Cohen I.M., Kundu P.K. Fluid Mechanics
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852 Appendix C: Founders of Modern Fluid Dynamics
He made notable innovations in the design of wind tunnels and other aerodynamic
equipment. His advocacy of monoplanes greatly advanced heavier-than-air aviation.
In experimental fluid mechanics he designed the Pitot-static tube for measuring veloc-
ity. In turbulence theory he contributed the mixing length theory.
Prandtl liked to describe himself as a plain mechanical engineer. So naturally he
was also interested in solid mechanics; for example, he devised a soap-film analogy
for analyzing the torsion stresses of structures with noncircular cross sections. In
this respect he was like G. I. Taylor, and his famous student von Karman; all three
of them did a considerable amount of work on solid mechanics. Toward the end of
his career Prandtl became interested in dynamic meteorology and published a paper
generalizing the Ekman spiral for turbulent flows.
Prandtl was endowed with rare vision for understanding physical phenomena. His
mastery of mathematical tricks was limited; indeed many of his collaborators were
better mathematicians. However, Prandtl had an unusual ability of putting ideas in
simple mathematical forms. In 1948, Prandtl published a simple and popular textbook
on fluid mechanics, which has been referred to in several places here. His varied
interest and simplicity of analysis is evident throughout this book. Prandtl died in
G
¨
ottingen 1953.
Geoffrey Ingram Taylor (1886−1975)
Geoffrey Ingram Taylor’s name almost always includes his initials G. I. in references,
and his associates and friends simply refer to him as “G. I.” He was born in 1886 in
London. He apparently inherited a bent toward mathematics from his mother, who was
the daughter of George Boole, the originator of “Boolean algebra.” After graduation
from the University of Cambridge, Taylor started to work with J. J. Thomson in pure
physics.
He soon gave up pure physics and changed his interest to mechanics of fluids
and solids. At this time a research position in dynamic meteorology was created at
Cambridge and it was awarded to Taylor, although he had no knowledge of meteo-
rology! At the age of 27 he was invited to serve as meteorologist on a British ship
that sailed to Newfoundland to investigate the sinking of the Titanic. He took the
opportunity to make measurements of velocity, temperature, and humidity profiles
up to 2000 m by flying kites and releasing balloons from the ship. These were the very
first measurements on the turbulent transfers of momentum and heat in the frictional
layer of the atmosphere. This activity started his lifelong interest in turbulent flows.
During World War I he was commissioned as a meteorologist by the British
Air Force. He learned to fly and became interested in aeronautics. He made the first
measurements of the pressure distribution over a wing in full-scale flight. Involvement
in aeronautics led him to an analysis of the stress distribution in propeller shafts.
This work finally resulted in a fundamental advance in solid mechanics, the “Taylor
dislocation theory.”
Taylor had a extraordinarily long and productive research career (1909–1972).
The amount and versatility of his work can be illustrated by the size and range of
his Collected Works published in 1954: Volume I contains “Mechanics of Solids”
(41 papers, 593 pages); Volume II contains “Meteorology, Oceanography, and
Supplemental Reading 853
Turbulent Flow” (45 papers, 515 pages); Volume III contains “Aerodynamics and
the Mechanics of Projectiles and Explosions” (58 papers, 559 pages); and Volume IV
contains “Miscellaneous Papers on Mechanics of Fluids” (49 papers, 579 pages).
Perhaps G. I. Taylor is best known for his work on turbulence. When asked, however,
what gave him maximum satisfaction, Taylor singled out his work on the stability of
Couette flow.
Professor George Batchelor, who has encountered many great physicists at
Cambridge, described G. I. Taylor as one of the greatest physicists of the century. He
combined a remarkable capacity for analytical thought with physical insight by which
he knew “how things worked.” He loved to conduct simple experiments, not to gather
data to understand a phenomenon, but to demonstrate his theoretical calculations; in
most cases he already knew what the experiment would show. Professor Batchelor
has stated that Taylor was a thoroughly lovable man who did not suffer from the
maladjustment and self-concern that many of today’s institutional scientists seem to
suffer (because of pressure!), and this allowed his creative energy to be used to the
fullest extent.
