Fluid Mechanics

Springer-Verlag Berlin Heidelberg GmbH

Joseph H. Spurk

Fluid Mechanics Problems and Solutions

With the assistance of H. Marschall

With 380 Figures

Springer

Prof. Dr.-Ing. Joseph H. Spurk

TH Dannstadt Institut für Technische Strömungslehre Petersenstraße 30 D - 64287 Darmstadt / Germany

Translation: Professor Taher Schobeiri Texas A & M University, Dept. of Mechanical Engineering College Station, 77843 31 2 Texas / USA

ISBN 978-3-642-63523-6

Die Deutsche Bibliothek - CIP-Einheitsaufuahme Spurk, Joseph H.: Fluid mechanics : problems and solutions / Joseph H. Spurk. With the assistance ofH. Marschall. (TransI.: Taher Schobeiri). -Berlin; Heidelberg; New York; Barcelona; Budapest; Hong Kong; London; Milan; Paris; Santa Clara; Singapur, Tokyo: Springer, 1997

Dt. Ausg. u. d. T.: Spurk, Joseph H.: Strömungslehre

ISBN 978-3-642-63523-6 ISBN 978-3-642-58277-6 (eBook) DOI 10.1007/978-3-642-58277-6

CIP data applied for

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© Springer-Verlag Berlin Heidelberg 1997 Softcover reprint of the hardcover I st edition 1997

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Dedicated to my Co-workers

Preface to the English Edition

This collection of exercises is meant as a companion volume to the textbook Fluid Mechanics. It is the translation of the second edition of Aufgaben zur Stromungslehre. The book contains about 200 problems worked out in detail. In selecting the exercises I have been guided by didactical consider­ations and included problems that demonstrate the application of the gen­eral principles of continuum mechanics to more or less classical problems in fluid mechanics. Most of these problems are found in other textbooks or collections. On the other hand, there is a good number of exercises designed to develop and further the ability to model and solve practical problems. Besides these worked examples, thirty examination problems with answers only are included. In addition there are also exercises for Cartesian tensor calculus. The book has been translated by Professor M. T. Schobeiri, Texas A & M University. I thank him and also Dorothee Sommer and Peter Pelz for their help with this book.

Darmstadt, March 1997 J. H. Spurk

Preface to the First German Edition

Since I have been lecturing in fluid mechanics at the Technische Hochschule, Darmstadt, there has always been a demand for a collection of worked ex­amples. One of the reasons it has taken so long for this to be realized, is my concern that active participation in tutorials would suffer if worked examples were at hand. Of course there are reasons which justify such a collection: it helps those who have difficulties keeping up with lectures and tutorials and those searching for access to the theoretical solution of prac­tical problems and to mathematical modelling. It can be of service while preparing for examinations, and is almost indispensible for self study of fluid mechanics.

In Darmstadt, an exercise sheet with five exercises is distributed before every tutorial. Whoever is responsible for tutorials that semester adds one new exercise to every exercise sheet. This is to bring forward the difference between creative thinking, which through the process of abstraction and simplification leads to a mathematical model and the simple recalculation of even difficult exercise problems. The reader will only be able to gain such

VIII

insight if he or she makes a serious attempt at an independent solution of the exercise, and only uses the worked solution as a check.

In the search for new exercises, we have gone through other available collections, and this book contains exercises from these, even though it is now difficult to trace their origin and give proper credit. However, many exercises have their origin in current research projects, or were inspired by industrial contacts and still contain the essence of the original problem. In this manner, a considerable number of exercises have arisen over the years. The basis for this book is a selection that Dr. Sauerwein prepared for the tutorials in the year 1987/88. The exercises are always adapted to the pace of the lectures, and assume knowledge of only up to the current lecture. This selection was thoroughly revised, and expanded by the inclusion of further exercises. The exercises now follow the same order as the chapters of my textbook (Fluid Mechanics, Springer-Verlag), and accordingly only the material of the earlier chapters is needed to tackle each example. Wherever it was deemed necessary, there is a reference (F. M. (xxx)) to formulae in Fluid Mechanics. But, the exercise collection can also be used in conjuction with other textbooks.

