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Introduction to Fluid Mechanics and Fluid Machines Revised Second Edition S K SOM Department of Mechanical Engineering Indian Institute of Technology Kharagpur G Biswas Department of Mechanical Engineering Indian Institute of Technology Kanpur Tata McGraw-Hill Publishing Company Limited NEW DELHI McGraw-Hill Offices New Delhi New York St Louis San Francisco Auckland Bogotá Caracas Kuala Lumpur Lisbon London Madrid Mexico City Milan Montreal San Juan Santiago Singapore Sydney Tokyo Toronto New Prelimes.p65 3/11/08, 4:21 PM Published by Tata McGraw-Hill Publishing Company Limited, West Patel Nagar, New Delhi 110 008 Copyright © 2008, by Tata McGraw-Hill Publishing Company Limited No part of this publication may be reproduced or distributed in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise or stored in a database or retrieval system without the prior written permission of the publishers The program listings (if any) may be entered, stored and executed in a computer system, but they may not be reproduced for publication This edition can be exported from India only by the publishers, Tata McGraw-Hill Publishing Company Limited ISBN-13: 978-0-07-066762-4 ISBN-10: 0-07-066762-4 Managing Director: Ajay Shukla General Manager: Publishing—SEM & Tech Ed: Vibha Mahajan Asst Sponsoring Editor: Shukti Mukherjee Jr Editorial Executive: Surabhi Shukla Executive—Editorial Services: Sohini Mukherjee Senior Production Executive: Anjali Razdan General Manager: Marketing—Higher Education & School: Michael J Cruz Product Manager: SEM & Tech Ed: Biju Ganesan Controller—Production: Rajender P Ghansela Asst General Manager—Production: B L Dogra Information contained in this work has been obtained by Tata McGraw-Hill, from sources believed to be reliable However, neither Tata McGraw-Hill nor its authors guarantee the accuracy or completeness of any information published herein, and neither Tata McGraw-Hill nor its authors shall be responsible for any errors, omissions, or damages arising out of use of this information This work is published with the understanding that Tata McGraw-Hill and its authors are supplying information but are not attempting to render engineering or other professional services If such services are required, the assistance of an appropriate professional should be sought Typeset at Script Makers, 19, A1-B, DDA Market, Paschim Vihar, New Delhi 110 063, Text and cover printed at India Binding House A-98, Sector 65, Noida RCLYCDLYRQRXZ New Prelimes.p65 3/11/08, 4:21 PM Contents Preface to the Revised Second Edition Preface to the Second Edition Preface to the First Edition xi xii xiii Introduction and Fundamental Concepts 1.1 1.2 1.3 1.4 Definition of Stress Definition of Fluid Distinction between a Solid and a Fluid Fluid Properties Summary 16 Solved Examples 18 Exercises 25 2 Fluid Statics 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 28 Forces on Fluid Elements 28 Normal Stresses in a Stationary Fluid 28 Fundamental Equation of Fluid Statics 30 Units and Scales of Pressure Measurement 33 The Barometer 34 Manometers 35 Hydrostatic Thrusts on Submerged Surfaces 40 Buoyancy 45 Stability of Unconstrained Bodies in Fluid 46 Summary 53 Solved Examples 54 Exercises 73 Kinematics of Fluid 3.1 3.2 3.3 3.4 New Prelimes.p65 78 Introduction 78 Scalar and Vector Fields 78 Flow Field and Description of Fluid Motion Existence of Flow 97 Summary 98 Solved Examples 99 Exercises 109 79 3/11/08, 4:21 PM Contents vi Conservation Equations and Analysis of Finite Control Volumes 4.1 4.2 4.3 4.4 4.5 4.6 System 108 Conservation of Mass—The Continuity Equation 109 Conservation of Momentum: Momentum Theorem 121 Analysis of Finite Control Volumes 127 Euler’s Equation: The Equation of Motion for an Ideal Flow Conservation of Energy 145 Summary 150 References 151 Solved Examples 151 Exercises 166 Applications of Equations of Motion and Mechanical Energy 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 