Fluid Mechanics

534 Pages·2007·8.7 MB·English

Fluid Mechanics
Second Edition
·
Joseph H. Spurk Nuri Aksel
Fluid Mechanics
Second Edition
123
ProfessorDr.JosephH.Spurk(em.) ProfessorDr.NuriAksel
TUDarmstadt UniversitätBayreuth
InstitutfürTechnischeStrömungslehre LehrstuhlfürTechnischeMechanik
Petersenstraße30 undStrömungsmechanik
64287Darmstadt Universitätsstraße30
Germany 95447Bayreuth
Germany
ISBN 978-3-540-73536-6 e-ISBN 978-3-540-73537-3
DOI10.1007/978-3-540-73537-3
LibraryofCongressControlNumber:2007939489
©2008,1997Springer-VerlagBerlinHeidelberg
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Preface to the First English Edition
ThistextbookisthetranslationofthefourtheditionofStrömungslehre, Ein-
führung in die Theorie der Strömungen. The German edition has met with
a favorable reception in German-speaking countries, showing that there was
a demand for a book that emphazises the fundamentals. In the English lit-
erature there are books of the same nature, some excellent, and these have
indeedinfluencedmetowritethisbook.However,theycoverdifferentground
and are not aimed primarily at mechanical engineering students, which this
book is. I have kept the originalconcept throughout all editions and there is
little tosaythathasnotbeensaidinthe prefacetothe firstGermanedition.
ThereisnowacompanionvolumeSolvedProblemsinFluidMechanics,which
alleviates the drawback of the first German edition, namely the absence of
problem exercises.
The book has been translated by Katherine Mayes during her stay in
Darmstadt, and I had the opportunity to work with her daily. It is for this
reason that I am solely responsible for this edition, too. My thanks also go
to Prof. L. Crane from Trinity College in Dublin for his assistance with this
book. Many people have helped, all of whom I cannot name, but I would
like to express my sincere thanks to Ralf Münzing, whose dependable and
unselfish attitude has been a constant encouragement during this work.
Darmstadt, January 1997 J. H. Spurk
Preface to the Second English Edition
ThefirstEnglisheditionwasthetranslationofthefourthGermanedition.In
the meantime the textbook has undergone several additions, mostly stimu-
latedbyconsultingactivitiesofthefirstauthor.Sincethetextbookcontinues
to receive favourable reception in German speaking countries and has been
translatedinotherlanguagesaswell,thepublishersuggestedasecondEnglish
edition. The additions were translated for the most part by Prof. L. Crane
from Trinity College in Dublin, who has accompanied this textbook from
the very beginning. Since the retirement of the first author, Prof. N. Aksel
from the University of Bayreuth, Germany, the second author, was actively
engaged in the sixth and the seventh edition. The additions were written by
the first author who accepts the responsibility for any mistakes or omissions
in this book.
Contents
1 The Concept of the Continuum and Kinematics .......... 1
1.1 Properties of Fluids, Continuum Hypothesis ............... 1
1.2 Kinematics ............................................ 7
1.2.1 Material and Spatial Descriptions .................. 7
1.2.2 Pathlines, Streamlines, Streaklines ................. 10
1.2.3 Differentiation with Respect to Time ............... 14
1.2.4 State of Motion, Rate of Change of Line, Surface
and Volume Elements ............................ 17
1.2.5 Rate of Change of Material Integrals ............... 29
2 Fundamental Laws of Continuum Mechanics.............. 35
2.1 Conservation of Mass, Equation of Continuity.............. 35
2.2 Balance of Momentum .................................. 37
2.3 Balance of Angular Momentum .......................... 44
2.4 Momentum and Angular Momentum
in an Accelerating Frame ................................ 46
2.5 Applications to Turbomachines........................... 54
2.6 Balance of Energy ...................................... 65
2.7 Balance of Entropy ..................................... 69
2.8 Thermodynamic Equations of State....................... 71
3 Constitutive Relations for Fluids ......................... 75
4 Equations of Motion for Particular Fluids ................ 95
4.1 Newtonian Fluids....................................... 95
4.1.1 The Navier-Stokes Equations ...................... 95
4.1.2 Vorticity Equation ............................... 98
4.1.3 Effect of Reynolds’ Number ....................... 100
4.2 Inviscid Fluids ......................................... 106
4.2.1 Euler’s Equations ................................ 106
4.2.2 Bernoulli’s Equation.............................. 107
4.2.3 Vortex Theorems ................................ 112
4.2.4 Integration of the Energy Equation................. 138
4.3 Initial and Boundary Conditions ......................... 141
4.4 Simplification of the Equations of Motion.................. 145
VIII Contents
5 Hydrostatics.............................................. 151
5.1 Hydrostatic Pressure Distribution ........................ 151
5.2 Hydrostatic Lift, Force on Walls.......................... 156
5.3 Free Surfaces .......................................... 162
6 Laminar Unidirectional Flows ............................ 167
6.1 Steady Unidirectional Flow .............................. 168
6.1.1 Couette Flow.................................... 168
