Egon Krause Fluid Mechanics
Egon Krause Fluid Mechanics With Problems and Solutions, and an Aerodynamic Laboratory With 607 Figures
Prof. Dr. Egon Krause RWTH Aachen Aerodynamisches Institut Wüllnerstr.5-7 52062 Aachen Germany ISBN 3-540-22981-7 Springer Berlin Heidelberg New York Library of Congress Control Number: 2004117071 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 other ways, 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-Verlag. Violations are liable to prosecution under German Copyright Law. SpringerisapartofSpringerScience+Business Media springeronline.com c Springer-Verlag Berlin Heidelberg 2005 Printed in Germany 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: Data conversion by the authors. Final processing by PTP-Berlin Protago-TEX-Production GmbH, Germany Cover-Design: design & production GmbH, Heidelberg Printed on acid-free paper 62/3020Yu -5 4 3 2 1 0
Preface During the past 40 years numerical and experimental methods of fluid mechanics were substantially improved. Nowadays time-dependent three-dimensional flows can be simulated on high-performance computers, and velocity and pressure distributions and aerodynamic forces and moments can be measured in modern wind tunnels for flight regimes, until recently not accessible for research investigations. Despite of this impressive development during the recent past and even 100 years after Prandtl introduced the boundary-layer theory, the fundamentals are still the starting point for the solution of flow problems. In the present book the important branches of fluid mechanics of incompressible and compressible media and the basic laws describing their characteristic flow behavior will be introduced. Applications of these laws will be discussed in a way suitable for engineering requirements. The book is divided into the six chapters: Fluid mechanics I and II, exercises in fluid mechanics, gas dynamics, exercises in gasdynamics, and aerodynamics laboratory. This arrangement follows the structure of the teaching material in the field, generally accepted and approved for a long time at German and foreign universities. In fluid mechanics I, after some introductory statements, incompressible fluid flow is described essentially with the aid of the momentum and the moment of momentum theorem. In fluid mechanics II the equations of motion of fluid mechanics, the Navier-Stokes equations, with some of their important asymptotic solutions are introduced. It is demonstrated, how flows can be classified with the aid of similarity parameters, and how specific problems can be identified, formulated and solved. In the chapter on gasdynamics the influence of variable density on the behavior of subsonic and supersonic flows is described. In the exercises on fluid mechanics I and II and on gasdynamics the material described in the previous chapters is elaborated in over 200 problems, with the solutions presented separately. It is demonstrated how the fundamental equations of fluid mechanics and gasdynamics can be simplified for the various problem formulations and how solutions can be constructed. Numerical methods are not employed. It is intended here, to describe the fundamental relationships in closed form as far as possible, in order to elucidate the intimate connection between the engineering formulation of fluid-mechanical problems and their solution with the methods of applied mathematics. In the selection of the problems it was also intended, to exhibit the many different forms of