Boundary-Layer Theory (eBook)

Hermann Schlichting (1907 – 1982) was a German fluid mechanics scientist. He studied the mathematics, physics and applied mechanics at the University of Jena, Vienne and Göttingen and was promoted 1930. From 1931 to 1935 he worked at the Kaiser Wilhelm Institute for Flow Research in Göttingen. After a short period at Dornier in Friedrichshafen where he was responsible for the new wind tunnel he joined the Technische Universität Braunschweig in 1937 and became professor in 1938 at the age of 30. Herman Schlichting became an Emeritus Professor in 1975. Klaus Gersten is a German mathematician, engineer and expert in fluid mechanics. He studied mathematics and physics at the Technical University Braunschweig from 1949 to 1953. In 1957 he completed his doctorate under supervision of Hermann Schlichting. He became head of the Department of Theoretical Aerodynamics and he was deputy director of the Institute of Aerodynamics of the German Research Institute for Aviation ( DFL ) in Braunschweig . After his habilitation in 1960, he was appointed in 1964 to the University of Bochum where he in the Institute of Thermodynamics and Fluid Dynamics until his retirement.

Preface to the Ninth English Edition 5
Preface to the Eighth English Edition 6
Preface to the Ninth German Edition 7
Contents 9
Introduction 19
Abstract 24
Part I Fundamentals of Viscous Flows 26
1. Some Features of Viscous Flows 27
1.1 Real and Ideal Fluids 27
1.2 Viscosity 28
1.3 Reynolds Number 30
1.4 Laminar and Turbulent Flows 36
1.5 Asymptotic Behaviour at Large Reynolds Numbers 38
1.6 Comparison of Measurements Using the Inviscid Limiting Solution 38
1.7 Summary 50
2. Fundamentals of Boundary–Layer Theory 52
2.1 Boundary–Layer Concept 52
2.2 Laminar Boundary Layer on a Flat Plate at Zero Incidence 53
2.3 Turbulent Boundary Layer on a Flat Plate at Zero Incidence 56
2.4 Fully Developed Turbulent Flow in a Pipe 59
2.5 Boundary Layer on an Airfoil 61
2.6 Separation of the Boundary Layer 62
2.7 Overview of the Following Material 71
3. Field Equations for Flowsof Newtonian Fluids 73
3.1 Description of Flow Fields 73
3.2 Continuity Equation 74
3.3 Momentum Equation 74
3.4 General Stress State of Deformable Bodies 75
3.5 General State of Deformation of Flowing Fluids 79
3.6 Relation Between Stresses and Rate of Deformation 84
3.7 Stokes Hypothesis 87
3.8 Bulk Viscosity and Thermodynamic Pressure 88
3.9 Navier–Stokes Equations 90
3.10 Energy Equation 91
3.11 Equations of Motion for Arbitrary Coordinate Systems (Summary) 95
3.12 Equations of Motion for Cartesian Coordinates in Index Notation 98
3.13 Equations of Motion in Different Coordinate Systems 101
4. General Propertiesof the Equations of Motion 105
4.1 Similarity Laws 105
4.2 Similarity Laws for Flow with Buoyancy Forces (Mixed Forced and Natural Convection) 108
4.3 Similarity Laws for Natural Convection 112
4.4 Vorticity Transport Equation 113
4.5 Limit of Very Small Reynolds Numbers 115
4.6 Limit of Very Large Reynolds Numbers 116
4.7 Mathematical Example of the Limit Re?? 118
4.8 Non–Uniqueness of Solutions of the Navier–Stokes Equations 121
5. Exact Solutionsof the Navier–Stokes Equations 122
5.1 Steady Plane Flows 122
5.1.1 Couette–Poiseuille Flows 122
5.1.2 Jeffery–Hamel Flows (Fully Developed Nozzle and Diffuser Flows) 125
5.1.3 Plane Stagnation–Point Flow 131
5.1.4 Flow Past a Parabolic Body 136
5.1.5 Flow Past a Circular Cylinder 136
5.2 Steady Axisymmetric Flows 137
5.2.1 Circular Pipe Flow (Hagen–Poiseuille Flow) 137
5.2.2 Flow Between Two Concentric Rotating Cylinders 138
5.2.3 Axisymmetric Stagnation–Point Flow 139
5.2.4 Flow at a Rotating Disk 140
5.2.5 Axisymmetric Free Jet 145
5.3 Unsteady Plane Flows 147
5.3.1 Flow at a Wall Suddenly Set into Motion (First Stokes Problem) 147