Introduction 175 Bernoulli’s Equation in Irrotational Flow Steady Flow Along Curved Streamlines Fluids in Relative Equilibrium 187 Principles of a Hydraulic Siphon 190 Losses Due to Geometric Changes 192 Measurement of Flow Rate Through Pipe Flow Through Orifices and Mouthpieces Summary 215 Solved Examples 217 Exercises 236 New Prelimes.p65 175 196 205 241 Introduction 241 Concept and Types of Physical Similarity 242 The Application of Dynamic Similarity— Dimensional Analysis 250 Summary 261 Solved Examples 261 Exercises 274 Flow of Ideal Fluids 7.1 7.2 7.3 7.4 138 176 179 Principles of Physical Similarity and Dimensional Analysis 6.1 6.2 6.3 108 277 Introduction 277 Elementary Flows in a Two-dimensional Plane Superposition of Elementary Flows 286 Aerofoil Theory 298 Summary 302 Solved Examples 302 References 310 Exercises 311 279 3/11/08, 4:21 PM Contents vii Viscous Incompressible Flows 8.1 8.2 8.3 8.4 8.5 314 Introduction 314 General Viscosity Law 315 Navier–Stokes Equations 316 Exact Solutions of Navier–Stokes Equations Low Reynolds Number Flow 337 Summary 342 References 342 Solved Examples 342 Exercises 355 324 Laminar Boundary Layers 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 9.11 9.12 359 Introduction 359 Boundary Layer Equations 359 Blasius Flow Over a Flat Plate 363 Wall Shear and Boundary Layer Thickness 369 Momentum-Integral Equations for Boundary Layer 372 Separation of Boundary Layer 373 Karman-Pohlhausen Approximate Method for Solution of Momentum Integral Equation over a Flat Plate 377 Integral Method for Non-Zero Pressure Gradient Flows 379 Entry Flow in a Duct 382 Control of Boundary Layer Separation 383 Mechanics of Boundary Layer Transition 384 Several Events of Transition 386 Summary 387 References 388 Solved Examples 388 Exercises 393 10 Turbulent Flow 398 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 10.10 Introduction 398 Characteristics of Turbulent Flow 398 Laminar–Turbulent Transition 400 Correlation Functions 402 Mean Motion and Fluctuations 403 Derivation of Governing Equations for Turbulent Flow 406 Turbulent Boundary Layer Equations 409 Boundary Conditions 410 Shear Stress Models 412 Universal Velocity Distribution Law and Friction Factor in Duct Flows for Very Large Reynolds Numbers 415 10.11 Fully Developed Turbulent Flow in a Pipe for Moderate Reynolds Numbers 419 New Prelimes.p65 3/11/08, 4:21 PM Contents viii 10.12 Skin Friction Coefficient for Boundary Layers on a Flat Plate Summary 424 References 425 Solved Examples 425 Exercises 430 11 Applications of Viscous Flows Through Pipes 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9 433 Introduction 433 Concept of Friction Factor in a Pipe Flow 433 Variation of Friction Factor 435 Concept of Flow Potential and Flow Resistance 439 Flow through Branched Pipes 441 Flow through Pipes with Side Tappings 448 Losses in Pipe Bends 450 Losses in Pipe Fittings 451 Power Transmission by a Pipeline 452 Summary 453 Solved Examples 454 Exercises 468 12 Flows with a Free Surface 12.1 12.2 12.3 12.4 12.5 472 Introduction 472 Flow in Open Channels 472 Flow in Closed Circular Conduits Only Partly Full Hydraulic Jump 489 Occurrence of Critical Conditions 492 Summary 491 Solved Examples 495 Exercises 503 487 13 Applications of Unsteady Flows 13.1 13.2 13.3 13.4 13.5 13.6 Introduction 505 Inertia Pressure and Accelerative Head 506 Establishment of Flow 507 Oscillation in a U-Tube 509 Damped Oscillation between Two Reservoirs Water Hammer 515 Summary 529 Solved Examples 530 Exercises 536 505 513 14 Compressible Flow 14.1 14.2 New Prelimes.p65 421 Introduction 538 Thermodynamic Relations of Perfect Gases 538 540 3/11/08, 4:21 PM Contents 14.3 14.4 14.5 14.6 14.7 ix Speed of Sound 546 Pressure Field due to a Moving Source 547 Basic Equations for One-Dimensional Flow 549 Stagnation and Sonic Properties 551 Normal Shocks 562 Summary 570 References 571 Solved Examples 571 Exercises 579 15 Principles of Fluid Machines 15.1 15.2 15.3 15.4 15.5 