Considerable analytical skill and indeed routine in the manipulation of mathematical expressions are required for the solution of these exer­cises, often absent today in many students, due to lack of sufficient prac­tise. Computer programs with symbolic computation capacity can help here. In revising these exercises we have made extensive use of the pro­gramming system Mathematica (Wolfram Research Inc.), without, how­ever, reproducing the appropriate commands. These can be taken from the handbook (Wolfram, Stephen: Mathematica 2nd ed. Addison-Wesley Publishing Company Inc.), practically without instruction. But other pro­gramming systems which work symbolically and graphically can also be used. In spite of this, all intermediate steps are given so that the exercises may also be solved by those without access to such programming systems, using the usual aids, such as collections of formulae, integral tables, etc. All exercises are developed from the fundamental balance laws, according to the principle "from the general to the specific" , even if a certain clumsiness had to be put up with in the presentation.

Current and former co-workers have carried out most of the work in this collection of exercises, and it is to them that I dedicate this book, for whose content and defects I, however, remain responsible.

Darmstadt, December 1993 J. H. Spurk

Contents

1 The Concept of Continuum and Kinematics 1.2 Kinematics . . . . . . . . . . . . . . . . .

Problem 1.2-1 Calculation of material coordinates for giv-

1 1

en pathlines ................ 1 Problem 1.2-2 Velocity and acceleration in material and

spatial coordinates with given pathlines. 2 Problem 1.2-3 Material description of a potential vortex

flow . . . . . . . . . . . . . . . . . . . .. 5 Problem 1.2-4 Material description of an axisymmetric stag-

nation point flow . . . . . . . . . . . . .. 7 Problem 1.2-5 Pathlines, streamlines, and streaklines of

an unsteady flow field . . . . . . . . . .. 9 Problem 1.2-6 Kinematics of an irrotational and diver-

gence free flow field . . . . . . . . . . .. 14 Problem 1.2-7 Kinematics of an unsteady, plane stagna-

tion point flow . . . . . . . . . . . . . .. 19 Problem 1.2-8 Streakline of a water jet . . . . . . . . .. 23 Problem 1.2-9 Streamlines and Streaklines in cylindrical

coordinates . . . . . . . . . . . . . . . .. 26 Problem 1.2-10 Streamlines and pathlines of standing grav-

ity waves . . . . . . . . . . . . . . . . .. 29 Problem 1.2-11 Change of material line elements in a Couette-

flow . . . . . . . . . . . . . . . . . . . .. 31 Problem 1.2-12 Change of material line elements in a three-

dimensional flow . . . . . . . . . . . . .. 34 Problem 1.2-13 Angular velocity vector and the change of

material line elements in a two-dimensional flow field . . . . . . . . . . . . . . . . .. 38

x Contens

Problem 1.2-14 Rate of deformation and spin tensors of an unsteady two-dimensional flow . . . . .. 43

Problem 1.2-15 Time change of the kinetic energy of a fluid body .................... 45

2 Fundamental Laws of Continuum Mechanics 50 2.1 Conservation of Mass, Equation of Continuity . . . . . .. 50

Problem 2.1-1 One-dimensional unsteady flow with given density field ................ 50

Problem 2.1-2 Plane, steady flow with a given density field 52 Problem 2.1-3 Velocity at the exit of a container ... 54 Problem 2.1-4 Steady flow through a circular channel. 56 Problem 2.1-5 Squeeze film flow. . . . . . . . . . . . 58 Problem 2.1-6 Moving Piston . . . . . . . . . . . . . 60 Problem 2.1-7 Flow between two inclined flat plates. 63 Problem 2.1-8 Oscillating journal bearing ...... 65 Problem 2.1-9 Effect of boundary layer displacement thick-

ness . . . . . . . . . . . . . . . . . . . .. 68 Problem 2.1-10 Flow through a diffuser with a linear ve-

locity change in flow direction. . . . . .. 71 Problem 2.1-11 Temperature boundary layer along a cold

wall . . . . . . . . . . . . 73 Problem 2.1-12 Flow in a lubrication gap . . . . 74