5.3.2 Flow at an Oscillating Wall (Second Stokes Problem) 150
5.3.3 Start–up of Couette Flow 151
5.3.4 Unsteady Asymptotic Suction 152
5.3.5 Unsteady Plane Stagnation–Point Flow 152
5.3.6 Oscillating Channel Flow 158
5.4 Unsteady Axisymmetric Flows 160
5.4.1 Vortex Decay 160
5.4.2 Unsteady Pipe Flow 160
5.5 Summary 162
Part IILaminar Boundary Layers 164
6. Boundary–Layer Equations in Plane Flow Plate Boundary Layer
6.1 Setting up the Boundary–Layer Equations 165
6.2 Wall Friction, Separation and Displacement 170
6.3 Dimensional Representation of the Boundary–Layer Equations 172
6.4 Friction Drag 175
6.5 Plate Boundary Layer 176
7. General Properties and Exact Solutions of the Boundary–Layer Equationsfor Plane Flows 185
7.1 Compatibility Condition at the Wall 186
7.2 Similar Solutions of the Boundary–Layer Equations 187
7.2.1 Derivation of the Ordinary Differential Equation 187
7.2.2 Wedge Flows 192
7.2.3 Flow in a Convergent Channel 194
7.2.4 Mixing Layer 195
7.2.5 Moving Plate 196
7.2.6 Free Jet 197
7.2.7 Wall Jet 200
7.3 Coordinate Transformation 202
7.3.1 G¨ortler Transformation 202
7.3.2 v. Mises Transformation 203
7.3.3 Crocco Transformation 204
7.4 Series Expansion of the Solutions 204
7.4.1 Blasius Series 204
7.4.2 G¨ortler Series 206
7.5 Asymptotic Behaviour of Solutions Downstream 207
7.5.1 Wake Behind Bodies 207
7.5.2 Boundary Layer at a Moving Wall 210
7.6 Integral Relations of the Boundary Layer 211
7.6.1 Momentum–Integral Equation 211
7.6.2 Energy–Integral Equation 212
7.6.3 Moment–of–Momentum Integral Equations 214
8. Approximate Methods for Solving the Boundary–Layer Equationsfor Steady Plane Flows 215
8.1 Integral Methods 216
8.2 Stratford’s Separation Criterion 222
8.3 Comparison of the Approximate Solutions with Exact Solutions 222
8.3.1 Retarded Stagnation–Point Flow 222
8.3.2 Divergent Channel (Diffuser) 224
8.3.3 Circular Cylinder Flow 225
8.3.4 Symmetric Flow past a Joukowsky Airfoil 227
9. Thermal Boundary Layers without Coupling of the Velocity Fieldto the Temperature Field 229
9.1 Boundary–Layer Equations for the Temperature Field 229
9.2 Forced Convection for Constant Properties 231
9.3 Effect of the Prandtl Number 235
9.4 Similar Solutions of the Thermal Boundary Layer 238
9.5 Integral Methods for Computing the Heat Transfer 243
9.6 Effect of Dissipation Distribution of the Adiabatic Wall Temperature
10. Thermal Boundary Layers with Coupling of the Velocity Fieldto the Temperature Field 251
10.1 Remark 251
10.2 Boundary–Layer Equations 251
10.3 Boundary Layers with Moderate Wall Heat Transfer (Without Gravitational Effects) 253
10.3.1 Perturbation Calculation 253
10.3.2 Property Ratio Method (Temperature Ratio Method) 257
10.3.3 Reference Temperature Method 260
10.4 Compressible Boundary Layers (Without Gravitational Effects) 261
10.4.1 Physical Property Relations 261
10.4.2 Simple Solutions of the Energy Equation 264
10.4.3 Transformations of the Boundary–Layer Equations 266
10.4.4 Similar Solutions 269
10.4.5 Integral Methods 278
10.4.6 Boundary Layers in Hypersonic Flows 283
10.5 Natural Convection 285
10.5.1 Boundary–Layer Equations 285
10.5.2 Transformation of the Boundary–Layer Equations 290
10.5.3 Limit of Large Prandtl Numbers ( 291
10.5.4 Similar Solutions 293
10.5.5 General Solutions 297
10.5.6 Variable Physical Properties 298
10.5.7 Effect of Dissipation 300
10.6 Indirect Natural Convection 301
10.7 Mixed Convection 304
11. Boundary–Layer Control(Suction/Blowing) 311
11.1 Different Kinds of Boundary–Layer Control 311
11.2 Continuous Suction and Blowing 315
11.2.1 Fundamentals 315
11.2.2 Massive Suction (vw ???) 317
11.2.3 Massive Blowing (vw ? +?) 319
11.2.4 Similar Solutions 322
11.2.5 General Solutions 327
11.2.6 Natural Convection with Blowing and Suction 330
11.3 Binary Boundary Layers 331
11.3.1 Overview 331
