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 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 and 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 which seamlessly merges into Chapter 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 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 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 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 MEEN622, 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 Introduction 1.1 1.2 1.3 Continuum Hypothesis Molecular Viscosity Flow Classification 1.3.1 Velocity Pattern: Laminar, Intermittent, Turbulent Flow 1.3.2 Change of Density, Incompressible, Compressible Flow 1.3.3 Statistically Steady Flow, Unsteady Flow 1.4 Shear-Deformation Behavior of Fluids References 10 Vector and Tensor Analysis, Applications to Fluid Mechanics 11 2.1 2.2 2.3 2.4 2.5 Tensors in Three-Dimensional Euclidean Space 2.1.1 Index Notation Vector Operations: Scalar, Vector and Tensor Products 2.2.1 Scalar Product 2.2.2 Vector or Cross Product 2.2.3 Tensor Product Contraction of Tensors Differential Operators in Fluid Mechanics 2.4.1 Substantial Derivatives 2.4.2 Differential Operator / Operator / Applied to Different Functions 2.5.1 Scalar Product of / and V 2.5.2 Vector Product 11 12 13 13 13 14 15 15 16 16 19 19 20 2.5.3 Tensor Product of / and V 2.5.4 Scalar Product of / and a Second Order Tensor 2.5.5 Eigenvalue and Eigenvector of a Second Order Tensor Problems References 21 21 25 27 29 X Contents Kinematics of Fluid Motion 31 3.1 Material and Spatial Description of the Flow Field 3.1.1 Material Description 3.1.2 Jacobian Transformation Function and Its Material Derivative 3.1.3 Velocity, Acceleration of Material Points 3.1.4 Spatial Description 3.2 Translation, Deformation, Rotation 3.3 Reynolds Transport Theorem 3.4 Pathline, Streamline, Streakline Problems References 31 31 32 36 37 38 42 44 46 49 Differential Balances in Fluid Mechanics 51 4.1 Mass Flow Balance in Stationary Frame of Reference 4.1.1 Incompressibility Condition 4.2 Differential Momentum Balance in Stationary Frame of Reference 4.2.1 Relationship between Stress Tensor and Deformation Tensor 4.2.2 Navier-Stokes Equation of Motion 4.2.3 Special Case: Euler Equation of Motion 4.3 Some Discussions on Navier-Stokes Equations 4.4 Energy Balance in Stationary Frame of Reference 4.4.1 Mechanical Energy 4.4.2 Thermal Energy Balance 4.4.3 Total Energy 4.4.4 Entropy Balance 4.5 Differential Balances in Rotating Frame of Reference 4.5.1 Velocity and Acceleration in Rotating Frame 4.5.2 Continuity Equation in Rotating Frame of Reference 4.5.3 Equation of Motion in Rotating Frame of Reference 4.5.4 Energy Equation in Rotating Frame of Reference Problems References 51 53 53 56 58 60 63 64 64 67 70 71 72 72 73 74 76 78 80 Integral Balances in Fluid Mechanics 81 5.1 5.2 5.3 5.4 Mass Flow Balance Balance of Linear Momentum Balance of Moment of Momentum Balance of Energy 81 83 88 94 A Tensor Operations in Orthogonal Curvilinear Coordinate Systems 485 (A.60) Introducing the non-zero Christoffel symbols into Eq (A.50), the components in gl, g2, and g directions are: (A.61) (A.63) (A.62) A.6.5 Introduction of Physical Components The physical components can be calculated from Eqs (A.21) and (A.24): (A.64) The Vi -components expressed in terms of V*i are: (A.65) Introducing Eqs.