15.6 Introduction 581 Classifications of Fluid Machines 581 Rotodynamic Machines 583 Different Types of Rotodynamic Machines Reciprocating Pump 629 Hydraulic System 635 Summary 637 Solved Examples 639 Exercises 658 581 594 16 Compressors, Fans and Blowers 16.1 16.2 16.3 New Prelimes.p65 661 Centrifugal Compressors 661 Axial Flow Compressors 672 Fans and Blowers 678 Summary 684 References 685 Solved Examples 685 Exercises 692 Appendix A—Physical Properties of Fluids 695 Appendix B—Review of Preliminary Concepts in Vectors and their Operations Index 698 708 3/11/08, 4:21 PM Preface to the Revised Second Edition The book was first released in 1998 and the second edition was publised in 2004 The book has been extensively used by the faculty members and the students across the country The present revised edition is based on the comments received from the users of the book We take this opportunity to thank the individuals in various colleges/universities/institutes who provided inputs for the improvements In the revised second edition, the typographical errors has been corrected During the revision, the focus was primarily on the chapters pertaining to Fluid Machinery (Chapter 15 and Chapter 16) Some discussions have been expanded to make a better connection between the fundamentals and the applications The illustrations have been improved We are grateful to Ms Surabhi Shukla and Ms Sohini Mukherjee of McGraw-Hill for the efficient production of the revised second edition We hope that our readers will find the revised second edition more usueful S K Som G Biswas New Prelimes.p65 11 3/11/08, 4:21 PM Preface to the Second Edition Many colleges, universities and institutions have used this book since the publication of its first edition The colleagues and students of the authors have made valuable suggestions for the improvement of the book The feedback of the students has influenced our style of presentation in the revised edition The suggestions received from Prof V Eswaran, Prof R P Chhabra and Prof P S Ghoshdastidar of IIT Kanpur are gratefully acknowledged A major revision has been brought about in Chapter 4, especially, following the suggestions of Prof V Eswaran on the earlier version of the chapter Prof B S Murty of IIT Madras provided valuable advice on the earlier version of Chapters 9, 11 and 12 Prof S N Bhattacharya, Prof S Ghosh Moulic, Prof P K Das and Prof Sukanta Dash of IIT Kharagpur prompted several important modifications Input from Prof P M V Subbarao of IIT Delhi was indeed extremely useful Prof B.S Joshi of Govt College of Engineering, Aurangabad, put forward many meaningful suggestions The authors have made use of this opportunity to correct the errors and introduce new material in an appropriate manner The text on Fluid Machines has been enhanced by adding an additional chapter (Chapter 16) Some exciting problems have been added throughout the book We sincerely hope that the readers will find this revised edition accurate and useful S K SOM G B ISWAS New Prelimes.p65 12 3/11/08, 4:21 PM Appendix B Review of Preliminary Concepts in Vectors and Their Operations B.1 DEFINITION OF VECTOR Definition of scalar and vector quantities has been provided in Sec 3.2 Vector quantities are denoted by symbols either with an arrow or a cap at the top, like HHH H A B C , etc or A B C , etc A vector quantity A is written in terms of its components in a rectangular cartesian coordinates system (Fig B.1) as H H H H A = i Ax + j Ay + k Az z y L o Fig B.1 Ax J A k Az Ay x i Magnitude and components of a vector Appendix B $'' H H H H where i , j and k are the unit vectors and Ax, Ay and Az are the components of A H H in x, y, z directions respectively | A | is the magnitude of A From Fig B.1 H [| A |]2 = L2 + A2z = A2x + A2y + A2z Therefore, B.2 H | A| = Ax2 + Ay2 + Az2 ADDITION OF VECTORS Vector quantities are added in consideration of both magnitude and direction H H Thus, for addition of two vectors A and B we have from the rule of