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SCHAUM’S OUTLINE OF FLUID MECHANICS This page intentionally left blank SCHAUM’S OUTLINE OF FLUID MECHANICS MERLE C POTTER, Ph.D Professor Emeritus of Mechanical Engineering Michigan State University DAVID C WIGGERT, Ph.D Professor Emeritus of Civil Engineering Michigan State University Schaum’s Outline Series McGRAW-HILL New York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto Copyright © 2008 by The McGraw-Hill Companies, Inc All rights reserved Manufactured in the United States of America Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher 0-07-159454-X The material in this eBook also appears in the print version of this title: 0-07-148781-6 All trademarks are trademarks of their respective owners Rather than put a trademark symbol after every occurrence of a trademarked name, we use names in an editorial fashion only, and to the benefit of the trademark owner, with no intention of infringement of the trademark Where such designations appear in this book, they have been printed with initial caps McGraw-Hill eBooks are available at special quantity discounts to use as premiums and sales promotions, or for use in corporate training programs For more information, please contact George Hoare, Special Sales, at george_hoare@mcgraw-hill.com or (212) 904-4069 TERMS OF USE This is a copyrighted work and The McGraw-Hill Companies, Inc (“McGraw-Hill”) and its licensors reserve all rights in and to the work Use of this work is subject to these terms Except as permitted under the Copyright Act of 1976 and the right to store and retrieve one copy of the work, you may not decompile, disassemble, reverse engineer, reproduce, modify, create derivative works based upon, transmit, distribute, disseminate, sell, publish or sublicense the work or any part of it without McGraw-Hill’s prior consent You may use the work for your own noncommercial and personal use; any other use of the work is strictly prohibited Your right to use the work may be terminated if you fail to comply with these terms THE WORK IS PROVIDED “AS IS.” McGRAW-HILL AND ITS LICENSORS MAKE NO GUARANTEES OR WARRANTIES AS TO THE ACCURACY, ADEQUACY OR COMPLETENESS OF OR RESULTS TO BE OBTAINED FROM USING THE WORK, INCLUDING ANY INFORMATION THAT CAN BE ACCESSED THROUGH THE WORK VIA HYPERLINK OR OTHERWISE, AND EXPRESSLY DISCLAIM ANY WARRANTY, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE McGraw-Hill and its licensors not warrant or guarantee that the functions contained in the work will meet your requirements or that its operation will be uninterrupted or error free Neither McGraw-Hill nor its licensors shall be liable to you or anyone else for any inaccuracy, error or omission, regardless of cause, in the work or for any damages resulting therefrom McGraw-Hill has no responsibility for the content of any information accessed through the work Under no circumstances shall McGraw-Hill and/or its licensors be liable for any indirect, incidental, special, punitive, consequential or similar damages that result from the use of or inability to use the work, even if any of them has been advised of the possibility of such damages This limitation of liability shall apply to any claim or cause whatsoever whether such claim or cause arises in contract, tort or otherwise DOI: 10.1036/0071487816 Professional Want to learn more? We hope you enjoy this McGraw-Hill eBook! If you’d like more information about this book, its author, or related books and websites, please click here PREFACE This book is intended to accompany a text used in that first course in fluid mechanics which is required in all mechanical engineering and civil engineering departments, as well as several other departments It provides a succinct presentation of the material so that the students more easily understand those difficult parts If an expanded presentation is not a necessity, this book can be used as the primary text We have included all derivations and