Fluid mechanics for engineers : A graduate textbook

Fluid Mechanics for Engineers

Meinhard T. Schobeiri

Fluid Mechanics for

Engineers

A Graduate Textbook

ABC

Prof.Dr.-Ing. Meinhard T. Schobeiri

Department of Mechanical Engineering

Texas A&M University

College Station TX, 77843-3123

USA

E-mail: [email protected]

ISBN 978-3-642-11593-6 e-ISBN 978-3-642-11594-3

DOI 10.1007/978-3-642-11594-3

Library of Congress Control Number: 2009943377

c 2010 Springer-Verlag Berlin Heidelberg

This work is subject to copyright. All rights are reserved, whether the whole or part of the mate￾rial is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,

broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Dupli￾cation of this publication or parts thereof is permitted only under the provisions of the German

Copyright Law of September 9, 1965, in its current version, and permission for use must always

be obtained from Springer. Violations are liable to prosecution under the German Copyright Law.

The use of general descriptive names, registered names, trademarks, etc. in this publication does

not imply, even in the absence of a specific statement, that such names are exempt from the relevant

protective laws and regulations and therefore free for general use.

Typesetting: Camera-ready by author, data conversion by Markus Richter, Heidelberg

Printed in acid-free paper

987654321

springer.com

Preface

The contents of this book covers the material required in the Fluid Mechanics

Graduate Core Course (MEEN-621) and in Advanced Fluid Mechanics, a Ph.D-level

elective course (MEEN-622), both of which I have been teaching at Texas A&M

University for the past two decades. While there are numerous undergraduate fluid

mechanics texts on the market for engineering students and instructors to choose

from, there are only limited texts that comprehensively address the particular needs

of graduate engineering fluid mechanics courses. To complement the lecture

materials, the instructors more often recommend several texts, each of which treats

special topics of fluid mechanics. This circumstance and the need to have a textbook

that covers the materials needed in the above courses gave the impetus to provide the

graduate engineering community with a coherent textbook that comprehensively

addresses their needs for an advanced fluid mechanics text. Although this text book

is primarily aimed at mechanical engineering students, it is equally suitable for

aerospace engineering, civil engineering, other engineering disciplines, and especially

those practicing professionals who perform CFD-simulation on a routine basis and

would like to know more about the underlying physics of the commercial codes they

use. Furthermore, it is suitable for self study, provided that the reader has a sufficient

knowledge of calculus and differential equations.

In the past, because of the lack of advanced computational capability, the subject

of fluid mechanics was artificially subdivided into inviscid, viscous (laminar,

turbulent), incompressible, compressible, subsonic, supersonic and hypersonic flows.

With today’s state of computation, there is no need for this subdivision. The motion

of a fluid is accurately described by the Navier-Stokes equations. These equations

require modeling of the relationship between the stress and deformation tensor for

linear and nonlinear fluids only. Efforts by many researchers around the globe are

aimed at directly solving the Navier-Stokes equations (DNS) without introducing the

Reynolds stress tensor, which is the result of an artificial decomposition of the

velocity field into a mean and fluctuating part. The use of DNS for engineering

applications seems to be out of reach because the computation time and resources

required to perform a DNS-calculation are excessive at this time. Considering this

constraining circumstance, engineers have to resort to Navier-Stokes solvers that are

based on Reynolds decomposition. It requires modeling of the transition process and

the Reynolds stress tensor to which three chapters of this book are dedicated.

The book is structured in such a way that all conservation laws, their derivatives

and related equations are written in coordinate invariant forms. This type of structure

enables the reader to use Cartesian, orthogonal curvilinear, or non-orthogonal body

fitted coordinate systems. The coordinate invariant equations are then decomposed

VI Preface

into components by utilizing the index notation of the corresponding coordinate

systems. The use of a coordinate invariant form is particularly essential in

understanding the underlying physics of the turbulence, its implementation into the

Navier-Stokes equations, and the necessary mathematical manipulations to arrive at

different correlations. The resulting correlations are the basis for the following

turbulence modeling. It is worth noting that in standard textbooks of turbulence, index

notations are used throughout with almost no explanation of how they were brought

about. This circumstance adds to the difficulty in understanding the nature of

turbulence by readers who are freshly exposed to the problematics of turbulence.

