fluid - scientific publishers · 2019-03-19 · 8.4.3 determination of coefficient of discharge 433...

FluidMechanics

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FluidMechanics

Dr. Kamlesh Purohit Professor Dept. of Mechanical Engineering, JNV University, Jodhpur

Dr. S.P. Harsha Assistant Professor Dept. of Mechanical & Industrial Engineering, IIT, Roorkee

Dr. R.K. Purohit Retd. Associate Professor Dept. of Mechanical Engineering, JNV University, Jodhpur

Published by: Scientific Publishers (India) 5-A, New Pali Road, P.O. Box 91, Jodhpur – 342 001 (India)

E-mail: [email protected] Website: www.scientificpub.com

© Authors, 2012

All rights reserved. No part of this publication or the information contained herein may be reproduced, adapted, abridged, translated, stored in a retrieval system, computer system, photographic or other systems or transmitted in any form or by any means, electronic, mechanical, by photocopying, recording or otherwise, without written prior permission from the authors/editors and the publishers.

Disclaimer: Whereas every effort has been made to avoid errors and omissions, this publication is being sold on the understanding that neither the author (or authors of chapters in edited volumes) nor the publishers nor the printers would be liable in any manner to any person either for an error or for an omission in this publication, or for any action to be taken on the basis of this work. Any inadvertent discrepancy noted may be brought to the attention of the publishers, for rectifying it in future editions, if published.

ISBN: 978-81-7233-757-5 eISBN: 978-93-8786-903-5

Lasertype set: Rajesh Ojha Printed in India

Preface

Since the publication of this book Foundation of Fluid Mechanics in 1996 has seen sea change in respect of contents and methodology conveying finer aspects of the subject. Hence, it demands through review and revision of the contents which should be in consonance with the changing worldview of the Engineering students. In order to make updated two renowned authorities in this field have been associated as co-authors. They are currently giving their best to budding generation of engineers. They are well versed in the modern contents and methodology. In this edition lot of changes have been made in all the chapters.

Jodhpur Dr. Kamlesh Purohit Dr. S.P. Harsha

Dr. R.K. Purohit

Contents

1. BASIC CONCEPT RELATING TO FLUIDS 1

1.1 INTRODUCTION 11.2 DEFINITION OF A FLUID 11.3 FLUID AS A CONTINUUM 11.4 INCOMPRESSIBLE AND COMPRESSIBLE FLOW 21.5 BASIC DEFINITIONS 2

1.5.1 Mass 21.5.2 Density 21.5.3 Specific Volume 21.5.4 Specific Weight 31.5.5 Relative density 3

1.6 VISCOSITY 31.6.1 Units of Viscosity 41.6.2 Dimensional Formula of Viscosity 51.6.3 Kinematic Viscosity 61.6.4 Units of Kinematic viscosity 61.6.5 Dimensional formula of kinematic viscosity 71.6.6 Newtonian and non Newtonian fluids 71.6.7 Effects of temperature and pressure on viscosity 81.6.8 Ideal Fluid 10

1.7 COMPRESSIBILITY AND ELASTICITY OF FLUIDS 111.8 SURFACE TENSION 111.9 CAPILLARITY 121.10 PRESSURE INSIDE A WATER DROPLET, SOAP AND BUBBLE 151.11 VAPOUR PRESSURE AND CAVITATION 16

vi Fluid Mechanics

ILLUSTRATIVE EXAMPLES 17EXERCISES

A. THEORY 35B. UNSOLVED PROBLEMS 36

2 STATIC PRESSURE AND ITS MEASUREMENT 39

2.1 INTRODUCTION 392.2 PRESSURE AT A POINT 392.3 PASCAL'S LAW 392.4 EULER'S DIFFERENTIAL EQUATIONS 41

2.4.1 The basic equation of hydrostatics 422.4.2 Pressure head 432.4.3 The hydrostatic paradox 44

2.5 ATMOSPHERIC PRESSURE 452.5.1 Fortin barometer 45

2.6 APPLICATION OF THE BASIC EQUATION OF FLUID STATICS 462.7 MANOMETERS 47

2.7.1 Piezometer 482.7.2 Simple manometers 482.7.3 Differential Manometers 50

2.8 MEASUREMENT OF SMALL PRESSURE DIFFERENCE 502.8.1 Inclined gauge 512.8.2 Micromanometers 52

2.9 PRESSURE VARIATION IN A COMPRESSIBLE FLUID 532.9.1 Variation under isothermal conditions 532.9.2 Variation under adiabatic conditions 542.9.3 Variation of pressure and density with altitude for a constant

