Experimental Fluid Mechanics
For further volumes: http://www.springer.com/series/3837
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L.P. Yarin
The Pi-Theorem Applications to Fluid Mechanics and Heat and Mass Transfer
L.P. Yarin Technion-Israel Institute of Technology Dept. of Mechanical Engineering Technion City 32000 Haifa Israel
ISBN 978-3-642-19564-8 e-ISBN 978-3-642-19565-5 DOI 10.1007/978-3-642-19565-5 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2011944650 # Springer-Verlag Berlin Heidelberg 2012 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
To the blessed memory of my parents Professor Peter Yarin and Mrs. Leah Aranovich
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Preface
The book is devoted to the Buckingham Pi-theorem and its applications to various phenomena in nature and engineering. The accent is made on problems characteristic of heat and mass transfer in solid bodies, as well as in laminar and turbulent flows of liquids and gases. Such choice is not accidental. It is dictated by the requirements of modern technology and encompasses a vast majority of important problems related with drag and heat transfer experienced by solid bodies moving in viscous fluids. These problems involve the evaluation of temperature fields in media with constant and temperature-dependent thermal diffusivity, heat and mass transfer in boundary layers, pipe and jet flows, as well as thermal processes occurring in reactive media. In all these cases a uniform approach to the corresponding complex thermohydrodynamical problems is used. It is based on the direct application of the Pi-theorem to the analysis of two types of problems: those which admit a rigorous mathematical formulation, as well as those for which such formulation is unavailable. For the former problems our attention will be focused on the establishment of self-similarity which reduces the governing partial differential equations to the ordinary ones by means of the Pi-theorem, whereas for the latter problems the Pi-theorem will be used to reveal a set of the governing dimensionless groups. To a certain degree the choice of the problems is subjective. However, it allows the evaluation of the range of possible applications of the Pi-theorem and the peculiarities characteristic of the complex thermohydrodynamical processes in continuous media. The book consists of nine chapters. They deal with the basics of the dimensional analysis, the application of the Pi-theorem to find self-similarities and reduce partial differential equations to the ordinary ones. Then, such interrelated topics as the drag force, laminar flows in channels, pipes and jets are covered in detail. The discussion also involves kindred heat and mass transfer in natural, forced and mixed convection and in situations with phase change and chemical reactions. Some problems of turbulence theory are also covered in the framework of the Pi-theorem. In addition to the in-depth exposition of the basic theory and the generic problems, a number of worked examples of problems related to the application of the Pi-theorem to different hydrodynamic, heat and mass transfer questions are presented in the end of each chapter. They can be interest to the engineering and physics students.
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The book is intended to scientists and engineers interested in hydrodynamic and heat and mass transfer problems. It could also be useful to graduate students studying mechanical, civil and chemical engineering, as well as applied physics. L.P. Yarin
Acknowledgment
I am especially grateful and deeply indebted to my son Professor Alexander Yarin for some special consultations related to the applications of the dimensional analysis to thermohydrodynamics problems, many insightful suggestions and discussions, as well as multiple comments on the contents of the book. I am deeply obligated to my daughter Mrs. Elena Yarin and my granddaughter Miss Inna Yarin. Without their help this book would not have materialized.
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Contents
1
The Overview and Scope of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2
Basics of the Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2.1 Dimensional and Dimensionless Parameters . . . . . . . . . . . . . . . . . . . . . 3 2.2.2 The Principle of Dimensional Homogeneity . . . . . . . . . . . . . . . . . . . . . 7 2.3 Non-Dimensionalization of the Governing Equations . . . . . . . . . . . . . . . . 11 2.4 Dimensionless Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.4.1 Characteristics of Dimensionless Groups . . . . . . . . . . . . . . . . . . . . . . 18 2.4.2 Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.5 The Pi-Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.5.1 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.5.2 Choice of the Governing Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3
Application of the Pi-Theorem to Establish Self-Similarity and Reduce Partial Differential Equations to the Ordinary Ones . . . . 3.1 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Flow over a Plane Wall Which Has Instantaneously Started Moving from Rest (the Stokes Problem) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Laminar Boundary Layer over a Flat Plate (the Blasius Problem) . . . 3.4 Laminar Submerged Jet Issuing from a Thin Pipe (the Landau Problem) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Vorticity Diffusion in Viscous Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Laminar Flow near a Rotating Disk (the Von Karman Problem) . . . . . 3.7 Capillary Waves after a Weak Impact of a Tiny Object onto a Thin Liquid Film (the Yarin-Weiss Problem) . . . . . . . . . . . . . . . . . . . . . . . 3.8 Propagation of Viscous-Gravity Currents over a Solid Horizontal Surface (the Huppert Problem) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Thermal Boundary Layer over a Flat Wall (the Pohlhausen Problem) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39 39 44 47 51 54 55 58 60 63 xi
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3.10 Diffusion Boundary Layer over a Flat Reactive Plate (the Levich Problem) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4
5
Drag Force Acting on a Body Moving in Viscous Fluid . . . . . . . . . . . . . . . 4.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Drag Action on a Flat Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Motion with Constant Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Oscillatory Motion of a Plate Parallel to Itself . . . . . . . . . . . . . . . . . 4.3 Drag Force Acting on Solid Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Drag Experienced by a Spherical Particle at Low, Moderate and High Reynolds Numbers . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 The Effect of Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 The Effect of Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 The Effect of the Free Stream Turbulence . . . . . . . . . . . . . . . . . . . . . 4.3.5 The Influence of the Particle-Fluid Temperature Difference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Drag of Irregular Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Drag of Deformable Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Drag of Bodies Partially Submerged in Liquid . . . . . . . . . . . . . . . . . . . . . . . 4.7 Terminal Velocity of Small Spherical Particles Settling in Viscous Liquid (the Stokes Problem for a Sphere) . . . . . . . . . . . . . . . . . 4.8 Sedimentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.1 Dimensionless Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.2 Terminal Velocity of Heavy Grains . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.3 The Critical State of a Fluidized Bed . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Thin Liquid Film on a Plate Withdrawn Vertically from a Pool Filled with Viscous Liquid (the Landau-Levich Problem of Dip Coating) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Laminar Flows in Channels and Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Flows in Straight Pipes of Circular Cross-Section . . . . . . . . . . . . . . . . . . . 5.2.1 The Entrance Flow Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Fully Developed Region of Laminar Flows in Smooth Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Fully Developed Laminar and Turbulent Flows in Rough Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Flows in Irregular Pipes and Ducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Microchannel Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Non-Newtonian Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71 71 73 73 75 76 76 79 80 81 82 82 84 86 87 90 90 91 92
93 96 101 103 103 106 106 109 109 111 112 113
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5.6 Flows in Curved Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Unsteady Flows in Straight Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
116 120 123 129
6
Jet Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 The Far Field of Submerged Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 The Dimensionless Groups of Jet Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Plane Laminar Submerged Jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Laminar Wake of a Blunt Solid Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Wall Jets over Plane and Curved Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Buoyant Jets (Plumes) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
131 131 136 139 141 143 146 149 154 156
7
Heat and Mass Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Conductive Heat and Mass Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Temperature Field Induced by Plane Instantaneous Thermal Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Temperature Field Induced by a Pointwise Instantaneous Thermal Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Evolution of Temperature Field in Medium with Temperature-Dependent Thermal Diffusivity (The Zel’dovich-Kompaneyets Problem) . . . . . . . . . . . . . . . . . . . . . 7.3 Heat and Mass Transfer Under Conditions of Forced Convection . . 7.3.1 Heat Transfer from a Hot Body Immersed in Fluid Flow . . . . 7.3.2 The Effect of Particle Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 The Effect of the Free Stream Turbulence . . . . . . . . . . . . . . . . . . . . 7.3.4 The Effect of Energy Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.5 The Effect of Velocity Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.6 Mass Transfer to Solid Particles and Drops Immersed in Fluid Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Heat and Mass Transfer in Channel and Pipe Flows . . . . . . . . . . . . . . . . . 7.4.1 Couette Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 The Entrance Region of a Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.3 Fully Developed Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Thermal Characteristics of Laminar Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Heat and Mass Transfer in Natural Convection . . . . . . . . . . . . . . . . . . . . . . 7.6.1 Heat Transfer from a Spherical Particle Under the Conditions of Natural Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.2 Heat Transfer from Spinning Particle Under the Condition of Mixed Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
159 159 160 160 161
162 165 165 169 171 173 174 176 178 178 180 181 183 186 186 187
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7.6.3 Mass Transfer from a Spherical Particle Under the Conditions of Natural and Mixed Convection . . . . . . . . . . . . . . . . 7.6.4 Heat Transfer From a Vertical Heated Wall . . . . . . . . . . . . . . . . . . 7.6.5 Mass Transfer to a Vertical Reactive Plate Under the Conditions of Natural Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Heat Transfer From a Flat Plate in a Uniform Stream of Viscous, High Speed Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 Heat Transfer Related to Phase Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8.1 Heat Transfer Due to Condensation of Saturated Vapor on a Vertical Wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8.2 Freezing of a Pure Liquid (The Stefan Problem) . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
199 202 205 209
8
Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Decay of Isotropic Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Turbulent Near-Wall Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Plane-Parallel Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Pipe Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3 Turbulent Boundary Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Friction in Pipes and Ducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Friction in Smooth Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Friction in Rough Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Turbulent Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Eddy Viscosity and Thermal Conductivity . . . . . . . . . . . . . . . . . . . . 8.5.2 Plane and Axisymmetric Turbulent Jets . . . . . . . . . . . . . . . . . . . . . . . 8.5.3 Inhomogeneous Turbulent Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.4 Co-flowing Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.5 Turbulent Jets in Crossflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.6 Turbulent Wall Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.7 Impinging Turbulent Jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
211 211 215 217 217 220 221 222 222 223 224 224 229 232 238 245 248 252 254 258
9
Combustion Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Thermal Explosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Combustion Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Combustion of Non-premixed Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Diffusion Flame in the Mixing Layer of Parallel Streams of Gaseous Fuel and Oxidizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
261 261 265 268 271
189 190 193 195 199
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9.6 Gas Torches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7 Immersed Flames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xv
280 288 294 296
Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
.
Nomenclature
Chapter 2: Ar Bi Bo Br C Cd c Ca cP Da Da De De D D Ec Ek Eu Fd Fr Fg f g Gr h hm Ja k kB Kn Ku L Lh
Archimedes number Biot number Bond number Brinkman number Speed of sound Drag coefficient Concentration Capillary number Specific heat Damkohler number Darcy number Dean number Deborah number Diffusivity Permeability coefficient of porous medium Eckert number Ekman number Euler number Drag force Froude number Gravity force Frequency Gravity acceleration Grashof number Heat transfer coefficient, or enthalpy Mass transfer coefficient Jacob number Thermal conductivity Boltzmann’s constant Knudsen number Kutateladze number Characteristic length scale Height of liquid layer
xvii
xviii
Nomenclature
l Le m M Nu P DP Pe Ped Pr Qv q R r Ra Re Ri Ro rv Sc Se Sh St St Ta T DT t tr t0 u v v Vm W We x, y, z
Length of a pipe Lewis number Mass of a particle Mach number Nusselt number Pressure Pressure drop Peclet number Peclet number (for diffusion) Prandtl number Volumetric flow rate Heat of reaction Gas constant, or radius of curvature Cross-sectional radius of a pipe Rayleigh number Reynolds number Richardson number Rossby number Latent heat of vaporization Schmidt number Semenov number Sherwood number Stanton number Strouhal number Taylor number Temperature, or torque Temperature difference Time Relaxation time Observation time Particle velocity Velocity vector with components u, v and w in projections to the Cartesian axes x, y and z Specific volume Mass flow rate Rate of chemical reaction rate, or power (Watt) Weber number Cartesian coordinates
Greek Symbols b g d dT y L l m n r s
Coefficient of bulk expansion Ratio of the specific heat at constant pressure to the specific heat at constant volume (the adiabatic index) Boundary layer thickness Thermal boundary layer thickness Dimensionless temperature Angle between the axis of Earth rotation and the direction of fluid motion Mean free path Viscosity Kinematic viscosity Density Surface tension
Nomenclature t tr t0 f o oe
Time Relaxation time Observation time Dissipation function Angular velocity Angular velocity of Earth’s rotation
Subscripts f v w 1
Fluid Vapor Wall Undisturbed fluid at infinity
Chapter 3: a D g h J j P Pr Q r r; y; ’ S Sc t U u v
Thermal diffusivity Diffusivity Gravity acceleration Thickness of liquid layer Total momentum flux in jet Diffusion flux Pressure Prandtl number Source strenght Radial coordinate Spherical coordinates Surface or surface area Schmidt number Time Plate or flow velocity in x direction Fluid velocity Velocity vector with components vr ; vy; and v’ in spherical coordinate system
Greek Symbols a G d # n r s t ’ O
Thermal diffusivity, exponent Strength of an infinitely thin vortex line Thickness of the boundary layer Dimensionless variable Dimensionless temperature Kinematic viscosity Density Surface tension Shear stress Polar angle, dimensionless function Vorticity component normal to the flow plane, or angular velocity
xix
xx
Nomenclature
Subscript 1
Undisturbed fluid
Chapter 4: Ac cd cl d fd fl g l P Q R T 0
u u; v; w v Fr Re We
Acceleration parameter Drag coefficient Lift coefficient Diameter Drag force Lift force Gravity acceleration Scale of turbulence, length of plate Pressure Volumetric flow rate Radius Dimensionless turbulence intensity Root-mean square of turbulent fluctuations Velocity components Velocity vector Froud number Reynolds number Weber number
Greak Symbols a m n r s t g o
Angle Viscosity Kinematic viscosity Density Surface tension Shear stress at the wall Dimensionless angular velocity Angular velocity
Subscripts d l p 1
Drag Lift Particle Ambient
Chapter 5: d FI Fc
Diameter Inertial force Centrifugal force
Nomenclature Fn Fo k K l l P DP Po Q R Re r0 t u; v; w u0 umax v w0 x, y, z r; y; x
Friction force Fouier number Dean number, or roughness Modified Dean number The entrance length of pipe Characteristic length of pipe Pressure Pressure drop Poiseuille number Volumetric flow rate Radius of curvature of a torus Reynolds number Cross-sectional radius of a pipe Time Velocity components Initial velocity Maximum velocity Velocity vector Mean velocity Cartesian coordinates Cylindrical coordinates
Greek Symbols a b g d l m m0 nQ r t t0
Large semi-axis of an ellipse Small semi-axis of an ellipse Shear rate Ratio of pipe radius to its curvature Friction factor Viscosity Viscosity of Binham fluid Kinematic viscosity Bingham number Density Shear stress, geometric torsion Yield stress
Chapter 6: h Ix Jx k Mx P Pr Red T u v
Enthalpy Kinematic momentum flux Momentum flux Thermal conductivity Total moment-of-momentum flux Pressure Prandtl number Local Reynolds number Temperature Longitudinal velocity component Transversal velocity component
xxi
xxii
Nomenclature
Greek Symbols b d m n # r
Thermal expansion coefficient Jet thickness Viscosity Kinematic viscosity Excessive temperature Density
Subscripts 1 m
Undisturbed fluid Jet axis
Chapter 7: c cP cv D d E g H h j k kB l P Q q ql r T Tu v; u ve0 Ec Gr M Nu Pe Pr Ra Re Reo Sh St
Specific heat capacity, concentration Specific heat at constant pressure Specific heat at constant volume Diffusivity Diameter Pointwise energy release Gravity acceleration Channel height Heat transfer coefficient, rate of heat transfer, enthalpy Mechanical equivalent of heat Thermal conductivity Boltzmann’s constant Turbulence scale Pressure Strength of thermal source Heat flux Latent heat of freezing Radius Temperature Turbulence intensity Velocity Velocity fluctuation Eckert number Grashof number Mach number Nusselt number Peclet number Prandtl number Rayleigh number Reynolds number Rotational Reynolds number Sherwood number Stephan number
Nomenclature
Greek Symbols a b g d w m n r
Thermal diffusivity Thermal expansion coefficient Ratio of specific heat at constant pressure to specific heat at constant volume (the adiabatic index) Delta function; boundary layer thickness Radiant thermal diffusivity Viscosity Kinematic viscosity Density
Subscripts en f P T W 1
Entrance Front of thermal wave Pressure Thermal Wall Undisturbed flow
Chapter 8: A C d0 dc Fr Gx H hc I0 Jx l Pr P Re T u,v um We
Cross-sectional area of a jet Concentration Nozzle diameter Nozzle width Froude number Total mass flux Distance between the nozzle exit and the unperturbed liquid surface Cavity depth The exit kinematic momentum flux Total momentum flux Characteristic length Prandtl number Pressure Reynolds number Temperature Velocity components Centerline velocity Weber number
Greek Symbols aT d m
Eddy thermal diffusivity Jet half-width Dimensionless variable Viscosity
xxiii
xxiv mT n nT r s
Nomenclature Eddy viscosity Kinematic viscosity Eddy kinematic viscosity Density Surface tension
Subscripts G L
Gas Liquid
Chapter 9: c cP D E h k k0 Le lf P Pe Q1 Q2 q R Re T uf u0 W Wj z
Reactant concentration Specific heat Diffusivity Activation energy Enthalpy Chemical reaction constant; thermal conductivity Pre-exponential Lewis number Flame length Pressure Peclet number Heat release Heat losses Heat of reaction The universal gas constant Reynolds number Temperature Speed of combustion wave Speed of reactive mixture at the nozzle exit Rate of chemical reaction Rate of conversion of the j-th species Pre-exponential
Greek Symbols a d m n r tk tD O
Thermal diffusivity Frank-Kamenetskii parameter Viscosity Kinematic viscosity Density Characteristic kinetic time Characteristic diffusion time Stoichiometric oxidizer-to-fuel mass ratio
Nomenclature
Subscripts f o m 0
Fuel Oxidizer Maximum; axis Initial state Gas-liquid interface
xxv
.
Chapter 1
The Overview and Scope of the Book
The present book deals with the concepts and methods of the dimensional analysis and their applications to various thermohydrodynamic phenomena in continuous media. A comprehensive exposition of the results of systematic analysis of a number of important problems in this area in the framework of the Pi-theorem is given in nine chapters. In Chap. 2 the basics of the dimensional analysis are discussed. In particular, the principle of dimensional homogeneity and nondimensionalization of the mass, momentum, energy and diffusion equations and the corresponding initial and boundary conditions are described in this chapter. This is complemented by the introduction of several dimensionless groups and similarity criteria characteristic of hydrodynamic and heat and mass transfer problems. The Buckingham Pi-theorem is also formulated in Chap. 2. In Chap. 3 the Pi-theorem is used to establish self-similarity if it is admitted by a particular problem and reduce the corresponding partial differential equations to the ordinary ones. This approach to the search of self-similarity is illustrated with a number of generic situations corresponding to the Stokes, Blasius, Landau, von Karman, Yarin-Wess and Huppert hydrodynamic problems and the Pohlhausen and Levich heat and mass transfer problems. Chapter 4 deals with the drag force acting on a body moving in viscous fluid. The attention is focused on drag experienced by spherical particles at low, moderate and high Reynolds numbers. Such additional effects on the drag force as particle rotation, free stream turbulence and particle-fluid temperature difference are also analyzed. Then, some problems related to sedimentation are considered in the framework of the dimensional analysis. Finally, the Landau-Levich withdrawal problem on the thickness of thin liquid film on a vertical plate in dip coating process is tackled. As before, the consideration is based on the Pi-theorem. Chapter 5 is devoted to laminar channel and pipe flows. In this chapter the Pitheorem is applied to study stationary flows of Newtonian and non-Newtonian fluids in straight smooth and rough pipes, as well as in curved channels and pipes. In addition, some transient flows of Newtonian fluids are considered.
L.P. Yarin, The Pi-Theorem, Experimental Fluid Mechanics, DOI 10.1007/978-3-642-19565-5_1, # Springer-Verlag Berlin Heidelberg 2012
1
2
1 The Overview and Scope of the Book
The application of the Pi-theorem to the laminar submerged viscous jets is discussed in Chap. 6. These include jets propagating in an infinite space, as well as wall jets and wakes of solid bodies. Chapter 7 deals with the heat and mass transfer phenomena. The results presented in this chapter are related to the application of the Pi-theorem to conductive heat transfer in media with constant and temperature-dependent thermal conductivity, convective heat and mass transfer in forced, natural and mixed convection. The Pi-theorem is also used for the analysis of heat and mass transfer associated with hot particles immersed in fluid flow, channel and pipe flows, high speed gas flows, as well as flows with phase changes. Chapter 8 is devoted to turbulence. Here the Pi-theorem is used to study problems related to a decay of the uniform and isotropic turbulence, turbulent near-wall flows and submerged and wall turbulent jets. The results are used to interpret the wide range experimental data. Chapter 9 is related with the application of the Pi-theorem to combustion processes. A number of important problems of the combustion theory are considered in this chapter. These include the thermal explosion, propagation of combustion waves and aerodynamics of gas torches.
Chapter 2
Basics of the Dimensional Analysis
2.1
Preliminary Remarks
In this introductory chapter some basic ideas of the dimensional analysis are outlined using a number of the instructive examples. They illustrate the applications of the Pi-theorem in the field of hydrodynamics and heat and mass transfer. The systems of units and dimensional and dimensionless quantities, as well as the principle of dimensional homogeneity are discussed in Sect. 2.2. Section 2.3 deals with non-dimensionalization of the mass and momentum balance equations, as well as the energy and diffusion equations. In Sect. 2.4 the dimensionless groups characteristic of hydrodynamic and heat and mass transfer phenomena are presented. Here the physical meaning of several dimensionless groups and similarity criteria is discussed, In addition, similitude and modeling characteristic of the experimental investigations of thermohydrodynamic processes are considered. The Pi-theorem is formulated in Sect. 2.5.
2.2 2.2.1
Basic Definitions Dimensional and Dimensionless Parameters
Momentum, heat and mass transfer in continuous media occur in processes characterized by the interaction and coupling of the effects of hydrodynamic and thermal nature. The intensity of these interactions and coupling is determined by the magnitudes of physical quantities involved which characterize the physical properties of the medium, its state, motion and interactions with the surrounding boundaries and penetrating fields. The magnitudes of these quantities are determined experimentally by comparing the readings of the measuring devices with some chosen scales, which are taken as units of the measured characteristics, L.P. Yarin, The Pi-Theorem, Experimental Fluid Mechanics, DOI 10.1007/978-3-642-19565-5_2, # Springer-Verlag Berlin Heidelberg 2012
3
4
2
Basics of the Dimensional Analysis
e.g. length, mass, time, etc. For example, an actual pipe diameter, fluid velocity or temperature are expressed as d ¼ nL ; v ¼ mV ; T ¼ kT
(2.1)
where n; m and k are some numbers, whereas L ; V and T are units of length, velocity and temperature, respectively. The quantities which characterize flow and heat and mass transfer of fluids are related to each other by certain expressions based on the laws of nature. For example, the volumetric flow rate Qv of viscous fluid through a round pipe of radius r, and the drag force Fd acting on a small spherical particle slowly moving with constant velocity in viscous fluid are expressed by the Poiseuille and Stokes laws pr 4 DP 8ml
(2.2)
Fd ¼ 6pmur
(2.3)
Qv ¼
In (2.2) and (2.3) DP is the pressure drop on a length l; m is the fluid viscosity, and u is the particle velocity. Equations 2.2 and 2.3 show that units of the volumetric flow rate Qv and drag force Fd can be expressed as some combinations of the units of length, velocity, viscosity and pressure drop. In particular, the unit of r coincides with the unit of length L, of u is expressed through the units of length and time as LT 1 , the unit of ½m ¼ L1 MT 1 in addition involves the unit of mass, as well as the unit of the pressure drop ½DP ¼ L1 MT 2 (cf. Table 2.1). Here and hereinafter symbol ½ A denotes units of a dimensional quantity A. It is emphasized that the units of numerous physical quantities can be expressed via a few fundamental units. For example, we have just seen that the units of volumetric flow rate and drag force are expressed via units of length, mass and time only, as ½Qv ¼ L3 T 1 ; and ½Fd ¼ LMT 2 . A detailed information the units of measurable quantities is available in the book by Ipsen (1960). The possibility to express units of any physical quantities as a combination of some fundamental units allows subdividing all physical quantities into two characteristic groups, namely (1) primary or fundamental quantities, and (2) derivative (secondary or dependent) ones. The set of the fundamental units of measurements that is sufficient for expressing the other measurement quantities of a certain class of phenomena is called the system of units. Historically, different systems of units were applied to physical phenomena (Table 2.2). In the present book we will use mainly the International System of Units (Table 2.3). In this system of units (hereinafter called SI Units) an amount of a substance is measured with a special unit- mole (mol). Also, two additional dimensionless units: one for a plane angle- radian (rad), and another one for a solid angle- steradian (sr), are used. A detailed description of the SI Units can be found in the books of Blackman (1969) and Ramaswamy and Rao (1971).
2.2 Basic Definitions Table 2.1 Physical quantities
5
Quantity A. (Mechanical quantities) Acceleration Action Angle (plane) Angle (solid) Angular acceleration Angular momentum Area Curvature Surface tension Density Elastic modulus Energy (work) Force Frequency Kinematic viscosity Mass Momentum Power Pressure Time Velocity Volume B. (Thermal quantities) Enthalpy Entropy Gas constant Heat capacity per unit mass Heat capacity per unit volume Internal energy Latent heat of phase change Quantity of heat Temperature Temperature gradient Thermal conductivity Thermal diffusivity Heat transfer coefficient
Dimensions
Derived units
LT 2 ML2 T 1 1 1 T 2 ML2 T 1 L2 L1 MT 2 ML3 ML1 T 2 ML2 T 2 MLT 2 T 1 L2 T 1 M MLT 1 ML2 T 3 ML1 T 2 T LT 1 L3
m s2 kg m2 s1 rad: sterad: rad s2 kg m2 s1 m2 m1 kg s2 kg m3 2 kg m1 s J N s1 m2 s1 kg kg m s1 W N m2 s m s1 m3
ML2 T 2 ML2 T 2 y1 L2 T 1 y1 L2 T 2 y1 ML1 T 2 y1 ML2 T 2 L2 T 2 ML2 T 2 y L1 y MT 3 Ly1 L2 T 1 MT 3 y1
J J K 1 J kg 1 K 1 1 J kg1 K 3 1 J m K J J kg1 J K K m1 1 W m1 K m2 s1 W m2 K 1
The numerical values of the physical quantities expressed through fundamental units depend on the scales of arbitrarily chosen for the latter in any given system of units. For example, the velocity magnitude of a solid body moving in fluid, which is 1 m/s in SI units is 100 cm/s in the Gaussian CGS (centimeter, gram, second) System of Units. The physical quantities whose numerical values depend on the
6
2
Table 2.2 Systems of units Absolute Quantity Mass Force Length Time
CGS Gram Dyne Centimeter Second
MKS Kilogram Newton Meter Second
Basics of the Dimensional Analysis
Technical FPS Pound Poundal Foot Second
Table 2.3 International system of units-SI Quantity Mass Length Time Temperature Electric current Luminous intensity
CGS 9.81 g Gram-force Santimeter Second
MKS 9.81 kg Kilogram-force Meter Second
Units Kilogram Meter Second Kelvin Ampere Candela
FPS Slug Pound-force Foot Second
Abbreviation kg m s K A cd
fundamental units are called dimensional. For such quantities, units are derivative and are expressed through the fundamental unites according to the physical expressions involved. For example, units of the gravity force Fg ¼ mg are expressed through the fundamental units bearing in mind the previous expression and the fact that ½m ¼ M; and ½g ¼ LT 2 as Fg ¼ LMT 2
(2.4)
In fact, units of any physical quantity can be expressed through a power law1 ½ A ¼ La1 Ma2 T a3
(2.5)
where the exponents ai are found by using the principle of dimensional homogeneity. The quantities whose numerical values are independent of the chosen units of measurements are called dimensionless. For example, the relative length of a pipe l ¼ dl (where l and d are the length and diameter of the pipe, respectively) is dimensionless. Formally this means that l ¼ 1: In the general case, physical quantities can be characterized by their magnitude and direction. Such quantities as, for example, temperature and concentration are scalar and are characterized only by their magnitudes, whereas such quantities as velocity and force are vectors and are characterized by their magnitudes and directions. Vectors can also be characterized by introducing a so-called vector length L (Williams 1892). Projections of the vector length L on, say, the axes of
1
A demonstration of this statement can be found in Sedov (1993).
2.2 Basic Definitions
7
a Cartesian coordinate system x; y and z are denoted as Lx; Ly and Lz , respectively. A number of instructive examples of application of vector length for studying different problems of applied mechanics are presented in the monographs by Huntley (1967) and Douglas (1969). The application of the idea of vector length in studying of drag and heat transfer at a flat plate subjected to a uniform flow of the incompressible fluid is discussed by Barenblatt (1996) and Madrid and Alhama (2005). The expansion of a number of the fundamental units allows a significant improvement of the results of the dimensional analysis. For this aim it is useful to consider different properties the mass: (1) mass as the quantity of matter Mm , and (2) mass as the quantity of the inertia Mi . Similarly, using projections of a vector L on the Cartesian coordinate axes as the fundamental units it is possible to express the units of such derivative (secondary) quantities as volume V and velocity vector v as ½V ¼ Lx Ly Lz and ½u ¼ Lx T 1 ; ½v ¼ Ly T 1 ; and ½w ¼ Lz T1 where u,v and w denote the projections of v on the coordinate axes as is traditionally done in fluid mechanics. It is emphasized that using two different quantities of mass and projections of a vector allows one to reveal more clearly the physical meaning of the corresponding quantities. For example, the dimensions of work W in a rectilinear motion and torque T in rotation system of units LMT are the same L2 MT 2 ; whereas in the system of unitsLx Ly Lz MT they are different, namely ½W ¼ L2x MT 2 ; whereas ½T ¼ Lx Ly MT2 :
2.2.2
The Principle of Dimensional Homogeneity
Principle of dimensional homogeneity expresses the key requirements to a structure of any meaningful algebraic and differential equations describing physical phenomena, namely: all terms of these equations must to have the same dimensions. To illustrate this principle, we consider first the expression for the drag force acting on a spherical particle slowly moving in highly viscous fluid. The Stokes formula describing Fd reads Fd ¼ 6pmur
(2.6)
Here ½Fd ¼ LMT 2 is the drag force, ½m ¼ L1 MT 1 is the viscosity of the fluid, ½u ¼ LT 1 and ½r ¼ L are the particle velocity and its radius, respectively. It is easy to see that (2.6) satisfies the principle dimensional homogeneity. Indeed, substitution of the corresponding dimensions to the left hand side and the right hand side of (2.6) results in the following identity LMT 2 ¼ ðL1 MT 1 ÞðLT 1 ÞðLÞ ¼ LMT 2
(2.7)
As a second example, we consider the Navier–Stokes and continuity equations. For flows of incompressible fluids they read
8
2
Basics of the Dimensional Analysis
@v 1 þ ðv rÞv ¼ rP þ nr2 v @t r
(2.8)
rv¼0
(2.9)
where v ¼ ½LT 1 is the velocity vector, ½r ¼ L3 M,½n ¼ L2 T 1 and ½P ¼ L1 MT 2 are the density, kinematic viscosity n and pressure, respectively. It is seen that all the terms in (2.8) have dimensions LT 2 and in (2.9) have dimensions T 1 . There are a number of important applications of the principle of the dimensional homogeneity. For example, it can be used for correcting errors in formulas or equations, which is advisable to students. Take the expression for the volumetric rate of incompressible fluid through a round pipe of radius r as pr2 DP Qv ¼ 8m l
(2.10)
where Qv is the volumetric flow rate, DP is the pressure drop over an arbitrary section of the pipe length of length l. The dimension of the term on the left hand side in (2.10) is L3 T 1 , whereas of the one on the right hand side of this equation is LT 1 . Thus, (2.10) does not satisfy the principle of dimensional homogeneity. In order to find the correct form of the dependence of the volumetric flow rate on the governing parameters, we present (2.10) as follows p a1 a2 DP a3 Qv ¼ r m 8 l
(2.11)
where ai are unknown exponents. Bearing in mind the dimensions of Qv ; r; m and DP l , we arrive at the following system of algebraical equations for the exponents ai a1 a2 2a3 ¼ 3 a2 þ a 3 ¼ 0 a2 2a3 ¼ 1
(2.12)
From (2.12) it follows that the exponents ai are equal a1 ¼ 4; a2 ¼ 1; and a3 ¼ 1. Then, the correct form of (2.10) reads as Qv ¼
pr4 DP 8m l
(2.13)
2.2 Basic Definitions
9
The third example concerns the application the principle of dimensional homogeneity to determine the dimensionless groups from a set of dimensional parameters. Consider a set of dimensional parameters a1 ; a2 ak ; akþ1 an
(2.14)
Assume that k parameters have independent dimensions. Accordingly, the dimensions of the other n k parameters can be expressed as 0
0
½akþ1 ¼ ½a1 a1 ½ak ak nk
nk
½an ¼ ½a1 a1 ½ak ak
(2.15)
Therefore, the ratios akþ1 0
a a11
a
0
ak k
¼ P1
an ¼ Pnk nk a1 ank k
(2.16)
are dimensionless. Requiring that the dimensions of the numerator and denominator in the ratios (2.16) will be the same, we arrive at the system of algebraical equations for the unknown exponents. In conclusion, we give one more instructive example of the application of the principle of dimensional homogeneity for the description of the equation of state of perfect gas. The general form of the equation of state reads (Kestin v.1 (1966) and v.2 (1968)): FðP; vs ; TÞ ¼ 0
(2.17)
where P; vs and T are the pressure, specific volume and temperature, respectively. Equation 2.17 can be solved (at least in principle), with respect to any one of the three variables involved. In particular, it can be written as P ¼ f ðvs ; TÞ
(2.18)
The set of the governing parameters involved in (2.18) is incomplete since the dimension of pressure ½P ¼ L1 MT 2 cannot be expressed in the form of any combination of dimensions of specific volume ½vs ¼ L3 M1 and temperature ½T ¼ y. Therefore, the function f on right hand side in (2.18) must include some dimensional constant c
10
2
Basics of the Dimensional Analysis
P ¼ f ðc; vs ; TÞ
(2.19)
It is reasonable to choose as such a constant the gas constant R that account for the physical nature of the gas, but does not depend on its specific volume, pressure and temperature. Assuming that c ¼ R=g (g is a dimensionless constant), we write the dimension of this constant as ½c ¼ L2 T 2 y1 : All the parameters in (2.19) have independent dimensions. Then, according to the Pi-theorem (see Sect. 2.5), (2.19) takes the form P ¼ g1 ca1 vas 2 T a3
(2.20)
where g1 is a dimensionless constant. Using the principle of the dimensional homogeneity, we find the values of the exponents ai as a1 ¼ 1; a2 ¼ 1; a3 ¼ 1: Assuming g ¼ g1 , we arrive at the Clapeyron equation P ¼ RrT
(2.21)
The equation of state of perfect gas can be also derived directly by applying the Pi-theorem to solve the problems of the kinetic theory and accounting for the fact pressure of perfect gas results from atom (molecule) impacts onto a solid wall.2 Considering perfect gas as an ensemble of rigid spherical atoms (or molecules) moving chaotically in the space, we can assume that pressure of such gas is determined by atom (or molecule) mass m, their number per unit volume N and the average velocity squared
P ¼ f ðm; N; Þ
(2.22)
The dimensions of P and the governing parameters m; N and are ½P ¼ L1 MT 2 ; ½m ¼ M; ½ N ¼ L3 ; ¼ L2 T 2
(2.23)
All the governing parameters have independent dimensions. Therefore, the difference between the number of the governing parameters n and the number of the parameters with independent dimensions k equals zero. In this case the pressure can be expressed as Sedov (1993); P ¼ gma1 N a2 a3
(2.24)
where g is a dimensionless constant.
2 This idea was expressed first by D. Bernoulli in 1727 who wrote that pressure of perfect gas is related to molecule velocities squared.
2.3 Non-Dimensionalization of the Governing Equations
11
Using the principle of dimensional homogeneity, we find the values of the exponents in (2.24) as a1 ¼ a2 ¼ a3 ¼ 1: Then, (2.24) takes the form P ¼ gmN
(2.25)
Bearing in mind that m is directly proportional kB T (m ¼ g1 kB T; where g1 is a dimensionless constant), we arrive at the following equation P ¼ ekB TN
(2.26)
Here e ¼ gg1 is a dimensionless constant, ½kB ¼ L2 MT 2 y1 is Boltzmann’s constant, ½T ¼ y is the absolute temperature. Applying (2.26) to a unit mole of a perfect gas, we can write the known thermodynamic relations as N ¼ Nm ; kB ¼
mR ; mvs ¼ constant Nm
(2.27)
Here Nm is the Avogadro number, m is the molecular mass, vs is the specific volume, and ½ R ¼ L2 T 2 y1 is the gas constant. Then, (2.27) takes the form P ¼ rRT
(2.28)
Summarizing, we see that the pressure of perfect gas is directly proportional to the product of the gas density, gas constant and the absolute temperature and does not depend on the mass of individual atoms (molecules). Note that (2.28) can be obtained directly from the functional equation P ¼ f ðm; N; T; kB Þ(Bridgman 1922).
2.3
Non-Dimensionalization of the Governing Equations
It is beneficial in the analysis complex thermohydrodynamic phenomena to transform the system of mass, momentum, energy and species balance equations into a dimensionless form. The motivation for such transformation comes from two reasons. The first reason is related with the generalization of the results of theoretical and experimental investigations of hydrodynamics and heat and mass transfer in laminar and turbulent flows by presentation the data of numerical calculation and measurements in the form of dependences between dimensionless parameters. The second reason is related to the problem of modeling thermohydrodynamic processes by using similarity criteria that determine the actual conditions of the problem. The procedure of non-dimensionalization of the continuity (mass balance), momentum, energy and species balance equations is illustrated below by transforming the following model equation
12
2 n X
Basics of the Dimensional Analysis
ðiÞ
Aj ¼ 0
(2.29)
j¼1 ðiÞ
where Aj includes differential operators, some independent variables, as well as constants; superscript i refers to the momentum ði ¼ 1Þ; energy ði ¼ 2Þ; species ði ¼ 3Þ and continuity ði ¼ 4Þ equations, n is the total number of terms in a given equation. The terms in (2.29) account for different factors that affect the velocity, temperature and species fields: the inertia features of fluid, viscous friction, conductive and ðiÞ convective heat transfer, etc. These terms are dimensional. The dimension of Aj in the system of units LMTy is h i ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ Aj ¼ Laj Mbj T gj yej
(2.30)
where the values of the exponents a; b; g and e are determined by the magnitude of i and j; all the terms that correspond to a given i have the same dimension: h
ðiÞ
A1
i
h i h i h i ðiÞ ðiÞ ¼ A2 ¼ Aj ¼ AðiÞ n
(2.31)
The variables and constants included in (2.29) may be rendered dimensionless by using some characteristic scales of the density ½r ¼ L3 M; velocity ½v ¼ LT 1 ; length ½l ¼ L; time ½t ¼ T, etc. Then, the dimensionless variables and constants of the problem are expressed as r¼ ¼
r v T c t P m k ;v ¼ ;T ¼ ;c ¼ ;t ¼ ;P ¼ ;m ¼ ; k ¼ ;D r v T c t P m k D g ;g ¼ D g
(2.32)
where the asterisks denote the characteristic scales, and the dimensionless parameters are denoted by bars. In addition, k ¼ LMT 3 y1 ; D ¼ ½L2 T 1 ; and g ¼ ½LT 2 are the characteristic scales of thermal conductivity, diffusivity and gravity acceleration, respectively. Taking into account (2.32), we can present all terms of (2.29) as follows ðiÞ
ðiÞ ðiÞ
Aj ¼ Aj Aj ðiÞ
(2.33)
where Aj is the corresponding dimensional multiplier comprised of the characterðiÞ ðiÞ ðiÞ istic scales, Aj ¼ Aj =Aj is the dimensionless form of the jth term in (2.29). ðiÞ The exact form of the multipliers Aj is determined by the actual structure of the ðiÞ terms Aj . For example, the multiplier of the first term of the momentum balance equation is found from
2.3 Non-Dimensionalization of the Governing Equations ðiÞ
A1 ¼ r
13
@v r v @ðv=v Þ ðiÞ ðiÞ ¼ A1 A1 ¼ t @ðt=t Þ @t
(2.34)
r v ðiÞ @v , A1 ¼ . t @t The substitution of the expression (2.33) into (2.29) yields ðiÞ
where A1 ¼
n X
ðiÞ ðiÞ
Aj Aj ¼ 0
(2.35)
j¼1 ðiÞ
Dividing the left and right hand sides of (2.35) by a multiplier Ak ð1 k nÞ, we arrive at the dimensionless form of the conservation equations ðiÞ Ak
þ
( ðiÞ k1 Y X j¼1
j
ðiÞ Aj
þ
ðiÞ n Y X
) Aj
ðiÞ
¼0
(2.36)
j¼kþ1 j
QðiÞ where j ¼ Aj =Ak are the dimensionless groups. To illustrate the general approach described above, we render dimensionless the Navier–Stokes equations, the energy and species balance equations, as well as the continuity equation. For incompressible fluids these equations read r
@v þ rðv rÞv ¼ rP þ mr2 v þ rg @t rcp
(2.37)
@T þ rcP ðv rÞT ¼ kr2 T þ f @t
(2.38)
@cx þ rðv rÞcx ¼ rDr2 cx @t
(2.39)
r
rv¼0
(2.40)
where r; v T; P and cx are the density, velocity vector, the temperature, pressure and the concentration of the species x. In particular, let us use the Cartesian coordinate system where vector v has components u; v and w in projections to the x; y and z axes. In addition, m; k and D are the viscosity, thermal conductivity and diffusivity which are assumed to be constant, g the magnitude of the gravity h acceleration g, f is the dissipation function f ¼ 2m ð@u=@xÞ2 þ ð@v=@yÞ2 þ ð@w=@zÞ2 þ mð@u=@y þ @v=@xÞ2 þ mð@v=@z þ @w=@yÞ2 þ mð@w=@x þ @u=@zÞ2 . ðiÞ The multipliers Aj in (2.37)–(2.40) are listed below ð1Þ
A1 ¼
r v ð1Þ r v2 ð1Þ P ð1Þ ; A2 ¼ ; A3 ¼ ; A4 ¼ r g t l l
(2.41)
14
2
ð2Þ
A1 ¼
Basics of the Dimensional Analysis
r cP T ð2Þ r cP v T ð2Þ k T ð2Þ m v2 ; A2 ¼ ; A3 ¼ ; A4 ¼ t l l l ð3Þ
A1 ¼
r c ð3Þ r c ð3Þ r D c ; A2 ¼ ; A3 ¼ t l l2 ð4Þ
A1 ¼ ð1Þ
v ð4Þ v ;A ¼ l 2 l
ð1Þ
ð2Þ
ð2Þ
ð3Þ
ð3Þ
ð4Þ
ð4Þ
Dividing the multipliers Aj by A2 ; Aj by A2 ; Aj by A2 and Aj by A2 , we arrive at the following system of dimensionless equations St
@v 1 2 1 þ ðv rÞv ¼ EurP þ r vþ @t Re Fr
(2.42)
@T 1 2 Br f þ ðv rÞT ¼ r Tþ @t Pe Re
(2.43)
@cx 1 2 þ ðv rÞcx ¼ r cx @t Ped
(2.44)
St
St
rv¼0
(2.45)
where St ¼ l =v t ; Eu ¼ P =r v2 ; Re ¼ v l =n ; Pe ¼ v l =a ; Ped ¼ v l =D , Fr ¼ v2 =g l ; Br ¼ m v2 =k T are the Strouhal, Euler and Reynolds numbers, as well as the thermal and diffusion Peclet numbers, and the Froude and Brinkman numbers, respectively, n and a are the kinematic viscosity diffusivity, h and thermal i 2 ; f ¼ f= mðv =l Þ v ¼ v=v ; P ¼ and the dimensionless dissipation function P rv2 ; T ¼ T =T and cx ¼ c=c are the dimensionless variables. The non-dimensionalization of the initial and boundary conditions is similar to the one described above. In that case each of the independent variables x; y; z and t, as well as the flow characteristics u; v; T and cx are also rendered dimensionless by using some scales that have the same dimensions as the corresponding parameters. For example, consider the non-dimensionalization of the initial and boundary conditions for the following three problems of the theory of viscous fluid flows: (1) steady flow in laminar boundary layer over a flat plate, (2) laminar flow about a flat plate which instantaneous started to move in parallel to itself, and (3) submerged laminar jet issued from a round nozzle. In case (1), let the velocity and temperature of the undisturbed fluid far enough from the plate be u1 , T1 , and the wall temperature be Tw ¼ const: Then, the boundary conditions read
2.3 Non-Dimensionalization of the Governing Equations
x ¼ 0; 0 y 1; u ¼ u1 ; T ¼ T1
15
(2.46)
x > 0, y ¼ 0, u ¼ v ¼ 0; T ¼ Tw ; y ! 1, u ! u1 , T ! T1 Introducing as the scales of length some L, velocity u1 and temperature Tw T1 , we rearrange (2.46) to the following dimensionless form3 x ¼ 0; 0 y 1 u ¼ 1; DT ¼ 1
(2.47)
x > 0, y ¼ 0 u ¼ v ¼ 0; DT ¼ 0; y ! 1 u ! 1; DT ! 1 where x ¼ x=L; y ¼ y=L; u ¼ u=u1 ; v ¼ v=u1 ; DT ¼ ðTw TÞ=ðTw T1 Þ. The equation for the heat flux at the wall is used to introduce the heat transfer coefficient h: @T hðTw T1 Þ ¼ k (2.48) @y y¼0 Being rendered dimensionless, the heat transfer coefficient is expressed in the following form @DT (2.49) Nu ¼ @y y¼0 where Nu ¼ hL=k is the dimensionless heat transfer coefficient is called the Nusselt number. In case (2), the initial and boundary conditions of the problem on a plate starting to move from rest with velocity U in the x-direction in contact with the viscous fluid read t ¼ 0; 0 y 1 u ¼ 0
(2.50)
t > 0, y ¼ 0 u ¼ U; y ¼ 1, u ¼ 0 Since no time or length scales are given, we use as the characteristic time scale t ¼ n=U 2 and as the characteristic length scale n=U. Then, (2.50) take the following dimensionless form t ¼ 0; 0 y 1 u ¼ 0; t > 0; y ¼ 0 u ¼ 1; y ! 1 u ! 0
(2.51)
In case (3), the boundary conditions for a submerged laminar jet are
3 It is emphasized that in the problem on flow in the boundary layer over a semi-infinite plate, a given characteristic scale L is absent. According to the self-similar Blasius solution of this problem, the dimensionless coordinate y ¼ y=ðnx=u1 Þ1=2 with ðnx=u1 Þ1=2 playing the role of the length scale (Sedov 1993).
16
2
Basics of the Dimensional Analysis
x ¼ 0; 0 y r0 ; u ¼ u0 ; T ¼ T0 ; y > r0 u ¼ 0; T ¼ T1 x > 0; y ¼ 0,
(2.52)
@u @T ¼ 0, ¼ 0; y ! 1, u ! 0, T ! T1 @y @y
where r0 is the nozzle radius. The dimensionless form of the conditions (2.52) is x ¼ 0; 0 y 1; u ¼ 1 DT ¼ 1; y > 1; u ! 0; DT ! 0
(2.53)
@u @DT ¼ 0, ¼ 0; y ¼ 1, u ! 0, DT ! 0 @y @y where x ¼ x=r0 ; y ¼ y=r0 ; u ¼ u=u0 ; DT ¼ ðT1 TÞ=ðT1 T0 Þ: At large enough distance from the jet origin at x=r0 >> 1, it is possible to use the R1 integral condition u2 ydy ¼ const; instead of the condition (2.52) at x ¼ 0. Note x > 0, y ¼ 0,
0
that there is another way of rendering the system of fundamental equations of hydrodynamics and heat and mass transfer theory dimensionless. It consists in rendering dimensionless each quantity in these equations using for this aim the scales of the density, velocity, temperature, etc. Requiring that the convective terms of these equations do not contain any dimensional multipliers, it is not easy to arrive at the equations identical to (2.42)–(2.45). To illustrate this approach to nondimensionalization of the mass, momentum, energy and species conservation equations, consider, for example, the system of equations describing flows of reactive gases
r
@r þ r ðrvÞ ¼ 0 @t
(2.54)
@v þ rðv rÞv ¼ rP þ r ðmrvÞ þ rg @t
(2.55)
@h þ rðv rÞh r ðkrTÞ ¼ qWk @t
(2.56)
@ck þ rðv rÞck r ðrDrck Þ ¼ Wk @t
(2.57)
r
r
P¼
g1 rh g
(2.58)
where v is the velocity vector, r; P; h and T are the density, pressure, enthalpy and temperature, ck ¼ rk =r is the relative concentration of the kth species, r ¼ Srk ; with rk being density of the kth species, Wk ðck ; TÞ and W are the chemical reaction rates, q is the heat of the overall reaction, and g ¼ cp =cv is the ratio of
2.3 Non-Dimensionalization of the Governing Equations
17
specific heat at constant pressure to the one at constant volume (the adiabatic index). Note that in the energy balance equation (2.56) the dissipation term is neglected. Introducing dimensionless parameters as follows a ¼ aa (the asterisk denotes the scale of a parameter a), we arrive at the following equations r @r r v þ r ðrvÞ ¼ 0 t @t L
(2.59)
r v @v r v P m v þ rðv rÞv ¼ rP þ 2 r ðmrvÞ þ r g rg t @t L L L
(2.60)
r h @h r v h k T r þ rðv rÞh 2 r ðkrTÞ ¼ qWk: W k t L L @t
(2.61)
r @ck r v r D þ r rðv rÞck 2 r ðrDrck Þ ¼ Wk: W k t @t L L
(2.62)
P¼
g 1 r h rh P g
(2.63)
where r ; v ; P ; T ; h and L are the scales of density, velocity, pressure, temperature, enthalpy and length, respectively. Requiring that the second terms on left hand sides in (2.59)–(2.62) do not contain any dimensionless multipliers and also accounting for the fact that for perfect gas r h =P ¼ g=ðg 1Þ, we obtain @r þ r ðrvÞ ¼ 0 @t
(2.64)
@v 1 1 þ rðv rÞv ¼ EurP þ r ðmrvÞ þ rg @t Re Fr
(2.65)
@T 1 þ rðv rÞT r ðkrTÞ ¼ Da3 W k @t Pe
(2.66)
@ck 1 r ðrDrck Þ ¼ Da1 W k þ rðv rÞck Ped @t
(2.67)
St
St
St
St
P ¼ rh
(2.68)
where in addition to previously introduced Strouhal, Reynolds, Euler, the thermal and diffusion Peclet numbers, and the Froude number, two Damkohler numbers Da1 ¼ Wk: L =r v ; and Da3 ¼ qWk: L =r v h (defined according to the Handbook of Chemistry and Physics,1968) appear.