Introducing the coordinate invariant approach makes it easier for the reader to follow

step-by-step mathematical manipulations, arrive at the index notation and the

component decomposition. This, however, requires the knowledge of tensor analysis.

Chapter 2 gives a concise overview of the tensor analysis essential for describing the

conservation laws in coordinate invariant form, how to accomplish the index notation,

and the component decomposition into different coordinate systems.

Using the tensor analytical knowledge gained from Chapter 2, it is rigorously

applied to the following chapters. In Chapter 3, that deals with the kinematics of flow

motion, the Jacobian transformation describes in detail how a time dependent volume

integral is treated. In Chapter 4 and 5 conservation laws of fluid mechanics and

thermodynamics are treated in differential and integral forms. These chapters are the

basis for what follows in Chapters 7, 8, 9, 10 and 11 which exclusively deal with

viscous flows. Before discussing the latter, the special case of inviscid flows is

presented where the order of magnitude of a viscosity force compared with the

convective forces are neglected. The potential flow, a special case of inviscid flow

characterized by zero vorticity , exhibited a major topic in fluid mechanics

in pre-CFD era. In recent years, however, its relevance has been diminished. Despite

this fact, I presented it in this book for two reasons. (1) Despite its major short

comings to describe the flow pattern directly close to the surface, because it does not

satisfy the no-slip condition, it reflects a reasonably good picture of the flow outside

the boundary layer. (2) Combined with the boundary layer calculation procedure, it

helps acquiring a reasonably accurate picture of the flow field outside and inside the

boundary layer. This, of course, is valid as long as the boundary layer is not

separated. For calculating the potential flows, conformal transformation is used where

the necessary basics are presented in Chapter 6, which is concluded by discussing

different vorticity theorems.

Particular issues of laminar flow at different pressure gradients associated with

the flow separation in conjunction with the wall curvature constitute the content of

Chapter 7 which seamlessly merges into Chapter 8 that starts with the stability of

laminar, followed by laminar-turbulent transition, intermittency function and its

implementation into Navier-Stokes. Averaging the Navier-Stokes equation that

includes the intermittency function leading to the Reynolds averaged Navier-Stokes

equation (RANS), concludes Chapter 8. In discussing the RANS-equations, two

quantities have to be accurately modeled. One is the intermittency function, and the

other is the Reynolds stress tensor with its nine components. Inaccurate modeling of

these two quantities leads to a multiplicative error of their product. The transition was

already discussed in Chapter 8 but the Reynolds stress tensor remains to be modeled.

Preface VII

This, however, requires the knowledge and understanding of turbulence before

attempts are made to model it. In Chapter 9, I tried to present the quintessence of

turbulence required for a graduate level mechanical engineering course and to

critically discuss several different models. While Chapter 9 predominantly deals with

the wall turbulence, Chapter 10 treats different aspects of free turbulent flows and

their general relevance in engineering. Among different free turbulent flows, the

process of development and decay of wakes under positive, zero, and negative

pressure gradients is of particular engineering relevance. With the aid of the

characteristics developed in Chapter 10, this process of wake development and decay

can be described accurately.

Chapter 11 is entirely dedicated to the physics of laminar, transitional and

turbulent boundary layers. This topic has been of particular relevance to the

engineering community. It is treated in integral and differential forms and applied to

laminar, transitional, turbulent boundary layers, and heat transfer.

Chapter 12 deals with the compressible flow. At first glance, this topic seems to

be dissonant with the rest of the book. Despite this, I decided to integrate it into this

book for two reasons: (1) Due to a complete change of the flow pattern from subsonic

to supersonic, associated with a system of oblique shocks makes it imperative to

present this topic in an advanced engineering fluid text; (2) Unsteady compressible

flow with moving shockwaves occurs frequently in many engines such as transonic

turbines and compressors, operating in off-design and even design conditions. A

simple example is the shock tube, where the shock front hits the one end of the tube

to be reflected to the other end. A set of steady state conservation laws does not

describe this unsteady phenomenon. An entire set of unsteady differential equations

must be called upon which is presented in Chapter 12. Arriving at this point, the

students need to know the basics of gas dynamics. I had two options, either refer the

reader to existing gas dynamics textbooks, or present a concise account of what is

most essential in following this chapter. I decided on the second option.