6.1.2 Couette-Poiseuille Flow........................... 169
6.1.3 Flow Down an Inclined Plane...................... 171
6.1.4 Flow Between Rotating Concentric Cylinders........ 174
6.1.5 Hagen-Poiseuille Flow ............................ 175
6.1.6 Flow Through Noncircular Conduits................ 180
6.2 Unsteady Unidirectional Flows ........................... 183
6.2.1 Flow Due to a Wall Which Oscillates
in its Own Plane................................. 183
6.2.2 Flow Due to a Wall Which is Suddenly Set in Motion 186
6.3 Unidirectional Flows of Non-Newtonian Fluids ............. 188
6.3.1 Steady Flow Through a Circular Pipe .............. 188
6.3.2 Steady Flow Between a Rotating Disk
and a Fixed Wall ................................ 190
6.3.3 Unsteady Unidirectional Flows
of a Second Order Fluid .......................... 191
6.4 Unidirectional Flows of a Bingham Material ............... 197
6.4.1 Channel Flow of a Bingham Material............... 197
6.4.2 Pipe Flow of a Bingham Material .................. 202
7 Fundamentals of Turbulent Flow.......................... 205
7.1 Stability and the Onset of Turbulence..................... 205
7.2 Reynolds’ Equations .................................... 207
7.3 Turbulent Shear Flow Near a Wall........................ 213
7.4 Turbulent Flow in Smooth Pipes and Channels............. 223
7.5 Turbulent Flow in Rough Pipes .......................... 226
8 Hydrodynamic Lubrication ............................... 229
8.1 Reynolds’ Equation of Lubrication Theory................. 229
8.2 Statically Loaded Bearing ............................... 232
8.2.1 Infinitely Long Journal Bearing.................... 232
8.2.2 Infinitely Short Journal Bearing ................... 238
8.2.3 Journal Bearing of Finite Length................... 239
8.3 Dynamically Loaded Bearings............................ 240
8.3.1 Infinitely Long Journal Bearing.................... 240
8.3.2 Dynamically Loaded Slider Bearing ................ 241
8.3.3 Squeeze Flow of a Bingham Material ............... 246
8.4 Thin-Film Flow on a Semi-Infinite Wall ................... 249
Contents IX
8.5 Flow Through Particle Filters............................ 252
8.6 Flow Through a Porous Medium ......................... 254
8.7 Hele-Shaw Flows ....................................... 258
9 Stream Filament Theory.................................. 261
9.1 Incompressible Flow .................................... 261
9.1.1 Continuity Equation.............................. 262
9.1.2 Inviscid Flow.................................... 263
9.1.3 Viscous Flow .................................... 266
9.1.4 Application to Flows with Variable Cross-Section .... 271
9.1.5 Viscous Jet...................................... 276
9.2 Steady Compressible Flow............................... 279
9.2.1 Flow Through Pipes and Ducts
with Varying Cross-Section........................ 279
9.2.2 Constant Area Flow.............................. 290
9.2.3 The Normal Shock Wave Relations................. 294
9.3 Unsteady Compressible Flow............................. 299
10 Potential Flows........................................... 315
10.1 One-Dimensional Propagationof Sound ................... 316
10.2 Steady Compressible Potential Flow ...................... 323
10.3 Incompressible Potential Flow............................ 324
10.3.1 Simple Examples of Potential Flows ................ 326
10.3.2 Virtual Masses .................................. 348
10.4 Plane Potential Flow.................................... 354
10.4.1 Examples of Incompressible, Plane Potential Flows... 354
10.4.2 Complex Potential for Plane Flows................. 358
10.4.3 Blasius’ Theorem ................................ 367
10.4.4 Kutta-JoukowskiTheorem ........................ 370
10.4.5 Conformal Mapping .............................. 372
10.4.6 Schwarz-ChristoffelTransformation................. 374
10.4.7 Free Jets........................................ 376
10.4.8 Flow Around Airfoils ............................. 382
10.4.9 Approximate Solution for Slender Airfoils
in Incompressible Flow ........................... 388
10.4.10Slender Airfoils in Compressible Flow............... 395
11 Supersonic Flow .......................................... 399
11.1 Oblique Shock Wave .................................... 400
11.2 Detached Shock Wave................................... 402
11.3 Reflection of Oblique Shock Waves........................ 403
11.4 Supersonic Potential Flow Past Slender Airfoils ............ 405
11.5 Prandtl-Meyer Flow .................................... 408
11.6 Shock Expansion Theory ................................ 414
X Contents
12 Boundary Layer Theory .................................. 417
12.1 Solutions of the Boundary Layer Equations ................ 421
12.1.1 Flat Plate....................................... 422
12.1.2 Wedge Flows .................................... 426
12.1.3 Unsteady Stagnation Point Flow................... 428
12.1.4 Flow Past a Body................................ 429
12.2 Temperature Boundary Layer in Forced Convection......... 431
12.3 Temperature Boundary Layer in Natural Convection........ 437
12.4 Integral Methods of Boundary Layer Theory ............... 440
12.5 Turbulent Boundary Layers.............................. 443