parallelogram (Fig B.2) A B C Fig B.2 Addition of vectors by the rule of parallelogram Hence, H H H H H H H C = A + B and C = i Cx + j Cy + k Cz H H H H H H = i Ax + j Ay + k Az + i Bx + j By + k Bz H H H = i (Ax + Bx) + j (Ay + By) + k (Az + Bz) Cx = Ax + Bx, Cy = Ay + By and Cz = Az + Bz H If a vector D equals to zero, then all its components are identically zero, i.e., Dx = Dy = Dz = B.3 PRODUCT OF VECTORS B.3.1 The Dot Product (or Scalar Product) H H The dot product of two vector quantities A and B is defined as H H H H A ◊ B = | A | | B | cos qAB where qAB is the angle between the vectors The H dot product is a scalar quantity H which physically represents the product of | A | with the component of | B | in the Introduction to Fluid Mechanics and Fluid Machines % H direction of A If qAB p/2, it is negative If qAB = p/2, A ◊ B = The dot products of unit vectors in a cartesian coordinate system are : H H H H H H j◊i =0 i◊i = k◊ i =0 H H H H H H j◊ j =1 i◊ j = k◊ j =0 H H H H H H i◊k = j ◊ k =0 k◊ k =1 H H H H H H H H Therefore, A ◊ B = ( i Ax + j Ay + k Az) ◊ ( i Bx + j By + k Bz) = Ax Bx + Ay By + Az Bz The following rules apply for the dot product H of H vectors: H H (i) The dot product is commutative, i.e A ◊ B = B ◊ A H H H H H H H (ii) The dot product is distributive, i.e A ◊ ( B + C ) = A ◊ B + A ◊ C H H H H H H (iii) The dot product is not associative, i.e., A ( B ◊ C ) π ( A ◊ B ) C B.3.2 Cross Product of Vectors H H H H The cross product of two vector quantities A and H B His written H Has A ¥ B It is a vector quantity whose Hmagnitude H is given by |HA ¥ HB | = | A | | B | sin qAB and is perpendicular toH both A and B HThe sense H ofH A ¥ B is given by the right-hand rule, that is, as A is rotated into B , then A ¥ B points in the direction of the right thumb This is shown in Fig B.3 z y A B B Rotation of A into B AB A o x Fig B.3 Cross product of vectors H H H H If A and B are parallel, then sin qAB = and A ¥ B = The cross products among unit vectors in a cartesian coordinate system are H H H H H H H H i ¥i = k ¥i = j j ¥i =–k Appendix B % H H H H H H H H i ¥ j = k k ¥ j = –i j ¥ j =0 H H H H H H H H i ¥k = – j k ¥k =0 j ¥k =i H H The cross product A ¥ B is usually written in a determinant form as H H H i j k H H A ¥ B = Ax Ay Az Bx By Bz By expanding the determinant we have H H H H H A ¥ B = i (Ay Bz – Az By) + j (Az Bx – Ax Bz) + k (Ax By – Ay Bx) The following properties apply for the cross product of H vectors: H H H (i) The cross product is not commutative, i.e A ¥ B π B ¥ A since H Hthe interchange of two rows changes the sign of a determinant, A ¥ B = H H – B ¥ A H H H H H L L (ii) The cross product is distributive, i.e., A ¥ ( B + C ) = A ¥ B + A + C H H H H L L (iii) The cross product is not associative, i.e., A ¥ ( B ¥ C ) π ( A ¥ B ) ¥ C B.4 DIFFERENTIATION OF VECTORS The derivative of a vector quantity is defined H H in the same way as it is done for a scalar quantity Let there be a vector A = A (t) then in rectangular coordinates Ax = Ax(t), Ay = Ay(t), Az = Az(t) H H H dA A (t + ∆ t ) − A (t ) = lim ∆ t →0 dt ∆t = lim Dt ®0 H H H i [ Ax ( t + Dt ) - Ax ( t ) + j [ Ay ( t + D t ) - Ay ( t )] + k [ Az ( t + D t ) - Az ( t )] Dt The limiting process applies to each term, and hence H Ay ( t + ∆ t ) − Ay (t ) H dA A (t + ∆ t ) − Ax ( t ) H = lim x j i + lim ∆ t →0 ∆t→0 dt ∆t ∆t A ( t + ∆ t ) − Az ( t ) H + lim z k ∆t→0 ∆t d Ax H d Ay H d Az H = i + j+ k dt dt dt H H Similarly, if A = A (x, y, z) that is, Ax = Ax (x, y, z), etc then H H ∂ Ax H ∂ Ay H ∂ Az ∂A = i + j +k ∂x ∂x ∂x ∂x H H ∂ Ax H ∂ Ay H ∂ Az ∂A = i + j +k ∂y ∂y ∂y ∂y Introduction to Fluid Mechanics and Fluid Machines % H H ∂ Ax H ∂ Ay H ∂ Az ∂A = i + j +k ∂z ∂z ∂z ∂z B.5 B.5.1 VECTOR OPERATOR — Definition of — The vector operator del, —, is defined as H H H —= i ∂ + j ∂ +k ∂ ∂x ∂y ∂z H ∂ H ∂ H ∂ +i + iz — = ir ∂ r θ r ∂θ ∂z (Cartesian coordinates) (cylindrical coordinates) H H H where, ir , iθ and iz are the unit vectors in r, q and z directions