At the end of each chapter, there is a section that entails problems and projects.

In selecting the problems, I carefully selected those from the book Fluid Mechanics

Problems and Solutions by Professor Spurk of Technische Universität Darmstadt

which I translated in 1997. This book contains a number of highly advanced problems

followed by very detailed solutions. I strongly recommend this book to those

instructors who are in charge of teaching graduate fluid mechanics as a source of

advanced problems. My sincere thanks go to Professor Spurk, my former Co-Advisor,

for giving me the permission . Besides the problems, a number of demanding projects

are presented that are aimed at getting the readers involved in solving CFD-type of

problems. In the course of teaching the advanced Fluid Mechanics course MEEN￾622, I insist that the students present the project solution in the form of a technical

paper in the format required by ASME Transactions, Journal of Fluid Engineering.

In typing several thousand equations, errors may occur. I tried hard to eliminate

typing, spelling and other errors, but I have no doubt that some remain to be found

by readers. In this case, I sincerely appreciate the reader notifying me of any mistakes

found; the electronic address is given below. I also welcome any comments or

suggestions regarding the improvement of future editions of the book.

VIII Preface

My sincere thanks are due to many fine individuals and institutions. First and

foremost, I would like to thank the faculty of the Technische Universität Darmstadt

from whom I received my entire engineering education. I finalized major chapters of

the manuscript during my sabbatical in Germany where I received the Alexander von

Humboldt Prize. I am indebted to the Alexander von Humboldt Foundation for this

Prize and the material support for my research sabbatical in Germany. My thanks are

extended to Professor Bernd Stoffel, Professor Ditmar Hennecke, and Dipl. Ing.

Bernd Matyschok for providing me with a very congenial working environment.

I am also indebted to TAMU administration for partially supporting my

sabbatical which helped me in finalizing the book. Special thanks are due to Mrs.

Mahalia Nix who helped me in cross-referencing the equations and figures and

rendered other editorial assistance.

Last, but not least, my special thanks go to my family, Susan and Wilfried for

their support throughout this endeavor.

M.T. Schobeiri

August 2009

College Station, Texas

[email protected]

Contents

1 Introduction ....................................... 1

1.1 Continuum Hypothesis .................................... 1

1.2 Molecular Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Flow Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3.1 Velocity Pattern: Laminar, Intermittent, Turbulent Flow ..... 4

1.3.2 Change of Density, Incompressible, Compressible Flow . . . . . . 8

1.3.3 Statistically Steady Flow, Unsteady Flow . . . . . . . . . . . . . . . . . 9

1.4 Shear-Deformation Behavior of Fluids . . . . . . . . . . . . . . . . . . . . . . . . 9

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Vector and Tensor Analysis, Applications to

Fluid Mechanics ................................... 11

2.1 Tensors in Three-Dimensional Euclidean Space . . . . . . . . . . . . . . . . 11

2.1.1 Index Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Vector Operations: Scalar, Vector and Tensor Products . . . . . . . . . . 13

2.2.1 Scalar Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.2 Vector or Cross Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.3 Tensor Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3 Contraction of Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4 Differential Operators in Fluid Mechanics . . . . . . . . . . . . . . . . . . . . . 15

2.4.1 Substantial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4.2 Differential Operator / . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.5 Operator / Applied to Different Functions . . . . . . . . . . . . . . . . . . . . 19

2.5.1 Scalar Product of / and V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.5.2 Vector Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.5.3 Tensor Product of / and V . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.5.4 Scalar Product of / and a Second Order Tensor . . . . . . . . . . . 21