temperature gradient 56

2.9.4 Variation of temperature and pressure in the atmosphere 56ILLUSTRATIVE EXAMPLES 58EXERCISES


3. FLUID STATICS 102

3.1 INTRODUCTION 102

Contents vii

3.2 TOTAL PRESSURE 1023.3 CENTRE OF PRESSURE 1023.4 TOTAL PRESSURE AND CENTRE OF PRESSURE ON A

VERTICAL PLANE SURFACE 102

3.5 TOTAL PRESSURE AND CENTRE OF PRESSURE FORINCLINED SURFACE

105

3.6 PRESSURE DIAGRAMS 1073.7 PRESSURE ON CURVED SURFACES 109

3.7.1 General case of pressure on curved surfaces 1093.7.2 Pressure on cylindrical surfaces 110

3.8 GRAVITY DAMS 1113.9 STATIC PRESSURE ON SLUICE GATES 1123.10 LOCK GATES 113

3.10.1. Purpose of Lock Gates 115ILLUSTRATIVE EXAMPLES 115EXERCISES


4. BUOYANCY AND FLOATATION 165

4.1. INTRODUCTION 1654.2. BUOYANCY 1654.3. CENTRE OF BUOYANCY 1664.4. EQUILIBRIUM OF FLOATING BODIES 1664.5. METACENTRE 1664.6. METACENTRIC HEIGHT 1674.7. STABILITY OF FLOATING BODIES-METACENTRE AND META-

CENTRIC HEIGHT 167

4.8 EXPERIMENTAL METHOD OF DETERMINATION OF META-CENTRIC HEIGHT

168

4.9 ANALYTICAL METHOD FOR METACENTRIC HEIGHT 1694.10 THE PERIOD OF ROLL OF A VESSEL 171ILLUSTRATIVE EXAMPLES 171EXERCISES


viii Fluid Mechanics

5. RELATIVE MOTIONS OF LIQUIDS 185

5.1 INTRODUCTION 1855.2 FLUID MASSES SUBJECTED TO ACCELERATION 1855.3 RELATIVE EQUILIBRIUM UNDER CONSTANT HORIZONTAL

ACCELERATION 187

5.4 RELATIVE EQUILIBRIUM UNDER CONSTANT VERTICALACCELERATION

188

5.5 RELATIVE EQUILIBRIUM IN A ROTATING CONTAINER 1905.6 HYDROSTATIC ACCELEROMETERS 191ILLUSTRATIVE EXAMPLES 192EXERCISES


6 KINEMATICS OF FLUID 215

6.1 INTRODUCTION 2156.2 DESCRIPTION OF FLUID MOTION 2156.3 FLUID FLOW CLASSIFICATIONS 216

6.3.1 Steady flow and unsteady flow 2166.3.2 Uniform flow and non uniform flow 2166.3.3 One, two and three dimensional flow 2176.3.4 Laminar flow and turbulent flow 2186.3.5 Rotational flow and Irrotational flow 218

6.4 FLOW LINES 2196.4.1 Streamline 2196.4.2 Stream tube 2206.4.3 Path line 2206.4.4 Streak line 221

6.5 VELOCITY AND ACCELERATION 2226.5.1 Convective acceleration and Local acceleration 2236.5.2 Tangential acceleration and normal acceleration 224

6.6 PRINCIPLE OF CONTINUITY-CONSERVATION OF MASS FLOW 2256.6.1 One dimensional continuity equation 2266.6.2 Continuity equation for three dimensional flow using

Cartesian Coordinates 227

6.6.3 Continuity equation for cylindrical coordinates 228

Contents ix

6.6.4 Continuity equation for spherical coordinates 2306.7 DEFORMATION OF A FLUID ELEMENT 2326.8 CIRCULATION AND VORTICITY 235