2.2 Balance of Momentum . . . . . . . . . . . . . . . 78 Problem 2.2-1 Principal axes of a stress tensor. 78 Problem 2.2-2 Fluid forces on a manifold. 80 Problem 2.2-3 Calculation of drag force 82 Problem 2.2-4 Force on a slender nozzle 85

2.3 Balance of Angular Momentum . . . . . . 87 Problem 2.3-1 Torque on pipe with slot 87 Problem 2.3-2 Moment exerted on the inlet guide vanes

of a water turbine . . . . . . . . . . . .. 90 Problem 2.3-3 Curvature radius of circular arc profiles of

a circular cascade ............. 93 2.4 Momentum and Angular Momentum in an Accelerating Frame 96

Problem 2.4-1 Fluid sprayed on a rotating disk . . . .. 96 Problem 2.4-2 Velocity of a moving container with a nozzle100 Problem 2.4-3 Acceleration and velocity of a rocket 107 Problem 2.4-4 Thrust reversal . . . . . . . . . 109 Problem 2.4-5 Torque on a rotating bent pipe 111 Problem 2.4-6 Thrust of a jet engine . . . . . 114

Contens XI

2.5 Applications to Turbomachines .. . . . . . . . . . . . .. 117 Problem 2.5-1 Circulation around a blade profile in a cir-

cular cascade . . . . 117 Problem 2.5-2 Axial turbine stage. 119 Problem 2.5-3 Kaplan turbine . . . 121 Problem 2.5-4 Torque converter . . Problem 2.5-5 Balancing of axial thrust

125 129

2.6 Conservation of Energy. . . . . . . . . . . 131 Problem 2.6-1 Cylinder with heat flux . 131 Problem 2.6-2 Energy balance in an axial turbine stage. 134

3 Constitutive equations 138 138 140 144 146 148

Problem 3-1 Velocity of a raft Problem 3-2 Problem 3-3 Problem 3-4 Problem 3-5

Energy balance in a journal bearing Pressure driven flow of paper pulp Flow of a non-Newtonian fluid

Extensional flow . . . . .

4 Equation of Motion for Particular Fluids 152 152 152

4.1 Newtonian Fluids ............. .

4.2

Problem 4.1-1 Poiseuillefiow ..... . Problem 4.1-2 Problem 4.1-3

Problem 4.1-4 Problem 4.1-5 Problem 4.1-6 Problem 4.1-7

Problem 4.1-8 Problem 4.1-9 Inviscid flow . Problem 4.2-1

Problem 4.2-2 Problem 4.2-3 Problem 4.2-4

Temperature distribution in a Poiseuille flow 156 Pressure driven flow in a channel with porous walls ............ 159 Boundary layer suction 161 Mixing of streams of fluids 165 Drag on a flat plate 168 Two-dimensional water jet impinging on a wedge. . . . . . . . . . . . .. 173 Rigid body rotation and potential vortex 175 Energy balance in a potential vortex flow 180

Pressure and energy increase of fluid in a centrifugal pump . ....... .. Pressure distribution within a spiral casing Free surface in a potential vortex . Circulation in a Couette flow . . . .

184

184 189 190 192

Problem 4.2-5 Velocity induced by a vortex ring. . . 193 Problem 4.2-6 Two infinitely long vortex filaments near a

wall ..... ........ 194

XII Contens

Problem 4.2-7 Wing with an elliptic spanwise distribution of circulation . . . . . . . . 198

Problem 4.2-8 Airfoil in parallel flow . . . . . . . . . .. 201 Problem 4.2-9 Jet angle in a Betz diffuser . . . . . . .. 204 Problem 4.2-10 Contraction coefficient of a Borda mouth-

pIece .................... 206 Problem 4.2-11 Pressure distribution in an inviscid and ax-

isymmetric flow .............. 208 Problem 4.2-12 Increase of static pressure in a Betz diffuser 210 Problem 4.2-13 Fluid flowing out of a tank . . . 212 Problem 4.2-14 Air bubble moving in a channel. . . . .. 214 Problem 4.2-15 Aircraft above the ground. . . . . . . .. 217 Problem 4.2-16 Flow between two rotating cylinders, cir-

culation and vorticity . . 223 Problem 4.2-17 Power of a Pelton turbine . . . . . . . .. 225