2.5.5 Eigenvalue and Eigenvector of a Second Order Tensor . . . . . . 25

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

X Contents

3 Kinematics of Fluid Motion ......................... 31

3.1 Material and Spatial Description of the Flow Field . . . . . . . . . . . . . . 31

3.1.1 Material Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.1.2 Jacobian Transformation Function and

Its Material Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.1.3 Velocity, Acceleration of Material Points . . . . . . . . . . . . . . . . 36

3.1.4 Spatial Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2 Translation, Deformation, Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3 Reynolds Transport Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.4 Pathline, Streamline, Streakline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4 Differential Balances in Fluid Mechanics .............. 51

4.1 Mass Flow Balance in Stationary Frame of Reference . . . . . . . . . . . . 51

4.1.1 Incompressibility Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2 Differential Momentum Balance in Stationary Frame of Reference . 53

4.2.1 Relationship between Stress Tensor and Deformation Tensor 56

4.2.2 Navier-Stokes Equation of Motion . . . . . . . . . . . . . . . . . . . . . . 58

4.2.3 Special Case: Euler Equation of Motion . . . . . . . . . . . . . . . . . 60

4.3 Some Discussions on Navier-Stokes Equations . . . . . . . . . . . . . . . . . 63

4.4 Energy Balance in Stationary Frame of Reference . . . . . . . . . . . . . . . 64

4.4.1 Mechanical Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.4.2 Thermal Energy Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.4.3 Total Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.4.4 Entropy Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.5 Differential Balances in Rotating Frame of Reference . . . . . . . . . . . . 72

4.5.1 Velocity and Acceleration in Rotating Frame . . . . . . . . . . . . . 72

4.5.2 Continuity Equation in Rotating Frame of Reference . . . . . . . 73

4.5.3 Equation of Motion in Rotating Frame of Reference . . . . . . . . 74

4.5.4 Energy Equation in Rotating Frame of Reference . . . . . . . . . . 76

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5 Integral Balances in Fluid Mechanics ..................... 81

5.1 Mass Flow Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.2 Balance of Linear Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.3 Balance of Moment of Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.4 Balance of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

Contents XI

5.4.1 Energy Balance Special Case 1: Steady Flow . . . . . . . . . . . . . 99

5.4.2 Energy Balance Special Case 2: Steady Flow,

Constant Mass Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.5 Application of Energy Balance to Engineering Components . . . . . . 100

5.5.1 Application: Pipe, Diffuser, Nozzle . . . . . . . . . . . . . . . . . . . 100

5.5.2 Application: Combustion Chamber . . . . . . . . . . . . . . . . . . . . 101

5.5.3 Application: Turbo-shafts, Energy Extraction, Consumption 102

5.5.3.1 Uncooled Turbine . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.5.3.2 Cooled Turbine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.5.3.3 Uncooled Compressor . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.6 Irreversibility, Entropy Increase, Total Pressure Loss . . . . . . . . . . . 106

5.6.1 Application of Second Law to Engineering Components . . . . . 107

5.7 Theory of Thermal Turbomachinery Stages . . . . . . . . . . . . . . . . . . . 110

5.7.1 Energy Transfer in Turbomachinery Stages . . . . . . . . . . . . . . 110

5.7.2 Energy Transfer in Relative Systems . . . . . . . . . . . . . . . . . . . 111

5.7.3 Unified Treatment of Turbine and Compressor Stages . . . . . 112

5.8 Dimensionless Stage Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.8.1 Simple Radial Equilibrium to Determine r . . . . . . . . . . . . . . 117