6.8.1 Circulation for the rectangular element 2366.8.2 Circulation for the circle 2376.8.3 Vorticity 237

6.9 STREAM FUNCTION AND VELOCITY POTENTIAL 2386.9.1 The stream function 2386.9.2 Velocity Potential 241

6.10 FLOW NET 2436.10.1 Methods of drawing flow nets 2456.10.2 Uses of flow net 2476.10.3 Limitations of flow net 248


A. Theory 289B. UNSOLVED PROBLEMS 292

7. DYNAMICS OF FLUID FLOW 297

7.1 INTRODUCTION 2977.2 EULER'S EQUATION OF MOTION 298

7.2.1 Bernoull's equation-Integration of Euler's equation along astreamline for steady flow

299

7.2.2 Limitations on Bernoulli's equations 3007.2.3 Modification to Bernoulli's equation 3017.2.4 The physical significance of Bernoulli's equation 301

7.3 APPLICATIONS OF BERNOULLI'S EQUATION 3027.3.1 Venturimeter 302

7.3.1.1 Venturimeter analysis 3047.3.1.2 Vertical/Inclined Venturimeter 3087.3.1.3 Use of differential manometer in venturimeter 310

7.3.2 Orifice meter 3117.3.3 Flow nozzle or Nozzle meter 3147.3.4. Flow tubes 3167.3.5 Pressure recovery 316

7.4 VELOCITY MEASUREMENTS 316

x Fluid Mechanics

7.4.1 Static, stangnation and dynamic pressures 3177.4.2 Pitot tube 3187.4.3 Pitot- static tube 319

7.5 MOMENTUM EQUATION 3207.5.1 Impulse-Momentum equation 3217.5.2 Momentum equation for two-and there dimensional flow

along a stream line 322

7.5.3 Momentum correction factor 3237.5.4 Application of the momentum equation 324

7.5.4.1 Forces on a pipe bend 3247.5.4.2 Force due to the diflection of a jet by a curved vane 3257.5.4.3 Force at a nozzle 3267.5.4.4 Reaction of a jet 326

7.6 MOMENTUM OF MOMENTUM OR ANGULAR MOMENTUMTHEOREM

327

7.7 VORTEX MOTION 3297.7.1 Energy variation normal to streamlines 3297.7.2 Forced vortex motion 3317.7.3 Free vortex motion 331



8. FLOW THROUGH ORIFICES AND MOUTH PIECES 426

8.1 INTRODUCTION 4268.2 SHARP EDGED ORIFICE DISCHARGING FREE 4268.3 HYDRAULIC COEFFICIENTS 428

8.3.1 Coefficient of contraction (Cc) 4288.3.2 Coefficient of velocity (Cv) 4298.3.3 Coefficient of discharge (Cd) 4298.3.4 Coefficient of resistance (Cr) 430

8.4 EXPERIMENTAL DETERMINATION OF HYDRAULIC COEFFICI-ENTS FOR AN ORIFICE

430

8.4.1 Experimental determination of coefficient of contraction 4308.4.2 Determination of coefficient of velocity Cv 431

Contents xi

8.4.2.1.Jet distance measurement method (Trajectory method) 4318.4.3 Determination of coefficient of discharge 433

8.5 SUBMERGED ORIFICE 4338.6 PARTIALLY SUBMERGED ORIFICE 4348.7 SHARP EDGED LARGE VERTICAL ORIFICE WITH RECTAN-

GULAR SHAPE 435

8.8 MOUTHPIECES OR TUBES 4368.8.1 External cylindrical mouthpiece running full 4378.8.2 Flow through convergent divergent mouthpiece 4408.8.3 Borda's or Re entrant mouthpiece 442

8.8.3.1 Borda's mouthpiece running free 4428.8.3.2 Borda's mouthpiece running full 443

8.9 FLOW THROUGH AN ORIFICE OR A MOUTHPIECE UNDERVARIABLE HEADS

445

8.9.1 General procedure for calculating time of emptying a tankthrough an orifice /mouthpiece at its bottom

445

8.9.2 Time of emptying cylindrical tank 4468.9.3 Determine the constant head under a head falling 447

8.10 TIME OF EMPTYING (OR FILLING) A TANK WITH INFLOW 4488.11 FLOW OF LIQUID FROM ONE VESSEL TO ANOTHER 449ILLUSTRATIVE EXAMPLES 450EXERCISES