4.3 Initial and Boundary Conditions. . . . . . . . . . . . . .. 230 Problem 4.3-1 Oscillation of an elliptic cylinder in fluid. 230 Problem 4.3-2 Flat plate with a pitching and oscillating

motion . . . . . . . . . . . . . . . . .. 231 Problem 4.3-3 Rotating cylinder moving through fluid 232 Problem 4.3-4 Vortical flow inside an elliptic cylinder. 234

5 Hydrostatics 236 5.1 Hydrostatic Pressure Distribution 236

Problem 5.1-1 U-tube manometer. . . 236 Problem 5.1-2 Hydraulic safety clutch 237 Problem 5.1-3 Rotating container filled with fluid 239 Problem 5.1-4 Centrifugal casting process 241 Problem 5.1-5 Depth gauge . . . . . . . . . . . . 242

5.2 Hydrostatic Lift, Force on Walls . . . . . . . . . . . 244 Problem 5.2-1 Force and moment on a throttle valve 244 Problem 5.2-2 Half sphere closing an orifice . . . . . 246 Problem 5.2-3 Force on a dam. . . . . . . . . . . . . 248 Problem 5.2-4 Half sphere cup sealing by its own weight 250 Problem 5.2-5 Cylindrical submarine 252 Problem 5.2-6 Car under water 254

6 Laminar Unidirectional Flow 257 Problem 6-1 Flow in an annular gap . . . . . . . 257 Problem 6-2 Crude oil transport through pipeline 261 Problem 6-3 Oscillating pipe flow . . . . . . . . . 264

Contens

Problem 6-4 Comparison of a Couette-Poiseuille flow of a Newtonian fluid, a Stokes fluid, and a

XIII

Bingham material . . . . . . . . . 267

7 Fundamentals of Turbulent Flows 274 274 Problem 7-1

Problem 7-2

Problem 7-3 Problem 7-4 Problem 7-5

Problem 7-6

Problem 7-7 Problem 7-8

Turbulent Couette flow Velocity distribution in turbulent Couette flow with given Reynolds number . Turbulent pipe flow ........... . Crystal growth on pipe walls . . . . . . . Comparison of momentum and energy flux in laminar and turbulent flow in a pipe Velocity distribution in a turbulent pipe flow resulting from the Blasius friction law Location of a pipe leakage . . . . . . . . . Cooling of superheated steam by water in-jection ................. .

277 278 280

282

285 287

289

8 Hydrodynamic Lubrication 293 Problem 8-1 Bearing with step slider . . . . . . . . .. 293 Problem 8-2 Friction torque transmitted by the shaft to

the journal . . . . . . . . . . . . . . . .. 297 Problem 8-3 Slider load in squeeze flow: Comparison

between different slider geometries 299

9 Stream filament theory 302 9.1 Incompressible Flow ............... 302

Problem 9.1-1 Rotating tube acting as pump 302 Problem 9.1-2 Volume flux through an orifice 305 Problem 9.1-3 Injector pump 306 Problem 9.1-4 Radial pump . 308 Problem 9.1-5 Bulb turbine 312 Problem 9.1-6 Problem 9.1-7 Problem 9.1-8 Problem 9.1-9 Problem 9.1-10 Problem 9.1-11 Problem 9.1-12 Problem 9.1-13 Problem 9.1-14

Coanda effect . Principle of a shaped charge Penstock and nozzle of a Pelton turbine Operating characteristic of a fan . . . Water power plant .......... . Flow through an exhaust gas analyser Flow deflection through a screen Hovercraft .. Wind turbine . . . . . . . . . . .