flows, observed in nature and technical applications. Because of the special importance of experiments in fluid mechanics, in the last chapter, aerodynamics laboratory, experimental techniques are introduced. It is not intended to give a comprehensive and complete description of experimental methods, but rather to explain with the description of experiments, how in wind tunnels and other test facilities experimental data can be obtained. A course under the same title has been taught for a long time at the Aerodymisches Institut of the RWTH Aachen. In the various lectures and exercises the functioning of low-speed and supersonic wind tunnels and the measuring techniques are explained in experiments, carried out in the facilities of the laboratory. The experiments comprise measurements of pressure distributions on a half body and a wing section, of the drag of a sphere in incompressible and compressible flow, of the aerodynmic forces and their moments acting on a wing section, of velocity profiles in a flat-plate boundary layer, and of losses in compressible pipe flow. Another important aspect of the laboratory course is to explain flow analogies, as for example the
VI Preface so-called water analogy, according to which a pressure disturbance in a pipe, filled with a compressible gas, propagates analogously to the pressure disturbance in supercritical shallow water flow. This book was stimulated by the friendly encouragement of Dr. M. Feuchte of B.G. Teubner- Verlag. My thanks go also to Dr. D. Merkle of the Springer-Verlag, who agreed to publish the English translation of the German text. Grateful acknowledgement is due to my successor Professor Dr.-Ing. W. Schröder, who provided personal and material support by the Aerodynamisches Institut in the preparation of the manuscript. I am indebted to Dr.-Ing. O. Thomer who was responsible for the preparatory work during the initial phase of the project until he left the institute. The final manuscript was prepared by cand.-ing. O. Yilmaz, whom I gratefully acknowledge. Dr.-Ing. M. Meinke offered valuable advice in the preparation of some of the diagrams. Aachen, July 2004 E. Krause
Table of Contents 1. Fluid Mechanics I 1 1.1 Introduction... 1 1.2 Hydrostatics... 2 1.2.1 Surface and Volume Forces... 2 1.2.2 Applications of the Hydrostatic Equation... 3 1.2.3 HydrostaticLift... 5 1.3 Hydrodynamics... 5 1.3.1 Kinematics of Fluid Flows... 5 1.3.2 Stream Tube and Filament... 7 1.3.3 Applications of Bernoulli s Equation... 8 1.4 One-Dimensional Unsteady Flow... 10 1.5 Momentum and Moment of Momentum Theorem... 12 1.5.1 Momentum Theorem... 12 1.5.2 Applications of the Momentum Theorem... 13 1.5.3 Flows in Open Channels... 17 1.5.4 Moment of Momentum Theorem... 18 1.5.5 Applications of the Moment of Momentum Theorem... 19 1.6 Parallel Flow of Viscous Fluids... 20 1.6.1 ViscosityLaws... 21 1.6.2 Plane Shear Flow with Pressure Gradient... 22 1.6.3 Laminar Pipe Flow... 24 1.7 Turbulent Pipe Flows... 25 1.7.1 Momentum Transport in Turbulent Flows... 26 1.7.2 Velocity Distribution and Resistance Law... 27 1.7.3 Pipes with Non-circular Cross Section... 29 2. Fluid Mechanics II 31 2.1 Introduction... 31 2.2 Fundamental Equations of Fluid Mechanics... 31 2.2.1 TheContinuityEquation... 31 2.2.2 TheNavier-StokesEquations... 32 2.2.3 The Energy Equation... 35 2.2.4 Different Forms of the Energy Equation... 36 2.3 Similar Flows... 38 2.3.1 Derivation of the Similarity Parameters with the Method of Dimensional Analysis... 38 2.3.2 The Method of Differential Equations... 40 2.3.3 Physical Meaning of the Similarity Parameters... 41 2.4 Creeping Motion... 42