9. FLOW OVER NOTCHES AND WEIRS 478

9.1 INTRODUCTION 4789.2 RECTANGULAR WEIRS 478

9.2.1 Flow over rectangular weir 4799.2.2 Flow over rectangular weir with velocity of approach 4809.2.3 Empirical formulae for discharge over rectangular weir 481

9.3 FLOW OVER A TRIANGULAR WEIR (V-NOTCH) 4839.4 FLOW OVER A TRAPEZOIDAL WEIR OR NOTCH 485

9.4.1 Cippoletti weir 4869.5 LONG BASED WEIRS 488

9.5.1 Broad crested weir 488

xii Fluid Mechanics

9.5.2 Round nosed weirs 4909.5.3 Crump weirs 490

9.6 SUBMERGED WEIRS 49419.7 OGEE WEIR 4929.8 PROPORTIONAL OR SUTRO WEIR 4939.9 VENTILATION OF WEIR 4939.10 TIME REQUIRED TO EMPTY A RESERVOIR 494

9.10.1 Rectangular weir 4959.10.2 Triangular weir 495



10. FLOW THROUGH PIPES 522

10.1 INTRODUCTION 52210.2 REYNOLDS EXPERIMENTS 522

10.2.1 Reynolds number and its Significance 52410.2.2 Laminar and turbulent flow 52410.2.3 Critical Reynolds number 526

10.3 FLUID FRICTION 52610.4 HEAD LOST DUE TO FRICTION IN PIPES-DARCY'S

WEISBACH EQUATION 527

10.4.1 Proof of the Darcy's Weisbach equation 52710.4.2 Chezy's formula 52910.4.3 Manning's formula 52910.4.4 Hazen William's formula 529

10.5 MINOR LOSSES 53010.5.1 Head loss due to sudden enlargement 53010.5.2 Head loss due to sudden contraction 53210.5.3 Head loss at entrance to pipe 53410.5.4 Exit loss 53410.5.5 Head loss due to obstruction 53510.5.6 Head loss due to bends, valves, Non symmetrical sections,

etc.536

10.6 TOTAL ENERGY LINE AND HYDRAULIC GRADIENT 537

Contents xiii

10.6.1 Total energy line (TEL) 53710.6.2 Hydraulic gradient line (HGL) 537

10.7 FLOW THROUGH PIPE LINES 53810.8 FLOW THROUGH PIPES CONNECTED IN SERIES 54010.9 METHOD OF EQUIVALENT LENGTHS 54210.10 FLOW THROUGH PIPES CONNECTED IN PARALLEL 54310.11 POWER TRANSMISSION THROUGH PIPES 544

10.11.1 Maximum power transmission efficiency 54610.12 FLOW THROUGH NOZZLE AT THE END OF A PIPE 547

10.12.1 Efficiency of Power transmission through nozzle 54810.12.2 Condition for maximum power transmission through nozzle 54810.12.3 Diameter of nozzle for maximum transmission of power

through nozzle 549

10.13 WATER HAMMER 55010.14 Water hammer analysis 55110.14.1 Rigid Water Column theory 55110.14.2 Elastic pipe theory 553


A. THEORY 629B.UNSOLVED PROBLEMS 634

11. LAMINAR VISCOUS FLOW 639

11.1 INTRODUCTION 63911.2 NAVIER-STOKES EQUATIONS 639

11.2.1 Navier-Stokes equation in vector form and meaning of each term

642

11.2.2 Navier-Stokes in cylindrical polar coordinates 64411.3. HAGEN-POISEUILLE FLOW 64511.4. PLANE POISEUILLE FLOW 65011.5 COUTTE FLOW 654ILLUSTRATIVE EXAMPLES 659EXERCISES


xiv Fluid Mechanics

12. TURBULENT FLOW THROUGH PIPES 695

12.1 INTRODUCTION 69512.2 TURBULENT SHEAR STRESS 69712.3 BOUSSINES EDDY VISCOSITY 69812.4. PRANDTL'S MIXING LENGTH THEORY 69812.5 SHEAR VELOCITY OR FRICTION VELOCITY 69912.6 PRANDTL'S UNIVERSAL VELOCITY DISTRIBUTION EQUAT-