respectively in a cylindrical coordinate system Three possible products and other functions can be formed with the operator — as follows: B.5.2 Gradient When — operates on a differentiable scalar function, the resulting term is known as the gradient of the scalar function Let y (x, y, z) be a scalar function, H ∂y H ∂ y H ∂y Then, —y = gradient y = grad y = i + j +k ∂x ∂y ∂z It has to be noted that though y is a scalar function, —y is a vector function (or field) The pressure gradient, —p, i.e., the gradient of a pressure field p = p (x, y, z), was used in Equation (4.26d) in Sec 4.3.1 and in Eq (8.21) in Sec 8.3 while describing the equation of motion for the ideal and real fluids respectively B.5.3 Divergence The dot product of — and a vector function (or field) results in a scalar function H (or field) known as divergence For a vector field A (x, y, z) in a rectangular Cartesian coordinate system, H H H — ◊ A = divergence A (or div A ) H ∂ H H ∂ H ∂ H H + j +k = i ( i Ax + j Ay + k Az ) ∂x ∂y ∂z FG H IJ K ∂ Ax ∂ Ay ∂ Az + + ∂x ∂y ∂z H H H ∂i ∂ j ∂ k = = =0 Since, ∂x ∂y ∂z H H In cylindrical coordinates, if A = A (r, q, z), then = F GH I JK Appendix B H —◊ A = = FG iH H r %! H ∂ H ∂ H H H ∂ + iq + iz ( ir Ar + iq Aq + iz Az ) r ∂q ∂r ∂z IJ K ∂ Ar Ar ∂ Ar ∂ Az + + + r r ∂θ ∂r ∂z H H H H ∂ ir ∂ iz ∂ ir H ∂ iq Since, = = and = iq , = 0, ∂r ∂z ∂q ∂r FG H H H H H H H H ∂ iq = - ir , ir ir = iq iq = iz iz = ∂q H H The divergence of velocity vector V , i.e., — ◊ V was used to describe the continuity equation [Eq (4.3)] in Sec 4.2 IJ K B.5.4 Curl The cross product between — and a vector function (or field) results in a vector H H function (or field) known as curl For a vector field A = A (x, y, z) in Cartesian coordinates, H H H i j k ∂ ∂ ∂ H H — ¥ A = Curl A = ∂ x ∂ y ∂ z Ax Ay Az H ∂ Ay ∂ Ax H H ∂ Az ∂ Ay H ∂ Ax ∂ Az − —¥ A = i + j − +k − ∂y ∂z ∂z ∂x ∂x ∂y H H In cylindrical coordinates, A = A (r, q, z) Then FG H or, H —¥ A = IJ FG K H FG iH H IJ FG K H IJ K H ∂ H ∂ H H H ∂ + iq + iz ¥ ( ir Ar + iq Aq + iz Az ) r ∂q ∂r ∂z H ∂ H H ∂ H H ∂ H × (ir Ar ) + ir × (iθ Aθ ) + ir × (iz Az ) = ir ∂r ∂r ∂r IJ K r H ∂ H H ∂ H H ∂ H + iθ × (ir Ar ) + iθ × (iθ Aθ ) + iθ × (iz Az ) r ∂θ r ∂θ r ∂θ H ∂ H H ∂ H H ∂ H + iz × (ir Ar ) + iz × (iθ Aθ ) + iz × (iz Az ) ∂z ∂z ∂z FG H H F +i ¥GA H IJ + iH ¥ iH ∂ A + iH ¥ FG A H ∂r K ∂i I H H ∂ A J + i ¥ i r ∂q ∂r K H H ∂ Ar= H ∂ i =0 = ir ¥ ir + ir ¥ Ar r ∂r ∂r H H ∂ Az + ir ¥ iz ∂r r z z r q q r q r r q H ∂ iq ∂r IJ K Introduction to Fluid Mechanics and Fluid Machines %" + iq H ∂ ir r ∂q F ¥GA H r H H ∂iq + iq ¥ Aq r ∂q F GH H H ∂ Ar + iz ¥ ir ∂z H H + iz ¥ iz = iq I + iH ¥ iH JK q q =0 r ∂ Aq ∂q I + iH ¥ iH ∂ A + i ¥ F A ∂iH I JK GH r ∂q JK r ∂q H H I H F H H F I ∂i ∂A ∂i +i ¥GA +i ¥i +i ¥GA J ∂z H ∂z K H ∂z JK H I H F ∂A ∂i +i ¥GA ∂z H ∂z JK = - ir q z =0 r r z z z z q =0 =0 =0 z z z z z q q z q q =0 H H ∂ Aq H ∂ Az H ∂ Ar H Aq H ∂ Az — ¥ A = iz - iq - iz + iz + ir ∂r ∂r r ∂q r r ∂q H ∂ Ar H ∂ Aq + iq - ir ∂z ∂z or, and finally, H H ∂ Az ∂ A H H ∂ Ar ∂ Az q — ¥ A = curl A = ir + iq ∂r ∂z ∂z r ∂q H ∂ ∂ Ar + iz r Aq r ∂r ∂q H H The curl of velocity vector V , i.e., — ¥ V was used in describing the rotation of a fluid element in Sec 3.2.5 FG H IJ K FG b g H B.5.5 FG H IJ K IJ K Laplacian The scalar function obtained by the dot product —◊ — is known as the Laplacian and is given by the symbol —2 Therefore, in cartesian coordinates, —2 = — ◊— = = F iH ∂ + Hj ∂ + kH ∂ I ◊F iH ∂ + Hj ∂ + kH ∂ I GH ∂x ∂y ∂z JK GH ∂x ∂y ∂z JK ∂2 ∂ x2 In cylindrical coordinates, ∂2 ∂r2 ∂2 ∂ y2 + ∂2 ∂z FG iH ∂ + iH ∂ + iH ∂ IJ ◊FG iH ∂ + iH ∂ + iH ∂ IJ H ∂r r ∂q ∂z K H ∂r r∂q ∂z K H H H F ∂i ∂ H ∂ ∂i ∂ H ∂ I ∂ +i G +i + +i J+ r H ∂q ∂r r ∂q K ∂ z ∂q ∂r ∂q r ∂q — = — ◊— = = + q r z q r r r q z q q 2 Appendix B = ∂ ∂2 ∂2 ∂2 + + + r ∂r r ∂q ∂r ∂z = ∂ ∂2 ∂ ∂2 r + + r ∂r ∂r r ∂q ∂z FH %# IK B.6 VECTOR IDENTITIES B.6.1 — x — q = 0, where q is Any Scalar Function This relation may be verified by expanding it into components Therefore, in cartesian