He thought of himself as an amateur, and worked for pleasure alone. He did not
take up a regular faculty position at Cambridge, had no teaching responsibilities, and
did not visit another institution to pursue his research. He never had a secretary or
applied for a research grant; the only facility he needed was a one-room laboratory
and one technical assistant. He did not “keep up with the literature,” tended to take up
problems that were entirely new, and chose to work alone. Instead of mastering tensor
notation, electronics, or numerical computations, G. I. Taylor chose to do things his
own way, and did them better than anybody else.
Supplemental Reading
Batchelor, G. K. (1976). “Geoffrey Ingram Taylor, 1886 – 1975.” Biographical Memoirs of Fellows of the
Royal Society 22: 565–633.
Batchelor, G. K. (1986). “Geoffrey Ingram Taylor, 7 March 1886–27 June 1975.” Journal of Fluid
Mechanics 173: 1–14.
Oswatitsch, K. and K. Wieghardt (1987). “Ludwig Prandtl and his Kaiser-Wilhelm-Institute.” Annual
Review of Fluid Mechanics 19: 1–25.
Von Karman, T. (1954). Aerodynamics, New York: McGraw-Hill.
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Appendix D
Visual Resources
Following is a list of films, all but the first by the National Committee for Fluid
Mechanics Films (NCFMF), founded in 1961 by the late Ascher H. Shapiro, then
Professor of Mechanical Engineering at M.I.T. Descriptive text for the films was
published separately as described below.
The Fluid Dynamics of Drag, Parts I, II, III, IV (1960)
Text: Ascher H. Shapiro, Shape and Flow:The Fluid Dynamics of Drag,
Doubleday and Co., New York (1961).
Vorticity, Parts I, II (1961)
The text for this and all following films is: NCFMF, Illustrated Experiments in
Fluid Mechanics, MIT Press, Cambridge, MA (1972).
Deformation of Continuous Media (1963)
Flow Visualization (1963)
Pressure Fields and Fluid Acceleration (1963)
Surface Tension in Fluid Mechanics (1964)
Waves in Fluids (1964)
*Boundary Layer Control (1965)
Rheological Behavior of Fluids (1965)
Secondary Flow (1965)
Channel Flow of a Compressible Fluid (1967)
Low-Reynolds-Number Flows (1967)
Magnetohydrodynamics (1967)
Cavitation (1968)
Eulerian and Lagrangian Descriptions in Fluid Mechanics (1968)
Flow Instabilities (1968)
Fundamentals of Boundary Layers (1968)
Rarefied Gas Dynamics (1968)
Stratified Flow (1968)
Aerodynamic Generation of Sound (1969)
855
©2010 Elsevier Inc. All rights reserved.
DOI: 10.1016/B978-0-12-381399-2.500 1-2 6
856 Appendix D: Visual Resources
Rotating Flows (1969)
Turbulence (1969)
Although these films are decades old, they remain excellent visualizations of the
principles of fluid mechanics. All but the one marked “*” are available for viewing
on the MIT web site as follows: http://web.mit.edu/fluids/www/Shapiro/ncfmf.html
It would be a very good idea to view the film appropriate to the corresponding section
of the text.