ION FOR TURBULENT PIPE FLOW 701

12.7 HYDRODYNAMICALLY SMOOTH AND ROUGH BOUNDARIES 70312.8 VELOCITY DISTRIBUTION FOR TURBULENT FLOW IN

SMOOTH AND ROUGH PIPES-PRANDTL KARMAN VELOCITYDISTRIBUTION EQUATION

705

12.8.1 Velocity distribution in smooth pipes 70612.8.2 Velocity distribution in rough pipes 708

12.9 VELOCITY DISTRIBUTION FOR TURBULENT FLOW INTERMS OF AVERAGE VELOCITY

709

12.9.1 Turbulent flow in smooth pipes 71012.9.2 Turbulent flow in rough pipes 71012.9.3 Difference between point velocity and average velocity for

smooth and rough pipes 710

12.10 TURBULENT PIPE COEFFICIENT 71212.11 THE CHRONOLOGICAL DEVELOPMENT OF TURBULENT

PIPE FLOW THEORIES 712

12.11.1 Smooth pipes and Blasius equation 71312.11.2 Stanton and Pannell 71412.11.3 Nikuradse experimental results using artificially rough

pipes 714

12.11.4 The smooth and rough laws of Prandtl and von Karman 71612.10.5 The Colebrook-White transition formula 71912.11.6 Moody diagram for commercial pipes 72212.11.7 Hydraulic Research station charts (HRS) Ackeres 72512.11.8 Barr explicit formula 72612.11.9 Murdock formula 72612.11.10 Swamee and Jain's explicit equation 72712.11.11 S.E. Haaland's formula 727

Contents xv

12.12 NON CIRCULAR PIPES 72712.13 ROUGHNESS OF PIPES WITH AGE (OLD PIPES) 728ILLUSTRATIVE EXAMPLES 728EXERCISES


13. DIMENSIONAL ANALYSIS AND SIMLITUDE 775

13.1 INTRODUCTION 77513.2 DIMENSIONS, DIMENSIONAL HOMOGENEITY AND UNITS 77613.3 DIMENSIONAL ANALYSIS 778

13.3.1 The Buckingham - theorem 77913.3.2 Selection of repeating variables 77913.3.3 Determining the groups 78013.3.4 Some additional comments about dimensional analysis 78113.3.5 Uniqueness of terms 78213.3.6 Limitations of dimensional analysis selection of variables -

superfluous and omitted variables 782

13.4 MODELING AND SIMILITUDE 78313.4.1 Geometric Similarity 78413.4.2 Kinematic similarity 78513.4.3 Dynamic similarity 78613.4.4 Standard dimensionless numbers 786

13.4.4.1 Reynold's number (Re) 78713.4.4.2 Froude's number ( RF ) 787

13.4.4.3 Mach's number (M) 78813.4.4.4 Euler's number (Eu) 78813.4.4.5 Weber's number (Wb) 789

13.5 MODEL LAWS 78913.5.1 Reynold's model law 79013.5.2 Froude's model law 79113.5.3 Mach model law 79313.5.4 Euler's model law 79313.5.5 Weber's model low 794

13.6 MODEL TESTS OF SYSTEMS DEPENDENT ON REYNOLD'SNUMBER AND FROUDE NUMBER-SHIP MODELS

794

xvi Fluid Mechanics

13.7 MODEL TESTS OF SYSTEMS DEPENDENT ON REYNOLD'SNUMBER AND MACH NUMBER

796

13.8 MODEL ANALYSIS OF TURBOMACHINES 79713.9 UNDISTORTED AND DISTORTED MODELS 79813.11 SCALE EFFECT 80013.12 COMMENTS ON MODEL TESTING 800ILLUSTRATIVE EXAMPLES 801EXERCISES


14. BOUNDARY LAYER THEORY 901

14.1 INTRODUCTION 90114.2 DESCRIPTION OF THE BOUNDARY LAYER 90214.3 BOUNDARY LAYER THICKNESSES 905

14.3.1 Boundary layer thicknesses 90514.3.2 Boundary layer displacement thickness 90514.3.3 Momentum thickness 90614.3.4 Energy thickness ** 907