coordinates, F GH H ∂q H ∂q H ∂q — ¥ —q = — ¥ i + j +k ∂x ∂y ∂z H i ∂ / ∂x I JK H k ∂ / ∂z H j ∂ / ∂y ∂q / ∂x ∂q / ∂y ∂q / ∂z H =i F ∂ q - ∂ q I + Hj F ∂ q ∂ q I + kH F ∂ q - ∂ q I GH ∂y∂z ∂z∂y JK GH ∂z∂x ∂x∂z JK GH ∂x∂y ∂y∂x JK 2 2 2 If q = q (x, y, z) is a continuous, differentiable function, then ∂ 2q ∂ 2q ∂ 2q ∂ 2q ∂ 2q ∂ 2q ; = and = = ∂x∂y ∂y∂x ∂x∂z ∂z∂x ∂y∂z ∂z∂y consequently, — ¥ —q = The proof of the identity in cylindrical coordinates is a more lengthy process and is left as an exercise for the readers H H B.6.2 For Two Vector Functions ) and * , H H H H H H H H H H — A◊ B = A ◊ — B + B ◊— A + A ¥ — ¥ B + B ¥ — ¥ A d i d i d i d i d i In cartesian coordinates, we can write H H A ◊ B = Ax Bx + Ay By + Az Bz and so, d i RST ∂∂x b A B g + ∂∂x d A B i + ∂∂x b A B gUVW U| H R| ∂ ∂ ∂ A B i + b A B gV + j S bA B g + d |T ∂y |W ∂y ∂y H H H H H — A◊ B = — A◊ B i d i x x x x y y y y z z z z Introduction to Fluid Mechanics and Fluid Machines %$ H +k RS ∂ b A B g + ∂ d A B i + ∂ b A B gUV ∂z ∂z T ∂z W x x y y z z (B.1) H ∂ ∂ ∂ + Ay + Az A◊— = Ax ∂x ∂z ∂y Again, F I d AH ◊—i BH = iH GH A ∂∂Bx + A ∂∂By + A ∂∂Bz JK H F ∂B ∂B ∂B I +A +A + jGA J ∂y ∂z K H ∂x H F ∂B ∂B ∂B I + k GA +A +A J ∂y ∂z K H ∂x and so, x x x y y x z x y y y z z y x z z z (B.2) In a similar, way, we can write F I d BH ◊—i AH = iH GH B ∂∂Ax + B ∂∂Ay + B ∂∂Az JK HF ∂A ∂A ∂A I +B +B + j GB J ∂y ∂z K H ∂x HF ∂A ∂A ∂A I + k GB +B +B J ∂y ∂z K H ∂x x x y x d y z z y x z y y z x H H and A ¥ — ¥ B = x y z z (B.3) H i H j H k Ax Ay Az i F∂B ∂B I F∂B ∂B I F∂B ∂B I GH ∂y - ∂z JK GH ∂z - ∂x JK GH ∂x - ∂y JK H F ∂B ∂B ∂B ∂B I +A -A -A = i GA J ∂x ∂y ∂z K H ∂x H F ∂B ∂B ∂B ∂B I – j GA +A -A -A J ∂z ∂y ∂y K H ∂x H F ∂B ∂B ∂B ∂B I +A -A -A + k GA J ∂z ∂x ∂y K H ∂z y z x y y x x z z y x z y y z x y y y x x x x z x z z y z z (B.4) Appendix B %% F i GH H H H ∂ Ay ∂ Az ∂ Ax ∂ Ax Similarly, B ¥ — ¥ A = i By + Bz - By - Bz ∂x ∂x ∂y ∂z d I JK F GH H ∂ Ay ∂ Ay ∂ Ax ∂ Az + Bz - Bx - Bz – j Bx ∂x ∂z ∂y ∂y F GH H ∂ Ay ∂ Ax ∂ Az ∂ Az + By - Bx - By + k Bx ∂z ∂z ∂x ∂y I JK I JK (B.5) Adding Eqs (B.2), (B.3), (B.4) and (B.5) we have H H H H H H H A ◊— B + B ◊— + A ¥ — ¥ B + B ¥ — ¥ A d i d i d i d i HR∂ U ∂ ∂ = i S b A B g + d A B i + b A B gV ∂x ∂x T ∂x W H |R ∂ ∂ ∂ |U + j S b A B g + d A B i + b A B gV |T ∂y |W ∂y ∂y HR∂ U ∂ ∂ + k S b A B g + d A B i + b A B gV ∂z ∂z T ∂z W x x y y z z x x y y z z x x y y z z (B.6) comparison of Eqs (B.1) and (B.6) proves that H H H H H H H H H H — A ◊ B = A ◊— B + B ◊— A + A ¥ — ¥ B + B ¥ — ¥ A c h c h c h c h c h The relation was used in deriving the Bernoulli’s equation for irrotational flow in Sec 5.2 Index Absolute pressure, see Pressure Absolute viscosity, see Viscosity Acceleration: convective 83, 84 local 84 material 84 Adhesion 12, 15 Adverse pressure gradient 373 Airfoils 298 Angular deformations, see Deformation Angular momentum 126, 127 Apparent stress 409 Apparent viscosity Archimedes principle 46 Atmosphere 32, 33 Average velocity 146 Barometer 34, 35 Bernoulli’s equation: along a streamline 148, 149 irrotational flow 94, 176, 177 with head loss 149 Bingham plastic Blasius frictional formula 419, 420 Blasius equation 365 Blowers 678 Body force 28, 316, 317 Boundary layer 359 displacement thickness 371 momentum thickness 371 skin friction coefficient 370, 433 Buckingham’s p theorem 252, 253 Bulk modulus 10 Buoyancy 45, 46 Capillarity 15, 16 Cavitation 192, 609, 610, 623, 624, 631 Centre of buoyancy 46 of pressure 41, 42 Chezy coefficient 479 Chezy equation 478, 479 Choked flow, see Flow Circulation definition 283 Closed system, see System Coefficient of compressibility 538 of contraction 195, 200, 207, 208 of discharge 199, 207, 208, 214 of friction 433, 434 of velocity 207, 208 of viscosity Cohesion 12, 15 Colebrook formula 418 Compressibility 538 Compressors centrifugal 661 axial flow 672 Concentric cylinder 335 Conjugate depth 491 Conservation of energy 145-148 of mass, see Continuity equation of momentum 121-127 Constitutive equation Contact