(A.65) into (A.61), (A.62), and (A.63) results in: (A.66) (A.67) (A.68) 486 A Tensor Operations in Orthogonal Curvilinear Coordinate Systems According to the definition: (A.69) the physical components of the velocity vectors are: (A.70) and insert these relations into Eqs (A.66) to (A.68), the resulting components in r, Ĭ, and z directions are: (A.71) A.7 Application Example 2: Viscous Flow Motion As the second application example, the Navier-Stokes equation of motion for a viscous incompressible flow is transferred into a cylindrical coordinate system, where it is decomposed in its three components r, ș, z The coordinate invariant version of the equation is written as: (A.72) The second term on the right hand side of Eq (A.72) exhibits the shear stress force It was treated in section A.5, Eq (A.39) and is the only term that has been added to the equation of motion for inviscid flow, Eq (A.40) A.7.1 Equation of Motion in Curvilinear Coordinate Systems The transformation and decomposition procedure is similar to the example in section A Therefore, a step by step derivation is not necessary (A.73) A Tensor Operations in Orthogonal Curvilinear Coordinate Systems 487 A.7.2 Special Case: Cylindrical Coordinate System Using the Christoffel symbols from section A.6.4 and the physical components from A.6.5, and inserting the corresponding relations these relations into Eqs (A.73), the resulting components in r, Ĭ, and z directions are: (A.74) (A.75) References Aris, R.: Vector, Tensors and the Basic Equations of Fluid Mechanics PrenticeHall, Englewood Cliffs (1962) Brand, L.: Vector and Tensor Analysis John Wiley and Sons, New York (1947) Klingbeil, E.: Tensorrechnung für Ingenieure Bibliographisches Institut, Mannheim (1966) Lagally, M.: Vorlesung über Vektorrechnung, 3rd edn Akademische Verlagsgesellschaft, Leipzig (1944) Vavra, M.H.: Aero-Thermodynamics and Flow in Turbomachines John Wiley & Sons, Chichester (1960) B Physical Properties of Dry Air Table B.1 Enthalpy h, specific heat at constant pressure cp, entropy s, viscosity ȝ and thermal conductivity ț as a function of temperature T pressure p = bar T h [C] [kJ/kg] 0.000 10.000 20.000 30.000 40.000 50.000 60.000 70.000 80.000 90.000 100.000 110.000 120.000 130.000 140.000 150.000 160.000 170.000 180.000 190.000 200.000 210.000 220.000 230.000 240.000 250.000 260.000 270.000 280.000 290.000 300.000 0.010 10.043 20.080 30.121 40.167 50.219 60.277 70.343 80.417 90.500 100.593 110.697 120.812 130.940 141.080 151.235 161.404 171.588 181.788 192.004 202.238 212.489 222.759 233.047 243.355 253.683 264.032 274.401 284.791 295.203 305.637 Cp [kJ/kg K] 1.003 1.003 1.004 1.004 1.005 1.005 1.006 1.007 1.008 1.009 1.010 1.011 1.012 1.013 1.015 1.016 1.018 1.019 1.021 1.022 1.024 1.026 1.028 1.030 1.032 1.034 1.036 1.038 1.040 1.042 1.044 s [kJ/kg K] 6.774 6.811 6.845 6.879 6.912 6.943 6.974 7.004 7.033 7.061 7.088 7.115 7.141 7.166 7.191 7.216 7.239 7.263 7.285 7.308 7.329 7.351 7.372 7.393 7.413 7.433 7.452 7.472 7.491 7.509 7.528 M.T Schobeiri: Fluid Mechanics for Engineers, pp 489–497 © Springer Berlin Heidelberg 2010 ȝ [kg/ms]106 17.294 17.744 18.190 18.632 19.069 19.503 19.933 20.359 20.781 21.199 21.613 22.024 22.431 22.834 23.234 23.630 24.023 24.412 24.798 