18
2.4 2.4.1
2
Basics of the Dimensional Analysis
Dimensionless Groups Characteristics of Dimensionless Groups
As was shown in Sect. 2.3, the dimensionless momentum, energy and diffusion equations contain a number of dimensionless groups, which represent themselves some combinations of the physical properties of fluid, acting forces, heat fluxes, etc. The physical meaning and number of these groups is determined by a specific situation, as well as by a particular model used for description of the physical phenomena characteristic of that situation (Table 2.4).4 Consider in detail some particular dimensionless groups. The Prandtl, Schmidt and Lewis numbers belong to a subgroup of dimensional groups that incorporate only quantities that account for the physical properties of fluid. They are expressed as the following ratios (cf. Table 2.4) n Pr ¼ ; a
Sc ¼
n a ; Le ¼ D D
(2.69)
where n; a and D are the kinematic viscosity, thermal diffusivity and diffusivity, respectively. Consider, for example the Prandtl number. It represents itself the ratio of kinematic viscosity to thermal diffusivity, i.e. of the characteristics of fluid responsible for the intensity of momentum and heat transfer. Accordingly, the Prandtl number can be considered as a parameter that characterizes the ratio of the extent of propagation of the dynamic and thermal perturbations. Therefore, at very low Prandtl numbers (for example, in flows of liquid metals), the thickness of the thermal boundary layer dT is much larger than the thickness of the dynamical one, d: In contrast, at Pr >> 1 (in flows of oils) the equality d >> dT is valid. The Schmidt number is the diffusion analog of the Prandtl number. It determines the ratio of the thicknesses of the dynamical and diffusion boundary layers. The Reynolds number belongs to the subgroup of the dimensionless groups which are ratios of the acting forces. It can be considered as the ratio of the inertia force Fi to the friction force Ff
4
Dimensionless groups can be also found directly by transformation of the functional equations of a specific problem using the Pi-theorem (see Sect. 2.5). A detailed list of dimensionless groups related to flows of incompressible and compressible fluids in adiabatic and diabatic conditions, flows of non-Newtonian fluids and reactive mixtures can be found in Handbook of Chemistry and Physics, 68th Edition, 1987–1988, CBC Inc. Boca Roton, Florida, and in Chart of Dimensionless Numbers, OMEGA Technology Company. See also Lykov and Mikhailov (1963) and Kutateladze (1986).
2.4 Dimensionless Groups
19
Table 2.4 Dimensionless groups Name Symbol Definition gL3 r Archimedes Ar m2 ðr rf Þ number
Biot number Bi
hL ks
Bond number
Bo
rgL2 s
Brinkman number Capillary number
Br
mv2 kDT
Ca
mv s
Damkohler number
Da1 Da3
WL Vm qWL rvcP DT
Darcy number Dean number
Da2
vL D
De
vRr m
Deborah number
De
tr t0
Eckert number Ekman number Euler number Grashof number Jacob number Knudsen number
Ec
v21 cP DT
Ek
qffiffiffi
R r
m 2roL2
1=2
Eu
rv2 DP
Gr
r2 gbL3 DT m2
Ja
cP rf DT rrV
Kn
l L
Kutateladze number
K
rv cP DT
Lewis number Mach number
Le
k rcP D
M
v C
Nu
hL k
Comparison ratio
Field of use
Gravity force to viscous force
Motion of fluid due to density differences (buoyancy) Heat transfer
Convection heat transfer to conduction heat transfer Gravitaty force to surface Motion of drops and tension bubbles. Atomization Heat dissipation to heat Viscous flows transferred Viscous force to surface Two-phase flow. tension force Atomization. Moving contact lines Chemical reaction rate to Chemical reactions, bulk mass flow rate. momentum, and Heat released to heat transfer convected heat Inertia force to permeation Flow in porous media force Centrifugal force to inertial Flow in curved force channels and pipes Non-Newtonian Relaxation time to the hydrodynamics. characteristic Rheology hydrodynamic time Kinetic energy to thermal Compressible flows energy (Viscous force to Coriolis Rotating flows force)1=2 Pressure drop to dynamic Fluid friction in pressure conduits Buoyancy force to viscous Natural convection force Heat transfer to heat of Boiling evaporation Mean free path to Rarefied gas flows characteristic dimension and flows in micro- and nanocapillaries Latent heat of phase change Combined heat and to convective heat mass transfer in transfer evaporation Thermal diffusivity to Combined heat and diffusivity mass transfer Flow speed to local speed of Compressible flows sound Forced convection
(continued)
20
2
Table 2.4 (continued) Name Symbol Definition Nusselt number Lrvcp Peclet Pe k number mcP Prandtl Pr k number gbL3 r2 cP Rayleigh Ra mk number
Richardson number
Ri
Rossby number
Ro
v oL sin L
Schmidt number Senenov number
Sc
m rD
Se
hm K
Sh
hm L D
St
h rvcP
St
fL v
Sherwood number Stenton number Strouhal number Taylor number Weber number
Ta We
g @P r @Lh
.
2
2oL2 r m
v2 rL s
Re ¼
@v @Lh w
Basics of the Dimensional Analysis
Comparison ratio Total heat transfer to conductive heat transfer Bulk heat transfer to conductive heat transfer Momentum diffusivity to thermal diffusivity Thermal expansion to thermal diffusivity and viscosity Gravity force to the inertia force
Field of use
Forced convection Heat transfer in fluid flows Natural convection
Stratified flow of multilayer systems The inertia force to Coriolis Geophysical flows. force Effect of earth’s rotation on flow in pipes Kinematic viscosity to Diffusion in flow molecular diffusivity Intensity of heat transfer to Reaction kinetics. intensity of chemical Convective heat reaction transfer. Mass diffusivity to Mass transfer molecular diffusitivy Heat transferred to thermal Forced convection capacity of fluid Time scale of flow to Unsteady flow. oscillation period Vortex shedding (Coriolis force to viscous Effect of rotation on force)2 natural convection The dynamic pressure to Bubble formation, capillary pressure drop impact
vL rv2 rv2 =L ¼ ¼ mðv=LÞ mðv=L2 Þ n
(2.70)
where r; m and L are the density, viscosity and the characteristic length. The dimensions of the and denominator in right hand side ratio in numerator (2.70) are ½rv2 =L ¼ m v L2 ¼ L2 MT 2 , i.e. the same as the dimensions of the terms r½@v=@t þ ðv rÞv and mr2 v accounting for the inertia and viscous forces in the momentum balance equation. The terms rv2 =L and mv=L2 can be treated as the specific inertia and viscous forces fi ¼ Fi =V and ff ¼ Ff =V , respectively, with the dimensions ½Fi ¼ LMT 2 , Ff ¼ LMT 2 , and ½V ¼ L3 . At small Reynolds numbers when the influence of viscosity is dominant, any chance perturbations of the flow field decay very quickly. At large Re such perturbations increase and result in laminar-turbulent transition. Therefore, the
2.4 Dimensionless Groups
21
Reynolds number is sensitive indicator of flow regimes. For example, in flows of an incompressible fluid in a smooth pipe, three kinds of flow regime can be realized depending on the value of the Reynolds number: (1) laminar (Re 2300), transitional (2300 Re 3500), and developed turbulent (Re > 3500). The Peclet number is an example of a dimensionless group that is a ratio of heat fluxes of different nature. It reads Pe ¼
vL rvcP DT ¼ DT a k L
(2.71)
where k and cP are the thermal conductivity and specific heat at constant pressure, DT is the characteristic temperature difference. The Peclet number is the ratio of the heat flux due to convection to the heat flux due to conduction. It can be considered as a measure of the intensity of molar to molecular mechanisms of heat transfer. We mention also the Damkohler number that characterize the conditions of chemical reaction which proceeds in a reactive mixture, i.e. in the process accompanied by consumption of the initial reactants, formation of the combustion products, as well as an intensive heat release. Under these conditions the evolution of the temperature and concentration fields is determined by two factors: (1) hydrodynamics of the flow of reacting mixture, and (2) the rate of chemical reaction. The contribution of each of these factors can be estimated by the ratio of the characteristic hydrodynamic time th W 1 to the chemical reaction time tr Vv1 i.e. by the Damkohler number Da1 ¼
th tr
(2.72)
If the Damkohler number is much less than unity, the influence of the chemical reaction on the temperature (concentration) field is negligible. At large values of Da1 the effect of the chemical reaction and its heat release is dominant.
2.4.2
Similarity
Before closing the brief comments on the dimensionless groups, we outline how such groups are used in modeling of hydrodynamic and thermal phenomena. For this aim, we turn back to (2.64)–(2.68) that describe the mass, momentum, heat and species transfer in flows of incompressible fluids with constant physical properties. These equations contain eight dimensionless groups, namely, St; Re; Pe; Ped ; Eu; Fr; Da1 and Da3 : If the initial and boundary conditions of a particular problem do not contain any additional dimensionless groups (as, for example, the conditions y ¼ 0 v ¼ 0; T ¼ 0; ck ¼ 0, y ! 1 v ¼ 1; T ¼ 1; ck ¼ 1), the velocity,
22
2
Basics of the Dimensional Analysis
temperature and concentration fields determined by (2.64)–(2.68) can be expressed as follows v ¼ fv ðx; y; z; St; Re; Eu; FrÞ
(2.73)
T ¼ ft ðx; y; z; St; Pe; Da1 Þ
(2.74)
ck ¼ fc ðx; y; z; St; Ped ; Da3 Þ
(2.75)
In (2.73) and (2.75) T ¼ ðT Tw Þ=ðT1 Tw Þ; and ck ¼ ðck ck;w Þ=ðck;1 ck;w Þ; subscripts w; and 1 correspond to the values at the wall and in undisturbed fluid. The expressions (2.73)–(2.75) are universal in a sense that the fields of dimensionless velocity, temperature and concentration determined by these expressions do not depend on the absolute values of the characteristic scales. That means that in geometrically similar systems (for example, cylindrical pipes of different diameter) values of dimensionless velocity, temperature and concentration at any similar point (with x1 ¼ x2 ¼ ¼ xi ; y1 ¼ y2 ¼ ¼ yi ; z1 ¼ z2 ¼ ¼ zi ) are the same if the values of the corresponding dimensionless groups are the same. Thus, the necessary conditions of the dynamic and thermal similarity in geometrically similar systems consist in equality of dimensionless groups (similarity numbers) relevant for the compared systems, i.e. St ¼ idem; Re ¼ idem; Eu ¼ idem; Fr ¼ idem; Pe ¼ idem; Ped ¼ idem; Da1 ¼ idem; Da3 ¼ idem
(2.76)
for a considered class of flows. It is emphasized that in geometrically similar systems the boundary conditions should also be identical in such comparisons. The conditions (2.76) allow modeling the momentum, heat and mass transfer processes in nature and technical applications by using the results of the experiments with miniature geometrically similar models. Note that among the totality of similarity numbers it is possible to select a family of dimensionless groups that contain combinations of only scales of the considered flow family and the physical parameters of a medium involved in a situation under consideration. Such similarity numbers are called similarity criteria (Loitsyanskii 1966). A number of similarity criteria can be less than the number of similarity numbers. For example, hydraulic resistance of cylindrical pipes with fully developed incompressible viscous fluid flow with a given throughput is characterized by two similarly numbers, namely, the Reynolds and Euler numbers. The first of them Re ¼ v0 d=n is the similarity criterion, since it contains known parameters: the average velocity of fluid v0 , its viscosity n and pipe diameter d. In contrast, the Euler number is not a similarity criterion, since it contains an unknown pressure drop which has to be found by solving the problem or measured experimentally (Loitsyanskii 1966).
2.5 The Pi-Theorem
2.5 2.5.1
23
The Pi-Theorem General Remarks
This whole book is devoted to the Buckingham Pi-theorem (1914), which is widely used in a number of important problems of modern physics and, in particular, mechanics. The proof of this theorem, as well as numerous instructive examples of its applications for the analysis of various scientific and technical problems are contained in the monographs by Bridgman (1922), Sedov (1993), Spurk (1992) and Barenblatt (1987). Referring the readers to these works, we restrict our consideration by applications of the Pi-theorem to problems of hydrodynamics and the heat and mass transfer only. The study of thermohydrodynamical processes in continuous media consists in establishing the relations between some characteristic quantities corresponding to a particular phenomenon and different parameters accounting for the physical properties of the matter, its motion and interaction with the surrounding medium. Such relations can be expressed by the following functional equation a ¼ f ða1 ; a2 an Þ
(2.77)
where a is the unknown quantities (for example, velocity, temperature, heat or mass fluxes, etc.), a1 ; a2 ; an are the governing parameters (the characteristics of an undisturbed fluid, physical constants, time and coordinates of a considered point). Equation 2.77 indicates only the existence of some relation between the unknown quantities and the governing parameters. However, it does not express any particular form of such relation. There are two approaches to determine an exact form of a relation of the type of (2.77): one is experimental, and the other one theoretical. The first approach is based on generalization of the results of measurements of unknown quantities a while varying the values of the governing parameters a1 ; a2 ; an : The second, theoretical, approach relies on the analytical or numerical solutions of the mass, momentum, energy and species balance equations. In both cases the establishment of a particular exact form of (2.77) does not entail significant difficulties while studying the simplest one-dimensional problems when (2.77) takes the form a ¼ f ða1 Þ: On the contrary, a comprehensive experimental and theoretical analysis of a multiparametric equation a ¼ f ða1 ; a2 an Þ is extremely complicated and often represents itself an insoluble problem. The latter can be illustrated by the problem on a drag force acting on a body moving with a constant velocity in an infinite bulk of incompressible viscous fluid. In this case the drag force Fd acting from the fluid to the body depends on four dimensional parameters, namely, the fluid density r and viscosity m, a characteristic size of the body d, and its velocity v. Then, the functional equation (2.77) takes the form Fd ¼ f ðr; m; d; vÞ
(2.78)
24
2
Basics of the Dimensional Analysis
In order to find experimentally the drag force, it is necessary to put the body into a wind tunnel and measure the drag force at a given velocity by an aerodynamic scale. That is the experimental way of solving the problem under consideration but only for one point on the parametric plane drag force-velocity. To determine the dependence of the drag force on velocity within a certain range of velocity v, it is necessary to reiterate the measurement of Fd at N values of v to determine the dependence Fd ¼ f ðvÞ within a range ½v1 ; v2 at fixed values of r; m and d. If we want to find the dependence Fd on all four governing parameters, we have to perform N 4 measurement.5 Therefore, if the number of data points forFd at varying one governing parameter is N ¼ 102 ; the total number of measurements that one needs will be equal to 108 ! It is evident that such number of measurements is practically impossible to perform. Moreover, even if we have an experimental data bank with 108 measurement points, we cannot say anything about the behavior of the function Fd ¼ f ðr; m; v; dÞ outside the studied range of the governing parameters. An analytical or numerical calculation of the dependence of drag force on density, viscosity, velocity and size of the body is also an extremely complicated problem in the general case (at the arbitrary values of r; m; v; and d) due to the difficulties involved in integrating the system of nonlinear partial differential equations of hydrodynamics. Essentially both approaches to study the dependence of drag force on density, viscosity, velocity and size of the body allow a significant simplification of the problem by using the Pi-theorem. The latter points at the way of transformation of the function of n dimensional variables into a function of m ðwith m < nÞ dimensionless variables. As a matter of fact, the Pi-theorem suggests how many dimensionless variables are needed for describing a given problem containing n dimensional parameters. The Pi-theorem can be stated as follows. Let some dimension physical quantities a depend on n dimensional parameters a1 ; a2 an ; where k of them have an independent dimension. Then the functional equation for the quantities a a ¼ f ða1 ; a2 ak ; akþ1 an Þ
(2.79)
can be reorganized to the form of the dimensionless equation P ¼ ’ðP1 ; P2 Pnk Þ
(2.80)
that contain n k dimensionless variables. The latter are expressed as P1 ¼
a1 0
0
a a a 1 1 a 22
0
a ak k
; P2 ¼
a2 00
00
a a a 11 a 22
00
a ak k
Pnk ¼
an ank ank a2nk 1 a1 a2 ak k
The dimensionless form of the unknown quantities a is
5
With an equal number of data points for each one of the four governing parameters.
(2.81)
2.5 The Pi-Theorem
25
P¼
aa11 aa22
a aak k
(2.82)
To illustrate the application of the Pi-theorem to hydrodynamic problems, return to the drag force acting on a body moving in viscous fluid. The unknown quantities and governing parameters of the corresponding problem have the following dimensions ½Fd ¼ LMT 2 ; ½r ¼ L3 M; ½m ¼ L1 MT 1 ; ½d ¼ L; ½v ¼ LT 1
(2.83)
Three from the four governing parameters of this problem have independent dimensions. That means that a dimension of any governing parameters in this case can be expressed as a combination of dimensions of the three others. The dimension of the unknown quantity is also expressed as a combination of the governing parameters having independent dimensions ½Fd ¼ LMT 2 ¼ ½rv2 d 2 ¼ ½m2 =r ¼ ½mvd : In accordance with the Pi-theorem, (2.78) takes the form P ¼ ’ðP1 Þ where P ¼ ra1 vFad2 da3 ; and P1 ¼
a
0
m a
0
r 1v 2d
a
0 3
(2.84)
:
Taking into account the dimension of the drag forceFd and governing parameters with independent dimension r; v and d and using the principle of the dimensional 0 homogeneity, we find the values of the exponents ai and ai 0
0
0
a1 ¼ 1; a2 ¼ 2; a3 ¼ 2; a1 ¼ 1; a2 ¼ 1; a3 ¼ 1
(2.85)
Then (2.84) reads Cd ¼ ’ðReÞ
(2.86)
where Cd ¼ Fd =rv2 d2 is the drag coefficient, and Re¼rvd=m is the Reynolds number. The exact form of the function ’ðReÞ cannot be determined by means of the dimensional analysis. However, this fact does not diminish the importance of the obtained result. Indeed, the dependence of the drag coefficient on only one dimensionless group (the Reynolds number) allows generalization of the experimental data on drag related to motions of bodies of different sizes moving with different velocities in fluids with different densities and viscosities. All this data can be presented in a collapsed form of a single curve Cd ðReÞ. Moreover, in some limiting cases corresponding to motion with low velocities (the so-called, creeping flows with Re << 1) or high speeds when Re >> 1, it is possible to determine the exact forms of the dependence of the drag coefficient on Re.
26
2
Basics of the Dimensional Analysis
In particular, at Re << 1 the inertia effects become negligible. Returning to (2.78), we can assume that the drag force depends on fluid viscosity, body size and its velocity Fd ¼ f ðm; v; dÞ
(2.87)
All the governing parameters in (2.87) have independent dimensions (n k ¼ 0). Therefore, in this case (2.87) reduces to Fd ¼ cma1 va2 d a3
(2.88)
where c is a dimensionless constant and a1 ¼ 1; a2 ¼ 1; a3 ¼ 1. Substituting the values of the exponents a1 ; a2 and a3 into (2.88) leads to the following expression for the drag coefficient Cd ¼
c Re
(2.89)
It is evident that to determine the dependence Cd ðReÞ at Re << 1 it is sufficient to perform only one measurement in order to establish the value of the constant c. It is emphasized that the efficiency of using the Pi-theorem in studies of physical phenomena is determined by the value of the difference n k, i.e. by the number of the governing dimensionless groups. In all cases (excluding k ¼ 0) the transformation of the functional equation by the Pi-theorem allows one to decrease number of variables. The most interesting two cases correspond to the difference n k being either 0 or 1. In the first case the functional equation takes the form a ¼ caa11 aa22 aann
(2.90)
In the second one it becomes P ¼ ’ðP1 Þ
(2.91)
where P represents itself the dimensionless group corresponding to the unknown parameter. Decreasing the number of dimensionless variables in (2.91) to only one is equivalent to the transformation of partial differential equations into the ordinary ones. A number of examples of transformation of the functional equations similar to (2.77) to a dimensionless form, as well as transformations of partial differential equations into the ordinary ones is given in the following sections.
2.5.2
Choice of the Governing Parameters
The theoretical study of hydrodynamic and heat and mass transfer processes is based on the system of partial differential equations that include the mass,
2.5 The Pi-Theorem
27
momentum, energy and species conservation balances. This system of equations is supplemented by an equation of state and correlations determining the physical properties of the medium. The exact and approximate solutions of hydrodynamic and heat transfer problems in the framework of the continuum approach yield comprehensive answers to different problems of the theory. In distinction the dimensional analysis of hydrodynamic and heat and mass transfer problems meets some difficulties that arise already at the first step of the investigation when choosing the governing parameter of the problem. They stem from certain vagueness in choosing the governing parameters beginning from a pure intuitive evaluation of the features of a phenomenon under consideration. In addition, such approach to choosing the governing parameters often involves a number of parameters whose influence will appear to be negligible at the end. The latter makes it difficult to foresee the results of the dimensional analysis from scratch in generalizing hydrodynamic and heat and mass transfer. In order to improve the procedure of choosing the governing parameters and simplify and the following analysis, it is possible to use the system of the mass, momentum, energy and species balance equations. Let us illustrate such an approach by the following examples.6 We begin with the drag force acting on a spherical particle moving with a constant velocity in an infinite bulk of viscous incompressible fluid. It is reasonable to assume that the force that acts on the particle depends on its size d; velocity v and physical properties of the fluid, namely its density r and viscosity m: In this case the functional equation for the drag force Fd reads Fd ¼ f ðr; m; d; vÞ
(2.92)
The dimensional analysis of (2.92) leads to the following transformation in the form of the drag coefficient Cd ¼ ’ ðReÞ
(2.93)
where Cd ¼ Fd =rv2 d 2 is the drag coefficient, and Re ¼ vd=n is the Reynolds number. It is emphasized that the function ’ ðReÞon the right hand side of (2.93) can be presented as c’ðReÞ, where c is a dimensionless constant with its value being chosen according to the experimental data. For example, for creeple motion of a small spherical particle when the drag force is given by the Stokes law Fd ¼ 3pmvd, constant c ¼ 8=p: Then the expression for the drag coefficient takes the form Cd ¼ 24=Re: To determine the exact form of the dependence (2.93), one needs to integrate the continuity and the Navier–Stokes equations subjected to the no-slip condition at the particle surface. In some limiting cases corresponding to special conditions of
6
A detailed analysis of these problems see in Chaps. 4 and 7
28
2
Basics of the Dimensional Analysis
particle motion (say, very slow or fast), it is possible to find the exact form of the function ’ ðReÞ in (2.93) using these equations only for determining the set of the governing parameters. For example, in the case of slow motion (creeping flows) the inertia terms on the left hand side of the Navier–Stokes equations rðv rÞv is much less than the terms in on the right hand side of these equations. That allows one to omit the inertial term and thereby exclude density from the governing parameters. As a result, the functional equation for the drag force reduces to the form of (2.6). Such simplification of the problem formulation is a key element which allows establishing an exact form of the dependence of the drag force on viscosity, velocity and diameter of the particle as Fd mvd that coincide (up to a numerical factor) with the exact result (the Stokes force) derived from the Navier–Stokes equations. In the second case corresponding to a rapid body motion (the case of a large Reynolds number) the dominant role belongs to the turbulent transfer. The average characteristics of fully developed turbulent flows are governed by the Reynolds equations (Hinze 1975; Loitsyanskii 1966) @vi @vi 1 @P 1 @ 0 0 þ vj ¼ þ nr2 vi þ ðrvi vj Þ r @xi r @xj @t @xj
(2.94)
(bars over parameters denote the average values). In high Reynolds number flows the term nr2 vi associated with the effect of the molecular momentum transfer through molecular viscosity mechanism can be omitted. Then, assuming steady state average turbulent flow, the drag force which does not depend on molecular viscosity and time is given by Fd ¼ f ðr; v; dÞ
(2.95)
Applying the Pi-theorem to (2.95), we can rearrange it to the following form v2 d 2 Fd r
(2.96)
which agrees with the Newton law for drag. Another example of employing the conservation equations to facilitate the dimensional analysis of complicated hydrodynamic and heat transfer problems is related to mass transfer to a vertical reactive plate in contact with a liquid solution of a reactive species (a reagent) which is initially at rest. When the rate of a heterogeneous reaction at the plate surface is much larger than the rate of diffusion transport of the reagent toward the surface, its concentration there equals zero, whereas far from the surface it is equal c1 : The gradient of the reagent concentration across the thickness of the diffusion boundary layer results in a non-uniform density field. That, in turn, triggers buoyancy force which results in liquid motion near the wall. It is reasonable to assume that the velocity and concentration of the reactive species in the dynamic and diffusion boundary layers are determined by four parameters r1 ; c1 ; n; g and two independent variables x and y
2.5 The Pi-Theorem
29
u ¼ fu ðr1 ; c1 ; n; g; x; yÞ
(2.97)
c ¼ fc ðr1 ; c1 ; n; g; x; yÞ
(2.98)
where we consider for brevity only one component of the velocity vector u; n and D are the kinematic viscosity and diffusivity, and gis the acceleration due to gravity. The functional equations (2.97) and (2.98) contain six governing parameters. Three of them have independent dimensions. Choosing r1 ; n and g or r1 ; D and g as parameters with the independent dimensions, we transform (2.97) and (2.98) to the following form Pu ¼ ’u ðP1 ; P2 ; P3 Þ
(2.99)
Pc ¼ ’c ðP1 ; P2 ; P3 Þ
(2.100)
where Pu ¼ u=ðgnÞ1=3 ; P1 ¼ c1 =r1 ; P2 ¼ x=ðn2 =gÞ1=3 ; P3 ¼ y=ðn2 =gÞ1=3 ; and Pc ¼ c=r1 ; P1 ¼ c1 =r1 ; P2 ¼ x=ðD2 =gÞ1=3 ; P3 ¼ y=ðD2 =gÞ1=3 : Equations (2.99) and (2.100) show that the dimensionless velocity and concentration of the reactive species are the function of three dimensionless groups, which makes the analysis of the problem under consideration difficult. Therefore, employ also the conservation equations. The momentum and species balance equations that describe flow in the boundary layer and mass transfer to the vertical reactive wall read (Levich 1962) (see Sect. 3.10) u
@u @u @2u þv ¼ n 2 þ g c @x @y @y
(2.101)
@c @c @ 2 c þv ¼D 2 @x @y @y
(2.102)
u
where g ¼ gðc1 =rÞð@r=@cÞc¼c1 ; c ¼ ðc1 cÞ=c1 ; and r ¼ rðcÞ. The boundary conditions for (2.101) and (2.102) are u¼v¼0
c ¼ 1 at y ¼ 0;
u¼v¼0
c ¼ 0 at y ! 1
(2.103)
Equations (2.101) and (2.102) and the boundary conditions (2.103) contain four parameters that determine the local velocity and concentration fields u ¼ fu ðx; y; n; g Þ
(2.104)
c ¼ fc ðx; y; D; g Þ
(2.105)
30
2
Basics of the Dimensional Analysis
Applying the Pi-theorem to transform (2.104) and (2.105) to the dimensionless form, we obtain Pu ¼ cu ðP1 Þ
(2.106)
Pc ¼ cc ðP1 Þ
(2.107)
u ¼ ðxg Þ1=2 cu ðÞ
(2.108)
pffiffiffiffiffi c ¼ cc ð ScÞ
(2.109)
or equivalently,
where
1=4
Pu ¼ u=ðxg Þ1=2 ; P1 ¼ yðg =xn2 Þ
; Pc ¼ c ; P1 ¼ yðg =xD2 Þ
1=4
,
2 1=4
¼ yðg =xn Þ ; and Sc ¼ n/D is the Schmidt number. A number of instructive examples of application of the mass, momentum, energy and species conservation equations for dimensional analysis of the hydrodynamic and heat and mass transfer problems can be found in Chap. 7.
Problems P.2.1. Transform the van der Waals equation (Kestin 1966; Jones and Hawkis 1986) to the dimensionless form. Show that such form is universal for any van der Waals gas if one uses the critical values of the pressure, volume and temperature as the characteristic scales. In order to transform equation the van der Waals equation ðP þ a=V 2 ÞðV bÞ ¼ RT(where a and b are constants) to dimensionless form, we present this equation as ðA1 þ A2 ÞðA3 þ A4 Þ ¼ A5
(P.2.1)
where A1 ¼ P; A2 ¼ a=V 2 ; A3 ¼ V; A4 ¼ b; A5 ¼ RT with P; V and T being pressure, molar volume and temperature, respectively. We can introduce some still undefined scales of pressure P0 ; volume V0 and temperature T0 and write the expressions for scales of Aj as A1 ¼ P ; A2 ¼
a ; A3 ¼ V ; A4 ¼ b; A5 ¼ RT V2
(P.2.2)
Then (P.2.1) reduces to the form ðA1 þ aA2 ÞðbA3 þ gAÞ ¼ eA5
(P.2.3)
Problems
31
. where A1 ¼ A1 =A1 ¼ P=P; A2 ¼ A2 A2; ¼ ðV =V Þ2 ; A3 ¼ A3 =A3 ¼ ðV=V Þ; A4 ¼ A4 =A4 ¼ 1 A5 ¼ A5 =A5 ¼ T=T are the dimensionless variables, and a ¼ A2 =A1 ¼ a= V2 P ; b ¼ A3 =A1 ¼ V =P ; g ¼ A4 =A1 ¼ b=P ; and e ¼ A5 =A1 ¼ RT =P are the dimensionless constants. Equation (P.2.3) is the dimensionless van der Waals equation. For its further transformation one should define the characteristic scales of pressure, volume and temperature. For that purpose, take as the scales P ; V and T the critical values of pressure, volume and temperature Pcr ; Vcr and Tcr , respectively. Bearing in mind that the critical point is the inflection point where ð@P=@VÞT ¼ 0; and ð@ 2 P=@V2 ÞT ¼ 0 , we find a¼ Pcr ¼
2 27 R2 Tcr Vcr ; b¼ 3 64 Pcr
1 a 8a ; Vcr ¼ 3b; Tcr ¼ 27 b2 27bR
(P.2.4)
(P.2.5)
Using as the characteristic scales the critical values of pressure, temperature and specific volume, we transform (P.2.3) to the following final form ðp þ
3 Þð3o 1Þ ¼ 8t o2
(P.2.6)
where p ¼ P=Pcr ; o ¼ V=Vcr ; and t ¼ T=Tcr . Equation (P.2.6) does not contain any constants accounting for the physical properties of any particular gas and, thus, is universal. It holds for any van der Waals gas. P.2.2. (i) Transform the momentum and continuity equations for laminar flow of incompressible fluid over a plane plate in the boundary layer approximation to the dimensionless form using the LMT and Lx Ly Lz MT systems of units. (ii) Show that the Lx Ly Lz MT system of units cannot be used for transformation of the Navier–Stokes equations to the dimensionless form. (i)-A: The LMTsystem of units. The boundary layer and continuity equations read u
@u @u @2u þv ¼n 2 @x @y @y
(P.2.7)
@u @v þ þ0 @x @y
(P.2.8)
where the dimensions of u; v; n; x and yare as follows ½u ¼ LT 1 ; ½v ¼ LT 1 ; ½n ¼ L2 T 1 ; ½ x ¼ L; ½ y ¼ L
(P.2.9)
32
2
Basics of the Dimensional Analysis
All the terms in (P.2.7) have the dimension LT 2 ; whereas the dimension of the terms in (P.2.8) is T 1 : That shows that (P.2.7) and (P.2.8) can be transformed to the dimensionless form by using the multipliers ½N1 ¼ ðLT 2 Þ1 and ½N2 ¼ ðT 1 Þ1 ; respectively. Introducing the scales of length L ; velocity V and kineðiÞ matic viscosity n ¼ n, we write the expressions for the coefficients Aj as follows ð1Þ
ð1Þ
A1 ¼ A2 ¼
V2 ð1Þ V ð2Þ V ð2Þ ; A ¼ n 2 ; A1 ¼ A2 ¼ L 3 L L
(P.2.10)
ðiÞ
Bearing in mind the dimensions of Aj , we express the multipliers N1 and N2 as N1 ¼
1 ð1Þ A1
; N2 ¼
1
(P.2.11)
ð2Þ
A1
Then (P.2.7) and (P.2.8) reduce to the following form u
@u @u 1 @2u ¼ þv @x @y Re @y2
(P.2.12)
@u @v þ ¼0 @x @y
(P.2.13)
where u ¼ u=V ; v ¼ v=V ; x ¼ x=L ; y ¼ y=L ; and Re ¼ V L =n . ðiÞ The coefficients Aj for the Navier–Stokes and continuity equations 2 @u @u @ u @2u u þv þ ¼n @x @y @x2 @y2
(P.2.14)
2 @u @v @ v @2v þ þv ¼n @x @y @x2 @y2
(P.2.15)
@u @v þ ¼0 @x @y
(P.2.16)
2 V2 ð1Þ V ð2Þ V ð3Þ ð2Þ V ð2Þ ð3Þ V ;A3 ¼ n ; A1 ¼ A2 ¼ ;A3 ¼ n ; A1 ¼ A2 ¼ L L L L L
(P.2.17)
v
are defined as follows ð1Þ
ð1Þ
A1 ¼ A2 ¼
ð1Þ
ð3Þ
Using the multipliers N1 ¼ 1=A1 ; N2 ¼ 1=A1 ð2Þ ; and N3 ¼ 1=A1 , we reduce (P.2.14)–(P.2.16) to the following dimensionless form @u @u 1 @2u @2u ¼ u þv þ @x @y Re @x2 @y2
(P.2.18)
Problems
33
v
@u @v 1 @2v @2v þ þv ¼ @x @y Re @x2 @y2
(P.2.19)
@u @v þ ¼0 @x @y
(P.2.20)
(i)-B: The Lx Ly Lz MT system of units. The dimensions of u; v; x and y are ½u ¼ Lx T 1 ; ½v ¼ Ly T 1 ; ½ x ¼ Lx ; ½ y ¼ Ly
(P.2.21)
where Lx and Ly are the scales of length in the x and ydirections. Introducing the characteristic scales of u; v; n; x and y as ½U ¼ Lx T 1 ; ½V ¼ Ly T 1 ; and ½n ¼ ½n, we transform first of all the boundary layer and continuity equations (P.2.7) and (P.2.8). To this aim, we write the expressions for the coeffiðiÞ cients Aj as ð1Þ
A1 ¼
U2 U V n U U V ð1Þ ð1Þ ð2Þ ð2Þ ; A2 ¼ ; A3 ¼ 2 ; A1 ¼ ; A2 ¼ Lx Ly Ly Lx Ly
(P.2.22)
Then (P.2.7) and (P.2.8) are transformed to @u V Lx @u ¼ u þ v U Ly @y @x
n Lx L2y U
!
@2u @y2
@u V Lx @v þ ¼0 U Ly @y @x
(P.2.23)
(P.2.24)
where u ¼ u=U ; v ¼ v=V ; x ¼ x=Lx ; y ¼ y=Ly ; and the multipliers before the second terms on left hand side of the boundary layer and continuity equations are dimensionless, i.e. V Lx =U Ly ¼ 1. In planar viscous flows in the x-direction with shear in the y-direction an important role is played by the shear component tyx of the stress tensor. The shear stress tyx can be presented as the ratio of the yx to the surface area force F2 F T ; and ½Szx ¼ Lz Lx : ¼ ML Szx which have the following dimensions: yx x 2 Then the dimension of the shear stress is tyx ¼ ML1 T : For viscous Newtonian z fluids tyx ¼ mdu=dy, where m is the viscosity. Then, we find the dimension of the viscosity in the Lx Ly Lz MT system of units as 1 1 ½m ¼ tyx =ðdu=dyÞ ¼ L1 x Ly Lz MT
(P.2.25)
34
2
Basics of the Dimensional Analysis
Bearing in mind that the dimension of density in the Lx Ly Lz MT system of units is 1 1 ½r ¼ L1 x Ly Lz M, we determine the dimension of the kinematic viscosity n as (P.2.26) ½n ¼ ½m=r ¼ L2y T 1
. Thus, the multiplier n Lx L2y U on the right hand is dimen side of (P.2.23) 2 , where Re ¼ U L n L sionless. It can be presented as Re1 is a modified = x y ; L ; U and Reynolds number. Taking into account that the characteristic scales L x y Vx are arbitrary, it is possible to assume that the ratio U Ly =V Lx ¼ 1. Then, (P.2.23) and (P.2.24) take the following form u
@u @u 1 @2u þv ¼ @x @y Re @y2
(P.2.27)
@u @v þ 0 @x @y
(P.2.28)
The Navier–Stokes and continuity equations (P.2.14) and (P.2.16) can be presented as ð1Þ ð1Þ
ð1Þ ð1Þ
ð1Þ ð1Þ
ð1Þ ð1Þ
ð2Þ ð2Þ
ð2Þ ð2Þ
ð2Þ ð2Þ
ð2Þ ð2Þ
A1 A1 þ A2 A2 ¼ A3 A3 þ A4 A4
(P.2.29)
A1 A1 þ A2 A2 ¼ A3 A3 þ A4 A4 ð3Þ ð3Þ
(P.2.30)
ð3Þ ð3Þ
A1 A1 þ A2 A2 ¼ 0 ð1Þ
ð1Þ
ð1Þ
(P.2.31) ð1Þ
ð2Þ
where A1 ¼ U2 =Lx ; A2 ¼ U V =Ly ; A3 ¼ n U =L2x ; A4 ¼ n U =L2y , A1 ¼ ð2Þ ð2Þ ð2Þ ð3Þ ð3Þ U V=Lx ; A2 ¼ V2 =Ly ; A3 ¼ n V =L2x ; A4 ¼ n V =L2y , A1 ¼ U =Lx ; A2 ¼ ð1Þ
ð1Þ
ð1Þ
ð1Þ
ð2Þ
V =Ly , A1 ¼ u@u=@x; A2 ¼ v@u=@y; A3 ¼ @ 2 u=@x2 ; A4 ¼ @ 2 u=@y2 , A1 ¼ v@u=@x, ð3Þ
ð2Þ
A2 ¼ v@v=@y,
ð2Þ
ð2Þ
A3 ¼ @ 2 v=@x2 ; A4 ¼ @ 2 v=@y2 ,
ð3Þ
A1 ¼ @u=@x; and
A2 ¼ @v=@y. Then (P.2.18)–(P.2.20) take the form @u V Lx @u ¼ u þ v U Ly @y @x
n Lx U L2y
! ( ) Ly 2 @ 2 u @ 2 u þ Lx @x2 @y2
@u V Lx @v v þ v ¼ U Ly @y @x
n Lx U L2y
! ( ) Ly 2 @ 2 v @ 2 v þ Lx @x2 @y2
@u V Lx @v þ ¼0 @x U Ly @y
(P.2.32)
(P.2.33)
(P.2.34)
Problems
35
The system of Eqs. (P.2.32)–(P.2.34) can be written as @u @u 1 u þv ¼ @x @y Re
( ) Ly 2 @ 2 u @ 2 u þ Lx @x2 @y2
@u @v 1 v þv ¼ @x @y Re
( ) Ly 2 @ 2 v @ 2 u þ Lx @x2 @y2
@u @v þ ¼0 @x @y
(P.2.35)
(P.2.36)
(P.2.37)
if we account for the fact that the dimension of the kinematic viscosity ½n ¼ L2y T 1 : However, even in this case (P.2.35) and (P.2.36) are not dimensionless, since the dimension of the ratio Ly =Lx is not 1. Moreover, (P.2.35) and (P.2.36) do not satisfy the principle of the dimensional homogeneity under any assumption on the dimension of the kinematic viscosity. The latter shows that applying the Lx Ly Lz MT system of units to transformation of the Navier–Stokes is incorrect. P.2.3. (Reynolds 1886) Determine the resistance force acting on each of two circular disks of radii R which approach each other along the joint axis of symmetry with a constant velocity u, while the gap between the disks and the surrounding space are filled with incompressible viscous fluid. The pressure in the surrounding fluid far from the disks is equal P . The liquid flow in the gap is axisymmetric. Therefore, we use cylindrical coordinates z; r; ’ with the origin at the center of the lower disk which is assumed to be motionless (z and r correspond to the vertical and radial directions, respectively). Consider the low velocity case when the inertial effects are negligible. The effect of the gravity force we also will neglected. Then, it is possible to assume that the pressure gradient DP=r ðDP ¼ P P Þ is determined by the speed of the upper disk u; liquid viscosity m; the instantaneous height of the gap h, and the radial position r DP ¼ f ðu; m; h; rÞ r
(P.2.38)
For analyzing the problem, we use two different systems of units with a single (L) and two (Lz ; Lr ) length scales. In the first case the dimensions of the pressure gradient and the governing parameters can be expressed as DP (P.2.39) ¼ L2 MT 2 ; ½u ¼ LT 1 ; ½m ¼ L1 MT 1 ; ½h ¼ L; ½r ¼ L r Three of the four governing parameters in (P.2.38) have independent dimensions. Choosing u; m; and r as the parameters with the independent dimensions, we reduce (P.2.38) according to the Pi-theorem to the following form
36
2
Basics of the Dimensional Analysis
P ¼ ’ðP1 Þ 0
(P.2.40) 0
0
where P ¼ ðDP=r Þ=ua1 ma2 r a3 and P1 ¼ h=ua1 ma2 ra3 . Using the principle of the dimensional homogeneity, we find the values of 0 0 0 0 the exponents ai and ai : a1 ¼ 1; a2 ¼ 1; a3 ¼ 2; a1 ¼ 0; a2 ¼ 0; and a3 ¼ 1. Accordingly, we arrive at the following expression DP h 2 ¼ umr ’ r r
(P.2.41)
The force acting at the disk is found as ZR Fd ¼ 2p
DPrdr
(P.2.42)
Substituting the expression (P.2.41) into (P.2.42), we obtain Z1 e dx Fd ¼ 2pumR ’ x
(P.2.43)
0
R1 where e ¼ h=R; x ¼ r=R; and ’ðe=xÞdx ¼ cðeÞ. Equation P.2.43 shows that0 the resistance force acting on a disk is directly proportional to its velocity, the radius of the disk, viscosity of the liquid, as well as a function of the ratio of the gap to the disk radius. Additionally we transform (P.2.38) using the system of units with the two length scales Lz and Lr in the z and r directions, respectively. First, we determine the dimensions of the governing parameters and pressure gradient. The dimensions of the velocity u, gap thickness h and r are ½u ¼ Lz T 1 ; ½h ¼ Lz ; ½r ¼ Lr
(P.2.44)
To determine the dimensions of viscosity m and pressure gradient DP=r, we take into account the fact that in flows of viscous fluids in a narrow gap the dominant role is played by the radial velocity component, since the axial one is typically much smaller, vz << vr : In this case the force acting in the r-direction is much larger than in the z-direction, so that its dimension is ½Fr ¼ MLr T 2 : Accordingly, the dimen2 sion of the shear stress tzr ¼ Fr =Srr ð½Srr ¼ L2r Þ is ½tzr ¼ ML1 r T : For Newtonian viscous fluids tzr ¼ mðdvr =dzÞ. As a result, we find the dimension of viscosity ½m ¼ 1 ML2 r Lz T : The dimensions of pressure and its gradient are ½DP ¼
Fr MLr T 2 2 ¼ ¼ ML1 z T Srz Lr L z
(P.2.45)
References
37
2 DP ML1 1 2 z T ¼ ML1 ¼ z Lr T Lr r
(P.2.46)
Thus, the dimensions of all the governing parameters are expressed in the system of units with two length scales are independent. Then, according to the Pi-theorem, (P.2.38) takes the form DP ¼ cua1 ma2 ha3 r a4 r
(P.2.47)
where c is a dimensionless constant. Determining the values of the exponents ai using the principle of the dimensional homogeneity as a1 ¼ 1; a2 ¼ 1; a3 ¼ 3 and a4 ¼ 1, we obtain DP r ¼ cum 3 r h
(P.2.48)
Then, the substitution of (P.2.48) into (P.2.42) yields 3 c R Fd ¼ pumR 2 h
(P.2.49)
The exact solution of this problem reads (Landau and Lifshitz 1987) 3 3 R Fd ¼ pmuR 2 h
(P.2.50)
The comparison of (P.2.49) and (P.2.50) shows that the exact solution and the result of the dimensional analysis agree up to a dimensionless numerical factor. At the same time, the dimensional analysis of the problem using the system of units with a single length scale yields a less informative result, since (P.2.43) contains an unknown function cðh=RÞ:
References Barenblatt GI (1987) Dimensional analysis. Gordon and Breach Science Publication, New York Barenblatt GI (1996) Similarity, self-similarity, and intermediate asymptotics. Cambridge University Press, Cambridge Blackman DR (1969) SI units in engineering. Macmillan, Melbourne Bridgman PW (1922) Dimension analysis. Yale University Press, New Haven Buckingham E (1914) On physically similar system: illustrations of the use of dimensional equations. Phys Rev 4:345–376 Chart of Dimensionless Numbers (1991) OMEGA Technology Company.
