steel E ¼ 2.1  1011 Pa. 6. Water flows at velocity 3 m/s in the steel pipe in Problem 5, of length 1000 m. Obtain the increase in pressure when the valve is shut instantaneously. 7. The steady-state pressure of water flowing in the pipeline in Problem 6, at a velocity of 3 m/s is 5  105 Pa. What is the maximum pressure pmax reached when the valve is shut in 5 s?

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Computational Fluid Dynamics For the flow of an incompressible fluid, if the continuity equation and the NaviereStokes equations are solved simultaneously under given boundary conditions, an exact solution should be obtained. However, since the NaviereStokes equations are non-linear, it is difficult to solve them analytically. Nevertheless, approximate solutions are obtainable, e.g., by omitting the inertia terms for a flow whose the Reynolds number Re is small, such as slow flow around a sphere or the flow of an oil film in a sliding bearing or alternatively by neglecting the viscosity term for a flow whose Re is large, such as a fast free-stream flow around a wing. But for intermediate Re, the equations cannot be simplified because the inertia term is roughly as large as the viscosity term. Consequently, there is no other way than to obtain the approximate solution by performing the numerical computation. For a compressible fluid, it is further necessary to solve the equation of state and the energy equation simultaneously with respect to the thermodynamical properties. Thus, multi-dimensional shock wave problems can only be solved by relying upon numerical computation. Of late, the improvement of computer performance is remarkable, and the computation of significantly complex phenomena can be done using a personal computer. Therefore, in the case of phenomena partially modelled by analytical methods, numerical computation becomes a very useful adjunct. The numerical computation is performed as per the following procedure: 1. Derive the basic equation 2. Decide what method and what algorithm (procedure of computation) to use 3. Discretizing the equation (convert the equation to an algebraic equation, such as solved by computer) and write a program 4. Set the computational domain and divide into a grid 5. Find a solution by setting the given initial and boundary conditions Recently, general purpose fluid analysis software has become available, which can be applied to significant problems as described in this chapter. 293 Introduction to Fluid Mechanics. https://doi.org/10.1016/B978-0-08-102437-9.00015-2 Copyright © 2018 Elsevier Ltd. All rights reserved.

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This field of engineering is referred to as computational fluid dynamics. Firstly, the main discretization technique is described, including examples of how the discretization method is used for different types of flow.

15.1

DISCRETIZATION METHOD

In the description method of a flow field, there are the Eulerian method and the Lagrangian method as mentioned in Chapter 4 and both methods have been developed in the discretization.

15.1.1

Solution Based on the Eulerian Method

This solution is a method of describing flow conditions such as velocity, pressure, density and temperature for the spatial point x, y, z and functions of time t. The basic equations of the flow field are approximated at each grid point by piece-wise low-order polynomials giving the differences of the spatial derivatives in a finite region surrounded by the grid (cell) at frequent time intervals to describe the physical quantity. Variants of this method, the finite difference method, the finite volume method, the finite element method and the boundary element method, etc., have been used.

Finite Difference Method The finite difference method is a method to solve by approximating the differential coefficient appearing in the differential equations by a difference quotient. Consider it in one-dimension. The function f(x) and its first-order derivative function f0 (x) shown in Fig. 15.1 is a one-valued function and is finite and continuous with respect to x. This permits a Taylor expansion at the point x and for two points on either side at a distance of Dx: x Dx and x þ Dx, f ðxþ DxÞ ¼ f ðxÞ þ ðDxÞf 0 ðxÞ þ f ðx

DxÞ ¼ f ðxÞ

ðDxÞf 0 ðxÞ þ

ðDxÞ2 00 ðDxÞ3 000 f ðxÞ þ f ðxÞ þ / 2! 3!

(15.1)

ðDxÞ2 00 f ðxÞ 2!

(15.2)

ðDxÞ3 000 f ðxÞ þ / 3!

Subtracting Eq. (15.2) from Eq. (15.1),
f ðx DxÞ ¼ 2ðDxÞf 0 ðxÞ þ O ðDxÞ3 (15.3)
Here, O ðDxÞ3 means the combination of terms of order (Dx)3 or less. f ðxþ DxÞ

From Eq. (15.3),

f 0 ðxÞ ¼

f ðxþ DxÞ f ðx 2Dx

DxÞ

þ O ðDxÞ2




(15.4)

15.1 Discretization Method

f (x) f (x)

f′ (x)

f (x + Δx)

f (x) f (x − Δx)

Δx x − Δx

Δx x

x + Δx

x

FIGURE 15.1 Finite difference approximation.

Since the finite difference approximation, omitting O(Dx2) of this equation, is approximated by functional values on either side of x, it is called the central difference. The central
difference is said to have second-order accuracy because it ignored O ðDxÞ2 . Next, solving Eq. (15.1) for f 0 (x),

f 0 ðxÞ ¼

f ðxþ DxÞ Dx

f ðxÞ

þ OðDxÞ

(15.5)

Since the finite difference approximation, omitting O(Dx) of this equation, is approximated by the functional values on the side of increasing x, it is called the forward difference. This finite difference indication is said to have firstorder accuracy. From Eq. (15.2) through similar process, f 0 ðxÞ ¼

f ðxÞ

f ðx Dx

DxÞ

þ OðDxÞ

(15.6)

Since this finite difference approximation is approximated by the functional values on the side of decreasing x, it is called the backward difference. Accuracy is first-order like the forward difference scheme. Next, the central difference for f Eq. (15.2). f 00 ðxÞ ¼

f ðxþ DxÞ

2f ðxÞ þ f ðx ðDxÞ

2

00

DxÞ

is obtainable by adding Eq. (15.1) to

þ O ðDxÞ2




(15.7)

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∆x

y

j+1 ∆y j

j–1

i–1

i

i+1

0

x

FIGURE 15.2 Finite difference grid.

00

The f obtained in this way is second-order accuracy. Consider next the method applied to the two-dimensional region. In the finite difference method, by dividing the region to be solved in a grid called a finite difference grid, the approximate value of the solution at the grid points is determined. Considering the x, y plane as shown in Fig. 15.2, subscripts i, j indicate the position of space points (grid) in x and y directions, respectively. When the mesh intervals in x and y directions are Dx and Dy, respectively, space points (i, j) mean (xi ¼ x0 þ iDx, yi ¼ y0 þ jDy). While f means a functional symbol, the central-differential for the first-order partial differential coefficient vf/vx is expressed by the following equation from Eq. (15.4).   fiþ1;j fi 1;j vf (15.8) z vx i;j 2Dx It is represented in the same way for the forward difference, the backward difference, the second-order partial differential coefficient and y direction. In this way, a partial differential coefficient is expressed in finite difference form as an algebraic equation. By substituting these coefficients, a partial differential equation can be converted to an algebraic equation. As this example, the flow in a sudden expansion of a channel as shown in Fig. 15.3 is obtained as a potential flow. In this case, the flow satisfies Laplace’s equation for the stream function j of Eq. (12.16). Since the flow is symmetric about the centre line, only the lower half of the channel is the computational region and it is covered by a square grid of interval h as shown in Fig. 15.4.

15.1 Discretization Method

Channel outlet

Channel inlet

l

2l 3 l 4

3l

FIGURE 15.3 Flow in a sudden expansion of a channel.

ψ = 0.5

ψ = 0.5

0.0

0.0

0.0

FIGURE 15.4 Mesh and boundary condition.

Discretizing Eq. (12.16) by Eq. (15.2), v2 j v2 j jiþ1;j 2ji;j þ ji þ ¼ vx2 vy2 ðDxÞ2

1;j

þ

ji;jþ1

2ji;j þ ji;j ðDyÞ

2

1

¼0

(15.9)

Here, since Dx and Dy are a square mesh, rearranging about ji,j as Dx ¼ Dy ¼ h ji;j ¼

jiþ1;j þ ji

1;j

þ ji;jþ1 þ ji;j 4

1

(15.10)

If the stream function through this sudden expansion of a channel is 1, it is 0 on the lower wall and is 0.5 on the centre axis as shown in Fig. 15.4. On the inlet and outlet of channel, it is assumed to take a value obtained by linearly changing the value of 0 to 0.5 at the location of each grid point. In this way, a condition set at the boundary is called a boundary condition, and using this value, simultaneous equations for all the inner grid points are obtained by Eq. (15.10). The value of the unknown stream function can be obtained by solving these equations. In general, because such equations have many unknowns, they are mostly solved by successive iteration methods [1]: of

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0.5

0.5 0.4

0.0

0.2

0.3

0.1 0

FIGURE 15.5 Streamlines of flow in a sudden expansion.

GausseSeidel method and successive over-relaxation method, etc. In other words, setting the initial value for the inner unknown grid points to zero, the value of all inner grid points from grid point of the one inside of the inlet boundary to grid point of the one inside of the outlet boundary is obtained sequentially by Eq. (15.10). If this operation is repeated several times, the value of grid points approaches to a solution, and when it satisfies the specified convergence condition, the computation is finished. The streamlines (contours of stream function) of a sudden expansion of a channel obtained through this procedure are shown in Fig. 15.5.

Finite Volume Method The finite volume method uses the equation which is integrated in a small region (the control volume shown in Fig. 15.6) and is the basic differential equation unlike the finite difference method in which the difference form Control volume

ψ i, j+1

j+1

ψ i, j+1/2 j+1/2

ψ i–1, j

ψ i+1, j

ψ i, j

ψ i–1/2, j

∆y

j

ψ i+1/2, j ψ i, j–1/2 j–1/2

ψ i, j–1 i–1

i–1/2

∆x

FIGURE 15.6 Control volume.

j –1

i

i+1/2

∆x

i+1

∆y

15.1 Discretization Method

replaces the differential coefficient. The physical quantity in the control volume is defined at the centre point of the region in an incompressible fluid and is the average value of the region in a compressible fluid. Since the values of a physical quantity at the boundary face of the control volume are obtained using the neighbouring grid points, and the integral equation is discretized, this method has the advantage that the physical quantity is conserved at the boundary face and is a method that is used most in recent multipurpose fluid analysis software [2]. This method can be explained in terms of the two-dimensional Laplace’s Eq. (12.16) for the stream function j used in the finite difference method. Integrating Eq. (12.16) in the control volume of Dx  Dy of Fig. 15.6,  ZZ  2 v j v2 j þ dxdy ¼ 0 (15.11) vx2 vy2 Here, defining the value of the control volume boundary surface as i þ 1/2, j þ 1/2, etc.,     ZZ Z v vj v vj dxdy ¼ Dy dx vx vx vx vx zDy

(

ðvj=vxÞiþ1=2;j

ðvj=vxÞi

1=2;j

Dx 

jiþ1;j ji;j ¼ Dy Dx

ji;j

)

ji Dx

(15.12)

$Dx

1;j



Integrating the second term in the same way, Dy j Dx iþ1;j

2ji;j þ ji

1;j




þ

Dx j Dy i;jþ1

2ji;j þ ji;j

1




¼0

(15.13)

In the case of Dx ¼ Dy, Eq. (15.13) is consistent with Eq. (15.10) and was obtained by the finite difference method, and it is possible to obtain the value of the stream function in each control volume in the same manner as the method described in the finite difference method.

Finite Element Method The finite element method divides the analytical region into elements as shown in Fig. 15.7, and by using physical approximations to discretize the differential equations, simultaneous algebraic equations are developed for the whole elements. Thus, an approximate solution of the differential equations satisfying the boundary conditions is obtained. The corners of the elements are called nodal points, at which such variables as coordinate x, y, velocity u, v and pressure p are defined.

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FIGURE 15.7 Two-dimensional elements.

For discretization by the finite element method, the variational principle or the method of weighted residuals is used. The variational principle is also called the minimum energy principle, which uses the principle that the potential energy is a minimum in the state of equilibrium. As this method has limited application, the method of weighted residuals is widely used. Consider the potential flow around a cylinder placed between flat plates as shown in Fig. 15.8. in region S containing fluid

v2 j v 2 j þ ¼0 vx2 vy2 (15.14)

At inlet and on wall surface S1 j ¼ j At outlet S2 which is free boundary

vj vj ¼ vn vn

where the bars above the letters indicate that the applicable values are those on the boundary.

Normal line n Boundary S1 — ψ =ψ y Region S

O

FIGURE 15.8 Flow around cylinder.

x

Boundary S2

∂ψ ∂ψ = ∂n ∂n

15.1 Discretization Method

Next, in order to obtain the stream function j, multiply a given function which is j* ¼ 0 on boundary S1 (and can be any value in other region) by Eq. (15.14). Then integrate for the whole region. The following equation is obtained:   Z  2 Z  v j v2 j  vj vj  þ 2 j dA þ (15.15) j dS ¼ 0 2 vy vn S vx S2 vn Here, function j* is called the weighting function. In Eq. (15.15), assume function j* and its derivative vj/vn are approximate values. The first term on the left expresses the quantity obtained by multiplying the error of the differential equation in the region (here, called the residual) by a given function and integrating for the whole region. Likewise, the second term expresses the quantity obtained by applying a similar process to the residual on boundary S2. This is called a weighted residual expression. When the right solution is obtained, this equation applies strictly to the given function j*. The approximate solution which distributes the error to satisfy the function j* ¼ 0 is called the method of weighted residuals. In the finite element method, improvement is made by applying an algebraic equation derived using the values at nodal points to approximate the unknowns in each element. This equation is called an interpolating function. Where a weighting function of the same type is chosen, it is called the Galerkin method. In the two-dimensional case, as shown in Fig. 15.9, by using triangular elements their size can be determined to the extent that the functions are expressible by a y 3(x3, y3)

2(x2, y2) (x, y)

1 O

FIGURE 15.9 Triangular element.

(x1, y1)

x

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one-dimensional function of coordinates according to how abruptly or gently the functional change is expected. In other words, j ¼ a1 þ a2 x þ a3 y

(15.16)

Assume the function values at the corners of triangle 1, 2 and 3 to be j1, j2 and j3, respectively, then 8 9 2 38 9 a1 > > > > > > > 1 x1 y1 7> > > > > 6 > > > > j > > > 1> > > > > 6 7 > > > = < = 6 7< > 6 7 (15.17) j2 ¼ 6 1 x2 y2 7 a2 > 6 > > 7> > > > > 6 > > > 7> > > > > > 5> > > > 4 > > j > > : > ; ; : 3> 1 x3 y3 > a3 From the above, 8 9 2 1 x1 > > > > a1 > > 6 > > > > < = 6 6 a 2 ¼ 6 1 x2 > > 6 > > > > 4 > > > > : a3 ; 1 x3

y1

3

7 7 7 y2 7 7 5 y3

18

9 j1 > > > > > > > > > < > = j2 > > > > > > > > > : > ; j3

(15.18)

Substitute Eq. (15.18) into Eq. (15.16), j ¼ f1 j1 þ f2 j2 þ f3 j3 ¼

3 X

fi ji

(15.19)

i¼1

In other words, j is the interpolating function expressed as the linear combination of nodal point values ji. Hence, in the following form, fi ¼ ai þ bi x þ ci y

ði ¼ 1; 2; 3Þ

(15.20)

is called the shape function, and ai, bi and ci are determined by the coordinates of the nodal points. Approximate the unknown function j and weighting function j*, respectively, in Eq. (15.15) by interpolating the functional Eq. (15.19) using the nodal point values in the element and the same equation with j changed to j*. Substituting these functions into the weighted residual equation, which is the deformed Eq. (15.15), gives the quantitative relation for each element. By overlapping them, a simulated linear equation covering the whole analytical region is developed. By solving these equations, it is possible to obtain the values at each nodal point and thus to draw the streamline of j ¼ constant. To compute the flow shown in Fig. 15.8, as this is the symmetrical flow about the centre of cylinder, the upper half of the flow is divided into large and small

15.1 Discretization Method

FIGURE 15.10 Mesh diagram of flow around cylinder (180 elements and 115 nodes) [3].

FIGURE 15.11 Flow around cylinder: (A) streamline; (B) velocity vector [3].

triangular elements as shown in Fig. 15.10 [3]. For the finite element method, it is enough, unlike the finite difference method, just to divide the flow section finely around the cylinder where the velocity changes abruptly. The computed streamline and velocity vector are shown in Fig. 15.11 [3].