315 316 319 321 325 328 329 331 333

XIV Contens

Problem 9.1-15 Discharge pipe of a reservoir: Comparison between different pipe geometries 337

Problem 9.1-16 Vibrating system consisting of a fluid col-umn and a spring suspended piston 339

Problem 9.1-17 Unsteady flow in a tube with flexible walls 343 Problem 9.1-18 Plunger pump . . . . . . . . . . 346 Problem 9.1-19 Flow within an urethra prothesis 350

9.2 Steady Compressible Flow . . . . . . . . . . . . . 352 Problem 9.2-1 Force on a plate in subsonic flow 352 Problem 9.2-2 Channel flow with heat addition 355 Problem 9.2-3 Normal shocks in an inlet guide vane. 358 Problem 9.2-4 Blunt body in supersonic flow. . . . . 363 Problem 9.2-5 Shock waves in the divergent part of a Laval

nozzle. . . . . . . . . . . . . . . 365 Problem 9.2-6 Supersonic nozzle in a spinneret 367 Problem 9.2-7 Ram jet in subsonic flow 370 Problem 9.2-8 High speed train in a tunnel .. 373 Problem 9.2-9 Labyrinth seal of a turbomachine . 376 Problem 9.2-10 Gas flow through an orifice . . . 379

9.3 Unsteady Compressible Flow. . . . . . . . . . . . 381 Problem 9.3-1 Traveling normal shock in a pipe 381 Problem 9.3-2 Shock tube . . . . . . . . . . . . 383 Problem 9.3-3 Motion of a piston in a tube .. 386 Problem 9.3-4 Reflection of a normal shock wave at the

open end of a tube . . . . . . . . . . . .. 389 Problem 9.3-5 Principle of an expansion tube . . . . .. 392 Problem 9.3-6 Propagation of acoustic waves in a closed

tube. . . . . . . . . . . . . . . . . . . .. 394

10 Potential Flow 399 10.3 Incompressible Potential Flow . . . . . . . . . 399

Problem 10.3-1 Expanding sphere . . . . . . 399 Problem 10.3-2 Sphere in a translational flow 402 Problem 10.3-3 Flow near the stagnation point of a body

in parallel flow . . . . . . . . . . . . . .. 406 Problem 10.3-4 Point source in a rotationally symmetric

stagnation point flow .......... 409 Problem 10.3-5 Point source above an impermeable wall. 412 Problem 10.3-6 Source distribution in parallel flow . . .. 414 Problem 10.3-7 Expanding sphere in an inviscid and in a

viscous flow. . . . . . . . . . . . . . . .. 416

Contens xv

Problem 10.3-8 Growth of a vapor filled cavity . . . . .. 420 Problem 10.3-9 Contraction coefficient for a circular orifice 423 Problem 10.3-10 Sphere rising in water . . . . . . . . . .. 427 Problem 10.3-11 Unsteady motion of a cylinder perpendic-

ular to its axis . . . . . . . . . . . . 430 Problem 10.3-12 Rotor oscillating in an inviscid fluid . .. 432

10.4 Plane Potential Flow . . . . . . . . . . . . . . . . . . . .. 436 Problem 10.4-1 Flow in the squeeze gap between a moving

piston and a wall. . . . . . . . . . . . .. 436 Problem 10.4-2 Sink distribution in a stagnation point flow 439 Problem 10.4-3 Circle theorem . . . . . . . . . . . . . 442 Problem 10.4-4 Half cylinder in stagnation point flow 447 Problem 10.4-5 Dipol flow around a circular cylinder . 451 Problem 10.4-6 Flow around a thin plate . . . 454 Problem 10.4-7 Airfoil over a fixed wall . . . . 457 Problem 10.4-8 Semi infinite body in a channel 461 Problem 10.4-9 Karman's vortex street .... 464 Problem 10.4-10 Joukowski mapping of a circular cylinder

in a uniform flow . . . . . . . . . . . . .. 467 Problem 10.4-11 Plane circular cascade. . . . . . . . . .. 470 Problem 10.4-12 Schwarz-Christoffel transformation of a wall

of infinite extent . . . . . . . . . . . . .. 473 Problem 10.4-13 Schwarz-Christoffel transformation of a con­

vergent channel. . . . . . . . . . . . . .. 476 Problem 10.4-14 Cavitation in a channel . . . . . . . . .. 480 Problem 10.4-15 Representation of a slender body by a source

distribution. . . . . . . . . . . . . . . .. 483 Problem 10.4-16 Distribution of vortex intensity and mean

camber line of a slender airfoil . . . . .. 488 Problem 10.4-17 Straight cascade . . . . . . . . . . . . .. 492 Problem 10.4-18 Vortex distribution of q flat-plate cascade 497 Problem lO.4-H) Compressible flow over a wavy wall. 503