Index
Ackeret, Jacob, 715, 758
Acoustic waves, 717
Adiabatic density gradient, 586, 605
Adiabatic process, 24, 737, 746
Adiabatic temperature gradient, 19, 605
Advection, 56
Advective derivative, 56
Aerodynamics
aircraft parts and controls, 680–683
airfoil forces, 684
airfoil geometry, 683
conformal transformation, 688–692
defined, 653
finite wing span, 695–697
gas, 679
generation of circulation, 687–688
incompressible, 679
Kutta condition, 684–686
lift and drag characteristics, 704–705
Prandtl and Lanchester lifting line
theory, 697–701
propulsive mechanisms of fish and birds,
706–708
sailing, 708–709
Zhukhovsky airfoil lift, 692–694
Air, physical properties of, 735
Aircraft, parts and controls, 680–683
Airfoil(s)
angle of attack/incidence, 683
camber line, 683
chord, 683
compression side, 684
conformal transformation, 688–692
drag, induced/vortex, 673–674, 697, 700
finite span, 695–697
forces, 684
geometry, 683
lift and drag characteristics, 704–705
stall, 694, 704
suction side, 684
supersonic flow, 758–761
thin airfoil theory, 688
Zhukhovsky airfoil lift, 692–695
Alston, T. M., 402, 405, 409
Alternating tensors, 36–37
Analytic function, 170
Anderson, John, D., Jr., 432, 465, 686,
688, 711
Angle of attack/incidence, 683, 698
Angular momentum principle/theorem, for
fixed volume, 98–100
Antisymmetric tensors, 40–41
Aorta
elasticity, 791–792
Arterioles, 779
resistance, 780–781
Aris, R., 51, 79, 101, 136
Ashley, H., 701, 711
Aspect ratio of wing, 681
Asymptotic expansion, 391–392
Atmosphere
properties of standard, 843
scale height of, 22
Attractors
aperiodic, 530
dissipative systems and, 527–528
fixed point, 527
limit cycle, 527
strange, 530–531
Autocorrelation function, 544
normalized, 545
of a stationary process, 544
Averages, 541–543
Axisymmetric irrotational flow, 201–202
Babuska–Brezzi stability condition, 438
Baroclinic flow, 147–148
Baroclinic instability, 665–673
Baroclinic/internal mode, 261
Barotropic flow, 118, 146, 147
Barotropic instability, 663–664
Barotropic/surface mode, 261–262, 633
Baseball dynamics, 380
Batchelor, G. K., 24, 105, 131, 133, 136, 163,
212, 298, 337, 410, 591, 600, 673,
711, 853
Bayly, B. J., 469, 515, 521, 528, 535
Becker, R., 740, 741, 763
Berg
´
e, P., 529, 531, 535
B
´
enard, H., 374
convection, 471
thermal instability, 470–479
Bender, C. M., 409
Bernoulli equation, 118–120
applications of, 122–123
energy, 121
one-dimensional, 723–724
steady flow, 119–120
857
858 Index
Bernoulli equation (continued)
unsteady irrotational flow, 121
β-plane model, 612
Bifurcation, 527
Biofluid mechanics
flow in blood vessels, 782–831
human circulatory system, 766–781
plants, 831–838
Biot and Savart, law of, 151
Bird, R. B., 294
Birds, flight of, 707–708
Blasius solution, boundary layer, 352–361
Blasius theorem, 184–185
Blocking, in stratified flow, 270
Blood
composition, 773–775
coronary circulation, 768
Fahraeus-Lindqvist effect, 775–776,
777–779, 785–787
flow, 782
flow in vessels, modelling of, 782–831
plasma, 774
pulmonary circulation, 767–768,
829–831
systemic circulation, 780–781
total peripheral resistance, 780–785
viscosity, 774–776
Blood vessels
bifurcation, 807–812
Casson fluid flow in rigid tube, 826–829
composition of, 775
flow in, 782–783
flow in collapsible tube, 818–819
flow in rigid walled curved tube,
812–818
Hagen-Poiseuille flow, 783–791
nature, 779–782
pulsatile flow, 791–796
Body forces, 88
Body of revolution