Boundary Element Method Instead of solving the difference equation which governs fluid movement under the given boundary conditions, the boundary element method uses an integral equation which must satisfy values on the boundary. To derive the integral equation, one can use the method using Green’s formula and also the method of weighted residuals. Green’s formula method has long been

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FIGURE 15.12 Mesh diagram by boundary element method of flow around cylinder [4].

used for analysing potential flow and has been systematized as the ‘panel method’, used for analysing external flows around aircraft, automobiles, etc. Brebbia derived an equation by the more general method of weighted residuals with wider applicability and named it the boundary element method [4]. It is often compared with the finite element method and has been used in many fields of application. In this method, the weighting function in the method of weighted residuals described in Finite Element Method section is selected so as to satisfy the Laplace Eq. (15.14) within region S and converted to an integral equation on boundary S2 surrounding the region as shown by the following equation: Z Z vj  vj dS dS ¼ 0 (15.21) j j vn S2 S2 vn Next, the boundary is divided into a number of line-segment elements. For example, in the case of the flow shown in Fig. 15.10, the mesh division is as shown in Fig. 15.12 [4]. Then, the value at a given point in the element is expressed in terms of the value of the nodal point by the interpolating Eq. (15.19) in the finite elements method. The simultaneous linear equation for the value at the nodal points can then be solved. The computational result for the case of Fig. 15.12 is shown in Fig. 15.13 [4]. Here, vj/vn expresses the flow velocity along the boundary.

15.1.2

Solution Based on the Lagrangian Method

This solution is a method to describe motion for fluid particles moving with time and attempts to approximate a continuous vorticity and velocity of the flow with the discrete elements, such as a finite number of very small vortices and fluid particles in place of space grids. As representative techniques of this method, the vortex method and the particle method have been used.

15.1 Discretization Method

∂ψ — ∂n

+

ψ

∂ψ —=0 ∂n

+

–

∂ψ — ∂n

FIGURE 15.13 Solution by boundary element method [4].

Vortex Method A continuous vorticity distribution of the flow field is represented discretely by a number of very small vortex elements, and the vorticity transport equation is solved numerically. Following the movement of vortex elements carried with the flow and monitoring the vorticity change of each vortex element at every moment provides a method to solve unsteady flow analytically.

Particle Method Analysis of the flow field is performed by tracking the movement of the virtual fluid mass in Lagrangian. In the particle method, the partial differential equation is discretized without using a grid. Particles in the particle method are not intended to be physical entities but are computation points that move with the stream. They are introduced in place of the grid points for the fluid computation. Therefore, a moving boundary, such as the free surface can be tracked directly by the particle movement.

15.1.3

Solution Based on the Cellular Automaton Method

Although both Eulerian and Lagrangian solutions are consistent with the basic equations of fluid mechanics, there are also attempts to express the movement of fluid by a completely different law. A typical example is the cellular automaton (CA: Cellular Automaton) method. The CA is a dynamical system that represents all of time, space and state variables at discrete values. The space is separated into cells, and the state of each cell develops with time under the influence of the state of adjacent cells. Representative techniques of this

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method are the lattice gas method and the lattice Boltzmann method. Because the latter is considered to be method by which the former was developed, only the lattice Boltzmann method is described here. The lattice Boltzmann method approximates the fluid flow by a number of virtual particles with a finite number of velocities creating a regular grid in the spatial region, and considering the velocity distribution function through a model in terms of the translation and collision of particles, it is a method of solving the evolution equations of the distribution function. It does not use the NaviereStokes equations, but theory supports the fact that the computational results are the solution of the NaviereStokes equations.

15.2 15.2.1

INCOMPRESSIBLE FLUID Finite Difference Method

In the case of an incompressible viscous fluid and a two-dimensional laminar flow, the equations governing the flow field are the continuity Eq. (6.2) and the NaviereStokes Eq. (6.13). In the solution of these equations, there is the method to solve the velocity u, v and pressure p in the two equations as unknown quantities directly and the method to solve the vorticity z and stream function j as unknown quantities using the vorticity transport Eq. (6.17) that eliminated the pressure term in order to reduce the number of unknown quantities. By using the stream function, this method satisfies the continuity Eq. (6.2) automatically.

Method Using Vorticity and Stream Function as Unknown Quantities Here, for simplicity considering a steady flow, the velocities u, v are expressed by the stream function j using a relational expression of Eq. (12.12), and non-dimensional vorticity transport Eq. (6.19) is expressed by the following equation:   vj vz vj vz 1 v 2 z v2 z ¼ þ (15.22) vy vx vx vy Re vx2 vy2 Also, the vorticity definition Eq. (4.7) is expressed in terms of the stream function j using the relationship in Eq. (12.12). v 2 j v2 j þ ¼ vx2 vy2

z

(15.23)

15.2 Incompressible Fluid

Discretize a partial differential coefficient in Eqs (15.22) and (15.23) using Eqs (15.7) and (15.8) as Dx ¼ Dy ¼ h, and rearrange for zij and jij, respectively, zi;j ¼

ji;j ¼


Re  1 ziþ1;j þ zi 1;j þ zi;jþ1 þ zi;j 1 þ jiþ1;j 4 16

ji;jþ1 ji;j 1 ziþ1;j zi 1;j 1 þ ji j 4 iþ1;j

1;j þ

ji;jþ1 þ ji;j

1þ

h2 zi;j

ji

1;j




zi;jþ1

zi;j

1




(15.24)




(15.25)

Using Eqs (15.24) and (15.25), the flow of a sudden expansion of a channel as described in finite difference method term of Section 15.1.1 is solved for twodimensional incompressible viscous laminar steady flow. In this case, similar to Fig. 15.4, the lower half of the channel is the computational region, and the flow is solved by a successive iterative method setting the boundary conditions for the stream function and vorticity. The boundary condition is similar to Fig. 15.4 for the stream function, is the specified value on the inlet and central axis for the vorticity and is determined from the interior value of the vorticity and stream function in the wall surface and outlet. Fig. 15.14 shows the streamlines and the equivorticity lines in the channel obtained through this procedure when Re ¼ 30. When the left-hand side of Eq. (15.22) is discretized using central differences, a stable convergent solution is hard to obtain for flow at high Reynolds number.

0.0 –1.0 –1.5 –5.0

–6.1 4.0

–2.0

–3.0

–1.0

ζ =0

l

ψ = 0.5

2l

0.4 0.3 0.2 0.1 –0.1

3 —l 4

–0.03

–0.06

0.0 –0.01

0.01 ψ = 0.0

3l

FIGURE 15.14 Equivorticity lines (upper half) and streamlines (lower half) of flow through sudden expansion (Re ¼ 30).

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In order to overcome this, the upwind difference method is mostly used for this finite difference method. This method is based upon the idea that most flow information comes from the upstream side. For example, if the upwind difference is applied to vz/vx, then the following equations are obtained.  vz zi;j zi 1;j  ¼ ji;jþ1  ji;j 1 ; then ui  0 vx h (15.26)  ziþ1;j zi;j  ¼ ji;jþ1 < ji;j 1 ; then ui < 0 h

However, Eq. (15.26) is first-order accuracy for the same formula as the forward-difference and backward-difference of Eqs (15.5) and (15.6). It should be noted that sometimes truncation errors can accumulate strongly in the numerical viscosity and may invalidate the solution. The upwind difference of high order accuracy is described in Section 15.4.2.

Method Using Velocity and Pressure as Unknown Quantities For the stream function used in the preceding section, it is difficult to establish a value on the boundary in the case of complex flow and cannot be defined in the case of three-dimensional flow. In such a case, computation must solve the flow velocity u, v and pressure p in the continuity Eq. (6.2) and the NaviereStokes Eq. (6.13) as unknown quantities. Typical of such methods is the MAC (marker and cell) method [5], which was developed as a numerical solution for a flow with a free surface following the movement of marker particles in the grid, but was later improved to be applicable to a variety of flows. At present, it is called the MAC method part of the algorithm of this numerical solution. Applications to flows with a free surface are described in Section 15.5.1. In this method, a staggered grid arranged the pressure at the cell centre surrounded by the grid and flow velocity at the cell boundary as shown in Fig. 15.15, is used. Therefore, the flow in the cell boundary can be computed, and it is possible to satisfy the continuity equation. Computation using the grid is performed as follows. Adding and rearranging by partial differentiation of the first equation of Eq. (6.13) by x and the second equation by y, the following equation is obtained. The body force term is omitted here.       2  v2 p v2 p vD v vu vu v vv vv v D v2 D þ þ v þ v þ ¼ r þ u þ u þ m vx2 vy2 vt vx vx vy vy vx vy vx2 vy2 (15.27) Here, since D ¼ vu/vx þ vv/vy is the same form as the continuity Eq. (6.2), it should be zero, but it allows for error in the numerical analysis. Eq. (15.27)

15.2 Incompressible Fluid

1 υ i, j + — 2

p i, j

1 ui – — ,j 2

1 ui + — ,j 2

y 1 υ i, j – — 2

O

x

FIGURE 15.15 Layout of variables in the marker and cell method.

is called the Poisson equation of pressure. Approximating the first term on the right side of this equation by the forward difference, Dnþ1 Dn Dt

(15.28)

Here, n denotes the time. Setting to Dnþ1 ¼ 0 in order to obtain the pressure field, so as to satisfy the continuity equation at time n þ 1, it is possible to obtain the pressure p at time n from Eq. (15.27) using the flow velocity u, v at time n. Using the obtained flow velocity u, v and pressure p at time n, it is possible to obtain the flow velocity at time n þ 1 from the equation approximating the time derivative term of the flow velocity of the first term on the left side of the NaviereStokes Eq. (6.13) similar to Eq. (15.28). Instead of solving the Poisson Eq. (15.27), the Highly Simplified MAC method [6] time-marching by modifying the velocity and pressure iteratively while satisfying the continuity equation has been developed. Fig. 15.16 shows comparison between the kaleidoscopic change of Kármán vortices in the flow behind a rectangular column computed by that method and the experiment [7].

15.2.2

Finite Volume Method

The finite volume method uses integration of the continuity equation and the NaviereStokes equation written in conservative form1 in the control volume similar to Fig. 15.6. The boundary values are obtained using the neighbouring

1

vu The method expressing the inertia term etc. of the NaviereStokes Eq. (6.13) by u vu vx ; v vy is called vðu2 Þ vðuvÞ nonconservative form, and the method expressed by vx ; vy is called conservative form.

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(A)

(B)

(C)

FIGURE 15.16 Time-sequenced change of Kármán vortex street: ( A ) start; ( B ) at 0.1 s; ( C ) at 0.2 s [7].

grid points. Then, the defining point of the flow velocity u, v and pressure p are used as a staggered grid as shown in Fig. 15.15. Discretizing and rearranging the integral equation similarly [Eq. (15.12)], it is possible to obtain the following algebraic equations. nþ1 nþ1 ai;j unþ1 ¼ aiþ1;j uiþ1;j þ ai;jþ1 ui;jþ1 þ ai 1;j uinþ1 i;j 1;j þ ai;j   DxDy nþ1 þ Dy piþ1;j pinþ1 un 1;j þ Dt i;j

nþ1 nþ1 nþ1 bi;j vi;j ¼ biþ1;j viþ1;j þ bi;jþ1 vi;jþ1 þ bi 1;j vinþ1 1;j þ bi;j   DxDy nþ1 nþ1 vn þ Dx pi;jþ1 pi;j 1 þ Dt i;j

nþ1 1 ui;j 1

(15.29) nþ1 1 vi;j 1

(15.30)

Here, a, b are coefficients that include u, v.

Since Eqs (15.29) and (15.30) hold for all grid points in the interior region, the simultaneous equations solving the flow velocity u, v on each grid point as

15.2 Incompressible Fluid

unknown quantities are constituted, it is possible to solve using an iterative method, if the pressure p is given somehow. Semi-implicit method for pressure-linked equation method [2] is a well-known method to solve Eqs (15.29) and (15.30). In this method, the pressure p is determined in the following manner. Defining the correct pressure and flow velocity to satisfy the continuity equation as e p; e u; ev and the provisional pressure as p and using p, the flow velocity computed from Eqs (15.29) and (15.30) as u, v, it is possible to express the following equations. e p ¼ p þ p0

e u ¼ u þ u0

ev ¼ v þ v 0

(15.31)

0

0

0

Here, p , u , v is a correction term of pressure and flow velocity. Substituting this into Eqs (15.29) and (15.30), e u; ev is obtained, and by substituting it into the 0 continuity equation, p is obtained, and it is possible to obtain e p. By repeating until convergence of this process, one time worth of computation is finished, and this is repeated until the necessary time steps. In Fig. 15.6, the grid showed a regular structured grid in a line. However, the finite volume method can also apply in the boundary-fitted grid following an irregular boundary or an unstructured grid. As an example, applying an unstructured grid of triangles, the mesh diagram that was used to compute the flow around a circular cylinder with grooves and the computed streamlines are shown in Fig. 15.17 [8].

15.2.3

Finite Element Method

In the solution by the finite element method, there is the method to solve the vorticity and stream function as unknown quantities and the method to solve the velocity and pressure as unknown quantities as described in the section of the finite difference method. When the continuity equation and the Naviere Stokes equation are discretized, the finite element method is used only for discretization of the spatial direction, and the finite difference method is often used for discretization of the time direction. The analysis method can be broadly classified into the direct method and the fractional step method. The direct method is a method to apply directly the finite element method for the continuity equation and the NaviereStokes equation. The fractional step method is a method that conforms to algorithms of the MAC method in the finite difference method, and we apply the finite element method for the Poisson Eq. (15.27).

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FIGURE 15.17 Flow around a circular cylinder with grooves: (A) whole mesh diagram; (B) mesh diagram near the surface of the circular cylinder; (C) streamlines [8] (see Videos 2.1e2.4 and 2.7).

As an application example of the direct method, the mesh diagram that was used to compute the flow around the stepping stones of the garden using the quadrangular elements and the computed streamlines are shown in Fig. 15.18 [9].

FIGURE 15.18 Flow around stones of garden: (A) mesh diagram; (B) streamlines [9].

15.2 Incompressible Fluid

15.2.4

Vortex Method

This technique replaces the successive distribution of vorticity produced in a flow field containing varied viscosity and density with discrete vortex elements. Each vortex motion is followed by the Lagrange method and thus analyses the unsteady flow. This technique is called the discrete vortex method. As an example, the modelling by vortex points for an unsteady flow around a square column in a uniform flow are shown in Fig. 15.19 [10]. In Fig. 15.20A and B, the left and right sides show, respectively, the distribution of developed vortex points and instantaneous streamlines [10]. In any of these cases, the positive vortex (clockwise rotation) develops from point A and the negative vortex (counterclockwise rotation) from point B or C. These vortices develop behind the rectangular column and the Kármán vortex street is formulated in the wake.

15.2.5

Lattice Boltzmann Method

Since the lattice Boltzmann method uses basically the grid of regular intervals, the particle velocity is constrained. Therefore, since an application is difficult for the boundary of complicated shape, the difference lattice Boltzmann method extended to the curvilinear grid is proposed [11]. As an example computed by this method, the mesh diagram that was used to flow around a circular cylinder and the computed instantaneous streamlines are shown in Fig. 15.21 [12].

Vortex point

U∞

A

D

B

C

α

y x

FIGURE 15.19 Modelling by vortex point for flow around a square column [10].

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(A) Ut/ b=28

Ut/ b=32

(B) Ut/ b=28

Ut/ b=32

FIGURE 15.20 Flow pattern around the rectangular column [10] (A) a ¼ 0 degree (B) a ¼ 30 degrees.

FIGURE 15.21 Flow around a circular cylinder (Re ¼ 200): (A) Mesh diagram; (B) instantaneous streamlines [12].