25.180 25.559 25.935 26.308 26.677 27.043 27.407 27.767 28.124 28.478 28.829 29.177 ț [J/msK]103 24.210 24.893 25.571 26.243 26.910 27.572 28.229 28.880 29.527 30.169 30.806 31.439 32.067 32.690 33.309 33.924 34.534 35.140 35.742 36.340 36.934 37.524 38.110 38.692 39.271 39.846 40.417 40.985 41.549 42.110 42.667 490 B Physical Properties of Dry Air T h [C] [kJ/kg] 300.000 310.000 320.000 330.000 340.000 350.000 360.000 370.000 380.000 390.000 400.000 410.000 420.000 430.000 440.000 450.000 460.000 470.000 480.000 490.000 500.000 510.000 520.000 530.000 540.000 550.000 560.000 570.000 580.000 590.000 600.000 610.000 620.000 630.000 640.000 650.000 660.000 670.000 680.000 690.000 700.000 305.637 316.093 326.572 337.074 347.598 358.146 368.718 379.313 389.932 400.575 411.242 421.933 432.648 443.388 454.151 464.939 475.751 486.587 497.448 508.332 519.240 530.172 541.128 552.107 563.110 574.135 585.184 596.256 607.351 618.468 629.607 640.769 651.952 663.157 674.384 685.631 696.900 708.190 719.500 730.830 742.180 Cp [kJ/kg K] 1.044 1.047 1.049 1.051 1.054 1.056 1.058 1.061 1.063 1.065 1.068 1.070 1.073 1.075 1.078 1.080 1.082 1.085 1.087 1.090 1.092 1.094 1.097 1.099 1.101 1.104 1.106 1.108 1.111 1.113 1.115 1.117 1.119 1.122 1.124 1.126 1.128 1.130 1.132 1.134 1.136 s [kJ/kg K] 7.528 7.546 7.564 7.581 7.598 7.615 7.632 7.649 7.665 7.681 7.697 7.713 7.729 7.744 7.759 7.774 7.789 7.804 7.818 7.833 7.847 7.861 7.875 7.889 7.902 7.916 7.929 7.942 7.955 7.968 7.981 7.994 8.006 8.019 8.031 8.044 8.056 8.068 8.080 8.091 8.103 ȝ [kg/ms]106 29.177 29.523 29.865 30.205 30.542 30.877 31.209 31.538 31.864 32.188 32.510 32.829 33.145 33.459 33.771 34.081 34.388 34.693 34.995 35.296 35.594 35.890 36.184 36.476 36.766 37.054 37.340 37.624 37.907 38.187 38.465 38.742 39.017 39.290 39.561 39.831 40.099 40.365 40.630 40.893 41.155 ț [J/msK]103 42.667 43.221 43.772 44.320 44.865 45.406 45.945 46.481 47.013 47.543 48.070 48.595 49.116 49.635 50.151 50.665 51.177 51.685 52.192 52.696 53.197 53.697 54.194 54.688 55.181 55.671 56.160 56.646 57.130 57.612 58.092 58.570 59.046 59.521 59.993 60.464 60.932 61.399 61.864 62.327 62.789 B Physical Properties of Dry Air T h [C] [kJ/kg] 710.000 720.000 730.000 740.000 750.000 760.000 770.000 780.000 790.000 800.000 810.000 820.000 830.000 840.000 850.000 860.000 870.000 880.000 890.000 900.000 910.000 920.000 930.000 940.000 950.000 960.000 970.000 980.000 990.000 1.000.000 1.010.000 1.020.000 1.030.000 1.040.000 1.050.000 1.060.000 1.070.000 1.080.000 1.090.000 1.100.000 753.550 764.940 776.349 787.777 799.223 810.689 822.172 833.674 845.193 856.730 868.284 879.855 891.443 903.047 914.669 926.306 937.959 949.627 961.311 973.011 984.725 996.454 1.008.198 1.019.956 1.031.728 1.043.515 1.055.315 1.067.129 1.078.956 1.090.796 1.102.650 1.114.516 1.126.395 1.138.287 1.150.191 1.162.108 1.174.036 1.185.977 1.197.929 1.209.893 Cp [kJ/kg K] 1.138 1.140 1.142 1.144 1.146 1.147 1.149 1.151 1.153 1.155 1.156 1.158 1.160 1.161 1.163 1.165 1.166 1.168 1.169 1.171 1.172 1.174 1.175 1.177 1.178 1.179 1.181 1.182 1.183 1.185 1.186 1.187 1.189 1.190 1.191 1.192 1.193 1.195 1.196 1.197 s [kJ/kg K] 8.115 8.126 8.138 8.149 8.160 8.172 8.183 8.194 8.204 8.215 8.226 8.237 8.247 8.258 8.268 8.278 8.289 8.299 8.309 8.319 8.329 8.339 8.348 8.358 8.368 8.377 8.387 