14.4 LOCAL SKIN FRICTION AND AVERAGE SKIN FRICTIONDRAG COEFFICIENT

908

14.4.1 Local skin friction drag coefficient Cf 90814.4.2 Average skin friction drag coefficient CD 908

14.5 THE PRANDTL BOUNDARY LAYER EQUATIONS 90814.6 BLASIUS SOLUTION 90914.7 MOMENTUM INTEGRAL BOUNDARY LAYER EQUATION FOR

FLAT PLATE OR VON KARMAN INTEGRAL EQUATION 914

14.7.1 Momentum integral equation for zero pressure gradient 91814.8 MOMENTUM INTEGRAL METHOD FOR LAMINAR FLOW

OVER A FLAT PLATE 919

14.9 TURBULENT BOUNDARY LAYER 92314.10 COMBINED LAMINAR & TURBULENT BOUNDARY LAYERS 92614.11 COEFFICIENT OF DRAG FOR TURBULENT BOUNDARY

LAYER FOR ROUGH PLATE 928

14.12 FLOW WITH A PRESSURE GRADIENT 92914.13 SEPARATION OF BOUNDARY LAYER FLOW 930

14.13.1 Examples of separation of boundary layer flow 933

Contents xvii

14.13.2 Separation control 935ILLUSTRATIVE EXAMPLES 937EXERCISES


15. FLOW OVER IMMERSED BODIES 992

15.1. INTRODUCTION 99215.2 FORCES ON IMMERSED BODIES-DRAG AND LIFT 99315.3 DRAG ON IMMERSED BODIES 99615.4 TYPES OF DRAG 996

15.4.1 Skin friction drag 99615.4.2 Pressure drag 99715.4.3 Profile drag 99815.4.4 Deformation drag 99815.4.5 Wave drag 99815.4.6 Induced drag 999

15.5 STREAMLINED AND BLUFF BODIES 99915.6 DRAG ON SPHERE 1000

15.6.1 Dynamics of sports ball 100415.7 DRAG ON CYLINDER 1005

15.7.1 von Karman vortex street 100815.7.2 Summary of flow regimes in flow past a circular cylinder 1010

15.8 LIFT 101115.8.1 Magus effect and the circulation theory of lift (Kutta-

Joukowski theorem) 1011

15.9 LIFT OF AN AEROFOIL 101215.10 AEROFOIL TERMINOLOGY 101315.11 INVISCID FLOW PAST A TWO DIMENSIONAL AEROFOIL 101415.12 REAL (VISCOUS) FLUID PAST A TWO DIMENSIONAL

AEROFOIL 1016

15.13 AEROFOIL CHARACTERISTICS 101715.14 THREE DIMENSIONAL AEROFOIL THEORY 101915.15 THE POLAR DIAGRAMS FOR LIFT AND DRAG 102215.16 TYPICAL PERFORMANCE CURVES 1023

xviii Fluid Mechanics

15.17 DECREASING THE DRAG AND INCREASING THE LIFTCOEFFICIENT

1025

15.18 MULTIELEMENT AIRFOIL SECTIONS FOR GENERATINGHIGH LIFT

1026

15.18.1 High lift military airfoils 102815.8.2 Multielement high lift configurations 1029

15.19 THE AIRCRAFT AND ITS CONTROL 103015.20 MEASUREMENT OF LIFT 103215.21 TWO DIMENSIONAL CASCADES 1034

15.21.1 Cascade of blades 103515.21.2 Annular cascades 1037


A.THEORY 1071B. UNSOLVED PROBLEMS 1073

16 OPEN CHANNEL FLOW 1079

16.1 INTRODUCTION 107916.2 FLOW CLASSIFICATION 1080

16.2.1 Steady uniform flow 108016.2.2 Steady non uniform flow 108016.2.3 Unsteady flow 108116.2.4 Laminar and turbulent flows 108116.2.5 Subcritical, critical and supercriticial flows 1082

16.3 GEOMETRIC ELEMENTS OF CHANNEL SECTION 108316.4 VELOCITY DISTRIBUTION IN A CHANNEL SECTION 108316.5 UNIFORM CHANNEL FLOW 1084

16.5.1 The Chezy's formula 108616.5.2 The Ganguillet and Kutter formula 108616.5.3 The Manning's formula 1086

16.6 MOST ECONOMICAL SECTION OF CHANNEL 108816.6.1 Rectangular channel section 108916.6.2 Trapezoidal channel section 109016.6.3 Triangular channel section 109316.6.4 Circular channel section 1095