15.3 Compressible Fluid

15.3

COMPRESSIBLE FLUID

The governing equations for compressible flows with dissipation such as viscosity are simultaneous partial differential equations called the NaviereStokes equation system, consisting of the conservation laws for mass, momentum and also energy with thermo-dynamic variables. The equations without dissipation are called the Euler equation system, and further the assumption of steady and non-vortical (no vortices) flow leads to the potential equation. The equations with fewer assumptions are stricter but these simpler ones each have advantages. Therefore, after taking into consideration the physics of the given problem and the restrictions of the computer used, it should be decided what kind of equations to solve. The difficulty for numerical methods applied to compressible flows is in suppressing the non-physical numerical oscillations in discontinuity waves such as shock waves and the divergent tendency of strong expansion waves.

15.3.1

Time-Marching Technique

Time-marching methods were proposed historically from numerical methods for the Euler equation system classified as hyperbolic partial differential equations. At the end of 1950s, the first-ordereaccurate Godunov method [13] was proposed, based on the Riemann problem, an example being the shock-tube problem. Later proposals from 1960 to 1970s were the second-order methods such as that by Lax-Wendroff [14] and that by MacCormack [15]. From the late 1970s as computers progressively developed, the BeameWarming method [16] was maturing, where the spatial differential of fluxes is approximated by the central difference and the fourth- and second-order numerical dissipations to suppress numerical oscillation, and the time integration is carried out by the Implicit Approximate Factorization method to reach a steady state faster with less memory capacity. The BeW scheme was applied to many practical computations such as viscous flows relating to airplanes. In the early 1980s, it was mathematically proved that for the nonlinear conservation law the numerical solution converges to the exact solution if TV stability (the total variation, TV, does not increase with time) and compatibility (when the flow becomes uniform, the numerical flux continuously reaches the exact flux) are satisfied. The so-called TVD (total variation diminishing) schemes [17] satisfying these conditions have become the mainstream in the numerical methods for compressible flows in 1990s, because they capture shock waves clearly without adjusting numerical dissipation. Usually the finite-volume methods are adopted for discretization. Plates 1 and 2 (see colour plate section) show their applications, respectively, to viscous flows about a fully configured aircraft model equipped in a transonic wind tunnel [18] and to unsteady viscous flows within a jet engine [19].

315

Computational Fluid Dynamics

ρ

(B)

(A) 5.0 4.0 3.0 2.0 1.0 0.0 0.0

ρ

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ρ

316

0.5

1.0

x

1.5

2.0

5.0 4.0 3.0 2.0 1.0 0.0 0.0 5.0 4.0 3.0 2.0 1.0 0.0 0.0

TVD scheme 0.5

1.0

x

1.5

2.0

WENO scheme

0.5

1.0

x

1.5

2.0

FIGURE 15.22 Solutions of Euler equations [total variation diminishing scheme and WENO scheme (non-dimensional density)] [23]. (A) initial condition (t ¼ 0) (B) numerical solution at t ¼ 0.47.

A weak point of the existing TVD schemes is that applying nonlinear conservation laws results in accuracy up to the second order, and therefore it is difficult to capture simultaneously both the weak pressure variation in sound waves and turbulence, etc., and the strong pressure change in shock waves. To improve the situation, higher resolution methods such as the ENO/WENO schemes [20], the ADER approach [21], and the discontinuous Galerkin methods [22] have been proposed. The ENO/WENO schemes principally make the spatial accuracy higher, and the ADER approach gives one-step schemes with high accuracy both in space and time. Developments and verifications of numerical methods for compressible flows are usually carried out for the Euler equation system. Fig. 15.22 shows distributions of the non-dimensional density for the test problem that a shock wave propagates to the flow field with small density perturbations (1) is the initial condition, and (2) shows comparison of numerical solutions, where the TVD scheme loses density oscillations that should exist physically, while the WENO scheme captures them. This reveals superiority of the high-accuracy WENO scheme to the TVD scheme. Further, it is reported that the ADER approach is superior to the WENO schemes in capturability of long-time propagating waves for a linear advection equation [23].

15.3.2

Boundary Element Method

As stated in Section 15.1.1 the boundary element method is popular for finding the velocity or pressure distribution on a body surface since the method only requires division of the boundary of the region into the elements. Fig. 15.23 is the mesh diagram for flow around a complete model of a transonic plane using the panel method. The computational result of the pressure distribution

15.3 Compressible Fluid

FIGURE 15.23 Mesh diagram for computing flow around full model of transonic plane [24].

obtained is shown in Plate 3A (see colour plate section), which coincides very well with the result of the wind tunnel experiment as shown in Plate 3B (see colour plate section) [24].

15.3.3

Method of Characteristics

Fig. 15.24 is a test rig for water hammer, which is capable of measuring the pressure response waveform by the pressure transducer set just upstream of the switching valve. When the switching valve is suddenly closed, pressure p increases and propagates along the pipe as a pressure wave. The wave phenomenon is expressed by a hyperbolic partial differential equation, and to obtain its numerical solution, the so-called method of characteristics [26] is used.

Camera

Oscilloscope

Amp.

Power source

Hydropower unit Electromagnetic switching valve

M D

b

Test pipe (20 m) l Semiconductor pressure transducer

FIGURE 15.24 Water hammer testing device [25].

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Computational Fluid Dynamics

Now, putting f as the friction coefficient of pipe and a as the propagation velocity of the pressure wave, linearly combine with l times the continuity equation and the momentum equation, which is the one-dimensionalized Eqs (6.1) and (6.13), to get       l vp a2 vp vv vv f þ vþ þ ðvþ lÞ vjvj ¼ 0 (15.32) þ þ ra2 vt vt vx 2D l vx Here, assume that vþ

a2 dx ¼ ; dt l

vþl¼

dx dt

ðl ¼ aÞ

(15.33)

and partial differential Eq. (15.32) is converted to an ordinary differential equation. Furthermore, discretize it, and, as shown in Fig. 15.25, v and p of point P after time interval Dt are obtained as the intersection of the curves Cþ(l ¼ a) and C (l ¼ a) which are expressed by Eq. (15.33) from the initial values of velocity v and pressure p at A and C. Fig. 15.26 shows the comparison between the pressure waves thus calculated and the experimental values [25]. The difference between them arises from the fact that the frequency-dependent pipe friction is not taken into account in Eq. (15.32).

15.4

TURBULENCE

Turbulence is a phenomenon that occurs at high Reynolds number, and the characteristics are believed to be expressed by the NaviereStokes Eq. (6.13). Therefore, if the afore-described analytical methods are applied, the turbulence phenomenon is theoretically computable. However, turbulence consists of

t0 + 4∆t t0 + 3∆t t0 + 2∆t

∆x

∆t P

t0 + ∆t t0

∆x

∆t

Boundary condition

t t0 + 4∆t

Boundary condition

CHAPTER 15:

Time

318

C–

C+

C–

C+ x

1

2

FIGURE 15.25 x-t grid for solution of single pipe line.

3

A

B C Position

N–1

N

N+1

Pressure p/p0

15.4 Turbulence

1

0

1

2

3

4

5

Time t/(2l/a) Computed value Measured value

FIGURE 15.26 Pressure response wave in water hammer action [25].

vortices of large and small scales, and the scale-width expands with the Reynolds number. To computationally deal with vortices over all scales, the computational grid is said to need a grid dependent on Reynolds number of degree Re9/4 . Since a grid number of 109 is required for turbulence of a relatively low Reynolds number Re ¼ 104, it becomes a great computational demand even in a current supercomputer. Therefore, in many cases, it is necessary to compute using some assumptions and models.

15.4.1

Direct Numerical Simulation

For turbulence of low Reynolds number of simple shapes, direct numerical simulation, solving directly the continuity Eq. (6.2) and the NaviereStokes Eq. (6.13) without using any model can be a feasible method. In this case, since it is necessary to perform high-precision computation, the spectral method and the higher-order precision central difference method are used. The spectral method [27] expands the solution of the physical space normally being carried out in a series by trigonometrical functions and polynomial expressions and is a method that obtains the solution by converting it into computation space (wave-number space) and substituting in the original differential equation. By this method, the computation of uniform isotropic turbulence (spatially uniform and virtual turbulence without solid walls) of intermediate Reynolds number was performed by Orszag and Patterson [28] for the first time in 1972, and they showed that a rational prediction is possible using the number of grid points 643. After that, Kim et al. [29] applied the method to turbulence between parallel plates with solid walls for the first time. In this computation, Reynolds number Rem non-dimensionalized by the average flow velocity in a crosssection of parallel plates Um, the half-width of distance between flat plate d and kinematic viscosity n was Rem ¼ 3300, and the number of grid points was 192  129  160.

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FIGURE 15.27 Vortex structure of turbulence between the parallel plates by direct numerical simulation (Res ¼ 1020. Flow is from the upper left to the lower right. Black: low velocity region, Grey: high velocity region) [30].

The spectral method is excellent in the computational accuracy. But, since it is not only applicable to simple shape from limitations of the method to give boundary conditions, higher-order accuracy finite difference method is used in recent years. As an example, the results of the computation of turbulence between parallel plates using a fourth-order accuracy central difference, the turbulent vortex structure and the large-scale structure that rises towards near the centre of the channel from the wall surface are shown in Fig. 15.27 [30]. Reynolds number non-dimensionalized by the wall friction velocity us and the half-width of distance between flat plates is Res ¼ 1020.

15.4.2

Method Using Numerical Viscosity

As a turbulence computation without using a physical model, this method simulates the movement of a large vortex by making the accuracy of the upwind difference scheme, shown in Eq. (15.26), of higher order to third-order accuracy [31] and fifth-order accuracy and also by making the numerical viscosity smaller. One such example is shown in Fig. 15.28 [32], the computed and visualized flows behind a step. It can be seen that the movement of the vortex behind the step with the passage of time is well simulated.

15.4.3

Large Eddy Simulation

Large eddy simulation (LES) computes large vortices as distinct flow fields. Small vortices within the size of the mesh are modelled in terms of the local mean mesh model. In this method, computation can follow the change in irregularly changing turbulence. Fig. 15.29A shows a solution for the flow between parallel walls [33]. Comparing this with Fig. 15.29B, a visualized photograph of bursting by the hydrogen bubble method [34], it is clear that they coincide well with each other. In Fig. 15.30, the turbulent flow over a step is computed and its time lines are

15.4 Turbulence

FIGURE 15.28 Translation of vortex in a circulation region (left: computation, right: visualisation by smoke wire method) [32] (A) t ¼ 5.0; (B) t ¼ 10.0; (C) t ¼ 15.0; (D) t ¼ 20.0.

FIGURE 15.29 Time line near the wall of a flow between parallel walls: (A) computation [33]; (B) experiment [34].

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FIGURE 15.30 Turbulent flow over step [35] (Reynolds number based on a channel width, Re ¼ 1.1  104).

shown graphically [35]. Plate 4 (see colour plate section) [36] for Kármán vortices around a rectangular column and Plate 5 (see colour plate section) [37] for the instantaneous value of the pressure of the body surface and the floor of the formula racing car were computed by LES.

15.4.4

Reynolds Averaged Model

As already stated in Section 6.4, making some assumption or simplification for computing the Reynolds stress st, expressed by Eq. (6.40), is called a Reynolds averaged model. It is mainly classified by the number of transport equations for the turbulence quantity used for computation. The method for which st is given by Eqs (6.41) or (6.44) is called a zero-equation model. The method for which the turbulent energy (the kinetic energy of turbulence) k is determined from the transport equation, while the length scale of turbulence l is given by an algebraic expression, is called a one-equation model. And the method by which both k and l are determined from the transport equation is called a two-equation model. The k-ε model, using the turbulence energy dispersion ε instead of l, is typical of the two-equation model. As an example, the turbulent energy of flow around a rectangular column and a rectangular column with corner cut-offs were calculated using the finite volume method, and the results of the comparison are shown in Fig. 15.31 [38]. Compared to the case without corner cutoffs, the rectangular column with corner cutoffs has smaller turbulent energy and less turbulence. Also, as an example calculated using the k-ε model by the finite element method, the turbulence velocity distribution of the flow in a clean room is shown in Plate 6 (see colour plate section) [39].

15.5 Free-Interface Flows

FIGURE 15.31 Comparison of turbulent energy of flow around rectangular column and rectangular column with corner cutoffs (Re ¼ 6  104) [38] (A) rectangular column (see Video 2.5) (B) rectangular column with corner cutoffs. (see Video 2.6)

15.5

FREE-INTERFACE FLOWS

The first method developed to solve the free interface flow accompanying the deformation of the gaseliquid interface is the MAC method described in Method Using Velocity and Pressure as Unknown Quantities section. Thereafter, the VOF (volume of fluid) method [40] was developed. These methods are Eulerian solutions, but there are particle methods with Lagrangian solutions.

15.5.1

Marker and Cell Method

In the early development of the MAC method, markers (which are weightless particles indicating the existence of fluid) were placed in the mesh unit called a cell, as shown in Fig. 15.32, and such particles were followed. One of the examples is shown in Fig. 15.33, where a comparison was made between the

FIGURE 15.32 Layout of cell and marker particles used for computing flow on inclined free surface.

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(A)

(B)

(C) (D)

Crown 42 FIGURE 15.33 Liquid drop falling onto thin liquid layer [41] (A) start; (B) at 0.0002 s; (C) at 0.0005 s; (D) at 0.0025 s.

photograph when a liquid drop fell onto a thin liquid layer and the computational result by the MAC method.

15.5.2

Volume of Fluid Method

The VOF method is a method for defining the free interface shape by defining the VOF function which is the volume occupancy of liquid for each cell in Fig. 15.32 and solving its transport equations. As an example, the change in the surface of the water near the breakwater opening due to tsunami is shown in Fig. 15.34 [43].

15.5.3

Particle Method

As a kind of particle method, there is the MPS (moving particle semi-implicit method) method proposed by Koshizuka [44]. In this method, the discrete computation of the continuum is performed through an inter-particle interaction model. Fig. 15.35 shows the collapse experiment of the water column and computation by the MPS method [44]. The water column starts to collapse after pulling

15.5 Free-Interface Flows

FIGURE 15.34 Change in the surface of the water near the breakwater opening due to the tsunami [43].

FIGURE 15.35 Collapse of the water column (time interval 0.2 s) [44] experiment (upper part), computation (lower part).

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up the supporting plate. Computations by the particle method can be computed not only by the large deformation of the free liquid surface but also by the occurrence of splitting or coalescence of the fluid, and it clearly reproduces the experiment qualitatively well.

SUPPLEMENTARY DATA Supplementary data related to this article can be found online at doi:10.1016/ B978-0-08-102437-9.00015-2.

References [1]

G.D. Smith, Numerical Solution of Partial Differential Equations, Oxford University Press, 1965.

[2]

S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere, New York, 1980.

[3]

K. Hayashi, et al., Flow Analysis by Personal Computer, Asakura-Shoten, Tokyo, 73, 1986.

[4]

C.A. Brebbia, The Boundary Element Method for Engineers, Pentech Press, London, 1978.

[5]

F.H. Harlow, J.E. Welch, Physics of Fluids 8 (1965) 2182.

[6]

C.W. Hirt, J.L. Cook, Journal of Computational Physics 10 (2) (1972) 324.

[7]

Y. Nakayama, K. Aoki, M. Oki, in: Proc. 3rd Asian Symp. on Visualization, 1994, p. 453.

[8]

Y. Yamagishi, M. Oki, Journal of Visualization 10 (2) (2007) 179.

[9]

Y. Nakayama, et al., in: Fifth Triennial International Symposium on Fluid Control, Measurement and Visualization, vol. 2, 1997, p. 785.

[10] T. Inamoto, et al., Finite Element Flow Analysis, University of Tokyo Press, 1982, p. 931. [11] N. Cao, et al., Physical Review E 55 (1997) 21. [12] M. Tsukahara, et al., Transaction of JSME B 69 (680) (2003) 841. [13] S.K. Gounov, Matematicheskii Sbornik 47 (1959) 357. [14] P. Lax, B. Wendroff, Communications on Pure and Applied Mathematics 13 (1960) 217. [15] R.W. MacCormack, in: AIAA Paper, 69-354, 1969. [16] R.M. Beam, R.F. Warming, AIAA Journal 16 (4) (1978) 393. [17] A. Harten, Journal of Computational Physics 49 (3) (1983) 357. [18] Y. Takakura, et al., AIAA Journal 33 (3) (1995) 557. [19] Nozaki, et al., Nikkei Science (2000e10) A18. [20] C.W. Shu, NASA/CR-97-206253, ICASE Report, 97-65, 1997. [21] E.F. Toro, V.A. Titarev, in: ECCOMAS CFD Conference, 2001. [22] H. Atkins, C.W. Shu, AIAA Journal 36 (1998) 775. [23] Y. Takakura, Journal of Computational Physics 219 (2) (2006) 855. [24] T. Kaiden, et al., in: Proc. 6th NAL Symp. on Aircraft Computational Aerodynamics, 1998, p. 141. [25] K. Izawa, (MS thesis), Faculty of Engineering, Tokai University, 1976. [26] V.L. Streeter, Fluid Mechanics, sixth ed., McGraw Hill, New York, 1975, p. 647. [27] C. Canuto, et al., Spectral Method in Fluid Dynamics, Springer-Verlag, 1987.