numerous applications, so it can be used with no supplemental material A solutions manual is available from the authors at MerleCP@sbcglobal.net We have included a derivation of the Navier– Stokes equations with several solved flows It is not necessary, however, to include them if the elemental approach is selected Either method can be used to study laminar flow in pipes, channels, between rotating cylinders, and in laminar boundary layer flow The basic principles upon which a study of fluid mechanics is based are illustrated with numerous examples, solved problems, and supplemental problems which allow students to develop their problem-solving skills The answers to all supplemental problems are included at the end of each chapter All examples and problems are presented using SI metric units English units are indicated throughout and are included in the Appendix The mathematics required is that of other engineering courses except that required if the study of the Navier– Stokes equations is selected where partial differential equations are encountered Some vector relations are used, but not at a level beyond most engineering curricula If you have comments, suggestions, or corrections or simply want to opine, please e-mail me at: merlecp@sbcglobal.net It is impossible to write an error-free book, but if we are made aware of any errors, we can have them corrected in future printings Therefore, send an email when you find one MERLE C POTTER DAVID C WIGGERT v Copyright © 2008 by The McGraw-Hill Companies, Inc Click here for terms of use This page intentionally left blank For more information about this title, click here CONTENTS CHAPTER I Basic Information 1.1 1.2 1.3 1.4 1.5 1.6 CHAPTER 3.3 3.4 Introduction Pressure Variation Manometers Forces on Plane and Curved Surfaces Accelerating Containers 20 20 22 24 27 39 Introduction Fluid Motion 3.2.1 Lagrangian and Eulerian Descriptions 3.2.2 Pathlines, Streaklines, and Streamlines 3.2.3 Acceleration 3.2.4 Angular Velocity and Vorticity Classification of Fluid Flows 3.3.1 Uniform, One-, Two-, and Three-Dimensional Flows 3.3.2 Viscous and Inviscid Flows 3.3.3 Laminar and Turbulent Flows 3.3.4 Incompressible and Compressible Flows Bernoulli’s Equation The Integral Equations 4.1 4.2 4.3 4.4 4.5 1 10 20 Fluids in Motion 3.1 3.2 CHAPTER Introduction Dimensions, Units, and Physical Quantities Gases and Liquids Pressure and Temperature Properties of Fluids Thermodynamic Properties and Relationships Fluid Statics 2.1 2.2 2.3 2.4 2.5 CHAPTER Introduction System-to-Control-Volume Transformation Conservation of Mass The Energy Equation The Momentum Equation vii 39 39 39 40 41 42 45 46 46 47 48 49 60 60 60 63 64 67 viii CHAPTER CONTENTS Differential Equations 5.1 5.2 5.3 5.4 CHAPTER CHAPTER 84 85 87 92 Dimensional Analysis and Similitude 97 6.1 6.2 6.3 97 97 102 Introduction Dimensional Analysis Similitude Internal Flows 7.1 7.2 7.3 7.4 7.5 7.6 7.7 CHAPTER Introduction The Differential Continuity Equation The Differential Momentum Equation The Differential Energy Equation 84 Introduction Entrance Flow Laminar Flow in a Pipe 7.3.1 The Elemental Approach 7.3.2 Applying the Navier –Stokes Equations 7.3.3 Quantities of Interest Laminar Flow Between Parallel Plates 7.4.1 The Elemental Approach 7.4.2 Applying the Navier –Stokes Equations 7.4.3 Quantities of Interest Laminar Flow between Rotating Cylinders 7.5.1 The Elemental Approach 7.5.2 Applying the Navier –Stokes Equations 7.5.3 Quantities of Interest Turbulent Flow in a Pipe 7.6.1 The Semi-Log Profile 7.6.2 The Power-Law Profile 7.6.3 Losses in Pipe Flow 7.6.4 Losses in Noncircular Conduits 7.6.5 Minor Losses 7.6.6 Hydraulic and Energy Grade Lines Open Channel Flow External Flows 8.1 8.2 8.3 Introduction Flow Around Blunt Bodies 8.2.1 Drag Coefficients 8.2.2 Vortex Shedding 8.2.3 Cavitation 8.2.4 Added Mass Flow Around Airfoils 110 110 110 112 112 113 114 115 115 116 117 118 118 120 120 121 123 123 125 127 127 129 130 145 145 146 146 149 150 152 