Contents xix

16.7 OPEN CHANNEL SECTION FOR CONSTANT VELOCITY ATALL DEPTHS OF FLOW

1098

16.8 SPECIFIC ENERGY AND CRITICAL DEPTH 110016.8.1 Total energy and specific energy 110016.8.2 Relationship between specific energy and depth 110116.8.3 Critical depth (yc) 110316.8.4 The general equation of critical flow 110316.8.5 An application of the critical depth line 110716.8.6 Channel Transitions 110716.8.7 Critical depth meters 1110

16.9 DETERMINATION OF AVERAGE-(MEAN) VELOCITY IN FLOWTHROUGH CHANNELS

1112

16.10 GRADUALLY VARIED FLOW 111416.10.1 General equation of gradually varied flow or Dynamic equ-

ation gradually varied flow 1115

16.10.2 Special values of varied flow equation 111716.10.3 Classification of surface profiles 111816.10.4 Methods of solution of the gradually varied flow equations 1123

16.11 RAPIDLY VARIED FLOW 112516.11.1 Hydraulic jump 112516.11.2 Forms of the hydraulic jump 112616.11.3 Uses of the hydraulic jump 112716.11.4 Analysis of the hydraulic jump 112816.11.5 Energy loss across a hydraulic jump 113016.11.6 Significance of the hydraulic jump equations 113216.11.7 Height and length of the hydraulic jump 113316.11.8 Location of hydraulic jump 1134

16.12 WAVES AND SURGES IN OPEN CHANNEL 113616.12.1 Waves and their classification 113616.12.2 Celerity of small solitary wave 113716.12.3 Monoclinal waves 113816.12.4 Surges 113916.12.5 The upstream positive surge 114016.12.6 The downstream positive surge 114116.12.7 Negative surge waves 1143

xx Fluid Mechanics



17 TWO DIMENSIONAL POTENTIAL FLOW THEORY 1217

17.1 INTRODUCTION 121717.2 STREAM FUNCTION AND VELOCITY POTENTIAL FOR POLAR

COORDINATES 1218

17.3 LAPLACE EQUATION FOR POLAR COORDINATES 121917.3.1 Methods of Solving Laplace's equation 1219

17.4 TWO DIMENSIONAL FLOW EXAMPLES 122017.4.1 Uniform flow 122017.4.2 Source and Sink flow 122117.4.3 Vortex 122417.4.4 Doublet 1226

17.5 SUPERPOSITION OF BASIC, PLANE POTENTIAL FLOWS 122917.5.1 Source with a uniform flow (Flow past a half body) 123017.5.2 Flow about a Rankine body 123117.5.3 Flow past a cylinder 123417.5.4 Flow past a circular cylinder with circulation 1237



18 COMPRESSIBLE FLUID FLOW 1265

18.1 INTRODUCTION 126518.2 THE SPEED OF SOUND 126518.3 MACH NUMBER 1268

18.3.1 Categories of compressible flow 126818.3.2 Mach angle, Mach line and Mach cone 1270

18.4 ONE DIMENSIONAL STEADY, ISENTROPIC FLOW OF ANIDEAL GAS

1271

18.4.1 Three reference speeds 1273

Contents xxi

18.4.2 Temperature, pressure and density ratios at the criticalstate

1274

18.4.3 The dimensionless velocity M* 127418.4.4 Effect of Mach number on compressibility 1277

18.5 VARIABLE AREA FLOW 127918.5.1 Effect of area variation on compressible flow 127918.5.2 Relation between area and Mach number for isentropic flow 128218.5.3 Mass flow rate as a function ratio or mach number 1285

18.6 FLOW THROUGH NOZZLES 128618.6.1 Convergent Nozzle 128618.6.2 Convergent- Divergent nozzle 1289

18.7 FLOW THROUGH DIFFUSERS 129418.7.1 Supersonic diffusers 129618.7.2 Wind tunnel operation with a supersonic diffusers 1296

18.8 RAYLEIGH AND FANNO FLOW 129718.8.1 Rayleigh curve 129718.8.2 The Fanno curves 129818.8.3 Important features of Fanno curve 1301

18.9 NORMAL SHOCK 130118.9.1 Governing relations of the normal shock 130218.9.2 Normal shock in a perfect gas 130318.9.3 Impossibility of a shock in subsonic flow 130818.9.4 Weak shock waves 1309



fluid - scientific publishers · 2019-03-19 · 8.4.3 determination of coefficient of discharge 433...

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