References

[28] S.A. Orszag, G.S. Patterson Jr., Physical Review Letters 28 (2) (1972) 76. [29] J. Kim, et al., Journal of Fluid Mechanics 177 (1987) 133. [30] H. Kawamura, Nagare 22 (6) (2003) 467. [31] T. Kawamura, K. Kuwahara, in: AIAA Paper, No. 84e0340, 1984. [32] M. Oki, et al., JSME International Journal B 36 (4) (1993) 577. [33] P. Moin, J. Kim, Journal of Fluid Mechanics 118 (1982) 341. [34] H.T. Kim, et al., Journal of Fluid Mechanics 50 (1971) 113. [35] T. Kobayashi, et al., Report IIS, University of Tokyo, 33-3, 1987, p. 25. [36] T. Kobayashi, Atlas of Visualization III, Plate 10, CRC Press, Boca Raton, FL, 1997. [37] M. Tsubokura, et al., in: SAE International Paper, No. 2007-01-0106, 2007. [38] Y. Yamagishi, et al., Journal of Visualization 13 (1) (2010) 61. [39] M. Ikegawa, et al., in: Proc. Int. Symp. on Supercomputers for Mechanical Engineering, JSME, 1988, p. 57. [40] C.W. Hirt, B.D. Nichols, Journal of Computational Physics 39 (1) (1981) 201. [41] B.D. Nichols, in: Proc. 2nd Int. Conf. on Numerical Methods in Fluid Dynamics, 1971, p. 371. [42] K. Fujii, H. Nakagome, Reading Physical Phenomena, Kodansha, Tokyo, 1978, p. 102. [43] H. Miyamoto, in: Abstract of the 50th Annual Scientific Lecture Meeting of the Japan Society of Civil Engineers, cs-64, 1995, p. 128. [44] S. Koshizuka, Y. Oka, Nuclear Science and Engineering 123 (1996) 421.

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Flow Visualization The flow of air cannot be seen by the naked eye. The flow of water can be seen but not its streamlines or velocity distribution. The consolidated science which analyses the behaviour of fluid invisible to the eye like this as image information is called ‘flow visualization’, and it is extremely useful for clarifying fluid phenomena. The saying ‘Seeing is believing’ most aptly expresses the importance of flow visualization. Analytical studies clarifying hitherto unclear flows and the developmental studies of flows in and around machinery have been much assisted by this science. In 1883, Reynolds made the great discovery of the law of similarity by visualization. Thereafter, Prandtl’s concept of the boundary layer and his ideas for its control, Kármán’s clarification of his vortex street, Kline’s discovery of the bursting phenomenon allied to developing the mechanism of turbulence and other major discoveries concerned with fluid phenomena were mostly achieved by flow visualization. Furthermore, in the clarification of turbulent structure, the establishment of mathematical models of turbulence, etc., which currently still pose big problems, flow visualization is furnishing extremely important information. With the progress of computers, its use has been enhanced by image processing. Also, computer-aided flow visualization, the visual presentation of numerical computation and measured results, is making great advances.

16.1

CLASSIFICATION OF TECHNIQUES

The visualization techniques are classified as shown on Table 16.1 and divided roughly into experimental methods and computer-aided visualization methods.

16.2 16.2.1

EXPERIMENTAL VISUALIZATION METHODS Wall Tracing Method

The oil-film method, typical of this technique, has long been used, so the technique is well established. There are many applications, and it is used for both Introduction to Fluid Mechanics. https://doi.org/10.1016/B978-0-08-102437-9.00016-4 Copyright © 2018 Elsevier Ltd. All rights reserved.

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Table 16.1 Classification of Visualization Techniques Visualization Technique

Air Flow

Water Flow

Explanation

Experimental Visualization Method 1. Wall surface tracing method Oil-film method, oil-dots method

Mass transfer method

￮

C

￮

C

Electrolytic corrosion method

C

Temperature-sensitive film method

￮

Pressure-sensitive paint method

￮

Pressure-sensitive paper method

￮

C

￮

C

Luminescent mini-tuft method

￮

C

3. Injected tracer method Injection streak line methoda

￮

C

Injection path line methodb

￮

C

Suspension methodc

￮

C

2. Tuft methods Various tuft methods

C

By attaching an oil film or oil dots to the body surface, from the stream pattern generated, the state including the direction of flow can be visualized. By utilizing the dissolution, evaporation or sublimation into the fluid of a film of a substance attached to the body, the flow state on the body surface can be visualized. By utilizing the corrosion due to electrolysis, the flow pattern on the body surface can be visualized. The surface temperature is visualized according to the colour distribution of a liquid crystal or such attached to the body. By utilizing the luminescence of a substance applied to the body surface, the pressure distribution on the surface can be visualized. By utilizing the colour density of the pressuresensitive paper, the pressure distribution on the body surface can be visualized. Method by which the flow direction is visualized from the flight behaviour of numerous short pieces of thread (tufts). By the surface tuft method, the flow near the surface is visualized; by the depth tuft method, the flow at a given point just off the surface is visualized; by the tuft grid method, the flow on a given section is visualized; and by the tuft stick method, the flow at a given point is visualized. Method by which, hardly having any effect on the flow, a single filament of nylon soaked in luminescent dye beforehand is photographed under highly luminous ultraviolet rays. Continuously inject tracers, capture the picture at a certain instant, and thus visualize the stream and streak lines. Intermittently inject tracers for some duration. Visualize the stream and path lines. Evenly suspended liquid or solid particle in the fluid in advance. Thus, visualize the stream and streak lines.

16.2 Experimental Visualization Methods

Table 16.1 Classification of Visualization Techniques Continued Visualization Technique

Air Flow

Surface floating tracer methodd Time line methode 4. Chemical reaction tracer method Non-electrolytic reaction method

Water Flow C

￮

C

￮

C

Electrolytic colouring method

C

5. Electric controlled tracer method Hydrogen bubble method

Explanation Let the tracer float on the liquid surface and thus visualize the stream and streak lines on the liquid surface. Inject the tracer vertically into the flow and thus visualize the time lines.

By utilizing the chemical reaction of a fluid with another specified substance, the flow behaviour on the solid surface or the boundary between two fluids can be visualized. By utilizing an electrolytically coloured substance as the tracer, the stream and streak lines can be visualized.

C

Utilizes as the tracer hydrogen bubbles developed through electrolysing with a fine metal wire as the negative pole. Visualize the stream, streak, path and time lines. Visualize the time line by means of groups of discharge sparks obtained one after another by a high-voltage pulse. Instantaneously heat an oiled fine metal wire to produce white smoke. Visualize the stream, streak, path and time lines using the white smoke as the tracer. Let a light emitted from a single point or parallel rays go through a flow region; the flow is visualized by means of the dark and grey shadow thus developed according to the changes in density. Parallel rays are made to deflect through a flow field with a difference in density. The deflected rays are cut with a knife edge, and the density gradient is visualized according to the difference in brightness thus developed. Parallel rays are divided into two, one of which is made to go through a flow with a difference in density. Quantitative judgement of density and pressure is made from the interference fringe developed by combining the two.

Spark tracing method

￮

Smoke wire method

￮

6. Optical method Shadowgraph method

￮

C

Schlieren photograph method

￮

C

MacheZehnder interferometer method

￮

C

Continued

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Table 16.1 Classification of Visualization Techniques Continued Air Flow

Water Flow

Laser holographic interferometer method

￮

C

Laser light sheet method

￮

C

Speckle method

￮

C

Infrared thermography

￮

C

Visualization Technique

Explanation A laser light is separated into two beams. Interference, the fringe pattern, from an object and another beam (reference beam) between the scattered beam is recorded on the hologram film. Illuminating this film by the reference beam, the object can be reconstructed. Laser rays are made to strike a cylindrical lens or a revolving or vibrating mirror to make a sheet-like ray, and the three-dimensional flow is visualized as a two-dimensional flow by light scattered from tracer particles. The flow velocity distribution is obtained by optically processing the speckle pattern obtained by instantaneously photo shooting at short intervals a fluid with suspended tracer particles. The temperature distribution is manifested in an image by detecting infrared radiant energy emitted from flow surface and transforming it to temperature.

Computer-Aided Visualization Method 7. Visualized image analysing method Hydrogen bubble method with image processing Particle Tracking Velocimetry (PTV) Particle Image Velocimetry (PIV) Stereo PIV

Micro PIV

Holographic PIV (HPIV) Spin-Tagging Magnetic Resonance Imaging

The velocity vector is obtained by visualizing streak lines and time lines at the same time, and then by digitizing and processing. The flow velocity distribution is obtained by taking photos of a tracer particle moving along with fluid flow at short intervals after suspending relatively low concentration of tracer particles in fluid. The flow velocity distribution is obtained by similarity of distribution patterns of tracer particles with high concentration at a short interval. Three velocity components in the measurement cross-section are obtained due to parallax by placing several cameras facing towards the measurement cross-section with change in view point. The flow velocity distribution is obtained by applying the PIV method in a microscopic fluid field by magnifying and photographing tracer particles to be measured by using a microscope. Three-dimensional velocity information is obtained by recording locational information on holograms and reconstructing it. The velocity field is visualized by deformation of a magnetic field formed in a grid shape.

16.2 Experimental Visualization Methods

Table 16.1 Classification of Visualization Techniques Continued Visualization Technique Molecular tagging velocimetry and thermometry (MTV&T) Computer tomography (CT)

8. Measured data visualization method Experimental data visualization method

Acoustic intensity method

9. Numerical analysis data visualization method 10. Numerical data manifestation method Contour manifestation method Area colouring manifestation method Isosurface manifestation method Volume-rendering method Vector manifestation method Animation method

Air Flow

Water Flow

Explanation

Molecular phosphorescence is made to follow fluid flow and laser light forming in grid shape is emitted into the fluid flow to excite the molecules. Both velocity and temperature distributions are obtained by visualizing deformation of the grid shape. Based on measurement of quantity of radiation, ultrasonic wave, electric field or laser light passing through a cross-section of an object, the cross-section image or the density and temperature distribution on the cross section in case of fluid is obtained through computation.

The method is practised such as those which utilize flow velocimeters, pressure gauges, temperature gauges, etc., where the velocity distribution, pressure distribution, temperature distribution, etc., obtained are to be manifested in an image by processing the data simultaneously. Size and direction of the sound energy at each sound field are manifested in an image by processing the cross spectrum of sound pressure signal from two microphones. The flow field is to be easily manifest in an image by numerous results of calculation from numerical analysis by a computer.

Physically equal values are connected by a contour. Manifestation is made by painting the areas in colours, respectively, corresponding to their levels of physical quantity. The values of physically equal value are three-dimensionally manifested in a surface. The levels of equi-value area manifestation are manifested by changing their degree of transparency. The size and direction of the flow velocity vectors, etc., are manifested by arrow marks. Still images on a display are developed into as-if-moving images by continuous shooting.

For injection tracer methods, the names of typical tracers are as follows: smoke (￮), colouring matter (C); b soap bubble (￮), air bubble (C), oil drop (C), luminescent particle (C); c metaldehyde (￮), air bubble (￮), cavitation (C), liquid tracer (C), aluminium powder (C), polystyrene particle (C); d aluminium powder (C), sawdust (C), foaming polystyrene (C); e smoke (￮), colouring matter (C). a

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FIGURE 16.1 Limiting streamlines of Wells turbine for wave power generator (revolving direction is counterclockwise) in water, flow velocity 3.2 m/s, angle of attack 11 degrees, chord length 73 mm.

water and air flow. The flows in the neighbourhood of a body surface, of a wall face inside fluid machinery, etc., have been observed. Fig. 16.1 shows the oilfilm pattern on the blade surface of a Wells turbine for a wave power generator.1 From this pattern the nature of the internal flow can be surmised. In addition to this method, flow, temperature, and pressure can be visualized by detecting changes in the oil-film pattern on a body surface.

16.2.2

Tuft Method

Although this is an unsophisticated method widely used for fluid experiments for some time, it has recently become easier to use and more informative as detailed experiments and analyses have been made of the static and dynamic tuft characteristics. It is utilized for visualizing flows near and around the surfaces of aircraft, hulls and automobiles as well as those behind them, the internal flows of pumps and blowers, and ventilation flows in rooms. Fig. 16.2 shows an example of the visualized flow behind an automobile,2 while Plate 7 (see colour plate section) shows that around a superexpress train.3 Fig. 16.3 shows an example of the utilization of extremely fine fluorescent mini-tufts which hardly disturb the flow.4

1

T. Tagori, Fluid Engineering, Plate 19, University of Tokyo Press, (1989). T. Tagori, et al., Proc. Flow Visualization Symp., (1980), p. 13. 3 JR Tokai. 4 T. Saga, T. Kobayashi, Flow Visualization 5 (Suppl.) (1985), p. 87. 2

16.2 Experimental Visualization Methods

FIGURE 16.2 Wake behind an automobile (tuft grid method) in water, flow velocity 1 m/s, length 530 mm (scale 1:8), Re ¼ 5  105.

FIGURE 16.3 Flow around an automobile (fluorescent mini-tuft method).

16.2.3

Injection Tracer Method

For water flow, the streak method using dye solution has widely been used for a long time. In the suspension method, aluminium powder or polystyrene particles are used, while in the surface floating tracer method, sawdust and aluminium power are used. The smoke method is used for air flows. In the application of these methods, there are many examples for visualizing the flow around or behind wings, hulls, automobiles, buildings and bridge piers, as well as for the internal flow of pipelines, blood vessels and pumps. Fig. 16.4 is a photograph where the flow around a double-delta wing aircraft is visualized by a water flow.5 It can be seen how the various vortices develop.

5

H. Werle, Proc. ISFV, Tokyo, (1977), p. 39.

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FIGURE 16.4 Flow around a double delta wing aircraft in water, angle of attack 15 degrees: (A) colour streak line method; (B) suspension method (air bubble method).

FIGURE 16.5 Flow around an automobile (smoke method).

These vortices act to increase the lift necessary for a high-speed aircraft to undertake low-speed flight. Plate 86 (see colour plate section) and Fig. 16.5 7 visualize the flow around an automobile by the smoke method. The flow pattern is clearly seen.

6 7

Flow Visualization Society, Flow Visualization Handbook, Plate 8, Asakura Shoten, (1986). W.H. Hucho, L.J. Janssen, Proc. ISFV, Tokyo, (1997), p. 103.

16.2 Experimental Visualization Methods

FIGURE 16.6 Flow in a bent divergent pipe (floating sawdust method) in water, flow velocity 0.4 m/s, Re ¼ 2.8  104.

Fig. 16.6 shows observation, by the floating method using sawdust, of the flow in a bent divergent pipe.8

16.2.4

Chemical Reaction Tracer Method

The electrolysis deposition method uses chemical reaction and electrolytic precipitation. There is negligible change in density due to chemical reaction, the settling velocity of the tracer is small and thus many compounds are suitable for visualizing low-velocity flow. The method has been used for visualizing the flow around and behind a flat board, wing and hull, the flow inside a pump and boiler and natural/thermal convection. As an example of the electrolysis deposition method, Fig. 16.7 is an observation of flow using the streaks developed by injecting saturated liquid ammonium sulphide through a fine tube onto a mixture of white lead and a quickdrying oil which has been applied to the surface of a model yacht.9

FIGURE 16.7 Flow on a model yacht surface (surface film colouring method) in water, flow velocity 1.0 m/s, length of model 1.5 m, Re ¼ 1.34  106, white lead and ammonium sulphide used.