152 Appendix B Vector Relationships A·B ẳ Ax Bx ỵ Ay By ỵ Az Bz A ã B ẳ Ay Bz Az By ịi ỵ Az Bx Ax Bz ịj ỵ Ax By Ay Bx ịk gradient operator : H ẳ @ @ @ iỵ jỵ k @x @y @z divergence of V ¼ H ·V ¼ curl of V ¼ H · V ẳ @u @v @w ỵ ỵ @x @y @z @w @v @u @w @v @u 2 iỵ jỵ k @y @z @z @x @x @y Laplaces equation: H2 f ¼ Irrotational vector field: H · V¼ 234 Copyright © 2008 by The McGraw-Hill Companies, Inc Click here for terms of use Appendix C Fluid Properties Table C.1 Properties of Water Temperature, T (–C) 10 15 20 30 40 50 60 70 80 90 100 Density, r (kg/m3) Viscosity, m [(N·s)/m2] Kinematic viscosity, n (m2/s) 999.9 1000.0 999.7 999.1 998.2 995.7 992.2 988.1 983.2 977.8 971.8 965.3 958.4 1.792 · 1023 1.519 1.308 1.140 1.005 0.801 0.656 0.549 0.469 0.406 0.357 0.317 0.284 · 1023 1.792 · 1026 1.519 1.308 1.141 1.007 0.804 0.661 0.556 0.477 0.415 0.367 0.328 0.296 · 1026 Table C.1E Temperature (–F) 32 40 50 60 70 80 90 100 120 140 160 180 200 212 Surface tension, s (N/m) Vapor pressure, pv (kPa) Bulk modulus, B (Pa) 0.0762 0.0754 0.0748 0.0741 0.0736 0.0718 0.0701 0.0682 0.0668 0.0650 0.0630 0.0612 0.0594 0.610 0.872 1.13 1.60 2.34 4.24 3.38 12.3 19.9 31.2 47.3 70.1 101.3 204 · 107 206 211 214 220 223 227 230 228 225 221 216 207 · 107 English Properties of Water Density (slug/ft3) Viscosity (lb·sec/ft2) Kinematic viscosity (ft2/sec) Surface tension (lb/ft) Vapor pressure (lb/in2) Bulk modulus (lb/in2) 1.94 1.94 1.94 1.94 1.94 1.93 1.93 1.93 1.92 1.91 1.90 1.88 1.87 1.86 3.75 · 1025 3.23 2.74 2.36 2.05 1.80 1.60 1.42 1.17 0.98 0.84 0.73 0.64 0.59 · 1025 1.93 · 1025 1.66 1.41 1.22 1.06 0.93 0.83 0.74 0.61 0.51 0.44 0.39 0.34 0.32 · 1025 0.518 · 1022 0.514 0.509 0.504 0.500 0.492 0.486 0.480 0.465 0.454 0.441 0.426 0.412 0.404 · 1022 0.089 0.122 0.178 0.256 0.340 0.507 0.698 0.949 1.69 2.89 4.74 7.51 11.53 14.7 293 000 294 000 305 000 311 000 320 000 322 000 323 000 327 000 333 000 330 000 326 000 318 000 308 000 300 000 235 Copyright © 2008 by The McGraw-Hill Companies, Inc Click here for terms of use 236 FLUID PROPERTIES Table C.2 [APPENDIX C Properties of Air at Atmospheric Pressure Temperature, T (–C) Density, r (kg/m3) Viscosity, m (N·s/m2) Kinematic viscosity, n (m2/s) 50 30 20 10 10 20 30 40 50 60 70 80 90 100 200 300 1.582 1.452 1.394 1.342 1.292 1.247 1.204 1.164 1.127 1.092 1.060 1.030 1.000 0.973 0.946 0.746 0.616 1.46 · l025 1.56 1.61 1.67 1.72 1.76 1.81 1.86 1.91 1.95 2.00 2.05 2.09 2.13 2.17 2.57 2.93 · 1025 0.921 · 1025 1.08 1.16 1.24 1.33 1.42 1.51 1.60 1.69 1.79 1.89 1.99 2.09 2.19 2.30 3.45 4.75 · 1025 Table C.2E Velocity of sound, c (m/s) 299 312 319 325 331 337 343 349 355 360 366 371 377 382 387 436 480 English Properties of Air at Atmospheric Pressure Temperature (–F) Density (slug/ft3) Viscosity [(lb·sec)/ft2] 220 20 40 60 68 80 100 120 160 200 300 400 1000 0.00280 0.00268 0.00257 0.00247 0.00237 0.00233 0.00228 0.00220 0.00213 0.00199 0.00187 0.00162 0.00144 0.000844 3.34 · 1027 3.38 3.50 3.62 3.74 3.81 3.85 3.96 4.07 4.23 4.50 4.98 5.26 7.87 · 1027 Kinematic viscosity (ft2/sec) 11.9 · 1025 12.6 13.6 14.6 15.8 16.0 16.9 18.0 18.9 21.3 24.1 30.7 36.7 93.2 · 1025 Velocity of sound (ft/sec) 1028 1051 1074 1096 1117 1125 1138 1159 1180 1220 1258 1348 1431 1839 APPENDIX C] Table C.3 Altitude (m) 500 1000 2000 4000 6000 8000 10 000 12 000 14 000 16 000 18 000 20 000 30 000 40 000 50 000 60 000 70 000 80 000 237 FLUID PROPERTIES Properties of the Standard Atmosphere Temperature (K) Pressure (kPa) Density (kg/m3) Velocity of sound (m/s) 288.2 284.9 281.7 275.2 262.2 249.2 236.2 223.3 216.7 216.7 216.7 216.7 216.7 226.5 250.4 270.7 255.8 219.7 180.7 101.3 95.43 89.85 79.48 61.64 47.21 35.65 26.49 19.40 14.17 10.35 7.563 5.528 1.196 0.287 0.0798 0.0225 0.00551 0.00103 1.225 1.167 1.112 1.007 0.8194 0.6602 0.5258 0.4136 0.3119 0.2278 0.1665 0.1216 0.0889 0.0184 4.00 · 1023 1.03 · 1023 3.06 · 1024 8.75 · 1025 2.00 · 1025 340 338 336 333 325 316 308 300 295 295 295 295 295 302 317 330 321 297 269 Table C.3E Altitude (ft) Temperature (–F) 1000 2000 5000 10 000 15 000 20 000 25 000 30 000 35 000 36 000 40 000 50 000 100 000 59.0 55.4 51.9 41.2 23.4 5.54 12.3 30.1 48.0 65.8 67.6 67.6 67.6 51.4 English Properties of the Atmosphere Pressure (lb/ft2) 2116 2014 1968 1760 1455 1194 973 785 628 498 475 392 242 23.2 Density (slug/ft3) Velocity of sound (ft/sec) 0.00237 0.00231 0.00224 0.00205 0.00176 0.00150 0.00127 0.00107 0.000890 0.000737 0.000709 0.000586 0.000362 3.31 · 1025 1117 1113 1109 1098 1078 1058 1037 1016 995 973 971 971 971 971 238 FLUID PROPERTIES [APPENDIX C Table C.4 Properties of Ideal Gases at 300 K (cv ¼ cp Gas Air Argon Carbon dioxide Carbon monoxide Ethane Helium Hydrogen Methane Nitrogen Oxygen Propane Steam Chemical formula Molar mass Ar CO2 k k ¼ cp =cv ) R cp k 28.97 39.94 44.01 ðft-lbÞ= slug-–R 1716 1244 1129 kJ= ðkg·KÞ 0.287 0.2081 0.1889 ðft-lbÞ= slug-–R 6012 3139 5085 kJ= ðkg·KÞ 1.004 0.5203 0.8418 1.40 1.667 1.287 CO 28.01 1775 0.2968 6238 1.041 1.40 C2H6 He H2 CH4 N2 O2 C3H8 H2O 30.07 4.003 2.016 16.04 28.02 32.00 44.10 18.02 1653 12 420 24 660 3100 1774 1553 1127 2759 0.2765 2.077 4.124 0.5184 0.2968 0.2598 0.1886 0.4615 10 700 31 310 85 930 13 330 6213 5486 10 200 11 150 1.766 5.193 14.21 2.254 1.042 0.9216 1.679 1.872 1.184 1.667 1.40 1.30 1.40 1.394 1.12 1.33 APPENDIX C] Table C.5 Liquid Properties of Common Liquids at Atmospheric Pressure and Approximately 16 to 21–C (60 to 70–F) Specific weight lb/ft Surface tension N/m 1.53 789 0.0015 0.022 Benzene 56.2 8828 1.75 902 0.0020 0.029 1.50 10.3 Carbon tetrachloride 99.5 15 629 3.09 1593 0.0018 0.026 12.50 86.2 Glycerin 78.6 12 346 2.44 1258 0.0043 0.063 · 1026 1.4 · 1025 50.5 7933 1.57 809 0.0017 0.025 — — 845.5 132 800 26.29 13 550 0.032 0.467 lb/in abs kPa abs — 25 2.31 · 10 1.59 · 1024 SAE 10 oil 57.4 9016 1.78 917 0.0025 0.036 — — SAE 30 oil 57.4 9016 1.78 917 0.0024 0.035 — — Water 62.4 9810 1.94 1000 0.0050 0.073 0.34 2.34 a FLUID PROPERTIES lb/ft 7744 Mercury kg/m Vapor pressure 49.3 a slug/ft Ethyl alcohol Kerosene N/m Density In contact with air 239 240 FLUID PROPERTIES [APPENDIX C Temperature (˚F) 20 60 100 140 2.0 220 Glycerine 1.0 1×10–2 Castor oil SAE-30 oil 2 1×10–3 SAE-10W-30 oil Viscosity (N·s/m2) 1×10–2 SAE-10W oil 1×10–4 Mercury Kerosene Carbon tetrachloride 1×10–3 1×10–5 Water 1×10–4 8 Viscosity (lb-sec/ft2) 1×10–1 180 Helium Octane Heptane Carbon dioxide 1×10–6 Methane Air 1×10–5 Hydrogen 20 40 60 80 100 2×10–6 120 Temperature (˚C) Figure C.1 Viscosity as a function of temperature (From R.W Fox and T.A McDonald, Introduction to Fluid Mechanics, 2nd ed., John Wiley & Sons, Inc., New York, 1978.) APPENDIX C] 241 FLUID PROPERTIES Temperature (˚F) 20 60 140 220 yc Gl eri ESA n 30 oil 10 E- SA 1×10–2 -3 W oi l Helium Hydrogen 1×10–4 SA E- 10 W oil Methane 1×10–5 Air 1×10–6 1×10–3 1×10–4 rbon dioxide Ca Kero sin e 2 Kinematic viscosity (m2/s) 180 m2/sec = 10.76 ft2/sec 1×10–3 100 Kinematic viscosity (ft2/sec) 1×10–2 Wate r Heptane Octane Carbon tetr achloride Mercury 1×10–5 1×10– 1×10–7 20 40 60 80 100 Temperature (˚C) Figure C.2 Kinematic viscosity as a function of temperature at atmospheric pressure (From R.W Fox and T.A McDonald, Introduction to Fluid Mechanics, 2nd ed., John Wiley & Sons, Inc., New York, 1978.) Appendix D Compressible Flow Table for Air Table D.1 M 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32 0.36 0.40 0.44 0.48 0.52 0.56 0.60 0.64 0.68 0.72 0.76 0.80 0.84 0.88 0.92 0.96 1.00 1.04 1.08 1.12 1.16 1.20 1.24 1.28 1.32 1.36 1.40 1.44 1.48 1.52 1.56 1.60 1.64 1.68 1.72 p/p0 1.0000 0.9989 0.9955 0.9900 0.9823 0.9725 0.9607 0.9470 0.9315 0.9143 0.8956 0.8755 0.8541 0.8317 0.8082 0.7840 0.7591 0.7338 0.7080 0.6821 0.6560 0.6300 0.6041 0.5785 0.5532 0.5283 0.5039 0.4800 0.4568 0.4343 0.4124 0.3912 0.3708 0.3512 0.3323 0.3142 0.2969 0.2804 0.2646 0.2496 0.2353 0.2217 0.2088 0.1966 T/T0 1.0000 0.9997 0.9987 0.9971 0.9949 0.9921 0.9886 0.9846 0.9799 0.9747 0.9690 0.9627 0.9560 0.9487 0.9410 0.9328 0.9243 0.9153 0.9061 0.8964 0.8865 0.8763 0.8659 