Cohen I.M., Kundu P.K. Fluid Mechanics
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x Contents
9. Flow past a Circular Cylinder without Circulation ................... 178
10. Flow past a Circular Cylinder with Circulation ...................... 180
11. Forces on a Two-Dimensional Body ................................ 184
12. Source near a Wall: Method of Images ............................. 189
13. Conformal Mapping .............................................. 190
14. Flow around an Elliptic Cylinder with Circulation ................... 192
15. Uniqueness of Irrotational Flows .................................. 194
16. Numerical Solution of Plane Irrotational Flow....................... 195
17. Axisymmetric Irrotational Flow ................................... 201
18. Streamfunction and Velocity Potential for Axisymmetric Flow ........ 203
19. Simple Examples of Axisymmetric Flows .......................... 205
20. Flow around a Streamlined Body of Revolution ..................... 206
21. Flow around an Arbitrary Body of Revolution ....................... 208
22. Concluding Remarks ............................................. 209
Exercises ........................................................ 209
Literature Cited .................................................. 212
Supplemental Reading ............................................ 212
Chapter 7
Gravity Waves
1. Introduction ..................................................... 214
2. The Wave Equation .............................................. 214
3. Wave Parameters ................................................. 216
4. Surface Gravity Waves ........................................... 219
5. Some Features of Surface Gravity Waves ........................... 223
6. Approximations for Deep and Shallow Water ....................... 229
7. Influence of Surface Tension ...................................... 234
8. Standing Waves .................................................. 237
9. Group Velocity and Energy Flux ................................... 238
10. Group Velocity and Wave Dispersion............................... 242
11. Nonlinear Steepening in a Nondispersive Medium ................... 246
12. Hydraulic Jump .................................................. 248
13. Finite Amplitude Waves of Unchanging Form in a
Dispersive Medium .............................................. 250
14. Stokes’ Drift..................................................... 253
15. Waves at a Density Interface between Infinitely Deep Fluids .......... 255
16. Waves in a Finite Layer Overlying an Infinitely Deep Fluid ........... 259
17. Shallow Layer Overlying an Infinitely Deep Fluid ................... 262
18. Equations of Motion for a Continuously Stratified Fluid .............. 263
19. Internal Waves in a Continuously Stratified Fluid .................... 267
20. Dispersion of Internal Waves in a Stratified Fluid .................... 270
21. Energy Considerations of Internal Waves in a Stratified Fluid ......... 272
Exercises ........................................................ 276
Literature Cited .................................................. 277
Contents xi
Chapter 8
Dynamic Similarity
1. Introduction ..................................................... 279
2. Nondimensional Parameters Determined from Differential
Equations ....................................................... 280
3. Dimensional Matrix .............................................. 284
4. Buckingham’s Pi Theorem ........................................ 285
5. Nondimensional Parameters and Dynamic Similarity ................ 287
6. Comments on Model Testing ...................................... 290
7. Significance of Common Nondimensional Parameters................ 292
Exercises ........................................................ 294
Literature Cited .................................................. 294
Supplemental Reading ............................................ 294
Chapter 9
Laminar Flow
1. Introduction ..................................................... 295
2. Analogy between Heat and Vorticity Diffusion ...................... 297
3. Pressure Change Due to Dynamic Effects ........................... 297
4. Steady Flow between Parallel Plates ............................... 298
5. Steady Flow in a Pipe ............................................ 302
6. Steady Flow between Concentric Cylinders ......................... 303
7. Impulsively Started Plate: Similarity Solutions ...................... 306
8. Diffusion of a Vortex Sheet........................................ 313
9. Decay of a Line Vortex ........................................... 315
10. Flow Due to an Oscillating Plate ................................... 317
11. High and Low Reynolds Number Flows ............................ 320
12. Creeping Flow around a Sphere.................................... 322
13. Nonuniformity of Stokes’ Solution and Oseen’s Improvement ........ 327
14. Hele-Shaw Flow ................................................. 332
15. Final Remarks ................................................... 334
Exercises ........................................................ 335
Literature Cited .................................................. 337
Supplemental Reading ............................................ 337
Chapter 10
Boundary Layers and Related Topics
1. Introduction ..................................................... 340
2. Boundary Layer Approximation ................................... 340