38
2
Basics of the Dimensional Analysis
Douglas JF (1969) An introduction to dimensional analysis for engineers. Isaac Pitman and Sons, London Hinze JO (1975) Turbulence, 2nd edn. McGraw-Hill, New York Huntley HE (1967) Dimensional analysis. Dover Publications, New York Ipsen DC (1960) Units, dimensions, and dimensionless numbers. McGraw-Hill, New York Jones JB, Hawkis GA (1986) Engineering thermodynamics. An introductory textbook. John Wiley & Sons, New York Kestin J (v.1, 1966; v.2, 1968) A course in thermodynamics. Blaisdell Publishing Company, New York Kutateladze SS (1986) Similarity analysis and physical models. Nauka, Novosibirsk (in Russian) Landau LD, Lifshitz EM (1987) Fluid mechanics, 2nd edn. Pergamon, Oxford Levich VG (1962) Physicochemical hydrodynamics. Prentice-Hill, Englewood Cliffs Loitsyanskii LG (1966) Mechanics of liquid and gases. Pergamon, Oxford Lykov AM, Mikhailov Yu A (1963) Theory of heat and mass transfer. Gosenergoizdat MoscowLeningrad (English translation 1965. Published by the Israel Program for Scientific Translation. Jerusalem) Madrid CN, Alhama F (2005) Discriminated dimensional analysis of the energy equation: application to laminar forced convection along a flat plate. Int J Thermal Sci 44:331–341 Ramaswamy GS, Rao VVL (1971) SI units. A source book. McGraw-Hill, Bombay Reynolds O (1886) On the theory of lubrication. Philos Trans R Soc 177:157–233 Sedov LI (1993) Similarity and dimensional methods in mechanics, 10th edn. CRC Press, Boca Raton Spurk JH (1992) Dimensionsanalyse in der Stromungslehre. Springer-Verland in Berlin, New York Weast RC Handbook of chemistry and physics, 68th edition. 1987–1988. CRC, Boca Raton, FL. Williams W (1892) On the relation of the dimensions of physical quantities to directions in space. Philos Mag 34:234–271
Chapter 3
Application of the Pi-Theorem to Establish Self-Similarity and Reduce Partial Differential Equations to the Ordinary Ones
3.1
General Remarks
Chapter 3 deals with the application of the Pi-theorem to reduce partial differential equations (PDEs) of certain hydrodynamic and heat transfer problems to the ordinary differential equations (ODEs). In the cases which allow for such transformation (including the initial and boundary conditions), solution of the problem reduces to a much simpler problem posed for an ODE, i.e. depends on a single compound variable. The latter represent itself a combination of variables and dimensional constants involved in the problem formulation. Such solutions are called self-similar, since a single fixed value of the compound single variable corresponds to numerous combinations of, say, coordinates or coordinates and time, which make them identical in the “space” of the single compound variable. The general consideration in Chap. 3 is followed by a number of examples illustrating the usage of the Pi-theorem for establishing self-similar solutions. They include flows of viscous incompressible fluid (the Stokes, Landau, and von Karman problems), hydrodynamic and thermal (diffusion) boundary layers (the Blasius, Pohlhausen and Levich problems), as well as some special hydrodynamical problems, e.g. propagation of viscous-gravity currents (the Huppert problem) and capillary waves on a thin liquid after a weak impact of a tiny droplet (the Yarin-Weiss problem). Below we consider the applications of the Pi-theorem for solving the differential equations of hydrodynamics and the theory of heat and mass transfer. In the frame of continuum approach, fluid flows and heat and mass transfer are described by a set of PDEs that include the continuity, momentum (the Navier–Stokes), energy and species balance equations (Kays and Crawford, 1993; Baehr and Stephan 1998; Schlichting 1979). Solving these equations is extremely difficult because of the non-linearity of the inertial terms of the Navier–Stokes equations and dependences of the velocity, temperature and species concentration fields on several variables: in the general case on three spatial coordinates and time. An essential simplification of the problem can be achieved by reducing the number of independent variables by L.P. Yarin, The Pi-Theorem, Experimental Fluid Mechanics, DOI 10.1007/978-3-642-19565-5_3, # Springer-Verlag Berlin Heidelberg 2012
39
40
3 Application of the Pi-Theorem to Establish Self-Similarity
means of a transformation of the problem PDEs to a set of ODEs if the initial and boundary conditional of a specific problem admit that as well. As a matter of fact, such a reduction can be considered as a typical problem of the dimensional analysis, namely, the transition from n dimensional variables to n k dimensionless ones. In the case where the number of dimensionless groups n k ¼ 1, a multi-dimensional problem reduces to a one-dimensional one. In order to demonstrate the transformation of PDEs into ODEs by using the Pi-theorem, we consider a process in which some unknown dimensional characteristics a (say, velocity, temperature, etc.) is expected to be determined by dimension parameters a1 ; a2 ; an (say, viscosity, density, velocity of the undisturbed flow, a characteristic size of a body, etc.) a ¼ f ða1 ; a2 ak an Þ
(3.1)
where a1 ; a2 ; ak are the parameters with independent dimensions. Assume that two parameters ai and aj among the n governing parameters are independent variables (say, a coordinate and time) and all the other n 2 parameters are constants (say, the undisturbed velocity, viscosity, etc.). Then the unknown characteristic a is described by PDE and the boundary and initial conditions which contain all the n 2 constants of the problem, the independent variables ai and aj , as well as the derivatives @a=@ai ; @a=@aj ; @ 2 a=@a2i ; @ 2 a=@a2j , etc. In order to reduce the problem to the integration of a set of ODEs with the appropriate boundary conditions, one should find such a transformation of the unknown characteristics a and the governing parameters a1; a2 ; an that the dimensionless unknown characteristics (denoted as P) will become a function of a single dimensionless group P1 formed from the governing parameters of the problem. The latter is possible when the number of the governing parameters equals k þ 1: Then (3.1) reduces to P ¼ ’ðP1 Þ a
0
(3.2) a
0
0
a
0
where P ¼ a=aa11 aa22 aak k , and P1 ¼ an =a11 a22 akk , with ai and ai (i ¼ 1, 2,. . . kÞ being some exponents. It is emphasized that P is described by an ODE (or a system of ODEs), since it is a function of a single variableP1 : The above consideration shows that transformation of a PDE into an ODE is determined by the dimensions of the governing parameters. For example, consider a particular case when the unknown characteristics a depends on four dimension parameters: two constants a1 ; a2 and two independent variables a3 ; a4 a ¼ f ða1 ; a2 ; a3 ; a4 Þ
(3.3)
Let the dimensions of the unknown characteristics a and governing parameters a1 ; a2 ; a3 and a4 be
3.1 General Remarks
41 0
0
0
00
00
00
½a ¼ Le1 Me2 T e3 ; ½a1 ¼ Le1 Me2 T e3 ; ½a2 ¼ Le1 Me2 T e3 ; ½a3 000
000
000
IV
IV
IV
¼ Le1 Me2 T e3 ; ½a ¼ Le1 Me2 T e3 0
00
(3.4)
000
where ei ; ei ; ei ; ei and eIV i (i ¼ 1,2,3) are some known exponents. When three governing parameters, for example, a1 ; a2 and a3 have independent dimensions, (3.3) takes the form of (3.2) with P ¼ a=aa11 aa22 aa33 and P1 ¼ 0
a
0
a
a
0
a4 =a11 a22 a33 : Bearing in mind the dimensions of the parameters a; a1 ; a2 ; a3 and a4 , we arrive at the following sets of the algebraic equations for the exponents 0 ai and ai : 0
00
000
0
00
000
0
00
000
e1 a1 þ e1 a2 þ e1 a3 ¼ e1 e2 a1 þ e2 a2 þ e2 a3 ¼ e2
(3.5)
e3 a1 þ e3 a2 þ e3 a3 ¼ e3 and 0
0
00
0
000
0
0
0
00
0
000
0
0
0
00
0
000
0
e1 a1 þ e1 a2 þ e1 a3 ¼ eIV 1 e2 a1 þ e2 a2 þ e2 a3 ¼ eIV 2 e3 a1 þ e3 a2 þ e3 a3 ¼
(3.6)
eIV 3
Solution of the systems of (3.5) and (3.6) exists when 0 e e00 e000 100 1000 10 e e e 6¼ 0 2 2 20 e e00 e000 3 3 3
(3.7)
The inequality (3.7) is the condition under which transformation of a PDE into an ODE is possible. This inequality determines only the necessary rather than the sufficient condition of the existence of self-similar solutions. In addition to satisfying the ODEs, such solutions should also satisfy the initial and boundary conditions of the problem. The transformation of a PDE into an ODE is also possible with n k ¼ 2; 3; l when all the constants in (3.1) have the same dimensions, which are different from the dimensions of the independent variables ai and aj : Indeed, (3.1) can be recast into the following form a ¼ f ða1 ; a2 ai ak aj an Þ
(3.8)
The latter equation is reduced to the dimensionless form when the Pi-theorem is applied P ¼ ’ðP1 ; P2 Pj Pnk Þ
(3.9)
42
3 Application of the Pi-Theorem to Establish Self-Similarity a
where a
00
00
a
0
a
0
a
00
00
aj
a
aj
aj
a
j
akþ2 =a11 a22 ai i akk ; Pj ¼ aj =a11 a22 ai i ak k ; ank ank a 1 1 a2 2
a
0
a
0
P ¼ a=aa11 aa22 aai i aakk ; P1 ¼ akþ1 =a11 a22 ai i ak k ; P2 ¼ ank ai i
and
Pnk ¼ ank =
ank ak k .
Due to the fact that the dimensions of all the constants are different from the 0 00 are equal to dimensions of variable ai ; we find that the exponents ai , ai ; ank i zero. Then (3.9) takes the following form P ¼ ’ðc1 ; c2 Pj cnk Þ ¼ ’ðPj Þ
(3.10)
are the dimensionless constants: c1 c1 ; c2 ; cnk 0 00 0 00 0 0 0 00 00 00 a1 a2 ai1 aiþ1 ak a 1 a2 ai1 aiþ1 ak akþ1 =a1 a2 ai1 aiþ1 ak , c2 ¼ akþ2 =a1 a2 ai1 aiþ1 ak ; . . ..cnk nk aiþ1 ank ank ank ank i1 aiþ1 ak k : ank =a11 a22 ai1 where
¼ ¼
It is emphasized that the possibility to transform the problem PDEs into ODEs can be recognized by a simple analysis of the dimensions of constants involved in the problem formulation, i.e. in the governing equations, the initial and boundary conditions, as well as the additional characteristics which are given, such as the invariant overall momentum flux in a submerged jet, etc. The rule that such a transformation is possible (and thus, a self-similar solution of the PDE-based problem exists) can be formulated as follows: the transition of PDEs into ODEs is possible when scales of the independent variables cannot be constructed using the scales of the problem constants (Loitsyanskii 1966). In order to illustrate this statement, we consider the radial flow in a plane wedge. The radial component of fluid velocity v in this case depends on the strength of a source Q located at the wedge apex, kinematic viscosity n of fluid, as well as the radial coordinate r and the polar angle ’ of the cylindrical coordinate system v ¼ f ðQ; n; r; ’Þ
(3.11)
The dimensions of the given constants Q and n are: ½Q ¼ L2 T 1 (per unit length of a source or sink normal to the wedge plane) and ½n ¼ L2 T 1 : It is easy to see that it is impossible to express the dimension of r via the dimensions of the constants Q and n: This fact points out at the possibility of transformation of the PDE describing the flow under consideration into an ODE. Indeed, accounting for the fact that two from the four governing parameters have independent dimensions, we can rewrite (3.11) as follows v ¼ f1
Q ; ’ ¼ f ð’Þ n
where v ¼ v= nr , and Q=n are dimensionless constants.
(3.12)
3.1 General Remarks
43
Substituting the expression (3.12) into the set of the governing equations 2 dw 1 @P @ v 1 @ 2 v 1 @v v þ þ ¼ þn v dr r @r @r2 r2 @’2 r @r r2
1 @P 2v @v þ ¼0 rr @’ r 2 @’ @ðrvÞ ¼0 @r
(3.13)
(3.14)
(3.15)
we arrive at the following ODE (Hamel 1917; see also Rosenhead 1963 and Loitsyanskii 1966) d2 f þ 4f þ 6f 2 ¼ 2c1 d’2
(3.16)
where c1 is a constant. The constant c1 , as well as the two additional constants c2 and c3 that appear when integrating (3.16), are determined by the no-slip boundary conditions at the wedge walls vða=2Þ ¼ 0 and the condition that a constant mass flow rate is given. It is emphasized that a similar analysis of a flow in a cone (instead of a wedge) where Q has the dimension L3 T 1 shows that transformation of the PDE describing this flow into an ODE is impossible since the dimension of L is provided by the ratio Q=n. Reduction of PDEs to ODEs is possible under a certain idealization of real phenomena. The absence of a characteristic scale (for example, the length scale is missing in flows about semi-infinite plates, in submerged jet flows issued from a point wise source of momentum flux, etc.) is a sign of the existence a self-similar solution of PDEs and a possibility of their transformation to ODEs (Sedov 1993). Solving such problems in dimensionless form requires introduction of some arbitrary scales instead of the missing ones. Only then the governing equations and the initial and boundary conditions can be transformed to the dimensionless form. As a result, the unknown characteristics can be expressed as a dimensionless function of dimensionless variables only. The requirement that the arbitrary length scale should not be involved in the dimensional solution of an idealized problem allows revealing the form of this function. It is emphasized that the absence of a characteristic length scale in the problem formulation is an essential but not a sufficient condition for the existence of a self-similar solution of a set of the corresponding PDEs. For example, the problem on flow in an axisymmetric diffuser with circle cross-section has no self-similar solution. Below we consider a number of examples of applications of the Pi-theorem for reduction of PDEs describing incompressible fluid flows and heat transfer in media at rest to ODEs, and thus finding self-similar solutions.
44
3.2
3 Application of the Pi-Theorem to Establish Self-Similarity
Flow over a Plane Wall Which Has Instantaneously Started Moving from Rest (the Stokes Problem)
Consider an upper half-space filled with a viscous incompressible fluid in contact with a flat plate corresponding to y ¼ 0 (Fig. 3.1). (Stokes 1851) This wall has instantaneously started to move horizontally with a constant velocity U. The wall motion is transmitted to the fluid due to the action of viscous forces. As a result, the fluid is entrained into horizontal motion as well. The fluid flow is subjected to the no-slip boundary condition at the wall surface and is described as follows @u @2u ¼n 2 @t @y
(3.17)
t ¼ 0 : 0 y 1 u ¼ 0; t>0 : y ¼ 0 u ¼ U; y ! 1 u ¼ 0
(3.18)
where u is the fluid velocity and n is the kinematic viscosity. Equation (3.17) with the conditions (3.18) show that fluid velocity depends on four parameters: two independent variables t and yand two constants n and U : u ¼ f ðU; n; y; tÞ
(3.19)
First of all, let us ascertain the possibility of reduction of (3.17) to an ODE. For this aim we make use of the above-mentioned signs pointing at such a transformation. The lack of a given characteristic length in (3.17) and the conditions (3.18) points at the possibility of reduction of (3.17) to an ODE. To apply the Pi-theorem to transform (3.17) into an ODE, it is necessary to consider the dimensions of the unknown characteristics u and the governing parameters n; U; y and t: In principle, it is possible to use different systems of fundamental units, in particular, the LMT system. Then, we have the dimensions as ½u ¼ LT 1 ; ½U ¼ LT 1 ; ½n ¼ L2 T 1 ; ½ y ¼ L; ½t ¼ T
y
y
t>0
t=0
U>0
U=0
Fig. 3.1 Scheme of flow over plane wall has instantaneous by started moving from rest
(3.20)
0
u
0
u
3.2 Flow over a Plane Wall Which Has Instantaneously Started Moving from Rest
45
Two from the four governing parameters have independent dimensions, so that n k ¼ 2: In accordance with that, we obtain u ¼ ’ðt; yÞ
(3.21)
where u ¼ u=U; t ¼ t=ðn=U 2 Þ; y ¼ y=ðn=UÞ. Equation (3.21) shows that u depends on two dimensionless groups that seemingly shows that it is impossible of transform (3.17) into an ODE. This result follows directly from the analysis of the dimensions of the parameters involved. Indeed, we can construct the length and time scales, L ¼ n=U, T ¼ n=U 2 , that shows that it is impossible to express u as a function of a single dimensionless variable. At the first sight this result contradicts to the expectations based on the absence of the characteristic length scale in the problem formulation as in the present case. The apparent contradiction can be explained as follows. The Pi-theorem determines only the number of dimensionless groups which can be constructed from n governing parameters including k parameters with independent dimensions. The number n is determined by the physical essence of the problem, whereas the number k can be changed depending on the system of units used. Thus, the difference n k that determines the number of dimensionless variables depends also on the system of units used. Let us extend the system of units by introducing three different length scales Lx and Ly for x and y directions (along and normal to the wall in Fig. 3.1), and Lz for the z direction normal to the xy plane. This means that the Lx Ly Lz MT system of units is used. Taking into account that the wall and the velocity component u, as well as the velocity of the unperturbed flow U are directed along the x-axis, we define their dimensions as ½u ¼ Lx T 1 ; ½U ¼ Lx T 1
(3.22)
where T is the time scale, ½t ¼ T: The dimension of the kinematic viscosity, is ½n ¼ L2y T 1
(3.23)
since in the case under consideration viscosity transmits information about the wall motion into the liquid bulk in the y direction. Indeed, the dimension of viscosity m can be found directly from the rheological constitutive equation of the Newtonian fluid tyx ¼ mðdu=dyÞ as the ratio of the shear stress to the velocity gradient (Huntley 1967; Douglas 1969). Bearing in mind that tyx ¼ Fyx =Sxz , we determine the dimension of tyx 2 tyx ¼ L1 (3.24) z MT where Fyx and Sxz are the force in the x direction acting at the surface element in the xz plane, respecticaly; Fyx ¼ Lx MT 2 ; ½Sxz ¼ Lx Lz .
46
3 Application of the Pi-Theorem to Establish Self-Similarity
1 Since the dimension of the velocity gradient du=dy is Lx L1 y T , the dimension of viscosity is expressed as 1 1 ½m ¼¼ L1 x Ly Lz MT
(3.25)
Then the dimension of the kinematic viscosity is
m ½n ¼ ¼ L2y T 1 r
(3.26)
1 1 where ½r ¼ L1 x Ly Lz M is the fluid density. As a result, we guaranteethat the of all the terms in (3.17) are the dimensions same: ½@u=@t ¼ Lx T 2 ; and n@ 2 u @y2 ¼ Lx T 2 . In the framework of the Lx Ly Lz MT system among the four governing parameters there are three parameters with independent dimensions e
0
0
e
0
e
00
e
00
000
00
e
e
000
000
½U ¼ Lx1 Ly2 T e3 ; ½n ¼ Lx1 Ly2 T e3 ; ½t ¼ Lx1 Ly2 T e3 0
0
0
00
00
00
000
(3.27) 000
where e1 ¼ 1; e2 ¼ 0; e3 ¼ 1; e1 ¼ 1; e2 ¼ 2; e3 ¼ 1; e1 ¼ 0; e2 ¼ 0; and 000 e3 ¼ 1. 0 00 000 At such values of the exponents ei ; ei and ei determinant (3.7) is not equal to zero. In this case (3.19) takes the form of (3.2) with P ¼ u=U a1 na2 ta3 and P1 ¼ 0
0
0
y=U a1 na2 ta3 : Taking into account the dimensions of u and U; n; t; y; we arrive 0 at the system of the algebraic equations for the exponents ai and ai (i ¼ 1; 2; 3Þ SLx 1 a1 ¼ 0; SLy a2 ¼ 0;
0
a1 ¼ 0 0
1 2a2 ¼ 0
ST a1 a3 1 ¼ 0;
0
0
a2 a3 ¼ 0
(3.28)
where the symbols SLx ; SLy and ST refer to the summation of the exponents of Lx ; Ly and T; respectively. From (3.28) it follows 1 1 0 0 0 a1 ¼ 1; a2 ¼ 0; a3 ¼ 0; a1 ¼ 0; a2 ¼ ; a3 ¼ 2 2 Then (3.19) reduces to
u y ¼ ’ pffiffiffiffi U nt
(3.29)
(3.30) 0
00
The substitution of the derivatives @u=@t ¼ U’ =2t and @ 2 u=@y2 ¼ U’ =nt into (3.17) leads to the following ODE determining the function ’ 0 00 ’ þ ’ ¼0 2
pffiffiffiffi 0 00 where ’ ¼ ’ðÞ; ’ ¼ d’=d, and ’ ¼ d 2 ’=d2 , with ¼ y= nt.
(3.31)
3.3 Laminar Boundary Layer over a Flat Plate (the Blasius Problem)
3.3
47
Laminar Boundary Layer over a Flat Plate (the Blasius Problem)
The previous example was related to flow development in fluid that has initially been at rest and started moving being entrained by a plate. Below we consider an application of the Pi-theorem for transformation of the boundary layers equations into an ODE in the case of fluid flow about a motionless wall. Consider a flow over a semi-infinite plate. The flow is assumed to be incompressible and fluid velocity is considered to be uniform far away from the plate surface (Fig. 3.2). (Blasius 1980) The system of the governing equations in this case reads
u
@u @v @2u þv ¼n 2 @x @y @y
(3.32)
@u @v þ ¼0 @x @y
(3.33)
Equations (3.32) and (3.33) should be integrated subjected to the no-slip boundary conditions at the plate surface, as well as and a given constant velocity of the stream parallel to the plate is prescribed far away from the plate y ¼ 0; u ¼ v ¼ 0; y ! 1; u ! U
(3.34)
We use the Blasius problem to demonstrate the efficiency of using the ordinary LMT and modified Lx Ly Lz MT systems of units for dimensional analysis of thermohydrodynamic problems. First of all, we consider the application of the Pi-theorem to the Blasius problem using the LMTsystem of units. Equations
y
u
d (x)
Fig. 3.2 The flow in the boundary layer over a flat plate
0
x
48
3 Application of the Pi-Theorem to Establish Self-Similarity
(3.32–3.33) and the boundary conditions (3.34) show that flow velocity within the boundary layer over a flat plate depends on four dimensional parameters: two independent variables x; y and two constants n and U: Therefore, we can write the following functional equation for the longitudinal velocity component u u ¼ f ðx; y; n; UÞ
(3.35)
where the dimensions of u; x; y; n and U are expressed as ½u ¼ LT 1 ; ½ x ¼ L; ½ y ¼ L; ½n ¼ L2 T 1 ; ½U ¼ LT 1
(3.36)
It is seen that of the four governing parameters, two parameters possess indepensdent dimensions. That means that the difference n k ¼ 2, so that the dimensionless velocity is a function of two dimensionless groups. Choosing n and U as the parameters with independent dimensions, we tramsform (3.35) to the dimensionless form using the Pi-theorem. As a result, we arrive at the following equation u ¼ ’ðx; Þ
(3.37)
where u ¼ u=U; x ¼ xU=n; ¼ yU=n: Consider (3.37) from the point of view of generalization of the experimental data for flows over flat plates, as well as the theoretical analysis of the corresponding problem. Assume that an experimental data bank for the velocity at a number of points within the boundary layer is available. According to (3.37), these data determine a surface in the parametric space u x : A section of this surface by a plane x ¼ const determines the velocity distribution in given cross-section of the boundaty layer. The totality of the velocity profiles corresponding to different values of x determines the flow field within the boundary layer. It is obvious that usefulness of such an approach for the generalization of the experimental data would be low, since it requires many diagrams corresponding to different crosssections of the boundary layer, which makes it extremely laborious. On the other hand, we apply now (3.37) for the theoretical analysis of the Blasius problem. For this aim we rewrite (3.32) and (3.33) and the boundary conditions (3.34) using the variables u; x and . Taking into account that @u u ¼ @x
3 3 2 U3 @u @u U @u @ 2 u U @ u u ;v v ;n 2¼ ¼ n n n @2 @x @y @ @y
(3.38)
and @u ¼ @x we arrive at the equations
2 2 U @u @v U @v ; ¼ n @x @y n @
(3.39)
3.3 Laminar Boundary Layer over a Flat Plate (the Blasius Problem)
u
49
@u @u @ 2 u þv ¼ @x @ @2
(3.40)
@u @v þ ¼0 @x @
(3.41)
Their solutions are subject to the following boundary conditions ¼ 0; u ¼ v ¼ 0; ! 1; u ! 1
(3.42)
were v ¼ v=U. Is easy to see that the transformation of (3.32) and (3.33) and the boundary conditions (3.34) using the LMTsystem of units does not lead to any simplification of the theoretical analysis of the Blasius problem. The latter still reduces to integrating the system of the partial differential equations (3.41) and (3.42). Accordingly, for the analysis of the planar boundary layer problems, it is convenient to use the modified LMT system that includes two different scales of length Lx , Ly and Lz for the x, y and z directions, respectively, where the x axis is parallel to the plate in the flow direction, while the y and z axes are normal to it (cf. Fig. 3.2). It is easy to show that the introduction of the two additional length scales does not affect the dimension uniformity of the terms of the boundary layer and continuity equations. Indeed, assuming that the dimensions of ½ x ¼ Lx and ½t ¼ T; we find that the corresponding dimension of the longitudinal velocity component is ½u ¼ Lx T 1
(3.43)
Requiring that both terms of the continuity equation (3.30) possess the same dimensions ½@u=@x ¼ T 1 ; and ½@v=@y ¼ T 1 ; we find the dimensions of v and y as ½n ¼ Ly T 1 ; ½ y ¼ Ly
(3.44)
Then the dimensions of the terms in the momentum equation (3.32) become
2
@u @u @ u 2 2 u ¼ Lx T ; v ¼ Lx T ; n 2 ¼ Lx T 2 @x @y @y
(3.45)
where the dimension of kinematic viscosity n is L2y T 1 . First we estimate the thickness of the boundary layers d: It is clear that d can be a function of a single independent variable x, as well as of the two constants of the problem: the kinematic viscosity n and the free stream velocity ½U ¼ Lx T 1 d ¼ fd ðU; n; xÞ
(3.46)
50
3 Application of the Pi-Theorem to Establish Self-Similarity
Since all the governing parameters in (3.46) possess independent dimensions, the difference n k ¼ 0 and the thickness of the boundary layer can be expressed as d ¼ cna1 xa2 U a3
(3.47)
where ½d ¼ Ly ; c is a dimensionless constant and the exponents a1 ; a2 and a3 are equal to 1=2; 1=2 and 1=2; respectively. As a result, we obtain rffiffiffiffiffi nx (3.48) d¼c U The velocity at any point in the boundary layer depends on the variables ½ x ¼ Lx ; ½ y ¼ Ly and constants n and U u ¼ fu ðU; n; x; yÞ
(3.49)
Three governing parameters in (3.49) possess independent dimensions. Therefore, in accordance with the p Pi-theorem, (3.49) can be reduced to the form of (3.2) ffiffiffiffiffiffiffiffiffiffiffi with P ¼ u=U and P1 ¼ y= nx=U , i.e. u 0 ¼ ’u ðÞ (3.50) U pffiffiffiffiffiffiffiffiffiffiffi 0 where ’u ¼ d’u =d, ¼ y= nx=U . Equation (3.50) shows that the dimensionless velocity u ¼ u=U is determined by a single variable : That allows one to generalize the experimental data for the velocity distribution in different cross-sections of the boundary layer over a flat plate in the form of a single curve uðÞ. Naturally such presentation of the results of experimental investigations has a significant advantage compared to the presentation of the experimental data in the form of a surface in the parametrical space u x discussed before. The theoretical analysis of the Blasius problem is also significantly simplified by using the Lx Ly Lz MT system of units, since the problem is reduced in this case to integrating an ordinary differential equation. Indeed, the substitution of the expression (3.50) into (3.32) and (3.33) results in the following ODE for the unknown function ’u ðÞ 000
0
00
2’u þ ’u ’u ¼ 0
(3.51)
with the boundary conditions 0
0
¼ 0; ’u ¼ 0 ’u ¼ 0; ! 1 ’u ¼ 1
(3.52)
pffiffiffiffiffiffiffiffiffiffiffiffiffi 0 The shear stress at the wall tw ¼ mð@u=@yÞ0 ¼ m U3 =nx’ ð0Þ; where 0 ’ ð0Þ ¼ du=dj0 . It is emphasized that there is another way of transforming (3.32) and (3.33) into the ODE. It is based on the assumption that velocity at any point of the boundary layer is determined by three governing parameters, namely, the free stream velocity
3.4 Laminar Submerged Jet Issuing from a Thin Pipe (the Landau Problem)
51
½U ¼ Lx T 1 ; the thickness of the boundary layer ½d ¼ Ly and the distance from the plate to a point under consideration ½ y ¼ Ly u ¼ fu ðU; y; dÞ
(3.53)
Since two of the three governing parameters in (3.53) possess independent dimensions, we obtain
y u 0 ¼ ’u (3.54) U d where the dependence dðxÞ is given by (3.48). The instructive examples of the applications of the Pi-theorem for the analysis of the Stokes and Blasius problems allow one to evaluate the true value of the LMT and Lx Ly Lz MT systems of units. The comparison of the results produced by both systems of units shows that the expansion of the system of units by introducing different length scales in the x; y and z directions allows one to reduce the number of the dimensionless groups and significantly simplifies generalization of the experimental data and theoretical analysis of these problems. As a matter of fact, the rationale for choosing a system of units (LMT or Lx Ly Lz MTÞ should be based on the comparison of the number of parameters with independent dimensions in the set of the governing parameters determining the problem. Indeed, since the total number of the governing parameters n does not depend on the system of units, the number of the dimensionless groups in any given problem, n k, is fully determined by the number of parameters with independent dimensions k. Therefore, the choice of the Lx Ly Lz MT system of units is desirable when k > k
(3.55)
where subscripts and correspond to the LMT and Lx Ly Lz MT systems of units, respectively. Thus, the LMT system of units should be used when ðn kÞ equals zero or unity. In the case when ðn kÞ > 1, it is preferable to use the Lx Ly Lz MT system of units. In future we will use both systems of units without an additional discussion of the reasons for choosing a given system.
3.4
Laminar Submerged Jet Issuing from a Thin Pipe (the Landau Problem)
Let an incompressible fluid be issued from a thin pipe into an infinite space filled with the same medium (with the same physical properties as those of the jet). As a result of the laminar jet flow, mixing of the issuing and the ambient fluids takes place
52
3 Application of the Pi-Theorem to Establish Self-Similarity
Fig. 3.3 Stream lines in flow is induced laminar jet issuing from a thin pipe
thin pipe
The flow is described by the Navier–Stokes and continuity equations 1 ðr vÞv ¼ rP þ nr2 v r
(3.56)
rv¼0
(3.57)
where v is the velocity vector with the components vr ; vy; and v’ , and r; y; and ’ are the spherical coordinates with the y axis (the polar axis) in the direction of the jet and centered at its origin; P is the pressure. The sketch of this flow is shown in Fig. 3.3 (Landau 1944). Let us assume that there is no swirl and v’ ¼ 0: In addition, due to the assumed axial symmetry of the flow about the polar axis (y ¼ 0Þ, the velocity components vr and vy are the function of only two variables: r and y: The velocity components also depend on viscosity, as well as on the kinematic momentum flux J ¼ I =r (I is the total momentum flux in the jet which is determined by the pipe flow and is given). Thus, we can write the functional equations for the velocity components vr and v’ and pressure P in the following form vr ¼ f1 ðr; y; n; JÞ
(3.58)
vy ¼ f2 ðr; y; n; JÞ
(3.59)
P ¼ f3 ðr; y; n; JÞ
(3.60)
where r; n, y and J have the following dimensions ½r ¼ L; ½n ¼ L2 T 1 ; ½y ¼ 1; ½ J ¼ L4 T 2
(3.61)
It is seen that two of the four dimensional parameters in (3.58–3.60) possess independent dimensions (n k ¼ 2Þ: In this case the Pi-theorem yields Pi ¼ ’i ðP1i ; P2i Þ
(3.62)
3.4 Laminar Submerged Jet Issuing from a Thin Pipe (the Landau Problem) 0
0
00
53
00
where Pi ¼ Ni =r a1i na2i ; P1:i ¼ y=ra1i na2i ; P2;i ¼ J=r a1i na2i ; N1 ¼ vr ; N2 ¼ vy ; and N3 ¼ P=r; with i ¼ 1; 2; 3. Bearing in mind the dimensions of vr , vy ; P; J and y; we find the values of the exponents in (3.62) a11 ¼ 1; a21 ¼ 1; a12 ¼ 1; a22 ¼ 1; a13 ¼ 2; a23 ¼ 2 0
0
0
0
0
0
a11 ¼ 0; a21 ¼ 0; a12 ¼ 0; a22 ¼ 0; a13 ¼ 0; a23 ¼ 0 00
00
00
00
00
00
a11 ¼ 0; a21 ¼ 0; a12 ¼ 0; a22 ¼ 2; a13 ¼ 0; a23 ¼ 0
(3.63)
Then, the dimensionless groups in (3.62) become P1i ¼ y; P21 ¼
J ¼ const: n2
(3.64)
for i ¼ 1; 2; 3; and P1 ¼
vr vy P ; P2 ¼ ; P3 ¼ 2 v v rv
(3.65)
where v ¼ n=r: Accordingly, we obtain the following expressions for the velocity components and pressure n n P n2 vr ¼ ’1 ðyÞ; vy ¼ ’2 ðyÞ; ¼ 2 ’3 ðyÞ r r r r
(3.66)
Substituting the expressions (3.66) into (3.56) and (3.57), we arrive at the following system of ODEs 00
0
’1 þ ’1 ðctgy ’2 Þ þ ’21 þ ’22 2’3 ¼ 0 0
0
0
’ 2 ’ 2 ’1 ’ 3 ¼ 0
(3.67) (3.68)
0
’1 þ ’2 ’2 ctgy ¼ 0
(3.69)
Excluding ’3 from (3.67–3.69), we obtain the following system of ODEs for the unknown functions ’1 and ’2 000
0
0
0
0
0
0
’1 þ ð’1 ctgyÞ þ ð’2 ’1 Þ þ 2’1 ’1 þ 2’1 ¼ 0 0
’1 þ ’1 ’2 ctgy ¼ 0
(3.70) (3.71)
A solution of (3.70) and (3.71) corresponding to the issuing viscous fluid from a thin pipe (a point wise source of momentum flux) was found by Landau (1944)
54
3 Application of the Pi-Theorem to Establish Self-Similarity
’1 ¼ 2 þ
2ðA2 1Þ 2
ðA þ cos yÞ
; ’2
2 sin y A þ cos y
(3.72)
where A is a constant of integration which is related to the total momentum flux of the jet J by the following expression J ¼ 16pn A 1 þ 2
A A Aþ1 ln 3ðA2 1Þ 2 A 1
(3.73)
A detailed analysis of the flow in a submerged jet issued from a thin pipe following the original work of Landau (1944). It can be also found in the monographs of Landau and Lifshitz (1987), Sedov(1993), Vulis and Kashkarov (1965).
3.5
Vorticity Diffusion in Viscous Fluid
Consider transformation of the PDE into an ODE in the problem which describes the evolution of an initially infinitely thin vortex line of strength G. Assume that the vortex line is normal to the flow plane (Fig. 3.4 a). The vorticity transport equation reads (Batchelor 1967) @O n @ @O ¼ r @t r @r @r
(3.74)
where O is the vorticity component (the only one which is non-zero and normal to the flow plane). The unknown characteristics ½O ¼ T 1 depends on two variables-time ½t ¼ T and the radial coordinate reckoned from the location of the initial vortex line
a
b
Ω
r
r ϕ
0
t
Fig. 3.4 Diffusion of vorticity in viscous fluid. (a) Stream lines. (b) The dependence of vorticity on time for different values of radial coordinate
3.6 Laminar Flow near a Rotating Disk (the Von Karman Problem)
55
½r ¼ L, as well as on two constants of the problem-the vortex strength ½G ¼ L2 T 1 and kinematic viscosity ½n ¼ L2 T 1 . From the initial condition of the problem GR ¼ G at t ¼ 0 (with GR being the circulation over a circle of radius r ¼ R where R is arbitrary) and the fact that (3.74) is linear, it follows that O is directly proportional to G (Sedov 1993) O ¼ Gf1 ðn; r; tÞ
(3.75)
Two from the three governing parameters in (3.75) have independent dimensions. Therefore, the difference n k ¼ 1: Then, in accordance with the Pi-theorem (3.75) takes the form P ¼ ’ðP1 Þ 0
(3.76)
0
where P ¼ O=Gna1 ta2 ; and P1 ¼ r=na1 ta2 . Bearing in mind the dimensions of O; G; n; r; and t, we find the values of 0 0 0 the exponents ai and ai as : a1 ¼ 1; a2 ¼ 1; a1 ¼ 1=2; and a2 ¼ 1=2: Then, (3.76) takes the form O¼
G r ’ pffiffiffiffi nt nt
(3.77)
Substituting the expression (3.77) into (3.74), we arrive at the ODE 0 0
0
2ð’ Þ þ ð2’ þ ’ Þ ¼ 0
(3.78)
pffiffiffiffi with ¼ r= nt. Its solution with the account for the initial condition yields the following wellknown vorticity distribution (cf. Sherman 1990) G r2 exp O¼ 4nt 4pnt
(3.79)
depicted in Fig. 3.4 b.
3.6
Laminar Flow near a Rotating Disk (the Von Karman Problem)
The flow sketch is presented in Fig. 3.5 (Karman 1921). The velocity vector of flow over a rotating disk has three projections u; v and w on the radial, azimuthal and axial axes of the cylindrical coordinate system associated with the center of the disk.
56
3 Application of the Pi-Theorem to Establish Self-Similarity
Fig.3.5 Flow over rotating disk in liquid at rest
z Ω
w
r P
v u
0 ϕ
The system of the governing Navier–Stokes and continuity equations corresponding to this flow takes the following form 2 @u v2 @u 1 @P @ u @ u @ 2 u ¼ þn þ þ u þw r @r @z r @r @r 2 @r r @z 2 @v uv @v @ v @ v @ 2 v þ þ þw ¼n þ @r r @z @r 2 @r r @z2
(3.81)
2 @w @w 1 @P @ w 1 @w @ 2 w þ þw ¼ þn þ @r @z r @z @r2 r @r @z2
(3.82)
@u u @w þ þ ¼0 @r r @z
(3.83)
u
u
(3.80)
The boundary conditions for (3.80–3.83) read z ¼ 0; u ¼ 0 v ¼ rO w ¼ 0; z ¼ 1; u ¼ v ¼ 0
(3.84)
where it is assumed that the disk rotates with the angular velocity O. Assume that velocity components or pressure at any point of a thin liquid layer over a rotating disk depend on some characteristic velocity (or pressure), the axial distance from the disk z and the layer thickness d. Then, the functional equations for the velocity components and pressure can be written as u ¼ f1 ðu ; z; dÞ
(3.85)
v ¼ f2 ðv ; z; dÞ
(3.86)
3.6 Laminar Flow near a Rotating Disk (the Von Karman Problem)
57
w ¼ f3 ðw ; z; dÞ
(3.87)
P ¼ f4 ðP ; z; dÞ
(3.88)
where the velocity component and pressure scales for a given radial position r are denoted with the asterisks. The dimensions of the governing parameters in (3.80–3.83) are ½u ¼ LT 1 ; ½v ¼ LT 1 ; ½w ¼ LT 1 ; ½z ¼ L; ½d ¼ L; ½P ¼ L1 MT 2 (3.89) It is seen that two of the three governing parameters on the right hand side in (3.85–3.88) possess independent dimensions. Accordingly, these equations can be presented in the form
z u ¼ ’1 u d
(3.90)
z v ¼ ’2 v d
(3.91)
z w ¼ ’3 w d
(3.92)
z P ¼ ’4 P d
(3.93)
It is easy p to ffiffiffiffiffiffiffiffi show that the thickness of the fluid layer carried by the disk d is of the order of n=O. Then, taking as the characteristic scales of u; v; w and P as u ¼ rO; v ¼ rO; w ¼
pffiffiffiffiffiffiffi nO; P ¼ rnO
(3.94)
we arrive at the following expressions u ¼ rO’1 ðÞ
(3.95)
v ¼ rO’2 ðÞ
(3.96)
pffiffiffiffiffiffi nO’3 ðÞ
(3.97)
P ¼ rnO’4 ðÞ
(3.98)
w¼
pffiffiffiffiffiffiffiffi where ¼ n=O. Using the expressions (3.95–3.98), we transform (3.80–3.83) into the following ODEs
58
3 Application of the Pi-Theorem to Establish Self-Similarity
2’1 þ ’3 ¼ 0 0
(3.99) 00
’21 þ ’1 ’3 ’22 ’1 ¼ 0 0
00
2’1 ’2 þ ’3 ’2 ’2 ¼ 0 0
00
’4 þ ’3 ’3 ’3 ¼ 0
(3.100) (3.101) (3.102)
where differentiation by is denoted by prime. The boundary conditions for (3.99–3.102) become ¼ 0; ’1 ¼ 0 ’2 ¼ 1 ’3 ¼ 0 ’4 ¼ 0; ! 1; ’1 ¼ 0 ’2 ¼ 0
(3.103)
Note that above approach dealing with the flow over an infinite disk can also be used for the evaluation of flow characteristics in the case of a finite radius disk if the latter is much larger than the thickness of the liquid layer adjacent to the disk surface (Schlichting 1979).
3.7
Capillary Waves after a Weak Impact of a Tiny Object onto a Thin Liquid Film (the Yarin-Weiss Problem)
The flow in a planar thin liquid film on a solid surface after an impact of a tiny wire (similarly to the axisymmetric case shown in Fig. 3.6) is governed by the following system of PDEs (the beam equations; Yarin and Weiss 1995) 4 @ 2w 2@ w ¼ a @t2 @x4
(3.104)
4 @2v 2@ v ¼ a @t2 @x4
(3.105)
where w ¼ Dh=h0 is the small dimensionless perturbation of the liquid layer thickness, with h0 and h being the unperturbed and perturbed thicknesses, Dh ¼ h h0 ; v is the liquid velocity in the x-direction (along the surface), a ¼ ðsh0 =rÞ1=2 , where s is the surface tension and r the density, t is time. Equations (3.104) and (3.105) correspond to the situations where gravity and viscous effects are negligible and perturbations of the liquid layer thickness and flow velocity are sufficiently small. All these assumptions are realized after impacts of tiny wire as in Fig. 3.6. Moreover, these objects should be assumed to be point wise. Then, a given length scale disappears from the problem, and there should exist a self-similar solution, which we are searching for below.
3.7 Capillary Waves after a Weak Impact of a Tiny Object
59
Fig. 3.6 A system of concentric waves propagating over a thin liquid film on a solid surface from the impact point of a thin stick seen at the center of the image Reprinted from Yarin and Weiss (1995) with permission
These equations show that w and v depends on two variables x and t and one constant a. Therefore, the functional equations for w and x read w ¼ f1 ða; x; tÞ
(3.106)
v ¼ f2 ða; x; tÞ
(3.107)
The dimensions of w; v; a; x and t are ½w ¼ 1; ½n ¼ L2 T 1 ; ½a ¼ L2 T 1 ; ½ x ¼ L; ½t ¼ T
(3.108)
It is seen that (3.106) and (3.107) contain three governing parameters, whereas two of them have independent dimensions. In accordance with the Pi-theorem, (3.106) and (3.107) can be presented in the following dimensionless form Pw ¼ ’w ðP1w Þ
(3.109)
Pv ¼ ’v ðP1v Þ
(3.110)
0
0
0
0
where Pw ¼ w=aa1 ta2 ; P1w ¼ x=aa1 ta2 ; Pv ¼ v=aa1 ta2 ; and P1v ¼ x=aa1 ta2 . 0 0 The exponents ai ; ai ; ai and ai found by applying the principle of the dimensional homogeneity are equal to 1 1 1 1 1 0 0 0 a1 ¼ 0; a2 ¼ 0; a1 ¼ ; a2 ¼ ; a1 ¼ ; a2 ¼ ; a1 ¼ ; 2 2 2 2 2
(3.111)
60
3 Application of the Pi-Theorem to Establish Self-Similarity
Accordingly, (3.106) and (3.107) take the form w ¼ ’w ðÞ rffiffiffi a ’ ðÞ v¼ t v
(3.112) (3.113)
pffiffiffiffi where ¼ x= at: Substituting the expressions (3.112) and (3.113) into (3.104) and (3.105) yields the following ODEs for the functions ’w ðÞ and ’v ðÞ 1 2 00 3 0 ’IV w þ ’w þ ’w ¼ 0 4 4
(3.114)
1 2 00 5 0 3 ’IV v þ ’v þ ’v þ ’v ¼ 0 4 4 4
(3.115)
Similarly, in the axisymmetric case corresponding to a weak impact of a tiny droplet or a stick (Fig. 3.6) the equation for the surface perturbation w 3 @ 2 w a2 @ @ w þ r 3 r @r @t2 @r
(3.116)
with r being the radial coordinate can be transformed to the following ODE 1 000 2 00 3 0 ’IV ’ þ ’ ¼ 0 w þ ’w þ 4 w 4 w
(3.117)
pffiffiffiffi where ¼ r= at. It is emphasized that the solutions corresponding to the self-similar capillary waves generated by impacts of poitwise objects in reality correspond to remote asymptotics of capillary waves generated by weak impacts of small but finite objects.
3.8
Propagation of Viscous-Gravity Currents over a Solid Horizontal Surface (the Huppert Problem)
Gravity currents belong to a wide class of flows in which one fluid with density r1 is intruding into another fluid with a different density r2 . Such flows are characteristic of many natural phenomena and various engineering processes (Hoult 1972; Simpson 1982). Below we consider one type of gravity currents, namely viscousgravity currents over a rigid surface (Fig. 3.7). (Huppert, 1982)
3.8 Propagation of Viscous-Gravity Currents over a Solid Horizontal Surface
61
Let a layer of a denser fluid of density r invades into a thicker layer of another fluid of a lower density r Dr under the action of gravity. Assume that the volume of the denser fluid increases as ta ; where t is time and ½a ¼ 1 is a constant. The system of the governing equations that describes such flow in lubrication approximation reads (Huppert, 1982) 0 @h 1 g @ 3 @h ¼0 h @t 3 n @x @x
(3.118)
xðN
hdx ¼ qta
(3.119)
0
where h is the thickness of the invading fluid layer, q is a constant, xN is the distance 0 from x ¼ 0 to the leading edge of the invading fluid layer, g ¼ ðDr=rÞg, and n is the kinematic viscosity of the invading denser fluid (cf. Fig. 3.7). The parameters that are involved in the problem formulation, (3.118) and (3.119) have the following dimensions h½ L; ½t ¼ T;
h 0i g ¼ LT 2 ; ½n ¼ L2 T 1 ; ½ x ¼ L; ½xN ¼ L; ½q
¼ L2 T a ; ½a ¼ 1
(3.120)
It is possible to reduce the number of parameters involved in (3.118) and (3.119) by introducing new generalized parameters: 0 3
h 1 gq ¼ L5 T ð3aþ1Þ ¼ L1 T a ; ½ A ¼ h ¼ q 3 n
(3.121)
Then, (3.118) and (3.119) take the following form
z P = P0 r – Δr, na h(x, t)
Fig. 3.7 Scheme of viscous gravity current over a solid horizontal surface
r, ν 0
xN(t) x
62
3 Application of the Pi-Theorem to Establish Self-Similarity
@h @ 3 @h h ¼0 A @t @x @x
(3.122)
xðN
hdx ¼ ta
(3.123)
0
The position of the leading edge of the gravity-driven current xN depends on one independent variable t and a generalized parameter A. Therefore, the functional equation for xN has the form xN ¼ f ðx; AÞ
(3.124)
The governing parameters in (3.124) have independent dimensions. In accordance with the Pi-theorem, (3.124) can be reduced to the form xN ¼ cAa1 ta2
(3.125)
where ½c ¼ 1 is a constant, and the exponents a1 and a2 are equal to: a1 ¼ 1=5; and a2 ¼ ð3a þ 1Þ=5, respectively. Accordingly, the coordinate of the leading edge xN can be expressed as xN ¼ cA1=5 tð3aþ1Þ=5
(3.126)
The thickness h of the gravity-driven current is determined by two independent variable x and t, as well as by the position of the leading edge of the denser layer xN [the latter involves the constants A and a, as per (3.126)]. Accordingly, the functional equation for h reads h ¼ f ðx; xN ; tÞ
(3.127)
Two from the three governing parameters involved in (3.127) possess independent dimensions. Applying the Pi-theorem to (3.127) we arrive at the following dimensionless equation P ¼ ’ðP1 Þ
(3.128)
where P ¼ h=xbN1 tb2 ; and P1 ¼ x=xN ; the exponents b1 and b2 are equal: b1 ¼ 1; and b2 ¼ a, respectively. Bearing in mind the values of the exponents b1 and b2 , as well as the expression (3.126), we rewrite (3.128) as a h ¼ x1 N t ’ðÞ
(3.129)
3.9 Thermal Boundary Layer over a Flat Wall (the Pohlhausen Problem) 0
63
0
where ¼ x=xN ¼ c xA1=5 t3ðaþ1Þ=5 ; and c ¼ 1=c. Substitution of the expression (3.126) into (3.129) yields 0
h ¼ c A1=5 tð2a1Þ=5 ’ðÞ
(3.130)
Calculating the derivatives @h=@t and @h=@x, we transform PDE (3.122) into the following ODE
0
c3 c
0
þ
o 0 1n ð3a þ 1Þc ð2a 1Þc ¼ 0 5
(3.131)
0 5=3 where c ¼ c ’. Equation (3.23) takes the form
0 5=3 ð1 cðÞ ¼ 1 c
(3.132)
0
Equations (3.131) and (3.132) manifest the fact that self-similar solutions of the nonlinear partial differential equations (3.118) and (3.119) do exist. The solutions of the ODEs corresponding to the plane and axisymmetric problems were found by Huppert (1982). Theoretical predictions were compared with the experimental data for the axisymmetric spreading of silicon oil puddles into air for the release rates corresponding to a ¼ 0 and a ¼ 1 in (2.219). Comparisons were also done between the results of the theoretical analysis and data for the axisymmetric spreading of salt water into sweet water in the experiments of Didden and Maxworthy (1982) and Britter (1979). A good agreement of the theoretical predictions with the experimental data was demonstrated.
3.9
Thermal Boundary Layer over a Flat Wall (the Pohlhausen Problem)
Consider the thermal field over a hot or cold semi-infinite flat wall subjected to a parallel uniform flow of an incompressible fluid of a different temperature T1 far from the wall (Pohlhausen 1921). We assume that the difference between the fluid and plate temperatures is sufficiently small, as well as neglect dissipation kinetic energy. We also neglect dependence of the physical properties of the fluid (the kinematic viscosity and thermal diffusivity) on temperature. In this case the system of the governing equations reads
64
3 Application of the Pi-Theorem to Establish Self-Similarity
@u @u @2u þv ¼n 2 @x @y @y
(3.133)
@u @v þ ¼0 @x @y
(3.134)
@DT @DT @ 2 DT þv ¼a @x @y @y2
(3.135)
u
u
where DT ¼ T T1 : The boundary conditions for (3.133–3.134) are as follows y ¼ 0; u ¼ v ¼ 0 DT ¼ DTw ; y ! 1; u ! U DT ¼ 0
(3.136)
if the wall temperature Tw is given. Another type of the thermal boundary condition at the plate might be that of thermal insulation. Then, the thermal boundary condition at the wall in (3.136) is replaced by @DT=@y ¼ 0 at y ¼ 0: Under the assumptions made, the dynamic and thermal problems are uncoupled. Then, the flow field is described by the self-similar Blasius solution of (3.133) and (3.134) (see Sect. 3.3 and Schlichting 1979) 1 u ¼ U’ ; v ¼ 2 0
rffiffiffiffiffiffi nU 0 ð’ ’Þ x
(3.137)
pffiffiffiffiffiffiffiffiffiffiffi where ’ ¼ ’ðÞ is the function determined by (3.51), ¼ y U=nx; and prime denotes differentiation by : The temperature at any point of the thermal boundary layer depends on the temperature difference DTw ¼ Tw T1 ; flow velocity, kinematic viscosity and thermal diffusivity of fluid, as well as on the location DT ¼ FðDTw ; U; x; y; n; aÞ
(3.138)
The dimensions of the governing parameters in (3.126) are ½DT ¼ y; ½U ¼ Lx T 1 ; ½ x ¼ Lx ; ½ y ¼ Ly ; ½n ¼ L2y T 1 ; ½a ¼ L2y T 1
(3.139)
Since four of the six governing parameters in (3.138) have independent dimensions, it can be reduced to the following dimensionless equation P ¼ #ðP1 ; P2 Þ where 00
P ¼ DT=DTwa1 Ua2 xa3 na4 ;
00 00 00 a a=DTw1 U a2 xa3 na4 .
(3.140) a
0
0
0
0
P1 ¼ y=DTw1 U a2 xa3 na4 ;
and
P2 ¼
3.10
Diffusion Boundary Layer over a Flat Reactive Plate (the Levich Problem)
65
Bearing in mind the dimensions of DT; DTw ; U; x; y; n and a, we find the 0 00 exponents ai ai and a0 as a1 ¼ 1; a2 ¼ 0; a3 ¼ 0; a4 ¼ 0 1 0 1 0 1 0 0 a1 ¼ 0; a2 ¼ ; a3 ¼ ; a4 ¼ 2 2 2 00 00 00 00 a1 ¼ 0; a2 ¼ 0; a3 ¼ 0; a4 ¼ 1
(3.141)
Accordingly, (3.140) takes the form DT ¼ #ð; PrÞ DTw
(3.142)
where Pr ¼ n=a is the Prandtl number. Substituting the expression (3.142) into (3.135), we arrive at the following ODE 00
# þ
1 0 Pr ’# ¼ 0 2
(3.143)
The boundary conditions for (3.143) read # ¼ 1at ¼ 0; # ¼ 0 at ! 1
(3.144)
Then, the solution (3.143) is found in the following form 1 Ð
#ð; PrÞ ¼
1 Ð
00
’ ðxÞ
Pr
dx (3.145)
½’00 ðxÞPr dx
0
where ’ðÞ is determined Eq (3.51). At Pr ¼1 0
#ðÞ ¼ 1 ’ ðÞ ¼ 1
u u1
(3.146)
i.e. the dimensionless excess temperature and velocity fields coincide.