angle 15 Continuity equation differential form 110-115 integral form 119, 120 Continuum 2-3 Control volume 109, 127 Convective acceleration, see Acceleration Couette flow 327-329 Critical depth 485 Critical flow in open channel 492 Cylindrical coordinates Navier-Stokes equations 321 Index Darcy-Weisbach formula 333 Deformation angular 92 linear 91 Degree of reaction 676 Density Depth(s) of flow alternative 486 conjugate 491 critical 485 Dialatant Diffuser 555, 615, 623, 666 Dimension 251, 252 Dimensional analysis 250-254 Dimensions of flow 89, 90 Discharge coefficient, see Coefficient Displacement thickness 371 Doublet two dimensional 286 Draft tube 602, 608, 609 Drag form 377 skin friction 377 Dynamic viscosity, see Viscosity Dynamic similarity, see Similarity Eddy viscosity 412 Efficiency of hydraulic turbines 589, 590 of pumps 589, 590 Elastic wave propagation of (in channels) 482-483 Energy: internal 147 kinetic 146 mechanical 149 potential 146 Energy gradient line 476, 477 Equation of motion, see Conservation of mass Euler’s number 247, 249 Eulerian description of fluid motion 80-81 Fan 678 Fan laws 683 Fanno line flow 564 Fittings losses in, see Losses Flow: around a cylinder 289-291 around a sphere 340 choked 559 compressible 11-12, 538-580 critical 487 establishment of 507, 508 gradually varied 474 ideal inviscid irrotational 94 non-uniform 82, 85 rapid 487 rapidly varied 474 steady 81-82 three-dimensional 90 tranquil 487 two dimensional 90 uniform 81, 85 uniformly accelerated 187-190 unsteady 81-82 varied 474 Flow behaviour index Flow consistency index Flow measurement venturimeter 196-199 orificemeter 199-201 flow nozzle 201-202 Flow meter, see Flow measurement Flow work 147 Fluid 1-2 Fluid properties 3-16 Force body 28 buoyancy 45 elastic 246, 249 gravity 246, 248 hydrostatic 40 inertia 245 surface 28 surface tension 12-15, 246, 248 Forced vortex, see Vortex flows Francis turbine 600-606 Free surface 12-13, 472 Free vortex, see Vortex flows Friction factor: for pipe flow 418, 433-437 Froude number 248, 487 Ganguillet-Kutter formula 479 Gauge pressure 33-34 %' % Gas constant: ideal gas equation of state 10 characteristic gas constant 10 Geometric similarity, see Similarity Gradually varied flow, see Flow Guide vane 600, 608 Head gross (across a turbine) 601, 602 net (across a turbine) 601, 602 potential 149 pressure 149 velocity 149 Head loss: major 192 minor 192-196 in open channel flow 475, 476 in pipe bends 450, 451 in valves and fittings 451 coefficient 451 Hydraulic diameter 334 Hydraulic gradient line 476, 477 Hydraulic jump: basic equation 489-491 increase in depth across 491 head loss across 491, 492 Hydraulic turbines, see Turbines Hydrostatic force, see Force Hydrostatic pressure, see Pressure Hypersonic flow 539 Ideal fluid Ideal gas 10 Impeller 615, 616, 662 Impeller eye 662, 666 Inertial control volume, see Control volume Irrotional flow, see Flow Irrotational vortex, see Vortex flows Jump Hydraulic, see Hydraulic jump Kaplan turbine 606-608 Kinematic similarity, see Similarity Kinematic viscosity, see Viscosity Kinetic energy, see energy correction Factor 146, 147, 334 Index Kutta-Joukowski theorem 297 Lagrangian description of fluid motion 79-80 Laminar Flow: between parallel plates 327 Couette flow 327 parallel flow 325-336 in a circular pipe 329-331 Lapse rate 32 Lift 292-296 cylinder, no circulation 293 cylinder, with circulation 293-297 Linear deformation, see Deformation Linear momentum, see Conservation of momentum Local acceleration, see Acceleration Loss, see Head loss Loss coefficient, see Head loss Loss frictional 668 incidence 668 clearance 668 Lubrication 337 Lumley, J.L 401 Mach angle 549 Mach cone 549 Mach number 12, 539 Major loss, see Head loss Manning’s formula 479 Manning’s roughness coefficient 479 Manometer: simple u tube 35-37 inclined tube 37-38 inverted tube 38 micro 39-40 Measurement of flow, see Flow measurement Mechanical energy, see Energy Metacentric height 49-52 Minor losses, see