Fluid Mechanics with Laboratory Manual SECOND EDITION Bireswar Majumdar Professor Department of Power Engineering Jadavpur University Kolkata PHI Learning 11)7°M@ Delhi-110092 2016 FLUID MECHANICS WITH LABORATORY MANUAL, Second Edition Bireswar Majumdar © 2016 by PHI Learning Private Limited, 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-978-81-203-5180-6 The export rights of this book are vested solely with the publisher. Second Printing (Second Edition) November, 2015 Published by Asoke K. Ghosh, PHI Learning Private Limited, Rimjhim House, 111, Patparganj Industrial Estate, Delhi-110092 and Printed by Rajkamal Electric Press, Plot No. 2, Phase IV, HSIDC, Kundli-131028, Sonepat, Haryana. To My Daughters Basupurna and Basumita CONTENTS Preface Preface to the First Edition 1. Introduction 1.1 Definition of Fluid 1.2 Fluid Continuum 3 1.3 Properties of Fluid 3 1.4 Units and Dimensions 8 1.5 Governing Equations of Fluid Flows 9 1.6 Forces on Fluid Elements 10 1.6.1 Internal Forces Acting on a Fluid Element 11 1.6.2 Internal Forces Acting on a Fluid Element in Presence of Shear Stresses 12 2. Fluid Static 15-53 2.1 Pressure 15 2.2 Euler's Equation of Motion 16 2.3 Absolute and Gauge Pressure 20 2.4 Pressure Measuring Devices 21 2.4.1 Tube Gauges 21 2.4.2 Mechanical Pressure Gauges (Bourdon Gauge) 25 2.4.3 Other Types of Pressure Gauges 25 2.5 Hydrostatic Force on Submerged Surfaces 26 2.5.1 Force on Submerged Plane Surface 26 2.5.2 Force on Submerged Curved Surface 28 2.6 Buoyancy and Stability of Submerged and Floating Bodies 29 2.6.1 Determination of Metacentric Height 30 2.7 Equilibrium of Moving Fluids 32 2.7.1 Static Fluid Subjected to Uniform Rotation 32 2.7.2 Static Fluid Subjected to Uniform Acceleration 34 Solved Examples 37 Problems 52 Vi Contents 3. Fluid Kinematics 54-77 3.1 Types of Fluid Flow 54 3.2 Deformation, Translation and Rotation 58 3.3 Circulation and Vorticity 58 3.4 Streamline, Pathline and Streak Line 60 3.4.1 Stream Filament 62 3.4.2 Stream Tube 63 3.5 Vortex Line 63 3.5.1 Vortex Tube 63 3.5.2 Vortex Filament 63 3.5.3 Vortex Sheet 64 3.6 Fluid Acceleration 64 Solved Examples 66 Problems 77 4. Inviscid Fluid Flow 78-156 4.1 Law of Conservation of Mass 78 4.2 Euler's Equation of Motion 79 4.3 Bernoulli's Equation of Energy 82 4.3.1 Physical Significance of Bernoulli's Equation 83 4.4 Application of Bernoulli's Equation 83 4.4.1 Flow of Liquid from a Tank Through a Small Orifice 84 4.4.2 Flow Through Obstruction Type Flow Meter 85 4.4.3 Flow over Notches/Weir 88 4.4.4 Pitot Tube 90 4.4.5 Application in Fluid Machines 91 4.5 Equation of State 92 4.5.1 Bernoulli's Equation for Isothermal Flow 92 4.5.2 Bernoulli's Equation for Adiabatic Flow 92 4.6 Potential Flow 93 4.6.1 Consequences of the Definition of 0 94 4.6.2 Euler's Equation for Potential Flow 95 4.6.3 Special Cases 98 4.7 Unsteady Potential Flow (Surface Waves) 100 4.7.1 Straight, Moving Waves 100 4.7.2 Particle Path 105 4.7.3 Effect of Surface Tension 106 4.7.4 Standing Waves and Group Velocity 107 4.8 Stream Function (v') 108 4.8.1 Consequences of the Definition of Stream Function yr 109 4.9 Different Type of Inviscid Flows 111 4.9.1 Uniform Rectilinear Flow 112 4.9.2 Source Flow 112 4.9.3 Sink Flow 114 4.9.4 Irrotational Vortex 114 4.10 Principle of Superimposition 115 4.10.1 Doublet 115 Contents Vii 4.11 Axi-symmetric Flow 127 4.11.1 Properties of yi 128 4.11.2 Uniform Flow Parallel to x-Axis 131 4.11.3 3-D Source Flow 132 4.11.4 Superimposition of Uniform Flow and a Source Flow 133 4.11.5 Uniform Flow with a Source and a Sink Flow 136 4.11.6 3-D Doublet 137 4.11.7 Uniform Rectilinear Flow with a Doublet at Origin 138 4.11.8 Line Source or Line Sink 140 4.11.9 Flow over a Streamlined Body 141 Solved Examples 142 Problems 155 5. Integral Analysis of Fluid Flow 157-186 5.1 Relation between Extensive Property N and Intensive Property ti 159 5.2 Reynolds Transport Theorem (RTT) 159 5.2.1 RTT Based on Inertial/Fixed Frame of Reference 159 5.3 Conservation of Mass 161 5.4 Conservation of Linear Momentum (Inertial/Non-accelerating Frame of Reference) 164 5.5 Principle of Angular Momentum (Inertial Frame of Reference) 170 5.6 Conservation of Linear Momentum (Non-inertial Frame of Reference) 173 5.7 Principle of Conservation of Energy 175 Solved Examples 