8.396 8.406 8.415 8.424 8.434 8.443 8.452 8.461 8.470 8.479 8.488 8.496 8.505 491 ȝ [kg/ms]106 41.415 41.673 41.930 42.186 42.440 42.692 42.944 43.193 43.442 43.689 43.935 44.180 44.423 44.665 44.906 45.146 45.384 45.621 45.857 46.093 46.326 46.559 46.791 47.022 47.251 47.480 47.708 47.934 48.160 48.385 48.609 48.832 49.054 49.275 49.495 49.714 49.932 50.150 50.367 50.583 ț [J/msK]103 63.249 63.707 64.163 64.618 65.071 65.522 65.972 66.420 66.866 67.311 67.754 68.196 68.636 69.075 69.511 69.947 70.381 70.813 71.243 71.672 72.100 72.526 72.950 73.373 73.794 74.213 74.631 75.047 75.462 75.875 76.286 76.696 77.104 77.511 77.915 78.318 78.719 79.119 79.516 79.912 492 B Physical Properties of Dry Air Enthalpy h , specific heat at constant pressure cp, entropy s, viscosity ȝ and thermal conductivity ț as a function of temperature T pressure p = 5.0 bar T h [C] [kJ/kg] 0.000 10.000 20.000 30.000 40.000 50.000 60.000 70.000 80.000 90.000 100.000 110.000 120.000 130.000 140.000 150.000 160.000 170.000 180.000 190.000 200.000 210.000 220.000 230.000 240.000 250.000 260.000 270.000 280.000 290.000 300.000 0.0100 10.043 20.080 30.121 40.167 50.219 60.277 70.343 80.417 90.500 100.593 110.697 120.812 130.940 141.080 151.235 161.404 171.588 181.788 192.004 202.238 212.489 222.759 233.048 243.356 253.684 264.032 274.401 284.791 295.203 305.637 Cp s [kJ/kg K] [kJ/kg K] 1,003 1.003 1.004 1.004 1.005 1.005 1.006 1.007 1.008 1.009 1.010 1.011 1.012 1.013 1.015 1.016 1.018 1.019 1.021 1.022 1.024 1.026 1.028 1.030 1.032 1.034 1.036 1.038 1.040 1.042 1.044 6.12 6.349 6.383 6.417 6.450 6.481 6.512 6.542 6.571 6.599 6.626 6.653 6.679 6.704 6.729 6.754 6.777 6.801 6.823 6.846 6.868 6.889 6.910 6.931 6.951 6.971 6.990 7.010 7.029 7.047 7.066 ȝ [kg/ms]106 17.294 17.744 18.190 18.632 19.069 19.503 19.933 20.359 20.781 21.199 21.613 22.024 22.431 22.834 23.234 23.630 24.023 24.412 24.798 25.180 25.559 25.935 26.308 26.677 27.043 27.407 27.767 28.124 28.478 28.829 29.177 ț [J/msK]103 24.210 24.893 25.571 26.243 26.910 27.572 28.229 28.880 29.527 30.169 30.806 31.439 32.067 32.690 33.309 33.924 34.534 35.140 35.742 36.340 36.934 37.524 38.110 38.692 39.271 39.846 40.417 40.985 41.549 42.110 42.667 B Physical Properties of Dry Air T h [C] [kJ/kg] 310.000 320.000 330.000 340.000 350.000 360.000 370.000 380.000 390.000 400.000 410.000 420.000 430.000 440.000 450.000 460.000 470.000 480.000 490.000 500.000 510.000 520.000 530.000 540.000 550.000 560.000 570.000 580.000 590.000 600.000 610.000 620.000 630.000 640.000 650.000 660.000 670.000 680.000 690.000 700.000 316.093 326.572 337.074 347.598 358.146 368.718 379.313 389.932 400.575 411.242 421.933 432.648 443.388 454.151 464.939 475.751 486.587 497.448 508.332 519.240 530.172 541.128 552.107 563.110 574.135 585.184 596.256 607.351 618.468 629.607 640.769 651.952 663.157 674.384 685.631 696.900 708.190 719.500 730.830 742.180 Cp [kJ/kg K] 1.047 1.049 1.051 1.054 1.056 1.058 1.061 1.063 1.065 1.068 1.070 1.073 1.075 1.078 1.080 1.082 1.085 1.087 1.090 1.092 1.094 1.097 1.099 1.101 1.104 1.106 1.108 1.111 1.113 1.115 1.117 1.119 1.122 1.124 1.126 1.128 1.130 1.132 1.134 1.136 s [kJ/kg K] 7.084 7.102 7.119 7.136 7.154 7.170 7.187 7.203 7.220 7.235 7.251 7.267 7.282 7.297 7.312 7.327 7.342 7.356 7.371 7.385 7.399 7.413 7.427 7.440 7.454 7.467 