3.10
Diffusion Boundary Layer over a Flat Reactive Plate (the Levich Problem)
Consider distribution of liquid (or gaseous) reactant in the boundary layer over a flat reactive plate (Levich, 1962). Assume that the rate of an exothermal hetorogenous reaction at the plate surface exceeds significantly the diffusion flux toward the
66
3 Application of the Pi-Theorem to Establish Self-Similarity
surface, and also neglect the influence of the heat release due on the flow field. Then, the field of the reactant concentration is described by the following problem @c @c @2c þv ¼D 2 @x @y @c
(3.147)
y ¼ 0 c ¼ 0; y ! 1 c ! c0
(3.148)
u
where the velocity components u and v are determined by the Blasius solution (Sect. 3.3), and c0 is the concentration of the reagent in the undisturbed flow, and D is the diffusion coefficient. The governing parameters that determined the concentration field at any point of the boundary layer are: the concentration c0 , the velocity of the undisturbed flow U, the kinematic viscosity of the liquid or gaseous carrier and diffusity n and D, respectively, as well as the coordinates of the point of interest x; and y: Accordingly, the functional equation for the reactant concentration c reads c ¼ f ðc0 ; U; x; y; n; DÞ
(3.149)
The dimensions of the governing parameters are as follows ½c0 ¼ L3 M; ½U ¼ LT 1 ; ½ x ¼ L; ½ y ¼ L; ½n ¼ L2 T 1 ; ½ D ¼ L2 T 1
(3.150)
Since four of the six governing parameters have independent dimensions, (3.49) takes the following dimensionless form P ¼ cðP1 ; P2 Þ a
0
0
0
0
(3.151) a
00
00
00
00
where P ¼ c=ca01 U a2 xa3 na4 ; P1 ¼ y=c01 U a2 xa3 na4 ; and P2 ¼ D=c01 U a2 xa3 na4 : Taking into account the principle of dimensional homogeneity, we find the 0 00 following values of the exponents ai ; ai and ai a1 ¼ 1; a2 ¼ 0; a3 ¼ 0; a4 ¼ 0 1 1 1 0 0 0 0 a1 ¼ 0; a2 ¼ ; a3 ¼ ; a4 ¼ 2 2 2 00 00 00 00 a1 ¼ 0; a2 ¼ 0; a3 ¼ 0; a4 ¼ 1
(3.152)
Then, (3.151) takes the following form c ¼ cð; ScÞ c0
(3.153)
Problems
67
pffiffiffiffiffiffiffiffiffiffiffi where ¼ y U=nx; and Sc ¼ n=D is the Schmidt number. Substituting the expression (3.153) into (3.47), we arrive at the following ODE 00 0 1 c þ Sc c ¼ 0 2
(3.154)
The solution of (3.154) with the corresponding boundary conditions following from (3.148) has the form 1 Ð
cð; ScÞ ¼
1 Ð
½’00 ðxÞSc dx (3.155) Sc
½’00 ðxÞ dx
0 00
where ’ ðÞis determined by Eq. (1.40). The diffusion flux of the reactant at the wall is found as rffiffiffiffiffi @c U ¼ Dc0 f ðScÞ j ¼ D @y 0 nx
(3.156)
where the function f ðScÞ equals to: 0.332Sc1/3 for 0.6<Sc<10, 0:564Sc1=2 for Sc ! 0, and 0:339Sc1=3 for Sc ! 1(Schlichting 1979).
Problems P.3.1. Show that flow of an incompressible fluid in a cone does not correspond to aself-similar solution. The steady-state flow in a cone is described by the Navier–Stokes and continuity equations 1 ðr vÞv ¼ rP þ nr2 v r
(P.3.1)
rv¼0
(P.3.2)
where v is the velocity vector with the radial and two azimuthal components vr ; vy ; and v’ , respectively, in the spherical coordinate system centered at the cone tip. The flow in the cone is axially symmetric relative to the axis y ¼ 0: Let the swirl is absent, so that v’ ¼ 0. Under these conditions the velocity components vr and vy depend on two given constants of the problem, namely, the kinematic viscosity of the fluid ½n ¼ L2 T 1 and the volumetric flow rate ½Q ¼ L3 T 1 ; as well as on two
68
3 Application of the Pi-Theorem to Establish Self-Similarity
coordinates r ½ L and y½1: Then, the functional equation for the velocity components vr and vy are vr ¼ fr ðn; Q; r; yÞ
(P.3.3)
vy ¼ fy ðn; Q; r; yÞ
(P.3.4)
From the three dimensional parameters n; Q and r one dimensionless group can be constructed, namely, R ¼ r=ðQ=nÞ: Accordingly, (P.3.3) and (P.3.4) take the the following form vr ¼ ’r ðR; yÞ
(P.3.5)
vy ¼ ’y ðR; yÞ
(P.3.6)
where vr ¼ vr ðQ=n2 Þ; and vy ¼ vy =ðQ=n2 Þ. Thus, vr and vy are the functions of two dimensionless variables R and y that does not allow to reduce PDEs (P.3.1) and (P.3.2) to ODEs. P.3.2. Find self-similarity for the flow in the boundary layer of an incompressible fluid near a solid wall in the case of a power-law velocity distribution far from the plate (Flakner and Skan 1931). In this case the velocity of the undisturbed flow far from the solid wall is given by the power law U ¼ cxm ; where c is a dimensional and m is a dimensionless constants, respectively. The boundary layer and continuity equations read u
@u @u @2u dU þv ¼n 2U @x @y @y dx
(P.3.7)
@u @v þ ¼0 @x @y
(P.3.8)
The boundary conditions are as following y ¼ 0; u ¼ v ¼ 0; y ! 1; u ¼ UðxÞ ¼ cxm
(P.3.9)
The governing parameters of the problem are: the dimensional constants ½c ¼ L1m T 1 ; and ½n ¼ L2y T 1 and the independent variables ½ x ¼ Lx and ½ y ¼ Ly : x Accordingly, the functional equation for u is u ¼ fu ðc; n; x; yÞ
(P.3.10)
It is seen that three governing parameters have independent dimensions, so that n-k ¼ 1. Then the dimensionless form of (P.3.10) is
References
69 0
P ¼ ’ ðP1 Þ 0
0
0
(P.3.11) 0
where P ¼ u=ca1 na2 xa3 ; P1 ¼ y=ca1 na2 xa3 ; ’ ¼ ’ðP1 Þ and ’ ¼ du=dP1 . Taking into account the dimensions of u; c; n; x and y; we find that 1 0 1 0 1m 0 a1 ¼ 1; a2 ¼ 0; a3 ¼ m; a1 ¼ ; a2 ¼ ; a3 ¼ 2 2 2
(P.3.12)
Accordingly, (P.3.11) takes the form u 0 ¼’ y U
rffiffiffiffiffiffiffiffiffiffiffiffi! cxm1 n
(P.3.13)
Substituting the expression (P.3.13) into (P.3.7) and (P.3.8), we obtain the following ODE m þ 1 00 02 ’’ ¼ mð’ 1Þ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where prime denotes differentiation by ¼ y cxm1 =n: 000
’ þ
(P.3.14)
References Baehr HD, Stephan K (1998) Heat and mass transfer. Springer, Heidelberg Batchelor GK (1967) An introduction to fluid dynamics. Cambridge University Press, Cambridge Britter RE (1979) The spread of a negatively plum in calm environment. Atmos Environ 13:1241–1247 Blasius H (1908) Grenzschichten in Flussigkeiten mit kleiner Reibung. Z Math Phys 56:1–37 Didden N, Maxworthy T (1982) The viscous spreading of plane and axisymmetric gravity currents. J Fluid Mech 121:27–42 Douglas JF (1969) An introduction to dimensional anlysis for engineers. Pitman, London Flakner VM, Skan SW (1931) Some approximate solutions of the boundary layer equiation. Phil Mag 12:865–896 Hamel G (1917) Spiralformige Bewegungen zahen Flussigkeiten. Jahr-Ber Dtsch Math Ver 25:34–60 Hoult DP (1972) Oil spreading on the sea. Ann Rev Fluid Mech 4:341–368 Huntley HE (1967) Dimensional analysis. Dover Publications, New York Huppert HE (1982) The propagation of two–dimensional and axisymmetric viscous- gravity currents over a rigid horizontal surface. J Fluid Mech 121:43–58 Karman Th (1921) Uber laminare and turbulente Reibung. ZAMM 1:233–252 Kays WM, Crowford ME (1993) Convective heat and mass transfer, 3rd edn. McGraw-Hill, New York Landau LD (1944) New exact solution of the navie-stokes equation. DAN SSSR 44:311–314 Landau LD, Lifshitz EM (1987) Fluid mechnics, 2nd edn. Pergamon, Oxford Levich VG (1962) Physicochemical hydrodynamics. Prentice Hall, Englewood Cliffs Loitsyanskii LG (1966) Mechanics of liquid and gases. Pergamon, Oxford
70
3 Application of the Pi-Theorem to Establish Self-Similarity
Pohlhausen E (1921) Der Warmeaustaush zwichen festen Korpern and Flussigkeiten mit kleiner Reibung and kleiner Warmeleitung. ZAMM 1:115 Rosenhead L (ed) (1963) Laminar boundary layers. Clarendon, Oxford Sedov LI (1993) Similarity and dimensional methods in mechanics, 10th edn. CRC, Boca Raton Schlichting H (1979) Boundary layer theory. McGraw-Hill, New York Sherman FS (1990) Viscous flow. McGraw-Hill, New York Simpson JE (1982) Gravity currents in the laboratory, atmosphere, and ocean. Ann Rev Fluid Mech 14:213–234 Stokes GC (1851) On the effect of internal friction of fluids on the motion of pendulums. Trans Cambridge Philos Soc 9:8–106 Vulis LA, Kashkarov VP (1965) Theory of viscous liquid jets. Nauka, Moscow (in Russian) Yarin AL, Weiss DA (1995) Impact of drops on solid surfaces: Self-similar capillary waves, and splashing as a new type of kinematic discontinuity. J Fluid Mech 283:141–173
Chapter 4
Drag Force Acting on a Body Moving in Viscous Fluid
4.1
Introductory Remarks
Drag force is one of the most important factors that determine dynamics of solid bodies moving in viscous fluids. The knowledge of this force is essential for a number of applications in engineering, in particular, for the evaluation the engine power to ensure a desirable velocity of the airplanes, ships, etc., as well as for analyzing the behavior of solid particles, droplets and bubbles in two-phase flows. Numerous experimental and theoretical investigations dealing with drag of bodies of different shapes moving with low and high velocities in viscous fluid were performed during the last three centuries. The results of these researches are generalized in monographs by Landau and Lifshitz (1987), Batchelor (1967), Happel and Brenner (1983), and Soo (1990). The detailed data on drag of bubbles, droplets and solid particles can be found in the monograph by Clift et al. (1978). The effect of heating, evaporation and combustion on drag force of small particles is discussed in the monograph by Yarin and Hetsroni (2004). The application of the dimensional analysis in studies of drag force of bodies moving in viscous fluid are discussed in Sedov (1993). The readers are referred to the above-mentioned works, whereas we discuss briefly some principles that are essential for problems dealing with drag force acting on bodies moving in viscous fluid and the related applications of the Pi-theorem. First we pose the problem on a steady-state linear translation of a solid body of a given shape with velocity v1 in an infinite uniform incompressible fluid. This problem is equivalent to the problem on the solid body subjected to a uniform fluid flow with the undisturbed and constant velocity v1 . The velocity and pressure fields in such flow are determined by the Navier-Stokes and continuity equations rðv rÞv ¼ rP þ mr2 v
(4.1)
rv¼0
(4.2)
L.P. Yarin, The Pi-Theorem, Experimental Fluid Mechanics, DOI 10.1007/978-3-642-19565-5_4, # Springer-Verlag Berlin Heidelberg 2012
71
72
4 Drag Force Acting on a Body Moving in Viscous Fluid
subjected to the no-slip conditions at the body surface y ðxÞ: v ¼ 0 at y ¼ y ðxÞ
(4.3)
v ¼ v1 at y ! 1
(4.4)
At low Reynolds number (the creeping flow with the Reynolds number Re 1) the convective terms in the momentum (4.1) can be omitted (the Stokes approximation) and the system of (4.1) and (4.2) takes the following form DP mr2 v ¼ 0
(4.5)
rv¼0
(4.6)
Equations (4.1) and (4.2) and the conditions (4.3) and (4.4) show that the situation under consideration is determined by the following four dimensional parameters r; m; v1 ; and d , which determine the velocity and pressure fields in fluid, as well as the drag force.1 Then, the functional equation for the drag force acting on a solid body moving in an unbounded fluid acquires the following form Fd ¼ f ðr; m; v1 ; d Þ
(4.7)
where ½Fd ¼ LMT 2 is the drag force. The set of the governing parameters corresponding to solid body motion in this case is ½r ¼ L3 M; ½m ¼ L1 MT 1 ; ½v1 ¼ LT 1 ; ½d ¼ L
(4.8)
Taking into account that three governing parameters in (4.8) have independent dimensions, we can transform the functional equation for the drag force (4.7) into the following dimensionless form P ¼ ’ ðP1 Þ
(4.9)
where P ¼ Fd =ðrv21 d2 Þ and P1 ¼ v1 d =n is the Reynolds number. On the other hand, when a solid body moves in liquid near a gas-liquid interface, it excites perturbations of the interface. The perturbed interface is wavy either due to the effect of gravity or/and surface tension. In this case the set of the governing parameters also includes gravity acceleration g or/and surface tension s. In this case the set of the parameters determining the drag force reads ½r ¼ L3 M; ½m ¼ L1 MT 1 ; ½v1 ¼ LT 1 ; ½d ¼ L; ½g ¼ LT 2 ; or=and, ½s ¼ MT 2
1
Here d is characteristic size of the body which is fully determined by its shape.
(4.10)
4.2 Drag Action on a Flat Plate
73
Then, the drag coefficient in the case of a body moving in liquid near a gas-liquid interface becomes (4.11) P ¼ ’1 ðP1 ; P2 Þ; or=and P ¼ ’2 ðP1 ; P2 ; P3 Þ pffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi where P2 ¼ v1 = gd and P3 ¼ m= rsd are the Froude and Ohnesorge numbers, respectively. In the general case, the set of the governing parameters can also include a number of parameters that account for roughness of the solid body surface, turbulence of the free stream, etc. In accordance with that the dimensionless form of the functional equation for the drag force (the functional equation for the drag coefficient) takes the following form P ¼ ’ðn1Þ ðP1 ; P2 ; :::Pn Þ
(4.12)
where ’ðn1Þ accounts for the effect of the nth factor. It is emphasized that the unknown function ’i in the expression for the drag coefficient can be presented as ’i ¼ a’i ; where a is a dimensionless constant. The numerical value of this constant is chosen in accordance with an accepted protocol of processing of experimental data. As a rule, the value of the coefficient a is assumed to be equal to 1=2. Then, (4.9), (4.11) and (4.12) determine the dependences of the drag coefficient cd ¼ P =ð1=2Þ ¼ Fd = ð1=2Þrv21 d2 on the dimensionless groups P1 ; P2 Pn characteristic of the considered problem. For a spherical body moving with small velocity ( the creeping flow Stokes approximation) when the drag force is expressed as Fd ¼ 3pmv1 d , it is assumed that a ¼ p=8: That leads to the generally accepted expression for the drag coefficient in the form cd ¼ P =ðp=8Þ ¼ 24=Re where cd is the drag coefficient, Re ¼ v1 d =n is the Reynolds number, v1 is the velocity of the undisturbed fluid (or a particle moving in fluid at rest), and n is the kinematic viscosity.
4.2 4.2.1
Drag Action on a Flat Plate Motion with Constant Speed
Consider viscous drag of a thin flat plate subjected to a parallel uniform fluid stream (Fig. 4.1). At sufficiently high values of the Reynolds number (Re>> 1), the boundary layers are formed over both sides of the plate. The thicknesses of the boundary layers increase downstream (cf. Fig. 4.1). The drag force that act on the plate is determined as ðl Fd ¼ 2b tw dx 0
(4.13)
74 Fig. 4.1 A thin flat plate subjected to a uniform parallel flow
4 Drag Force Acting on a Body Moving in Viscous Fluid y
δ(x)
Plate
u∞
x
0
where Fd is the drag force, tw is the shear stress at the plate surface, and b and l are the width and length of the plate. Bearing in mind the character of flow over the plate, we can assume that drag force is determined by density and viscosity of the fluid, the undisturbed flow velocity u1 , as well as the length and width of the plate. Then, we can present (4.13) as follows Fd ¼ 2bf ðr; m; u1 ; lÞ
(4.14)
where ½Fd ¼ LMT 2 ; ½r ¼ L3 M; ½m ¼ L1 MT 1 ; ½u1 ¼ LT 1 ; ½l ¼ L. Applying the Pi-theorem to (4.14), we arrive at the following expression for the drag coefficient cd ¼ ’ðReÞ
(4.15)
where cd ¼ Fd =ru1 2 2bl, u1 is the velocity of the undisturbed fluid is the drag coefficient, and Re ¼ u1 l=n is the Reynolds number. In order to reveal an explicit form of the dependence cd ðReÞ, we use the expression for the shear stress at the plate surface that was found in Chap. 3 by via the dimensional analysis of laminar flow over a plate rffiffiffiffiffiffi u3 00 tw ¼ m 1 ’ ð0Þ nx
(4.16)
where ’ðÞ is determined by solving the Blasius equation (3.51), and pffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ y= nx=u1 is the dimensionless variable. Substitution of the expression (4.16) into (4.13) yields 00
Fd ¼ 4b’ ð0Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffi mrlu31
(4.17)
4.2 Drag Action on a Flat Plate
75
and 00
4’ ð0Þ cd ¼ pffiffiffiffiffiffi Re
(4.18)
00
where ’ ð0Þ is constant.
4.2.2
Oscillatory Motion of a Plate Parallel to Itself
The flow in the vicinity of an oscillating plate is determined by the unsteady boundary layer equations. Transient and oscillatory flows in the boundary layers are discussed in the monographs of Schlichting (1979) and Loitsyanskii (1967). In the framework of the dimensional analysis the problem on a drag Fd experienced by an oscillating plate involves choosing a set of the governing parameters and subsequent transformation of the functional equation for Fd to a dimensionless form using the Pi-theorem. The set of the governing parameters in this case includes the parameters responsible for the physical properties of the fluid (its density and viscosity r and mÞ; sizes of the plate (l and bÞ, as well as such flow characteristics as its period ½t ¼ T (or frequency) of the oscillations and the maximum velocity um that plays the role of the velocity scale. Then, the functional equation for Fd takes the form Fd ¼ 2bf1 ðr; m; um ; l; tÞ
(4.19)
It is seen that the present problem contains five governing parameters, three of which have independent dimensions. Therefore, according to the Pi-theorem (4.19) can be reduced to the following form P ¼ ’ðP1 ; P2 Þ 0
0 a2
0
(4.20) 00
00 a2
00
where P ¼ Fd =2bra1 uam2 la3 ; P1 ¼ m=ra1 um la3 ; and P2 ¼ t=ra1 um la3 . 0 00 Determining the values of the exponents ai ; ai and ai with the help of the principle of dimensional homogeneity, we find that a1 ¼ 1; a2 ¼ 2; a3 ¼ 1; 0 0 0 00 00 00 a1 ¼ 1; a2 ¼ 1; a3 ¼ 1; a1 ¼ 0; a2 ¼ 1; and a3 ¼ 1. Then, we arrive at the following expression for the drag coefficient cd ¼ ’1 ðRe; KsÞ
(4.21)
where cd ¼ Fd 2bru2m l is the drag coefficient, Re ¼ um l=n is the Reynolds number, Ks ¼ St1 ; where St is the Strouhal number determined by the maximum velocity and length of the plate, while Ks ¼ Ks b; with Ks ¼ tum =b being the KeuleganCarpenter number, and b ¼ b=l. Shih and Buchanan (1971) studied experimentally the dependence cd ¼ ’1 ðRe; KsÞ. It was shown that the drag coefficient of an
76
4 Drag Force Acting on a Body Moving in Viscous Fluid
oscillating plate decreases as the Reynolds number increases. An increase in Ks also leads to decreasing cd . For the engineering applications the following empirical correlation is useful
1:88 Re0:547
cd ¼ 15ðKsÞ exp
(4.22)
where Re ¼ um b=n. The forces acting on cylinders in viscous oscillatory flow are also determined by Ks at low values of the Keulegan-Carpenter numbers (Graham 1980; Bearman et al 1985).
4.3
Drag Force Acting on Solid Particles
4.3.1
Drag Experienced by a Spherical Particle at Low, Moderate and High Reynolds Numbers
In Sect. 4.1 we discussed briefly the application of the Pi-theorem for evaluating drag force experienced by a solid body moving in viscous fluid. In the present section we consider this problem in more detail, in particular, dealing with the drag force acting on a spherical particle at low, moderate and high Reynolds numbers. The drag of a spherical particle moving in viscous fluid represents the total force exerted by the surrounding fluid on the particle surface. This force depends on the physical properties of the fluid, as well as on particle size and its velocity fd ¼ f ðr; m; d; uÞ
(4.23)
where fd is the drag force, d is the particle diameter, and uis the particle velocity relative to fluid at infinity. The drag force fd and the governing parameters r; m; d; and u have the following dimensions ½ fd ¼ LMT 2 ; ½r ¼ L3 M; ½m ¼ L1 MT 1 ; ½d ¼ L; ½u ¼ LT
1
(4.24)
It is seen that three from the four governing parameters have independent dimensions. Then, in accordance with the Pi-theorem we transform (4.23) into the following dimensionless form P ¼ ’ ðP1 Þ 0
0
(4.25) 0
0
where P ¼ fd =ra1 ua2 d a3 and P1 ¼ m=ra1 ua2 d a3 , and the exponents ai and ai are determined from the principle of dimensional homogeneity. They are found as
4.3 Drag Force Acting on Solid Particles
77
0
0
0
follows: a1 ¼ 1; a2 ¼ 2; a3 ¼ 2; a1 ¼ 1; a2 ¼ 1; and a3 ¼ 1. Then, we obtain that P ¼ fd =ru2 d 2 ; and P1 ¼ m=rud ¼ Re1 and (4.25) takes the following form cd ¼ ’ðReÞ
(4.26)
where cd ¼ P =ðp=8Þ and Re are the drag coefficient and the Reynolds number, respectively. The explicit forms of the dependence (4.26) can be found in the framework of the dimensional analysis for two limiting cases corresponding to very small and very large Reynolds number (see Problems P.4.1 and P.4.2). Equation 4.26 indicates that the drag coefficient of a spherical particle depends on a single dimensionless group, namely, the Reynolds number. In order to determine an exact form of the dependence cd ðReÞ, it is necessary to either solve the hydrodynamic problem on flow of viscous fluid about the particle, or to study it experimentally. The structure of such flow determines the normal and shear stresses at the particle surface, i.e. the total drag force. The flow about a spherical particle that moves rectilinearly with a constant velocity in fluid is described by the system of the Navier-Stokes and continuity equations rðv rÞv ¼ rP þ mr2 v
(4.27)
rv¼0
(4.28)
which are subjected to the following boundary conditions v ¼ 0; r ¼ R; v1 ¼ u; r ¼ 1
(4.29)
where r and m are the density and viscosity of the fluid, R is the particle radius, v is fluid velocity relative the spherical coordinate system associated with the center of the moving particle, u is the absolute particle velocity, r is the radial coordinate, P is the pressure; the boldface symbols represent vector quantities. The inertial term rðv rÞv on the left-hand side of (4.27) is negligible at low Reynolds numbers. The problem is thus can be simplified significantly and reduced to the integration of the linear Stokes equations rP mr2 v ¼ 0
(4.30)
rv¼0
(4.31)
subjected to the boundary conditions (4.29). The solution of (4.30) and (4.31) results in the following Stokes expression for the drag force fd ¼ ff þ fp ¼ 3pmud
(4.32)
78
4 Drag Force Acting on a Body Moving in Viscous Fluid
Ðp Ðp where ff ¼ tRy sin y 2pa2 sin ydy ¼ 2pmud, fd ¼ P cos y 2pa2 sin ydy ¼ 0
0
pmud are the contributions to the total drag force from the viscous friction (the shear stresses) and pressure, respectively, tRy ; is the shear stress at the surface, P is the pressure at the surface, a is the sphere radius, R and y are the radial and angular coordinates in the spherical coordinate system (Stokes 1851)). The drag coefficient for a spherical particle becomes accordingly cd ¼
24 Re
(4.33)
The Stokes’ law (4.32) and (4.33) is valid only for low Reynolds numbers Re 0:1 (cf. Fig. 4.2). The deviation of the predicted values of cd from the experimental data for the drag coefficient does not exceed 2% at Re 0.24 and 20% at Re 0.75. The experimental data show that the dependence of cd ¼ cd ðReÞ has a rather complicated shape when a wider range of the Reynolds number values is considered (Fig. 4.2). In the range of 1 < Re < 800 the drag coefficient is accurately expressed by the empirical Schiller and Naumann law (Clift et al. 1978) cD ¼ ð24=ReÞ 1 þ 0:15Re0:687
(4.34)
Significant deviations from Stokes’ law are related to the growth of the so-called form drag component of the drag force at higher Reynolds numbers. It is associated
Fig. 4.2 Drag coefficient of a spherical particle: the solid line – the dependence of the drag coefficient on the Reynolds number, the dotted line – the Stokes’ law
4.3 Drag Force Acting on Solid Particles
79
with the development of the boundary layer near the particle surface and its separation at the rear part. The later results in a stagnation zone behind the particle and a reduced pressure at the rear compared to the full dynamic pressure acting at the front part of the particle. In range of the Reynolds number 750 Re 3 105 the drag coefficient is close to a constant value of 0.445 (the Newton law). At higher Re, the drag coefficient reveals a dimple at about Re 2 105 . The latter is a result of the change in the flow structure, when transition to turbulence happens in the boundary layer at the sphere surface, which leads to the flow reattachment to the surface and diminishes the form drag.
4.3.2
The Effect of Rotation
Particle rotation is a cause of lift force fl , which is directed normally to the plane formed by the particle velocity and angular velocity vectors v and o, respectively. The magnitude of this force, which is the cause of the Magnus effect, depends on the physical properties of the fluid and diameter of a spherical particle, as well as on its velocity (relative to the fluid) u ¼ jvj and the magnitude of the angular velocity o. Therefore, the functional equation for the lift force reads fl ¼ f ðr; m; u; d; oÞ
(4.35)
The problem at hand involves five governing parameters, three of them with independent dimensions. Then, in accordance with the Pi-theorem, we find that the lift force coefficient cl ¼ 4f =ðru2 pd 2 =2Þ is given by the following expression cl ¼ ’ðRe; gÞ
(4.36)
where g ¼ od=2u is the dimensionless angular velocity. A lift force also acts at a spherical particle moving in a simple shear flow characterized by velocity gradient du=dy (Saffman 1965, 1968). In this case the functional equation for the Saffman lift force flS is
du flS ¼ f r; m; u; d; dy
(4.37)
Applying the Pi theorem, we arrive at the following dimensionless expression for the Saffman lift force coefficient clS normalized as cl before clS ¼ ’ðRe; gÞ where g ¼ ðdu=dyÞd 2 =n.
(4.38)
80
4 Drag Force Acting on a Body Moving in Viscous Fluid
The important results regarding the Saffman lift force were obtained by Dandy and Dwyer (1990), McLaughlin (1991), Anton (1987) and Mei (1992). In particular, Dandy and Dwyer (1990) showed that at a fixed shear rate the lift and drag coefficients for a spherical particle, normalized using the uniform flow velocity are approximately constant over the range 40 Re 100. On the other hand, the drag and lift coefficients cd and cl increase sharply as the Reynolds number decreases in the range Re<10.
4.3.3
The Effect of Acceleration
Transient motions of particles result in additional forces imposed on them by the surrounding fluid. These forces are related to the acceleration of the surrounding fluid (the added mass force), as well as to the viscous effects due to delay in flow development as the velocity changes with time (the Basset force). Transient motions of spherical particles in an incompressible fluid at rest were studied by Boussinesq (1903), Oseen (1910, 1927) and Basset (1961). Recently a number of modified model equations describing transient particle motions in steady-state or weakly fluctuating flows were proposed (Maxey and Riley 1983; Berlemont et al. 1990; Mei et al. 1991; Mei 1994; Chang and Maxey 1994, 1995). These modifications were mostly dealing with the history term in the drag force. In particular, they accounted for the effect of the initial velocity difference between the fluid and a particle. At the same time, some other approaches to determine the unsteady drag force are based on the empirical correlations constructed in accordance with the Pi-theorem (Odar and Hamilton 1964; Karanfilian and Kotas 1978). They account for the influence of particle acceleration du/dt on drag coefficient. Accordingly, assuming that the transient drag force depends not only on particle velocity but also on its acceleration, we present the functional equation for fd in the following form
du fd ¼ f r; m; u; d; dt
(4.39)
It is easy to see that three from the five governing parameters involved have independent dimensions. Then, in accordance with the Pi-theorem, we obtain the drag coefficient based on the previous non-dimensionalization in the following dimensionless form cd ¼ ’ðRe; AcÞ where Ac ¼ u2 =½ðdu=dtÞd is the acceleration parameter. An explicit expression for cd ðRe; AcÞ reads (Karanfilian and Kotas 1978)
(4.40)
4.3 Drag Force Acting on Solid Particles
81
n cd ¼ cd ðReÞ 1 þ Ac1
(4.41)
where n ¼ 1:2 0:03. The expression (4.41) gives a fairly good agreement with experiments in the range 102 Re 104 and 0 Ac<1:05. The forces exerted on an “infinite” cylindrical particle in an oscillatory flow normal to its axis were studied by Kenlegan and Carpenter (1958), Graham (1980) and Bearmen et al. (1985). There are three factors that determine the drag force: (1) the inertia of the accelerating outer flow, (2) the influence of the viscous boundary layers, and (3) separation of these boundary layers leading to vortex shedding. The drag coefficient of such particles is determined by two dimensionless groups: the Reynolds and Kenlegan-Carpenter numbers.
4.3.4
The Effect of the Free Stream Turbulence
There is a number of additional factors which affect particle drag. Turbulence of free stream impinging on a particle is among them. In order to account for the effect of turbulence intensity on particle drag, it is necessary to extend the set of the governing parameters by including it in the set. For example, in the case when a particle moves in a turbulent flow, the functional equation for the drag force fd reads 0
fd ¼ f ðr; m; u; u ; lÞ
(4.42) p ffiffiffiffiffi ffi 0 where u ¼ u0 2 is the root-mean square of turbulent fluctuations of the carrier 0 fluid, l is the turbulence length scale (it is implied that u and l are some characteristic values of the turbulent fluctuations and of the turbulence length scale). Then, the dimensional analysis yields the following equation for the drag coefficient cd ¼ ’ðRe; Tu ; lÞ 0
(4.43)
where Tu ¼ u =u is the dimensionless turbulence intensity, and l ¼ l=d is the dimensionless turbulence scale. 0 At a given l , the effect of u on the drag coefficient depends on the Reynolds number. At sufficiently high Re, close to transition to turbulence in the particle boundary layer, an increase in the turbulence intensity of the free stream is accompanied by a decrease in cd . This is a result of a shift of the boundary layer separation point towards the rear stagnation point. In the range of relatively low 0 Reynolds numbers, the drag coefficient slightly increase with u . This effect is due to the intensified viscous dissipation. The effect of the turbulence scale on the drag coefficient depends on l . At l<<1 the effect of the turbulence scale is negligible, whereas at l>1, the drag coefficient increases with l.
82
4.3.5
4 Drag Force Acting on a Body Moving in Viscous Fluid
The Influence of the Particle-Fluid Temperature Difference
A difference between the particle and surrounding temperature can also affect the drag force. For the flow of viscous incompressible fluid the functional equation for the drag force fd in this case reads fd ¼ f ðr1 ; m1 ; d; u1 ; T1 ; TP Þ
(4.44)
where subscripts P and 1 refer to the particle and the ambient parameters. Applying the Pi-theorem to transform (4.44) to dimensionless form, we arrive at the following equation cd ¼ ’ðRe; cÞ
(4.45)
where cd ¼ 4fd =pð1=2Þr1 u21 d 2 is the drag coefficient, and c ¼ TP =T1 is the temperature ratio. It is emphasized that when transforming (4.44) to the dimensionless form, we do not account for the effect of temperature on the physical properties of the fluid. At large enough values of the temperature difference TP T1 , a significant variation of density and viscosity of fluid within the boundary layer can take place. In this case the situation becomes complicated. In accordance with that, a set of the governing parameters should includes thermal conductivity and heat capacity. The effect of variation of the physical properties of the fluid within the boundary layer on the drag force was studied by Fendell et al. (1966), Kassoy et al. 1966) and Dwyer (1989). In particular, Kassoy et al. (1966) evaluated the drag coefficient of a spherical particle moving in a high-velocity flow of a perfect gas. Assuming a linear dependence of the viscosity and thermal conductivity on temperature, as well as constant specific heat and Prandtl number, they derived the following relation for the drag coefficient cd ¼
24 16C K Re 3K 3
(4.46)
where K ¼ OðO þ 2Þ and C is tabulated function of the parameter O ¼ ðc 1Þ. According to (4.46), the particle drag coefficient increases almost linearly with O. When O is of the order of one, the drag coefficient at low Reynolds numbers increases up to 70% over the isothermal value corresponding to the Stokes law.
4.4
Drag of Irregular Particles
Consider the factors which determine drag force of an irregular body moving in an unbounded viscous incompressible fluid with a constant velocity. From the physical point of view it is clear that the drag force of such a body should depend on the fluid
4.4 Drag of Irregular Particles Fig. 4.3 Orientation of an irregular body in a uniform flow directed along the x-axis
83 y
d
α 0
x
properties, the body center-of-mass velocity and orientation relatively to the velocity vector, as well as the body geometry. The latter (at fixed configuration of an irregular body) can be determined by one characteristic length, for example, the length of the wing chord (Fig. 4.3). The pressure of the undisturbed fluid P1 does not affect the flow field in the incompressible fluid. Indeed, in this case instead of the total pressure P it is always possible to consider the difference P P1 : Thus, the value of P1 is immaterial. It can be excluded from the set of the governing parameters of the problem (Sedov 1993). Therefore, we can write the functional equation for the drag force as follows fd ¼ f ðr; m; u; d; aÞ
(4.47)
where d is the characteristic length, and a is the angle of inclination (cf. Fig. 4.3). By using the Pi-theorem, (4.47) can be transformed to into the following dimensionless form cd ¼ ’ðRe; aÞ
(4.48)
where cd ¼ 4fd =ð1=2Þru2 d2 is the drag coefficient. The Reynolds number characterizes the ratio of the inertia and viscous forces. The contribution of the viscous forces to the drag force decreases as the viscosity m decreases, the Reynolds number grows. At large values of Re, the dominant role is played by fluid inertia. In such cases it is possible to neglect the effect of viscosity and to reduce the number of the governing parameters to four. Then we obtain fd ¼ rd 2 u2 ’ðaÞ
(4.49)
Equation 4.49 shows that in an ideal (inviscid) fluid the drag force is proportional to the velocity squared. Note that the effect of particle shape becomes important only at sufficiently high Reynolds numbers when a vortex zone forms behind the particle. In creeping flow this effect is less expressed. For example, for a thin disk oriented normally to the flow, the factor in the Stokes’ law equals 8, whereas for a spherical particle it equals 3 p (Lamb 1959). In the general case, the drag coefficient of irregular particles cd
84
4 Drag Force Acting on a Body Moving in Viscous Fluid
Fig. 4.4 The drag coefficient of an irregular particle
depends on two dimensionless groups: the Reynolds number and the shape factor (Boothroyd 1971) cd ¼ cd ðRe; fÞ
(4.50)
where f ¼ Seq =S is the shape factor, Seq is the surface area of a volume-equivalent sphere, and S is the actual surface area. The surface area and diameter of the volume-equivalent sphere are defined as Seq ¼ p1=3 ð6VÞ2=3 and deq ¼ ð6V=pÞ1=3 ; respectively, where V is the particle volume. In (4.50) the Reynolds number is based on the equivalent diameter deq. The dependence cd ðRe; fÞ is presented in Fig. 4.4 as a family of curves cd ðReÞ corresponding to different values of f Boothroyd (1971). It is seen that the drag coefficient of irregular particles is larger than the one for a volume-equivalent spherical particle. The difference increases significantly with the Reynolds number.
4.5
Drag of Deformable Particles
In distinction from rigid particles, the drag of drops and bubbles depends not only on the outside velocity distribution but also on their deformation and the inside flow. The particle deformation is controlled by the competition between the surface tension and the hydrodynamic forces arising as a result of the particle-fluid interaction. For sufficiently small Weber numbers (We ¼ ru2 d=s; where s is the surface tension), particles remain spherical for any finite Reynolds number. At Re 1 the functional equation for the drag force of a droplet (a bubble) with viscosity m2 moving in a fluid with viscosity m1 has the following form fd ¼ f ðm1 ; m2 ; d; uÞ
(4.51)
4.5 Drag of Deformable Particles
85
Three governing parameters from the four in (4.51) have independent dimensions. Therefore, (4.51) can be reduced to P ¼ ’ðP1 Þ
(4.52)
0
0 0 a m2 =m11 da2 ua3 .
where P ¼ fd =ma11 d a2 ua3 ; and P1 ¼ Using the principle of the dimensional homogeneity, we find the values of the 0 0 0 exponents in (4.52) as a1 ¼ a2 ¼ a3 ¼ 1; and a1 ¼ 1; a2 ¼ a3 ¼ 0. Then, we arrive at the following equation for the drag force for a small deformable particle (a drop or a bubble fd ¼ m1 du’ðm21 Þ
(4.53)
where m21 ¼ m2 =m1 . Equation 4.53 can be transformed to the following form for the drag coefficient cd ¼
’ðm21 Þ Re
(4.54)
The function ’ðm21 Þ is strongly dependent on the fluid circulation inside a drop, which is determined by the ratio of fluid viscosities inside and outside it m21 . An increase in the value of m21 is accompanied by flow weakening inside the drop. In the limiting case of an infinitely large m21 , the flow of the inner fluid completely stops and the drag coefficient approaches to the drag coefficient of a rigid spherical particle. Therefore, ’ð1Þ ¼ 24. In the second limiting case corresponding to a very small viscosity of the inner fluid (m21 0), the drag coefficient approaches to the drag coefficient of a bubble and thus ’ð0Þ ! const: An explicit form of (4.54) can be found by integrating the following set of the momentum balance (in the creeping flow, inertialess approximation) and continuity equations rP þ mðiÞ r2 vðiÞ ¼ 0
(4.55)
r vðiÞ ¼ 0;
(4.56)
where i ¼ 1 and 2 for the outer and inner fluids, respectively. The solution of (4.55) and (4.56) is subject to the following boundary conditions: (1) a given uniform flow at infinity, (2) finite velocity everywhere inside the particle. Also, at the drop interface with the surrounding fluid: (3) continuity of tangential stress components, and a jump of the normal stresses due to surface tension; (4) continuity of the velocity components of the inner and outer liquids. It is also assumed that the drop keeps its spherical shape during its motion in an immiscible fluid with different physical properties. This is possible when the drop diameter is sufficiently small for the surface tension force being dominant compared to the hydrodynamic tractions tending to distort the droplet shape. The problem (4.55) and (4.56) was solved for a perfectly spherical drop by Hadamard (1911) and Rybczynski (1911). They derived the following relation for the drag force acting on a drop moving in uniform fluid
86
4 Drag Force Acting on a Body Moving in Viscous Fluid
fd ¼ 3pm1 u1 d
2m1 þ 3m2 3ðm1 þ m2 Þ
(4.57)
Accordingly, the drag coefficient becomes cd ¼
16 1 þ 3m21 =2 Re 1 þ m21
(4.58)
Equation 4.58 acquires the simplest forms corresponding to the drag coefficient for a rigid particle as m2i ! 1 and cd ¼ 24=Re. Another important limit corresponds to bubbles when m21 ! 0 and cd ¼ 16=Re. Note that the drag on a spherical particle given by (4.57) and (4.58) does not depend on the density ratio of the inner and outer fluids, because the creeping flow approximation was employed.
4.6
Drag of Bodies Partially Submerged in Liquid
Consider the drag of a spherical body moving with a constant velocity u along an air-water interface. Let the density and viscosity of water and air be r1 ; r2 and m1 ; m2 , respectively, and the vertical size of the underwater part be l. The latter is determined by the body weight G ¼ r1 gD, where g and D are the gravity acceleration and body displacement, respectively. Thus the set of the governing parameters in this case includes seven dimensional parameters r1 ; m1 ; r2 ; m2 ; D; g, and u. The contribution to the drag force from the air-body interaction, as a rule, is negligible in comparison with the effect of water-body interaction. This makes the effect of the air density and viscosity r2 ; and m2 immaterial. Therefore, the number of the governing parameters can be reduced to only five and, correspondingly, the functional equation for the total drag force acting on a partially submerged body moving along the air-water interface can be written as fb ¼ f ðr; m; g; D; uÞ
(4.59)
Here and hereinafter subscripts 1 for r and m are omitted. Three from the five governing parameters in (4.59) have independent dimensions. Then, according to the Pi-theorem the dimensionless form of (4.59) reads P ¼ ’ðP1 ; P2 Þ, which is equivalent to fb ¼ ’ðFr; ReÞ
(4.60)
pffiffiffiffiffiffi where fb ¼ fb =L3=2 mg1=2 is the dimensionless drag force, Fr ¼ u= gL and Re ¼ uL=n are the Froude and Reynolds numbers, and L ¼ D1=3 is the characteristic size of the body.
4.7 Terminal Velocity of Small Spherical Particles Settling
87
Equation 4.60 shows that the drag force of a partially submerged body is determined by the dimensionless groups with different dependences on the characteristic scale L. Namely, the Reynolds is proportional to L, whereas the Froude number is inverse to L. This circumstance does not allow modeling of drag force acting on a partially submerged body (for example, a ship) when using the same liquid in laboratory experiments as in reality (Sedov 1993). It is emphasized that drag acting on a solid body partially submerged in viscous fluid is determined by two different phenomena. (1) Namely, by viscous friction resulting from body-water interaction, and (2) generation of gravitational waves on the account of the kinetic energy of motion. Accordingly, the total drag force can be presented approximately as a sum of the friction ff and wave fw components, as fb ¼ ff þ fw
(4.61)
Introducing the corresponding drag coefficients as cf ¼
ff ; ð1=2Þru2 S
cw ¼
fw rgL
(4.62)
where S is the cross-sectional area of the body, one can present (4.61) as follows fb ¼ cf
ru2 S þ cw rgL 2
(4.63)
where cf ¼ cf ðReÞ; and cw ¼ cw ðFrÞ. A detailed consideration of the applications of the dimensional analysis to motion of partially submerged solid bodies (in particular, ships) can be found in the monograph by Sedov (1993).
4.7
Terminal Velocity of Small Spherical Particles Settling in Viscous Liquid (the Stokes Problem for a Sphere)
The problem on a small heavy spherical particle settling in viscous liquid was solved by Stokes (1851). Later on, Bridgman (1922) studied this problem in the framework of the dimensional analysis. He considered the steady-state stage of settling (which follows the initial transient stage) when the equilibrium between the weight, buoyancy and viscous drag has already been established. Under such conditions particles settle steadily with the so-called terminal velocity v, which is determined by the particle and liquid densities, particle radius, liquid viscosity and gravity acceleration g. Then, the functional equation for the terminal velocity reads v ¼ f ðd; r1 ; r2; m; gÞ
(4.64)
88
4 Drag Force Acting on a Body Moving in Viscous Fluid
where r is the particle radius, r1 and r2 are the particle and liquid density, respectively, and m is the liquid viscosity. The terminal velocity v and the governing parameters d; r1 ; r2 ; m and g have the following dimensions in the LMT System ½v ¼ LT 1 ; ½d ¼ L; ½r1 ¼ L3 M; ½r2 ¼ L1 M; ½m ¼ L1 MT 1 ; ½g ¼ LT 2
(4.65)
It is seen that three of the five governing parameters in (4.64) have independent dimensions, so that the difference n k ¼ 2: Then, in accordance with the Pi-theorem, (4.64) reduces to the following expression P ¼ ’ðP1 ; P2 Þ
(4.66)
where P ¼ v=ðm=r1 dÞ; P1 ¼ ðr1 =mÞ2 gd 3 ; and P2 ¼ r2 =r1 . Regarding (4.66) Bridgman noticed that “we evidently can say nothing about the effect on the velocity of any of the elements taken themselves, since they all occur under the arbitrary functional symbol”. In order to further simplify the analysis, Bridgman uses the system of units, which includes units of the length, mass, time and force. At small Reynolds numbers when the effect of liquid inertia is negligible, the force is treated as its own compensating dimensional constant. In this case the dimensions of viscosity and gravity acceleration are expressed as it follows from their definitions: ½m ¼ F=Sðdu=dyÞ0 ¼ FL2 T and ½g ¼ ½F=m ¼ FM1 , where, m is the particle mass, S is its surface area, and ðdu=dyÞ0 is the velocity gradient at the surface. Then, the set of the governing parameters includes five parameters of which four parameters have independent dimensions, i.e. the difference n k ¼ 1: Accordingly, (4.64) takes the form P ¼ ’ðP1 Þ
(4.67)
where P ¼ v=ðr 2 r1 g=mÞ; and P1 ¼ r2 =r1 . Huntley (1967) obtained (4.67) by using the Lx Ly Lz MT System of Units for the dimensional analysis the present problem. If the z axis is in the direction of gravity, it is possible to say that the viscous drag depends on the particle cross-section normal to the direction of its motion, i.e. on the scales Lx and Ly : Since, the diameter with the dimension Lx has an equal status with the diameter with the dimension Ly (owing to the axial symmetry of the problem), one can define the characteristic 1=2 particle size ½d as L1=2 x Ly . Bearing in mind that viscosity is defined as the ratio of the shear stress to the corresponding velocity gradient, it is possible to determine the expressions for the “directional” viscosities mx and my , the x and y directions, respectively, 1 1 1 1 1 ½mx ¼ Lx L1 y Lz MT ; my ¼ Lx Ly Lz MT
(4.68)
4.7 Terminal Velocity of Small Spherical Particles Settling
89
Taking into account the isymmetry of the h problem, we write the dimension of 1=2 1 . Thus, the dimensions of the governing ¼ L1 viscosity as ½m ¼ m1=2 x my z MT parameters are 1
1
1 1 1 1 1 1 1 2 2 ½r1 ¼ L1 x Ly Lz M; ½d ¼ Lx Ly ; ½r2 ¼ Lx Ly Lz M; ½m ¼ Lz MT ½g
¼ Lx T 2
(4.69)
Four of the five governing parameters in (4.69) have independent dimensions, so that the difference n k ¼ 1: In this case, in accordance with the Pi-theorem, (4.64) reduces to the following one P ¼ ’ðP1 Þ 0 a1
(4.70) 0
0
0
where P ¼ v=ra11 d a2 ma3 ga4 and P1 ¼ r2 =r1 d a2 ma3 ga4 . 0 The exponents ai and ai are equal to: a1 ¼ 1; a2 ¼ 2; a3 ¼ 1; a4 ¼ 1; 0 0 0 0 a1 ¼ 1; a2 ¼ a3 ¼ a4 ¼ 0: Accordingly, (4.70) takes the form of (4.67) Equation 4.67 determines the dimensionless settling velocity as a function of only one dimensionless parameter, namely, the ratio of the liquid density to the particle density. However, the explicit form of this dependence is unknown, which is a drawback inherent in both (4.66) and (4.67). It should be also emphasized that the previous consideration is related to a particular case of very-low-Reynolds number particles when r1 dv1 =dt ¼ 0; and r2 ð@v2i =@t þ v2i @v2i =@xk Þ ¼ 0: In fact, in this case the influence of the inertial forces is completely neglected. Obviously, in this case densities of the settling particle and the surrounding liquid do not influence the process and have to be excluded from the set of the governing parameters. The above-mentioned drawbacks can be removed by a correct choice of the set of the governing parameters beginning from the analysis of the actual physical factors determining settling of particles in viscous liquid. The latter means that particle and liquid densities should be accounted for even when the inertial effects are expected to be small. In particular, settling of a particle in viscous fluid is due to the excess of particle weight compared to its buoyancy force. Then, the force responsible for the particle settling is the net weight-buoyancy force 4 Fnet ¼ pðr1 r2 Þgr 3 3
(4.71)
Equation (4.71) shows that the difference of the particle and fluid densities ðr1 r2 Þ, as a factor in the product ðr1 r2 Þg (but not separately r1 and r2 or their ratio) is one of the governing parameters that affect the driving force and thus, particle settling. In the set of the governing parameters one should also include fluid viscosity and particle radius as the factors which the drag and driving forces. Therefore, the functional equation for the settling velocity v takes the following form
90
4 Drag Force Acting on a Body Moving in Viscous Fluid
v ¼ f ðr; m; gDrÞ
(4.72)
where Dr ¼ r1 r2 , and all the governing parameters of the problem have independent dimensions in the LMT System of Units: ½r ¼ L; ½m ¼ L1 MT 1 ; and ½gDr ¼ L2 MT 2 . Applying the Pi-theorem to (4.72), we obtain v ¼ cr a1 ma2 ðgDrÞa3
(4.73)
where c is a dimensionless constant. Finding the values of the exponents using the principle of dimensional homogeneity, as a1 ¼ 2; a2 ¼ 1; and a3 ¼ 1, we arrive at the following expression v¼c
r 2 gðr1 r2 Þ m
(4.74)
It agrees (up to a dimensionless factor c) with the exact analytical solution of the problem. In conclusion, we mention an additional simple approach to determine terminal velocity from the equilibrium of forces acting on a particle settling in fluid (Lamb 1959). The dimensional analysis shows that the drag force acting on a spherical particle at Re < 1 can be expressed as Fd ¼ cmrv (with c being a dimensionless constant). The driving force Fnet is determined by (4.71). The equality of Fd and Fnet results in (4.74).