3. Different Measures of Boundary Layer Thickness ................... 346
4. Boundary Layer on a Flat Plate with a Sink at the Leading Edge:
Closed Form Solution ............................................ 348
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5. Boundary Layer on a Flat Plate: Blasius Solution ................... 352
6. von Karman Momentum Integral .................................. 362
7. Effect of Pressure Gradient ....................................... 364
8. Separation ...................................................... 366
9. Description of Flow past a Circular Cylinder ....................... 368
10. Description of Flow past a Sphere ................................. 375
11. Dynamics of Sports Balls ......................................... 376
12. Two-Dimensional Jets ........................................... 381
13. Secondary Flows ................................................ 388
14. Perturbation Techniques .......................................... 389
15. An Example of a Regular Perturbation Problem ..................... 394
16. An Example of a Singular Perturbation Problem .................... 396
17. Decay of a Laminar Shear Layer .................................. 401
Exercises ........................................................ 407
Literature Cited .................................................. 409
Supplemental Reading ............................................ 410
Chapter 11
Computational Fluid Dynamics
1. Introduction ..................................................... 411
2. Finite Difference Method ......................................... 413
3. Finite Element Method ........................................... 418
4. Incompressible Viscous Fluid Flow ................................ 426
5. Three Examples.................................................. 440
6. Concluding Remarks ............................................. 461
Exercises ........................................................ 463
Literature Cited .................................................. 464
Chapter 12
Instability
1. Introduction ..................................................... 467
2. Method of Normal Modes ......................................... 469
3. Thermal Instability: The B
´
enard Problem ........................... 470
4. Double-Diffusive Instability ....................................... 482
5. Centrifugal Instability: Taylor Problem ............................. 486
6. Kelvin–Helmholtz Instability ...................................... 493
7. Instability of Continuously Stratified Parallel Flows.................. 500
8. Squire’s Theorem and Orr–Sommerfeld Equation.................... 507
9. Inviscid Stability of Parallel Flows ................................. 510
10. Some Results of Parallel Viscous Flows ............................ 514
11. Experimental Verification of Boundary Layer Instability ............. 520
12. Comments on Nonlinear Effects ................................... 522
Contents xiii
13. Transition ....................................................... 523
14. Deterministic Chaos .............................................. 525
Exercises ........................................................ 533
Literature Cited .................................................. 535
Chapter 13
Turbulence
1. Introduction ..................................................... 537
2. Historical Notes.................................................. 539
3. Averages ........................................................ 541
4. Correlations and Spectra .......................................... 543
5. Averaged Equations of Motion .................................... 547
6. Kinetic Energy Budget of Mean Flow .............................. 554
7. Kinetic Energy Budget of Turbulent Flow .......................... 556
8. Turbulence Production and Cascade ................................ 559
9. Spectrum of Turbulence in Inertial Subrange ........................ 562
10. Wall-Free Shear Flow ............................................ 564
11. Wall-Bounded Shear Flow ........................................ 570
12. Eddy Viscosity and Mixing Length ................................ 580
13. Coherent Structures in a Wall Layer ................................ 584
14. Turbulence in a Stratified Medium ................................. 586
15. Taylor’s Theory of Turbulent Dispersion ............................ 591
16. Concluding Remarks ............................................. 598
Exercises ........................................................ 598
Literature Cited .................................................. 600
Supplemental Reading ............................................ 601
Chapter 14
Geophysical Fluid Dynamics
1. Introduction ..................................................... 603
2. Vertical Variation of Density in Atmosphere and Ocean .............. 605
3. Equations of Motion ............................................. 607
4. Approximate Equations for a Thin Layer on a Rotating Sphere ........ 610
5. Geostrophic Flow ............................................... 613
6. Ekman Layer at a Free Surface .................................... 617
7. Ekman Layer on a Rigid Surface .................................. 622
8. Shallow-Water Equations ......................................... 625
9. Normal Modes in a Continuously Stratified Layer ................... 628
10. High- and Low-Frequency Regimes in Shallow-Water
