Table Of Content

ENGINEERING FLUID MECHANICS P. Balachandran ENGINEERING FLUID MECHANICS P. BALACHANDRAN Scientist Indian Space Research Organisation (ISRO) Trivandrum, Kerala Formerly, Assistant Professor TKM College of Engineering Kollam, Kerala New Delhi-110001 2011 ENGINEERING FLUID MECHANICS P. Balachandran © 2011 by PHI Learning Private Limited, New Delhi. All rights reserved. No part of this book may be reproduced in any form, by mimeograph or any other means, without permission in writing from the publisher. ISBN-978-81-203-4072-5 The export rights of this book are vested solely with the publisher. Published by Asoke K. Ghosh, PHI Learning Private Limited, M-97, Connaught Circus, New Delhi-110001 and Printed by Mohan Makhijani at Rekha Printers Private Limited, New Delhi-110020. To My Parents and Teachers Contents Preface xix 1. Fundamental Concepts and Fluid Properties 1–30 1.1 Introduction 1 1.1.1 Fluid 1 1.1.2 Liquids, Gases and Vapours 2 1.2 Fluid Statics, Kinematics and Dynamics 2 1.3 Fluid Properties 3 1.3.1 Mass Density 3 1.3.2 Specific Weight 4 1.3.3 Specific Gravity 4 1.3.4 Specific Volume 5 1.3.5 Viscosity 5 1.3.6 Compressibility and Elasticity 9 1.3.7 Surface Tension and Capillarity 11 1.3.8 Vapour Pressure 14 1.4 Compressible Fluids 15 1.4.1 Equation of State 16 1.4.2 Thermodynamic Process 16 1.4.3 Isothermal and Isentropic Bulk Modulus 16 Illustrative Examples 17 Review Questions 26 Numerical Problems 27 v vi Contents 2. Analysis of Fluid at Rest 31–111 2.1 Introduction 31 2.2 Fluid Pressure 32 2.2.1 Intensity of Pressure 32 2.2.2 Pascal’s Law 32 2.2.3 Hydrostatic Law 33 2.2.4 Pressure Head 34 2.2.5 Atmospheric Pressure 36 2.2.6 Absolute and Gauge Pressure 36 2.2.7 Pressure Variation in a Compressible Fluid 37 2.2.8 Manometers 40 2.2.9 Differential Manometers 44 Illustrative Examples 47 2.3 Hydrostatic Force Analysis 56 2.3.1 Total Pressure and Centre of Pressure 57 2.3.2 Force on Immersed Planar Aligned (Vertical/ Horizontal) Surface 57 2.3.3 Force on Immersed Planar Non-aligned (Inclined) Surface 59 2.3.4 Force on Immersed Curved Surface 61 2.3.5 Pressure Diagram 62 2.3.6 Engineering Applications of Hydrostatic Force Analysis 62 Illustrative Examples 66 2.4 Analysis of Floating and Submerged Bodies 80 2.4.1 Forces on Immersed Bodies 80 2.4.2 Buoyant Force and Centre of Buoyancy 80 2.4.3 Archimedes Principle 80 2.4.4 Principle of Floatation 82 2.4.5 Metacentre and Metacentric Height 82 2.4.6 Evaluation of Metacentric Height 83 2.4.7 Experimental Method for Metacentric Height 85 2.4.8 Period of Oscillation 86 2.4.9 Stability of Floating and Submerged Bodies 87 Illustrative Examples 89 2.5 Liquid in Relative Rest 95 2.5.1 Liquid Subjected to Uniform Horizontal Acceleration 95 2.5.2 Liquid Subjected to Uniform Vertical Acceleration 97 2.5.3 Liquid Subjected to Uniform Acceleration along Inclined Plane 98 2.5.4 Liquid Subjected to Uniform Rotation 99 Illustrative Examples 101 Review Questions 104 Numerical Problems 107 Contents vii 3. Kinematic Analysis of Fluid Flow 112–169 3.1 Introduction 112 3.2 Types of Flow 112 3.2.1 Lagrangian and Eulerian Approaches 112 3.2.2 