4.8 4.8.1
Sedimentation Dimensionless Groups
Sedimentation of granular materials in fluid flows exerts considerable influence on various physical processes in nature and engineering. In the present Section we consider in accordance with Yalin (1972) only one aspect of sedimentation, namely the application of the Pi-theorem in the sedimentation context. We begin with the parameters that significantly affect flows carrying heavy granules. Namely, the density and viscosity of the carrier fluid, the density of granules and their sizes, the average flow depth h and its slope j ( j ¼ sin a; with a being the slope angle), as well as gravity acceleration. Correspondingly, the set of the governing parameters that determine characteristics of uniform two-phase stationary flows carrying cohesionless granular material consists of the following list r; m; rs ; d; h; j; g
(4.75)
4.8 Sedimentation
91
where subscript s corresponds to granular material. Sedimentation inevitably involves the compound parameter involved in weight and buoyancy forces (cf. Sect. 4.7), gs ¼ gðrs rÞ, as well as a characteristic velocity v ¼ ðgjhÞ1=2 , which also accounts for the flow inclination. Then, the set of the governing parameters takes the following form r; m; rs ; d; h; v ; gs
(4.76)
In this case, the functional equation for a flow characteristics A reads A ¼ f ðr; m; rs ; d; h; v ; gs Þ
(4.77)
where A is, for example, the average velocity of the fluid, or the rate of the granule transport, etc. The dimensions of the governing parameters in (4.77) are as follows ½r ¼ L3 M; ½m ¼ L1 MT 1 ; ½rs ¼ L¼3 M; ½d ¼ L; ½h ¼ L; ½gs ¼ L2 MT 2
(4.78)
Three of the seven governing parameters have independent dimensions, so that the difference n k ¼ 4: In accordance with that, (4.77) can be reduced to the following dimensionless form P ¼ ’ðP1 ; P2 ; P3 ; P4 Þ
(4.79)
where P ¼ A=ra1 va2 d a3 ; P1 ¼ dv =n; P2 ¼ rv2 =gs d; P3 ¼ h=d; and P4 ¼ rs =r, and the exponents ai are determined for any given A by using the principle of dimensional homogeneity. The first dimensionless group P1 represents itself the grain-size-based Reynolds number, which expresses the ratio of the inertia and viscous forces acting on an individual grain. The second group P2 characterizes the ratio of the hydrodynamic and buoyancy forces acting on a grain. This group is referred to as the mobility number and it plays an important role in studies grain motion. The limiting case P2 ! 0(gs ! 1Þ corresponds to a rough rigid bed. The dimensionless groups P3 and P4 express the influence of the flow depth and the grain material density on sedimentation, respectively.
4.8.2
Terminal Velocity of Heavy Grains
In the general case the sedimentation velocity of a heavy grain depends on the physical properties of the carrier fluid (its density and viscosity r and m, respectively), the grain density rs , gravity acceleration g, as well as on time t:
92
4 Drag Force Acting on a Body Moving in Viscous Fluid
As it was established earlier, sedimentation is characterized rather by the product gs ¼ gðrs rÞ than by g. In this case the functional equation for the terminal velocity of sedimentation vt takes the form vt ¼ f ðr; m; rs ; gs ; tÞ
(4.80)
At sufficiently large values of time t, the grain velocity reaches its stationary value called the terminal velocity. In such a non-accelerating, steady-state motion, the grain density rs and time tshould be excluded from (4.80). Then, it takes the following form vt ¼ f ðr; m; d; gs Þ
(4.81)
Since, three of the four governing parameters in (4.81) have independent dimensions, it can be reduced to the following dimensionless form P ¼ ’ðP1 Þ
(4.82)
where P ¼ vt dr=m; P1 ¼ gs rd 3 =m2 ¼ ðgs =gÞðd3 g=n2 Þ; and g ¼ rg: Measurements of terminal sedimentation velocity show that all the experimental data for grains of different types ð1bgs =g<38Þ collapse onto a single curve P ¼ ’ðP1 Þ (Yalin 1972). For sedimentation of very small grains, when the effect of the fluid inertia is negligible (in the Stokes creeping flow regime), (4.81) inevitably takes the scaling form vt ¼ cma1 d a2 gas 3
(4.83)
where the exponents ai are equal a1 ¼ 1; a2 ¼ 2; and a3 ¼ 1, as it follows from the principle of the dimensional homogeneity. As a result, we arrive at (4.74).
4.8.3
The Critical State of a Fluidized Bed
Consider detachment of an individual grain from a fluidized bed assuming that the height of the sand roughness equals grain size and the lower surface of the fluidized bed is approximately a horizontal plane. The critical state that corresponds to the onset of the instability refers to a situation, in which grains are beginning to settle down from the lower surface of the bed. In order to find a relation between the critical parameters corresponding to onset of such settling, we use (4.79) written for the sedimentation transport rate Pqs ¼ fqs ðP1 ; P2 ; P3 ; P4 Þ
(4.84)
4.9 Thin Liquid Film on a Plate Withdrawn Vertically from a Pool Filled
93
where Pqs ¼ qs =rv2 ; with qs being the total transport rate (the total load per unit width of the flow). The positive values of the parameter Pqs ðPqs >0Þ correspond to all possible states of two-phase flow in the fluidized bed, whereas the case Pqs ¼ 0 corresponds to the critical state. Since the granular material does not settle from the lower side of the fluidized bed in the critical state, the dimensionless group P4 ¼ rs =r can be excluded from the set of the governing parameters. In addition, it is also possible to omit the dimensionless group P3 ¼ h=d; since at the critical state grain motion is entirely due to the action of the flow in the vicinity of the lower surface of the fluidized bed. Accordingly, we arrive at the following equation ’ðP1;cr ; P2;cr Þ ¼ 0
(4.85)
P2;cr ¼ ’ðP1;cr Þ
(4.86)
Which is equivalent to
For small and large values of P1;cr , the following expressions are valid const P1;cr
(4.87)
P2;cr ¼ FðP1;cr Þ ¼ const
(4.88)
P2;cr ¼ FðP1;cr Þ ¼
Note, that the first of these equations follows directly from the assumption that the critical stage at small P1;cr corresponds to the equality of the hydrodynamic lift force of Saffman (1965) and the grain weight, i.e to the condition Fl =G ¼ 1; where Fl and G are the lift force and the grain weight, respectively. At large values of P1;cr the fluid viscosity, and accordingly, P1;cr , are no longer the characteristic parameters. In this case corresponding to very small m, it is possible to assume that P2;cr is constant. The predictions of (4.86–4.88) are in a good agreement with the experimental data (Yalin 1972).
4.9
Thin Liquid Film on a Plate Withdrawn Vertically from a Pool Filled with Viscous Liquid (the Landau-Levich Problem of Dip Coating)
In dip coating process a thin film of viscous liquid is withdrawn by a flat plate from a pool filled with that liquid, as shown in Fig. 4.5. The film is withdrawn by the moving vertical plate due to the action of viscous forces, which overbear counteraction of gravity and surface tension. As can be seen in Fig. 4.5, it is possible to
94
4 Drag Force Acting on a Body Moving in Viscous Fluid
Fig. 4.5 Withdrawn liquid film. 1-Liquid pool, 2-the free surface, 3-plate moving upwards
x h(x) 2 g 3 1 0
y
distinguish two characteristic flow domains located at different distances from the moving plate. In the first of these domains the free surface elevations are small and the dominant role is played by the capillary force and gravity forces. On the other hand, in the second domain close to the plate, viscous forces are dominant. An asymptotic solution of the dip coating problem consists in the calculations within each one of the above-mentioned domains 1 and 2 and matching the results, which allows determination of the withdrawn film thickness (Landau and Levich 1942; Levich 1962). Referring the readers to the original work by Landau and Levich, we restrict our discussion to the analysis of several particular cases of film withdrawal in dip coating in the framework of the Pi-theorem. We will consider plate motion with sufficiently small velocities, as specified below, when the inertia effect is negligible. In this case the thickness of the withdrawn liquid film h depends on plate velocity v, gravity acceleration g, liquid specific weight, viscosity and surface tension rg (with r being density), m and s, respectively, as well as on the coordinate along the plate x. Accordingly, the functional equation for the thickness of the withdrawn liquid film reads h ¼ f ðrg; m; v; s; xÞ
(4.89)
It is emphasized that the set of the governing parameters does not contain liquid density alone because the inertia effect is assumed to be negligibly small. The dimensions of the governing parameters that determine the film thickness are ½ðrgÞ ¼ L2 MT 2 ; ½m ¼ L1 MT 1 ; ½v ¼ LT 1 ; ½s ¼ MT 2 ; ½ x ¼L
(4.90)
Three of the five governing parameters have independent dimensions. Choosing as these parameters (rgÞ; v and s, we transform (4.89) using the Pi-theorem to the following dimensionless form P ¼ ’ðP1 ; P2 Þ
(4.91)
4.9 Thin Liquid Film on a Plate Withdrawn Vertically from a Pool Filled 0
0
0
95 00
00
00
where P ¼ h=ðrgÞa1 va2 sa3 ; P1 ¼ m=ðrgÞa1 va2 sa3 ; and P2 ¼ x=ðrgÞa1 va2 sa3 . The corresponding exponents are found using the principle of the dimensional 0 0 0 homogeneity as a1 ¼ 1=2; a2 ¼ 0; a3 ¼ 1=2; a1 ¼ 0; a2 ¼ 1; a3 ¼ 1; and 00 00 00 a1 ¼ 1=2; a2 ¼ 0; a3 ¼ 1=2: Then (4.91) takes the following form
h¼
s rg
1=2 "
2 1=2 # mv x rg ’ ; s s
(4.92)
In the limit x>>1; it is possible to assume that the effect of the dimensionless 1=2 has already been saturated and it does not affect the group P2 ¼ ðx2 rg=sÞ withdrawn film thickness anymore. Then, (4.92) reduces to the following expression
1=2 s mv ’ h¼ (4.93) rg s It is convenient to present (4.93) as a dependence of the dimensionless withdrawn film h ¼ h=l; where l ¼ ðs=rgÞ1=2 is the capillary length, on the capillary number Ca ¼ mv=s h ¼ ’ðCaÞ
(4.94)
The function ’ðCaÞ can be either found experimentally, or through the original asymptotic solution of Landau and Levich. It has the following form (Levich 1962) ’ðCaÞ ¼ 0:93Ca1=6
(4.95)
’ðCaÞ 1
(4.96)
at Ca<<1; and
at Ca 1. At a high withdrawal velocity the thickness of the withdrawn liquid film h should not depend on the surface tension, since both the effects of gravity and viscous forces become dominating. Therefore, the functional equation for the thickness of the withdrawn liquid layer becomes h ¼ f ðrg; m; vÞ
(4.97)
All the parameters in (4.97) have independent dimensions. Therefore, it takes the following form h ¼ cðrgÞa1 ma2 va3 where c is a dimensionless constant.
(4.98)
96
4 Drag Force Acting on a Body Moving in Viscous Fluid
The exponents ai are found from the principle of the dimensional homogeneity as a1 ¼ 1=2; a2 ¼ 1=2; and a3 ¼ 1=2. Accordingly, (4.98) becomes
1=2 mv h¼c rg
(4.99)
The numerical value of the constant c can be found experimentally and approximately equals to 2/3 (Derjaguin and Levi 1964).
Problems P.4.1. Show that at low Reynolds numbers the drag coefficient of a spherical particle moving with a constant velocity in an incompressible viscous fluid is inversely proportional to the Reynolds number. The terms rðv rÞv and mr2 v in the Navier-Stokes equation have the order 2 2 x =l of rv x and mv x =lx , respectively, whereas and their ratio is of the order of 2 2 rvx =lx = mvx =lx ¼ Re<<1 (vx and lx are the characteristic scales of the velocity and length in the direction of the particle motion). That allows to omit the term rðv rÞv from left hand side of the Navier-Stokes equation and thus eliminate the fluid density from the set of the governing parameters. Then the functional equation for the drag force takes the following form fd ¼ f ðm; u; dÞ
(P.4.1)
All the governing parameters in (P.4.1) have independent dimensions, so that the difference n k ¼ 0. Therefore, in accordance with the Pi-theorem we obtain fd ¼ Cma1 ua2 d a3
(P.4.2)
where C is a dimensionless constant. Taking into account the dimensions of fd ; m; u and d, we find the values of the exponents ai from the principle of dimensional homogeneity as a1 ¼ a2 ¼ a3 ¼ 1. Accordingly, the drag force and the drag coefficient are expressed as fd ¼ Cmdu cd ¼
C1 Re
(P.4.3) (P.4.4)
Equations P.4.3 and P.4.4 coincide with the Stokes law up to a dimensionless factor. The values of the constants C and C1 , which can be either determined from
Problems
97
the exact solution of the Stokes equations or measured experimentally are 3 and 24, respectively. P.4.2. Show that the drag coefficient of a spherical particle does not depend on Re at the high values of the Reynolds number. At high values of the Reynolds number the inertia forces play dominant role. Under these conditions it is possible to omit fluid viscosity from the set of the governing parameters. That leads to (4.48) for the drag force. For a spherical particle ’ðaÞ is constant. Then, we obtain from (4.48) the following expression for the drag coefficient cd ¼
fd ¼ const: ð1=2Þru2 d 2
(P.4.5)
P.4.3. Derive a dimensionless equation for the drag force acting on an infinitely long cylinder in subjected to a uniform laminar flow of incompressible fluid normal to its axis. Let the fluid density and viscosity be r and m, respectively, and velocity of the undisturbed flow and radius of cylinder be u and R, respectively. Then the drag force fc is determined by the following fourth governing parameters fc ¼ f ðr; m; u; RÞ
(P.4.6)
That have the dimensions as listed: ½r ¼ L3 M; ½m ¼ L1 MT 1 ; ½u ¼ LT 1 and R½ L. The dimension of the drag force acting at a unit length of the cylinder is ½fc ¼ MT 2 . Since three of the four governing parameters have independent dimensions, (P.4.6) reduces to the following form P ¼ ’ðP1 Þ 0
0
(P.4.7) 0
where P ¼ fc =ra1 ma2 ua3 ; and P1 ¼ R=ra1 ma2 ua3 . Bearing in mind the dimensions of the drag force and the governing parameters and applying the principle of the dimensional homogeneity, we find values of the 0 0 0 0 exponents ai and ai as: a1 ¼ 0; a2 ¼ 1; a3 ¼ 1; a1 ¼ 1; a2 ¼ 1; and a3 ¼ 1. Then, the drag force fc can be expressed as fc ¼ mu’ðReÞ
(P.4.8)
where Re ¼ ruR=m is the Reynolds number. Note, that the exact solution of the problem based on Oseen’s equation reads (Lamb 1959) fc ¼
4pmu 4pmu ¼ 1=2 C þ lnðReÞ lnð37=ReÞ
(P.4.9)
98
4 Drag Force Acting on a Body Moving in Viscous Fluid
where C ¼ 0:577::::is the Euler constant. Comparing the expressions (P.4.8) and (P.4.9), we find that ’ðReÞis ’ðReÞ ¼
4p lnð37=ReÞ
(P.4.10)
P.4.4. Show that in the frame of Stokes’ approximation it is impossible to determine an steady-state drag force acting on a unit length of an infinitely long cylinder. Stokes’ approximation is based on the assumption that the inertial term rðv rÞv in (P.4.6) can be neglected and, thus, the drug force acting on a unit length of a cylinder is written as fc ¼ f ðm; u; RÞ
(P.4.11)
Applying the Pi-theorem to (P.4.11) leads to an unrealistic result: the drag force does not depend on the body size, i.e. its radius (Landau and Lifshitz. 1987). P.4.5. Find the drag force acting on two round coaxial disks submerged in a fluid and moving towards each other. Let the lower disk is motionless, whereas the upper one moves with a constant velocity u towards the lower one. The approach of the disks inevitable squeezes the fluid from the gap in between. The force acting on the disks due to the difference of pressure in the gap P and in the surrounding fluid P0 is ðR fdisk ¼ 2p ðP P0 Þrdr
(P.4.12)
0
The force fc ¼ LMT 2 is determined by the following four governing parameters: the fluid viscosity ½m ¼ L1 MT 1 , the velocity of the moving disk ½u ¼ LT 1 , the disk radii ½ R ¼ L, and the current gap width between the disks ½h ¼ L. Therefore, it is given by the following equation fc ¼ f ðm; u; R; hÞ
(P.4.13)
Three of the four governing parameters have independent dimensions, which allows us to reduce (P.4.13) to the following dimensionless form P ¼ ’ðP1 Þ 0
0
(P.4.14) 0
where P ¼ fdisk =ma1 ua2 Ra3 ; and P1 ¼ h=ma1 ua2 Ra3 . Taking into account the dimensions of fdisk and h; m; u; and R and applying the 0 principle of the dimensional homogeneity, we find values of the exponents ai and ai 0 0 0 as: a1 ¼ a2 ¼ a3 ¼ 1; a1 ¼ a2 ¼ 0; and a3 ¼ 1. Then, we obtain the following expression for the drag force
Problems
99
fdisk
h ¼ muR’ R
(P.4.15)
It is seen that the drag acting on a disk is directly proportional to the product of the viscosity, velocity of the disk, its radius, as well as a function of the ratio h/R. This function cannot be established in the framework of the dimensional analysis. An exact solution of the problem yields fdisk
3 3p R ¼ muR 2 h
(P.4.16)
P.4.6. Determine the dependence of the friction force acting on a sphere rotating in an infinite viscous fluid. The friction force ff ¼ LMT 2 depends on the following three governing parameters: the fluid viscosity ½m ¼ L1 MT 1 , sphere radius ½ R ¼ L, and the angular velocity of rotation ½o ¼ T 1 . These parameters have independent dimensions. According to the Pi-theorem, we have the following functional equation for the drag force ff ¼ Cma1 Ra2 oa3
(P.4.17)
where C is a constant. Bearing in mind the dimensions of ff ; m; R and o, we arrive at the following result ff ¼ CmR2 o
(P.4.18)
P.4.7. Determine the dependence of the semi-axes ratio of a droplet moving in air, which acquires an approximately spheroidal shape, on the physical properties of liquid and velocity of motion. The surface of a relatively small droplet moving in air approximately resembles a spheroid. We adopt the Cartesian axes OX; OY; and OZ with the center O at the droplet center and the axis OZ directed with the undisturbed flow experienced by an observer moving with the droplet. The droplet surface is approximated as a spheroid with semi-axes a and b (with a>b) x2 þ y2 z2 þ 2¼1 a2 b
(P.4.19)
The deviation of the droplet surface from a sphere is characterized by the difference a b that depends on the characteristic size of the droplet, as well as the density, velocity and surface tension of the liquid a b ¼ f ðb; r; u; sÞ
(P.4.20)
100
4 Drag Force Acting on a Body Moving in Viscous Fluid
The three governing parameters in (P.4.20) have independent dimensions. Therefore, the dimensionless form of (P.4.20) reads P ¼ ’ðP1 Þ
(P.4.21) 0
0
0
Where P ¼ ða bÞ=ra1 ua2 sa3 ; and P1 ¼ b=ra1 ua2 sa3 . 0 Determining the values of the exponents ai and ai , we arrive at the following equation w ¼ 1 þ cðWeÞ
(P.4.22)
where w ¼ a=b, and We ¼ ru2 b=s is the Weber number. The latter can equally be based on the volume-equivalent droplet diameter d. P.4.8. Determine the moment of force required for a slow steady-state rotation of an infinitely long solid cylinder of radius R1 with the angular velocity o about its axis. Consider two cases: (1) the cylinder rotates in an infinite viscous fluid; and (2) the cylinder rotates inside a cylindrical shell of radius R2 , which is sufficiently larger than R1 , filled with a viscous fluid. 1. The moment of force required for swirling of a cylinder in viscous fluid is given by the expression 2ðp
Mf ¼
tSd’
(P.4.23)
0
where t is the surface traction, S ¼ 2pR1 is the area of the lateral surface of the cylinder of unit length. The functional equation for the moment of force in the inertialess case of slow swirling considered here reads Mf ¼ f ðm; o; RÞ
(P.4.24)
All the governing parameters in (P.4.24) have independent dimensions, namely ½m ¼ L1 MT 1 ; ½o ¼ T 1 ; and ½ R ¼ L. Then, in accordance with the Pi-theorem, (P.4.24) reduces to the following one Mf ¼ cma1 oa2 Ra13
(P.4.25)
where c is a constant. Accounting for the dimension of the moment of force Mf ¼ LMT 2 , we find the values of the exponents as a1 ¼ 1; a2 ¼ 1; and a3 ¼ 2: Then, (P.4.25) takes the following form Mf ¼ cmoR2
(P.4.26)
References
101
2. In the second case the functional equation for the moment of force reads Mf ¼ f ðm; o; R1 ; e; eÞ
(P.4.27)
where e ¼ R2 R1 is the gap between the cylinder and the surrounding shell, and e is the eccentricity ðe½ L; and e½ LÞ: Three of the five governing parameters in (P.4.27) possess independent dimensions, so that the difference n k ¼ 2: Therefore, the dimensionless form of (P.4.27) reads P ¼ ’ðP1 ; P2 Þ
(P.4.28)
where P ¼ Mf =moR21 ; P1 ¼ e=R1 ; and P2 ¼ e=R1 : In the particular case of concentric cylinder and shell ðe ¼ 0Þ, (P.4.28) takes the form
e 2 (P.4.29) Mf ¼ moR1 ’ R1 In the general case the expression for Mf has the form (Loitsyanskii 1966) 4pmoR3 cðlÞ e h pffiffiffiffiffiffiffiffiffiffiffiffiffii where cðlÞ ¼ 1 þ 2l2 = 2 þ l2 1 l2 ; and l ¼ e=e. Mf ¼
(P.4.30)
References Anton TR (1987) The lift force on a spherical body in rotation flow. J Fluid Mech 183:199–218 Batchelor GK (1967) An introduction to fluid dynamics. Cambridge University Press, Cambridge Basset AB (1961) A treatise on hydrodynamics, vol 2. Dover, New York Bearman PW, Dowirie MJ, Graham JMP, Obasaju ED (1985) Forces on cylinders in viscous oscillatory flow at low Kenlegan-Carpenter numbers. J Fluid Mech 154:337–356 Berlemont A, Desjonqueres P, Gouesbet G (1990) Particle Lagrangian simulation in turbulent flow. Int J Multiphas Flow 16:19–34 Boothroyd RG (1971) Following gas-solids suspensions. Chapman and Hall, London Boussinesq J (1903) Theorie Analytique de la Chaleur. L’ Ecole Polytechnique, Paris Bridgman PW (1922) Dimensional analysis. Yale University Press, New Haven Chang EJ, Maxey MR (1994) Unsteady flow about a sphere at low to moderate Reynolds number. Part 1. Oscillatory motion. J Fluid Mech 277:347–379 Chang EJ, Maxey MR (1995) Unsteady flow about a sphere at low to moderate Reynolds number. Part 2. Accelerated motion. J Fluid Mech 303:133–153 Clift R, Grace JR, Weber ME (1978) Bubbles, droplets and particles. Academic, New York Dandy DS, Dwyer HA (1990) A sphere in shear flows at finite Reynolds number: effect of shear on particle lift, drag and heat transfer. J Fluid Mech 216:821–828 Derjaguin BM, Levi SM (1964) Film coating theory. The Focal Press, London
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4 Drag Force Acting on a Body Moving in Viscous Fluid
Dwyer HA (1989) Calculation of droplet dynamics in high temperature environments. Prog Energ Combust Sci 15:131–158 Fendell FE, Sprankle ML, Dodson DS (1966) Thin-flame theory for a fuel drop in slow viscous flow. J Fluid Mech 26:267–280 Graham JMR (1980) The forces on sharp-edged cylinders in oscillatory flow at low KenleganCarpenter numbers. J Fluid Mech 97:331–346 Hadamard JS (1911) Movement permanent lend d’une sphere liquid et visqueuse dans un liquid visqueux. C.R. ACAD Sci Paris 152:1735–1738 Happel J, Brenner H (1983) Low Reynolds number hydrodynamics. Martinus Nijhoft, The Hague Huntley HE (1967) Dimensional analysis. Dover Publications, New York Karanfilian SK, Kotas TJ (1978) Drag on sphere in steady motion in a liquid at rest. J Fluid Mech 87:85–96 Kassoy DR, Adamson TC, Messiter AF (1966) Compressible low Reynolds number flow around a sphere. Phys Fluids 9:671–681 Kenlegan GH, Carpenter LH (1958) Forces on cylinders and plates in an oscillating fluid. J Res Nat Bur Standard 60:423–440 Lamb H (1959) Hydrodynamics, 6th edn. Cambridge University Press, Cambridge Landau LD, Lifshitz EM (1987) Fluid mechanics, 2nd edn. Pergamon, New York Landau LD, Levich VG (1942) Dragging of a liquid by moving plate. Acta Physicochimica USSR 17:42–54 Levich VG (1962) Physicochemical hydrodynamics. Prentice-Hall, Englewood Cliffs Loitsyanskii LG (1967) Laminar Grenzschichten. Academic-Verlag, Berlin Loitsyanskii LG (1966) Mechanics of liquids and gases. Pergamon, Oxford Maxey MR, Riley JJ (1983) Equation of motion for a small rigid sphere in a nonuniform flow. Phys Fluids 26:863–888 McLaughlin JB (1991) Inertia migration of small sphere in linear shear flows. J Fluid Mech 224:261–274 Mei R, Lawrence CJ, Adrian RJ (1991) Unsteady drag on sphere of finite Reynolds number with small fluctuations of the free stream velocity. J Fluid Mech 223:613–631 Mei R (1992) An approximate expression for the shear lift force on a spherical particle at the finite Reynolds number. Int J Multiphas Flow 18:145–147 Mei R (1994) Flow due to an oscillating sphere: an expression for unsteady drag on the sphere at finite Reynolds number. J Fluid Mech 270:133–174 Odar F, Hamilton WS (1964) Forces on a sphere accelerating in viscous fluid. J Fluid Mech 18:302–314 Oseen CW (1910) Uber die Stokes’ Formuel, und uber eine verwen die Aufgabe in der Hydrodynamik. Ark Mth Astronom Fus 6(29):1–20 Oseen CW (1927) Hydrodynamik. Akademise Verlagsgesellichaft, Leipzig Rybchynski W (1911) Uber die fortschreitendl Bewegung einer flussingen Kugelin einem Zahen Medium. Bull Inst Acad Sci Cracovie ser. A 1:40–46 Saffman PS (1965) The lift on small sphere in a shear flow. J Fluid Mech 22:385–400 Saffman PS (1968) Corrigendum to ‘the lift on a small sphere in a slow shear flow’. J Fluid Mech 31:624–624 Schlichting H (1979) Boundary layer theory. McGraw-Hill, New York Sedov LI (1993) Similarity and dimensional methods in mechanics, 10th edn. CRC Press, Boca Raton Shih CC, Buchanan HJ (1971) The drag on oscillating flat plates in liquids at low Reynolds numbers. J Fluid Mech 48:229–239 Soo SL (1990) Multiphase fluid dynamics. Science Press and Gower Technical, Beijing Stokes GC (1851) On the effect of internal friction of fluids on the motion of pendulums. Trans Cambridge Philos Soc 9:8–106 Yalin MS (1972) Mechanics of sediment transport. Pergamon Press, Oxford Yarin LP, Hetsroni G (2004) Combustion of two-phase reactive media. Springer, Berlin
Chapter 5
Laminar Flows in Channels and Pipes
5.1
Introductory Remarks
Fluid flow in pipes and ducts was a subject of numerous experimental and theoretical investigations performed during the last two centuries. Beginning from the seminal works of Hagen (1839) and Poiseuille (1840), a detailed data on flows of incompressible viscous fluids in pipes and ducts of different geometry was obtained. These results are presented in many review articles, monographs and textbooks. A comprehensive analysis of problems related to laminar and turbulent flows in pipes and ducts (the physical foundations of the theory and its mathematical formulation) can be found in such widely known books as Schlichting (1979), Landau and Lifshitz (1987), Loitsyanskii (1966) and Ward-Smith (1980). Refering the readers to these monographs, we focus on the applications of the Pi-theorem for the analysis of pipe and duct flows. First we discuss some specific features of laminar flows of incompressible viscous fluid outflowing from a large tank through a conical nozzle into a straight pipe or duct. A sketch of such flow within the entrance and fully developed sections of a pipe located, correspondingly, near and far from the inlet is shown in Fig. 5.1. It is seen that the velocity profile which is uniform at the entrance cross-section of the pipe becomes non-uniform at x>0 as a consequence of fluid-wall interaction at the flow periphery. Under the conditions corresponding to relatively large Reynolds numbers determined by the mean velocity, pipe diameter and fluid viscosity, laminar flows in straight pipes can be schematically presented as follows. Near the pipe inlet, the boundary layer forms over the wall. The thickness of the layer increases downstream (cf. the left part of Fig. 5.1). The longitudinal velocity component u increases within the boundary layer from zero at the wall to the velocity of the undisturbed fluid flow in the potential core. The velocity in the potential core gradually increases from an initial velocity u0 at x ¼ 0 up to the maximum one, umax , in the cross-section x ¼ len where the boundary layers from different sides merge. Approximately at x ¼ len the velocity profile approaches to the parabolic Poiseuille profile, the flow becomes fully L.P. Yarin, The Pi-Theorem, Experimental Fluid Mechanics, DOI 10.1007/978-3-642-19565-5_5, # Springer-Verlag Berlin Heidelberg 2012
103
104
5 Laminar Flows in Channels and Pipes
Fig. 5.1 Velocity profiles in laminar flows in straight cylindrical pipes
developed, and does not vary with x anymore. Thus, the following two flow regions in the pipe flows are distinguished. Namely, (1) the entrance flow region (xlen ). The flow characteristics within the entrance region are mostly determined by the boundary layer, whereas within the fully developed region all fluid elements move parallel to the pipe axis x with the longitudinal velocity u ¼ uðyÞ; and v ¼ w ¼ 0, where x and y are the longitudinal and transversal coordinates, respectively. Schlichting (1979). Stationary flows of incompressible viscous fluid in straight pipes are described by the Navier–Stokes and continuity equations 1 v rv ¼ rP þ nr2 v r
(5.1)
rv¼0
(5.2)
where r; P and v are the density, pressure and velocity vector, respectively, n is the kinematic viscosity. Solutions of (5.1) and (5.2) should satisfy the no-slip boundary conditions at the wall, as well as to correspond to given velocity distributions v0 and v00 at the inlet (x ¼ 0) and outlet (x ¼ L) of the pipe v ¼ v0 at x ¼ 0; v ¼ v00 at x ¼ L; v ¼ 0 at the pipe wall
(5.3)
The approximate analytical solutions of the problem on the flow in the entrance region of a plane channel was obtained by Schlichting (1934) and Schiller (1922). In this cases the length of the entrance region len is determined by the expression len ¼c l Re
(5.4)
5.1 Introductory Remarks
105
where l is the characteristic size of a channel or a pipe, a semi-height or a radius, respectively. Re is the Reynolds number based on the channel height or pipe diameter, and c is a dimensionless constant. The numerical solution of the Navier–Stokes equations (Friedmann et al. 1968), as well as the experimental studies of Emery and Chen (1968) and Fargie and Martin (1971) show that c in (5.4) depends on the Reynolds number at Re<500: Within this range of the Reynolds number, c decreases monotonously as Re increases. In the range Re>500 c is practically constant (cf. Fig. 5.2). Accordingly, in the framework of the two region model described above, the problem on the laminar flow in a pipe of circular cross-section amounts to solving the following system of equations (Loitsyanskii 1966) ru
@u @u dP 1 @ @u þ rv ¼ þm k yk @x @y dx y @y @y @u 1 @vyk ¼0 þ @x yk @y
(5.5)
(5.6)
for the entrance region (in the boundary layer approximation), and dP ¼ mr2 u dx
(5.7)
for the fully developed region. In (5.5) and (5.6) k ¼ 0 or 1 for the plane or axisymmetric flows, respectively.
Fig. 5.2 The dependence of the length of the entrance section on the Reynolds number in laminar flow in a straight cylindrical pipe. 1-Calculations by Friedman et al. (1968), 2-Data by Emery and Chen (1968)
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5 Laminar Flows in Channels and Pipes
The boundary conditions for (5.5) and (5.6) read y ¼ 0 u ¼ v ¼ 0; y ¼ yd u ¼ ud
(5.8)
(with yd being the boundary layer thickness and u ¼ ud ðxÞ the velocity of the external flow velocity in the potential core), and for (5.7) y¼0
@u ¼ 0; y ¼ r0 u ¼ 0 @y
(5.9)
In (5.5), (5.6) and (5.8) y is understood as the distance from the wall to a point within the boundary layer. On the other hand, in (5.7) and (5.9) y is reckoned from the channel axis and r0 is the cross-sectional radius of the pipe. It is emphasized that within the entrance region of the flow the pressure gradient is a function on x, whereas within the fully developed flow region it is constant. That allows us to understand, as usual for the Poiseuille flows, that in (5.7) dP=dx ¼ DP=L; where DP is the pressure drop over a pipe section of length l. As usual, two types of questions can be posed regarding flows in pipes (and similarly, in planar channels). In the first one, a volumetric flow rate Q is given, while the corresponding pressure gradient required to sustain such flow, DP=l should be found. In the second one, a pressure drop DP=l is given, while the corresponding volumetric flow rate Q is to be found.
5.2 5.2.1
Flows in Straight Pipes of Circular Cross-Section The Entrance Flow Region
The flow of viscous fluid within the entrance region of a pipe is illustrated in Fig. 5.1. As a result of the fluid friction at the wall, the boundary layer is formed. The thickness of the boundary layer dðxÞ increases along the pipe, with the longitudinal coordinate x. At some distance from the inlet x ¼ len , the boundary layer fills the whole pipe cross-section and thus dðlen Þ ¼ d=2: Having in mind the two-region pattern of pipe flows, we evaluate their characteristics using the dimensional analysis. First, we evaluate the dependence of the length of the entrance region len on the flow parameters. It is natural to assume that the thickness of the boundary layer depends on the fluid density and viscosity r and m, velocity in the undisturbed core u, as well as the distance from the inlet to a cross-section under consideration. Then, we have the functional equation for dðxÞ as follows dðxÞ ¼ f ðr; m; u; xÞ
(5.10)
In the framework of the boundary layer theory expediently it is natural to use the System of Units Lx Ly Lz MT with three different length scales Lx ,Ly and Lz in the x,y
5.2 Flows in Straight Pipes of Circular Cross-Section
107
and z directions, respectively. Using this system of units, we express the 1 1 dimensions of the governing parameters as ½r ¼ L1 x Ly Lz M; ½m ¼ 1 1 1 1 Lx Ly Lz MT ; ½u ¼ Lx T ,½ x ¼ L, and the thickness of the boundary layer as ½d ¼ Ly . Taking into account that all the governing parameters have independent dimensions, we rewrite (5.10) as follows dðxÞ ¼ cra1 ma2 ua3 xa4
(5.11)
where c is a dimensionless constant and the exponents ai found using the principle of the dimensional homogeneity, read a1 ¼ 1=2; a2 ¼ 1=2; a3 ¼ 1=2; and a4 ¼ 1=2. Then (5.11) takes the form
mx dðxÞ ¼ c ru
1=2 (5.12)
or nx 1=2 dðxÞ ¼c 2 d d u
(5.13)
Bearing in mind that dðlen Þ ¼ d=2, we obtain len ¼ c Red d
(5.14)
1
where1 Red ¼ ud=n; and c ¼ ð2cÞ2 . The dependence (5.14) qualitatively agrees with the theoretically predicted value of len ¼ 0:04dRe (Schlichting 1979). The results of the numerical investigation of the dependence len ðReÞ for flows in straight cylindrical pipes are presented in Fig. 5.2. They show that the value of the ratio len =dRe is approximately constant at Re > 500 when the boundary layer approximations are valid. In addition, we address the pressure drop over the entrance region of the pipe flow. Equation (5.14) shows that the length of the entrance region len depends on pipe diameter and the Reynolds number. That allows us to use len as a generalized parameter of the problem. Namely, assume that the ratio ðdP=dxÞ=ru2 depends on a single dimensional parameter len and one variable x. Then, the functional equation for this ratio becomes dP 2 ru ¼ f ðlen ; xÞ dx
1
½Re ¼ 1 since it is defined by the parameter magnitudes.
(5.15)
108
5 Laminar Flows in Channels and Pipes
In (5.15) one of the two governing parameters has an independent dimension. Then, in accordance with the Pi-theorem, (5.15) reduces to the dimensionless form P ¼ ’ðP1 Þ
(5.16)
0
where P ¼ ðdP=dxÞ=ðru2 Þ=laen ; and P1 ¼ x=laen . 1 2 Bearing in mind the dimension of ½dP=dx ¼ L1 y LZ MT , we find the 0 0 values of the exponents ai and ai as a ¼ 1; a ¼ 1. Accordingly, (5.16) takes the form ðdP=dxÞlen x ¼’ 2 len ru
(5.17)
Substituting the expression (5.14) for len into (5.17), we arrive at the dependence Po ¼ ’ðXÞ
(5.18)
where the Poiseuille number Po ¼ c lRed ; l ¼ ðdP=dxÞd=ru2 ; and X ¼ x=ðc dRed Þ. Equation (5.18) shows that the Poiseuille number is a function of one dimensionless variable X alone (Fig. 5.3).
Fig. 5.3 The dependence of the Poiseuille number on X
5.2 Flows in Straight Pipes of Circular Cross-Section
5.2.2
109
Fully Developed Region of Laminar Flows in Smooth Pipes
Consider pressure drop in the fully developed flow region. To choose the governing parameters for this flow, it is necessary to account for such the specific features as (1) constancy of the pressure gradient DP=l; and (2) fluid motion parallel to the pipe axis with a constant mean velocity when density does not affect the flow. Under these conditions the functional equation for the pressure drop takes the form DP ¼ f ðm; u0 ; dÞ l
(5.19)
where ½m ¼ L1 MT 1 is the dimension of the viscosity, ½u0 ¼ LT 1 is dimension of the mean velocity and ½d ¼ L (with d ¼ 2r0 being the cross-sectional diameter), i.e. they have independent dimensions. Then, using the Pi-theorem, we can transform (5.19) to the following one DP ¼ cma1 ua02 da3 l
(5.20)
where c is a dimensionless constant, and the exponents ai are equal to a1 ¼ 1; a2 ¼ 1 and a3 ¼ 2: As a result, we arrive at the expression DP mu0 ¼c 2 d l
(5.21)
Dividing both sides of (5.21) by ru2 , we obtain the following expression for the friction factor l l¼
c Re
(5.22)
where l ¼ 2ðDP=lÞ d=ru20 is the friction factor, c ¼ 2c is a dimensionless constant. As follows from the exact analytical solution of the Hagen-Poiseuille problem on laminar flows in round smooth pipes, the factor c in (5.22) equals 64.
5.2.3
Fully Developed Laminar and Turbulent Flows in Rough Pipe
Turbulent pipe flows were studied experimentally in Nikuradse (1933) who used uniform sand grains of different sizes glued to the inside walls of pipes to vary roughness in a controlled manner. It is emphasized that in stable laminar pipe flows
110
5 Laminar Flows in Channels and Pipes
at sufficiently low values of the Reynolds number (Re 2000) the wall roughness was assumed to have minor effect. In stable pipe flows perturbations introduced by roughness elements were expected to fade sufficiently fast to affect flow field globally (Schlichting 1979). Later on, Kandlicar (2005) revisited the results of Nikuradse’s experiments in the light of the recent studies of laminar flows in micro-channels with rough walls and concluded that surface roughness plays an important role in fluid flow in micro and mini-channels. The numerical calculations by Hezwig et al. (2008) showed that in laminar flows the resistance of rough-wall channels with regular roughness significantly depends on the relative roughness size. A number of experimental studies of laminar flows in micro-channels also suggest some influence of wall roughness on resistance to flow in such channels (Yarin et al. 2009). Nevertheless, in spite of the existence of a number of theoretical and experimental works devoted to laminar flows in rough pipes and channels, the role of roughness and mechanisms of its influence on the resistance of such channels is not clear yet. Below we attempt to reveal some special features characteristic of resistance of rough straight pipes on fully developed laminar flow by using the dimensional analysis. Suppose that a rough pipe is characterized by two dimensional parameters, namely, its effective diameter def and the height of roughness elements k. In addition, the set of the governing parameters of such flows should include three dimensional parameters that characterize the physical properties of fluid-its density r and viscosity m; as well as the average flow velocity u: (A discussion concerning the choice of the characteristic size of rough elements see in Hezwig et al. 2008). Then, the functional equation for pressure drop over pipe section of length l reads DP ¼ f ðr; m; u; def ; kÞ l
(5.23)
The governing parameters in (5.23) have the following dimensions: ½r ¼ L3 M; ½m ¼ L1 MT 1 ; ½u ¼ LT 1 ; def ¼ L; ½k ¼ L: Three of the five governing parameters have independent dimensions. Therefore, according to the Pi-theorem, (5.23) can be transformed to the form P ¼ ’ðP1 ; P2 Þ
(5.24)
where P ¼ 2DPd=lru2 ¼ l is the friction factor, P1 ¼ m=rudef ¼ Re1 ; with Re ¼ rudef =m, which is the Reynolds number, and P2 ¼ k=def ¼ k is the relative roughness of pipe wall. The dimensional analysis does not allow finding an exact form of the dependence of the drag friction factor on the dimensionless groups- the Reynolds number and relative roughness of the wall. However, it is possible to evaluate some particular trends of the friction factor when Re and k are changing. Indeed. At large values of the Reynolds number corresponding to fully developed turbulence, the inverse Reynolds number P1 is much less than the value of the relative
5.3 Flows in Irregular Pipes and Ducts
111
roughness P2 . In this case the effect of viscosity is negligible, and the friction factor is determined mostly by the relative roughness l ’ðkÞ. In the second corresponding to relatively small Reynolds number (laminar flows), the inverse Reynolds number P1 can be much large than the relative roughness P2 . Therefore, in the latter case the roughness effects are weak. It is emphasized that at any finite values of Re the influence of wall roughness on the friction factor of a pipe exists. Only at Re ! 0, (5.24) reduces to the form l ¼ ’ðReÞ, whereas within the range of moderate Reynolds numbers the friction factor depends on both dimensionless groups, namely l ¼ lðRe; kÞ. The latter form practically encompasses the whole bulk of the available experimental data (Moody 1948; Schlichting 1979).
5.3
Flows in Irregular Pipes and Ducts
At applying the Pi-theorem to flows in irregular (non-circular) pipes or channels, it is convenient to use an effective size def ½ L expressed through the cross-section geometry. For example, the so-called hydraulic diameter dh defined as 4S=P, where S is the cross-sectional area and P the wetted perimeter is used as an effective size of irregular pipes and channels. In this case the form of the functional equation for the pressure drop in irregular pipes and ducts remains the same as the regular ones (with circular cross-section). However, as the parameter d in (5.19) the effective size of an irregular pipe def is to be used. In particular, the Reynolds number Re is defined based on def . As an example, we mention that the effective size of pipes with the elliptic crosssection is defined as follows pffiffiffi def ¼ 2 2
b
(5.25)
ð1 þ e2 Þ1=2
where e ¼ b=a; and a and b are the large and small semi-axes of the ellipse. The detailed data on pressure drop in the irregular pipes and ducts is available in monographs of Shah and London (1978), Ward-Smith (1980), Ma and Peterson (1997) and White (2008). Some results that display the effect of pipe shape on resistance to laminar flows in the irregular pipes are presented in Table 5.1. Table 5.1 Characteristics of irregular pipes Shape of pipe Caracteristic size Circular def ¼ d pffiffiffi Elliptic def ¼ 2 2 ab 1=2 ða2 þb2 Þ
Equilateral triangular Square
def ¼ h def ¼ H
Constant in (5.22) 64 64
Reynolds number
160 128=f ðxÞ
uh n u2H n
ud n udef n
112
5 Laminar Flows in Channels and Pipes
In this Table in addition to the previously introduced notation, h is a side of an equilateral triangle, H is a side of a square, and f ðxÞ is a tabulated function of the height to width ratio [ f ðxÞ changes from 2.253 to 5.333 when x varies from 1 to 1 (Loistyanskii, 1966)]. Note that the factor c in (5.22) for the friction factor for laminar pipe flows is equal to 128, 160, and 64 for pipes with rectangular, equilateral triangular and elliptic cross-sections, respectively (Table 5.1). The friction factor for laminar flows in concentric and eccentric annuli can be also presented in the form of (5.22). In these cases the factor c depends on the ratio of the effective diameters of the external and internal pipes, as well as on the value of their eccentricity (see the Problems at the end of this Chapter).
5.4
Microchannel Flows
A detailed data on flows in microchannels with circular, rectangular, triangular and trapezoidal cross-sections with hydraulic diameters in the range from 105 m to 103 m is available in a number of monographs (Incropera 1999; Celata 2000; Kakas et al. 2005; Yarin et al. 2009) and review articles (Gad-el-Hak 1999; Garimella and Sobhan 2003; Hetsroni et al. 2005a, b). In spite of the existence of the numerous experimental and theoretical studies devoted to hydrodynamics and heat transfer in laminar flows in microchannels, there is no consensus, in particular, between the data on the friction factor of rough pipes and ducts, the effect of energy dissipation, thermal conductivity of the wall and fluid, etc. Such situation complicates understanding of the physical phenomena occurring in flows in microchannels and sometimes leads to questionable ‘discoveries’ of specific ‘micro-effects’ (Duncan and Peterson, 1994; Ho and Tai 1988; Plam 2000; Hezwig 2002; Hezwig and Hausner 2003; Gad-el-Hak 2003). The shaky foundations of a number of such results were revealed by comparison of the experimental data with predictions of the conventional theory based on the Navier–Stokes equations. In the framework of this theory, solutions corresponding to developed laminar flow in straight microcannels involve a number of assumptions on flow conditions. They are as follows: (1) the flow is driven by a static pressure drop in the fluid, (2) the flow is stationary and fully developed, i.e. strictly axial, (3) the flow is laminar, (4) the Knudsen number is small enough so that the fluid can be considered a continuum, (5) there is no slip at the wall, (6) the fluid is incompressible Newtonian with constant viscosity, (7) there is no heat transfer to (from) the ambient medium, (8) the energy dissipation is negligible, (9) there is no fluid-wall interaction (except purely viscous, so the electric, dispersive London and van der Waals forces are neglected), (10) the micro-channel walls are smooth. Under these conditions problems on flows in microchannels reduces to integrating (5.7) with no-slip conditions over the wetted perimeter of the channel. The corresponding solutions lead to the following results: the Poiseuille number Po is a constant, which is determined by the micro-channel shape. The discrepancy between this result and
5.5 Non-Newtonian Flows
113
the experimental data on microchannel resistance to flow are interpreted sometimes as a manifestation of some new effects inherent to microchannel flows. The puzzling situation was considered in Hetsroni et al. (2005a) where the influence of different factors that affect hydrodynamics in microchannels (e.g. the change of the physical properties of fluid, the energy dissipation, etc.) was evaluated. In addition, the conformity of the actual experimental conditions with those assumed in the theoretical description of micro-channel flows was considered. It was shown that only in certain cases the experimental conditions were consistent with the theoretical assumptions. The experimental results corresponding to these cases agree fairly well with the theory. In particular, in single-phase fluid flow in smooth micro-channels of hydraulic diameter in the range from 15 mm to 4010 mm the Poiseuille number is independent on the Reynolds number and equals 64. In singlephase gas flows in micro-channels of hydraulic diameter in the range from 10.1 mm to 4010 mm and the Knudsen number 0:001 Kn 0:38; the friction factor agrees fairly well with the theoretical predictions for fully developed laminar flows (Yarin et al. 2009). Therefore, it was concluded that the experiments on flows in smooth micro-channels show no difference with macro-scale flow and no new physical effects should be invoked. From the point of view of the dimensional that means that the approach which is used for study of friction factors in macroscopic channels can be equally used for micro-scale channels. In both cases the problem consists in choosing as the governing parameters of fluid viscosity, mean velocity and macroor micro-channels diameter followed by transformation of the functional equation for pressure drop to dimensionless form using the Pi-theorem. It is emphasized that this statement refers to the application of the Pi-theorem to flows in smooth microchannels only. A number of experiments show that the Poiseuille number in rough micro-tubes exceeds significantly the one in smooth micro-tubes (Qu et al. 2000; Pfund et al. 2000; Li et al. 2003; Bahrami et al. 2006; Wang and Wang 2007; Li et al. 2007). The latter shows that the set of the governing parameters responsible for the friction factor of rough micro-channels has to include roughness of the wall.
5.5
Non-Newtonian Flows
Non-Newtonian fluids are those for which the rheological constitutive equation differs from the Newton-Stokes one. In particular, it means that such fluids can possess a yield stress or/and in simple shear flow the dependence of shear stress on shear rate becomes nonlinear (i.e. the shear viscosity of such fluids is not constant at given pressure and temperature as Fig. 5.4 shows). The rheological constitutive equations of non-Newtonian fluids are more complex than that of Newtonian ones. They can contain a number of parameters that account for such specific features of non-Newtonian fluids as consistency index, the exponent, yields stress which should be exceeded before flow starts, and viscoelastic relaxation time (Wilkinson and Chen 1960; Astarita and Marrucci 1974; Bird et al. 1977).