177 Problems 185 6. Differential Analysis of Fluid Flow 187-214 6.1 Conservation of Mass 187 6.2 Newton's Second Law of Motion 188 6.2.1 Stresses on a Fluid Element and Their Nature 189 6.2.2 Properties of Stress Tensor 190 6.3 Equation of Motion 192 6.4 Volumetric Deformation 193 6.5 Angular Deformation (Shear Deformation/Volumetric Strain) 194 6.6 Rotation of the Fluid Element 196 6.7 Vorticity () 197 6.8 Circulation (F) 197 6.9 Constitutive Equation Relating ru, cif, co;, (Navier—Stokes' Equation) 199 6.10 Energy Equation 203 6.11 Fundamental Equations in Cylindrical and Spherical Co-ordinates 207 Solved Examples 208 Problems 214 Viii Contents 7. Exact Solutions of Newtonian Fluid Flow 215-230 7.1 Parallel Flows 215 7.2 Flow between Two Parallel Plates 215 7.3 Flow Through Circular Pipe 218 7.4 Flow Through Annulus Pipe 223 7.5 Flow between Concentric Rotating Cylinders 224 7.6 Suddenly Accelerating Plate (Rayleigh's First Problem) (Unsteady Flow) 226 7.7 Flow with Thermal Parameters 228 8. Low Reynolds Number Flow 231-242 8.1 Flow Past a Sphere 233 8.2 Drag on the Sphere 235 8.3 Ossen Approximation (1910) 236 8.4 Hydrodynamic Theory of Lubrication 237 8.4.1 Pressure Distribution 241 9. Large Reynolds Number Flow 243-291 9.1 Governing Equation for Boundary Layer and Order of Magnitude 245 9.2 Properties of Boundary Layer Equation 248 9.3 Solution of Laminar Boundary Layer Equation (Blausius Solution, 1908) 250 9.3.1 Flow over a Flat Plate 250 9.3.2 Characteristics of Blasius Solution 255 9.3.3 Blasius Solution is Exact but Boundary Layer Equations are not Exact 259 9.3.4 Drag on the Plate 260 9.3.5 Flow over Surfaces for dp/dx # 0 261 9.4 Integral Approach for Boundary Layer Flows 263 9.4.1 Procedure for Solving Momentum Integral Equation 268 9.4.2 Solution for Boundary Layer Equation for a Body for Arbitrary Shape 272 9.5 Solution Procedure 274 9.5.1 Marching Technique 274 9.5.2 Energy Integral Method 276 Solved Examples 279 10. Transition and Turbulent Flows 292-321 10.1 Transition Phenomenon in Various Practical Flow Situations 293 10.1.1 Flow Through a Circular Pipe (Reynolds Experiment) 293 10.1.2 Transition Process in Boundary Layer 293 10.1.3 Flow over Bluff Bodies 294 10.1.4 Why Transition Takes Place? 295 10.1.5 How to Represent Disturbances Mathematically? 295 10.2 Fully Developed Turbulent Flows 298 10.2.1 Governing Equation of Motion for Turbulent Flows 301 Contents ix 10.3 Phenomenological Theories of Turbulence 304 10.3.1 Boussinesq Hypothesis 304 10.3.2 Prandtl's Mixing Length Hypothesis 305 10.4 Example 308 10.4.1 Channel Flow 308 10.4.2 No-slip Condition at the Edge of Laminar Sub-layer 308 10.4.3 Turbulent Flows in Absence of the Wall 310 10.5 Examples of Turbulent Flows in Practical Situation 311 10.5.1 Couette Flow 311 10.5.2 Turbulent Flow through a Circular Pipe 312 10.6 When a Pipe Surface is Considered as Smooth? 316 10.7 Correlation for 1-,„ 318 10.8 Criteria for Treating a Surface as Smooth 320 10.9 Comparison of Laminar and Turbulent Boundary Layers 321 11. Fundamentals of Compressible Fluid Flow 322-373 11.1 Review of Basic Thermodynamic Principle 322 11.2 Elastic Waves in a Compressible Fluid 326 11.3 Mach Number and Its Effect 329 11.3.1 Incompressible Medium 329 11.3.2 Compressible Medium 329 11.4 Isentropic Stagnation State 331 11.4.1 Stagnation Enthalpy 331 11.4.2 Stagnation Temperature 332 11.4.3 Isentropic Temperature Ratio for a Perfect Gas (To/T) 332 11.4.4 Isentropic Pressure Ratio for a Perfect Gas (po/p) 333 11.4.5 Compressibility Correction Factor (CCF) 334 11.5 Steady 1-D Isentropic Flow of a Perfect Gas Through a Variable Area Duct 335 11.5.1 Flow Through Nozzle 339 11.5.2 Flow Through Convergent—Divergent Nozzle (De-Laval nozzle) 344 11.6 Shock Wave 345 11.6.1 Normal Shock Wave 346 11.6.2 Oblique Shock Wave 350 11.7 Compressible Flow in Pipe of Constant Cross-Section 353 11.7.1 Adiabatic Flow with Friction 353 11.7.2 Steady Isothermal Flow in a Constant Area Duct with Friction 361 11.7.3 Comparison of the Results Obtained for Adiabatic Flow and Isothermal Flow 364 11.8 Steady One-dimensional Flow with Heat Transfer in a Constant Area Duct 364 11.8.1 Physical Interpretation of Rayleigh Line 366 11.8.2 Intersection of a Fanno Line and a Rayleigh Line 367 Solved Examples 368 Problems 371