13 Creeping Flows ........................................... 451
13.1 Plane and Axially-Symmetric Flows ...................... 451
13.1.1 Examples of Plane Flows.......................... 453
13.1.2 Plane Creeping Flow Round a Body
(Stokes’s Paradox) ............................... 465
13.1.3 Creeping Flow Round a Sphere .................... 465
A Introduction to Cartesian Tensors ........................ 471
A.1 Summation Convention ................................. 471
A.2 Cartesian Tensors ...................................... 472
B Curvilinear Coordinates .................................. 481
B.1 Cartesian Coordinates .................................. 488
B.2 Cylindrical Coordinates ................................. 490
B.3 Spherical Coordinates................................... 493
C Tables and Diagrams for Compressible Flow .............. 497
D Physical Properties of Air and Water..................... 515
References.................................................... 519
Index......................................................... 521
1 The Concept of the Continuum
and Kinematics
1.1 Properties of Fluids, Continuum Hypothesis
Fluid mechanics is concerned with the behavior of materials which deform
withoutlimitunderthe influenceofshearingforces.Evenaverysmallshear-
ing forcewilldeformafluidbody,but the velocityofthe deformationwillbe
correspondingly small. This property serves as the definition of a fluid: the
shearing forces necessary to deform a fluid body go to zero as the velocity
of deformation tends to zero. On the contrary, the behavior of a solid body
is such that the deformation itself, not the velocity of deformation, goes to
zero when the forces necessary to deform it tend to zero. To illustrate this
contrasting behavior, consider a material between two parallel plates and
adhering to them acted on by a shearing force F (Fig. 1.1).
IftheextentofthematerialinthedirectionnormaltotheplaneofFig.1.1
andinthe x-directionis muchlargerthanthatinthe y-direction,experience
shows that for many solids (Hooke’s solids), the force per unit area τ =
F/A is proportional to the displacement a and inversely proportional to the
distance between the plates h. At least one dimensional quantity typical for
the material must enter this relation, and here this is the shear modulus G.
The relationship
τ =Gγ (γ (cid:2)1) (1.1)
between the shearing angle γ = a/h and τ satisfies the definition of a solid:
the force per unit area τ tends to zero only when the deformation γ itself
Fig. 1.1. Shearing between two parallel plates
2 1 The Concept of theContinuum and Kinematics
goes to zero. Often the relation for a solid body is of a more general form,
e.g. τ =f(γ), with f(0)=0.
Ifthematerialisafluid,thedisplacementoftheplateincreasescontinually
withtimeunderaconstantshearingforce.Thismeansthereisnorelationship
between the displacement, or deformation, and the force. Experience shows
here that with many fluids the force is proportional to the rate of change of
the displacement, that is, to the velocity of the deformation.Again the force
is inversely proportional to the distance between the plates. (We assume
that the plate is being dragged at constant speed, so that the inertia of the
material does not come into play.) The dimensional quantity required is the
shear viscosity η, and the relationship with U =da/dt now reads:
U
τ =η =ηγ˙ , (1.2)
h
or, if the shear rate γ˙ is set equal to du/dy,
du
τ(y)=η . (1.3)
dy
τ(y) is the shear stress on a surface element parallelto the plates atpoint y.
In so-called simple shearing flow (rectilinear shearing flow) only the x-
component of the velocity is nonzero, and is a linear function of y.
The above relationship was known to Newton, and it is sometimes in-
correctly used as the definition of a Newtonian fluid: there are also non-
Newtonian fluids whichshowalinearrelationshipbetweenthe shearstressτ
andthe shearrateγ˙ inthis simplestateofstress.Ingeneral,therelationship
for a fluid reads τ =f(γ˙), with f(0)=0.
Whiletherearemanysubstancesforwhichthisclassificationcriterionsuf-
fices, there are some which show dual character. These include the glasslike
materials which do not have a crystal structure and are structurally liquids.
Underprolongedloadsthesesubstancesbegintoflow,thatistodeformwith-
out limit. Under short-term loads, they exhibit the behavior of a solid body.
Asphalt is an oftquoted example: you can walk on asphalt without leaving
footprints(short-termload),butifyouremainstandingonitforalongtime,
youwillfinallysinkin.Underveryshort-termloads,e.g.ablowwithaham-
mer, asphalt splinters, revealing its structural relationship to glass. Other
materials behave like solids even in the long-term, provided they are kept
below a certain shear stress, and then above this stress they will behave like
liquids. A typical example of these substances (Bingham materials) is paint:
it is this behavior which enables a coat of paint to stick to surfaces parallel
to the force of gravity.
Theabovedefinitionofafluidcomprisesbothliquidsandgases,sincenei-
ther showany resistanceto changeof shape when the velocityof this change
tends to zero. Now liquids develop a free surface through condensation, and
in general do not fill up the whole space they have available to them, say