8 9

K. Akashi, et al., Symp. on Flow Visualization (first), (1973), p. 109. S. Matsui, Nishinihon Ryutaigiken Co., Nagasaki, Japan.

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FIGURE 16.8 Kármán vortex street behind a cylinder (electrolytic precipitation method) in water, flow velocity 10 mm/s, diameter of cylinder 10 mm, Re ¼ 105.

Fig. 16.8 visualizes a Kármán vortex street using as the tracer the white condensation produced when water is electrolysed with the cylinder as the positive pole.10

16.2.5

Electrically Controlled Tracer Method

Included in this method are three categories: the hydrogen bubble method, spark tracing method and smoke wire method. Each one of them is capable of providing quantitative measurement. By these methods, the flow around and the vortex behind a cylinder, flat board, sphere, wing, aircraft and hull, the flow in a cylinder, the flow around a valve and the flow in a blower/ compressor have been observed. Plate 9 (see colour plate section) is a picture visualizing the flow around a cylinder by the hydrogen bubble method,11 while Plate 10 (see colour plate section) shows the flow around a sphere by the spark tracing method.12 Fig. 16.9 shows the flow around a wing by the same method,13 and Fig. 16.10 shows the flow around an automobile by the smoke wire method.14

16.2.6

Optical Visualization Method

This method, whose most significant characteristic is the capability of complete visualization without affecting the flow, is widely used. The Schlieren method utilizes the change in diffraction rate due to the change in density (temperature). The interference method, which uses the fact that the number of interference fringes is proportional to the difference in density, is mostly applied to air flow. For free surface water flow, the stereophotography method is used. The

10

S. Taneda, Fluid Mechanics Learned from Pictures, Asakura Shoten, (1988), p. 92. H. Endo, et al., Symp. of Flow Visualization (second), (1974), p. 135. 12 Y. Nakayama, Flow Visualization 8e28 (1988), p. 14. 13 Y. Nakayama, et al., Symp. on Flow Visualization (fourth), (1976), p. 105. 14 Y. Nakayama, Faculty of Engineering, Tokai University, (1982). 11

16.2 Experimental Visualization Methods

FIGURE 16.9 Flow around a wing (spark tracing method) in air, flow velocity 28 m/s, angle of attack 10 degrees, Re ¼ 7.4  104.

FIGURE 16.10 Flow around an automobile (smoke wire method).

unevenness of a liquid surface is stereophotographed to determine the difference in the height of the liquid surface and thus the state of flow is known. The Moiré method is also used for water flows. The state of the flow is checked by obtaining as light and dark stripes the contours indicating the unevenness of the liquid surface. A new technique, the laser holography method, has been used. An optical reference path is added to the optical system of the shadowgraph method or the Schlieren method. In addition, the laser speckle method and the infrared thermography method have been employed. In the laser speckle method, the flow velocity distribution is obtained by optically processing speckle pattern and in

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the infrared thermography, the temperature distribution is obtained by detecting infrared radiation energy. Various actual examples of the optical visualization method are shown in Plates 1115, 1216 and 1317 (see colour plate section) and Figs. 16.1118, 16.1219 and 16.1320.

FIGURE 16.11 Flow at bottom dead point of vertically vibrating wing (Schlieren method) in air, flow velocity 5 m/s, chord length 100 mm, Re ¼ 3  104, vibration frequency 90 Hz, single amplitude 4 mm.

Normal shock wave

Oblique shock wave

Spike

FIGURE 16.12 Flow at air inlet of supersonic aircraft engine (colour Schlieren method), M ¼ 2.0, Re ¼ 1.0  107.

15

K. Fujii, Journal of the Visualization Society of Japan 15e57 (1995), p. 142. N. Hara, T. Yoshida, Proc. of FLUCOME Tokyo ’85, vol. II, (1986), p. 725. 17 M. Kawahashi, K. Hosoi, Experiments in Fluids 11 (1991), p. 278. 18 H. Ohashi, N. Ishikawa, Journal of the JSME 74e634 (1974) p. 500. 19 T. Asanuma, et al., Report of Aerospace Research Institute of University of Tokyo, 9e2, (1973), p. 499. 20 T. Nagayama, T. Adachi, Joint Gas Turbine Congress, Paper No. 36, (1977). 16

16.3 Computer-Aided Visualization Methods

FIGURE 16.13 Equidensity interference fringe photograph of driven blade on low-pressure stage in steam turbine (MacheZehnder interferometer method) in air, inlet Mach number 0.275, outlet Mach number 2.123, pitch 20 mm, chord length 33.6 mm.

16.3 16.3.1

COMPUTER-AIDED VISUALIZATION METHODS Visualized Image Analysis

In this method, a visualized image is put into a still or video camera so that its density values are digitized. It is then put into a computer to be processed analytically, statistically in colour distribution and otherwise and thus is made much easier to interpret. Various techniques for this method have been developed. As an example of the hydrogen bubble image analysis method, Fig. 16.14 shows the hydrogen bubble technique where the time line and the streak line are simultaneously visualized. The visualized image is captured by a CCD camera, converted to binary codes and fine lines and thus the velocity vector is obtained.21 As an example of particle tracking velocimetry, Plate 14 (see colour plate section) shows the velocity vectors obtained for flow over a cylinder by following, from time to time, the spherical plastic tracer particles of diameter 0.5 mm suspended in the water.22 Plate 15 (see colour plate section) is an example of PIV

21

Y. Nakayama, et al., Report of Research Results, Faculty of Engineering, Tokai Univ. (fifth), (1987), p. 1. 22 R.F. Boucher, M.A. Kamala, Atlas of Visualization, vol. 1, (1992), p. 297.

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FIGURE 16.14 Kármán vortex street behind a cylinder (hydrogen tube method): (A) visualized image; (B) binarization; (C) change to fine line; (D) velocity vector; (E) velocity vector at grid point.

(Particle Image Velocimetry). The image was obtained by injecting a smoke tracer into the room from the floor under the chair on which a man was sitting and natural convection around a human body was visualized.23 Plate 16 (see colour plate section) shows flow in a bent pipe visualized by PIV.24

23 24

T. Kobayashi, et al., Journal of the Visualization Society of Japan 17e66 (1977) p. 204. Oishi et al., The 13th JSME Bioengineering Conference and Seminar, No. 02e26, (2009), p. 65.

16.3 Computer-Aided Visualization Methods

In Plate 17A (see colour plate section), the flow in an intracranial aneurysm is visualized by the stereo PIV25 and in Plate 17B (see colour plate section), the brain vascular network is shown.26 In Plate 18 (see colour plate section), the velocity distribution of two-phase flow of water mixing with ethyl alcohol in a very small Y-shaped flow channel is visualized by the micro-PIV.27 In Plate 19 (see colour plate section), the three-dimensional flow velocity field in a pipe at Re ¼ 6000 is measured at high resolution by the holographic PIV and the three-dimensional unsteady flow velocity field by eliminating average flow velocity component is shown.28 In Plate 20 (see colour plate section), an example of flow visualization in the flow in a pipe is shown by the Spin-Tagging Magnetic Resonance Imaging (Spin-Tagging MRI)29 and in Plate 21 (see colour plate section), the flow velocity and temperature distribution in the wake behind a heated circular column is visualized by the Molecular Tagging Velocimetry and Thermometry.30 Furthermore, in Plate 22 (see colour plate section), the density distribution of catalyst particles in air flow in a two-phase flow in a pipe is visualized by the Computer Tomography.31

16.3.2

Measured Data Visualization

The data obtained by the simultaneous use of many instruments such as a Pitot tube, hot-wire anemometer, laser Doppler velocimeter, pressure gauge, thermometer, etc. can be processed by computer, and thus the phenomena are visualized as images. In Plate 23 (see colour plate section) pressure-sensitive, light-emitting diodes are placed transversely. The total pressure pattern of a wake of an airplane wing is then obtained by photographing the diode emissions, whose colours change with total pressure.32 Fig. 16.15 shows the measured result of the flow velocity in the area behind a model passenger car obtained using a three-dimensional laser Doppler velocimeter, presented as a velocity vector diagram.33 In Fig. 16.16 the acoustic power flow from a cello is visualized by the acoustic intensity method. The size and direction of the energy flow at

25

Y. Akedo, et al., Proceedings of the eighth Asian Symposium on Visualisation, No. 46, (2005). H. Gibo, et al., CHUGAI-IGAKUSHA, (2006), p. 23. 27 Y. Sugii, et al., Journal of Visualisation 8 (2) (2005), p. 117. 28 D.H. Barnhart, et al., Applied Optics 33 (30) (1994), p. 7159. 29 K.W. Moser, et al., Annals of Biomedical Engineering 29 (1) (2001), p. 9. 30 H. Hu, Koochesfahani, Measurement Science and Technology 17 (6) (2006), p. 1269. 31 T. Zhao, (Ph.D. thesis), Nihon University, (2010). 32 Sakai et al., Journal of the Japan Society for Aeronautical and Space Sciences 31 (350) (1988), p. 205. 33 Sato and Takagi, Journal of the Visualization Society of Japan 12 (Suppl. 1) (1992), p. 87. 26

343

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(A)

Z

0–300 mm

CHAPTER 16:

277 mm

344

Y

0

0–400 mm

(B) Z 25 m/s 300

200

100

0

100

200

300

400

Y

FIGURE 16.15 Flow behind an automobile with spoiler (laser Doppler velocimeter method): (A) measured region; (B) mean velocity vector.

each point is obtained through a computational process from the cross-vector of the sonic pressure signal on a microphone.34

16.3.3

Numerical Analysis Data Visualization Method

As explained in Chapter 15, enormous computational output calculated by numerical fluid dynamics is visualized in an easy-to-understand figure, image or animation. Plate 1e6 (see colour plate section), Figs 15.14, 15.16e15.18, 15.20, 15.21, 15.27e15.31 and 15.33e15.35 are their examples.

34

H. Tachibana, et al., Atlas of Visualization, vol. 2, (1996), p. 203.

16.3 Computer-Aided Visualization Methods

FIGURE 16.16 Radiating power flow of a cello (acoustic intensity method). (A) Magnitude of energy flow and (B) direction of energy flow.

16.3.4

Numerical Data Manifestation Method

In visualized image analysis, measured data visualization, and numerical calculation and data visualization, various kinds of presentation by computer graphics techniques are used. The kinds of presentation include contours, where physically equal values are connected by a curve; area colouring, where areas are painted in colours respectively corresponding to the physical quantity level of areas; isosurface, where physically equal values are three-dimensionally manifested in surfaces; volume rendering, where the levels expressed in an isosurface are manifested by changing the degree of transparency; and vectorial, where sizes and directions of flow velocity, etc., are manifested by arrow marks. Presentation can also be as graphs or animations. Examples of contour presentation are shown in Fig. 15.4, where streamlines (which are the contours of stream function) and contours of vorticity are manifested, Plate 12 (see colour plate section), where contours of density are shown, and Fig. 16.16A where the contours of sound wave energy are depicted. Examples of area colouring are Plates 1B, 3, 4 and 5 (see colour plate section), where the pressure distribution is shown, Plate 22 (see colour plate section)

345

346

CHAPTER 16:

Flow Visualization

where density distribution of particles in two-phase flow is depicted, Fig. 15.31 where turbulent energy is shown, and so forth. And an example of isosurface presentation is Plate 24 (see colour plate section) where the temperature distribution is shown,35 examples of the vector presentation are Plates 6, 14, 16, 18 and 21 (see colour plate section) and Figs 15.11B, 16.15B, 16.16B and so on. An example of volume rendering is shown in Plate 25 (see colour plate section) where horseshoe vortex and vortex structure in the wake in the flow around a cube on a flat plate is shown.36

35 36

H. Miyachi, How to Visualize your Data using AVS, Fig. 5.28, Kubota Co., Tokyo, (1995). Ono et al., Nikkei Science 26 (1996), p. A10.

Answers to Problems

2. CHARACTERISTICS OF FLUID 1. 2. 3. 4. 5. 6. 7. 8. 9.

kg m/s2 Viscosity: Pa s, kinematic viscosity: m2/s v ¼ 0.001 m3/kg 2.13  107 Pa cos q, h ¼ 1.48 cm h ¼ 2Trgb 291 Pa 9.15  10 3 N 1.38 N 1461 m/s

3. FLUID STATICS 1. 6.57  107 Pa 2. (a) p ¼ p0 þ rgH (b) p ¼ p0 rgH (c) p ¼ p0 þ r0 gH0 rgH 3. (a) p1 p2 ¼ ðr0 rÞgH þ rgH1 (b) p1 p2 ¼ ðr r0 ÞgH 4. 50 mm 5. Total pressure P ¼ 9.56  105 N, hc ¼ 6.62 m 6. 2.94  104 N, 5.87  104 N 7. 9.84  103 N 8. Force acting on the unit width: 1.28  106 N, Action point located along the wall from the water surface: 11.6 m 9. 7700 N m 10. Horizontal component Px ¼ 1.65  105 N, vertical component Py ¼ 1.35  105 N, total pressure P ¼ 2.13  105 N, acting in the direction of 39.3 from a horizontal line 11. 976 m3 12. h ¼ 0.22 m, T ¼ 0.55 s

347

348

Answers to Problems

13. u ¼

1 r0

pffiffiffiffiffiffiffiffiffi 2gh0 rad=s, u ¼ 14 rad/s at h0 ¼ 10 cm, speed of revolution

when the cylinder bottom begins to appear n ¼ 4.23s

1

¼ 254 rpm

4. FUNDAMENTALS OF FLOW 1. (a) steady , density opened

Velocity , time

pressure ,

unsteady flow

velocity

pressure

position

closed

(b) proportional 2. 3. 4. 5. 6. 7.

density

discharge inversely

proportional

 G ¼ 0:493 m2 s Re ¼ 6  104, turbulent flow dy dx ¼ x y , namely xy ¼ constant (a) Rotational flow, (b) irrotational flow, (c) irrotational flow Water: vc ¼ 23.3 cm/s, air: vc ¼ 3.5 m/s G ¼ 82 m2/s

5. ONE-DIMENSIONAL FLOW 1. 2. 3. 4.

See text. v1 ¼ 6:79m=s; v2 ¼ 4:02m=s; v3 ¼ 1:70m=s p2 ¼ 39.5 kPa, p3 ¼ 46.1 kPa p0: atmospheric pressure, p: pressure at the point of arbitrary radius r   rQ2 1 1 p0 p ¼ 8p2 h2 r 2 r22    r12 rQ2 r2 1 1 Total pressure (upward direction) P ¼ 4ph log . 2 r1 2 r2 2

5. vr ¼ 5.75 m/s, pr pffiffiffi 2Apffiffiffiffi H 6. t ¼ Ca 2g

p0 ¼

4

1.38  10 Pa 

2

pv pffiffiffiffi Ca 2g

7. Condition of section shape H ¼ R ¼ 12.9 cm, d ¼ 1.29 mm 8. H ¼ 2.53 m, 1 þ cos q 1 cos q 9. Q1 ¼ Q; Q2 ¼ Q; 2 2   Q1 ¼ 0:09m3 s; Q2 ¼ 0:03m3 s;

10. 7.49 m H2O (¼ 0.76  10 3 Pa) 11. n ¼ 6.89s 1 ¼ 413 rpm, torque 8.50  10 12. F ¼ 749 N 13. Cc ¼ 0.64, Cv ¼ 0.95, C ¼ 0.61

r4

F ¼ rQv sin q F ¼ 2:53  104 N 2

Nm

Answers to Problems

6. FLOW OF VISCOUS FLUID 1. See text vðrvÞ vv v vu 2. 1r vr þ vu vx ¼ 0; or vr þ r þ vx ¼ 0    2 y 1 u 3. (a) u ¼ 6v hy , (b) u0 ¼ 1:5 max h (c) Q ¼

h3 Dp 12m l ,

4. (a) u ¼ 2u0 (c) Q ¼

"

(d) Dp ¼ 12mlQ h3  2 # r 1 ; (b) u0 ¼ 12umax r0

pd4 Dp 128m l ;

(d) Dp ¼

128mlQ pd4

5. (a) u0 ¼ 0.82umax, (b) r ¼ 0.76r0

6. 7. 8. 9. 10.

nt ¼ 4.57  10 3 Dp Q ¼ pdh 12m l h2 ¼ 0.72 mm LT 1 8.16 N

5

m2/s, l ¼ 2.01 cm

7. FLOW IN PIPES 1, 5. 6. 7. 8. 9. 10.

2, 3, 4.See applicable texts. See applicable text. Error of loss head h is 5a (%) h ¼ 733 m at diameter 50 mm, h ¼ 26.4 m at diameter 100 mm 24.6 kW Pressure loss Dp ¼ 508 Pa 3.2 cm H2O (¼ 0.33  10 5 Pa) hs ¼ 6.82 cm, h ¼ 0.91

8. FLOW IN WATER CHANNEL 4:56 1. i ¼ 1000 2. From Chézy’s equation Q ¼ 40.4 m3/s, from Manning’s equation Q ¼ 40.9 m3/s 3. Q ¼ 19.3 m3/s 4. Flow velocity becomes maximum at q ¼ 257.5 degrees, h ¼ 2.44 m and discharge becomes maximum at q ¼ 308 degrees, h ¼ 2.85 m 5. Tranquil flow, E ¼ 1.52 m 6. hc ¼ 0.972 m, vc ¼ 3.09 m/s 7. Qmax ¼ 14.4 m3/s 8. 1.18 m 9. See applicable text.