0.8552 0.8444 0.8333 0.8222 0.8108 0.7994 0.7879 0.7764 0.7648 0.7532 0.7416 0.7300 0.7184 0.7069 0.6954 0.6840 0.6726 0.6614 0.6502 0.6392 0.6283 A/A* 14.4815 7.2616 4.8643 3.6727 2.9635 2.4956 2.1656 1.9219 1.7358 1.5901 1.4740 1.3801 1.3034 1.2403 1.1882 1.1452 1.1097 1.0806 1.0570 1.0382 1.0237 1.0129 1.0056 1.0014 1.000 1.001 1.005 1.011 1.020 1.030 1.043 1.058 1.075 1.094 1.115 1.138 1.163 1.190 1.219 1.250 1.284 1.319 1.357 M 1.76 1.80 1.84 1.88 1.90 1.92 1.96 2.00 2.04 2.08 2.12 2.16 2.20 2.24 2.28 2.32 2.36 2.40 2.44 2.48 2.52 2.56 2.60 2.64 2.68 2.72 2.76 2.80 2.84 2.88 2.92 2.96 3.00 3.04 3.08 3.12 3.16 3.20 3.24 3.28 3.32 3.36 3.40 3.44 p/p0 0.1850 0.1740 0.1637 0.1539 0.1492 0.1447 0.1360 0.1278 0.1201 0.1128 0.1060 0.9956 0.9352 0.8785 0.8251 0.7751 0.7281 0.6840 0.6426 0.6038 0.5674 0.5332 0.5012 0.4711 0.4429 0.4165 0.3917 0.3685 0.3467 0.3263 0.3071 0.2891 0.2722 0.2564 0.2416 0.2276 0.2146 0.2023 0.1908 0.1799 0.1698 0.1602 0.1512 0.1428 Isentropic Flow 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 T/T0 0.6175 0.6068 0.5963 0.5859 0.5807 0.5756 0.5655 0.5556 0.5458 0.5361 0.5266 0.5173 0.5081 0.4991 0.4903 0.4816 0.4731 0.4647 0.4565 0.4484 0.4405 0.4328 0.4252 0.4177 0.4104 0.4033 0.3963 0.3894 0.3827 0.3761 0.3696 0.3633 0.3571 0.3511 0.3452 0.3393 0.3337 0.3281 0.3226 0.3173 0.3121 0.3069 0.3019 0.2970 A/A* 1.397 1.439 1.484 1.531 1.555 1.580 1.633 1.688 1.745 1.806 1.869 1.935 2.005 2.078 2.154 2.233 2.316 2.403 2.494 2.588 2.686 2.789 2.896 3.007 3.123 3.244 3.370 3.500 3.636 3.777 3.924 4.076 4.235 4.399 4.570 4.747 4.930 5.121 5.319 5.523 5.736 5.956 6.184 6.420 242 Copyright © 2008 by The McGraw-Hill Companies, Inc Click here for terms of use M 3.48 3.52 3.56 3.60 3.64 3.68 3.72 3.76 3.80 3.84 3.88 3.92 3.96 4.00 4.04 4.08 4.12 4.16 4.20 4.24 4.28 4.32 4.36 4.40 4.44 4.48 4.52 4.54 4.58 4.62 4.66 4.70 4.74 4.78 4.82 4.86 4.90 4.94 4.98 6.00 8.00 10.00 p/p0 0.1349 0.1274 0.1204 0.1138 0.1076 0.1018 0.9633 0.9116 0.8629 0.8171 0.7739 0.7332 0.6948 0.6586 0.6245 0.5923 0.5619 0.5333 0.5062 0.4806 0.4565 0.4337 0.4121 0.3918 0.3725 0.3543 0.3370 0.3288 0.3129 0.2978 0.2836 0.2701 0.2573 0.2452 0.2338 0.2229 0.2126 0.2028 0.1935 0.0633 0.0102 0.0236 21 21 21 21 21 21 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 23 T/T0 0.2922 0.2875 0.2829 0.2784 0.2740 0.2697 0.2654 0.2613 0.2572 0.2532 0.2493 0.2455 0.2418 0.2381 0.2345 0.2310 0.2275 0.2242 0.2208 0.2176 0.2144 0.2113 0.2083 0.2053 0.2023 0.1994 0.1966 0.1952 0.1925 0.1898 0.1872 0.1846 0.1820 0.1795 0.1771 0.1747 0.1724 0.1700 0.1678 0.1219 0.0725 0.0476 A/A* 6.664 6.917 7.179 7.450 7.730 8.020 8.320 8.630 8.951 9.282 9.624 9.977 10.34 10.72 11.11 11.51 11.92 12.35 12.79 13.25 13.72 14.20 14.70 15.21 15.74 16.28 16.84 17.13 17.72 18.32 18.94 19.58 20.24 20.92 21.61 22.33 23.07 23.82 24.60 53.19 109.11 535.94 APPENDIX D] Table D.2 M1 1.00 1.04 1.08 1.12 1.16 1.20 1.24 1.28 1.30 1.32 1.36 1.40 1.44 1.48 1.52 1.56 1.60 1.64 1.68 1.72 1.76 1.80 1.84 1.88 1.92 1.96 2.00 2.04 2.08 2.12 2.16 2.20 2.24 2.28 2.30 2.32 2.36 2.40 2.44 2.48 2.52 2.56 2.60 2.64 2.68 2.72 2.76 2.80 2.84 2.88 2.92 2.96 3.00 3.04 3.08 M2 1.000 0.9620 0.9277 0.8966 0.8682 0.8422 0.8183 0.7963 0.7860 0.7760 0.7572 0.7397 0.7235 0.7083 0.6941 0.6809 0.6684 0.6568 0.6458 0.6355 0.6257 0.6165 0.6078 0.5996 0.5918 0.5844 0.5774 0.5707 0.5643 0.5583 0.5525 0.5471 0.5418 0.5368 0.5344 0.5321 0.5275 0.5231 0.5189 0.5149 0.5111 0.5074 0.5039 0.5005 0.4972 0.4941 0.4911 0.4882 0.4854 0.4827 0.4801 0.4776 0.4752 0.4729 0.4706 243 COMPRESSIBLE FLOW TABLE FOR AIR p2/p1 1.000 1.095 1.194 1.297 1.403 1.513 1.627 1.745 1.805 1.866 1.991 2.120 2.253 2.389 2.529 2.673 2.820 2.971 3.126 3.285 3.447 3.613 3.783 3.957 4.134 4.315 4.500 4.689 4.881 5.077 5.277 5.480 5.687 5.898 6.005 6.113 6.331 6.553 6.779 7.009 7.242 7.479 7.720 7.965 8.213 8.465 8.721 8.980 9.243 9.510 9.781 10.06 10.33 10.62 10.90 T2/T1 1.000 1.026 1.052 1.078 1.103 1.128 1.153 1.178 1.191 1.204 1.229 1.255 1.281 1.307 