7.480 7.493 7.506 7.519 7.532 7.545 7.557 7.569 7.582 7.594 7.606 7.618 7.630 7.641 493 ȝ [kg/ms]106 29.523 29.865 30.205 30.542 30.877 31.209 31.538 31.864 32.188 32.510 32.829 33.145 33.459 33.771 34.081 34.388 34.693 34.995 35.296 35.594 35.890 36.184 36.476 36.766 37.054 37.340 37.624 37.907 38.187 38.465 38.742 39.017 39.290 39.561 39.831 40.099 40.365 40.630 40.893 41.155 ț [J/msK]103 43.221 43.772 44.320 44.865 45.406 45.945 46.481 47.013 47.543 48.070 48.595 49.116 49.635 50.151 50.665 51.177 51.685 52.192 52.696 53.197 53.697 54.194 54.688 55.181 55.671 56.160 56.646 57.130 57.612 58.092 58.570 59.046 59.521 59.993 60.464 60.932 61.399 61.864 62.327 62.789 494 B Physical Properties of Dry Air T h [C] [kJ/kg] 710.000 720.000 730.000 740.000 750.000 760.000 770.000 780.000 790.000 800.000 810.000 820.000 830.000 840.000 850.000 860.000 870.000 880.000 890.000 900.000 910.000 920.000 930.000 940.000 950.000 960.000 970.000 980.000 990.000 1.000.000 1.010.000 1.020.000 1.030.000 1.040.000 1.050.000 1.060.000 1.070.000 1.080.000 1.090.000 1.100.000 753.550 764.940 776.349 787.777 799.223 810.689 822.172 833.674 845.193 856.730 868.284 879.855 891.443 903.047 914.669 926.306 937.959 949.627 961.311 973.011 984.725 996.454 1.008.198 1.019.956 1.031.728 1.043.515 1.055.315 1.067.129 1.078.956 1.090.796 1.102.650 1.114.516 1.126.395 1.138.287 1.150.191 1.162.108 1.174.036 1.185.977 1.197.929 1.209.893 Cp [kJ/kg K] 1.138 1.140 1.142 1.144 1.146 1.147 1.149 1.151 1.153 1.155 1.156 1.158 1.160 1.161 1.163 1.165 1.166 1.168 1.169 1.171 1.172 1.174 1.175 1.177 1.178 1.179 1.181 1.182 1.183 1.185 1.186 1.187 1.189 1.190 1.191 1.192 1.193 1.195 1.196 1.197 s [kJ/kg K] 7.653 7.664 7.676 7.687 7.698 7.710 7.721 7.732 7.743 7.753 7.764 7.775 7.785 7.796 7.806 7.816 7.827 7.837 7.847 7.857 7.867 7.877 7.887 7.896 7.906 7.916 7.925 7.934 7.944 7.953 7.963 7.972 7.981 7.990 7.999 8.008 8.017 8.026 8.035 8.043 ȝ ț [kg/ms]106 [J/msK]103 41.415 41.673 41.930 42.186 42.440 42.692 42.944 43.193 43.442 43.689 43.935 44.180 44.423 44.665 44.906 45.146 45.384 45.621 45.857 46.093 46.326 46.559 46.791 47.022 47.251 47.480 47.708 47.934 48.160 48.385 48.609 48.832 49.054 49.275 49.495 49.714 49.932 50.150 50.367 50.583 63.249 63.707 64.163 64.618 65.071 65.522 65.972 66.420 66.866 67.311 67.754 68.196 68.636 69.075 69.511 69.947 70.381 70.813 71.243 71.672 72.100 72.526 72.950 73.373 73.794 74.213 74.631 75.047 75.462 75.875 76.286 76.696 77.104 77.511 77.915 78.318 78.719 79.119 79.516 79.912 B Physical Properties of Dry Air 495 Enthalpy h , specific heat at constant pressure cp, entropy s, viscosity ȝ and thermal conductivity ț as a function of temperature T pressure p = 10 bar T h [C] [kJ/kg] 0.000 10.000 20.000 30.000 40.000 50.000 60.000 70.000 80.000 90.000 100.000 110.000 120.000 130.000 140.000 150.000 160.000 170.000 180.000 190.000 200.000 210.000 220.000 230.000 240.000 250.000 260.000 270.000 280.000 290.000 300.000 0.010 10.043 20.080 30.121 40.167 50.219 60.277 70.343 80.417 90.500 100.593 110.697 120.812 130.940 141.080 151.235 161.404 171.588 181.788 192.004 202.238 212.489 222.759 233.047 243.355 253.683 264.032 274.401 284.791 295.203 305.637 Cp [kJ/kg K] 1.003 1.003 1.004 1.004 1.005 1.005 1.006 1.007 1.008 1.009 1.010 1.011 1.012 1.013 1.015 1.016 1.018 