349

350

Answers to Problems

9.

DRAG AND LIFT 1. Using Stokes equation, terminal velocity U ¼ 2. 3. 4.

5. 6, 7. 8. 9. 10.

10.

d2 g 18n



rs rw

1



where, d is diameter of a spherical sand particle and rw, rs are density of water and sand, respectively D ¼ 1450 N, Maximum bending moment Mmax ¼ 3620 Nm D ¼ 2.70 N dmax ¼ 3.2 cm at wind velocity 4 km/h, dmax ¼ 4.1 cm at wind velocity 120 km/h T ¼ 722 Nm, L ¼ 4.54  104 Nm/s See texts. Df ¼ 88.9 N, required power L ¼ 133 Nm/s L ¼ 3.57 N D ¼ 134 N

DIMENSIONAL ANALYSIS AND LAW OF SIMILARITY

1. Consider v, g, H as the physical influencing quantities and perform pffiffiffiffiffiffiffi dimensional analysis. v ¼ C gH 2. D ¼ CmUd qffiffiffi

3. a ¼ C

K r

4. D ¼ rL2 v2 f 4



pvffiffiffiffi Lg



5. Q ¼ C dm Dp l  6. d ¼ xf Ux n  pffiffiffiffiffiffiffiffi d 2rDp ¼ f ðReÞ 7. C ¼ f m 8. (a) 167 m/s, (b) 33.3 m/s, (c) 11.1 m/s 9. Towing velocity for the model vm ¼ 2.88 m/s 1 10. 2:36

11.

MEASUREMENT OF FLOW VELOCITY AND FLOW RATE 1. v ¼ 4.44 m/s 2. v ¼ 28.5 m/s

Answers to Problems

3. 4, 5, 6. 7. 8. 9.

12.

Mass flow rate m ¼ 0.325 kg/s See applicable texts. U ¼ 50 cm/s See applicable texts. Error for rectangular weir is 3%, error for triangular is 5%

FLOW OF IDEAL FLUID

1. f ¼ u0x þ v0y, j ¼ u0y v0x 2. See applicable text. 3. Flow in counterclockwise rotary motion, vq ¼ G/(2pr), vr ¼ 0, around the origin q q 4. f ¼ 2p log r; j ¼ 2p q 5. Putting r ¼ r0, j ¼ 0, the circumference becomes one streamline. Velocity distribution vq ¼ 2Usin q, pressure distribution p pN ¼ 1 4 sin2 q rU2 =2

6. The flow around a rectangular corner. 7. Flow in clockwise rotary motion, vq ¼ 8. w ¼ Uze ia 9.

G 2pr ;

vr ¼ 0, around the origin.

351

352

Answers to Problems

10.

13.

FLOW OF A COMPRESSIBLE FLUID p RT

 ¼ 1:226kg m3 pffiffiffiffiffiffiffiffiffi 2. a ¼ kRT ¼ 1297m=s  1 k 1 1 2 u22 ¼ 418 K u 3. T2 ¼ T1 þ 2 k R 1 t2 ¼ 145 C  k=ðk 1Þ T1 ¼ 3:4  105 Pa p2 ¼ p1 T2 1. r ¼

4. T0 ¼ 278.2 K, t0 ¼ 5.1 C p0 ¼ 6.81  104 Pa r0 ¼ 0.85 kg/m3 5. u ¼ 444 m/s pffiffiffiffiffiffiffiffiffi 6. M ¼ 0:73; a ¼ kRT ¼ 325m=s; u ¼ aM ¼ 237m=s 7. u ¼ 272 m/s 8. pp0 ¼ 0:455 < 0:528; m ¼ 0:0154 kg=s

9. AA ¼ 1:66 10. A2 ¼ 2354 cm2 11. 2.35  105 N 12. Mach number ¼ 0.58, flow velocity ¼ 246 m/s, pressure ¼ 2.25  105 Pa

Answers to Problems

14. 1.

dz dt

UNSTEADY FLOW ¼ 1:39m=s; T ¼ 1:57 s

2. 0.69 m/s qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3. T ¼ 2p gðsin q1 lþsin q2 Þ 4. 5. 6. 7.

t ¼ 1 min 20 s a ¼ 837 m/s Dp ¼ 2.51  106 Pa pmax ¼ 1.56  106 Pa

353

List of Symbols

A a a0 B b C Cc CD Cf CL CM Cp Cv c cp cv D Df Dp d E e F Fr f g H h I i J K k kd L l

Area, constant of integration Area (relatively small), velocity of sound The propagation velocity of a pressure wave in an elastic pipe Width Width, thickness Coefficient of discharge, integration constant Coefficient of contraction Drag coefficient Frictional drag coefficient Lift coefficient Moment coefficient Pressure coefficient Coefficient of velocity Integration constant, coefficient of Pitot tube, flow velocity coefficient Specific heat at constant pressure Specific heat at constant volume Diameter, drag, friction resistance Friction drag Pressure drag, form drag Diameter Specific energy, Young’s modulus Internal energy Force, resistance Froude number Coefficient of friction, vortex shedding frequency Gravitational acceleration Head Head, clearance, loss of head, depth, enthalpy Geometrical moment of inertia Slope Moment of inertia, momentum Bulk modulus Interference factor, turbulence energy dispersion Cavitation number Length, power, lift Length, mixing length, representative size

355

356

List of Symbols M m n P p p0 ps pt pN Q q R Re r St s T t U u u* V v W w w(z) a b G g d d* 3 z h q k l m n x r s s 4 j u

Mass, Mach number, moment, mole Mass flow rate, mass (relatively small), strength of doublet, hydraulic mean depth Polytropic exponent Total pressure, supporting load Pressure Stagnation pressure, total pressure, atmospheric pressure Static pressure Total pressure Pressure unaffected by body Volumetric flow rate Discharge quantity per unit time, quantity of heat per unit mass Gas constant Reynolds number Radius(at any position) Strouhal number Specific gravity, entropy, wetted perimeter Tension, absolute temperature, torque, thrust, period, time, temperature Time Velocity unaffected by body, representative velocity Velocity (x-direction), peripheral velocity Friction velocity Volume Specific volume, mean velocity, velocity (y-direction), absolute velocity Weight Velocity (z-direction), relative velocity Complex potential Acceleration, angle, coefficient of discharge Compressibility, angle, throttle diameter ratio Circulation, strength of vortex Strain angle Boundary layer thickness Displacement thickness Height of wall surface roughness, expansion coefficient of gas, turbulence energy dispersion Vorticity, loss factor Efficiency, pressure recovery efficiency Angle, momentum thickness Ratio of specific heat Friction coefficient of pipe, wavelength Coefficient of viscosity, dynamic viscosity Kinematic viscosity Loss factor Density Stress, shock wave angle Shear stress Angle, velocity potential Stream function Angular velocity

Colour Plate

Plate 1 Pressure distribution on the surface of an aircraft model in a transonic wind tunnel (TVD method with finite volume method): (A) mesh diagram; (B) pressure distribution simulated (lower pressure in blue and higher pressure in red) (Ref. 18 in Chapter 15).

357

358

Colour Plate

Plate 2 Total pressure distribution on a fan and bypass duct of a jet engine (TVD method with finite volume method), outer diameter 1555.75 mm (Ref. 19 in Chapter 15).

Plate 3 Pressure distribution on the surface of a transonic aircraft: (A) numerical simulation (boundary element method); (B) experimental result (measured pressure value); Mach number 0.6, angle of attack 0 degree (blue: low, red: high) (Ref. 24 in Chapter 15).

Colour Plate

Plate 4 Turbulent flow around rectangular column [large eddy simulation (LES), pressure distribution, streak line, time line, Re ¼ 20,000] (Ref. 35 in Chapter 15).

Plate 5 Pressure distribution on the surface of body and floor of a formula car (LES), Running speed 45 m/s, Re ¼ 6.6  106 (Ref. 36 in Chapter 15).

359

360

Colour Plate

Plate 6 Turbulent velocity distribution in a clean room (finite element method); flow velocity from air ventilation system 38 cm/s (Ref. 38 in Chapter 15).

Plate 7 Flow around N700 type Shinkansen bullet train in wind tunnel (Tuft method) (A) Shinkansen bullet train at speed of 300 km/h; (B) wind tunnel: air velocity of 55.5 m/s, 1/20 scale model, Re ¼ 6.6  105 [provided by JR Tokai].

Colour Plate

Plate 8 Flow around an automobile (smoke method) in air, flow velocity 6 m/s, model 1:5, Re ¼ 2  105 (Footnote 6 in Chapter 16).

Plate 9 Karman vortex street behind a circular cylinder (hydrogen bubble method) in water, flow velocity ¼ 2.6 m/s, diameter of cylinder ¼ 8 mm, Re ¼ 195 (Footnote 11 in Chapter 16).

361

362

Colour Plate

Plate 10 Comparison of air flow around a golf ball and a sphere of equal size (spark tracing method) in air, flow velocity ¼ 23 m/s, diameter of ball ¼ 42.7 mm, Re ¼ 7  104; (A) sphere; (B) golf ball (Footnote 12 in Chapter 16).

Plate 11 Supersonic flow around a simplified supersonic aircraft (AGARD-B model), (colour Schlieren method) (Footnote 15 in Chapter 16).

Colour Plate

Plate 12 Density distribution of flow around a conical body flying at supersonic velocity, Mach number 2, angle of attack 20 degrees (blue: low, red: high) (laser holographic interferometer þ computer tomography) (Footnote 16 in Chapter 16).

363

364

Colour Plate

Plate 13 Flow velocity distribution in mixing region of aerosol jet (Laser Speckle method) (Footnote 17 in Chapter 16).

Plate 14 Velocity vectors of flow over a circular cylinder (PTV) in water, flow velocity 1.2 m/s, diameter of cylinder 38.3 mm, Re ¼ 3545, tracer: plastic particles of diameter 0.5 mm, velocity regions shown by colour (in mm/s) (Footnote 22 in Chapter 16).

Colour Plate

Plate 15 Natural convection around a human body (density correlation method) Maximum ascending velocity is about 0.2 m/s (Footnote 23 in Chapter 16).

Plate 16 Flow in a curved pipe (PIV), R ¼ 1.5d (R: radius of curvature, d: inner diameter ¼ 10 mm), flow velocity ¼ 81 mm/s, Re ¼ 345 (Footnote 24 in Chapter 16).

365

366

Colour Plate

Plate 17 Blood flow in a human brain visualised by the stereo PIV. (A) Flow in an intracranial aneurysm (Footnote 25 in Chapter 16). (B) Brain vascular network (Footnote 26 in Chapter 16).

Plate 18 Velocity distribution of two-phase flow of water and ethanol in a very small Y-shaped flow channel (micro-PIV). Flow channel with semicircle section of radius of 35 mm (Footnote 27 in Chapter 16).

Plate 19 Three-dimensional flow velocity field in a pipe measured by high resolution (holographic PIV) (Footnote 28 in Chapter 16).

Plate 20 Flow in a pipe visualised by the Spin-Tagging Magnetic Resolution Imaging (Spin-Tagging MRI) (Footnote 29 in Chapter 16).

368

Colour Plate

Plate 21 Flow velocity and temperature distribution in the wake behind a heated circular column (MYV&T), air flow velocity ¼ 0.026 m/s, cylinder diameter ¼ 4.76 mm, Re ¼ 130 (Footnote 30 in Chapter 16).

Colour Plate

Plate 22 Density distribution of catalyst particles in a two-phase flow with air and catalyst particle in a pipe (Computer tomography), air flow velocity ¼ 8.247 m/s, pipe diameter ¼ 270 mm, Re ¼ 352.6 (Footnote 31 in Chapter 16).

369

370

Colour Plate

Plate 23 Total head pattern behind horizontal tail (pressure sensors and light-emitting diodes combination method) (Footnote 32 in Chapter 16).

Colour Plate

Plate 24 Temperature distribution in an air-conditioned room (isosurface manifestation method); red colour shows warmer areas (Footnote 35 in Chapter 16).

371

372

Colour Plate

Plate 25 Horseshoe vortex and vortex structure in the wake behind a cube on a flat plate (volume rendering), air flow velocity ¼ 0.15 m/s, 0.01-m cube, Re ¼ 1000 (Footnote 36 in Chapter 16).

Index

‘Note: Page numbers followed by “f” indicate figures, “t” indicate tables and “b” indicate boxes.’