flow around arbitrary, 208–209
flow around streamlined, 206–208
Bohlen T., 364, 409
Bond number, 289
Boundary conditions, 129–133, 669
geophysical fluids, 630
at infinity, 168
kinematic, 220
on solid surface, 168
Boundary layer
approximation, 340–346
Blasius solution, 352–361
breakdown of laminar solution, 360–361
closed form solution, 348–351
concept, 340
decay of laminar shear layer, 401–406
displacement thickness, 346–347
drag coefficient, 358
dynamics of sports balls, 376–381
effect of pressure gradient, 364–366,
517–518
Falkner–Skan solution, 358–360
flat plate and, 348–352
flow past a circular cylinder, 368–375
flow past a sphere, 375–376
instability, 520–522
Karman momentum integral, 362–364
momentum thickness, 347–348
perturbation techniques, 389–394
secondary flows, 388–389
separation, 366–368
simplification of equations, 319–324
skin friction coefficient, 357–358
technique, 2, 154
transition to turbulence, 367
two-dimensional jets, 381–388
u = 0.99U thickness, 346
Bound vortices, 697
Boussinesq approximation, 73, 86, 108–109
continuity equation and, 125–126
geophysical fluid and, 583–585
heat equation and, 127–128
momentum equation and, 126–127
Bradshaw, P., 588, 600
Brauer, H., 465
Breach, D. R., 337
Bridgeman, P. W., 294
Brooks, A. N., 429, 464
Brunt–V
¨
ais
¨
al
¨
a frequency, 265
Buckingham’s pi theorem, 285–287
Buffer layer, 575
Bulk strain rate, 60
Bulk viscosity, coefficient of, 102
Buoyancy frequency, 265, 606
Buoyant production, 558, 587
Bursting in turbulent flow, 585
Buschmann, M. H., 576, 600
Camber line, airfoil, 683
Cantwell, B. J., 584, 600
Capillarity, 9
Capillary number, 289
Capillary waves, 234, 237
Cardiac cycle, 768
net work done by ventricle on blood in
one, 772–773
Cardiac output, 773
Cardiovascular system (human), functions, 766
Carey G. F., 438, 465
Cascade, enstrophy, 674
Casson fluid, laminar flow in a rigid walled
tube, 826–829
Casson model, 776, 777
Casten R. G., 415
Index 859
Castillo, L., 576, 600–601
Cauchy–Riemann conditions, 167, 170
Cauchy’s equation of motion, 93
Cavitation, 834
Centrifugal force, effect of, 108–109
Centrifugal instability (Taylor), 486–493
Chandrasekhar, S., 137, 469, 490, 500,
533, 535
Chang G. Z., 464
Chaos, deterministic, 525–533
Characteristics, method of, 246
Chester, W., 331, 337
Chord, airfoil, 683
Chorin, A. J., 430, 436, 464
Chow C. Y., 688, 711
Circular Couette flow, 303
Circular cylinder
flow at various Re, 368–375
flow past, boundary layer, 368–375
flow past, with circulation, 180–184
flow past, without circulation, 178–179
Circular Poiseuille flow, 302–303
Circulation, 62–64
Kelvin’s theorem, 144–149
Clausius-Duhem inequality, 103
Cnoidal waves, 252, 253
Coefficient of bulk viscosity, 102
Cohen I. M., iii, xviii, 402, 405, 409, 741, 763
Coherent structures, wall layer, 584–586
Coles, D., 492, 535
Collapsible tubes
flow in, 818
one-dimensional steady flow in,
820–826
Starling resistor experiment, 819–820
Comma notation, 49, 152
Complex potential, 170
Complex variables, 169–171
Complex velocity, 171
Compressible flow
classification of, 715–716
friction and heating effects, 747–749
internal versus external, 714
Mach cone, 750–752
Mach number, 714–715
one-dimensional, 721–724, 730–733
shock waves, normal, 734–742
shock waves, oblique, 752–757
speed of sound, 717–720
stagnation and sonic properties, 724–730
supersonic, 756–758
Compressible medium, static equilibrium of,
18–19