1.019 1.021 1.022 1.024 1.026 1.028 1.030 1.032 1.034 1.036 1.038 1.040 1.042 1.044 s [kJ/kg K] 6.114 6.150 6.184 6.218 6.251 6.282 6.313 6.343 6.372 6.400 6.427 6.454 6.480 6.506 6.530 6.555 6.578 6.602 6.624 6.647 6.669 6.690 6.711 6.732 6.752 6.772 6.792 6.811 6.830 6.848 6.867 ȝ [kg/ms]106 17.294 17.744 18.190 18.632 19.069 19.503 19.933 20.359 20.781 21.199 21.613 22.024 22.431 22.834 23.234 23.630 24.023 24.412 24.798 25.180 25.559 25.935 26.308 26.677 27.043 27.407 27.767 28.124 28.478 28.829 29.177 ț [J/msK]103 24.210 24.893 25.571 26.243 26.910 27.572 28.229 28.880 29.527 30.169 30.806 31.439 32.067 32.690 33.309 33.924 34.534 35.140 35.742 36.340 36.934 37.524 38.110 38.692 39.271 39.846 40.417 40.985 41.549 42.110 42.667 496 B Physical Properties of Dry Air T h [C] [kJ/kg] 310.000 320.000 330.000 340.000 350.000 360.000 370.000 380.000 390.000 400.000 410.000 420.000 430.000 440.000 450.000 460.000 470.000 480.000 490.000 500.000 510.000 520.000 530.000 540.000 550.000 560.000 570.000 580.000 590.000 600.000 610.000 620.000 630.000 640.000 650.000 660.000 670.000 680.000 690.000 700.000 316.093 326.572 337.074 347.598 358.146 368.718 379.313 389.932 400.575 411.242 421.933 432.648 443.388 454.151 464.939 475.751 486.587 497.448 508.332 519.240 530.172 541.128 552.107 563.109 574.135 585.184 596.256 607.350 618.468 629.607 640.768 651.952 663.157 674.383 685.631 696.900 708.190 719.500 730.830 742.180 Cp [kJ/kg K] 1.047 1.049 1.051 1.054 1.056 1.058 1.061 1.063 1.065 1.068 1.070 1.073 1.075 1.078 1.080 1.082 1.085 1.087 1.090 1.092 1.094 1.097 1.099 1.101 1.104 1.106 1.108 1.111 1.113 1.115 1.117 1.119 1.122 1.124 1.126 1.128 1.130 1.132 1.134 1.136 s [kJ/kg K] 6.885 6.903 6.920 6.938 6.955 6.971 6.988 7.004 7.021 7.037 7.052 7.068 7.083 7.098 7.113 7.128 7.143 7.158 7.172 7.186 7.200 7.214 7.228 7.241 7.255 7.268 7.281 7.295 7.307 7.320 7.333 7.346 7.358 7.370 7.383 7.395 7.407 7.419 7.431 7.442 ȝ [kg/ms]106 29.523 29.865 30.205 30.542 30.877 31.209 31.538 31.864 32.188 32.510 32.829 33.145 33.459 33.771 34.081 34.388 34.693 34.995 35.296 35.594 35.890 36.184 36.476 36.766 37.054 37.340 37.624 37.907 38.187 38.465 38.742 39.017 39.290 39.561 39.831 40.099 40.365 40.630 40.893 41.155 ț [J/msK]103 43.221 43.772 44.320 44.865 45.406 45.945 46.481 47.013 47.543 48.070 48.595 49.116 49.635 50.151 50.665 51.177 51.685 52.192 52.696 53.197 53.697 54.194 54.688 55.181 55.671 56.160 56.646 57.130 57.612 58.092 58.570 59.046 59.521 59.993 60.464 60.932 61.399 61.864 62.327 62.789 B Physical Properties of Dry Air T h [C] [kJ/kg] 710.000 720.000 730.000 740.000 750.000 760.000 770.000 780.000 790.000 800.000 810.000 820.000 830.000 840.000 850.000 860.000 870.000 880.000 890.000 900.000 910.000 920.000 930.000 940.000 950.000 960.000 970.000 980.000 990.000 1000.000 1010.000 1020.000 1030.000 1040.000 1050.000 1060.000 1070.000 1080.000 1090.000 1100.000 1.110.000 753.550 764.940 776.349 787.776 799.223 810.688 822.172 833.673 845.193 856.730 868.284 879.855 891.443 903.047 914.668 926.305 937.958 949.627 961.311 973.010 984.725 996.454 1008.198 1019.956 1031.728 1043.514 1055.315 1067.128 1078.955 1090.796 1102.650 1114.516 1126.396 1138.287 1150.191 1162.108 1174.037 1185.977 1197.929 1209.893 1.221.869 Cp [kJ/kg K] 1.138 1.140 1.142 1.144 1.146 1.147 1.149 