A Absolute pressure, 26 Absolute streamlines, 52, 53f Absolute system of units, 9e10 CGS system of units, 10 MKS system of units, 10 SI prefixes, 10 SI units, 10 Acceleration, 51 Acoustic intensity method, 330te333t Acoustic power flow from cello, 343e344, 345f Actual head, 157 Adiabatic change, 21e22 Aerofoil. See Wing Aerofoil section, wing, 193, 193f, 195f cavitation development, 200f development of circulation around, 197f inside flow, 199f Air, 1 bubble method, 336f solubility in water, 200t viscosity and kinematic viscosity, 17t Airship, 208, 208f Allievi’s equation, valve closure, 286e287 Aluminium powder, 335 Anemometre constant-voltage type, 218, 219f CTA, 218, 219f hot-wire, 218 interference fringe, 219, 220f

laser Doppler, 218e219, 219f reference beam, 219, 220f single-beam, 219, 220f Angle of attack, wing, 193 Angular momentum conservation of, 90e92 equation, 90e91 Animation method, 330te333t Aqua, spread of, 46b Aqueducts, 163, 163f Arbitrary particle, 51 Archimedes’ principle, 41e43, 41f Area colouring manifestation method, 330te333t Area flowmetre, 223e224 Area force, 103 Aspect ratio, 193 Atmospheric pressure, 25 Attachment amplifier, 229 Automatic power control, 277 Average flow velocity, 135e136, 138 Axial symmetry, 106 Axially symmetric flow, 100, 101f

B Base units, 9e10 Bazin equation, 165, 165t BeameWarming method, 315 Bend, pipes, 151 loss factor, 152t Bernoulli, Daniel, 72f Bernoulli’s equation, 68e74, 89e90, 180e181, 262 application of, 74e82 Pitot tube, 76e82 Venturi tube, 74e76

conservation of fluid energy, 69f force acting on fluid, 70f hydraulic grade line and energy line, 74f movement of roller-coaster, 68f sensing static pressure, 73f Bernoulli’s principle, 198 Bernoulli’s theorem, 69, 73 Bingham fluid, 17 Blade. See Wing Blasius equation, 140 Body force, 103 Boeing 747, 210f BordaeCarnot head loss, 88 Boundary condition, 297e298, 297f Boundary element method, 294, 303e304, 304fe305f, 316e317, 317f Boundary layer, 123e128, 177, 233 development, 124e127 motion equation, 127e128 separation, 128, 128f Bourdon tube, 34, 34f ‘Boyle’seCharles’ law, 9, 22 Branch pipes, 152e153, 153f Buckingham’s p theorem, 204 Buoyancy force, 41 Butterfly valve, 154, 156f

C CA method. See Cellular automaton method (CA method) Camber, 193 Camber line, 193 Candela (cd), 10 Capillarity, 19e20, 20f

373

374

Index

Cascade, 196, 198, 198f CauchyeRiemann equations, 238 Cavitation, 198e200 development in aerofoil section, 200f number, 199e200 Cellular automaton method (CA method), 305e306 solution on, 305e306 Centipoise (cP), 15 Centistokes (cSt), 15e16 Centripetal acceleration, 44 Characteristic curves, 159, 193e198, 194fe195f Chattock tilting micromanometer, 33e34 Chemical reaction tracer method, 330te333t, 337e338 Chemical vapour deposition method, 36 Chezy’s formula, 165 Choke, 146e147, 147f coefficient of discharge, 148f Chord length, wing, 193 Circular pipes, flow in, 111e114 Circular water channel, 166e167, 166f Circulation, 61e64, 62f City water system, 2 Cock, 155, 157f Coefficient of contraction, 7, 79 of diffusion, 107e108 of discharge, 80 of kinematic viscosity, 15 Pitot tube, 215 of velocity, 80 Collecting vessel, method using, 220 Complex potential, 238e239, 240f Compressibility, 9, 21e22, 255 Compressible fluid, 293, 315e318. See Incompressible fluid boundary element method, 316e317 method of characteristics, 317e318, 318fe319f time-marching technique, 315e316

Compressible fluid flow. See Ideal fluid flow; Unsteady flow basic equations for onedimensional compressible flow, 262e264 Fanno flow and Rayleigh flow, 273e274 isentropic flow, 264e268 Mach number, 260e262 shock waves, 268e273 sonic velocity, 258e260 thermodynamic characteristics, 255e258 Compression wave, 268, 270f Compressive fluid, 58 Computational fluid dynamics, 294 compressible fluid, 315e318 discretization method, 294e306 free-interface flows, 323e326 incompressible fluid, 306e313 turbulence, 318e322 Computer tomography, 330te333t Computer-aided flow visualization, 329 Computer-aided visualization methods, 330te333t measured data visualization, 343e344 numerical analysis data visualization method, 344 numerical data presentation method, 345e346 visualized image analysis, 341e343 Cone-type Venturi tube, 222 Conformal mapping, 249e253, 250f, 253f corresponding mesh on z and z planes, 250f Conservation of angular momentum angular momentum equation, 90e91 power of water pump or wheel, 92 of energy application, 74e82 Bernoulli’s equation, 68e74 of mass, 67

of momentum application of, 85e90 momentum equation, 83e85 Constant discharge, 169e171, 170f specific energy, 171, 172f water depth, 171, 172f Constant-temperature anemometre (CTA), 218, 219f Constant-voltage type anemometre, 218, 219f Continuity equation, 67, 68f, 99e100, 104e106, 127, 186, 234e235, 258e259, 293, 308e309, 311 Contour manifestation method, 330te333t Control volume, 298e299, 298f, 309e310 Convection accelerations, 106 Convergent nozzle, 265e267, 266f Convergent pipe, 151 Convergentedivergent nozzle, 173, 267e268, 268f Coriolis flowmetre, 228, 228f Correlation, 117e118 Couette flow, 13, 13f, 111 CouetteePoiseuille flow, 111, 112f Critical differential pressure (pcr), 288 Critical Reynolds number, 58 Critical water depth, 170 Cross-sectional area, 135 CTA. See Constant-temperature anemometre (CTA) Cylinder cylinder-type Pitot tube, 215e217, 216f drag of, 179e185 lift acting on rotating, 191f Cylindrical coordinates, 100

D d’Alembert’s paradox, 181e182 da Vinci, Leonardo, 4e5, 5f DarcyeWeisbach equation, 138 Data visualization, 343e344 De Laval nozzle, 267

Index

Deadweight tester, 36 Deceleration, 136e137 Degree of compressibility, 255 Density, 12, 13t Depth, 25, 28e30 Derived units, 9e10 Diaphragm pressure gauge, 35, 35f Differential manometer, 31e32, 32fe33f Diffuser, 147e150, 148fe149f Diffusion, 107, 233 Dimensional analysis, 203 Buckingham’s p theorem, 204 flow resistance of sphere, 204e206 pressure loss caused by pipe friction, 206e207 Dimensions, 9e11, 12t Direct numerical simulation, 319e320, 320f Discrete vortex method, 313 Discretization method, 294e306 solution on cellular automaton method, 305e306 solution on Eulerian method, 294e304 solution on Lagrangian method, 304e305 Divergent pipe, 147e150 divergent flow, 148f loss factor for, 149f separation occurring in, 150f velocity distribution in, 150f Doppler effect, 261 Double-delta wing aircraft, 335e336, 336f Doublet, 246, 246f Drag, 99, 124e126, 178e179, 178f. See Lift acting on wing, 194 coefficient, 179, 180t, 194 cylinder, 185f of sphere, 186f contributions for various shapes, 179t for cylinder, 179e185 of flat plate, 186e189 laminar boundary layer, 187e188

turbulent boundary layer, 188e189 friction torque acting on revolving disc, 189e191 ideal fluid, 179e182 of sphere, 185e186 viscous fluid, 182e185 Dynamic pressure, 72, 199e200

E Eddies, 177, 182 Elastic modulus, 282e284 Elastic-type pressure gauge, 34e35 Elasticity force, 207 Elbow, pipes, 152, 152f loss factor, 153t Electric controlled tracer method, 330te333t Electric-type pressure transducer, 35e36 Electrically controlled tracer method, 338 Electrolysis deposition method, 337, 337f Electrolytic colouring method, 330te333t Electrolytic corrosion method, 330te333t Elongation transformation, 104e108 Energy cascade, 118 conservation of, 68e82 line, 74 ENO/WENO schemes, 316 Equiaccelerated straight-line motion, 43e44 Equipotential line, 246 Euler, Leonhard, 5e6, 71, 71f Euler equation system, 315, 316f equation of motion, 71, 233e235, 234f in steady state, 262 Eulerian method, 51, 294 solution on, 294e304 boundary element method, 303e304

finite difference method, 294e298 finite element method, 299e303 finite volume method, 298e299 Expansion loss, 88 Experimental data visualization method, 330te333t Experimental visualization method, 330te333t External flow, 177

F Fanno flow, 273e274, 274f Finite difference approximation, 295, 295f grid, 296, 296f Finite difference method, 294e298, 306e309 flow in sudden expansion of channel, 297f mesh and boundary condition, 297f streamlines of flow in sudden expansion, 298f velocity and pressure as unknown quantities, 308e309 vorticity and stream function as unknown quantities, 306e308 Finite element method, 294, 299e303, 311e312 flow around cylinder, 300f, 303f flow around stones of garden, 312f mesh diagram of flow around cylinder, 303f triangular element, 301f two-dimensional elements, 300f Finite volume method, 294, 298e299, 309e311 flow around circular cylinder with grooves, 312f time-sequenced change of Kármán vortex street, 310f First law of thermodynamics, 256 First-ordereaccurate Godunov method, 315 Five-hole spherical Pitot tubes, 217, 217f

375

376

Index

Flat plate changes in boundary layer thickness and friction stress, 189f drag of, 186e189 friction drag coefficients, 190f Float-type area flowmetre, 223e224, 223f Flow discharge measurement. See Flow velocity measurement area flowmetre, 223e224 coriolis flowmetre, 228, 228f fluidic flowmetre, 229, 230f magnetic flowmetre, 227, 227f methods using collecting vessel, 220 using flow restrictions, 220e222 positive displacement flowmetre, 224, 224f thermal mass flowmetre, 229, 229f turbine flowmetre, 224, 225f ultrasonic flowmetre, 225e227, 226f vortex shedding flowmetre, 225, 226f weirs, 230, 230f Flow velocity measurement. See Flow discharge measurement hot-wire anemometre, 218 laser Doppler anemometre, 218e219, 219f Pitot tube, 215e218 Flow visualization, 51, 329 classification of techniques, 329, 330te333t computer-aided visualization methods, 341e346 experimental visualization methods chemical reaction tracer method, 337e338 electrically controlled tracer method, 338 injection tracer method, 335e337 optical visualization method, 338e340

tuft method, 334 wall tracing method, 329e334 Flow(s), 1, 5e6, 67 in bent divergent pipe, 337, 337f around body, 177, 177f circulation, 61e64 of compressible fluid basic equations for onedimensional compressible flow, 262e264 Fanno flow and Rayleigh flow, 273e274 isentropic flow, 264e268 Mach number, 260e262 shock waves, 268e273 sonic velocity, 258e260 thermodynamic characteristics, 255e258 contraction, 145e147 around cylinder, 183fe184f, 246e249, 248fe249f, 300f expansion, 144 external, 177 around flat plate, 187f of ideal fluid complex potential, 238e239 conformal mapping, 249e253 Euler’s equation of motion, 233e235 example of potential flow, 239e249 stream function, 236e238 velocity potential, 235e236 incompressible and compressive fluid, 58 of incompressible fluid, 293 internal, 177 laminar flow, 55e57 methods using flow restrictions, 220e222 movement, 51 one-dimensional, 54 in open water channel, 164e166 circular water channel, 166e167, 166f hydraulic jump, 172e174, 173f rectangular water channel, 168, 168f

specific energy, 168e171 path line, 51e54 pipes in, 135, 136f, 143f, 264e265, 264f, 265t cock, 155, 157f flow in inlet region, 136e138 frictional loss on pipes, 141e143 loss by pipe friction, 138e141, 139f losses in pipe lines, 144e157 pipe branch, 152e153, 153f pipe junction, 153, 154f pumping to higher levels, 157e159 total loss along pipe line, 157 valve, 154e155, 155te156t rate, 99e100, 114f, 129e130 resistance of sphere, 204e206 Reynolds number, 58 rotation and spinning of liquid, 58e61 steady flow, 54 streak line, 51e54 stream tube, 51e54 streamline, 51e54 three-dimensional, 54 turbulent flow, 55e57 two-dimensional, 54 unsteady flow, 54 of viscous fluid boundary layer, 123e128 continuity equation, 99e100 lubrication theory, 129e132 NaviereStokes equation, 101e108 velocity distribution of laminar flow, 108e114 velocity distribution of turbulent flow, 115e123 Fluid(s), 1, 9, 99, 255, 277, 282e284 characteristics of perfect gas, 22e23 compressibility, 21e22 density, 12 dynamics, 203 flow, 51 kinetic energy, 137 mass, 99e100

Index

mechanics, 1e7 movement, 25 Newtonian fluid, 17e18 non-Newtonian fluid, 17e18 oscillation, 287e289 phenomena, 329 pressure, 41 specific gravity, 12 specific volume, 12 statics, 25 Archimedes’ principle, 41e43 forces acting on liquid vessel, 36e41 pressure, 25e36 relatively stationary state, 43e46 surface tension, 18e21 units and dimensions, 9e11 velocity, 259e260 viscosity, 13e16 Fluidic(s) flowmetre, 229, 230f Fluidic vortex chamber oscillator, 287e288, 287f successfully appling to ‘Shinkansen’ bullet train, 289b Fluorescent mini-tuft method, 335f Force(s), 207 acting on body, 178e179, 178f acting on liquid vessel, 36e41 force to tear cylinder, 40e41 water pressure acting on bank or sluice gate, 37e40 area, 103 body, 103 of jet, 85e87 perpendicular to surface, 104e108 tangential to surface, 103e104 viscous, 103e108 Forced vortex flow, 61 Form drag, 178e179 Fractional step method, 311 Free vortex, 242e243, 243f flow, 61 Free-interface flows, 323e326 MAC method, 323e324 particle method, 324e326 VOF method, 324

Friction coefficient of circular pipe, 141f drag, 178e179 loss by pipe friction, 138e141, 139f pressure loss caused by pipe, 206e207 torque acting on revolving disc, 189e191 velocity, 121 Frictional coefficient, 164 Frictional loss on pipes, 141e143 Frictional resistance, 277 Froude, William, 174f Froude number (Fr), 173, 203, 208e209 Froude’s law of similarity, 209 Full width weir, 229, 230f Functional relationship, 203e207

G Galerkin method, 301 Galvanometre, 218 GanuilleteKutter equation, 165, 165t Gate valve, 154, 155f Gauge pressure, 26 GausseSeidel method, 297e298 Gettingen-type micromanometer, 33e34 Globe valve, 154, 156f Gravitational acceleration, 44 Gravity force, 207 Green’s formula method, 303e304 Gyrostatics, 44

H Hagen, Gotthilf Heinrich Ludwig, 113f, 114 HagenePoiseuille Formula, 114 Head curves. See Characteristic curves Head loss in valves, 154 Helmholtz theory, 107 Henry’s law, 198e199 High-pressure fluid, 277 Higher critical Reynolds number, 58 Holographic PIV, 330te333t

Hot-wire anemometre, 218 Hydraulic(s) development, 1e2, 4e5 grade line, 74 jump, 172e174, 173f mean depth, 164 Hydrodynamics, 5e6 Hydrogen bubble method, 330te333t, 338 image analysis method, 341, 342f

I Ideal fluid, 9, 179e182 Ideal fluid flow. See Compressible fluid flow; Unsteady flow complex potential, 238e239 conformal mapping, 249e253 Euler’s equation of motion, 233e235 example of potential flow, 239e249 stream function, 236e238 velocity potential, 235e236 Ideal gas, 9 Inclined manometer, 33e34, 34f Incompressible flow, 236 Incompressible fluid, 58, 99e100, 127. See Compressible fluid finite difference method, 306e309 finite element method, 311e312 finite volume method, 309e311 Incompressible viscous fluid, 135 lattice Boltzmann method, 313 vortex method, 313 Inertia terms, 293 Inertial force, 207 Infrared thermography, 330te333t Injected tracer method, 330te333t Injection path line method, 330te333t Injection streak line method, 330te333t Injection tracer method, 335e337 Inlet flow at, 340f flow in inlet region, 136e138 of pipe line, 146 region, 136e138

377

378

Index

Instantaneous valve closure, 285e286 Interference coefficient, 198 fringe type laser Doppler anemometre, 219, 220f Internal flow, 177 International System of Units (SI), 10 basic and supplementary units, 10t derived units, 11t Interpolating function, 301 Irrotational flow, 58e60, 235 ISA 1932 Nozzle, 222, 222f Isentropic flow convergent nozzle, 265e267 convergentedivergent nozzle, 267e268 flow in pipe, 264e265 Isentropic index, 23 Isosurface manifestation method, 330te333t Itaya equation, 140

J Jet force, 85e87 Jet pump, 88e89, 88f Jomon pot, 4 Joukowski’s hypothesis. See Kutta condition Joukowski’s transformation, 251, 252f Journal bearing, 131e132 Junction pipes, 153, 154f

K Kaleidoscopic changes, 51 Kármán, Theodor von, 123f, 131f, 131be132b Kármán vortex street, 182, 183f, 338, 338f, 342f KarmaneNikuradse equation, 140 KármánePrandtl 1/7 power law, 122 Kinematic viscosity, 15 Kinetic energy of fluid, 137 Kirchhoff, G. (German physicist), 7 Kline, Stephen Jay, 119e120, 120f Kutta condition, 196 KuttaeJoukowski equation, 192 k-ε model, 322

L La Systeme International d’Unites. See International System of Units (SI) Lagrangian method, 51, 294 solution on, 304e305 particle method, 305 vortex method, 305 Laminar boundary layer, 127 drag of flat plate, 187e188 incompressible fluid in, 127 flow, 55e57, 136f, 137e139 velocity distribution, 108e114 frictional resistance, 278e279, 280f sublayer, 119e120 Laplace’s equation, 236 Large Eddy Simulation (LES), 320e322, 321f Laser Doppler anemometre, 218e219, 219f interference fringe type, 219, 220f reference beam type, 219, 220f single-beam type, 219, 220f Laser holographic interferometer method, 330te333t Laser holography method, 339e340 Laser light sheet method, 330te333t Laser speckle method, 339e340 Lattice Boltzmann method, 306, 313, 314f Law of momentum, 85e86, 89e90, 125e126 Law of similarity, 207e213 model testing, 211e213 non-dimensional Groups, 208e210 LES. See Large Eddy Simulation (LES) Lift, 178e179, 178f. See Drag acting on rotating cylinder, 191f coefficient, 194e195 development, 191e193 wing, 193e198 Linear momentum, 83e84 Liquids, 9 forces acting on liquid vessel, 36e41 rotation and spinning, 58e61, 62f