114
5 Laminar Flows in Channels and Pipes
Fig. 5.4 Flow curves for time-independent nonNewtonian fluids. 1-Bingham plastic, 2-pseudoplastic fluid, 3-Newtonian viscous fluid, 4 dilatant fluid
As an example that demonstrate the application of the Pi-theorem in hydrodynamics of non-Newtonian fluids, we consider laminar flow in a straight cylindrical pipe of Bingham fluid possessing yield stress. Our consideration will be restricted to the analysis of fully developed flows in the two cases corresponding to: (1) a given volumetric flow rate (i.e. a given mean velocity in the pipe), and (2) a given pressure drop along the pipe. In the first case the problem consists in finding the friction factor, whereas in the second one in finding the volumetric flow rate. In both cases we use the Pi-theorem to eatablish dependences of the unknown characteristics on flow and geometrical parameters. The shear behavior of Bingham fluids is given by the following equation (Wilkinson and Chen 1960) t t0 ¼ m0 g_ ;
t>t0
(5.26)
where t is the shear stress, t0 is the yield stress, m0 is the viscosity, and g_ is the rate of shear. We assume that flow of Bingham fluid in a pipe is dominated by shear near the walls and thus consider (5.26) to be an adequate representative of the overall (tensorial) rheological constitutive equation of the fluid. It is seen that (5.26) contains two characteristic parameters: t0 and m0 which have the following dimensions ½t0 ¼ L1 MT 2 ;
½m ¼ L1 MT 1
(5.27)
The pressure drop in flows of Bingham fluids in straight cylindrical pipes should depend on the mean velocity u and pipe diameter d in addition to the rheological parameters. Then, the functional equation for the pressure drop DP=l reads
5.5 Non-Newtonian Flows
115
DP ¼ f ðm0 ; t0 ; u; dÞ l
(5.28)
Three governing parameters in (5.28) have independent dimensions, so that the difference n k ¼ 1: Then, in accordance with the Pi-theorem (5.28) reduces to the following one P ¼ ’ðPÞ a
(5.29) 0
0
0
where P ¼ ðDP=lÞ=ma01 ua2 d a3 ; and P ¼ t0 =m01 ua2 da3 . Bearing in mind that ½DP=l ¼ L2 MT 2 and ½u ¼ LT 1 , we find that 0 0 0 the exponents ai and ai are as follows: a1 ¼ 1; a2 ¼ 1; a3 ¼ 2; a1 ¼ 1; a2 ¼ 1; 0 and a3 ¼ 1: In accordance with that, (5.29) takes the form f¼
c ’ ðPÞ Re
(5.30)
where f ¼ 2ðDP=lÞd=ru2 is the friction factor, c ’ ¼ 2’, c is dimensionless constant equels 64 for flow in round pipe, and P ¼ t0 d=m0 u is the dimensionless yield stress parameter called the Bingham number. The dependences of the friction factor on the Reynolds number in (5.30) shows that friction factor f is a function of two dimensionless groups, namely, the Reynolds number Re and the Bingham number P. This dependence is shown in Fig. 5.5 in log-log coordinates for different values of the Bingham number P. For the lowest line P ¼ 0, which corresponds to Newtonian fluid. The kink on the lowest curve corresponds to laminar-turbulent transition. In the second case when the pressure drop DP=l is given, the volumetric flow rate depends on the parameters m0; t0 and r0 (r0 is the radius of pipe)
1000 Π
Friction factor
100
Fig. 5.5 Friction factor versus Reynolds number for Bingham plastic fluids in laminar regime
104
10
5.103 1.0 Rough
0.1
0.01
Smooth 0.1
1.0
10
100 .104
116
5 Laminar Flows in Channels and Pipes
DP Q ¼ f ð ; m0 ; t0 ; r0 Þ l
(5.31)
Three governing parameters in (5.31) have independent dimensions. Then, according to the Pi-theorem, (5.31) transforms to the following one P ¼ ’ðP1 Þ
(5.32)
where P ¼ Q= ðDP=lÞa1 mao2 r0a3 ; P1 ¼ t0 =fðDP=lÞ m0 r0 g::. Bearing in mind the dimensions of Q; r0 ; mo ; DP=l and t0 , we find the exponents 0 0 0 0 ai and ai : They are as follows: a1 ¼ 1; a2 ¼ 1; a3 ¼ 4; a1 ¼ 1; a2 ¼ 0; a3 ¼ 1: Then (5.32) takes the form 0 a1
0 a2
0 a3
DP ro4 to l ’ Q¼ l m0 ro DP
(5.33)
The analytical solution of this problem leads to Buckingham’s equation (Wilkinson and Chen 1960) p DP r04 Q¼ 8 l m0
5.6
(
) 4 2lt0 1 2lt0 4 þ 1 3 r0 DP 3 r0 DP
(5.34)
Flows in Curved Pipes
Flows in curved toroidal pipes demonstrate a number of characteristic features, which result from the action of the centrifugal forces. The centrifugal-force– induced pressure gradient generates the secondary flow in the pipe cross-section. In the general case flows in curved pipes depend on the inertia, centrifugal and viscous friction forces, which, in its turn, are determined by the pipe geometry, as well as the physical properties of fluid. At a given pressure gradient the local velocity in a curved pipe depends on the fluid density and viscosity, radius of the pipe and its curvature, as well as the coordinates of a given point. The mean flow characteristics, in particular, the pressure drop DP=l, are determined by the physical properties of fluid, pipe geometry and mean velocity w0 DP ¼ f ðr; m; r0 ; R; w0 Þ l
(5.35)
where r0 and R are the radius of the pipe cross-section and the radius of curvature. Equation (5.35) contains five governing parameters, whereas (5.15) corresponding to flows in straight pipes only three of them: m; r0 ; and u0 : The additional parameters
5.6 Flows in Curved Pipes
117
r and R are included in the set of the governing parameters in (5.35) in order to account for the effect of the centrifugal acceleration of fluid in a curved pipe and actual pipe geometry. The governing parameters in (5.35) have the following dimensions ½r ¼ L3 M; ½m ¼ L1 MT 1 ; ½r0 ¼ L; ½ R ¼ L; ½w0 ¼ LT 1
(5.36)
It is seen that in the present case the difference n k ¼ 2, and (5.35) reduces to the following form P ¼ f ðP1 ; P2 Þ 0
0 a2
0 a3
(5.37) 00
00 a2
00 a3
where P ¼ ðDP=lÞ=ra1 wa02 r0a3 ; P1 ¼ R=ra1 w0 r0 ; and P2 ¼ m=ra1 w0 r0 . 0 00 In addition, the exponents ai ; ai and ai are found from the principle of dimen0 0 0 00 sional homogeneity as a1 ¼ 1; a2 ¼ 2; a3 ¼ 1; a1 ¼ 0; a2 ¼ 0; a3 ¼ 1; a1 ¼ 1; 00 00 a2 ¼ 1; a3 ¼ 1 Then, (5.37) takes the form l ¼ ’ðd; ReÞ
(5.38)
where l ¼ 2ðDP=lÞð2r0 Þ=rw20 is the friction factor, Re ¼ w0 r0 =n is the Reynolds number, and d ¼ r0 =R is the ratio of the cross-sectional radius of pipe to its radius of curvature. The above result is obviously very weak. Indeed, in writing the functional equation (5.35), we implicitly assumed that flows in curved pipes are determined by the radius of curvature R. Under this assumption, (5.35) is identical to the functional equation for flows in straight irregular ducts, in particular, to the one for rectangular channels where the friction factor depends on two dimensionless groups: (1) the Reynolds number, and (2) the ratio of channel width to its depth. In contrast with the flow in a straight pipe, when studying flows in curved pipes it is necessary to account for a number of different factors that affect such flows through the interaction of the inertial, centrifugal and viscous forces. Accordingly, the functional equation for the pressure drop in a curved pipe as a dependence of ðDP=lÞ on the acting forces reads
DP l
¼ f ðfi ; fc ; ff Þ
(5.39)
where fi ; fc and f f are the specific inertial, centrifugal and viscous forces, respectively, with fj ¼ L2 MT 2 and subscript j ¼ i; c; f . All the governing parameters in (5.39) have independent dimensions. Then, according to the Pi-theorem, this equation takes the form
DP l
¼ cfia1 fca2 ffa3
(5.40)
118
5 Laminar Flows in Channels and Pipes
or l ¼ c1 fia1 fca2 ffa3
(5.41)
where l ¼ 2ðDP=lÞfi1 is the friction factor, ci ¼ 2c; and a1 ¼ a1 1. Since the dimension of the friction factor ½l ¼ 1; the dimension of the right hand side of (5.41) has to be equal 1. There is a number of different combinations of fi ; fc and ff that are dimensionless. However, only one of them, namely, 1=2 1=2 fi fc =ff has a clear physical meaning. It can be interpreted as the ratio of the inertial forces to the viscous ones, i.e. as the natural analog of the Reynolds number for flows in curved pipes. Then, we can present (5.41) in the following form l ¼ c1
1=2 1=2 fc
fi
!n
ff
(5.42)
with n being a dimensionless constant. Assuming that all the velocity components are proportional to the mean velocity w0 , which is defined as the ratio of the flow rate to the cross-sectional area of a curved pipe, we can estimate fi ; fc and ff as follows fi
rw20 rw2 w0 ; fc 0 ; ff m 2 r0 R r0
(5.43)
Substituting (5.43) into (5.42), one finds the following equation for the friction factor l ¼ ’ðkÞ
(5.44)
where k ¼ d1=2 Re is the Dean number, and ’ðkÞ is the function of the Dean number. Thus, we arrive at a very impotent result: the friction factor of a slightly curved pipe is determined by a single dimensionless group-the Dean number. Naturally, this result can be obtained directly by transforming the Navier–Stokes equations to the dimensionless form that contain two dimensionless groups, d and Re (Berger et al. 1983). A number of dimensionless groups that characterize flows in curved pipes can be reduced to a single one in the particular case of a flow in a slightly curved pipe, i.e. d<<1 (Dean 1927, 1928). When the order of magnitude of the inertial, centrifugal and viscous forces is the same, the system of dimensionless equations for the fully developed flows in slightly curved pipes takes the form. @u u 1 @v þ þ ¼0 @r r r @a
(5.45)
5.6 Flows in Curved Pipes
119
@u v @u v2 @P1 2 1 @ @v v 1 @u 2 u þ w cos a ¼ þ @r @r r @a r k r @a @r r r @r u
(5.46)
@v v uv 1 @P1 2 @ @v v @u þ þ þ w2 sin a ¼ þ þ @r r r r @a k @r @r r @a u
@w v @w @P0 2 þ þ ¼ @z @r r @a k
0
(5.47)
@ 1 @w 1 @ 2 w þ þ 2 @r r @r r @a2
(5.48)
0
in the toroidal u ¼ u =w0 ; v ¼ v0 =w0 ; w ¼ w =w0 are the velocity components 0 coordinate system r ; a; y (cf. Fig. 5.6), P ¼ P rw20 is the pressure 0 0 0 ½P ¼ PðzÞ þ Pðr; z; aÞ, r ¼ r =r0 ; s ¼ s =r0 ¼ Ry=r0 ¼ d1=2 z, r denotes the distance from the center of circular pipe cross-section in its plane, a is the angle 0 between the radius vector r and the plane of symmetry, y is the angular distance of the cross-section from the pipe entrance. Therefore, according to (5.45–5.48) the velocity components and pressure will depend on the sole dimensionless group- the Dean number. As a result, all the integral characteristics of flow, in particular, the axial pressure drop are also functions of the Dean number. The friction factor of a curved pipe lc can be expressed at small values of the Dean number k as (White 1929) l ¼ 1 þ 0:00306
K 576
2
þ 0:0110
K 576
4 þ :::
(5.49)
where l ¼ lc =ls ; with lc and ls being the friction factors of the flow in curved and the corresponding straight pipe, respectively, and K ¼ 2dRe2 . At large values of the Dean number the asymptotic expressions for the ration of the friction factors l are given by (Adler 1934; Ito 1959; Barua 1963; Mori and Nakayama 1965) l 0:1k1=2
(5.50)
z
R v′ r′
0
Fig. 5.6 Pipe cross-section and the toroidal coordinate system
θ
α
w′
u′ P(r′, a, q)
r0 S¢ = Rq
120
5 Laminar Flows in Channels and Pipes
and (Van Dyke 1978) l ¼ 0:4713k1=2
(5.51)
The correlation (5.49) agrees fairly well with the measurements of White (1929), Adler (1934) and Ito (1959) at relatively small values of the Dean number ðk<102 Þ and small values of d ðd<103 Þ. At a fixed d, an increase in the Dean number (i.e. an increase in Re) results in a significant disagreement of the theoretical predictions with experiment. The asymptotic formula of Van Dyke (1978) corresponding to very large Dean numbers (k ! 1) agrees well with the experimental data at d<4 103 and k<102 : In all cases the disagreement of the theoretical and experimental results stems from the change in the flow structure as the Reynolds number increases. In flows in helical pipes (Fig. 5.7) the set of the governing parameters is supplemented by an additional dimensional parameter, the geometric torsion ½t ¼ L1 : Accordingly, the flow in helical pipes is determined by two dimensionless groups (Germano 1989) K ¼ 2dRe2 ; T ¼
t=r Re
(5.52)
where r ¼ R=ðR2 þ l2 Þ is the modified curvature, and t ¼ l=ðR2 þ l2 Þ is the modified torsion.
5.7
Unsteady Flows in Straight Pipes
Consider axisymmetric flows of incompressible viscous fluids in straight cylindrical pipes driven by given pressure gradient DP=l ¼ const, which is imposed instantaneously on fluid at rest at t ¼ 0 (the startup flow). The reduced form of the Navier–Stokes equations, which corresponds to this flow reads (Loitsyanskii 1966) 2 @u @ u 1 @u 1 DP þ ¼ n @t @r 2 r @r r l
(5.53)
The boundary and initial conditions corresponding in this case are u ¼ 0 at r ¼ r0 ; u ¼ 0 at t ¼ 0
(5.54)
Equation (5.53) subjected to the conditions (5.54) shows that the velocity at any point of pipe cross-section depends on six dimensional parameters. Four of the are the given constants, namely, DP=l; r; m and r0 , whereas the two others are
5.7 Unsteady Flows in Straight Pipes
121
Fig. 5.7 Helical pipe with the modified curvature R=ðR2 þ l2 Þ and torsion t ¼ l=ðR2 þ l2 Þ
variables r and t (the radial coordinate in the cross-section and time). Accordingly, the mean flow characteristics, in particular, the volumetric flow rate, depend on five dimensional parameters (without r). Therefore, we can write the following functional equations for the volumetric rate ½Q ¼ L3 T 1 and local velocity ½u ¼ LT 1 DP Q ¼ f ð ; m; r0 ; r; tÞ l
(5.55)
DP u ¼ f1 ð ; m; r0 ; r; r; tÞ l
(5.56)
The governing parameters in (5.55) and (5.56) have the following dimensions DP ¼ L2 MT 2 ; l
½m ¼ L1 MT 1 ;
½r0 ¼ L;
½r ¼ L3 M; ½t ¼ T (5.57)
122
5 Laminar Flows in Channels and Pipes
First we consider (5.55). It contains five governing parameters, three of them have independent dimensions. Therefore, the difference n k ¼ 2: In this case (5.55) reduces to the following dimensionless equation P ¼ ’ðP1 ; P2 Þ
(5.58)
with the dimensionless groups being given by the following expressions: h i 0 00 0 a0 00 00 a1 a2 a3
P ¼ Q= ðDP=lÞ m r0a ; P1 ¼ r= ðDP=lÞa1 ma2 ro 2 ; P2 ¼ t= ðDP=lÞa1 ma2 r0 : Bearing in mind the dimensions of the volumetric flow rate and the governing 0 00 parameters, we find the values of the exponents ai ; ai and ai as 0
0
0
00
00
a1 ¼ 1; a2 ¼ 1; a3 ¼ 4; a1 ¼ 1; a2 ¼ 2; a3 ¼ 3; a1 ¼ 1; a2 00
¼ 1; a3 ¼ 1
(5.59)
Then, the expressions for the dimensionless groups P; P1 and P2 become P¼
Qm ; ðDP=lÞr04
P1 ¼
rðDP=lÞr03 tðDP=lÞr0 ¼ c1 ; P2 ¼ ¼ c1 Fo m2 m
(5.60)
For given conditions (the physical properties of fluid, the pipe radius and the pressure gradient DP=lÞ the dimensionless group P1 ¼ c1 is a constant (½c1 ¼ 1) whereas the dimensionless group P2 ¼ c1 Fo where Fo ¼ tn=r02 is kindred to the Fourier number, with nbeing kinematic viscosity. Accordingly, (5.58) takes the form 4 DP r0 Q¼ ’ðFoÞ L m
(5.61)
In order to compare the expression (5.61) with the known analytical solution corresponding to this case, recast (5.58) as follows Q¼
DP r04 ½1 cðFoÞ m l
(5.62)
where cðFoÞ ! 0as Fo ! 1, which means the asymptotic approach to the Poiseuille law, as expected on the physical grounds. The exact analytical solution of the problem is available and yields (Loitsyanskii 1966) # " 1 X p DP r04 expðl2k FoÞ 1 32 Q¼ 8 l m l4k k¼1
(5.63)
Problems
123
where lk are the roots of the equation J0 ðlk Þ ¼ 0; with J0 being the Bessel function of the zero order. It is seen that the structure of (5.62) resembles that of (5.63). In both cases the factors on the right hand side of these equations correspond to the fully developed laminar flow in a straight pipe, whereas the factor in the parentheses approaches one at large Fo as expected when transient effects fade. Applying the Pi-theorem to (5.56), we arrive at P ¼ ’ðP1 ; P2 ; P3 Þ
(5.64)
where P ¼ u= ðDP=lÞr04 =m ; P1 ¼ r=r0 ; P2 ¼ c1 ; with ½c1 ¼ 1 being a constant. Then, we obtain the following expression for the velocity profile u 2 DP r0 (5.65) u¼ ’ðr; FoÞ l m where r ¼ r=r0 : For comparison, the analytical solution for u is (Loitsyanskii 1966) ) 2 ( 1 X DP r0 J0 ðlk rÞ 2 expðlk FoÞ 1 ðrÞ 8 u¼ l 4m J1 ðlk Þ k¼1
(5.66)
where J1 is the Bessel function of the first order.
Problems P.5.1. Determine the dependence of the friction factor of the concentric and eccentric annuli with fully developed stationary flows on the governing parameters. A concentric annulus (Fig. 5.8) is characterized by two geometric parameters, namely: (1) the internal diameter d1 and (2) the external diameter d2 . Then, the functional equation for the pressure gradient becomes DP ¼ f ðm; u ; d1 ; d2 Þ l
(P.5.1)
where u is the mean velocity. Three of the four governing parameters in (P.5.1) have independent dimensions. Then, according to the Pi-theorem, (P.5.1) reduces to P ¼ ’ðP1 Þ
(P.5.2) 0
a
0
a
0
where P ¼ ðDP=lÞ=ma1 ua2 d2a3 ; and P1 ¼ d1 =ma1 u2 d2 3 :
124
5 Laminar Flows in Channels and Pipes
Fig. 5.8 Concentric (a) and eccentric (b) annuli
a
d1
d2
b
d1
d2
e
Taking into account the dimensions of the parameters involved, ½DP=l ¼ L2 MT 2 ; ½m ¼ L1 MT 1 ; ½u ¼ LT 1 ; ½d1 ¼ L; and ½d2 ¼ L, we find the 0 0 0 values of the exponents ai and ai as a1 ¼ 1; a2 ¼ 1; a3 ¼ 2; a1 ¼ 0; a2 ¼ 0; 0 and a3 ¼ 1: Then, (P.5.2) takes the following form l¼
1 d1 ’ d2 Re
(P.5.3)
where l ¼ 2ðDP=lÞd2 =ru2 ; and Re ¼ u d2 =n: It is emphasized that the equivalent diameter de defined as de ¼
4S ¼ d2 d1 P
(P.5.4)
where S is the cross-sectional area, and P the wetted perimeter, can be used as one of the governing parameters for flows in concentric annuli. In the latter case the form of the dependence of the friction factor on the two dimensionless groups involved retains the same form as (P.5.3), albeit the Reynolds number Re should be replaced by Ree based on the equivalent diameter de : The analytical solution of this problem can be found by integrating the Navier– Stokes equations, which in the present case reduce to
Problems
125
dP m d du ¼ r dx r dr dr
(P.5.5)
subjected to the no-slip boundary conditions u ¼ 0 at r ¼ r1 ; u ¼ 0 at r ¼ r2
(P.5.6)
The solution leads to the following expression for the friction factor (WardSmith 1980) l¼
64 d1 ’ d2 Ree
(P.5.7)
where
h i n o ’ðd1 =d2 Þ ¼ ½1 d1 =d2 2 lnðd1 =d2 Þ= 1 ðd1 =d2 Þ2 þ 1 þ ðd1 =d2 Þ2 lnðd1 =d2 Þ . In the case of a fully developed stationary flow in a straight eccentric annuls (Fig. 5.8 b) the functional equation for the dependence of the pressure gradient on flow and geometric parameters reads DP ¼ f1 ðm; u ; d1 ; de ; eÞ l
(P.5.8)
where e is eccentricity (Fig. 5.8b). Applying the Pi-theorem to (P.5.8) we arrive at the following expression for the friction factor 1 r1 e ’ ; l¼ Ree 1 r2 re
(P.5.9)
where r1 and r2 are the radii involved, and re is the equivalent radius equal to r 2 r1 : The analytical solution describing flows in eccentric annuli is readily available for comparison (Ward-Smith 1980) l¼
64 r1 e ’ ; r2 re Ree
(P.5.10)
In the limit r1 =r2 ! 1 the function ’ on the right hand side of (P.5.10) takes the form ’1 ¼ fð2=3Þ½1 þ ð3=2Þðe=re Þg2 . P.5.2. Determine the velocity profile in the cross-section of a cylindrical pipe with fully developed laminar flow of Newtonian fluid and a given pressure gradient. Also, determine the relation between the volumetric flow rate and fluid viscosity, pressure drop and pipe radius.
126
5 Laminar Flows in Channels and Pipes
At a given pressure gradient DP=l, the functional equation which determines the velocity profile in the cross-section of a cylindrical pipe reads DP ; r; r0 u ¼ f m; l
(P.5.11)
where u is the longitudinal velocity component corresponding to the radial coordinate r, r0 is the pipe radius. Applying the Pi-theorem to (P.5.11), we arrive at the following dimensionless equation P ¼ ’ðP1 Þ
(P.5.12)
0 0
0 a where P ¼ u ma1 ðDP=lÞa2 r0a3 ; and P1 ¼ r=fma1 ðDP=l:Þa2 r0 3 g:. Taking into account the dimensions of ½u ¼ LT 1 ; ½m ¼ L1 MT 1 ; ½DP=l ¼ 2 L MT 2 ; ½r ¼ L and ½r0 ¼ L, we arrive at the equations 0 0 0 a1 2a2 þ a3 1 ¼ 0; a1 2a2 þ a3 1 ¼ 0 0
0
a1 þ a2 ¼ 0; a1 þ a2 ¼ 0 0
(P.5.13) 0
a1 2a2 þ 1 ¼ 0; a1 2a2 ¼ 0 From (P.5.13) it follows that a1 ¼ 1;
a2 ¼ 1;
0
a3 ¼ 2;
a1 ¼ 0;
0
a2 ¼ 0;
0
a3 ¼ 1
(P.5.14)
Thus, (P.5.12) takes the following form u r ¼’ 2 r0 DPr0 =lm
(P.5.15)
For the axial (maximum) velocity u ¼ um at r ¼ 0, (P.5.11) yields the following functional equation
DP um ¼ f m; ; r0 l
(P.5.16)
Since all the governing parameters in (P.5.16) possess independent dimensions, (P.5.16) takes the form um ¼ c where c is a constant.
2 DP r0 m l
(P.5.17)
Problems
127
Comparing (P.5.15) with (P.5.17), we obtain u r ¼ ’ um r0
(P.5.18)
where ’ ðr=r0 Þ ¼ c’ðr=r0 Þ: Calculating the volumetric flow rate using the expression for u rð0
Q ¼ 2p urdr
(P.5.19)
0
we arrive at the following formula Q ¼ 2pc
4 ð1 DP r0 ’ðrÞrdr m l
(P.5.20)
0
where r ¼ r=r0 : The comparison of (P.5.20) with the exact analytical solution of the present Ð1 problem shows that c ’ðrÞrdr ¼ 1=16: 0
It is emphasized that the form of the dependence Q on the DP=L; m and r can be directly revealed by applying the Pi-theorem to the functional equation DP Q¼f ; m; r0 (P.5.21) l Bearing in mind the dimensions of the volumetric flow rate, pressure drop, fluid viscosity and pipe radius, we arrive at the following expression a2 a1 DP ma 3 (P.5.22) Q ¼ cr0 l where the factor c is a constant and a1 ¼ 4; a2 ¼ 1; and a3 ¼ 1. Thus, for the volumetric flow rate we have the expression DP 1 Q ¼ cr04 l m
(P.5.23)
The exact analytical solution of the present problem yields a numerical value for the constant c: It is equal to p=8: P.5.3. Determine the dependence of the friction factor on the Reynolds number for fully developed flows of viscous incompressible fluid in rough cylindrical pipes. Let the characteristic sizes of the pipe and its roughness are d and ks ; the mean velocity of the fluid u; and fluid density and viscosity r and m; respectively. Then the functional equation for the pressure gradient is
128
5 Laminar Flows in Channels and Pipes
DP ¼ f ðd; k; r; m; uÞ l
(P.5.24)
The set of the governing parameters in (P.5.24) includes three parameters that have independent dimensions. Then, in accordance with the Pi-theorem, (P.5.24) can be transformed to the following dimensionless form l ¼ ’1
1 ;k Re
(P.5.25)
where l ¼ 2ðDP=lÞd=ru2 is the friction factor, Re ¼ ud=n is the Reynolds number, and k ¼ k=d is the relative roughness. In the case of fully developed laminar flows the number of the governing parameters reduces to four (d; k; m; and u), since density of the fluid does not affect such flows. Then, (P.5.25) takes the following form DP ¼ f ðd; k; m; uÞ l
(P.5.26)
Applying the Pi-theorem to (P.5.26), we obtain l¼
1 ’2 k Re
(P.5.27)
In the fully developed (in average) turbulent flows the effect of molecular viscosity is negligible, and m can be excluded from the set of governing parameters. Accordingly, the functional equation for the pressure gradient reads DP ¼ f ðd; k; r; uÞ l
(P.5.28)
The dimensionless form of (P.5.28) becomes l ¼ ’3 ðkÞ
(P.5.29)
In connection with the above results, it is necessary to add the following remarks. An explicit form of the dependences ’i ðkÞ cannot be revealed in the framework of the dimensional analysis. To determine the dependences ’i ðkÞ, it is necessary to recall some additional physical considerations or the experimental data. The data of Nikuradse (1930) and Schiller (1923) show that the friction factor of the conventional macroscopic rough pipes does not depend on k in fully developed laminar flows and corresponds to the one following from the Poiseuille law. Thus, the function ’2 ðkÞ in (P.5.27) can be assumed to be constant equal to 64. In the fully developed (in average) turbulent flow (at high values of Re) the friction factor does not depend on Re and is fully determined by relative roughness.
References
129
Therefore, the dependence of l on Re for rough conventional pipes can be selected separately for three characteristic flow regimes corresponding to laminar ðReRecr2 Þ flows, for which (P.5.25), (P.5.27) and (P.5.28) are valid, respectively. For micro-channels ð10 d 103 mmÞ the recent measurements show that the friction factor corresponding to fully developed laminar flows depends significantly on the value of relative roughness (Yarin et al. 2009).
References Adler M (1934) Stromung in gekruiimmeten. Rohren Z Angew Math 14:257–275 Astarita G, Marrucci G (1974) Principies of non-newtonian fluid mechanics. McGraw-Hill, New York Bahrami M, Yovanovich MM, Culham JR (2006) Pressure drop of fully developed laminar flow in rough microtybes. J Fluids Eng Trans ASME 128:632–637 Barua SN (1963) On secondary flow in stationary curved pipes. QJ Mech Appl Math 16:61–77 Berger SA, Tabol L, Yao L-S (1983) Flow in curved pipes. Annu Rev Fluid Mech 15:461–512 Bird RB, Armstrong RC, Hassager O (1977) Dynamics of polymeric liquids. In: Fluid mechanics, vol 1. Wiley&Sons, New York Celata GP (2000) Heat transfer and fluid flow in microchannels. Begell Hause, New York Dean WR (1927) Note on the motion of fluid in a curved pipe. Philos Mag 20:208–223 Dean WR (1928) The streamline motion of fluid in a curved pipe. Philos Mag 30:673–693 Dunkan AB, Peterson GP (1994) Review of micro-scale heat transfer. App Mech 47:397–428 Van Dyke M (1978) Extended Stokes series: Laminar flow through a loosely coiled pipe. J Fluid Mech 36:129–145 Emery AE, Chen CS (1968) An experimental investigation of possible methods to reduce laminar entry length. Trans ASME Ser D 90:134–137 Fargie D, Martin BW (1971) Developing laminar flow in a pipe of circular cross-section. Proc Roy Soc 321A:461–476 Friedmann M, Gilis J, Liron N (1968) Laminar flow in a pipe at low and moderate Reynolds numbers. App Sci Res 19:426–438 Gad-el-Hak M (1999) The fluid mechanics of micro-devices. The Freeman Scholar Lecture. J Fluid Eng 121:5–33 Gad-el-Hak M (2003) Comments or “critical” view on new results in micro-fluid mechanics. Int J Heat Mass Transf 46:3941–3945 Garimella S, Sobhan C (2003) Transport in microchannels: critical review. Annu Rev Heat Transf 13:1–50 Germano M (1989) The Dean equations extended to a helical pipe flow. J Fluid Mech 203:289–305 Hagen G (1839) Uber die Bewegung des Wassers in engen zylindrisghen Rohren. Pogg Ann 46:423–442 Hezwig H (2002) Flow and heat transfer in micro systems. Everything different or just smaller? ZAMM 82(9):579–586 Hezwig H, Hausner O (2003) Critical view on new results in micro-fluid mechanics: an example. Int J Heat Mass Transf 46:935–937 Hezwig H, Gloss D, Wenterodt T (2008) A new approach to understanding and modeling the influence of wall roughness on friction factors for pipe and channel flows. J Fluid Mech 613:35–53 Hetsroni H, Mosyak A, Pogrebnyak E, Yarin LP (2005a) Fluid flow in microchannels. Int J Heat Mass Transf 48:1982–1998
130
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Hetsroni G, Mosyak A, Pogrebnyak E, Yarin LP (2005b) Heat transfer in micro-channels: comparison of experiments with theory and numerical results. Int J Heat Mass Transf 48:5580–5601 Ho C-M, Tai Y-C (1988) Micro-electro-mechanical systems (MEMS) and fluid flows. Annu Rev Fluid Mech 30:579–612 Incropera FP (1999) Liquid cooling of electronic devices by single-phase convection. John Wiley&Sons, New York Ito H (1959) Friction factors for turbulent flow in curved pipes. Trans ASME J Basic Eng 81:123–134 Kakas S, Vasiliev LL, Bayazitoglu Y, Yener Y (2005) Micro-scale heat transfer. Springer, Berlin Kandlicar SG (2005) Roughness effects at microscale-reassessing Nikuradse’s experiments on liquid flow in rough tubes. B Pol Acad Sci Tech Sci 53:343–349 Landau LD, Lifshitz EM (1987) Fluid mechanics, 2nd edn. Pergamon, New York Li ZX, Du DX, Guo ZY (2003) Experimental study on flow characteristics of liquid in circular micro-tubes. Microscale Thermophys Eng 7:253–265 Li Z, He Y-L, Tang G-H, Tao W-Q (2007) Experimental and numerical studies of liquid flow and heat transfer in microtubes. Int J Heat Mass Transf 50:3442–3460 Loitsyanskii LG (1966) Mechanics of liquids and gases. Pergamon Press, Oxford Ma HB, Peterson GP (1997) Laminar friction factor in micro-scale ducts of irregular cross-section. Micro-scale Thermophys Eng 1:253–265 Moody LF (1948) Friction factors for pipe flow. Trans ASME 66:671–684 Mori Y, Nakayama W (1965) Study on forced convective heat transfer in curved pipes (1st Report, Laminar flow). Int J Heat Mass Transf 8:67–82 Nikuradse J (1930) Turbulente stromung in nicht kreisfozmigen rohren. Ing Arch 1:306–332 Nikuradse J (1933) Stromungsgesetze in rauhen. Rihren Forschg Arb Ing-Wes 361. Translated in NACA Memo. N1292, (1950) Pfund D, Rector D, Shekarriz A (2000) Pressure drop measurements in micro-channel. AIChE 46:1496–1507 Plam B (2000) Heat transfer in micro-channels. In: Heat transfer and transport phenomena in microscale. Banff Oct, pp 54–64 Poiseuille J (1840) Recherches experimentelles sur le mouvements des liquides dans les tubes de tres petits diameters. Comptes Rendus 11:961–967, 1041–1048 Qu W, Mala GM, Li D (2000) Pressure driven water flows in trapezoidal silicon micro-channels. Int J Heat Mass Transf 43:353–364 Schlichting H (1979) Boundary layer theory, 8th edn. Springer, Berlin Schiller L (1923) Uber den Stromungswiderrstand von Rohren verschiedenen Querschnitts-und Rauhigkeitsgrades. ZAMM 3:2–13 Shah RK, London AL (1978) Laminar flow forced convection in duct. Academic, New York Wang HL, Wang Y (2007) Flow in microchannels with rough walls: flow pattern and pressure drop. J Micromech Microeng 17:586–596 Ward-Smith AS (1980) Internal fluid flow (The fluid dynamics of flow in pipes and ducts). Clarendon, Oxford White CM (1929) Streamline flow through curved pipes. Proc R Soc London Ser 123A:645–663 White FM (2008) Viscous fluid flow, 7th edn. McGraw-Hill, New York Wilkinson WL, Chen AML (1960) Non-Newtonian fluids (Fluid mechanics, mixing and heat transfer). Pergamon Press, New York Yarin LP, Mosyak A, Hetsroni G (2009) Fluid flow, Heat transfer and boiling in micro-channels. Springer, Berlin
Chapter 6
Jet Flows
6.1
Introductory Remarks
The subject of the present chapter is the hydrodynamics of laminar submerged jets in the light of the dimensional analysis. Submerged jets are discussed in detail in special monographs devoted to the jet theory (Pai 1954; Abramovich 1963; Vulis and Kashkarov 1965), boundary layer theory (Schlichting 1979), as well as the theory of turbulent flows (Townsend 1956; Hinze 1959). Submerged jets belong to the vast class of laminar and turbulent shear flows. They originate from issuing of viscous fluid into an infinite or semi-infinite space filled with the same fluid. Jet flows are encountered in numerous engineering applications such as steam and gas turbines, ejectors, jet engines, industrial furnaces, pneumatic and ventilation systems, as well as in the phenomena typical for the environmental flows. A wide variety of flow configurations is characteristic of jet flows. There are, for example, mixing layers due to the interaction of two parallel streams moving with different velocities, or jets of viscous fluid issued from a nozzle or a slit into fluid which is moving or at rest. Wakes behind solid bodies moving in fluid or plumes due to the action of buoyancy forces also belong to the class of jet flows. Three main types of jet flows depending on conditions of their formation can be distinguished: (1) submerged jets, (2) wakes, (3) plumes and thermals (Fig. 6.1). The existence of convective motion that is predominantly directed along the flow axis, as well as an intensive transversal mass and momentum transfer due to strong shear in cross-section is typical for all kinds of jet flows. Submerged jets can be classified by a number of characteristic features: the flow regime (laminar, turbulent), geometry (plane, axisymmetric), mutual direction of the jet spreading and motion of the surrounding fluid (co- and counter-flows, a jet in crossflow), the interaction with the ambient fluid and solid walls (without any contact with walls, or the wall and impinging jets). A detailed sketch of a planar submerged jet of an incompressible fluid is shown in Fig. 6.2. It forms as a result of mixing of viscous fluid issued from a slit into the same fluid as rest. The horizontal lines a a subdivide the domains of jet issued from the nozzle and the ambient L.P. Yarin, The Pi-Theorem, Experimental Fluid Mechanics, DOI 10.1007/978-3-642-19565-5_6, # Springer-Verlag Berlin Heidelberg 2012
131
132
6
Jet Flows
Fig. 6.1 Jet flows. (a) Submerged jet: 1-constant velocity core, 2-the far field of the jet. (b) Wake behind a solid body 1. (c) Buoyant plume rising from a hot body 1
fluid at the initial cross-section x ¼ 0 where the jet originates. These are the lines of a discontinuity of the tangential velocity component. In viscous fluid such a discontinuity cannot be sustained. As a result of the interaction of the fluid in the jet with the ambient fluid, a thin mixing layer with a steep continuous velocity profile forms in the vicinity of the lines a a. The viscous stresses developed at the jet periphery lead to the expansion of the shear layers downstream. The inner boundaries din ðxÞ of the shear layers reach the jet axis at some distance xin from the nozzle exit encasing the core with a plug velocity profile which originated in the slit. Outside the core domain the velocity profile in the jet cross-section decreases from its value in the core and asymptotically approaches to zero at the outer boundaries of the shear layers. In the cross-section x ¼ xin ; the shear layers merge. At x>xin smooth velocity profiles with characteristic maximum at the axis are observed. The size and geometry of the slit or a nozzle, as well as the velocity distribution at its exit affect the flow field in submerged jets. The influence of the issuing
6.1 Introductory Remarks
133
Fig. 6.2 Sketch of a submerged jet. The inner and outer boundaries of the boundary layer are denoted as din and dout , respectively
conditions is significant only in a relatively short section of the jet adjacent to the slit or the nozzle. The length of this section is about 20–30 times the slit or nozzle cross-sectional size for plane and axisymmetric jets, whereas in three-dimensional (non-axisymmetric) jets it is significantly larger (Trentacoste and Sforzat 1967; Krothapalli et al. 1981; Ho and Gutmark 1987; Hussain and Hussain 1989). Therefore, at large x the flow field of submerged jets practically does not depend on the details of the issuing conditions and is fully determined by the integral characteristics of the jets. The latter express conservation of the momentum and energy fluxes along the jets. For a submerged jet issuing from a thin tube the total momentum flux is (Landau and Lifshitz 1987) þ Jx ¼ Prr cos yds
(6.1)
Q where rr ¼ P þ rvr 2 2m@vr =@r is the component of the tensor of the momentum flux in the r-direction, dS ¼ r2 sin ydyd’ is the surface element, r and P are the density and pressure, vr is the r-component of the velocity, m is the viscosity; r; y and ’ are the spherical coordinate frame with the center located at the jet axis at the tube exit. The total energy flux in the jet can be expressed as (Vulis and Kashkarov 1965) Q¼
þ @T ds rvr ðh h1 Þ k @r
(6.2)
134
6
Jet Flows
where h and T are the enthalpy and temperature, respectively, k is the thermal conductivity, subscript 1 corresponds to the undisturbed fluid far away from the jet axis. The submerged jets represent themselves boundary layers. Therefore, the boundary layer theory can be used to study hydrodynamics of submerged jets. For laminar planar submerged jets issuing from a slit, or a non-swirling (about the jet axis) axisymmetric jets issuing from a pipe, the boundary layers equations have the following form @u @u m @ @u þ rv ¼ j yj @x @y y @y @y
(6.3)
@h @h k @ j @T ¼ y ru þ rv @x @y yj @y @y
(6.4)
ru
Here (6.3) expresses the longitudinal momentum balance, (6.4)-the corresponding energy balance; u and v are the longitudinal and transversal velocity components, j ¼ 0 or 1 for the plane and axisymmetric jets, respectively; pressure drop is omitted in (6.3), since pressure in the jets is practically constant over the entire flow field. In (6.4) viscous dissipation is omitted, since it is negligible at sufficiently small flow velocities. The corresponding continuity equation reads @ruyj @rvyj þ ¼0 @x @y
(6.5)
Using (6.5), one can transform the momentum and energy balance Eqs. 6.3 and 6.4 to the following form @ru2 yj @ruvyj @ j @u þ ¼m y @x @y @y @y
(6.6)
@ruDhyj @rvDhyj @ @T þ ¼k yj @x @y @y @y
(6.7)
where Dh ¼ h h1 . The boundary conditions for planar and axisymmetric submerged jets read y ¼ 0:
@u @T @h ¼ ¼ ¼ 0; v ¼ 0 @y @y @y
y ! 1 : u ! 0; T ! T1 ; Dh ! 0
(6.8) (6.9)
6.1 Introductory Remarks
135
Integration of (6.6) and (6.7) across the jet cross-section yields the following invariants 1 ð
Jx ¼
ru2 yj dy ¼ const:
(6.10)
0 1 ð
Qx ¼
ruDhyj ¼ const:
(6.11)
0
It is emphasized that (6.10) expresses the fact that the total momentum flux Jx does not vary along the jet. Similarly, (6.11) expresses the fact that the total excessive enthalpy flux Qx does not vary along the jet. Note that the expressions for Jx and Qx for submerged turbulent jets are similar to those in (6.10) and (6.11). However, in that case r; u; T and h imply their average values. In addition, it 1 Ð 0 should be assumed that u02 þ P yj dy ¼ 0; where u is the velocity fluctuation. 0
Swirling motion about the jet axis results in a non-uniform pressure distribution across it. In this case the system of Eqs. 6.3 and 6.4 is supplemented by the following two equations describing the balance between the centrifugal force due to swirling and the corresponding pressure gradient across the jet and the azimuthal momentum balance allowing for finding the swirling velocity component w j
ru
w2 1 @P ¼ y r @y
@w @w vw 1 @ @u w 2 þ rv þr ¼m j yj @x @y y y @y @y y
(6.12)
(6.13)
In addition, the longitudinal momentum balance of Eq. 6.6 is now modified by the inclusion the longitudinal pressure gradient @P=@x on its right-hand side. Then, the following set of the integral invariants for the axisymmetric swirling jet is obtained via integration of the modified momentum balance equation, as well as (6.7), (6.12) and (6.13) 1 ð
Jx ¼
P þ ru2 ydy
(6.14)
0 1 ð
Mx ¼
ruwy2 dy 0
(6.15)
136
6
Jet Flows
1 ð
Qx ¼
ruDhydy
(6.16)
0
where Mx is the total moment-of-momentum flux. The integral invariants for different types of submerged jets are presented in Table 6.1. The integral invariants (6.1) and (6.10) express conservation of the momentum flux which is brought by the fluid issued from the jet origin. The relations (6.1) and (6.10) are not identical. The first one accounts for the full momentum flux around the momentum source, i.e. represent itself the momentum flux through a closed surface S (Fig. 6.3a), whereas the second one-only the momentum flux through a cross-section AA normal to the jet axis (Fig. 6.3b). The invariant (6.1) and the closed surface are used to solve the problem of the submerged jet in the framework of the full Navier–Stokes equations, whereas the invariant (6.10) and the crosssections of the type AA are used to solve the problem in the framework of the boundary layer theory. It is emphasized that in the framework of the boundary layer theory the integral invariant (6.10) does not account for the flow ‘history’ (Konsovinous 1978). (Schneider 1985) Nevertheless, the relation (6.10) is a fairly good approximation of the relation (6.1) and is widely used in the theory of jets.
6.2
The Far Field of Submerged Jets
The understanding that the far field of a jet flow does not depend on the issuing conditions and the related simplifications plays a pivotal role in the boundary layer theory of submerged jets. It allows one to decrease the number of factors which should be accounted for and to develop an asymptotic model of the flow. Namely, when a jet is assumed to be issued from a pointwise source, it is natural to assume that the far field of velocity in the jet is determined by physical properties of fluid and total momentum flux (but not by flow details at the nozzle/pipe exit). Therefore, we can write the functional equation for the longitudinal velocity component u as u ¼ fu ðr; m; Jx ; x; yÞ
(6.17)
The functional equation (6.17) contains the set of parameters which determine the velocity at any point of the far field of a submerged jet. Any change of the flow conditions (for example the appearance of a co- or counter flow of the surrounding fluid) requires an expansion of the set of the determining parameters on the right hand side of Eq. (6.17). In particular, when r and m are constant, the fluid properties are characterized by a single parameter, namely, the kinematic viscosity n ¼ m=r, as it follows from the boundary layer equations. In this case (6.17) reduces to
Wall jet
Radial jet
Axisymmetric jet
Plane jet
Jet geometry
Table 6.1 Jet invariants
Jet’s scheme
0
Jx ¼
0
1 Ð
ru2
0
y Ð
1
rudy dy
ru2 dy
ru2 ydy
þ1 Ð
1 Ð
Jx ¼ 2px
Jx ¼ 2p
0
Dynamic problem 1 Ð Jx ¼ ru2 dy
Integral invariant
QxPr¼1 ¼
þ1 Ð
0
1 ð
Qxad ¼ 0
1 ð
rucp ðT T1 Þdy
0
0y 1 ð rucp ðT T1 Þ@ rudyAdy
rucp ðT T1 Þdy
rucp ðT T1 Þdy
1
0
1 Ð
Qx ¼ 2px
Qx ¼ 2p
0
Thermal problem 1 Ð Qx ¼ rucp ðT T1 Þdy
6.2 The Far Field of Submerged Jets 137
138
6
Jet Flows
Fig. 6.3 Sketch of a jet flow. (a) Flow in a jet issued from a thin tube (the Navier–Stokes approximation). (b) Jet from a pointwise momentum source (the boundary layer approximation)
u ¼ fu ðn; Ix ; x; yÞ
(6.18)
where Ix ¼ Jx =r is the kinematic momentum flux. Similarly we write the functional equation for the jet thickness as d ¼ fd ðr; m; Jx ; xÞ
(6.19)
d ¼ fd ðn; Ix ; xÞ
(6.20)
or
Since at y ¼ 0; u ¼ um , with um being the axial maximal velocity in a jet crosssection, we obtain from (6.17) and (6.18) the following equations for the axial velocity in the planar and axisymmetric (round) jets um ¼ fu ðr; m; Jx ; xÞ
(6.21)
um ¼ fu ðn; Ix ; xÞ
(6.22)
or
In a cross-section of an axisymmetric jet it is possible to select two characteristic parameters: the jet (boundary layer) thickness and the axial velocity. These parameters can be taken as the governing ones. Then, we can assume that velocity at any point of the jet depends on um ðxÞ; d and the distance from the jet axis to the point under consideration y: Then, (6.18) is replaced with the following one u ¼ fum ðum ; d; yÞ
(6.23)
6.3 The Dimensionless Groups of Jet Flows
6.3
139
The Dimensionless Groups of Jet Flows
The dimensions of the governing parameters can be expressed in the framework of any system of fundamental units. For the dimensional analysis of jet flows it is convenient to use the modified LMT system that includes three different length scales Lx , Ly ; and Lz , respectively for the x, y and z directions. In this case the dimensions of the characteristic parameters are 1 1 ½u ¼ Lx T 1 ; ½v ¼ Ly T 1 ; ½ x ¼ Lx ; ½ y ¼ Ly ; ½r ¼ L1 x Ly Lz M; 1 1 2 1 ½m ¼ L1 x Ly Lz MT ; ½n ¼ Ly T ; ½d ¼ Ly
(6.24)
In addition, the dimension of the momentum flux is ½Jx ¼ Lex1 Ley2 Lez3 M e4 T e5
(6.25)
where ei 6¼ 0 (i ¼ 1, 2, 3, 4, 5) are constants which depend on the jet configuration (Table 6.2). Applying the Pi-theorem to (6.17)–(6.23), we can transform them to the following canonical form P ¼ ’j ðP1 ; P2 :::Pnk Þ
(6.26)
where n k is the number of dimensionless groups P1 ; P2 :::; whereas n and k are the number of the governing parameters and the parameters with independent dimensions, subscript j ¼ u; d refers to the velocity or thickness of the jet, respectively. Among the numerous cases corresponding to different values of n k, the most important ones are the following two: (i) n k ¼ 1; and n k ¼ 0: The former corresponds to a self-similar flow in which dimensionless velocity depends on a single dimensionless variable, whereas the latter – to a constant dimensionless velocity or jet thickness. In the latter case u and d can be expressed in the form of an explicit function of the governing parameters. As can be seen from (6.24) and (6.25), the parameters r; m; Jx and x on the right hand side in Eqs. (6.17), (6.19) and (6.21) have independent dimensions. Accordingly, we obtain the following equations for the dimensionless velocity and jet thickness Table 6.2 Exponents ei for different jets Jet geometry e1 Plane 1 Axisymmetric 2 Radial 2 Wall 1
e2 0 0 0 0
e3 1 1 1 2
e4 1 1 1 2
e5 2 2 2 3
140
6
Jet Flows
Pu ¼ ’ðP1 Þ
(6.27)
Pum ¼ c1
(6.28)
Pd ¼ c 2
(6.29)
where Pu ¼
u ra1 ma2 Jxa3 xa4 Pu;m ¼ Pd ¼
; P1 ¼
y 0 a1
0 a2
a
0
um ra1 ma2 Jxa3 xa4
0 a2
a
0
(6.30)
(6.31)
d 0 a1
0
r m J x 3 x a4
(6.32)
0
r m Jx 3 xa4
and c1 , c2 are constants. 0 The unknown exponents ai and ai (i ¼ 1; 2; 3; 4) are found from the expressions (6.30)–(6.32) accounting for the dimensions of the parameters on the left and right hand sides of these there. As a result, we arrive at the following sets of algebraic equations X
: a1 a2 þ e1 a3 þ a4 ¼ 1
Lx
X
: a1 þ a2 þ e2 a3 ¼ 0
Ly
X
: a1 a2 þ e3 a3 ¼ 0
Lz
X
: a1 þ a2 þ e4 a3 ¼ 0
M
X
: a2 þ e5 a3 ¼ 1
T
and X Lx
0
0
0
0
: a1 a2 þ e1 a3 þ a4 ¼ 0
(6.33)
6.4 Plane Laminar Submerged Jet
141
X
0
0
0
0
0
0
: a1 þ a2 þ e2 a3 ¼ 1
(6.34)
Ly
X
: a1 a2 þ e3 a3 ¼ 0
Lz
X
0
0
0
: a1 þ a2 þ e4 a3 ¼ 0
M
X
0
0
: a2 þ e5 a3 ¼ 0
T
where the symbols
P P P P P ; ; ; ; refer to summation of the exponents of Lx
LY
Lz
M
T
lengths and mass time scales, respectively. Bearing in mind (6.30)–(6.32), we present (6.27)–(6.29) as follows u ¼ ’u ðÞ
(6.35)
um ¼ c1 ra1 ma2 Jxa3 xa4
(6.36)
0
0
a
0
0
d ¼ c2 ra1 ma2 Jx 3 xa4
(6.37)
0
where u ¼ u=um ; ¼ y=d; and ai and ai depend on ei which is different for different jet geometry.