11 Supersonic Flow 509 11.1 Oblique Shock Waves. . . . . . . . . . . . . . . . . . . 509

Problem 11.1-1 Wedge with a thin plate in front of it 509 Problem 11.1-2 Inlet of a plane channel . . . . . . . . 511

11.3 Reflection of Oblique Shock Waves ........... 514 Problem 11.3-1 Flow over a wedge in a supersonic wind

tunnel. . . . . . . . . . . . . . . . . .. 514 Problem 11.3-2 Supersonic flow in a convergent channel . 516

XVI Contens

11.5 Prandtl-Meyer Flow ..................... 518 Problem 11.5-1 Centered expansion wave in a divergent

channel . . . . . . . . . . 518 11.6 Shock Expansion Theory. . . . . . . . . . . . . 522

Problem 11.6-1 Airfoil in supersonic flow ... 522 Problem 11.6-2 Inlet of a supersonic jet engine 526

12 Boundary Layer Theory 530 Problem 12-1 Boundary layer momentum equation 530 Problem 12-2 Flow over a wedge . . . . . . . . . . 533 Problem 12-3 Diffuser with discontinuous change of the

cross-section . . . . . . . . . . . . . 537 Problem 12-4 Drag coefficient of a diamond airfoil . .. 543

A Tensor calculus 551 Problem A-I 551 Problem A-2 551 Problem A-3 552 Problem A-4 553 Problem A-5 554 Problem A-6 555 Problem A-7 555 Problem A-8 556 Problem A-9 557 Problem A-I0 558 Problem A-11 560

B Examination problems 562 Problem B-1 Streamlines and pathlines 562 Problem B-2 Drag of a half cylinder shell. 563 Problem B-3 A wning in a storm . . . . 564 Problem B-4 Stretching of a foil . . . . . . 565 Problem B-5 Single stage, axial blower . . 566 Problem B-6 Blade profile for given pressure distribution 567 Problem B-7 Combustion chamber of a piston engine . 568 Problem B-8 Two-dimensional oblique stagnation point

flow . . . . . . . . . . . . . . . . . . .. 569 Problem B-9 Problem B-I0 Problem B-11 Problem B-12

Generalized Hagen-Poiseuille flow ... Induced velocity of a horse-shoe vortex Open channel flow through a weir Safety valve. . . . . . . . . . . . . . . .

570 571 572 573

Contens

Problem B-13 Problem B-14 Problem B-15

Problem B-16

Problem B-17 Problem B-18 Problem B-19 Problem B-20

Problem B-21 Problem B-22 Problem B-23 Problem B-24 Problem B-25 Problem B-26 Problem B-27 Problem B-28 Problem B-29 Problem B-30 Problem B-31

ProblemB-32 Problem B-33 Problem B-34

Problem B-35

Problem B-36

XVII

Liquid in container. . . . . .. .... 574 Sluice gate .. .......... 575 Pressure driven flow in the radial gap be-tween two concentric ring plates 576 Pressure driven channel flow with variable viscosity .... 577 Temperature induced flow. . . . . 578 Shock absorber. . . . . . . . . . . 579 Frequency of a Helmholtz resonator 580 Chamber and exhaust pipe of an internal combustion engine . . 582 Pump-turbine storage plant 583 Overexpanded Laval nozzle 584 Nozzle inlet . . . . . . . . 585 Solid propellant rocket engine . 586 Ram jet. . . . . . . . . 587 Ludwieg-tube. .. .... 588 Dipol above an impermeable wall . 589 Virtual mass of a thin plate . . . 591 Removal of liquid through a plane channel 592 Unsteady flow over a wavy wall. .. 593 Wing section for given source and vortex distribution. ........ 594 Infinitely thin plate with aileron .... 595 Supersonic inlet .... ........ 596 Infinitely thin, flat plate in two-dimensional supersonic flow. . . . . . . . . . . . . .. 598 Guide vane cascade of a supersonic com­pressor .. .... 599 Boundary layer on a foil . . . . . . . . .. 600