potential temperature and density, 20–22
scale height of atmosphere, 22
Compression waves, 214
Computational fluid dynamics (CFD)
advantages of, 412–413
conclusions, 462
defined, 411
examples of, 440–463
finite difference method, 413–418
finite element method, 418–426
incompressible viscous fluid flow,
426–439
sources of error, 412
Concentric cylinders, laminar flow between,
303–306
Conformal mapping, 190–192
application to airfoil, 688–692
Conservation laws
Bernoulli equation, 118–124
boundary conditions, 129–134
Boussinesq approximation, 124–128
differential form, 82
integral form, 82
of mass, 84–86
mechanical energy equation, 111–115
of momentum, 92–93
Navier-Stokes equation, 104–105
rotating frame, 105–111
thermal energy equation, 115–116
time derivatives of volume integrals,
82–84
Conservative body forces, 88, 147
Consistency, 415–418
Constitutive equation, for Newtonian fluid,
100–104
Continuity equation, 73–75, 84, 86
Boussinesq approximation and, 125–126
one-dimensional, 722
Continuum hypothesis, 4–5
Control surfaces, 82
Control volume, 82
Convection, 57
-dominated problems, 427–429
forced, 589
free, 589
sloping, 673
Convergence, 415–418
Conversion factors, 841
Corcos G. M., 504, 536
Coriolis force, effect of, 109–111
Coriolis frequency, 611
Coriolis parameter, 611
Coronary arteries, 768
Coronary circulation, 766, 768
Correlations and spectra, 543–547
Couette flow
circular, 303
plane, 300, 516
Courant, R., 764
Cramer, M. S., 751, 763
Creeping flow, around a sphere, 322–327
860 Index
Creeping motions, 321
Cricket ball dynamics, 377–379
Critical layers, 512–514
Critical Re
blood flow, 783
Critical Re for transition
over circular cylinder, 373–374
over flat plate, 360
over sphere, 375–376
Cross-correlation function, 547
Cross product, vector, 38
Curl, vector, 40
Curtiss, C. F., 294
Curvilinear coordinates, 845–850
D’Alembert’s paradox, 179, 188
D’Alembert’s solution, 215
Davies, P., 526, 533, 535
Dead water phenomenon, 258
Dean number, 815–818
Decay of laminar shear layer, 401–406
Defect law, velocity, 573
Deflection angle, 754
Deformation
of fluid elements, 112–113
Rossby radius of, 643
Degree of freedom, 527
Delta wings, 705
Dennis, S. C. R., 464
Density
adiabatic density gradient, 586, 605
potential, 20–22
stagnation, 725
Derivatives
advective, 56
material, 56–57
particle, 56
substantial, 56
time derivatives of volume integrals,
82–84
Deviatoric stress tensor, 100
Diastole, 770, 771
Differential equations, nondimensional
parameters determined from,
280–284
Diffuser flow, 729–733
Diffusion of vorticity
from impulsively started plate, 306–312
from line vortex, 315–317
from vortex sheet, 313–315
Diffusivity
eddy, 580–584
effective, 597–598
heat, 297
momentum, 297
thermal, 116, 128
vorticity, 147, 313–315
Dimensional homogeneity, 284
Dimensional matrix, 284–285
Dipole. See Doublet
Dirichlet problem, 195
Discretization error, 412
of transport equation, 414–415
Dispersion
of particles, 591–595
relation, 223, 654–656, 660–664
Taylor’s theory, 591–598
Dispersive wave, 203, 221–225, 248–250
Displacement thickness, 340–341
Dissipation
of mean kinetic energy, 513
of temperature fluctuation, 545
of turbulent kinetic energy, 517
viscous, 112–113
Divergence
flux, 111–112
tensor, 39–40
theorem, 45, 85
vector, 39
Doormaal, J. P., 411, 413, 450
Doppler shift of frequency, 219
Dot product, vector, 37
Double-diffusive instability, 482–486
Doublet