11.3.2 Basic Equations 332
11.3.3 Analogy Between Heat and Mass Transfer 336
11.3.4 Similar Solutions 337
12. Axisymmetric and Three–DimensionalBoundary Layers 341
12.1 Axisymmetric Boundary Layers 341
12.1.1 Boundary–Layer Equations 341
12.1.2 Mangler Transformation 343
12.1.3 Boundary Layers on Non–Rotating Bodies of Revolution 344
12.1.4 Boundary Layers on Rotating Bodies of Revolution 347
12.1.5 Free Jets and Wakes 351
12.2 Three–Dimensional Boundary Layers 355
12.2.1 Boundary–Layer Equations 355
12.2.2 Boundary Layer at a Cylinder 361
12.2.3 Boundary Layer at a Yawing Cylinder 362
12.2.4 Three–Dimensional Stagnation Point 364
12.2.5 Boundary Layers in Symmetry Planes 365
12.2.6 General Configurations 365
13. Unsteady Boundary Layers 368
13.1 Fundamentals 368
13.1.1 Remark 368
13.1.2 Boundary–Layer Equations 369
13.1.3 Similar and Semi–Similar Solutions 370
13.1.4 Solutions for Small Times (High Frequencies) 371
13.1.5 Separation of Unsteady Boundary Layers 372
13.1.6 Integral Relations and Integral Methods 373
13.2 Unsteady Motion of Bodies in a Fluid at Rest 374
13.2.1 Start–Up Processes 374
13.2.2 Oscillation of Bodies in a Fluid at Rest 381
13.3 Unsteady Boundary Layers in a Steady Basic Flow 384
13.3.1 Periodic Outer Flow 384
13.3.2 Steady Flow with a Weak Periodic Perturbation 386
13.3.3 Transition Between Two Slightly Different Steady Boundary Layers 388
13.4 Compressible Unsteady Boundary Layers 389
13.4.1 Remark 389
13.4.2 Boundary Layer Behind a Moving Normal Shock Wave 390
13.4.3 Flat Plate at Zero Incidence with Variable Free Stream Velocity and Wall Temperature 392
14. Extensions to the Prandtl Boundary–LayerTheory 395
14.1 Remark 395
14.2 Higher Order Boundary–Layer Theory 397
14.3 Hypersonic Interaction 407
14.4 Triple–Deck Theory 410
14.5 Marginal Separation 421
14.6 Massive Separation 426
Part IIILaminar–Turbulent Transition 430
15. Onset of Turbulence (Stability Theory) 431
15.1 Some Experimental Results on the Laminar–Turbulent Transition 431
15.1.1 Transition in the Pipe Flow 431
15.1.2 Transition in the Boundary Layer 435
15.2 Fundamentals of Stability Theory 440
15.2.1 Remark 440
15.2.2 Fundamentals of Primary Stability Theory 441
15.2.3 Orr–Sommerfeld Equation 443
15.2.4 Curve of Neutral Stability and the Indifference Reynolds Number 450
15.2.4a Plate Boundary Layer 452
15.2.4b Effect of Pressure Gradient 461
15.2.4c Effect of Suction 473
15.2.4d Effect of Wall Heat Transfer 476
15.2.4e Effect of Compressibility 479
15.2.4f Effect of Wall Roughness 483
15.2.4g Further Effects 488
15.3 Instability of the Boundary Layer for Three–Dimensional Perturbations 489
15.3.1 Remark 489
15.3.2 Fundamentals of Secondary Stability Theory 492
15.3.3 Boundary Layers at Curved Walls 497
15.3.4 Boundary Layer at a Rotating Disk 501
15.3.5 Three–Dimensional Boundary Layers 503
15.4 Local Perturbations 509
Part IVTurbulent Boundary Layers 513
16. Fundamentals of Turbulent Flows 514
16.1 Remark 514
16.2 Mean Motion and Fluctuations 516
16.3 Basic Equations for the Mean Motion of Turbulent Flows 519
16.3.1 Continuity Equation 519
16.3.2 Momentum Equations (Reynolds Equations) 520
16.3.3 Equation for the Kinetic Energy of the Turbulent Fluctuations ( 522
Equation) 522
16.3.4 Thermal Energy Equation 525
16.4 Closure Problem 526
16.5 Description of the Turbulent Fluctuations 527
16.5.1 Correlations 527
16.5.2 Spectra and Eddies 528
16.5.3 Turbulence of the Outer Flow 530
16.5.4 Edges of Turbulent Regions and Intermittence 530
16.6 Boundary–Layer Equations for Plane Flows 531
17. Internal Flows 534
17.1 Couette Flow 534
17.1.1 Two–Layer Structure of the Velocity Field and the Logarithmic Overlap Law 534
17.1.2 Universal Laws of the Wall 539
17.1.3 Friction Law 551
17.1.4 Turbulence Models 553
17.1.5 Heat Transfer 556
17.2 Fully Developed Internal Flows (A = const) 558