Equations ....................................................... 634
11. Gravity Waves with Rotation ...................................... 636
12. Kelvin Wave ..................................................... 639
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13. Potential Vorticity Conservation in Shallow-Water
Theory ......................................................... 644
14. Internal Waves .................................................. 647
15. Rossby Wave ................................................... 657
16. Barotropic Instability ............................................. 663
17. Baroclinic Instability ............................................. 665
18. Geostrophic Turbulence .......................................... 673
Exercises ........................................................ 676
Literature Cited .................................................. 677
Chapter 15
Aerodynamics
1. Introduction ..................................................... 679
2. The Aircraft and Its Controls ...................................... 680
3. Airfoil Geometry................................................. 683
4. Forces on an Airfoil .............................................. 684
5. Kutta Condition .................................................. 684
6. Generation of Circulation ......................................... 687
7. Conformal Transformation for Generating Airfoil Shape ............. 688
8. Lift of Zhukhovsky Airfoil ........................................ 692
9. Wing of Finite Span .............................................. 695
10. Lifting Line Theory of Prandtl and Lanchester ...................... 697
11. Results for Elliptic Circulation Distribution ......................... 701
12. Lift and Drag Characteristics of Airfoils ............................ 704
13. Propulsive Mechanisms of Fish and Birds .......................... 706
14. Sailing against the Wind .......................................... 708
Exercises ........................................................ 709
Literature Cited .................................................. 711
Supplemental Reading ............................................ 711
Chapter 16
Compressible Flow
1. Introduction ..................................................... 713
2. Speed of Sound .................................................. 717
3. Basic Equations for One-Dimensional Flow......................... 721
4. Stagnation and Sonic Properties ................................... 724
5. Area–Velocity Relations in One-Dimensional Isentropic Flow ........ 729
6. Normal Shock Wave.............................................. 733
7. Operation of Nozzles at Different Back Pressures.................... 741
8. Effects of Friction and Heating in Constant-Area Ducts .............. 747
9. Mach Cone ...................................................... 750
10. Oblique Shock Wave ............................................. 752
11. Expansion and Compression in Supersonic Flow .................... 756
Contents xv
12. Thin Airfoil Theory in Supersonic Flow ............................ 758
Exercises ........................................................ 761
Literature Cited .................................................. 763
Supplemental Reading ............................................ 763
Chapter 17
Introduction to Biofluid Mechanics
1. Introduction ..................................................... 765
2. The Circulatory System in the Human Body ........................ 766
3. Modelling of Flow in Blood Vessels ................................ 782
4. Introduction to the Fluid Mechanics of Plants ....................... 831
Exercises ........................................................ 837
Acknowledgment ................................................ 838
Literature Cited .................................................. 838
Appendix A
Some Properties of Common Fluids
A1. Useful Conversion Factors ........................................ 841
A2. Properties of Pure Water at Atmospheric Pressure ................... 842
A3. Properties of Dry Air at Atmospheric Pressure ...................... 842
A4. Properties of Standard Atmosphere ................................ 843
Appendix B
Curvilinear Coordinates
B1. Cylindrical Polar Coordinates ..................................... 845
B2. Plane Polar Coordinates .......................................... 847
B3. Spherical Polar Coordinates ....................................... 847
Appendix C
Founders of Modern Fluid Dynamics
Ludwig Prandtl (1875–1953) ...................................... 851
Geoffrey Ingram Taylor (1886–1975) .............................. 852
Supplemental Reading ............................................ 853
Appendix D
Visual Resources 855
Index 857
New in This Book
We are pleased to include a free copy of the DVD Multimedia Fluid Mechanics,
2/e, with this copy of Fluid Mechanics, Fourth Edition
.
You will find it in a
p
lastic
sleeve on the inside back cover of the book. If you are purchasing a used copy, be
aware that the DVD might have been removed by a previous owner.
Additional new features:
xvi
Inspired by the reception of the first edition, the objectives in Multimedia
Fluid
M
echanics
,
2/e, remain to exploit the moving image and interactivity of multi
–
media to improve the teaching and learning of fluid mechanics in all disciplines by
illustrating fundamental phenomena and conveying fascinating fluid flows fo
r
generations to come.