One, Two and Three-dimensional Flows 113 3.2.3 States of Flow 114 3.2.4 Steady and Unsteady Motion 114 3.2.5 Uniform and Non-uniform Flow 115 3.3 Description of Fluid Flow 115 3.3.1 Velocity and Acceleration of Fluid Particles 116 3.3.2 Streamlines and Stream Tube 117 3.4 Flow Rate and Continuity Equation 120 3.4.1 Principle of Conservation of Mass 120 3.4.2 Continuity Equation for One-Dimensional Steady Incompressible Flow 120 3.4.3 General Continuity Equation in Differential Form 121 3.4.4 Continuity Equation for Two-dimensional Flow 123 Illustrative Examples 124 3.5 Rotational and Irrotational Flows 128 3.5.1 Circulation and Vorticity 128 3.5.2 Possible Motions of a Fluid Element 131 3.5.3 Component of Rotation 132 3.6 Velocity Potential and Stream Function 133 3.6.1 Velocity Potential Function 133 3.6.2 Stream Function 134 3.6.3 Cauchy-Riemann Equations 136 3.6.4 Equi-potential Lines and Streamlines 137 Illustrative Examples 137 3.7 Flow of an Ideal Fluid (Potential Flow) 144 3.7.1 Flow Net 145 3.7.2 Potential Flow Patterns 146 3.7.3 Uniform Rectilinear Flow (Free Stream Flow) 146 3.7.4 Flow from a Source 148 3.7.5 Flow to a Sink 149 3.7.6 Free Vortex Flow 149 3.7.7 Source Flow with Uniform Rectilinear Flow (Flow Around Rankine Half Body) 150 3.7.8 Source and Sink of Equal Strength 152 3.7.9 Source and Sink with Uniform Rectilinear Flow (Flow Past Rankine Body) 154 3.7.10 Doublet (Dipole) 156 3.7.11 Doublet with Uniform Rectilinear Flow (Flow Past Cylinder) 156 viii Contents 3.7.12 Doublet with Uniform Flow and Vortex (Flow Past Cylinder with Circulation) 158 Illustrative Examples 160 Review Questions 165 Numerical Problems 166 4. Dynamic Analysis of Flow 170–229 4.1 Introduction 170 4.2 Energy Equation 170 4.2.1 Energy of a Flowing Fluid 170 4.2.2 Bernoulli’s Theorem 172 4.2.3 Bernoulli’s Equation in Terms of Energy Head 173 4.2.4 Assumptions in Bernoulli’s Equation 175 4.2.5 Proof of Bernoulli’s Equation 175 4.2.6 Kinetic Energy Correction Factor 177 4.3 Equations of Motion 178 4.3.1 Forces Influencing Fluid Motion 179 4.3.2 Simplified Forms of Equations of Motion 179 4.3.3 Euler’s Equations of Motion 180 4.3.4 Euler’s Equation along a Streamline 182 4.3.5 Bernoulli’s Equation Derived from Euler’s Equation 184 4.3.6 Hydrostatic Equation from Euler’s Equation 184 4.3.7 Bernoulli’s Equation for a Compressible Fluid 185 4.4 Practical Use of Bernoulli’s Theorem 186 4.4.1 Venturimeter 186 4.4.2 Nozzlemeter or Flow Nozzle 190 4.4.3 Orificemeter or Pipe Orifice 191 4.4.4 Pitot Tube 193 4.4.5 Free Liquid Jet 194 Illustrative Examples 196 4.5 Impulse Momentum Theorem 205 4.5.1 Impulse–Momentum Principle 205 4.5.2 Impulse–Momentum Equations for Steady Flow 205 4.5.3 Momentum Correction Factor 207 4.5.4 Force Exerted on Pipe Bend 208 4.5.5 Force Exerted on Nozzle 209 4.5.6 Force Exerted by a Jet on Plane Surface 209 Illustrative Examples 210 4.6 Angular Momentum Principle 214 4.7 Vortex Motion 216 4.7.1 Types of Vortex Motion 216 4.7.2 Fundamental Equation for Vortex Flow 216 4.7.3 Forced Vortex 218 4.7.4 Free Vortex 221 Contents ix Illustrative Examples 222 Review Questions 225 Numerical Problems 