1.151 1.153 1.155 1.156 1.158 1.160 1.161 1.163 1.165 1.166 1.168 1.169 1.171 1.172 1.174 1.175 1.177 1.178 1.179 1.181 1.182 1.183 1.185 1.186 1.187 1.189 1.190 1.191 1.192 1.193 1.195 1.196 1.197 1.198 s [kJ/kg K] 7.454 7.465 7.477 7.488 7.499 7.511 7.522 7.533 7.544 7.554 7.565 7.576 7.586 7.597 7.607 7.617 7.628 7.638 7.648 7.658 7.668 7.678 7.688 7.697 7.707 7.717 7.726 7.736 7.745 7.754 7.764 7.773 7.782 7.791 7.800 7.809 7.818 7.827 7.836 7.844 7.853 497 ȝ [kg/ms]106 41.415 41.673 41.930 42.186 42.440 42.692 42.944 43.193 43.442 43.689 43.935 44.180 44.423 44.665 44.906 45.146 45.384 45.621 45.857 46.093 46.326 46.559 46.791 47.022 47.251 47.480 47.708 47.934 48.160 48.385 48.609 48.832 49.054 49.275 49.495 49.714 49.932 50.150 50.367 50.583 50.798 ț [J/msK]103 63.249 63.707 64.163 64.618 65.071 65.522 65.972 66.420 66.866 67.311 67.754 68.196 68.636 69.075 69.511 69.947 70.381 70.813 71.243 71.672 72.100 72.526 72.950 73.373 73.794 74.213 74.631 75.047 75.462 75.875 76.286 76.696 77.104 77.510 77.915 78.318 78.719 79.119 79.516 79.912 80.306 Index Acceleration 36 Algebraic model 311 Baldwin-Lomax 311 Cebeci-Smith 310 Prandtl mixing length 304 Anemometer 248 Averaging 286 conservation equations 287 continuity equation 287 mechanical energy equation 288 Navier-Stokes equation 287 total enthalpy equation 291 Axial moment 91 Axial vector 41 Bernoulli equation 61, 310 Bingham fluids 10 Bio-Savart law, 193 Blade forces 130 drag 130 inviscid flow field 124 viscous flow 129 Blasius equation of laminar flow 363 Blending function 316 Buffer layer 201, 306 Boundary layer 369 displacement thickness, displacement 370 energy deficiency thickness 370 integral equation 373 length scale 407 logarithmic layer 306 momentum thickness 371 outer layer 307 re-attachment 404, 405, 407, 408 separation 404 viscous sublayer 306 similarity requirement 365 transitional flow 307 von Karman constant 309 Wake function 307 Wall influence 392 Boundary layer theory 357 Blasius 362, 363 concept of 357 laminar 361 viscous layer 357 Boussinesq relationship 303 Calmed region 258 Cascade process 272 Cauchy-Poisson law 57 Cauchy-Riemann equations, 143 Chebyshev polynomial 243 Christoffel symbols 202, 479 Circulation, 147, 226 Combustion chamber 102 Complex amplitude 240 Conformal transformation, 143, 167 basic principles, 167 Continuum hypothesis Contravariant components 202, 475 Convergent divergent 432 Convergent exit nozzle 428 Cooled turbine 104 Correlations 275 autocorrelation 275 coefficients 274 osculating parabola 279 single point 274 tensor 274 two-point correlation 274 Covariance 275 Critical Reynolds number 6, 233 500 Critical State 425 density 428 pressure 428 pressure ratio 428 Cross-Section change 430 Curved channel 201 negative pressure gradient 207 positive pressure gradient 208, 213 Curvilinear coordinate system 53 continuity balance in 53 Navier-Stokes equation in 59 Deformation 25, 38 Deformation tensor 3, 38, 56 Deformed state 35 Degree of reaction 119, 121 effect of 121 Derivatives material 16 substantial 16 temporal 16 Descriptions Euler, spatial 37 Lagrangian 32 material 31 Detached shock 454 Deterministic 237 Diabatic systems 100 Differential operator / 15, 16 Diffusion 301 Diffusivity 272 Direct Navier-Stokes Simulations 303 Dissipation function 65 Dissipation 