Fluid Mechanics for Engineers

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: tschobeiri@mengr-tamu.org 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 material 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. Duplication 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 tschobeiri@mengr.tamu.edu

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 5.10.1Blade Force in an Inviscid Flow Field... 124 5.10.2Blade Forces in a Viscous Flow Field... 128 5.10.3Effect 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 10.3.1Plane Free Jet Flows... 333 10.3.2Straight 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 10.4.4Near 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 11.2.2Wedge Flows... 364 11.2.3Polhausen 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 11.4.1Universal Wall Functions... 378 11.4.2Velocity 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 11.6.1Experimental 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 11.7.1Steady 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 - 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 12.1.2Fluid Density, Mach Number, Critical State... 425 12.1.3Effect 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

XVI Contents 12.1.4 Supersonic Flow... 450 12.1.4.1 The Oblique Shock Wave Relations...451 12.1.4.2 Detached Shock Wave...454 12.1.4.3 Prandtl-Meyer Expansion...456 12.2 Unsteady Compressible Flow... 458 12.2.1One-dimensional Approximation... 459 12.3 Numerical Treatment... 466 12.3.1Unsteady Compressible Flow: Example: Shock Tube... 467 12.3.2Shock Tube Dynamic Behavior... 468 12.3.2.1 Pressure Transients...468 12.3.2.2 Temperature Transients...469 12.3.2.3 Mass Flow Transients...470 Problems and Projects... 471 References... 473 A Tensor Operations in Orthogonal Curvilinear Coordinate Systems... 475 A.1 Change of Coordinate System... 475 A.2 Co- and Contravariant Base Vectors, Metric Coefficients... 475 A.3 Physical Components of a Vector... 478 A.4 Derivatives of the Base Vectors, Christoffel Symbols... 479 A.5 Spatial Derivatives in Curvilinear Coordinate System... 480 A.5.1 Application of to Tensor Functions... 480 A.6 Application Example 1: Inviscid Incompressible Flow Motion... 482 A.6.1 Equation of Motion in Curvilinear Coordinate Systems... 482 A.6.2 Special Case: Cylindrical Coordinate System... 483 A.6.3 Base Vectors, Metric Coefficients... 483 A.6.4 Christoffel Symbols... 484 A.6.5 Introduction of Physical Components... 485 A.7 Application Example 2: Viscous Flow Motion... 486 A.7.1 Equation of Motion in Curvilinear Coordinate Systems... 486 A.7.2 Special Case: Cylindrical Coordinate System... 487 References... 487 B Physical Properties of Dry Air... 489 Index... 499

Nomenclature A acceleration vector b wake width c complex eigenfunction, c = c r + ic i c speed of sound c p, c v specific heat capacities C von Kármán constant C D drag coefficient C f friction coefficient C p pressure coefficient D deformation tensor D total differential operator in absolute frame of reference D van Driest s damping function D R total differential operator in relative frame of reference e specific total energy e i orthonormal unit vector E Source (+), sink (-) strength E total energy E(k) energy spectrum f S sampling frequency F force F(z) complex function g i, g i co-, contravariant base vectors in orthogonal coordinate system g ij, g ij co-, contravariant metric coefficients G i transformation vector h, H specific static, total enthalpy H 12 boundary layer momentum form factor, H 12 = 1 / 2 H 13 boundary layer energy form factor, H 32 = 3 / 2 q heat flux I(x,t) intermittency function I 1, I 2, I 3 principle invariants of deformation tensor J Jacobian transformation function k thermal conductivity k wave number vector K specific kinetic energy l m Prandtl mixing length turbulence length scale m mass

XVIII Nomenclature ṁ mass flow M Mach number M vector of moment of momentum M a axial vector of moment of momentum n normal unit vector N Navier-Stokes operator Nu Nusselt number p static pressure deterministic pressure fluctuation p + dimensionless pressure gradient random pressure fluctuation P, p o total (stagnation) pressure, P = p + V 2 /2 Pr Prandtl number Pr e effective Prandtl number Pr t turbulent Prandtl number q specific thermal energy Q thermal energy heat flux vector R radius in conformal transformation Re Reynolds number Re crit critical Re correlation second order tensor s specific entropy St Stanton number Str Strouhal number S, S(t) fixed, time dependent surface t time t tangential unit vector turbulence time scale T static temperature T stress tensor, T = e i e j ij T o stagnation or total temperature Tr trace of second order tensor T n (y) Chebyshev polynomial of first kind u specific internal energy u velocity u wall friction velocity u + dimensionless wall velocity, u + = u/u U undisturbed potential velocity U rotational velocity vector time averaged wake velocity defect time averaged wake momentum defect

Nomenclature XIX v V V 0 V(t) V V L V T w m W W x i y + z maximum velocity defect specific volume volume fixed volume time dependent volume absolute velocity vector velocity vector, laminar solution velocity vector, turbulent solution deterministic velocity fluctuation vector mean velocity vector random velocity fluctuating vector co- and contravariant component of a velocity vector ensemble averaged velocity vector specific shaft power mechanical energy mechanical energy flow (power) shaft power relative velocity vectors coordinates dimensionless wall distance, y + = u y/ complex variable Greek Symbols i r i jk min, max, 1, 2, 3, heat transfer coefficient real quantity in disturbance stream function disturbance amplification factor circular disturbance frequency time averaged intermittency factor, ensemble averaged intermittency at a fixed position ensemble averaged maximum intermittency at a fixed position ensemble averaged minimum intermittency at a fixed position circulation strength relative intermittency circulation vector Christoffel symbol minimum, maximum intermittency Kronecker delta boundary layer displacement, momentum, energy thickness

XX h m ijk o, W, Nomenclature turbulence dissipation eddy diffusivity eddy viscosity permutation symbol dimensionless periodic parameter Kolmogorov s length scale total pressure loss coefficient shock expansion angle one-dimensional spectral function isentropic exponent, ratio of specific heats von Kármán constant disturbance wave length eigenvalue Taylor micro length scale tangent unit vector absolute viscosity Mach angle expansion angle kinematic viscosity dimensionless coordinate, = x/l position vector in material coordinate system dimensionless coordinate, = y/l Kolmogorov s length scale pressure ratio stress tensor, = e i e j density dimensionless correlation coefficient Kolmogorov s time scale wall sear stress Kolmogorov s velocity scale dimensionless wake velocity defect dissipation function potential, stream function spectral tensor mass flow function complex function angular velocity vorticity vector Rotation tensor

Nomenclature XXI Subscripts, Superscripts freestream a, t axial, tangential ex exit in inlet max maximum min minimum s isentropic t turbulent w wall S time averaged random fluctuation deterministic fluctuation * dimensionless + wall functions