226 5. Analysis of Incompressible Flow 230–326 5.1 Introduction 230 5.1.1 Ideal and Real Flows 230 5.1.2 Laminar and Turbulent Flows 231 5.2 Flow Regimes and Reynolds Number 232 5.2.1 Reynolds Experiment 232 5.2.2 Determination of Critical Reynolds Number 233 5.2.3 Reynolds Number 234 5.3 Laminar Flow 235 5.3.1 Navier–Stokes’ Equations of Motion 235 5.3.2 Pressure Gradient in Laminar Flow 239 5.3.3 Flow with Very Small and Very Large Reynolds Number 240 5.3.4 Steady Laminar Flow in a Circular Pipe (Hagen–Poiseuille Flow) 240 5.3.5 Laminar Flow in Inclined Pipe 246 5.3.6 Laminar Flow between Parallel Plates (Couette Flow) 247 5.3.7 Laminar Flow around Sphere (Stokes’ Law) 250 Illustrative Examples 251 5.4 Turbulent Flow 259 5.4.1 Characteristics of Turbulent Flow 259 5.4.2 Intensity and Scale of Turbulence 260 5.4.3 Reynolds Equations of Motion 264 5.4.4 Turbulence Modelling 264 5.4.5 Boussinesq Eddy Viscosity Concept 265 5.4.6 Prandtl’s Mixing-length Concept 265 5.4.7 Von–Karman Similarity Concept 266 5.4.8 Prandtl Universal Velocity Distribution for Turbulent Pipe Flow 266 5.4.9 Karman–Prandtl Velocity Distribution in Turbulent Pipe Flow 268 5.4.10 Power Law for Velocity in Smooth Pipes 272 5.4.11 Friction Factor for Smooth and Rough Pipes 272 5.4.12 Charts for Friction Factor in Pipe Flow 273 Illustrative Examples 275 5.5 Flow in Boundary Layer 278 5.5.1 Description of Boundary Layer—Growth of Boundary Layer over Flat Plate 278 5.5.2 Thickness of Boundary Layer 281 5.5.3 Pandtl’s Boundary Layer Equations 284 5.5.4 Von–Karman Momentum Integral Equation 285 x Contents 5.5.5 Shear Stress and Drag Force 288 5.5.6 Laminar Boundary Layer 288 5.5.7 Turbulent Boundary Layer 292 5.5.8 Pressure Distribution in the Boundary Layer 294 5.5.9 Boundary Layer Separation 295 Illustrative Examples 295 5.6 Flow around Submerged Bodies 301 5.6.1 Drag and Lift Force 302 5.6.2 Drag and Lift Coefficients 302 5.6.3 Drag on a Cylinder 306 5.6.4 Drag on an Airfoil 309 5.6.5 Lift on Cylinder 309 Illustrative Examples 315 Review Questions 319 Numerical Problems 323 6. Analysis of Flow in Pipes, Ducts, Orifices and Mouthpieces 327–398 6.1 Introduction 327 6.2 Loss of Energy in Pipes and Ducts 328 6.2.1 Major Loss—Loss of Energy due to Friction 329 6.2.2 Minor Losses 333 6.3 Graphical Reprsentation of Energy 337 6.3.1 Total Energy Line 338 6.3.2 Hydraulic Gradient Line 338 6.3.3 Hydraulic Gradient and Energy Gradient 339 Illustrative Examples 339 6.4 Fluid Circuit 345 6.4.1 Reservoirs Connected by Long Pipes 345 6.4.2 Pipe Discharging to Atmosphere 347 6.4.3 Pipe Connected in Series 347 6.4.4 Dupuits Equation for Equivalent Pipe 348 6.4.5 Pipes Connected Parallel 348 6.4.6 Siphon Pipe 349 6.4.7 Branching Pipe System 350 6.4.8 Analysis of Pipe Networks 351 6.5 Hydraulic Transmission of Power 352 6.5.1 Power Transmission Through Pipe 352 6.5.2 Pipe with Nozzle at Exit 353 Illustrative Examples 355 6.6 Hammer Blow (Water Hammer) 364 6.6.1 Pressure Rise due to Gradual Closure of Valve 365 6.6.2 Pressure Rise due to Rapid Closure of Valve 366 Illustrative Examples 368 6.7 Flow Through Orifice 370 6.7.1 Vena Contracta 370