1.334 1.361 1.388 1.416 1.444 1.473 1.502 1.532 1.562 1.592 1.624 1.655 1.688 1.720 1.754 1.787 1.822 1.857 1.892 1.929 1.947 1.965 2.002 2.040 2.079 2.118 2.157 2.198 2.238 2.280 2.322 2.364 2.407 2.451 2.496 2.540 2.586 2.632 2.679 2.726 2.774 Normal Shock Flow p02/p01 1.000 0.9999 0.9994 0.9982 0.9961 0.9928 0.9884 0.9827 0.9794 0.9758 0.9676 0.9582 0.9476 0.9360 0.9233 0.9097 0.8952 0.8799 0.8640 0.8474 0.8302 0.8127 0.7948 0.7765 0.7581 0.7395 0.7209 0.7022 0.6835 0.6649 0.6464 0.6281 0.6100 0.5921 0.5833 0.5745 0.5572 0.5401 0.5234 0.5071 0.4991 0.4754 0.4601 0.4452 0.4307 0.4166 0.4028 0.3895 0.3765 0.3639 0.3517 0.3398 0.3283 0.3172 0.3065 M1 3.12 3.16 3.20 3.24 3.28 3.30 3.32 3.36 3.40 3.44 3.48 3.52 3.56 3.60 3.64 3.68 3.72 3.76 3.80 3.84 3.88 3.92 3.96 4.00 4.04 4.08 4.12 4.16 4.20 4.24 4.28 4.32 4.36 4.40 4.44 4.48 4.52 4.56 4.60 4.64 4.68 4.72 4.76 4.80 4.84 4.88 4.92 4.96 5.00 6.00 7.00 8.00 9.00 10.00 M2 0.4685 0.4664 0.4643 0.4624 0.4605 0.4596 0.4587 0.4569 0.4552 0.4535 0.4519 0.4504 0.4489 0.4474 0.4460 0.4446 0.4433 0.4420 0.4407 0.4395 0.4383 0.4372 0.4360 0.4350 0.4339 0.4329 0.4319 0.4309 0.4299 0.4290 0.4281 0.4272 0.4264 0.4255 0.4247 0.4239 0.4232 0.4224 0.4217 0.4210 0.4203 0.4196 0.4189 0.4183 0.4176 0.4170 0.4164 0.4158 0.4152 0.4042 0.3974 0.3929 0.3898 0.3875 0.3780 p2/p1 11.19 11.48 11.78 12.08 12.38 12.54 12.69 13.00 13.32 13.64 13.96 14.29 14.62 14.95 15.29 15.63 15.98 16.33 16.68 17.04 17.40 17.76 18.13 18.50 18.88 19.25 19.64 20.02 20.41 20.81 21.20 21.61 22.01 22.42 22.83 23.25 23.67 24.09 24.52 24.95 25.39 25.82 26.27 26.71 27.16 27.62 28.07 28.54 29.00 41.83 57.00 74.50 94.33 116.50 T2/T1 2.823 2.872 2.922 2.972 3.023 3.049 3.075 3.127 3.180 3.234 3.288 3.343 3.398 3.454 3.510 3.568 3.625 3.684 3.743 3.802 3.863 3.923 3.985 4.047 4.110 4.173 4.237 4.301 4.367 4.432 4.499 4.566 4.633 4.702 4.771 4.840 4.910 4.981 5.052 5.124 5.197 5.270 5.344 5.418 5.494 5.569 5.646 5.723 5.800 7.941 10.469 13.387 16.693 20.388 p02/p01 0.2960 0.2860 0.2762 0.2668 0.2577 0.2533 0.2489 0.2404 0.2322 0.2243 0.2167 0.2093 0.2022 0.1953 0.1887 0.1823 0.1761 0.1702 0.1645 0.1589 0.1536 0.1485 0.1435 0.1388 0.1342 0.1297 0.1254 0.1213 0.1173 0.1135 0.1098 0.1062 0.1028 0.994821 0.962821 0.93202l 0.902221 0.873521 0.845921 0.819221 0.793421 0.768521 0.744521 0.721421 0.699121 0.677521 0.656721 0.636621 0.617221 0.296521 0.153521 0.084921 0.049621 0.030421 244 COMPRESSIBLE FLOW TABLE FOR AIR Table D.3 M 1.00 1.04 1.08 1.12 1.16 1.20 1.24 1.28 1.32 1.36 1.40 1.44 1.48 1.52 1.56 1.60 1.64 1.68 1.72 1.76 1.80 1.84 1.88 1.92 1.96 2.00 2.04 2.08 2.12 2.16 2.20 2.24 2.28 2.32 2.36 2.40 2.44 2.48 2.52 2.56 2.60 2.64 2.68 2.72 2.76 2.80 2.84 2.88 2.92 2.96 3.00 y 0.3510 0.9680 1.735 2.607 3.558 4.569 5.627 6.721 7.844 8.987 10.146 11.317 12.495 13.677 14.861 16.043 17.222 18.397 19.565 20.725 21.877 23.019 24.151 25.271 26.380 27.476 28.560 29.631 30.689 31.732 32.763 33.780 34.783 35.771 36.746 37.708 38.655 39.589 40.509 41.415 42.307 43.187 44.053 44.906 45.746 46.573 47.388 48.190 48.980 49.757 [APPENDIX D Prandtl – Meyer Function m 90.00 74.06 67.81 63.23 59.55 56.44 53.75 51.38 49.25 47.33 45.58 43.98 42.51 41.14 39.87 38.68 37.57 36.53 35.55 34.62 33.75 32.92 32.13 31.39 30.68 30.00 29.35 28.74 28.14 27.58 27.04 26.51 26.01 25.53 25.07 24.62 24.19 23.78 23.38 22.99 22.62 22.26 21.91 21.57 21.24 20.92 20.62 20.32 20.03 19.75 19.47 M 3.04 3.08 3.12 3.16 3.20 3.24 3.28 3.32 3.36 3.40 3.44 3.48 3.52 3.56 3.60 3.64 3.68 3.72 3.76 3.80 3.84 3.88 3.92 3.96 4.00 4.04 4.08 4.12 4.16 4.20 4.24 4.28 4.32 4.36 4.40 4.44 4.48 4.52 4.56 4.60 4.64 4.68 4.72 4.76 4.80 4.84 4.88 4.92 4.96 5.00 y 50.523 51.277 52.020 52.751 53.470 54.179 54.877 55.564 56.241 56.907 57.564 58.210 58.847 59.474 60.091 60.700 61.299 61.899 62.471 63.044 63.608 64.164 64.713 65.253 65.785 66.309 66.826 67.336 67.838 68.333 68.821 69.302 69.777 70.245 70.706 71.161 71.610 72.052 72.489 72.919 73.344 73.763 74.176 74.584 74.986 75.383 75.775 76.162 76.544 