in axisymmetric flow, 205
in plane flow, 174–175
Downwash, 698–699
Drag
characteristics for airfoils, 704–705
on circular cylinder, 374
coefficient, 288, 357–358
on flat plate, 357–358
force, 684
form, 367, 705
induced/vortex, 697, 700
pressure, 684, 705
profile, 705
skin friction, 357–358, 684, 705
on sphere, 375–376
wave, 290–291, 700, 760
Drazin, P. G., 469, 471, 481, 491, 498, 513,
515, 535
Dussan, V., E. B., 137
Dutton J. A., 581, 586–587, 600
Dynamic pressure, 123, 297–298
Dynamic similarity
nondimensional parameters and,
287–290
role of, 279–280
Dynamic viscosity, 7
Eddy diffusivity, 580–584
Eddy viscosity, 580–584
Index 861
Effective gravity force, 109
Eigenvalues and eigenvectors of symmetric
tensors, 41–44
Einstein summation convention, 28
Ekman layer
at free surface, 617–622
on rigid surface, 622–625
thickness, 619
Ekman number, 615
Ekman spiral, 619–620
Ekman transport at a free surface, 620
Elastic waves, 214, 717
Element point of view, 424–426
Elliptic circulation, 701–703
Elliptic cylinder, ideal flow, 192–194
Elliptic equation, 168
End diastolic volume (EDV), 772
End systolic volume (ESV), 773
Energy
baroclinic instability, 671–673
Bernoulli equation, 121–122
spectrum, 546
Energy equation
integral form, 82
mechanical, 111–115
one-dimensional, 721–723
thermal, 115–116
Energy flux
group velocity and, 238–241
in internal gravity wave, 272–276
in surface gravity wave, 229
Ensemble average, 541–542
Enstrophy, 673
Enstrophy cascade, 674
Enthalpy
defined, 14
stagnation, 724
Entrainment
in laminar jet, 381
turbulent, 566
Entropy
defined, 15
production, 116–118
Epsilon delta relation, 37, 105
Equations of motion
averaged, 547–554
Boussinesq, 126, 607–608
Cauchy’s, 93
for Newtonian fluid, 100–104
in rotating frame, 105–111
for stratified medium, 607–610
for thin layer on rotating sphere,
610–612
Equations of state, 13–14
for perfect gas, 16–18
Equilibrium range, 563
Equipartition of energy, 228
Equivalent depth, 627
Eriksen, C. C., 504, 535
Euler equation, 104, 118, 317
one-dimensional, 723–724
Euler momentum integral, 96–97, 188
Eulerian description, 55
Eulerian specifications, 54–55
Exchange of stabilities, principle of, 470
Expansion coefficient, thermal, 16–18
Fahraeus effect, 777–778
Fahraeus-Lindqvist (FL) effect, 775, 776–779
mathematical model, 785–787
Falkner, V. W., 358, 409
Falkner–Skan solution, 358–360
Feigenbaum, M. J., 531, 535
Fermi, E., 131, 137
Feynman, R. P., 595, 600
Fick’s law of mass diffusion, 6
Finite difference method, 413–418, 421–424
Finite element method
element point of view, 424–426
Galerkin’s approximation, 420–421
matrix equations, 421–424
weak or variational form, 418–420
First law of thermodynamics, 13
thermal energy equation and, 115–116
Fish, locomotion of, 706–708
Fixed point, 527
Fixed region, mechanical energy equation and,
114
Fixed volume, 83
angular momentum principle for,
98–100
momentum principle for, 93–97
Fjortoft, R., 673, 674, 677
Fjortoft’s theorem, 511–512
Flat plate, boundary layer and
Blasius solution, 352–361
closed form solution, 348–351
drag coefficient, 358
Fletcher, C. A. J., 426, 429, 464
Flow limitation, 819
Fluid mechanics, applications, 1–2
Fluid mechanics, visual resources, 855–856
Fluid statics, 9–12
Flux divergence, 111–112
Flux of vorticity, 64
Force field, 88
Force potential, 88
Forces
conservative body, 88, 147
Coriolis, 109–111
on a surface, 33–36
Forces in fluid
body, 88
line, 89