5.8.2 Effect of Degree of Reaction on the Stage Configuration . . . 121

5.8.3 Effect of Stage Load Coefficient on Stage Power . . . . . . . . . 121

5.9 Unified Description of a Turbomachinery Stage . . . . . . . . . . . . . . . 122

5.9.1 Unified Description of Stage with Constant Mean Diameter . 123

5.10 Turbine and Compressor Cascade Flow Forces . . . . . . . . . . . . . . . . 124

5.10.1 Blade Force in an Inviscid Flow Field . . . . . . . . . . . . . . . . . . 124

5.10.2 Blade Forces in a Viscous Flow Field . . . . . . . . . . . . . . . . . . 128

5.10.3 Effect of Solidity on Blade Profile Losses . . . . . . . . . . . . . . . 134

Problems, Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

6 Inviscid Flows .................................... 139

6.1 Incompressible Potential Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

6.2 Complex Potential for Plane Flows . . . . . . . . . . . . . . . . . . . . . . . . . 142

6.2.1 Elements of Potential Flow . . . . . . . . . . . . . . . . . . . . . . . . . . 145

6.2.1.1 Translational Flows . . . . . . . . . . . . . . . . . . . . . . . . . . 145

6.2.1.2 Sources and Sinks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

6.2.1.3 Potential Vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

6.2.1.4 Dipole Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

6.2.1.5 Corner Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

6.3 Superposition of Potential Flow Elements . . . . . . . . . . . . . . . . . . . . 150

XII Contents

6.3.1 Superposition of a Uniform Flow and a Source . . . . . . . . . . 150

6.3.2 Superposition of a Translational Flow and a Dipole . . . . . . . 151

6.3.3 Superposition of a Translational Flow, a Dipole and a Vortex 154

6.3.4 Superposition of a Uniform Flow, Source, and Sink . . . . . . . 159

6.3.5 Superposition of a Source and a Vortex . . . . . . . . . . . . . . . . 160

6.4 Blasius Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

6.5 Kutta-Joukowski Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

6.6 Conformal Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

6.6.1 Conformal Transformation, Basic Principles . . . . . . . . . . . . . 167

6.6.2 Kutta-Joukowsky Transformation . . . . . . . . . . . . . . . . . . . . . 169

6.6.3 Joukowsky Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 170

6.6.3.1 Circle-Flat Plate Transformation . . . . . . . . . . . . . . . . 171

6.6.3.2 Circle-Ellipse Transformation . . . . . . . . . . . . . . . . . . 172

6.6.3.3 Circle-Symmetric Airfoil Transformation . . . . . . . . . . 172

6.6.3.4 Circle-Cambered Airfoil Transformation . . . . . . . . . . 173

6.6.3.5 Circulation, Lift, Kutta Condition . . . . . . . . . . . . . . . . 175

6.7 Vortex Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

6.7.1 Thomson Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

6.7.2 Generation of Circulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

6.7.3 Helmholtz Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

6.7.4 Vortex Induced Velocity Field, Law of Bio -Savart . . . . . . . . 190

6.7.5 Induced Drag Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

7 Viscous Laminar Flow ............................ 201

7.1 Steady Viscous Flow through a Curved Channel . . . . . . . . . . . . . . . 201

7.1.1 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

7.1.2 Solution of the Navier-Stokes Equation . . . . . . . . . . . . . . . . . 205

7.1.3 Curved Channel, Negative Pressure Gradient . . . . . . . . . . . . 207

7.1.4 Curved Channel, Positive Pressure Gradient . . . . . . . . . . . . . 208

7.1.5 Radial Flow, Positive Pressure Gradient . . . . . . . . . . . . . . . . 209

7.2 Temperature Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

7.2.1 Solution of Energy Equation . . . . . . . . . . . . . . . . . . . . . . . . . 211

7.2.2 Curved Channel, Negative Pressure Gradient . . . . . . . . . . . . 213

7.2.3 Curved Channel, Positive Pressure Gradient . . . . . . . . . . . . . 213

7.2.4 Radial Flow, Positive Pressure Gradient . . . . . . . . . . . . . . . . 214

7.3 Steady Parallel Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