VIII Table of Contents 2.5 Vortex Theorems... 44 2.5.1 Rotation and Circulation... 44 2.5.2 Vorticity Transport Equation... 45 2.6 Potential Flows of Incompressible Fluids... 46 2.6.1 Potential and Stream Function... 46 2.6.2 Determination of the Pressure... 48 2.6.3 The Complex Stream Function... 48 2.6.4 Examples for Plane Incompressible Potential Flows... 49 2.6.5 Kutta-Joukowski Theorem... 53 2.6.6 PlaneGravitationalWaves... 54 2.7 Laminar Boundary Layers... 55 2.7.1 Boundary-Layer Thickness and Friction Coefficient... 56 2.7.2 Boundary-Layer Equations... 56 2.7.3 The von KármánIntegralRelation... 58 2.7.4 Similar Solution for the Flat Plate at Zero Incidence... 59 2.8 Turbulent Boundary Layers... 61 2.8.1 Boundary-Layer Equations for Turbulent Flow... 61 2.8.2 Turbulent Boundary Layer on the Flat Plate at Zero Incidence... 62 2.9 Separation of the Boundary Layer... 64 2.10 Selected References... 66 2.11 Appendix... 66 3. Exercises in Fluid Mechanics 69 3.1 Problems... 69 3.1.1 Hydrostatics... 69 3.1.2 Hydrodynamics... 71 3.1.3 Momentum and Moment of Momentum Theorem... 76 3.1.4 Laminar Flow of Viscous Fluids... 80 3.1.5 PipeFlows... 83 3.1.6 Similar Flows... 86 3.1.7 Potential Flows of Incompressible Fluids... 88 3.1.8 Boundary Layers... 91 3.1.9 Drag... 92 3.2 Solutions... 96 3.2.1 Hydrostatics... 96 3.2.2 Hydrodynamics... 99 3.2.3 Momentum and Moment of Momentum Theorem... 106 3.2.4 Laminar Flow of Viscous Fluids... 113 3.2.5 PipeFlows... 119 3.2.6 Similar Flows... 123 3.2.7 Potential Flows of Incompressible Fluids... 126 3.2.8 Boundary Layers... 132 3.2.9 Drag... 134 4. Gasdynamics 139 4.1 Introduction... 139 4.2 Thermodynamic Relations... 139
Table of Contents IX 4.3 One-Dimensional Steady Gas Flow... 141 4.3.1 ConservationEquations... 141 4.3.2 TheSpeedofSound... 142 4.3.3 Integral of the Energy Equation... 143 4.3.4 Sonic Conditions... 144 4.3.5 The Limiting Velocity... 145 4.3.6 Stream Tube with Variable Cross-Section... 145 4.4 NormalCompressionShock... 147 4.4.1 The Jump Conditions... 147 4.4.2 IncreaseofEntropyAcrosstheNormalCompressionShock... 149 4.4.3 NormalShockinTransonicFlow... 150 4.5 Oblique Compression Shock... 151 4.5.1 Jump Conditions and Turning of the Flow... 151 4.5.2 WeakandStrongSolution... 153 4.5.3 Heart-CurveDiagramandHodographPlane... 154 4.5.4 WeakCompressionShocks... 155 4.6 ThePrandtl-MeyerFlow... 156 4.6.1 IsentropicChangeofVelocity... 157 4.6.2 CornerFlow... 158 4.6.3 InteractionBetweenShockWavesandExpansions... 159 4.7 LiftandWaveDraginSupersonicFlow... 160 4.7.1 TheWaveDrag... 161 4.7.2 LiftofaFlatPlateatAngleofAttack... 161 4.7.3 Thin Profiles at Angle of Attack... 161 4.8 Theory of Characteristics... 163 4.8.1 The Crocco Vorticity Theorem... 163 4.8.2 The Fundamental Equation of Gasdynamics... 164 4.8.3 Compatibility Conditions for Two-Dimensional Flows... 166 4.8.4 Computation of Supersonic Flows... 167 4.9 Compressible Potential Flows... 170 4.9.1 Simplification of the Potential Equation... 170 4.9.2 Determination of the Pressure Coefficient... 171 4.9.3 Plane Supersonic Flows About Slender Bodies... 172 4.9.4 Plane Subsonic Flow About Slender Bodies... 174 4.9.5 Flows about Slender Bodies of Revolution... 175 4.10 Similarity Rules... 178 4.10.1 Similarity Rules for Plane Flows After the Linearized Theory... 178 4.10.2 Application of the Similarity Rules to Plane Flows... 180 4.10.3 Similarity Rules for Axially Symmetric Flows... 182 4.10.4 Similarity Rules for Plane Transonic Flows... 183 4.11 Selected References... 184 5. Exercises in Gasdynamics 185 5.1 Problems... 185 5.1.1 One-Dimensional Steady Flows of Gases... 185 5.1.2 NormalCompressionShock... 188 5.1.3 Oblique Compression Shock... 191 5.1.4 ExpansionsandCompressionShocks... 193 5.1.5 Lift and Wave Drag Small-Perturbation Theory... 196