76.920 m 19.20 18.95 18.69 18.45 18.21 17.98 17.75 17.53 17.31 17.10 16.90 16.70 16.51 16.31 16.13 15.95 15.77 15.59 15.42 15.26 15.10 14.94 14.78 14.63 14.48 14.33 14.19 14.05 13.91 13.77 13.64 13.51 13.38 13.26 13.14 13.02 12.90 12.78 12.67 12.56 12.45 12.34 12.23 12.13 12.03 11.92 11.83 11.73 11.63 11.54 INDEX A Absolute pressure, Accelerating containers, 27 Acceleration, 2, 40 Added mass, 152 Adiabatic, 10 Airfoils, 152 Angular velocity, 2, 42 Archimedes principle, 26 Area, Aspect ratio, 118 Atmospheric pressure, Completely turbulent zone, 126 Complex velocity potential, 155 Compressible flow, 48, 181 Conservation: of energy, 60, 62, 64 of mass, 60, 62, 63 of momentum, 61, 62, 67 Constitutive equations, 44, 63 Continuity equation, 44, 63 differential, 85 Continuum, Control volume, 61 Convective acceleration, 41 Couette flow, 116, 117, 121 Critical area, 186 Critical Reynolds number, 48, 160 B Bernoulli’s equation, 50, 87 Blades, 66 Blasius formula, 164 Boiling, Boundary conditions, 84 Boundary layer, 48, 159 Boundary-value problem, 84 Bubble, Buckingham p-theorem, 99 Bulk modulus, Buoyancy, 26 D Darcy – Weisbach equation, 114 Deflection angle, 192 Deflectors, 67 Demand curve, 211 Density, 2, Detached shock, 193 Developed flow, 456, 110 Differential continuity equation, 86 Diffuser, 185 Dilitant, Dimensional analysis, 97 Dimensional homogeneity, 97 Dimensions, table, Discharge, 63 Displacement thickness, 162, 167 Divergence, 86 Divergence theorem, 94 Doublet, 156 Drag, 146 on airfoil, 153 Drag coefficient, 146 curves, 147 table, 148 Droplet, Dynamic similarity, 102 C Capillary tube, Cauchy– Riemann equations, 155 Cavitation, 9, 150 Cavitation number, 150 Celsius scale, Center of pressure, 24 Centrifugal pump, 212 Centroid, 24 Channel flow, 115 Characteristic pump curve, 74, 142, 211 Characteristic dimension, 4, 47 Choked flow, 186 Chord, 146 Chezy – Manning coefficient, 131 Chezy – Manning equation, 131 Circulation, 156 Coefficient of thermal expansion, 16 245 Copyright © 2008 by The McGraw-Hill Companies, Inc Click here for terms of use 246 INDEX E Efficiency, 65 Energy, conservation of, 61 Energy equation, 11, 64, 92 Energy grade line, 129 Enthalpy, 10 Entrance flow, 110 turbulent, 111 Entrance length, 110 Equivalent length, 129 Eulerian description, 40 Euler number, 101 Euler’s equation, 88 Expansion fan, 195 Expansion waves, 196 External flow, 46, 145 I Ideal gas law, 10 Incompressible flow, 48, 86 Induced velocity, 188 Inertial reference frame, 41 Initial conditions, 84 Integral equations, 60 Interior nodes, 215 Interior loops, 215 Internal energy, 10 Internal flow, 110 Inviscid flow, 46, 145 Inviscid core length, 110 Irrotational flow, 42, 154 Isentropic, 10 nozzle flow, 184 Isotropic fluid, 88 F Fahrenheit scale, First law, 10 Flow field, 40 Flow rate, 2, 63 Force, 1,2 on a curved surface, 25 on a plane surface, 24 Free-stream flow, 48, 145, 160 Free-stream fluctuation intensity, 160 Frequency, Friction factor, 114 Froude number, 101 J Joukowsky equation, 221 G Gage pressure, Gas constant, 10 Gases, Gauss’ theorem, 94 Geometric similarity, 102 H Hardy Cross Method, 216 Harmonic functions, 155 Head, 51, 65 pump, 65 total, 51, 65 turbine, 65 velocity, 51, 65 Head loss, 65, 115 Heat rate, Heat transfer, 11 Homogeneous fluid, 90 Hydraulic grade line, 129 Hydraulic jump, 71 Hydraulic radius, 127 Hydrofoil, 150 K Karman integral equation, 162 Kelvin scale, Kinematic similarity, 102 Kinematic viscosity, 2, 8, 47 Kinetic-energy correction factor, 65 Kutta– Joukowski theorem, 159 L Lagrangian description, 40 Laminar boundary layer, 162 Laminar flow, 47 in a channel, 116 in a pipe, 112 Laplace’s equation, 154 Laplacian, 90 Length, Lift, 146 on an airfoil, 153 Line coefficient, 146 Line source, 155 Liquids, Local acceleration, 41 Local skin friction coefficient, 163 Loss coeffiecient, 64, 129 M Mach angle, 183 Mach cone, 183 Mach number, 49, 101, 183 Mach waves, 183 Manning n, 131 Manometer, 22 INDEX Mass, conservation of, 