6.4
Plane Laminar Submerged Jet
Consider the application of the Pi-theorem to study characteristics of plane laminar submerged jets (Fig. 6.4). We deal with only the far-field region of the jet, as explained before, and select the following governing parameters: r; m; Jx ; x and y which have dimensions listed in (6.24) and (6.25). Four of the five governing parameters have independent dimensions, which yields (6.34)–(6.37) that determine velocity distribution in jet cross-section, the velocity variation along the jet axis, and the boundary layer thickness. Taking into account the values of the exponents ei ði ¼ 1; 2; 3; 4Þ in (6.25) (see Table 6.2), we find from (6.33) and 0 (6.34) the values of the exponents ai and ai 1 1 2 1 0 1 0 2 0 1 0 a1 ¼ ; a2 ¼ ; a3 ¼ ; a4 ¼ ; a1 ¼ ; a2 ¼ ; a3 ¼ ; a4 3 3 3 3 3 3 3 2 ¼ (6.38) 3
142
6
Jet Flows
Fig. 6.4 Plane laminar jet
In accordance with (6.36) and (6.37), we obtain
Jx2 r2 n
um ¼ c 1 d ¼ c2
rn2 Jx
1=3
x1=3
(6.39)
x2=3
(6.40)
1=3
Using (6.35), (6.39) and (6.40), it is possible to estimate several important characteristics of plane laminar submerged jets, in particular, the local Reynolds number. The latter is defined by the average velocity, the boundary layer thickness and the kinematic viscosity Red ¼ where ¼ 1d
d=2 Ð
d n
(6.41)
udy is the average velocity in jet cross-section.
d=2
Taking into account (6.35), (6.39) and (6.40), we arrive at the following expression for the local Reynolds number Red ¼ c3
where c3 ¼ c1 c2 :
Jx x rn2
1=3 1=2 ð ’ðÞd 1=2
(6.42)
6.5 Laminar Wake of a Blunt Solid Body
143
Fig. 6.5 Laminar-turbulent transition (in cross-section A-A) in a submerged jet
Equation 6.42 shows that the local Reynolds number in plane laminar submerged jet increases downstream as x1=3 : At sufficiently large values of x the local Reynolds number exceeds the critical value corresponding to laminar-turbulent transition (Fig. 6.5). A higher total momentum flux or a lower fluid viscosity increase the local Reynolds number Red and the laminar-turbulent transition cross-section approaches the jet origin. The observations show that the laminar sections of plane submerged jets exist up to Re0 ’ 30 (Re0 is based on the outflow jet velocity and the slit width, Andrade 1939).
6.5
Laminar Wake of a Blunt Solid Body
As the second example application of the Pi-theorem to jet flows consider plane laminar wake behind a blunt solid body in a uniform stream of viscous fluid (Fig. 6.1b). Similarly to flows in submerged jets, flows in laminar wakes can be characterized by an integral invariant which accounts for the body characteristics. In order to find this invariant we use the boundary layer equation ru
@u @u @2u þ rv ¼m 2 @x @y @y
(6.43)
144
6
Jet Flows
Consider the relative longitudinal velocity component u 1 ¼ u1 u
(6.44)
where u1 is the free stream velocity; the body is considered to be at rest, while the stream impinges on it with velocity u1 . At a large distance from the body u1 becomes sufficiently small. Then (6.43) can be linearized and takes the following form ru1
@u1 @ 2 u1 ¼m 2 @x @y
(6.45)
Integrating (6.45) from y ¼ 1 to y ¼ þ1 across the wake and accounting for the fact that @u1 =@y ! 0 at y ! 1, we obtain 1 ð
Jx ¼
ru1 dy ¼ const:
(6.46)
1
Below it will be shown that um depends on the physical properties of fluid, velocity of the free stream, as well as on the integral invariant Jx u1 ¼ f ðr; m; Jx ; u1 x; yÞ
(6.47)
where the dimensions of r; m; Jx ; u1 ; x; and y in the Lx Ly Lz MT system of units are 1 1 1 1 1 1 1 1 L1 x Ly Lz M; Lx Ly Lz MT ; Lz MT ; Lx T ; Lx and Ly ; respectively. We also write the functional equations for u1m and dw u1m ¼ fw ðr; m; Jx ; u1 ; xÞ
(6.48)
dw ¼ fdw ðr; m; Jx ; u1 ; xÞ
(6.49)
The functional equation for u1 can be rewritten as follows u1 ¼ f ðu1m ; dw ; yÞ
(6.50)
Equation 6.50 for u1 expresses the physically realistic assumption that velocity at any point of a wake cross-section is determined by three parameters: velocity at the wake axis u1m ; the boundary layer (wake) thickness dw and the distance from the wake axis to the point under consideration y: Applying the Pi-theorem to (6.48), we obtain Pm ¼ ’m ðP1 Þ
(6.51)
6.5 Laminar Wake of a Blunt Solid Body
145 0
0
a
0
0
where Pm ¼ u1m =ra1 ma2 ua13 xa4 ; P1 ¼ Jx =ra1 ma2 u13 xa4 ; and a1 ¼ a2 ¼ a4 ¼ 0; 0 0 0 0 a3 ¼ 1; a1 ¼ 1; a1 ¼ a2 ¼ a3 ¼ a4 ¼ 1=2: Therefore, (6.51) takes the form ( ) u1:m Jx ¼ ’m u1 ðrmu1 xÞ1=2 u1
(6.52)
The application of the Pi-theorem to (6.50), yields u1 ¼ ’w ðÞ
(6.53)
where u1 ¼ u1 =u1m ; and ¼ y=dw : In order to determine the dependence of Jx on the drag force acting on a blunt solid body, we use the overall momentum and mass balances for the rectangular control volume ABCD encompassing the body (Fig. 6.1b). They read þh ð
ru2 2h W
ru2 dy ¼ 0
(6.54)
h þh ð
ru1 2h
rudy ¼ 0
(6.55)
h
where W is the drag force acting on the solid body, and 2h is height of the contour ABCD. From (6.54) and (6.55) it follows that W equals to þh ð
ruðu1 uÞdy
W¼
(6.56)
h
Taking into account that in the limit h ! 1, we can rewrite (6.56) as 1 ð
1 ð
rðu1 uÞu1 dy u1
W¼ 1
ru1 dy
(6.57)
1
The comparison of (6.46), (6.56) and (6.57), reveals that Jx ¼
W u1
(6.58)
146
6
Jet Flows
Accordingly, (6.52) takes the form ( ) u1m W ¼ ’m u1 ðrmu1 xÞ1=2
(6.59)
The wake thickness can be found directly from (6.49) or (6.53) and (6.57). From (6.53) and (6.47) it follows that dw ¼ ru1 u1m
6.6
W 1 Ð 1
(6.60) ’w ðÞd
Wall Jets over Plane and Curved Surfaces
Laminar wall jets represent themselves a “compound” boundary layers, which are similar to the free boundary layer of submerged jets on the one side, and to the nearwall (Blasius) boundary layers on the other side (close to the wall). The presence of the wall friction results in variation of the total momentum flux downstream the wall jets. Akatnov (1953) and Glauert (1956) showed that there is an integral y 1 Ð Ð 2 invariant Jx ¼ ru rudy dy which remains constant along the laminar wall 0
0
jets over a plane wall. As in the previous cases of jet flows considered in the present chapter, we begin our consideration with formulation of the functional equations for the maximal longitudinal velocity and the boundary layer thickness in the jet. These equations are identical to (6.19) and (6.21). Since all the governing parameters in the functional equation for the maximal longitudinal velocity and the boundary thickness have independent dimensions, we obtain um ¼ cu ra1 ma2 Jxa3 xa4 0
0
a
0
(6.61)
0
d ¼ cd ra1 ma2 Jx 3 xa4
(6.62) 0
where cu and cd are constants, and the exponents ai and ai are given by 1 1 1 1 0 1 0 3 0 1 0 a1 ¼ ; a2 ¼ ; a3 ¼ ; a4 ¼ ; a1 ¼ ; a2 ¼ ; a3 ¼ ; a4 2 2 2 2 4 4 4 3 ¼ 4 Then the expressions for um and dd read
(6.63)
6.6 Wall Jets over Plane and Curved Surfaces
147
Table 6.3 Characteristic constants for laminar and turbulent jetsa Laminar jet Turbulent jet Jet geometry Plane jet
Ai pffi 3J2 1 ½34r2xn 2
Axisym-metric jet
3Jx 8prn
Radial jet Wall jet
pffi 9J 2 1 ½32p2 rx2 n 4 qffiffiffiffiffiffiffiffiffiffi Jx 4r2 nJ1
Bi pffi J 1 ½36rnx 2 2 qffiffiffiffiffiffiffiffiffi 3Jx 8prn2
pffi 3J2 1 ½34prnx 2 2 pffiffiffi3Jx2 J1 1 1 2 4 4r2 n2
ai
bi
13
23
1
1
1
1
12
34
3Jx 8rk1=2
1 2k1=2
12
bi ’ðÞ 1 ’P ðÞ
3Jx 8prk
1 k1=2
1
1 ’A ðÞ
3Jx 16prk 1=2
p1ffiffiffiffi 2k
1
1 ’R ðÞ
Ai qffiffiffiffiffiffiffiffiffi
Bi
qffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffi
ai
a
’W ðÞ
A; B; a and b are the coefficients in the expressions for um and um ¼ Axa ; ’P ðÞ ¼ ð1 tanh2 Þ; ’A ¼ ð1 þ 18 Þ2 ; ’R ðÞ ¼ ð1 tanh2 Þ, 3 1 ’W ðÞ ¼ ðF1 2 F2 F2 Þ, the function F is determined by the transcendental equation
um ¼ cu
Jx rm
dd ¼ cd
1=2
m3 rJx
x1=2
’:
(6.64)
1=4 x3=4
(6.65)
It is emphasized that (6.39) and (6.40), as well as (6.64) and (6.65) coincide with the expressions obtained as the exact analytical solutions of the boundary layer equations (Table 6.3) (Vulis and Kashkarov 1965). pffiffiffi pffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 3 1 F þ FF1 þ F1 2 F þ F1 1 pffiffiffiffiffiffiffiffiffi arctan pffiffiffiÞ, ¼ ln pffiffiffiffiffiffiffi pffiffiffi 2 þ ðarctan F 2F1 3F 3 1 1 ð F1 FÞ 1 Ð 0 ¼ Byxb , F1 ¼ 1:7818, the dimensionless value J1 ¼ FF d: 0
The approach outlined above can be also used for the analysis of wall jets over curved surfaces. This type of jets was studied theoretically by Wygnanski and Champagne (1968). They found the integral invariant of the problem and the selfsimilar solutions for both concave and convex surfaces in the case when the local radius of curvature varies as x3=4 (with x being reckoned along the surface). Below we consider wall jets over curved surfaces in the framework of the dimensional analysis. In the case when the ratio of the characteristic jet thickness to the local radius of curvature R is sufficiently small and curvature variations occur in such a way that dR=dx 1; the set of the governing boundary layer equations has the following form ru
@u y @u uv @P y @ 2 u m @u þ þ 1þ v þr ¼ þm 1þ @x R @y R @X R @y2 R @r ru2 @P ¼ @y R
(6.66)
(6.67)
148
6
@ru @ n y o þ 1 þ rv ¼ 0 @x @y R
Jet Flows
(6.68)
where x and y are the coordinates along and normal to the surface, u and v are the velocity components in the x and y directions. When the local radius of curvature R>0, the wall is convex outwards, whereas R<0 the wall is concave. Equations (6.66)–(6.68) are subjected to the following boundary conditions at the wall (y ¼ 0) and far from it y ¼ 0; u ¼ v ¼ 0; y ¼ 1; u ¼ 0
(6.69)
Integrating (6.66)–(6.68) yields the invariant 8 <
1 ð
Jx ¼
ru 0
:
1þ
1 ð y
R 0
ru2 dy
1 R
1 ð
9 = ru2 ydy dy ¼ const ;
(6.70)
0
2 3 where ½Jx ¼ Lx L2 z M T : At R ! 1, the invariant (6.70) takes the form of the Akatnov’s invariant mentioned above. It is emphasized that there are two identical forms of the integral invariant of the ! wall jet near a straight wall: (i) Akatnov’s (1953) invariant! 1 1 1 1 Ð Ð Ð Ð 2 2 Jx ¼ ru rudy dy; and Glauert’s (1956) invariant-Jx ¼ ru ru dy dy: 0
y
0
y
It is easy to show that they are identical. From the physical point of view, we note that the velocity and the jet thickness in the far field of the wall jet over a curved surface should depend on the fluid properties, the integral invariant of the jet, as well as on the radius of curvature of the wall u ¼ fu ðr; m; Jx ; R; x; yÞ
(6.71)
um ¼ fum ðr; m; Jx ; R; xÞ
(6.72)
d ¼ fd ðr; m; Jx ; R; xÞ
(6.73)
where u is the longitudinal velocity, um its local maximal value, and d is the jet thickness. Equation 6.71 can be presented in the form u ¼ fu ðum ; d; R; yÞ
(6.74)
Since two of the four governing parameters in (6.74) have independent dimensions, it reduces to
6.7 Buoyant Jets (Plumes)
149
u ¼ ’u ð; rÞ
(6.75)
where u ¼ u=um ; ¼ y=d; and r ¼ R=d: Applying the Pi-theorem to (6.72), we arrive at the following expression um ¼
1=2 Jx 1=2 1 R pffiffiffi ’um rm ðm3 x3 =rJx Þ x
(6.76)
The self-similar solution of the present problem exists when R ¼ cx3=4 ; where c is a constant (Wygnanski and Champagne 1968). Substitution this expression in (6.76) yields ( ) 1=2 Jx 1 c pffiffiffi ’um (6.77) um ¼ rm x ðm3 =rJx Þ1=4
6.7
Buoyant Jets (Plumes)
The buoyant jets belong to a special class of jet flows developing due to the influence of buoyancy forces on fluid motion (Fig. 6.1c). They are typical for a number of phenomena in nature and engineering: mass, momentum and heat transfer in the atmosphere and oceans, pollutant dispersal, etc. (Turner 1969; 1986; Jaluria 1980). The buoyant jets were a subject of a number of theoretical and experimental investigations of complex flows developing under the influence of the inertial, viscous and buoyancy forces (Zel’dovich 1937; Batchelor 1954; Morton et al. 1956). Below we consider only a few simplest examples of buoyant jets that illustrate the application of the Pi-theorem to buoyancy-driven jet flows. Following Zel’dovich (1937), we consider laminar vertical buoyant jets (plumes) over a pointwise horizontal wire or a pointwise sphere treated as sources of heat. In the framework of the Boussinesq approximation, the boundary layer equations describing flows in laminar buoyant jet, which are generated by such sources read @u @u n @ @u u þv þ bg# (6.78) ¼ k yk @x @y y @y @y
u
@u 1 @vyk ¼0 þ @x yk @y
(6.79)
@# @# a @ @# þv ¼ k yk @x @y y @y @y
(6.80)
150
6
Jet Flows
where n and a; are the kinematic viscosity and thermal diffusivity, respectively, b is the thermal expansion coefficient, u and v are the longitudinal and transverse components of velocity in the vertical and horizontal directions x and y; respectively (cf. Fig. 6.1c), # ¼ T T1 ; with T1 being the ambient temperature, g is the gravity acceleration, k ¼ 0 or 1 for the plane or axisymmetric problems, respectively. The solutions of (6.78)–(6.80) are subjected to the following boundary conditions y¼0
@u @# ¼ 0; v ¼ 0; ¼ 0; y ! 1 u ! 0; v ! 0 @y @y
(6.81)
Rewriting (6.80) in the divergent form @u# 1 @v#yk a @ k @# ¼ k þ k y @x y @y y @y @y
(6.82)
and integrating it in y from the wake center y ¼ 0 to its edge y ¼ 1, we arrive at the following integral invariant expressing conservation of the excessive convective heat flux along the plume 1 ð
u#yk dy ¼ const:
Q¼
(6.83)
0
Equations 6.78–6.80 with the boundary conditions (6.81) and the integral invariant (6.83) show that velocity and temperature in buoyant laminar vertical jets depend on six dimensional parameters, namely, ½gb ¼ Ly1 T 2 ; ½n ¼ L2 T 1 ; ½a ¼ L2 T 1 ; ½Q ¼ L2þk yT 1 ; ½ x ¼ L; ½ y ¼L
(6.84)
Therefore, the functional equations for the velocity and temperature fields can be presented by the following functions u ¼ f1 ðgb; n; a; Q; x; yÞ
(6.85)
# ¼ f2 ðgb; n; a; Q; x; yÞ
(6.86)
In order to reduce the number of the governing parameters it is convenient to introduce new variables ½e v ¼
h v i h y i 1=2 ¼ T ¼ T 1=2 ; ½y ¼ n1=2 n1=2
(6.87)
6.7 Buoyant Jets (Plumes)
151
Then (6.78)–(6.80) take the form u
@u @u 1 @ @u þ bg# þ ve ¼ k ye @x @e y ye @e y @e y
(6.88)
@u 1 @e vyk þ k ¼0 @x ye @e y
(6.89)
@# @# 1 1 @ k @# ye u þ ve ¼ @x @e y Pr yek @e y @e y
(6.90)
The boundary and integral conditions in new variables read ye ¼ 0
@u @# ¼ 0; ve ¼ 0; ¼0 @e y @e y
ye ! 1 u ! 0; # ! 0 Q1
(6.91) (6.92)
1 ð
u#e y k de y
¼
(6.93)
0
where Q1 ¼ LyT ðk1Þ=2 . Then, the velocity and temperature fields are given by the following functional equations (where it is assumed that the Prandtl number equals one) u ¼ f1 ðbg; Q1 ; x; ye Þ
(6.94)
# ¼ f2 ðbg; Q1 ; x; ye Þ
(6.95)
The axial velocity in the buoyant jet corresponds to ye ¼ 0, which reduces (6.94) to the following equation um ¼ f1 ðbg; Q1 ; xÞ
(6.96)
All the governing parameters in (6.96) have independent dimensions. Therefore, in accordance with the Pi-theorem, (6.96) takes the form um ¼ cðbgÞa1 ðQ1 Þa2 ðxÞa3
(6.97)
where c is a constant and the exponents ai are determined using the principle of the dimensional homogeneity as
152
6
a1 ¼ a2 ¼
2 1k ; a3 ¼ 5k 5k
Jet Flows
(6.98)
Accordingly, (6.97) reads um ¼ c
n
bgQ1
2=ð5kÞ
xð1kÞ=ð5kÞ
o (6.99)
Since Q1 ¼ Q=nðkþ1Þ=2 ; (6.99) takes the form n o um ¼ ðbgQÞ2=ð5kÞ nðkþ1Þ=ð5kÞ xð1kÞ=ð5kÞ
(6.100)
Taking in (6.100) k ¼ 0 or 1, we arrive at the following expressions for the axial velocity in plane and axisymmetric buoyant laminar jets (plumes) n o (6.101) um;plane ¼ c ðbgQÞ2=5 n1=5 x1=5 n o um;axis: ¼ c ðbgQÞ1=2 n1=2
(6.102)
According to the Pi-theorem (6.94) corresponds to the case of nk ¼ 1. Therefore, it can be reduced to the following dimensionless form P ¼ ’ðP1 Þ
(6.103) 0
0
0
where P ¼ u=ðbgÞa1 ðQ1 Þa2 ðxÞa3 ; P ¼ y =ðbgÞa1 ðQ1 Þa2 ðxÞa3 ; and a1 ¼ a2 ¼ 0 0 0 2=ð5 kÞ; a3 ¼ ð1 kÞ=ð5 kÞ; a1 ¼ a2 ¼ 1=ð5 kÞ; a3 ¼ 2=ð5 kÞ. For k ¼ 0 or 1, (6.103) yields the longitudinal velocity profiles in plane and axisymmetrical buoyant jets as n o uplane ¼ ðbgQÞ2=5 n1=5 x1=5 ’plane yðbgQÞ1=5 =n3=5 x2=5 n o uaxis: ¼ ðbgQÞ1=2 n1=2 ’axis: yðbgQÞ1=4 =n3=4 x1=2
(6.104) (6.105)
Similarly the temperature distributions in buoyant laminar jets are given by n o yplane ¼ Q4=5 ðbgÞ1=5 n2=5 x3=5 ’plane yðbgQÞ1=5 =x2=5 y3=5 n o yaxis: ¼ Qn1 x1 ’axis: yðbgQÞ1=4 =x1=2 n3=4
(6.106) (6.107)
The expressions (6.104)–(6.107) determine only the form of the self-similar solutions for the velocity and temperature distributions in laminar buoyant jets.
6.7 Buoyant Jets (Plumes)
153
They can be also used to establish the exact analytical solution of this problem. In particular, substitution of the expressions (6.104)–(6.107) into (6.78)–(6.80) allows transformation of the system of the governing partial differential equation into the system of ordinary differential equation for the unknown functions ’plane ; ’axisymmetric ; ’plane ; and ’axisymmetric : It is emphasized that that the analytical solution of this problem is found without any particular hypothesis about the mechanism of mixing of hot fluid with the surrounding fluid in the buoyant jets. It is interesting to note that the Reynolds numbers in plane and axisymmetric laminar buoyant jets increase with x Replane ¼ bgQx3 =n3
(6.108)
Reaxisymmetric ¼ bgQx2 =n3
(6.109)
That means that there is some critical distance xcr from the source where transition of laminar flow into the turbulent one inevitably occur (Zel’dovich 1937). The developed approach also allows the analysis of the velocity and temperature fields in turbulent buoyancy jets. In this case the system of the governing equations is written for the average flow characteristics and should be closed using a semiempirical model of turbulence, in particular, the Prandtl mixing length model. Applying the Pi-theorem it is not possible to obtain the following expressions for the longitudinal velocity and temperature distributions in plane and axisymmetric turbulent jets. In particular, for plane turbulent jets uT ¼
bgQ x
1=3
’plane ðy=xÞ; yT ¼ Q2=3 ðbgÞ1=3 x5=3 ’plane ðy=xÞ
(6.110)
and for the axisymmetric jets uT ¼ ðbgQ=xÞ1=3 ’axis: ðy=xÞ; yT ¼ Q2=3 ðbgÞ1=3 x5=2 ’axis: ðy=xÞ
(6.111)
Note that the expressions (6.110) and (6.111), as any expressions derived using semi-empirical models of turbulence incorporate some constants involved in these models. Studies of turbulent buoyant jets widely use models based on the so-called entrainment hypothesis which assumes a certain relation of the mean inflow velocity across the edge of a turbulent jet with some characteristic velocity in a given cross-section (Batchelor 1954; Morton et al. 1956). In the framework of such models the following system of the “entrainment” equations can be formulated d 2 b um ¼ 2abu um dz u
(6.112)
154
6
Jet Flows
d 1 2 2 b u ¼ l2 b2u gym dz 2 u u
(6.113)
d l2 b2u um gym ¼ b2u um N 2 ðzÞ dz 1 þ l2
(6.114)
where a is the entrainment constant, l ¼ by =bu is the ratio of the widths of the temperature and velocity profiles (l is assumed to be constant-l 1:2Þ; NðzÞ ¼ ðg=r1 Þðdr0 =dzÞ; with r0 and r1 being the density at a given height z and the reference density in the environment. The change of the characteristic scales of the problem with height z can be easily found by considering the dimensions of the parameters involved. As a result, the velocity at the axis of an axisymmetric jet um is determined as um ¼ cM1=2 z1 where rM ¼ 2p
1 Ð 0
(6.115)
ru2 z¼0 rdr is the momentum flux at z ¼ 0; and c is a constant.
A number of instructive examples of the application of the entrainment model to flows in buoyant jets of different types can be found in the surveys of Turner (1969) and List (1982), as well as in the original works of Morton (1957, 1959), Turner (1966, 1986), Papanicolaou and List (1982), Baines et al. (1990), and Bloomfield and Kerr (2000).
Problems P.6.1. Determine the velocity distribution along the axis of plane laminar submerged jet. The governing parameters in the present case are: ½n ¼ L2y T 1 ; ½Ix ¼ 2 Lx Ly T 2 ; and ½ x ¼ Lx : Therefore, the functional equation for the axial velocity reads um ¼ fm ðn; Ix ; xÞ
(P.6.1)
Since the parameters n; Ix ; and x have independent dimensions, (P.4.1) reduces to the following one um ¼ c1 na1 Ixa2 xa3 where c1 is constant.
(P.6.2)
Problems
155
Taking into account the dimensions of um ; n, Ix and x, we find the values of the exponents ai : a1 ¼ 1=3; a2 ¼ 2=3; a3 ¼ 1=3. Substitution of the values of a1 ; a2 and a3 in (P.6.2) yields rffiffiffiffiffi 2 3 I um ¼ c 1 x nx
(P.6.3)
P.6.2. Show that the local Reynolds number Red ¼ d=n in the axisymmetric laminar submerged jet is independent of x. The functional equations for u; um and d have the following form u ¼ fu ðr; m; Jx ; x; yÞ
(P.6.4)
um ¼ fu:m ðr; m; Jx ; xÞ
(P.6.5)
d ¼ fd ðr; m; Jx ; xÞ
(P.6.6)
where the governing parameters have the dimensions 1 1 1 1 1 1 2 ½r ¼ L1 x Ly Lz M; ½m ¼ Lx Ly Lz MT ; ½Jx ¼ Lx Ly Lz MT ; ½ x
¼ Lx ; ½ y ¼ Ly :
(P.6.7)
Equation (P.6.4) can be presented in the form u ¼ fu ðum ; d; yÞ
(P.6.8)
Applying the Pi-theorem to (P.6.8), we reduce it to the dimensionless form u ¼ ’u ðÞ
(P.6.9)
where u ¼ u=um , and ¼ y=d: Bearing in mind that r; m; Jx and x have independent dimensions, we present (P.6.5) and (P.6.6) as follows um ¼ c1 ra1 ma2 Jxa3 xa4 0
0
a
0
(P.6.10)
0
d ¼ c2 ra1 ma2 Jx 3 xa4
(P.6.11) 0
where c1 and c2 are constants, and the exponents ai and ai are equal to a1 ¼ 0; 0 0 0 0 a2 ¼ 1; a3 ¼ 1; a4 ¼ 1; a1 ¼ 1=2; a2 ¼ 1; a3 ¼ 1=2, and a4 ¼ 1: Then, we obtain
156
6
um ¼ c 1 d ¼ c2 Taking into account that ¼ um
Jx rnx
rnx ðrJx Þ1=2
þ1=2 Ð 1=2
1=2 Ð 1=2
(P.6.12)
(P.6.13)
’ðÞd; we find
1=2 Jx ¼ const Red ¼ A rn2 where A ¼ c1 c2
Jet Flows
(P.6.14)
’ðÞd:
P.6.3 Describe the velocity variation along the axis of a radial laminar submerged jet (Table 6.1). The axial velocity of radial jets depends on four governing parameters: ½r ¼ 1 1 1 1 1 2 1 2 L1 and ½x ¼ Lx that have x Ly Lz M; ½m ¼ Lx Ly Lz MT ; ½Jx ¼ Ly Lz MT independent dimensions. According to the Pi-theorem, um ¼ cra1 ma2 Jxa3 xa4
(P.6.15)
where c is a constant, and the exponents ai have the following values: a1 ¼ 1=3; a2 ¼ 1=3; a3 ¼ 2=3; and a4 ¼ 1: As a result, we obtain the following expression for the axial velocity in radial laminar submerged jets sffiffiffiffiffiffiffi 2 3 Jx 1 um ¼ c x 2 r n
(P.6.16)
References Abramovich GN (1963) The theory of turbulent jets. MIT Press, Boston Akatnov NI (1953) Development of two-dimensional laminar incompressible jet near a rigid wall. Proc Leningrad Polytec Inst 5:24–31 Andrade EN (1939) The velocity distribution in liquid-into-liquid jet. The plane jet. Proc Phys Soc London 51:748–793 Baines WD, Turner JS, Campbell IH (1990) Turbulent fountains in an open chamber. J Fluid Mech 212:557–592 Batchelor GK (1954) Heat convection and buoyancy effects in fluid. Quart J Roy Meteor Soc 80:339–358 Bloomfield LJ, Kerr RC (2000) A theoretical model of a turbulent fountain. J Fluid Mech 424:197–216
References
157
Glauert MB (1956) The wall jet. J Fluid Mech 1:625–643 Hinze JO (1959) Turbulence. An introduction to its mechanism and theory. McGraw Hill Book Company, New York Ho CM, Gutmark E (1987) Vortex induction and mass entrainment in a small-aspect-ratio elliptic jet. J Fluid Mech 179:383–405 Hussain F, Hussain HS (1989) Elliptic jets. Part 1. Characteristics of unexcited and excited jets. J Fluid Mech 208:259–320 Jaluria Y (1980) Natural convective heat and mass transfer. Pergamon, Oxford Konsovinous NS (1978) A note on the conservation of the axial momentum of turbulent jet. J Fluid Mech 87:55–63 Korthapalli A, Baganoff D, Karamcheti K (1981) On the mixing of rectangular jet. J Fluid Mech 107:201–220 Landau LD, Lifshitz EM (1987) Fluid mechanics, 2nd edn. Pergamon, London List EJ (1982) Turbulent jets and plumes. Annu Rev Fluid Mech 14:189–212 Morton BR (1957) Buoyant plumes in moist atmosphere. J Fluid Mech 2:127–144 Morton BR (1959) Forced plumes. J Fluid Mech 5:151–163 Morton BR, Taylor GI, Turner JS (1956) Turbulent gravitational convection from maintained and instantaneous sources. Proc Roy Soc London A 234:1–23 Pai SI (1954) Fluid dynamics of jets. D. Van Nastrand Company, New York Papanicolaou PN, List EJ (1982) Investigations of round vertical turbulent buoyant jets. J Fluid Mech 195:341–391 Schlichting H (1979) Boundary layer theory, 8th edn. Springer, Berlin Schneider W (1985) Decay of momentum flux in submerged jets. J Fluid Mech 154:91–110 Townsend AA (1956) The Structure of turbulent shear flow. Cambridge University Press, Cambridge Trentacoste N, Sforzat P (1967) Further experimental results for three-dimensional free jets. AIAA J 5:885–891 Turner JS (1966) Jets and plumes with negative or reversing buoyancy. J Fluid Mech 26:729–792 Turner JS (1969) Buoyant plumes and thermals. Annu Rev Fluid Mech 1:29–44 Turner JS (1986) Turbulent entrainment: the development of the entrainment assumption and its application to geophysical flows. J Fluid Mech 173:431–471 Vulis LA, Kashkarov VP (1965) The theory of viscous fluid jets. Nauka, Moscow (in Russian) Wygnanski I, Champagne FH (1968) The laminar wall jet over curved surface. J Fluid Mech 31:459–465 Zel’dovich YaB (1937) Limiting laws of free-rising convective flows. J Exp Theoret Phys 7:1463–1465. The English translation in: Ya.B. Zel’dovich: Selected Works of Ya.B. Zel’dovich, vol. 1. Chemical Physics and Hydrodynamics. Limiting laws of freely rising convective currents (Princeton Univ. Press, Princeton, 1992).
.
Chapter 7
Heat and Mass Transfer
7.1
Introductory Remarks
This Chapter deals with processes of heat and mass transfer in solid, liquid and gaseous media. They have important implications for understanding various natural phenomena as well as technological processes. A vast literature is devoted to heat and mass transfer in motionless and moving media. Tens of thousands of journal publications contain detailed information on modern methods of investigation of heat and mass transfer problems (e.g. mathematical modeling, measuring thermohydrodynamical quantities, etc.). The results of numerous theoretical and experimental researches in this field are generalized in a number of well-known monographs on fluid dynamics (Landau and Lifshitz 1987; Schlichting 1979; Levich 1962), heat and mass transfer (Kutateladze 1963; Spalding 1963; Kays 1975; White 1988), as well as in reference books (Rohsenow et al. 1998; Kaviany 1994). Heat and mass transfer in various media occur under the interaction of different factors, which involve the effects of physical properties of the matter, its equation of state and regime of motion, contact with the surrounding fluid or solid surfaces, etc. In the general case the relevant physical processes are described by a system of coupled non-linear partial differential equations that include the continuity, momentum, energy and species conservation equations. This system of equations should be supplemented by the equation of state and correlations determining the dependences of physical properties of the matter on temperature and species concentrations. Solving such complicated system of equations entails great difficulties. Therefore, studing heat and mass transfer processes, as a rule, employs several simplifying assumptions that lead to an approximate solution of the problem at hand. A qualitative analysis of such complex phenomena as heat and mass transfer in continous media is signicantly facilitated by applying the dimensional analysis, in particular, the approach based on the Pi-theorem. The results discuss in the present chapter demonstrate the applications of the Pi-theorem to conductive heat transfer in media with constant and temperature-dependent thermal diffusivity, convective heat transfer under the conditions of forced, natural and mixed L.P. Yarin, The Pi-Theorem, Experimental Fluid Mechanics, DOI 10.1007/978-3-642-19565-5_7, # Springer-Verlag Berlin Heidelberg 2012
159
160
7 Heat and Mass Transfer
convection, as well as heat and mass transfer in laminar boundary layers and laminar pipe and jet flows. Using the Pi-theorem for studing heat and mass transfer under the conditions of phase change is also discussed in this Chapter.
7.2 7.2.1
Conductive Heat and Mass Transfer Temperature Field Induced by Plane Instantaneous Thermal Source
Consider the evolution of the temperature field due to a plane instantaneous thermal source of strength Q releasing heat at t ¼ 0 and x ¼ 0 in an infinite medium with constant (temperature-independent) physical properties (Fig. 7.1). The source can be presented as q ¼ QdðxÞdðtÞ
(7.1)
where Q is a constant, and dðÞ is the delta-function. The medium is at rest, heat is transferred by conduction only, and the thermal balance equation that describe the one-dimensional excessive (relative to the initial one) temperature field Tðx; tÞ at t>0 reads @T @2T ¼a 2 @x @x
(7.2)
where a is the thermal diffusivity. Integrating (7.2) by x from 1 to þ 1 and accounting for the boundary þ1 Ð conditions at infinity @T=@xj1 ¼ @T=@xj1 ¼ 0; we find that Q ¼ Tdx 1
¼ const, i.e. Q is the problem invariant. At the initial time moment t ¼ 0 the excessive temperature equals zero at any x>0. That means that the previous integral is indeed equal to the heat source strength, while at t¼0 the temperature field is
Fig. 7.1 Temperature distribution corresponding to a plane instantaneous thermal source acting at t ¼ 0 at x ¼ 0 in an infinite medium with constant properties. The snapshots shown correspond to ti>0
7.2 Conductive Heat and Mass Transfer
161
given as Tðx; 0Þ ¼ QdðxÞ: Then, the functional equation for the temperature field reads T ¼¼ f ða; Q; t; xÞ
(7.3)
Analyzing the dimensions of the governing parameters (½a ¼ L2 T 1 ; ½Q ¼ Ly; ½t ¼ T; ½ x ¼ L, with y being the temperature scale of temperature and applying the Pi-theorem, we arrive at the following equation Q x T ¼ pffiffiffiffiffi ’ pffiffiffiffi ax at
(7.4)
pffiffiffiffi where ’ ¼ ’ðÞ; and ¼ x= at:
7.2.2
Temperature Field Induced by a Pointwise Instantaneous Thermal Source
The approach of the previous sub-section can be also applied to the evolution of the excessive temperature field triggered by a pointwise thermal source of strength Q, which acted at t¼0 and r¼0 in an infinite medium with constant thermal diffusivity q ¼ QdðrÞdðtÞ
(7.5)
where Q is a constant, and r is the radial coordinate in the spherical coordinate system centered at the heat source. The thermal balance equation that describe the evolution of the temperature field at t>0 reads1 @T 1 @ @T ¼a 2 r2 @r r @r @r
(7.6)
Integrating (7.6) by r from r ¼ 0 to r ¼ 1 and accounting for the boundary conditions @T=@rjr¼0 ¼ @T=@rjr¼1 ¼ 0, yields the following invariant 1 ð
Tr2 dr ¼ Q ¼ const: 0
where ½Q ¼ L3 y:
1
Equation 7.6 accounts for the spherical symmetry of the temperature field.
(7.7)
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7 Heat and Mass Transfer
Then, the excessive temperature field satisfies the following functional equation T ¼ f ða; Q; t; rÞ
(7.8)
Taking into account the dimensions of the governing parameters in (7.8) and using the Pi-theorem, we arrive at the dimensionless equation P ¼ ’ðP1 Þ 0
0
(7.9)
0
where P ¼ T=aa1 Qa2 ta3 ; and P1 ¼ r=aa1 Qa2 ta3 . 0 0 Determining the exponent as a1 ¼ 3=2; a2 ¼ 1; a3 ¼ 3=2; a1 ¼ 1=2; a2 ¼ 0 0 and a3 ¼ 1=2, we arrive at the following expression for the temperature field T¼
7.2.3
Q ðatÞ3=2
’
r
!
ðatÞ1=2
(7.10)
Evolution of Temperature Field in Medium with Temperature-Dependent Thermal Diffusivity (The Zel’dovich-Kompaneyets Problem)
The present sub-section is devoted to the evolution of temperature field in response to an instantaneous plane energy source in medium which thermal diffusivity depending on temperature. Very strong heat release in a substance is accompanied by temperature rise of the order of tens or even hundred of thousands degrees. In such cases the energy transport occurs mainly by radiation. Under these conditions the radiant thermal diffusivity coefficient depends on temperature and can be expressed as (Zel’dovich and Kompaneyets 1970; Zel’dovich and Raizer 2002) w ¼ aT n
(7.11)
where a and n are given constants, in particular, a is dimensional, ½a ¼ L2 T 1 yn and n is dimensionless, ½n ¼ 1: According to (7.11), the radiant thermal diffusivity coefficient w approaches to zero at T ! 0: At high temperature in a heated zone Th T1 ðTh and T1 are the temperatures in the heated zone and the surrounding medium, respectively) it is possible to assume that ambient temperature equals zero, i.e. T1 ¼ 0: In this case heat can not be transferred instantaneously to large distances from the thermal source. It spreads over substance with finite speed, so that there exists some boundary that separate the heated zone from the cooled undisturbed one. In this case head spreads in the form of a thermal wave as is shown in Fig. 7.2.
7.2 Conductive Heat and Mass Transfer
163 T
Fig. 7.2 Temperature distribution in response to a plane instantaneous thermal source at t ¼ 0 at x ¼ 0 in medium with temperaturedependent thermal diffusivity
t1 t2 t3
0
x
Let at t¼0 in plane x ¼ 0 thermal energy of E (say, Joule) is released per 1m2 of surface. The evolution of the temperature field at t>0 is described by the thermal balance equation @T @ @T ¼ w @t @x @x
(7.12)
with the boundary conditions x ! 1; T ! 0; x ¼ 0;
@T ¼0 @x
(7.13)
where the dependence of the radiant thermal diffusivity coefficient on temperature is given by (7.11). Integrating (7.12) in x from 1 to 1, we obtain the invariant of the present problem 1 ð
Q¼
Tdx
(7.14)
1
where ½Q ¼ ½E=rcP ¼ Ly, where r and cP are density and the specific heat at constant pressure of the matter, respectively. From (7.11), (7.12) and (7.14) it follows that there are the following governing parameters of the problem: two constants a and Q and two variables x and t T ¼ f ða; Q; x; tÞ
(7.15)
It is seen that three of the four governing parameters have independent dimensions. Then, in accordance with the Pi-theorem (7.15) reduces to the following dimensionless form P ¼ ’ðP1 Þ 0
0
0
where P ¼ T=aa1 Qa2 ta3 ; and P1 ¼ x=aa1 Qa2 ta3 .
(7.16)
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7 Heat and Mass Transfer
Taking into account the dimensions of the parameters involved, we arrive at the 0 system of the six algebraic equations for the exponents ai and ai : 0
0
2a1 þ a2 ¼ 0; 2a1 þ a2 ¼ 1 0
0
0
0
a1 þ a3 ¼ 0 ; a1 þ a3 ¼ 0
(7.17)
na1 þ a2 ¼ 1; na1 þ a3 ¼ 0 From (7.17) it follows that 1 2 1 0 ; a2 ¼ ; a3 ¼ ; a1 nþ2 nþ2 nþ2 1 n 1 0 0 ; a ¼ ; a ¼ ¼ nþ2 2 nþ2 3 nþ2
a1 ¼
(7.18)
Then (7.16) takes the form
Q2 T¼ at
) 1=ðnþ2Þ ( x ’ ðaQn tÞ1=ðnþ2Þ
(7.19)
Substituting the expression (7.19) into (7.12) yields the following ODE for the unknown function ’ d d’ n d’ þx ’ þ’¼0 ðn þ 2Þ dx dx dx
(7.20)
The boundary conditions for (7.20) are ’ðxÞ ¼ 0 at x ! 1; where x ¼
x ðaQn tÞ1=ðnþ2Þ
d’ðxÞ ¼ 0 at x ¼ 0 dx
(7.21)
:
The solution of (7.20) and (7.21) is (Zel’dovich and Raizer 2002) ’ðxÞ ¼ x20
n 2ðn þ 2Þ
"
2 #1=n x 1 x0
(7.22)
at x<x0 ; and ’ðxÞ ¼ 0
(7.23)
7.3 Heat and Mass Transfer Under Conditions of Forced Convection
165
at x ¼ 0, where x0 is a constant which is found from the energy invariant (7.14) ( x0 ¼
)1=ðnþ2Þ ðn þ 2Þ1þn 21n Gn ð1=2 þ 1=nÞ Gn ð1=nÞ npn=2
(7.24)
In (7.24) GðÞis the gamma function. The position and velocity of the thermal wave front are given by the following expressions xf ¼ x0 ðaQn tÞ1=ðnþ2Þ vf ¼ x 0
n 1=ðnþ2Þ 1 aQ n þ 2 tnþ1
(7.25) (7.26)
where xf ðtÞ and vf ðtÞ are the current coordinate and velocity of the thermal wave front.
7.3
7.3.1
Heat and Mass Transfer Under Conditions of Forced Convection Heat Transfer from a Hot Body Immersed in Fluid Flow
The first attempt of theoretical investigation of this problem by applying the dimensional analysis dates back to Lord Rayleigh (1915). He employed the Pitheorem for studying heat transfer from a hot body moving in an incompressible fluid. Rayleigh assumed that five dimensional parameters, namely, (1) the characteristic size of a body, say, a spherical particle, d; (2) its velocity relative to the surrounding medium v; (3) the temperature difference between the body and the undisturbed fluid far away from it DT; as well as (4) the fluid heat capacity c and (5) thermal conductivity k determine the rate of heat transfer h h ¼ f ðd; v; DT; c; kÞ
(7.27)
where DT ¼ Tw T1 ; Tw and T1 are the body and undisturbed fluid temperature, respectively. The unknown rate of heat transfer h and the governing parameters of the problem d; v; DT; c and k have the following dimensions ½h ¼ JT 1 ; ½d ¼ L; ½v ¼ LT 1 ; ½DT ¼ y; ½c ¼ JL3 y1 ; ½k ¼ JL1 T 1 y1 where J and y are the independent units of heat and temperature.
(7.28)
166
7 Heat and Mass Transfer
The set of the five governing parameters contains four parameters with independent dimension, so that n k ¼ 1: Choosing as the parameters with the independent dimensions d; v; DT; and k, we write in accordance with the Pi-theorem the dimensionless form of (7.27) as P ¼ ’ðP1 Þ 0
(7.29) 0
0
0
where P ¼ h =d a1 va2 DT a3 ka4 ; and P1 ¼ c=d a1 va2 DT a3 ka4 : Using the principle of dimensional homogeneity, we find values of the exponents ai 0 0 0 0 0 and ai : a1 ¼ 1; a2 ¼ 0; a3 ¼ 1; a4 ¼ 1; a1 ¼ 1; a2 ¼ 1; a3 ¼ 0; and a4 ¼ 1: Then (7.29) takes the form h dvc ¼’ dkDT k
(7.30)
Defining heat flux as q ¼ h=S; with S being the surface of the body, we rewrite (7.30) as Nu ¼ ’ðPeÞ
(7.31)
where Nu ¼ qd=kDT; and Pe ¼ dvc=k are the Nusselt and Peclet numbers, respectively. Equation 7.31 shows that the Nusselt number depends on a single dimensionless group Pe. At a fixed Peclet number the rate of heat transfer h is directly proportional to the temperature difference DT and the characteristic size of the body d, whereas the heat flux q is inversely proportional to d: The results of Rayleigh demonstrated the efficiency of application of the dimensional analysis to problems related to convective heat transfer, which attracted a significant attention to this approach which was thoroughly discussed in the following references: Riabouchinsky (1915), Brigman (1922) and Sedov (1993). In particular, Riabouchinsky made a remark about the choice of the system of units of in description of convective heat transfer phenomena. The system of units that Rayleigh used includes three mechanical units (length L; mass M and time TÞ and two independent thermal units for quantity of heat (understood as thermal energy) J and temperature y: It is emphasized that it is possible to express the dimension of temperature, and accordingly the other thermal quantities, by means of the basic mechanical units LMT: Indeed, temperature can be related with the average kinetic energy of molecules with the dimension ½ML2 T 2 in the LMT system of units. According to the First Law of thermodynamics, the dimension of heat in the LMT system of units ss J ½ML2 T 2 : Then the dimensions of the governing parameters in (7.27) are as follows ½d ¼ L;
½v ¼ LT 1 ;
½y ¼ ML2 T 2 ;
½c ¼ L3 ;
½k ¼ L1 T 1
(7.32)
7.3 Heat and Mass Transfer Under Conditions of Forced Convection
167
Three parameters of the five in (7.32) have independent dimensions, so that n k ¼ 2: In this case the dimensionless form of (7.27) reads P ¼ ’ðP1 ; P2 Þ
(7.33)
where P ¼ h =kDTd; P1 ¼ dvc=k; and P2 ¼ cd3 : Thus, instead of (7.29) that determined the dimensionless rate of heat transfer as a function of a single dimensionless group P1 ; we arrive at (7.33) where the unknown quantity P is a function of two dimensionless groups (P1 and P2 ) that makes its much less valuable. Essentially one sees that under the same conditions (7.29) and (7.33) determine different dependences of the dimensionless heat transfer rate P on the governing parameters. In particular, according to (7.29) the dimensionless heat transfer rate P does not depend on the heat capacity c at vc ¼ const, whereas (7.33) shows the existence of such dependence. The seeming contradiction related to using two different systems of units in the above-mentioned problem deserves the following comments: 1. By choicing governing parameters, one, in fact, makes some assumptions on the structure of substance involved. In the frame of the continuum approach and thermodynamics thermal phenomena are described in the macroscopic approximation fully the molecular structure of substance. Accoordingly, in this case Rayleigh’s approach is correct, while Riabouchinsky’s counter-example is illegal. 2. The expression of dimensions of thermal quantities via mechanical units is based on the First Law of thermodynamics that postulate the equivalence of all kinds of energy, in particular, the thermal and mechanical ones. Therefore, in the general case the set of governing parameters that determine the heat transfer rate should be supplemented by two constants characterizing the relation of thermal energy to mechanical one. These are the mechanical equivalent of heat ½ j ¼ ML2 T 2 J 1 and Boltzmann’s constant ½kB ¼ ML2 T 2 y1 . Accordingly, Rayleigh’s set of the governing parameters read d;
v;
DT;
c;
k;
j;
kB
(7.34)
Then the apppplication of the Pi-theorem leads to the following expression for the heat transfer rate (Sedov 1993) h dvc jcd 3 ¼ ’ð ; Þ kDTd k kB
(7.35)
When the effect of transformation of mechanical energy into heat (dissipation) is negligible, (7.35) reduces to (7.30). 3. Many additional questions were raised by Rayleigh’s analysis (Brigman 1922). In particular, concerns were driven by the fact that density and viscosity are missing in the set of governing parameters. The lack of these quantities in the set of the governing parameters, seemingly diminishes the value of the result of the dimensional analysis in this case.