290, 301, 302, 303 energy 271 equation 280 exact derivation of 303 kinetic energy 302 parameter 283 range 282 turbulence 280, 290 viscous 290 Eddy viscosity 304 Einstein summation convention 59 Index Einstein's summation 12 Energy cascade process 272, 273 Energy spectrum 281 dissipation range 282 large eddies 281 Energy spectral function 284 Energy extraction, consumption 102 Energy balance in stationary frame 64 dissipation function 65 mechanical energy 64 thermal energy 67 Entropy balance 71 Entropy increase 106 Equation of motion 344 Equation 296 turbulence kinetic energy 296 Euler turbine equation 114 Euler equation of motion 60 Falkner-Skan equation 366 Fanno process 437 Fluctuation kinetic energy 292 Fluids 1, Bingham 10 Newtonian fluids pseudoplastic 10 Frame indifference 56 Frame indifferent quantity 56 Frame of reference 51 Free turbulent flow 271, 327 characteristic quantities 332 free jet 327 free wakes 327 Gaussian function 332 momentum defects 330 velocity defect and wake width 327 velocity defect 329 Friction stress tensor 57 Gaussian distribution 259 Heat transfer Nusselt number 397 Stanton number 397 thermography 397 Index Helmholtz first theorem, 186 second theorem, 186 third theorem, 186 Holomorphic 143 Homogeneous gases liquids saturated superheated vapors unsaturated Hot wire anemometry 391 aliasing effect 393 analog/digital converter 394 constant current mode 391 constant-temperature mode 391 cross-wire 391 folding frequency 394 Nyquist-frequency 394 sample frequency 393 sampling rate 393 signal conditioner 394 single, cross and three-wire probes 391 single wire 391 three-wire 391 Hugoniot relation 446 Hypothesis frozen turbulence 277 G.I Taylor 277 Kolmogorov 272 mixing length hypothesis 306 Incompressible 8, 202, 203, 210, 229 Incompressibility condition 53 Index notation 12 Induced drag 195 Induced velocity 190 Integral balances balance of energy 94 balance of linear momentum 83 balance of moment of momentum 88 mass flow balance 81 Intermittency factor Intermittency function 390 Intermittency 6, 258 averaged 259 ensemble-averaged 259 function 390 maximum 259 minimum 259 Inviscid 4, 208, 226, 227 Inviscid flows, 139 Irreversibility 106 Irrotational flow 161, 140 Irrotational 227, 228 Isotropic turbulence 286 Isotropy 273 Jacobian functional determinant 35 transformation 32, 95 Joukowski airfoil 172 base profiles 172 lift equation 163, 165 transformation 169-171 theorem 157 Kinetic energy 282, 285, 286, 292 Kolmogorov 272 eddies 272 first hypothesis 282 hypothesis 272 inertial subrange 281, 282 length scale 281 scales 281 second hypothesis 283 time scale 281 universal equilibrium 272 velocity scale 281 Kronecker tensor 57 Kutta condition, 175 Kutta-Joukowski lift equation 163, 165 transformation 169-171 base profiles 172 theorem 157 Laminar flow 4, 201 Laminar flow stability 233 501 .. .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... exhibits a linear shear- deformation behavior are called Newtonian Fluids There are, however, many fluids which exhibit a nonlinear shear- deformation behavior Fig 1.11 shows qualitatively the behavior... Jacobian transformation describes in detail how a time dependent volume integral is treated In Chapter and conservation laws of fluid mechanics and thermodynamics are treated in differential and