17.2.1 Channel Flow 558
17.2.2 Couette–Poiseuille Flows 559
17.2.3 Pipe Flow 564
17.3 Slender–Channel Theory 569
18. Turbulent Boundary Layers without Coupling of the Velocity Fieldto the Temperature Field 572
18.1 Turbulence Models 572
18.1.1 Remark 572
18.1.2 Algebraic Turbulence Models 574
18.1.3 Turbulent Energy Equation 575
18.1.4 Two–Equation Models 577
18.1.5 Reynolds Stress Models 580
18.1.6 Heat Transfer Models 583
18.1.7 Low–Reynolds–Number Models 585
18.1.8 Large–Eddy Simulation and Direct Numerical Simulation 586
18.2 Attached Boundary Layers (?w = 0) 587
18.2.1 Layered Structure 587
18.2.2 Boundary–Layer Equations Using the Defect Formulation 589
18.2.3 Friction Law and Characterisitic Quantities of the Boundary Layer 592
18.2.4 Equilibrium Boundary Layers 595
18.2.5 Boundary Layer on a Plate at Zero Incidence 597
18.3 Boundary Layers with Separation 604
18.3.1 Stratford Flow 604
18.3.2 Quasi–Equilibrium Boundary Layers 606
18.4 Computation of Boundary Layers Using Integral Methods 609
18.4.1 Direct Method 609
18.4.2 Inverse Method 612
18.5 Computation of Boundary Layers Using Field Methods 613
18.5.1 Attached Boundary Layers (?w = 0) 613
18.5.2 Boundary Layers with Separation 616
18.5.3 Low–Reynolds–Number Turbulence Models 618
18.5.4 Additional Effects 619
18.6 Computation of Thermal Boundary Layers 622
18.6.1 Fundamentals 622
18.6.2 Computation of Thermal Boundary Layers Using Field Methods 624
19. Turbulent Boundary Layers with Coupling of the Velocity Fieldto the Temperature Field 626
19.1 Fundamental Equations 626
19.1.1 Time Averaging for Variable Density 626
19.1.2 Boundary–Layer Equations 628
19.2 Compressible Turbulent Boundary Layers 632
19.2.1 Temperature Field 632
19.2.2 Overlap Law 634
19.2.3 Skin–Friction Coefficient and Nusselt Number 636
19.2.4 Integral Methods for Adiabatic Walls 638
19.2.5 Field Methods 640
19.2.6 Shock–Boundary–Layer Interaction 640
19.3 Natural Convection 642
20. Axisymmetric and Three–DimensionalTurbulent Boundary Layers 645
20.1 Axisymmetric Boundary Layers 645
20.1.1 Boundary–Layer Equations 645
20.1.2 Boundary Layers without Body Rotation 646
20.1.3 Boundary Layers with Body Rotation 649
20.2 Three–Dimensional Boundary Layers 651
20.2.1 Boundary–Layer Equations 651
20.2.2 Computation Methods 655
20.2.3 Examples 657
21. Unsteady Turbulent Boundary Layers 658
21.1 Averaging and Boundary–Layer Equations 658
21.2 Computation Methods 661
21.3 Examples 662
22. Turbulent Free Shear Flows 665
22.1 Remark 665
22.2 Equations for Plane Free Shear Layers 667
22.3 Plane Free Jet 671
22.3.1 Global Balances 671
22.3.2 Far Field 672
22.3.3 Near Field 677
22.3.4 Wall Effects 677
22.4 Mixing Layer 679
22.5 Plane Wake 681
22.6 Axisymmetric Free Shear Flows 683
22.6.1 Basic Equations 683
22.6.2 Free Jet (U? = 0, = 8?(x ? x0)) 684
22.6.3 Wake (|UN| U?, = ?(x ? x0)1/3) 685
22.7 Buoyant Jets 687
22.7.1 Plane Buoyant Jet 687
22.7.2 Axisymmetric Buoyant Jet 688
22.8 Plane Wall Jet 689
Part V Numerical Methods in Boundary–LayerTheory 693
23. Numerical Integrationof the Boundary–Layer Equations 694
23.1 Laminar Boundary Layers 694
23.1.1 Remark 694
23.1.2 Note on Boundary–Layer Transformations 695
23.1.3 Explicit and Implicit Discretisation 696
23.1.4 Solution of the Implicit Difference Equations 700
23.1.5 Integration of the Continuity Equation 702
23.1.6 Boundary–Layer Edge and Wall Shear Stress 702
23.1.7 Integration of the Transformed Boundary–Layer Equations Using the Box Scheme 703
23.2 Turbulent Boundary Layers 706
23.2.1 Method of Wall Functions 706
23.2.2 Low–Reynolds–Number Turbulence Models 711
23.3 Unsteady Boundary Layers 712
23.4 Steady Three–Dimensional Boundary Layers 714
List of Frequently Used Symbols 719
Indices 725
Other Symbols 726
References and Index of Authors 727
Index 808