Head loss Modified Bernoulli’s equation 487 Modulus of elasticity’s see Bulk modulus Momentum angular, see Angular momentum, Conservation of momentum Index linear, see Conservation of momentum Momentum equation: differential form 138-142 integral form 125, 126 Momentum thickness 371 Moody diagram 435, 437 Navier-Stokes equations 316-324 cylinderical coordinates 321 Net positive suction head 624 Newtonian fluid 5, Newton’s law of viscosity Nikuradse, J 417, 435 Non-Newtonian fluid Normal shock 562 Normal stress 1, 29 Mo slip condition Nozzle converging 559, 560 converging-diverging 561 DeLaval 555 Oblique shocks 568, 569 One-dimensional flow, see Dimensions of flow Open channel flow, see Channel flow Open system, see System Orifice 205-209 Orificemeter, see Flow measurement Oscillation of liquid column in a U tube 509-513 of liquid column between two reservoirs 513-515 Parallel flow 325-336 Pascal 33 Pascal’s law 29 Pathline 87, 88 Pelton wheel 594-600 Perfect gas 10 Piezometer 34 Pipe system: series 441, 442 parallel 443, 444 Pi-theorem 252, 253 Pitot static tube 205 Pitot tube 204 Poiseullie flow 329-331 Potential flow 278 % doublet 286-289 sink 280, 281 source 280, 281 with circulation 293-297 without circulation 289-293 Power law model Prandtl, L 359 Prandtl boundary layer 363 Prandtl mixing length 412-415 Pressure absolute 33 hydrostatic 29 stagnation 202-204 static 201-202 thermodynamic 29 Pressure wave 481, 482, 546 Properties of fluids, see Fluid properties Pseudoplastic Pump: centrifugal 615-624 axial 624-625 reciprocating 629-635 Rank of matrix 253 Rate of deformation, see Deformation Rayleigh’s indical method 260, 261 Rayleigh line flow 565, 566 Reynolds number 247, 333, 337, 361, 437 Rotating cylinders 335, 336 Runge-Kutta method 366 Schlichting, H 369, 388, 423 Separation 373-376 Shock normal 562 oblique 568, 569 Shooting technique 366 Skin friction coefficient 370 Slip factor 664 Stagnation properties 551-553 Stokes hypothesis 316 Stokes law of viscosity 316 Streak line 88-89 Stream function 116 Streamline 86-87 Streamtube 87 Stress % Sublayer 401 Subsonic flow 539 Supersonic flow 539 Surface force, see Force Surcace tension, 12-15 Surge tank 507-508 Surging 671 System: closed 108, 109 isolated 108, 109 open 108, 109 Tennekes, H 401 Tensor 318 Thermodynamic pressure, see Pressure Thoma’s cavitation parameter 610, 624 Tranquil flow, see Flow Translation 90 Turbine: impulse, see Pelton wheel reaction 600-612 Turbulent flows 398-432 rought plates 423 smooth plates 421, 422 mean-time averages 403 pipe flow 419-421 Navier-Stockes equations 406-409 Two-dimensional flow, see Flow Uniform flow, see Flow Uniformly accelerated flow, see Flow Universal velocity profile 417 Universal gas constant 540 Index Unsteady flow, see flow Varied flow, see Flow Vector, definition 79 Velocity defect law 416 Vena contracts 199, 207 Venturimeter, see Flow measurement Viscosity: apparent dynamic (or absolute) 4-5 eddy 412 kinematic Newton’s law 315 Stokes’ law 315 Von Karman, T momentum integral equation 377 Vortex street 385 Vortex flow 181-186, 283-285 forced 184, 185, 281, 285 free or irrotational vortex 181- 183, 283-284 spiral vortex 184 Vortex shedding frequency 385 Vorticity 94-96 Wake 374, 375 Water hammer 515-526 Weber number 248 Wetted perimeter 472 Zone of action 548, 549 Zone of silence 548, 549 ... shear stress, however small the shear stress may be As such, this continuous deformation under the application of shear stress constitutes a flow For example (Fig 1.2), if a shear stress t is... viscosity or simply the dynamic viscosity Another coefficient of viscosity known as kinematic viscosity is defined as n= m r Its dimensional formula is L2T–1 and is expressed as m2 /s in SI units The... the shear stressdeformation rate relationship is linear 1.4.6 Kinematic Viscosity The coefficient of viscosity m which has been discussed so far is known as the coefficient of dynamic viscosity
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