The completely new edition on the DVD includes the following:
• Twice the coverage with new modules on turbulence, control volumes,
interfacial phenomena, and similarity and scaling
• Four times the number of fluids videos, now more than 800
• Now more than 20 Virtual labs and simulations
• Dozens of new interactive demonstrations and animations
• Improved navigation via side bars that provide rapid overviews of
modules and guided browsing
• Media libraries for each chapter that give a snapshot of videos, each with
descriptive labels
• Ability to create movie playlists, which are invaluable in teaching
• Higher-resolution graphics, with full or part screen viewing options
• Operates on either a PC or a Mac OSX
Preface
Fluid mechanics has a vast scope and touches every aspect of our lives. Just look at
the contents of the 39 volumes of the Annual Review of Fluid Mechanics (1969–2007)
for validation of that statement. We cover only a tiny fraction of that scope in this
book.
This Fourth Edition continues to evolve due to the kindness of readers and users
who write to me suggesting corrections. Specifically, Roger Berlind of Columbia
University is responsible for the revisions to the Thermal Wind subsection of
Chapter 14. Howard Hu has revised, streamlined, and updated his chapter on Com-
putational Fluid Dynamics, and P. S. Ayyaswamy has contributed a new chapter (17)
on Introduction to Biofluid Mechanics. It is an excellently written contribution
and unique in that its level is appropriate for this book. It is between the advanced
treatises and overly simplified treatments available elsewhere. I have tried to update
much of the remaining material, particularly on turbulence, where so many new
papers appear each year. Reference to the collection of the National Committee for
Fluid Mechanics Films, now available for viewing via the Internet, is made in a new
Appendix D. These films may be old but remain an excellent resource for visualization
of flows.
On a more personal note, the bladder cancer (transitional cell carcinoma) diag-
nosed in the Fall of 2003, and attacked by a sequence of surgeries and a regimen of
chemotherapy, was never completely killed and grew back to visibility in the Fall
of 2006. The minimum visible spot on MRI can contain 70,000 TCC. A new and
harsher regimen of chemotherapy was prescribed through the Spring of 2007, during
the period when the updates were prepared. The irony of the fact that all the chemical
poisons infused into my veins are fluids is not lost on me. Because of the fatigue,
I accomplished less than I had hoped. Since radiologists cannot distinguish viable
living TCC from those that have been killed and remain in place, the only means of
discerning living cancer is to image again and see if there is new growth. The image
in early June confirmed the message that my body had already sent me: the cancer
was growing back and causing pain. Radiation was tried for a while to shrink the
painful tumor but that was unsuccessful. In late July a new regimen of chemotherapy
began, to last perhaps through the end of the year and beyond. The initial response
was positive and, although I have come to realize that my condition is incurable, I was
ever hopeful. After three infusions, however, the toxic effects of the chemotherapy
were more destructive than the aggressiveness of the cancer. I was left with no lung
capacity, no muscle strength, and was very ill and weak when page proofs came back
for checking. I remain so at the time of this writing.
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xviii Preface
I appreciate all the help provided by Ms. Susan Waddington in the preparation
of the final stages of this manuscript.
I am very grateful to my family, friends, and colleagues for their support through-
out this ordeal.
Ira M. Cohen
Preface to Third Edition
This edition provided me with the opportunity to include (almost) all of the additional
material I had intended for the Second Edition but had to sacrifice because of the
crush of time. It also provided me with an opportunity to rewrite and improve the
presentation of material on jets in Chapter 10. In addition, Professor Howard Hu
greatly expanded his CFD chapter. The expansion of the treatment of surface tension
is due to the urging of Professor E. F. “Charlie” Hasselbrink of the University of
Michigan.
I am grateful to Mr. Karthik Mukundakrishnan for computations of boundary
layer problems, to Mr. Andrew Perrin for numerous suggestions for improvement
and some computations, and to Mr. Din-Chih Hwang for sharing his latest results
on the decay of a laminar shear layer. The expertise of Ms. Maryeileen Banford in
preparing new figures was invaluable and is especially appreciated.
The page proofs of the text were read between my second and third surgeries
for stage 3 bladder cancer. The book is scheduled to be released in the middle of my
regimen of chemotherapy. My family, especially my wife Linda and two daughters
(both of whom are cancer survivors), have been immensely supportive during this
very difficult time. I am also very grateful for the comfort provided by my many
colleagues and friends.
Ira M. Cohen
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