X Table of Contents 5.1.6 Theory of Characteristics... 198 5.1.7 Compressible Potential Flows and Similarity Rules... 199 5.2 Solutions... 203 5.2.1 One-Dimensional Steady Flows of Gases... 203 5.2.2 NormalCompressionShock... 208 5.2.3 Oblique Compression Shock... 211 5.2.4 ExpansionsandCompressionShocks... 214 5.2.5 Lift and Wave Drag Small-Perturbation Theory... 217 5.2.6 Theory of Characteristics... 219 5.2.7 Compressible Potential Flows and Similarity Rules... 221 5.3 Appendix... 225 6. Aerodynamics Laboratory 233 6.1 Wind Tunnel for Low Speeds (Göttingen-Type Wind Tunnel)... 233 6.1.1 Preliminary Remarks... 233 6.1.2 Wind Tunnels for Low Speeds... 234 6.1.3 Charakteristic Data of a Wind Tunnel... 235 6.1.4 Method of Test and Measuring Technique... 237 6.1.5 Evaluation... 243 6.2 Pressure Distribution on a Half Body... 247 6.2.1 Determination of the Contour and the Pressure Distribution... 247 6.2.2 Measurement of the Pressure... 248 6.2.3 The Hele-Shaw Flow... 249 6.2.4 Evaluation... 251 6.3 Sphere in Incompressible Flow... 253 6.3.1 Fundamentals... 253 6.3.2 Shift of the Critical Reynolds Number by Various Factors of Influence... 257 6.3.3 MethodofTest... 258 6.3.4 Evaluation... 259 6.4 Flat-Plate Boundary Layer... 262 6.4.1 Introductory Remarks... 262 6.4.2 MethodofTest... 263 6.4.3 Prediction Methods... 265 6.4.4 Evaluation... 267 6.4.5 Questions... 268 6.5 Pressure Distribution on a Wing... 271 6.5.1 Wing of Infinite Span... 271 6.5.2 Wing of Finite Span... 273 6.5.3 MethodofTest... 278 6.5.4 Evaluation... 280 6.6 Aerodynamic Forces Acting on a Wing... 283 6.6.1 Nomenclature of Profiles... 283 6.6.2 Measurement of Aerodynamic Forces... 283 6.6.3 Application of Measured Data to Full-Scale Configurations... 287 6.6.4 Evaluation... 295 6.7 Water Analogy Propagation of Surface Waves in Shallow Water andofpressurewavesingases... 299 6.7.1 Introduction... 299
Table of Contents XI 6.7.2 The Water Analogy of Compressible Flow... 299 6.7.3 The Experiment... 304 6.7.4 Evaluation... 305 6.8 Resistance and Losses in Compressible Pipe Flow... 307 6.8.1 Flow Resistance of a Pipe with Inserted Throttle (Orificee, Nozzle, Valve etc.)... 307 6.8.2 FrictionResistanceofaPipeWithoutaThrottle... 307 6.8.3 Resistance of an Orifice... 311 6.8.4 Evaluation... 313 6.8.5 Problems... 316 6.9 Measuring Methods for Compressible Flows... 320 6.9.1 Tabular Summary of Measuring Methods... 320 6.9.2 Optical Methods for Density Measurements... 320 6.9.3 OpticalSetup... 325 6.9.4 Measurements of Velocities and Turbulent Fluctuation Velocities... 326 6.9.5 Evaluation... 327 6.10 Supersonic Wind Tunnel and Compression Shock at the Wedge... 329 6.10.1 Introduction... 329 6.10.2 Classification of Wind Tunnels... 329 6.10.3 Elements of a Supersonic Tunnel... 332 6.10.4 The Oblique Compression Shock... 333 6.10.5 Description of the Experiment... 337 6.10.6Evaluation... 339 6.11 Sphere in Compressible Flow... 342 6.11.1 Introduction... 342 6.11.2 The Experiment... 342 6.11.3 Fundamentals of the Compressible Flow About a Sphere... 344 6.11.4Evaluation... 348 6.11.5Questions... 348 Index 351