61, 63 Mass flux, 2, 63 Material derivative, 42 Mean free path, 4, 12 Method of images, 175 Minor losses, 129 Momentum, conservation of, 61 equation, 67 Momentum correction factor, 67 Momentum thickness, 162, 167 Moody diagram, 125 static, 51 total, 51 vapor, Pressure head, 51, 65 Pressure pulse, 223 Pressure waves, 224 Profile development region, 110 Pseudoloops, 215 Pseudoplastics, Pump curve, 74, 142, 211 Pump efficiency, 65 Pump head, 65 N Navier – Stokes equations, 44, 90 Network analysis, 219 Newtonian fluid, 7, 88 Newton’s second law, 1, 67, 87 Noncircular conduits, 127 Normal pressure variation, 52 Normal shock waves, 188 Normal stress, 4, 44, 87 No-slip condition, 8, 46 Nozzle flow, 186 Q Quasi-equilibrium, 10 O Oblique shock waves, 192 One-dimensional flow, 46 Open channel flow, 130 Outer region, 124 P Pascal, Pathline, 40 Piezometer, 51 Piezometric head, 207 Pipe flow, 112 Pipe networks, 215 Pipe systems, 208 Pitot-static probe, 51 Pitot tube, 51 Power, 2, 64 Power-law profile, 123 Plane flow, 46 Plastics, Poiseuille flow, 113, 114 Potential flow, 154 around a cylinder, 158 Power-law profile, 164 Prandtl boundary layer equation, 166 Prandtl– Meyer function, 196 Pressure, 2, 3, 5, 20 absolute, atmospheric, gage, R Raleigh pitot-tube formula, 199 Rankine oval, 175 Rankine scale, Ratio of specific heats, 10 Relative roughness, 125 Repeating variables, 100 Reynolds number, 48, 101 critical, 48, 160 Reynolds transfer theorem, 62 Roughness, relative, 125 Rotating containers, 27 Rotating cylinders, 119 S Scale, 103 Separated region, 52, 145 Shaft work, 64 Shear stress, 4, 87 Shear velocity, 123, 165 Shock waves, 183, 188 oblique, 192 strong, 193 weak, 193 Similitude, 97, 102 Sink, 159 Skin friction coefficient, 163 Sound wave, 182 Source strength, 156 Specific energy, 11 Specific gravity, Specific heat, Specific internal energy, 10 Specific weight, 2, Speed of sound, 8, 49, 182 Stagnation point, 51 Stagnation pressure, 51 Standard atmosphere, Static head, 210 247 248 Static pressure, 51 Statics, 20 Steady flow, 40 Steady turbulent flow, 47 Stokes flow, 146 Stokes hypothesis, 88 Streakline, 40 Stream function, 154 Streamline, 40 Streamtube, 40 Stress, Stress vector, Strouhal number, 101, 149 Subsonic, 2, 8, 183 Substantial derivative, 42 Supersonic, 183 Surface tension, 2, Superposition, 157 Surging, 219 Swamee and Jain formulas, 126 System, 10, 61 T Temperature, Thermal conductivity, 92 Thermal diffusivity, 92 Three-dimensional flow, 46 Time, Time average, 122 Torque, 2, Total head, 51, 65 Total pressure, 51 Transition zone, 126 Turbine efficiency, 65 Turbine head, 65 Turbulent boundary layer, 164 Turbulent flow, 47, 121 Turbulent zone, 122 Two-dimensional flow, 46 U Uniform flow, 46, 155 INDEX Units, table, Unsteady flow, 46 V Vacuum, Vanes, 67, 69 Vapor pressure, Velocity, Velocity field, 40 Velocity gradient, Velocity head, 51, 65 Velocity potential, 154 Velocity vector, 41 Venturi meter, 73 Venturi tuve, 202 Viscometer, Viscosity, 2, Viscous flow, 47 Viscous wall layer, 110, 122, 162 Volume, Von Karman integral equation, 162 Vortex, 156 Vortex shedding, 149 Vortex strength, 156 Vorticity, 43 W Wake, 145 Wall region, 123 Water hammer, 221 Waves, 224 Wave speed, 10 Weber number, 101 Wedge angle, 192 Wetted perimeter, 127 Wing span, 153 Work, 2, 10, 64 Work rate, 64 Z Zone of silence, 183 ...SCHAUM’S OUTLINE OF FLUID MECHANICS This page intentionally left blank SCHAUM’S OUTLINE OF FLUID MECHANICS MERLE C POTTER, Ph.D Professor Emeritus of Mechanical Engineering... corrected in future printings Therefore, send an email when you find one MERLE C POTTER DAVID C WIGGERT v Copyright © 2008 by The McGraw-Hill Companies, Inc Click here for terms of use This page intentionally... Introduction Fluid Motion 3.2.1 Lagrangian and Eulerian Descriptions 3.2.2 Pathlines, Streaklines, and Streamlines 3.2.3 Acceleration 3.2.4 Angular Velocity and Vorticity Classification of Fluid Flows
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