7.3.1 Couette Flow between Two Parallel Walls . . . . . . . . . . . . . . 216

Contents XIII

7.3.2 Couette Flow between Two Concentric Cylinders . . . . . . . . . 218

7.3.3 Hagen-Poiseuille Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

7.4 Unsteady Laminar Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

7.4.1 Flow Near Oscillating Flat Plate, Stokes-Rayleigh Problem . 223

7.4.2 Influence of Viscosity on Vortex Decay . . . . . . . . . . . . . . . . 226

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

8 Laminar-Turbulent Transition ......................... 233

8.1 Stability of Laminar Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

8.2 Laminar-Turbulent Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

8.3 Stability of Laminar Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

8.3.1 Stability of Small Disturbances . . . . . . . . . . . . . . . . . . . . . . . 237

8.3.2 The Orr-Sommerfeld Stability Equation . . . . . . . . . . . . . . . . 239

8.3.3 Orr-Sommerfeld Eigenvalue Problem . . . . . . . . . . . . . . . . . . 241

8.3.4 Solution of Orr-Sommerfeld Equation . . . . . . . . . . . . 243

8.3.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

8.4 Physics of an Intermittent Flow, Transition . . . . . . . . . . . . . . . . . . . . 247

8.4.1 Identification of Intermittent Behavior of Statistically

Steady Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

8.4.2 Turbulent/non-turbulent Decisions . . . . . . . . . . . . . . . . . . . . . 250

8.4.3 Intermittency Modeling for Steady Flow at Zero Pressure

Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

8.4.4 Identification of Intermittent Behavior of Periodic

Unsteady Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

8.4.5 Intermittency Modeling for Periodic Unsteady Flow . . . . . . 258

8.5 Implementation of Intermittency into Navier Stokes Equations . . . . 261

8.5.1 Reynolds-Averaged Equations for Fully Turbulent Flow . . . 261

8.5.2 Intermittency Implementation in RANS . . . . . . . . . . . . . . . . . 265

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

9 Turbulent Flow, Modeling .............................. 271

9.1 Fundamentals of Turbulent Flows . . . . . . . . . . . . . . . . . . . . . . . . . . 271

9.1.1 Type of Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

9.1.2 Correlations, Length and Time Scales . . . . . . . . . . . . . . . . . . 274

9.1.3 Spectral Representation of Turbulent Flows . . . . . . . . . . . . . 281

9.1.4 Spectral Tensor, Energy Spectral Function . . . . . . . . . . . . . . 284

9.2 Averaging Fundamental Equations of Turbulent Flow . . . . . . . . . . 286

XIV Contents

9.2.1 Averaging Conservation Equations . . . . . . . . . . . . . . . . . . . . 287

9.2.1.1 Averaging the Continuity Equation . . . . . . . . . . . . . . 287

9.2.1.2 Averaging the Navier-Stokes Equation . . . . . . . . . . . . 287

9.2.1.3 Averaging the Mechanical Energy Equation . . . . . . . 288

9.2.1.4 Averaging the Thermal Energy Equation . . . . . . . . . . 289

9.2.1.5 Averaging the Total Enthalpy Equation . . . . . . . . . . . 291

9.2.1.6 Quantities Resulting from Averaging to be Modeled . 294

9.2.2 Equation of Turbulence Kinetic Energy . . . . . . . . . . . . . . . . . 296

9.2.3 Equation of Dissipation of Kinetic Energy . . . . . . . . . . . . . . . 302

9.3 Turbulence Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

9.3.1 Algebraic Model: Prandtl Mixing Length Hypothesis . . . . . . 304

9.3.2 Algebraic Model: Cebeci-Smith Model . . . . . . . . . . . . . . . . . 310

9.3.3 Baldwin-Lomax Algebraic Model . . . . . . . . . . . . . . . . . . . . . 311

9.3.4 One- Equation Model by Prandtl . . . . . . . . . . . . . . . . . . . . . . 312

9.3.5 Two-Equation Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

9.3.5.1 Two-Equation k-g Model . . . . . . . . . . . . . . . . . . . . . . 313

9.3.5.2 Two-Equation k-ω-Model . . . . . . . . . . . . . . . . . . . . . . 315

9.3.5.3 Two-Equation k-ω-SST-Model . . . . . . . . . . . . . . . . . . 316

9.3.5.4 Two Examples of Two-Equation Models . . . . . . . . . . 318

9.4 Grid Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