Logarithmic velocity distribution, 121e122 Long wave, 171 Loss factor ball valve, 158t bend, 152t elbow, 153t valves, 158t Loss(es) convergent pipe, 151 divergent pipe or diffuser, 147e150, 148fe149f flow contraction, 145e147 flow direction changes, 151e152 flow expansion, 144 by pipe friction, 138e141, 139f flow velocity and loss head relationship, 139f laminar flow, 138e139 turbulent flow, 139e141 in pipe lines, 144e157 total loss along pipe line, 157 Lower critical Reynolds number, 58 Lubrication theory, 129e132 bird stalls, 131be132b Prandtl experiment, 131b Luminescent mini-tuft method, 330te333t

M MAC method. See Marker and cell method (MAC method) Mach, Ernst, 210f mysteries solved by, 212be213b Mach number (M), 58, 209e210, 260e262, 261f MacheZehnder interferometer method, 330te333t, 341f Magnetic flowmetre, 227, 227f Magnus effect, 192 Manning equation, 165, 165t Manometer, 31e34, 32f Marker and cell method (MAC method), 308, 309f, 323e324, 323fe324f Mass flow rate, 67 velocity, 144, 146, 154, 159

Index

Mass transfer method, 330te333t Measured data visualization method, 330te333t Measurement of flow discharge, 220e230 of flow velocity, 215e219 measuring technology, 122e123 Metacentric height, 43 Micro PIV, 330te333t Minimum energy principle. See Variational principle Mixing length, 118 Model testing in circulating flume or towing tank, 211 in wind tunnel, 211e213 Moiré method, 338e339 Molecular Tagging Velocimetry and Thermometry, 330te333t, 343 Moment acting on wing, 194 coefficient, 194 Momentum equation, 83e85 application, 85e90 efficiency of propeller, 89e90 force of jet, 85e87 jet pump, 88e89 loss in suddenly expanding pipe, 87e88 conservation, 83e90 Moody diagram, 141, 142f Motion equation of boundary layer, 127e128 Movement of flow, 51 of fluid, 123e124 Moving particle semi-implicit method (MPS method), 324

N Nabla (Note 3), 58e60 Navier, Louis Marie Henri, 108f NaviereStokes equation, 7, 101e108, 127, 186, 293, 309e311, 315 area force, 103

balance of forces on fluid element, 102f body force, 103 viscous force, 103e108 Newton, Isaac, 14f Newton’s law of gravity, 4e5 second law of motion, 69e71, 83e84, 277e278 third law of motion, 4e5 of viscosity, 14e15 Newtonian fluid, 17e18 Bingham fluid, 17 Nikuradse equation, 140 Non-dimensional groups, law of similarity Froude number, 208e209 Mach number, 209e210 Reynolds number, 208 Weber number, 209 Non-electrolytic reaction method, 330te333t Non-linear equation, 108 Non-Newtonian fluid, 17e18 pseudoplastic fluid, 18 Normal shock wave, 271e272, 271f Nozzle tube, 220 Nozzle Venturi tube, 222 Nozzle-type Venturi tube, 222, 222f Numerical analysis data visualization method, 330te333t, 344 Numerical data presentation method, 330te333t, 345e346 Numerical viscosity method, 320, 321f

O Oil-dots method, 330te333t Oil-film method, 329e334, 330te333t One-dimensional compressible flow, 262e264 correction to Pitot tube, 263e264 One-dimensional flow, 54, 67 conservation of angular momentum, 90e92

of energy, 68e82 of momentum, 83e90 continuity equation, 67 One-equation model, 322 Open water channel, 163, 164f, 169f circular water channel, 166e167, 166f flow in, 164e166 hydraulic jump, 172e174, 173f rectangular water channel, 168, 168f specific energy, 168e171 Optical method, 330te333t Optical visualization method, 338e340, 340f equidensity interference fringe photograph of driven blade, 341f flow at air inlet of supersonic aircraft engine, 340f Orifice plate, 221 with pressure tappings, 221f Orifice tube, 220 Oscillation of liquid column in Utube, 277e279, 278f laminar frictional resistance, 278e279 Oval gear type positive displacement flowmetres, 224, 224f

P Parallel axis theorem, 39 Parallel flow, 239e241, 240f Parallel plates, flow between, 108e111 Particle image velocimetry (PIV), 330te333t, 341e342 Particle method, 305, 324e326, 325f Particle tracking velocimetry, 341e342 Pascal, Blaise, 27, 28f Pascal’s law, 27 Path line, 51e54 Perfect fluid, 9 Perfect gas, 9, 255e256 characteristics, 22e23

379

380

Index

Photographing diode emissions, 343e344 Physical vapour deposition method, 36 Pipe(s) bend, 151 branch, 152e153, 153f convergent, 151 divergent, 147e150 elbow, 152 flow in, 135, 136f friction head loss, 138e141, 139f pressure loss, 206e207 junction, 153, 154f propagation of pressure, 279e281 transient flow, 281e282 velocity of pressure waves in, 282e284 Pitot, Henri de, 76e77, 77f Pitot tube, 76e82, 77fe78f, 215e218, 216f correction to, 263e264, 263t cylinder-type, 215e217 for measuring flow velocity, 218 spherical, 217 weir, 82, 83f Pitot-static tube. See Pitot tube PIV. See Particle Image Velocimetry (PIV) Plastic fluid, 17 Poise unit (P), 15 Poiseuille, Jean Louis, 114, 114f Poiseuille’s law, 15 Polystyrene particles, 335 Portable-type deadweight tester, 36, 36fe37f Positive displacement flowmetre, 224, 224f Potential flow, 235 free vortex, 242e243 parallel flow, 239e241 source, 241e242, 241f synthesizing of flows, 244e249 Power transmission, 277 of water pump or wheel, 92 Prandtl, Ludwig, 115f

Prandtl’s mixing length hypothesis, 118 PrandtleMeyer expansion, 271e272 Pressure, 25e36, 135, 138 absolute, 26 characteristics of, 26e28 coefficient, 180e181 critical, 266 distribution, 181e182, 181f drag, 178e179 of fluid at rest, 28e31 force, 207 gauge, 26 loss, 99, 143 measurement deadweight tester, 36 elastic-type pressure gauge, 34e35 electric-type pressure transducer, 35e36 manometer, 31e34 propagation in pipeline, 279e281, 280f recovery efficiency, 149e150 resistance, 181e182 units of, 25, 26t as unknown quantities, 308e309 Pressure-sensitive paint method, 330te333t Pressure-sensitive paper method, 330te333t Propeller, efficiency of, 89e90 Pseudoplastic fluid, 18 Pump discharge, 158e159 total head and load curve, 160f Pumping to higher levels, 157e159

Q Quasi-static state, 43

R RankineeHugoniot equations, 271, 273 Rapid flow, 171e172, 173f Rayleigh flow, 273e274, 274f

Rectangular water channel, 168, 168f Rectangular weir, 229, 230f Reference beam type laser Doppler anemometre, 219, 220f Relative rest, 25 Relatively stationary state, 43e46 equiaccelerated straight-line motion, 43e44 rotational motion, 44e46 Resistance curve, 159 Restoring forces, 43 Revolving disc, 190f friction torque acting on, 189e191 Reynolds, Osborne, 55, 57f, 64b Reynolds averaged model, 322 Reynolds law of similarity, 208 for Shinkansen, 213b Reynolds number (Re), 58, 107e108, 115, 121e122, 163, 182, 203, 206, 208, 233, 293, 318e319, 322f Reynolds stress, 116e117 Right-angled triangular weir, 229, 230f Roots type positive displacement flowmetres, 224, 224f Rotametre. See Float-type area flowmetre Rotation of vector (V), 58e60 Rotational motion, 44e46 Rough circular pipe, turbulent flow, 140e141 Roughness of commercial pipe, 141, 142f

S Saturation pressure for water, 200t Schlieren method, 339e340, 340f Schlieren photograph method, 330te333t Sectional change effect, 264e265 Semi-implicit method for pressurelinked equation method, 311 Separation point of flow, 128 Shadowgraph method, 330te333t Shaft horsepower, 159

Index

Shear deformation, 103e104 flow, 14e15 Shearing stress, 126 Ship, 209, 209f Shock waves, 268e273, 269f, 271fe274f SI. See International System of Units (SI) SI prefixes, 10, 11t Similarity of Reynolds, 107e108 Single-beam type laser Doppler anemometre, 219, 220f Sink, 242, 244e246, 245f Slow valve closure case, 286e287 Smoke method, 335e336, 336f Smoke wire method, 330te333t, 338, 339f Smooth circular pipe, turbulent flow, 140 Solubility of air in water, 200t Sonic flow, 209e210 Sonic velocity, 258e260 propagation of pressure wave, 259f Source, 241e242, 241f, 244e246, 244f Spark tracing method, 330te333t, 338, 339f Specific energy, 168e171 constant discharge, 169e171, 170f constant specific energy, 171, 172f constant water depth, 171, 172f Specific gravity, 12 Specific volume, 12 Speckle method, 330te333t Sphere drag of, 185e186 flow resistance of, 204e206 in uniform flow, 205f Spherical Pitot tubes five-hole, 217, 217f thirteen-hole, 217, 217f Spin-tagging magnetic resonance imaging (Spin-tagging MRI), 330te333t, 343 Stagnation point, 177 Standard atmospheric pressure, 25 Starting-up vortex, 196, 197f

Static pressure, 137 Steady flow, 54, 67 Steady-state turbulent boundary layer, 127e128 Stereo PIV, 330te333t Stokes, George Gabriel, 63, 63fe64f Stokes (St), 15e16 equation, 186 flow, 186 theorem, 63 Storage pump, 157, 159f Streak line, 51e54 Stream function, 236e238, 238f as unknown quantities, 306e308, 307f tube, 51e54 Streamline, 51e54, 53f, 69e71 Streamlined shape, 185 Strength of sink, 242 of source, 242 vortex, 243 Strouhal number (St), 184e185 Subcritical flow. See Tranquil flow Subsonic flow, 209e210, 255, 265, 265t Successive over-relaxation method, 297e298 Supercavitation, 199 Supercritical flow. See Rapid flow Supersonic flow, 209e210, 255, 265, 265t velocity, 268, 269f Surface floating tracer method, 330te333t force perpendicular to, 104e108 force tangential to, 103e104 Surface tension, 18e21, 18t balance between pressure increases within liquid drop and, 19f capillarity, 20f change of liquid surface, 20f dewdrop on taro leaf, 19f force, 207 Suspension method, 330te333t Synthesizing of flows, 244e249

combining of source and sink, 244e246 flow around cylinder, 246e249

T Taylor equation, 184e185 Tear cylinder, force to, 40e41 Temperature critical, 267 dynamic, 263 static, 263 total, 263 Temperature-sensitive film method, 330te333t Thermal mass flowmetre, 229, 229f Thermodynamics, 203 characteristics, 255e258 first law of, 256 second law of, 258 Thirteen-hole spherical Pitot tubes, 217, 217f Three-dimensional flow, 54, 233 Three-dimensional laser Doppler velocimeter, 343e344, 344f Throttle, 146e147 Time line method, 330te333t Time-marching technique, 315e316 Torque, 90, 92 Torricelli’s theorem, 79 Total head, 157 Total variation diminishing (TVD), 315 Tracers, 51 Tranquil flow, 171 Transient flow in pipeline, 281e282 , 281f, 283f Transition zone, 127 Tuft method, 330te333t, 334, 335f Turbine flowmetre, 224, 225f Turbulence, 318e322, 323f direct numerical simulation, 319e320 LES, 320e322 numerical viscosity method, 320 Reynolds averaged model, 322 Turbulent boundary layer, 127, 182 drag of flat plate, 188e189

381

382

Index

Turbulent flow, 55e57, 136f, 137, 139e141, 163 rough circular pipe, 140e141 smooth circular pipe, 140 velocity distribution, 115e123 Turbulent kinematic viscosity, 117 Turbulent shearing stress, 115 TVD. See Total variation diminishing (TVD) Twin vortices, 182 Two-dimensional flow, 54, 233 Two-dimensional Poiseuille flow, 111

U U-tube manometer, 32f, 33 oscillation of liquid column in, 277e279 Ultrasonic flowmetre, 225e227, 226f Units, 9e11 absolute system of, 9e10 base, 9e10 derived, 9e10 of pressure, 25, 26t Unsteady flow, 54. See Compressible fluid flow; Ideal fluid flow fluid oscillation, 287e289 oscillation of liquid column in U-tube, 277e279 propagation of pressure in pipeline, 279e281 transient flow in pipeline, 281e282 velocity of pressure waves in pipe, 282e284 water hammer, 284e287

V Valve butterfly, 154, 156f disc, 158t flow in pipes, 154e155, 155te156t gate, 154, 155f globe, 154, 156f head loss in, 154 loss factor, 158t

Variational principle, 300 Vector manifestation method, 330te333t Velocity, 135 average flow, 135e136, 138 critical, 58, 267 fluctuating, 115 mean flow, 144, 146, 154, 159 potential, 235e236 of pressure waves in pipe, 282e284 as unknown quantities, 308e309 of viscous fluid, 136e137 Velocity distribution, 136e138, 141 in divergent pipe, 150f of laminar flow, 108e114 flow between parallel plates, 108e111 flow in circular pipes, 111e114 of turbulent flow, 115e123 momentum transport and energy transport, 116f shearing stresses distribution of flow between parallel flat plates, 117f smoke vortices from chimney, 119f turbulence, 116f velocity distribution in circular pipe, 122f velocity distribution of turbulent flow, 123f viscous sublayer, 120f Vena contracta, 79 Venturi, Giovanni Battista, 75f Venturi tube, 74e76, 75f, 220 Viscosity, 293 law of, 14e15 Viscous fluid, 17, 182e185 flow of boundary layer, 123e128 continuity equation, 99e100 lubrication theory, 129e132 NaviereStokes equation, 101e108 velocity distribution of laminar flow, 108e114 velocity distribution of turbulent flow, 115e123

Viscous force, 207 force perpendicular to surface, 104e108 force tangential to surface, 103e104 Viscous frictional resistance, 277e279 Viscous sublayer, 119e120, 120f Visualisation of flow, 5 methods, 330te333t Visualized image analysing method, 330te333t Visualized image analysis, 341e343 Volume of fluid method (VOF method), 324, 325f Volume-rendering method, 330te333t Volumetric flow rate, 67, 221 Vortex flow, 61f method, 305, 313, 313fe314f shedding flowmetre, 225, 226f theory of Helmholtz, 107 Vorticity, 58e60, 233 transport equation, 107 as unknown quantities, 306e308, 307f

W Wake with eddies, 177, 182 Wall tracing method, 329e334, 330te333t, 334f Water flows in pipe, 284e285 hammer, 284e287, 285fe286f instantaneous valve closure, 285e286 slow valve closure, 286e287 testing device, 317, 317f horsepower, 158 pressure acting on bank or sluice gate, 37e40 pump, power of, 92 saturation pressure for, 200t solubility of air in, 200t Wave power generator, 329e334 Weber number (We), 209

Index

Weber’s law of similarity, 209 Weighted residual expression, 301 Weirs, 82, 83f, 230, 230f flow computation formulae, 231t Wetted perimeter, length of, 143 Wheel, power of, 92 Wind tunnel, 211

Wing, 193e198 angle of attack, 193 cavitation, 198e200 characteristic curves, 193e198, 194fe195f flow around stalled, 196f liftedrag polar, 196 Wire strain gauge, 35e36

Y Yawmetre, 215e217

Z Zero lift angle, 194e195 Zero-equation model, 322

383