Problems and Projects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

10 Free Turbulent Flow .............................. 327

10.1 Types of Free Turbulent Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

10.2 Fundamentals Equations of Free Turbulent Flows . . . . . . . . . . . . . . 328

10.3 Free Turbulent Flows at Zero-Pressure Gradient . . . . . . . . . . . . . . . 329

10.3.1 Plane Free Jet Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

10.3.2 Straight Wake at Zero Pressure Gradient . . . . . . . . . . . . . . . . 333

10.3.3 Free Jet Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338

10.4 Wake Flow at Non-zero Lateral Pressure Gradient . . . . . . . . . . . . . 340

10.4.1 Wake Flow in Engineering, Applications, General Remarks . 340

10.4.2 Theoretical Concept, an Inductive Approach . . . . . . . . . . . . . 344

10.4.3 Nondimensional Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 347

10.4.4 Near Wake, Far Wake Regions . . . . . . . . . . . . . . . . . . . . . . . 349

10.4.5 Utilizing the Wake Characteristics . . . . . . . . . . . . . . . . . . . . 350

Computational Projects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356

Contents XV

11 Boundary Layer Theory ................................ 357

11.1 Boundary Layer Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . 358

11.2 Exact Solutions of Laminar Boundary Layer Equations . . . . . . . . . 361

11.2.1 Laminar Boundary Layer, Flat Plate . . . . . . . . . . . . . . . . . . . 362

11.2.2 Wedge Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364

11.2.3 Polhausen Approximate Solution . . . . . . . . . . . . . . . . . . . . . . 368

11.3 Boundary Layer Theory Integral Method . . . . . . . . . . . . . . . . . . . . . 369

11.3.1 Boundary Layer Thicknesses . . . . . . . . . . . . . . . . . . . . . . . . . 369

11.3.2 Boundary Layer Integral Equation . . . . . . . . . . . . . . . . . . . . . 372

11.4 Turbulent Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375

11.4.1 Universal Wall Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 378

11.4.2 Velocity Defect Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 381

11.5 Boundary Layer, Differential Treatment . . . . . . . . . . . . . . . . . . . . . 386

11.5.1 Solution of Boundary Layer Equations . . . . . . . . . . . . . . . . . 390

11.6 Measurement of Boundary Flow, Basic Techniques . . . . . . . . . . . . 391

11.6.1 Experimental Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391

11.6.1.1 HWA Operation Modes, Calibration . . . . . . . . . . . . 391

11.6.1.2 HWA Averaging, Sampling Data . . . . . . . . . . . . . . 393

11.7 Examples: Calculations, Experiments . . . . . . . . . . . . . . . . . . . . . . . 394

11.7.1 Steady State Velocity Calculations . . . . . . . . . . . . . . . . . . . . 394

11.7.1.1 Experimental Verification . . . . . . . . . . . . . . . . . . . . . 396

11.7.1.2 Heat Transfer Calculation, Experiment . . . . . . . . . . . 397

11.7.2 Periodic Unsteady Inlet Flow Condition . . . . . . . . . . . . . . . . 398

11.7.2.1 Experimental Verification . . . . . . . . . . . . . . . . . . . . . 401

11.7.2.2 Heat Transfer Calculation, Experiment . . . . . . . . . . . 403

11.7.3 Application of ț-Ȧ Model to Boundary Layer . . . . . . . . . . . . 404

11.8 Parameters Affecting Boundary Layer . . . . . . . . . . . . . . . . . . . . . . 404

11.8.1 Parameter Variations, General Remarks . . . . . . . . . . . . . . . . 405

11.8.2 Effect of Periodic Unsteady Flow . . . . . . . . . . . . . . . . . . . . . . 409

Problems and Projects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418

12 Compressible Flow ................................ 423

12.1 Steady Compressible Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423

12.1.1 Speed of Sound, Mach Number . . . . . . . . . . . . . . . . . . . . . . . 423

12.1.2 Fluid Density, Mach Number, Critical State . . . . . . . . . . . . . 425

12.1.3 Effect of Cross-Section Change on Mach Number . . . . . . . . 430

12.1.3.1 Flow through Channels with Constant Area . . . . . . 437

12.1.3.2 The Normal Shock Wave Relations . . . . . . . . . . . . . 445