168
7
Heat and Mass Transfer
Consider Rayleigh’s problem with the account for the fluid density and viscosity. Then the functional equation for the heat flux q reads q ¼ f ðd; v; DT; k; m; cP ; rÞ
(7.36)
where ½d ¼ L; ½v ¼ LT 1 ; ½r ¼ L3 M; ½m ¼ L1 MT 1 ; ½cP ¼ JMy1 ; and ½q ¼ JT 1 L2 ; ½k ¼ LMT 3 y1 ; ½DT ¼ y The set of the governing parameters in (7.36) contains seven parameters, five of them with independent dimensions, so that n k ¼ 2: Take as the parameters with independent dimensions d; DT; k; m and r and using the Pi-theorem transform (7.36) to the following dimensionless form P ¼ ’ðP1 ; P2 Þ
(7.37)
0
0
0
0
0
where P ¼ q=da1 DT a2 ka3 ma4 ra5 ; P1 ¼ v=d a1 DT a2 ka3 ma4 ra5 ; 00
00
00
00
00
and P2 ¼ cP =d a1
DT a2 ka3 ma4 ra5 : Using the principle of dimensional homogeneity, we find the values of the 0 00 exponents ai ; ai and ai as a1 ¼ 1; 0
a1 ¼ 1; 00
a1 ¼ 0;
a2 ¼ 1; 0
a3 ¼ 1; 0
a2 ¼ 0;
a3 ¼ 0;
00
00
a2 ¼ 0;
a3 ¼ 1;
a4 ¼ 0; 0
a4 ¼ 1; 00
a4 ¼ 1;
a5 ¼ 0 0
a5 ¼ 1
(7.38)
00
a5 ¼ 0
Then (7.37) takes the form Nu ¼ ’ðRe; PrÞ
(7.39)
where Nu ¼ qd=kDT; Re ¼ vdr=m; and Pr ¼ cp m=k are the Nusselt, Reynolds and Prandtl numbers, respectively. In the particular case corresponding to creeping flows with small Reynolds numbers (Re<1) the inertial effects are negligible. Then it is possible to omit density from the set of the governing parameters and write (7.36) as follows q ¼ f ðd; v; DT; k; m; cP Þ
(7.40)
The dimensionless form of (7.40) is Nu ¼ ’ðPrÞ
(7.41)
where Pr is the Prandtl number. The explicit form of (7.31) and (7.37) is determined either experimentally or by solving the Navier-Stokes and energy equations. For flows of incompressible fluids
7.3 Heat and Mass Transfer Under Conditions of Forced Convection
169
with constant physical properties the momentum and continuity equations can be uncoupled and integrated independently of the energy equation. In this case the heat transfer problem reduces to solving the energy equation with a known velocity field. For creeping flows the expressions for the Nusselt number are typically presented in the form of the series of the Peclet number. For example, Acrivos and Taylor (1962) obtained the following expression for the Nusselt number for a spherical particle at Re 1 and Pr 1 using the method of matched asymptotic expansions 1 Nu ¼ 2 þ Pe þ Pe2 ln Pe þ 0:829Pe2 þ Pe3 ln Pe 2
(7.42)
For calculating heat transfer from spherical bodies in a wide range of the Reynolds and Prandtl numbers a number of empirical correlations have been proposed (Soo 1990). In particular, the correlation valid in the range 1
Nu ¼ 2 þ 0:459Re0:55 Pr
(7.43)
It is seen that the form of the dependences NuðPeÞ and NuðRr; PrÞobtained in the framework of the dimensional analysis is identical to the form of correlations (7.42) and (7.43) resulting from the analytical solution of the problem and experimental data.
7.3.2
The Effect of Particle Rotation
The effect of rotation of a spherical particle on heat transfer is due to the influence of the centrifugal force on fluid around the particle (Kreith 1968). The particle rotation about an axis through its two poles promotes secondary currents that are directed toward the poles and outward from the equatorial region. The superposition of the secondary flows on the main flow (driven by the inertial, buoyancy and pressure forces) determines the hydrodynamic structure of the overall flow and its evolution, as well as the general characteristics of heat and mass transfer. In the general case the intensity of heat transfer from a spinning spherical particle depends on nine dimensional parameters accounting for the effect of the inertial, buoyancy and centrifugal forces: r; m; d; u1 ; k; cP ; DT; ðbgÞ; and o. Accordingly, the functional equation for the heat flux from the particle reads q ¼ f ðr; m; d; u1 ; k; cP ; DT; bg; oÞ
(7.44)
where ½b ¼ y1 is the thermal expansion coefficient, ½g ¼ LT 2 is the gravity acceleration, and ½o ¼ T 1 is the angular speed of rotation.
170
7 Heat and Mass Transfer
Five governing parameters of the nine in (7.44) have independent dimensions so that the difference n k ¼ 4: Then, according to the Pi-theorem, (7.44) reduces to the following form P ¼ ’ðP1 ; P2 ; P3 ; P4 Þ where 00 a2
00 a3
00 a4
P ¼ q=ra1 ma2 d a3 ka4 DT a5 ; 00 a5
000 a1
000 a2
(7.45) 0
000 a3
0
0
0
0
00
P1 ¼ u=ra1 ma2 d a3 ka4 DT a5 ,P2 ¼ cP =ra1 000 a4
000 a5
1V
1V
m d k DT ; P3 ¼ bg=r m d k DT ; and P4 ¼ o=ra1 ma2 1V 1V a a1V a d 3 k 4 DT 5 . Taking into account the dimension of the heat flux q and those of the parameters 0 00 000 governing it, we find the values of the exponents ai ; ai ; ai ; ai and a1V i : They are equal to a1 ¼ 0; 0
a1 ¼ 1; 00
a1 ¼ 0; 000
a2 ¼ 0;
a3 ¼ 1;
0
a3 ¼ 1;
00
a3 ¼ 0;
000
a2 ¼ þ1; a2 ¼ 1;
0
00
000
a1 ¼ 2;
a2 ¼ 2;
a3 ¼ 3;
a1V 1 ¼ 1;
a1V 2 ¼ 1;
a1V 3 ¼ 2;
a4 ¼ 1;
a5 ¼ 1
0
0
a4 ¼ 0; 00
a4 ¼ 1; 000
a4 ¼ 0; a1V 4 ¼ 0;
a5 ¼ 0 00
a5 ¼ 0
(7.46)
000
a5 ¼ 1 a1V 5 ¼ 0
Bearing in mind (7.46), we rewrite (7.45) as follows Nu ¼ ’ðRe; Pr; Gr; Reo Þ
(7.47)
where Nu¼hd=k;Re¼u1 d=n;Pr¼n=a;Gr¼bgðTw T1 Þd3 =n2 ;and Reo ¼od 2 =n are the Nusselt, Reynolds, Prandtl, Grashof and particle rotational Reynolds numbers, respectively, h¼q=DT is the heat transfer coefficient, and a¼k=rcP is the thermal diffusivity. The approaches based on the boundary layer theory are used for theoretical description of heat transfer of a spinning particle (Dorfman 1967; Banks 1965; Chao and Greif 1974; Lee et al. 1978). Under the conditions of forced convection due to particle rotation the heat transfer coefficient depends on the rotational Reynolds and Prandtl numbers, as is anticipated from (7.47). The dependence on Reo may be expressed as (Kreith 1968) 0:4 Nu ¼ 0:43Re0:5 o Pr
(7.48)
Nu ¼ 0:066Reo 0:67 Pr0:4
(7.49)
for Reo <5 105 ; and
for Reo >5 105 .
7.3 Heat and Mass Transfer Under Conditions of Forced Convection
171
The comparison of a number of theoretical predictions with experimental data on heat transfer of a spherical particle under the conditions of forced convection due to its rotation shows that all the dependences NuðReo ; PrÞ proposed have the form Nu ¼ ARe0:5 Pr0:4 ; with only the values of the factor A being different (Hussaini and Sastry 1976). The empirical correlations for the average Nusselt number of a spherical particle rotating in air at rest or in air stream were suggested by Eastop (1973). In the former case the correlation reads Nu ¼ 0:353Re0:5 o
(7.50)
whereas in the latter one (at Reo =Re>0:54) it has the following form Reo 0:54 Nu ¼ 0:288Re0:6 1 þ 0:167 Re
7.3.3
(7.51)
The Effect of the Free Stream Turbulence
The free stream turbulence tends to enhance heat transfer from a particle to the surrounding fluid due to the eddies penetrating from the external flow into the particle boundary layer. The disturbances of the near-wall flow facilitate transition to turbulence and shift the boundary layer separation point downstream over the particle surface. The intensity of heat transfer from a heated spherical particle immersed into turbulent flow depends on the physical properties of fluid: its density ½r ¼ L3 M; viscosity ½m ¼ L1 MT 1 ; thermal conductivity ½k ¼ JT 1 y1 L1 ; and specific heat ½cP ¼ Jy1 M1 . It also depends on particle diameter ½d ¼ L and its velocity ½v1 ¼ LT 1 ; the temperature difference between the particle and surrounding fluid ½DT ¼ y; the characteristic velocity of turbulent fluctuations h i pffiffiffiffiffi ve0 ¼ LT 1 (ve0 ¼ v02 is the root mean square of the turbulence velocity fluctuations) and the integral scale of turbulence ½l ¼ L: Accordingly, the functional equation for the heat flux ½q ¼ JT 1 L2 reads q ¼ f ðr; m; k; cP ; v1 ; DT; d; ve0 ; lÞ
(7.52)
Equation 7.52 contains nine dimensional parameters, five of them having independent dimensions. Then, according with the Pi-theorem, the dimensionless form of (7.52) appears to be P ¼ ’ðP1 ; P2 ; P3 ; P4 Þ
(7.53)
172
7 Heat and Mass Transfer 0
0
0
0
0
00
00
00
where P ¼ q=ra1 ma2 ka3 DT a4 d a5 ; P1 ¼ cP =ra1 ma2 ka3 DT a4 d a5 ; P2 ¼ v1 =ra1 ma2 ka3 00 00 000 000 000 000 000 1V 1V 1V 1V 1V DT a4 d a5 ; P3 ¼ ve0 =ra1 ma2 ka3 DT a4 d a5 ; and P4 ¼ l=ra1 ma2 ka3 DT a4 da5 and the 0
00
000
exponents ai ; ai ; ai ; ai and a1V i are equal to a1 ¼ 0; 0
a1 ¼ 0;
a2 ¼ 0; 0
a2 ¼ 1;
00
a2 ¼ 1;
000
a1 ¼ 1; a1 ¼ 1; a1V 1 ¼ 1;
a3 ¼ 1; 0
a3 ¼ 1;
00
a3 ¼ 0;
a2 ¼ 1;
000
a1V 2 ¼ 0;
a4 ¼ 1; 0
a4 ¼ 0;
00
a4 ¼ 0;
00
a3 ¼ 0;
000
a4 ¼ 0;
a1V 3 ¼ 0;
a1V 4 ¼ 0;
000
a5 ¼ 1; 0
a5 ¼ 0; 00
a5 ¼ 1;
(7.54)
000
a5 ¼ 1; a1V 5 ¼ 1; 0
00
000
Taking into account the values of the exponents ai ; ai ; ai ; ai and a1V i , it is possible to present (7.52) in the following form Nu ¼ ’ðPr; Re; ReT ; lÞ
(7.55)
where Nu ¼ qd=kDT; Pr ¼ cP m=k; Re ¼ v1 dr=m; and ReT ¼ ve0 =v1 v1 dr=m e0 ¼ Tu Re (Tu ¼ vv1 is the turbulence intensity) are the Nusselt, Prandtl, Reynolds and turbulent Reynolds number, respectively, l ¼ l=d is the dimensionless turbulence scale. Equation 7.55 shows that the contribution of turbulence to heat transfer is determined by two dimensionless parameters accounting for the turbulence intensity and the size of turbulent eddies. Depending on values of these parameters, different conditions of heat transfer may be realized. When l 1; the particle experiences a time-dependent flow, whereas when l<<1, the flow becomes quasi-steady. In the latter case the heat transfer coefficient depends on the Reynolds and Prandtl numbers, as well as on the turbulence intensity Tu (Kestin 1966) Nu ¼ ’ðPr; Re; Tu Þ
(7.56)
In particular, for a spherical particle immersed in the turbulent flow with 2 103
(7.57)
for ReT <103 and f ðReT Þ ¼ 0:145Re0:25 for Re>103 : where f ðReT Þ ¼ 0:629Re0:035 T T The difference in the dependences f ðReT Þcorresponding to small and large values of ReT are related to the peculiarities of flow about a particle at low and high turbulence intensity. The influence of the free stream turbulence on the particle boundary layer is weak enough at low Tu : When turbulence disturbances reach
7.3 Heat and Mass Transfer Under Conditions of Forced Convection
173
some critical level, laminar-turbulent transition in the particle boundary layer occurs. It is accompanied by significant changes in the flow structure, reducing dramatically the particle drag.
7.3.4
The Effect of Energy Dissipation
The internal friction of viscous fluid near the surface of a solid body moving in it is accompanied by the mechanical energy dissipation. The latter leads to heating the body, which can result in a significant temperature increase of the body temperature Tw above the temperature of the undisturbed fluid T1 : In flows of viscous incompressible fluid heating of solid bodies depends on physical properties of fluid, relative velocity of the body to that of fluid and its characteristic size. These considerations allow us to present the functional equation for equilibrium temperature difference DT ¼ Tw T1 as follows DT ¼ f ðp; m; k; cP ; u; d Þ
(7.58)
where p; m and k are the viscosity, thermal conductivity, cp is the specific heat of fluid, u is the relative velocity of the body, and d its characteristic size. Since the set of the five governing parameters of the problem contains three parameters possessing independent dimensions, we can present (7.58) in the following form P ¼ ’ðP1 ; P2 Þ 0
0
a
0
(7.59) 0
00
00
a
00
00
where P ¼ DT=pa1 ma2 caP3 ua4 ; P1 ¼ k=pa1 ma2 cP3 ua4 ; and P2 ¼ d =pa1 ma2 cP3 ua4 . 0 00 Determining the values of the exponents ai ; ai and ai using the principle of dimensional homogeneity, we find a1 ¼ 0; a2 ¼ 0; a3 ¼ 1; a4 ¼ 2; 0 0 0 0 00 00 00 00 a1 ¼ 0; a2 ¼ 1; a3 ¼ 1; a4 ¼ 0; a1 ¼ 1; a2 ¼ 1; a3 ¼ 0 and a4 ¼ 1: Then, we can transform (7.59) to the following form DT ¼
u2 ’ðPr; ReÞ cP
(7.60)
The expression (7.60) shows that the temperature difference due to viscous dissipation is determined by the relative velocity of the body, as well as values of the Reynolds and Prandtl numbers. The evaluations show that in the two limiting cases corresponding to small and large Reynolds numbers heating of a body moving in fluid can be expressed as (Landau and Lifshitz 1987) Tw T1 ¼ c Pr
u2 cP
(7.61)
174
7 Heat and Mass Transfer
Tw T 1 ¼
u2 ’ ðPrÞ cP
(7.62)
respectively. In (7.61) c is a constant that depend on the body shape; ’ ðPrÞ is a function of the Prandtl number. Essentially, that in both cases corresponding to small and large Reynolds number body heating due to viscous dissipation is directly proportional to the square of the relative velocity.
7.3.5
The Effect of Velocity Gradient
Consider heat flux from a spherical particle immersed in a uniform flow of hot incompressible fluid with linear velocity distribution. Assume that the total heat flux is a result of superposition of two components: one – due to the rectilinear translational motion relative to the surrounding fluid, and the other one – due to shear. In this case it is possible to write the equation for the total heat flux in the form qt ¼ qc þ qsh
(7.63)
where qt ; qc and qsh are the total heat flux, heat flux due to rectilinear translational motion, and heat flux due to shear, respectively, [qt] ¼ [qc] ¼ [qsh] ¼ JT1L2. The functional equations for each component of the total heat flux read qc ¼ fc ðr; m; d; u; k; cP ; DTÞ
(7.64)
qsh ¼ fsh ðr; m; d; g; k; cP ; DTÞ
(7.65)
where g½T 1 is the shear rate, and u is the velocity of the particle, and d is the diameter of a spherical particle. Equations 7.64 and 7.65 contain seven governing parameters, five of which having independent dimensions. According to the Pi-theorem, (7.64) and (7.65) can be presented as qc ¼ ra1 ma2 d a3 ka4 DT a5 ’
qsh ¼ r m d k DT a1 a2 a3 a4
a5
u
!
cP
0 ; 0 0 0 0 0 0 0 0 0 ra1 ma2 d a3 ka3 DT a5 ra1 ma2 d a3 ka4 DT a5
’sh
g
cP
(7.66) !
0 ; 00 0 00 0 0 0 00 00 00 ra1 ma2 da3 ka4 DT a5 ra1 ma2 d a3 ka4 DT a5 (7.67)
7.3 Heat and Mass Transfer Under Conditions of Forced Convection
175
Divide the left and right hand sides of (7.63) by the product ra1 ma2 d a3 ka4 DT a5 . As a results, we arrive at the following dimensionless equation (accounting for the equality ai ¼ ai Þ P ¼ Pc þ Psh where
(7.68)
P ¼ qt =ra1 ma2 d a3 ka4 DT a5 ; Pc ¼ qc =ra1 ma2 da3 ka4 DT a5 ; Psh ¼ qsh =ra1 ma2
da3 ka4 DT a5 ; Pc ¼ ’c ðP1 ; P2 Þ, Psh ¼ ’sh ðP1 ; P2 Þ, 00
00
00
00
00
0
0
0
0
0
0
0
0
0
P1 ¼ u=ra1 ma2 d a3 ka4 DT a5 , 0
00
P2 ¼ cP =ra1 ma2 d a3 ka4 DT a5 , P1: ¼ g=ra1 ma2 d a3 ka4 DT a5 , and P2 ¼ cp =ra1 00
00
00
00
ma2 d a3 ka4 DT a5 . Taking into account the dimensions of qt ; qc ; qsh ; u; g and cP ; as well as the fact that the governing parameters r; m; d; k and DTpossess independent 0 00 0 00 dimensions, we find the values of the exponents ai ; ai ; ai ; ai ; ai ; and ai as a1 ¼ 0; a2 ¼ 0; a3 ¼ 1; a4 ¼ 1; a5 ¼ 1 0
0
0
0
0
a1 ¼ 1; a2 ¼ 1; a3 ¼ 1; a4 ¼ 0; a5 ¼ 0 00
00
00
00
00
a1 ¼ 0; a2 ¼ 1; a3 ¼ 0; a4 ¼ 1; a5 ¼ 0 0
0
0
00
00
0
0
00
00
(7.69)
a1 ¼ 1; a2 ¼ 1; a3 ¼ 2; a4 ¼ 0; a5 ¼ 0 00
a1 ¼ 0; a2 ¼ 1; a3 ¼ 0; a4 ¼ 1; a5 ¼ 0 According to (7.69), (7.68) takes the form Nut ¼ Nuc þ Nush
(7.70)
Where Nut ¼ qt d=kDT; Nuc ¼ qc d=kDT; Nush ¼ qsh =kDT; P1 ¼ Re ¼ udr=m; P2 ¼ Pr ¼ cP m=k; P1 ¼ d 2 grcP =k and P2 ¼ Pr ¼ cP m=k. At low Reynolds number the Nusselt number approaches the value of 2. In this case (7.70) reads Nut ¼ 2 þ ’sh ðPe1 ; PrÞ
(7.71)
The analytical solution of the problem yields the following expression for the Nusselt number which is valid for low Reynolds and Peclet numbers (Frankel and Acrivos 1968) Nu ¼ 2 þ
0:9104 ð2pÞ1=2
Pe
where Pe ¼ r 2 grcP =k and r is the particle radius.
(7.72)
176
7.3.6
7 Heat and Mass Transfer
Mass Transfer to Solid Particles and Drops Immersed in Fluid Flow
From the physical point of view the diffusion flux to a reactive spherical particle immersed in an infinite reactant flow is determined by the physical properties of liquid (its density, viscosity and diffusivity), particle size, reactant concentration in the undisturbed flow, fluid velocity, as well as the kinetics of heterogeneous chemical reaction at the particle surface. When the rate of the surface reaction exceeds the rate of reactant diffusion in the carrier fluid, the concentration of the reactant at the particle surface equals zero. In this case the influence of the kinetic characteristics of the heterogeneous chemical reaction at the particle surface on the mass transfer rate in the fluid bulk is negligible and these characteristics should be excluded from the set of the governing parameters. At small values of the Reynolds number corresponding to the Stokes creeping flow, the fluid density does not affect the flow field and can be safely excluded from the governing parameters. Assume that the carrier phase is liquid. Taking into account the fact that typically in liquids the admixture (e.g., reactant) diffusivities are much less than viscosity (D<
(7.73)
where r0 is the radius of particle, u1 is the undisturbed flow velocity, and c1 is the reactant concentration in the liquid carrier far from the particle. The dimensions of the governing parameters are ½ D ¼ L2 T 1 ;
½r0 ¼ L;
½u1 ¼ LT 1 ;
½c1 ¼ L3 M
(7.74)
Three governing parameters from the set (7.74) have independent dimensions, so that the difference n k ¼ 1: Then, accoring to the Pi-theorem, (7.73) transform as P ¼ ’ðP1 Þ
(7.75)
where P ¼ qm r0 =Dc1 ¼ Sh is the Sherwood number, and P1 ¼ u1 r0 =D ¼ Ped is the Peclet number. Small droplets immersed in liquid flow stay spherical and effectively undeformable due to the action of the interfacial tension. However, the mass transfer to the surface of a small reactive droplet of viscosity m2 immersed into a reactant liquid solution solution of viscosity m1 can differ significantly from the mass transfer to a solid reactive particle. This stems from the different hydrodynamic conditions at the liquid-liquid and liquid-solid interfaces. The existence of a non-zero interfacial tangential velocity at the liquid-liquid interface determined by
7.3 Heat and Mass Transfer Under Conditions of Forced Convection
177
viscosities of two liquids determines the interfacial velocity as characteristic scale of the problem. Then the functional equation for the diffusion flux qm takes the form qm ¼ f ðD; r0 ; v ; c1 Þ
(7.76)
where v is the absolute value of the interfacial velocity at the drop equator. In this case the dimensionless form of (7.76) reads P ¼ ’ðeP1 Þ
(7.77)
where e ¼ cðm2 =m1 Þ. The explicit forms of the dependences (7.75) and (7.77) can be found only experimentally or via theoretical solutions of the corresponding problem (Levich 1962). When the rate of the heterogeneous reaction at the particle surface is much larger than the diffusion rate, the problem reduces to the integration of the species balance equation. In the framework of the diffusion boundary layer, and for the creeping flow velocity distribution it reads vr
2 @c v’ @c @ c 2 @c þ þ ¼D r @y @r @r 2 r @r
(7.78)
with the boundary conditions c ! c1 at y ! 1; c ¼ 0 at r ¼ r0
(7.79)
where r; ’; and y are the spherical coordinates with the origin at the particle center. The Levich solution results in the following dependence of the Sherwood number on the Peclet number Sh ¼ 7:85Pe1=3
(7.80)
for a solid particle, and in the dependence 1=2 2 m1 1=2 Sh ¼ pffiffiffiffiffiffi Pe m1 þ m2 6p
(7.81)
for a liquid droplet, where Sh ¼ r0 =Dc1 is the Shrwood number, ¼ I=r02 is the average diffusion flux, I ¼ qm ds is the total diffusion flux at the particle or droplet surface. The comparision of the results of the dimensional analysis given by (7.75) and (7.77) with the analytical solution of the problem corresponding to (7.80) and (7.81) reveals the benefits of applying the dimensional anaysis to study mass transfer to solid particles and droplets in flows of liquid reactant solutions. It is seen that results
178
7 Heat and Mass Transfer
of the dimensional analysis correctly represent the qualitative character of the dependence of the Sherwood number on the Peclet number, but do not allow finding the exact form of this dependence. The knowledge of an exact form of the dependence ShðPeÞ is important for understanding of the laws of mass transfer to solid particles and droplets. As can be seen from (7.80) and (7.81), two different scaling laws in the dependence on the Peclet number exist, with the exponents being one third and one half for particles and droplets, respectively. That shows that ratio of the mass transfer intensity to a droplet to that of a comparable solid particle Shd Shp depends weekly on the Peclet number and strongly on the viscosities of the droplet and surroundind medium.
7.4 7.4.1
Heat and Mass Transfer in Channel and Pipe Flows Couette Flow
Consider heat transfer in fully laminar flow of incompressible fluid between two parallel closely located infinite plates having different temperature. Let the lower plate is motionless, whereas the upper one moves with a constant velocity U. Assuming that temperatures of the lower and upper plates being equal to T0 and T1 , respectively (with T1 >T0 ). In the case when T0 and T1 are constant, the fluid temperature changes in y-th direction only (coordinate y is normal to the plates). The set of the governing parameters that determine heat flux from the upper wall to the fluid q, as well as the fluid temperature # ¼ T T0 reads ½k ¼ LMT 3 y1 ; ½m ¼ L1 MT 1 ; ½ H ¼ L; ½U ¼ LT 1 ; ½# ¼ y; ½ y ¼ L
(7.82)
where H is the gap between the plates, and # ¼ T1 T0 . Accordingly, the functional equations for q and # have the following form q ¼ fq ðk; m; H; U; # Þ
(7.83)
# ¼ f# ðk; m; H; U; # ; yÞ
(7.84)
Four governing parameters listed in (7.83) and (7.84) possess independent dimension, so that the difference n k for (7.83) and (7.84) equals 1 and 2, respectively. Then the dimensionless form of (7.83) and (7.84) is Pq ¼ ’q ðP1q Þ
(7.85)
P# ¼ ’# ðP1# ; P2q Þ
(7.86)
7.4 Heat and Mass Transfer in Channel and Pipe Flows 0
0
wherePq ¼ q=ka1 H a2 U a3 #a4 ; P1q ¼ m=ka1 Ha2 0
179
0
a
0
0
0
0
U a3 #4 ; P# ¼ #=ka1 Ha2 Ua3 #a4 ; P1# ¼ m=ka1 Ha2 U a3 #a4 00 a1
00 a2
00 a3
and
00
a #4 .
P2# ¼ y=k H U Bearing in mind the dimensions of heat flux, temperature and the governing 0 0 00 parameters, we find the values of the exponents ai ; ai ; ai ; ai and ai as a1 ¼ 1; a2 ¼ 1; a3 ¼ 0; a4 ¼ 1 0
0
0
0
a1 ¼ 1; a2 ¼ 0; a3 ¼ 2; a4 ¼ 1 a1 ¼ 0; a2 ¼ 0; a3 ¼ 0; a4 ¼ 1 0
0
0
(7.87)
0
a1 ¼ 1; a2 ¼ 0; a3 ¼ 2; a4 ¼ 1 00
00
00
00
a1 ¼ 0; a2 ¼ 1; a3 ¼ 0; a4 ¼ 0 Using (7.87), (7.85) and (7.86) take the following form Nu ¼ ’q ðPr EcÞ
(7.88)
# ¼ ’# ð; Pr EcÞ
(7.89)
where Nu ¼ qH=kðT1 T0 Þ; Pr ¼ mcP =k; and Ec ¼ U 2 =cP ðT1 T0 Þ are the Nusselt, Prandtl and Eckert numbers, respectively, # ¼ #=# ; and ¼ y=H: The analytical solution of the problem on heat transfer in Couette flow reads (Bayley et al. 1972) 1 # ¼ 1 þ Pr Ecð1 Þ 2
(7.90)
The heat flux from the top plate to the fluid is expressed using the analytic solution as @T Ts T0 1 q¼k 1 Pr Ec ¼k H @y y¼0 2
(7.91)
The comparison of the results of the exact analytical solution of the problem with its analysis based on the Pi-theorem indicates a certain insufficiency of the latter approach. In particular, the dimensional analysis allows finding the dimensionless groups only, but cannot reveal a number of the important peculiarities of the process, for example, the reverse of the heat flux at high values of the product PrEcwhen q changes sign.
180
7.4.2
7 Heat and Mass Transfer
The Entrance Region of a Pipe
The evaluation of the thickness of the thermal boundary layer within the entrance region of heated pipe is tackled in the present subsection. Considering flows of incompressible fluids it is possible to assume that the local thickness of the thermal boundary layer dT ðxÞ depends on five dimensional parameters: the density r, thermal conductivity k and specific heat of the fluid cP ; its velocity in the undisturbed core u and the cross-section coordinate x:2 Accordingly, the functional equation for the thickness of the thermal boundary layer is written as dT ðxÞ ¼ fT ðr; k; cP ; u; x; Þ
(7.92)
In the system of units Lx Ly Lz MTJy the dimensions of the governing parameters and the thickness of the thermal boundary layer are expressed as h i 1 1 1 1 ; ½cP ¼ JM1 y1 ; ½u r L1 L L M ; k L1 z1 x y x Ly Lz JT y
(7.93) ¼ Lx T 1 ; ½ x ¼ L; ½d ¼ Ly dT Ly It is seen that all the governing parameters have independent dimensions. Therefore, in accordance with the Pi-theorem, (7.92) takes the form dT ðxÞ ¼ cT ra1 ka2 cap3 ua4 xa5
(7.94)
where cT is a constant. Accounting for the fact that ½dT ¼ Ly , we find the values of the exponents ai . They are: a1 ¼ 1=2; a2 ¼ 1=2; a3 ¼ 1=2; a4 ¼ 1=2 and a5 ¼ 1=2. Then (7.94) takes the form dT ðxÞ ¼ cT
kx rucP
1=2 (7.95)
or 1=2
1=2
dT ðxÞ ¼ cT x Pr Red
(7.96)
where Pr ¼ n=a and Re ¼ ud=n are the Prandtl and Reynolds numbers with n and a being the kinematic viscosity and thermal diffusivity, respectively. Taking into account that thickness of the thermal boundary layer in the crosssection that correspond to the end of entrance region equals d=2, we find the thermal entrance lenth lenT as lenT ¼ cT Red Pr d where cT ¼ ð2cT Þ2 .
(7.97)
7.4 Heat and Mass Transfer in Channel and Pipe Flows
181
Equations (5.12) and (7.95) show that the dynamic and thermal entrance lengths of a pipe are determined by its diameter, the physical properties of fluid and flow velocity. Therefore, it is possible to use len and lenT as some generalized parameters in studies of flow resistance and heat transfer in the of pipes.
entrance section Introducing the dimensional parameters as ðdP=dxÞ ðru2 Þ ¼ L1 and ½qs =ðkDT Þ ¼ L1 (where qs is the heat flux at the pipe wall), we write the functional equations for these parameters as follows ðdP=dxÞ ¼ fP ðlen ; xÞ ru2 qs ¼ fq ðlenT ; xÞ kDT
(7.98) (7.99)
In both expressions the difference between the number of dimensional parameters n and the number of parameters having independent dimensions k is equal to 1. Then, according to the Pi-theorem, (7.98) and (7.99) are deduced to ðdP=dxÞlen x (7.100) ¼ ’ P ru2 len qs len:T x ¼ ’q ð Þ DT len:T
(7.101)
Substituting (4.14) and (6.65) into (7.100) and (7.101), we arrive at the following dimensionless expressions l¼ Nu ¼
1 ’ ðXÞ Red P
1 ’ ðXT Þ Pr Red q
(7.102) (7.103)
where l ¼ ½ðdP=dxÞdc =ru2 and Nu ¼ qs lenT =kDT are the friction factor and the Nusselt number, as well as X ¼ x=dc Red and XT ¼ X=dcT Pr Red . Comparing (7.95) with (5.12), we find the relation between the thicknesses of the thermal and dynamical boundary layers as dT 1=2
Pr d
7.4.3
(7.104)
Fully Developed Flow
Consider heat transfer in laminar pipe flow of incompressible fluid. Let pipe diameter be denoted as d. We will use the LMTJy system of units. Assume that physical properties of fluid are constant and flow is fully developed dynamically
182
7 Heat and Mass Transfer
and thermally. Let heat flux at the wall qs ¼ hðtw tm Þ is constant along the tube with ½h ¼ JT 1 L2 y1 being the heat transfer coefficient, and tw and tm being the wall and mean (bulk) fluid temperature, respectively. It is plausible to assume that heat flux at the pipe wall depends on the physical properties of fluid (½m ¼ L1 MT 1 ; ½k ¼ JT 1 L1 y1 ; ½cP ¼ JM1 y1 ; ½r ¼ L3 MÞ; the mean velocity of flow ½u ¼ LT 1 ; the difference between the wall and bulk temperature ½Dt ¼ tw tm ¼ y and the diameter ½d ¼ L qs ¼ fq ðr; m; k; cP ; u; Dt; dÞ
(7.105)
Equation 7.105 contains seven governing parameters, five of them with independent dimensions. Applying the Pi-theorem, we transform (7.105) to the following dimensionless form P ¼ ’q ðP1 ; P2 Þ 0
(7.106) 0
0
0
0
00
where P ¼ qs =ra1 ma2 ka3 d a4 Dta5 ; P1 ¼ u=ra1 ma2 ka3 d a4 Dta5 and P2 ¼ cP =ra1 00 a2
00 a3
00 a4
00 a5
m k d Dt : 0 00 The exponents ai ; ai and ai are equal to a1 ¼ 0; a2 ¼ 0; a3 ¼ 1; a4 ¼ 1; a5 ¼ 1 0
0
0
0
0
a1 ¼ 1; a2 ¼ 1; a3 ¼ 0; a4 ¼ 1; a5 ¼ 0 00
0
00
00
(7.107)
00
a1 ¼ 0; a2 ¼ 1; a3 ¼ 1; a4 ¼ 0; a5 ¼ 0 Then, we obtain Nu ¼ ’d ðRe; PrÞ
(7.108)
where Nu ¼ qs d=kDt; Re ¼ udr=m and Pr ¼ cP m=k are the Nusselt number based, which represents itself in this case the dimensionless heat flux at the pipe wall, the Reynolds and Prandtl numbers, respectively. In the particular case of creeping flows corresponding to very small Reynolds numbers (Re ! 0Þ it is possible to omit u and m from the set of the governing parameters and write the functional equation for the heat flux in the form qs ¼ fq ðr; k; cP ; d; DtÞ
(7.109)
All the governing parameters in (7.109) have independent dimensions. Accordingly, this equation should take the following form qs ¼ cra1 ka2 caP3 da4 Dta5 where c is a constant.
(7.110)
7.5 Thermal Characteristics of Laminar Jets
183
Taking into account the dimension ½qS ¼ JT 1 L2 , we determine values of the exponents ai as a1 ¼ 0; a2 ¼ 1; a3 ¼ 0; a4 ¼ 1 and a5 ¼ 1. Accordingly, the Nusselt number is expressed as Nu ¼
qs d ¼ c ¼ const: kDt
(7.111)
The numerical value of the constant c in (7.111) that follows from the analytical solution of this problem is 4.364 (Kays and Crawford 1980).
7.5
Thermal Characteristics of Laminar Jets
Consider thermal characteristics of plane submerged laminar jet when a heated (or cooled) fluid is issuing from a narrow slit in the same cooler (or warmer) fluid, which is at rest far from the jet origin. We assume that the difference between the fluid temperature in the jet and the surrounding medium is sufficiently small, which allows us to consider the fluid density and viscosity being constant. In this case the flow and temperature fields of fluid in the jet are determined by two parameters: (1) the kinematic viscosity n, and (2) thermal diffusivity a. Non-isothermal jets, in fact, form two boundary layers, the dynamic and thermal one. In the general case their thicknesses d and dT are not equal. The relation between d and dT depends on the relative intensity of the momentum and heat transfer which is characterized by the Prandtl number Pr ¼ n=a: Only when Pr ¼ 1; d ¼ dT ; whereas when Pr <1 and Pr >1; dT >d and dT
Fig. 7.3 Plane jet. Thicknesses of the dynamic and thermal boundary layers at different values of the Prandtl number
184
7 Heat and Mass Transfer
component of the heat transfer, as well as the thermal diffusivity characterizing the conductive component. Then, the functional equation for the temperature field reads DT ¼ f ðu; a; qx ; x; yÞ
(7.112)
where ½DT ¼ T T1 ¼ y; T1 is the temperature of the undisturbed fluid, ½a ¼ L2y T 1 is the thermal diffusivity, ½qx ¼ Qx =ðrcP Þ ¼ Lx Ly T 1 y where the axial 1 Ð 1 1 convective enthalpy flux Qx ¼ ruDhdy ¼ L1 being z T J with ½h ¼ JM 0
the enthalpy, in the Lx Ly Lz MTJy system of units. For the axial excess (maximal) temperature DTm , and the thermal boundary thickness dT the following functional equations are valid DTm ¼ fT ðum ; a; qx ; xÞ
(7.113)
dT ¼ fd:T ðum ; a; qx ; xÞ
(7.114)
Since the governing parameters um (the axial maximal velocity in jet crosssection), a, qx and x have independent dimensions, (7.113) and (7.114) take the form DT ¼ c1T uam1 aa2 qax 3 xa4 a
a
dT ¼ c2T um1 aa2 qx3 xa4
(7.115) (7.116)
where c1T and c2T are constants. Then the exponents ai and ai are found as 1 1 1 a1 ¼ ; a2 ¼ ; a3 ¼ 1; a4 ¼ ; a1 2 2 2 1 1 1 ¼ ; a2 ¼ ; a3 ¼ 0; a4 ¼ 2 2 2
(7.117)
Therefore, (7.115) and (7.116) take the form DTm ¼ c1T
1=2 Qx 1 Pr x1=3 rcP ðJx n=rÞ1=6
1=2 1 2=3 dT ¼ c2T 1=3 Pr x 2 2 Jx =r n
(7.118)
(7.119)
where c1T ¼ c1T =c1 and c2T ¼ c2T =c1 with c1 being a constant from (5.28).
7.5 Thermal Characteristics of Laminar Jets
185
Comparing (6.40) and (7.119), we estimate the ratio of the thicknesses of the dynamic and thermal boundary layers pffiffiffiffiffi d
Pr dT
(7.120)
Using the Pi-theorem, we transform (7.112) to the following form DT y ¼ ’T DTm dT
(7.121)
Bearing in mind (7.120), we present (7.121) in the form
DT ¼ ’T Pr1=2 DTm
(7.122)
where ¼ y=d. The temperature distribution in cross-sections of plane laminar jet is shown in Fig. 7.4. It is seen that profiles DTðÞ=DTm are wider than profiles uðÞ=um at Pr <1; and narrower at Pr >1:
Fig. 7.4 Velocity and temperature distributions in cross-sections of plane laminar submerged jet. 1- Velocity and temperature distributions for Pr ¼ 1; 2-Temperature distribution for Pr < 1; 3Temperature distribution for Pr > 1
186
7 Heat and Mass Transfer
7.6
Heat and Mass Transfer in Natural Convection
7.6.1
Heat Transfer from a Spherical Particle Under the Conditions of Natural Convection
Consider heat transfer from a warm spherical particle immersed into still fluid. The temperature difference between the particle and surrounding fluid is the reason of natural convection, which arises due to the existence of a non-uniform density field in fluid with temperature distribution and thermal expansion. Under the conditions of natural convection it is plausible to assume that heat flux from a warm particle is determined by the physical properties of fluid- its density r; thermal conductivity k; viscosity m; specific heat cp ; the particle and fluid temperature difference Dt ¼ tp t1 ; as well as the product of gravity acceleration g and thermal expansion coefficient b: Accordingly, the functional equation for the heat flux reads qt ¼ f ðr; m; k; cP ; Dt; gb; r0 Þ
(7.123)
In the system of units LMTJy the dimensions of the governing parameters are ½r ¼ L3 M; ½m ¼ L1 MT 1 ; ½k ¼ JT 1 L1 y1 ; ½cP ¼ JM1 y1 ½Dt ¼ y; ½gb ¼ y1 LT 2 ; ½r0 ¼ L
(7.124)
while ½qt ¼ JL2 T1 . Five of the seven governing parameters in (7.123) possess independent dimensions. Then, according to the Pi-theorem the dimensionless form of (7.123) is P ¼ ’ðP1 ; P2 Þ 0 a1
0 a2
(7.125) 0 a3
0 a4
0 a5
00 a1
00
00
where P ¼ qt =ra1 ma2 ka3 Dta4 r0a5 ; P1 ¼ cp =r m k Dt r0 and P2 ¼ gb=r ma2 ka3 00
00
a
Dta4 r0 5 : Bearing in mind the dimension of the heat flux qt ½JT 1 L2 , we find values of the 0 00 exponents ai ; ai and ai a1 ¼ 0; 0
a1 ¼ 0; 00
a1 ¼ 2;
a2 ¼ 0;
a3 ¼ 1;
0
a2 ¼ 1; 00
a2 ¼ 2;
a4 ¼ 1;
0
0
a3 ¼ 1; 00
a3 ¼ 0;
a4 ¼ 0; 00
a4 ¼ 1;
a5 ¼ 1 0
a5 ¼ 0
(7.126)
00
a5 ¼ 3
Accordingly, (7.125) takes the form Nu ¼ ’ðPr; Gr Þ
(7.127)
where Nu ¼ qt r0 =kDt; Pr ¼ cp m=k and Gr ¼ r2 gbDtr03 =m2 are the Nusselt, Prandtl and Grashof numbers, respectively.
7.6 Heat and Mass Transfer in Natural Convection
187
The explicit form of the dependence of the Nusselt number on Pr and Gr for a warm spherical particle found experimentally has the form (Raithby and Hollands 1998) Nu ¼ 0:878ARa1=4
(7.128)
where A is a weak function of the Prandtl number, namely, 0.086
(7.129)
Five governing parameters in (7.129) have independent dimensions. Then the Pi-theorem allows transformation of (7.129) to the following dimensionless form P ¼ ’ðP1 Þ
(7.130) 0
a
0
0
0
a
0
where P ¼ qt =ra1 cap2 ka3 Dta4 r0a5 and P2 ¼ gb=ra1 cp2 ka3 Dta4 r0 5 . Using the principle of dimensional homogeneity, we find the values of the 0 exponents ai and ai as a1 ¼ 0; 0
a1 ¼ 2;
a2 ¼ 0; 0
a3 ¼ 1;
a2 ¼ 2;
a4 ¼ 1;
0
0
a3 ¼ 2;
a4 ¼ 1;
a5 ¼ 1
(7.131)
0
a5 ¼ 3
Then (7.129) takes the form 2
Nu ¼ ðGrPrÞ
7.6.2
(7.132)
Heat Transfer from Spinning Particle Under the Condition of Mixed Convection
Consider the general case of heat transfer from a spherical spinning particle under conditions of mixed convection when both natural and forced convection are essential. We assume fluid to be incompressible and possess invariable physical properties. The factors that determine the heat transfer intensity in the present case are: (1) the physical properties of fluid-its density r, thermal conductivity k, viscosity m and specific heat
188
7 Heat and Mass Transfer
cP ; (2) particle linear and angular velocities relative to fluid u1 and o, respectively, particle diameter d, and temperature difference between the particle and fluid Dt ¼ tP t1 . In the case when the effect of buoyancy force is significant, it is necessary to supplement the system of the governing parameters with the product bg(where b is the thermal coefficient and g gravity acceleration). Therefore, the functional equation for the heat flux from the particle surface reads q ¼ f ðr; k; m; cP ; u1 ; o; d; bg; DtÞ
(7.133)
In (7.133) the set of the nine governing parameters contains five parameters with independent dimensions. Then, according to the Pi-theorem, the number of dimensionless groups that determine the dimensionless heat transfer coefficient (the Nusselt number) equals four. Choosing as parameters with independent dimensions r; k; m; d and Dt, we arrive at the following equation P ¼ ’ðP1 ; P2 ; P3 ; P4 Þ
(7.134)
where 0
0
0
0
0
1V
1V
00
00
00
00
00
P ¼ q=ra1 ka2 ma3 d a4 Dta5 ; P1 ¼ u1 =ra1 ka2 ma3 d a4 Dta5 ; P2 ¼ o=ra1 ka2 ma3 da4 Dta5 ; 000
000
000
000
000
1V
1V
1V
P3 ¼ cP =ra1 ka2 ma3 d a4 Dta5 ; and P4 ¼ bg=ra1 ka2 ma3 d a4 Dta5 : 0 00 000 The values of the exponents ai ; ai ; ai ; ai and a1V i are equal to a2 ¼ 1;
a ¼ 0; 0
0
a1 ¼ 1;
a2 ¼ 0;
00
00
a1 ¼ 1; 000
a1 ¼ 0; a1V 1 ¼ 2;
a2 ¼ 0; 000
a2 ¼ 1; a1V 2 ¼ 0;
a3 ¼ 0; 0
a3 ¼ 1; 00
a3 ¼ 1; 000
a3 ¼ 1; a1V 3 ¼ 2;
a4 ¼ 1; 0
a4 ¼ 1; 00
a4 ¼ 2; 000
a4 ¼ 0; a1V 4 ¼ 3;
a5 ¼ 1 0
a5 ¼ 0 00
a5 ¼ 0
(7.135)
000
a5 ¼ 0 a1V 5 ¼ 1
Then (7.134) takes the form Nu ¼ ’ðRe; Reo ; Pr; GrÞ
(7.136)
where Nu ¼ qd=kDt; Re ¼ u1 dr=m; Reo ¼ od 2 r=m; Pr ¼ mcP =k; and Gr ¼ bgDtd 3 r2 =m2 are the Nusselt, Reynolds, rotational Reynolds, Prandtl and Grashof numbers, respectively. In the particular case corresponding to heat transfer in motionless fluid at an invariable Prandtl number, the heat transfer coefficient is determined by two
7.6 Heat and Mass Transfer in Natural Convection
189
dimensionless groups, namely, the rotational Reynolds and Grashof numbers. The measurements show that buoyancy effects are dominant at Reo <103 when the heat transfer intensity is determined by natural convection (Tieng and Yan 1993). In the range 103 9 103 the forced convection is dominant. In this case the contribution of the jet eruptions to the heat transfer is essential. For calculations of the average Nusselt number for mixed convection, Tieng and Yan (1993) suggested the following empirical relation which is valid at Pr ¼ 0:71 : 3
3
3
Nu ¼ Nun þ Nuf
(7.137)
where Nu; Nun and Nuf are the overall Nusselt number and the average Nusselt numbers for the natural and forced convection, respectively which are given by Nun ¼ 2 þ 0:392Gr0:31
(7.138)
Nuf ¼ 2 þ 0:175Re0:583 o
(7.139)
at 1
at Gr ¼ 0:10
7.6.3
Mass Transfer from a Spherical Particle Under the Conditions of Natural and Mixed Convection
Heterogeneous reactions taking place at the liquid-solid interface are the reason of formation of non-uniform concentration fields in liquid solutions. That leads to changing density of liquid solution and arising of buoyancy forces which drive natural convection. When the dependence of liquid density on concentration of reactant in solution is weak, the following approximation for solution density is valid (Levich 1962) @r ðc c1 Þ (7.140) rðcÞ rðc1 Þ þ @c c¼c1 where c and c1 are the local concentration and concentration far from the particle. In this case the buoyancy force can be evaluation as follows fb ¼ gfrðc1 Þ rðcÞg gðc1 cÞ
@r @r agc 1 @c c¼c1 @c c¼c1
(7.141)
where g is the gravity acceleration and a is a known dimensionless constant (a<1Þ.
190
7 Heat and Mass Transfer
Under the conditions of natural convection, the mass flux to a particle is determined by the buoyancy force, concentration of the reagent in solution, reagent diffusivity and particle size. Accordingly, the functional equation for the mass flux reads qm ¼ f ðfb ; c1 ; D; r0 Þ
(7.142)
Three governing parameters in (7.142) have independent dimensions. Therefore, it follows from the Pi-theorem that (7.142) can be transformed to the following dimensionless form P ¼ ’ðP1 Þ
(7.143)
where P ¼ qm r=Dc1 and P1 ¼ fb r03 =D2 c1 agr03 ð@r=@cÞjc¼c1 =D2 are the Sherwood and Grashof numbers, respectively. The functional equation for the mass transfer to a particle under the conditions of mixed convection reads qm ¼ f ðfb ; c1 ; D; r0 ; u1 Þ
(7.144)
where u1 is the velocity far from particle. In this case the set of the five governing parameters contains three parameters with independent dimensions. Therefore, the dimensionless form of (7.144) is P ¼ ’ðP1 ; P2 Þ
(7.145)
where P; P1 and P2 ¼ u1 r0 =D are the Sherwood, Grashof and Peclet numbers, respectively.
7.6.4
Heat Transfer From a Vertical Heated Wall
Consider heat transfer from a vertical heated wall under the conditions of natural convection. In the framework of the boundary layer approximation, the system of the governing equations describing the flow and heat transfer near the vertical wall reads u
@u @u @2u þv ¼ n 2 þ gbðT T1 Þ @x @y @y
(7.146)
@u @v þ ¼0 @x @y
(7.147)
@T @T @2T þv ¼a 2 @x @T @y
(7.148)
u
7.6 Heat and Mass Transfer in Natural Convection
191
where n and a are the viscosity and thermal diffusivity, b ¼ 1=T1 is the thermal expansion coefficient, g gravity acceleration, x and y are the coordinate axes directed along the wall and normal to it and subscript 1 denotes the ambient conditions. The boundary conditions for (7.146)–(7.148) are as follows y ¼ 0; u ¼ v ¼ 0 T ¼ Tw ; y ! 1; u ! 0 T ! T1
(7.149)
where Tw is the wall temperature. It is convenient to introduce the excess of temperature DT ¼ T T1 and rewrite (7.146) and (7.148) in the following form u
@u @u @2u þv ¼ n 2 þ g y @x @y @y
(7.150)
@y @y @ 2 y þv ¼ @x @y @y2
(7.151)
u
where y ¼ ðT T1 Þ=ðTw T1 Þ and g ¼ gðTw T1 Þ=T1 . The boundary conditions for (7.150) and (7.151) read y ¼ 0 u ¼ v ¼ 0; y ! 1 u ! 1 y ! 0
(7.152)
Equations 7.147, 7.150 and 7.151 and conditions (7.152) contain three dimensional constants ðn; a; and g Þand two variables ðx and yÞ that determine the velocity and temperature fields. Therefore, one can assume that the following functional equations are valid u ¼ fu ðx ; y; n; g Þ
(7.153)
y ¼ fy ðx ; y; a; g Þ
(7.154)
where x ¼ ex with e being a dimensionless constant (its numerical value is determined from the consideration of dimensions and the condition of the existence of a self-similar solution), x is the longitudinal coordinate. The dimensions of velocity u, temperature #, as well as of the governing parameters x ; y; n; a; and g (in the system of units Lx Ly Lz MTy) are ½u ¼ Lx T 1 ; ½y ¼ 1; ½x ¼ Lx ; ½ y ¼ Ly ; ½n ¼ L2 T 1 ; ½a ¼ L2y T 1 ; ½g ¼ Lx T 2
(7.155)
The set of the governing parameters in 7.153 and 7.154 contains three parameters with independent dimensions, so that the difference n k ¼ 1. Therefore, the dimensionless forms of these equations read
192
7 Heat and Mass Transfer
Pu ¼ ’ðP1u Þ
(7.156)
Py ¼ #ðP1y Þ
(7.157)
0 a1
0 a3
0
Pu ¼ u=xa1 na2 ga3 ; P1u ¼ y=x na2 g ,
where
0 a1
Py ¼ y=xa1 aa2 ga3
0 a3
0 a2
and
P1y ¼ y=x a g . Taking into account the dimensions of u; #; x ; y; n and a, we find the values of 0 the exponents ai and ai 1 a1 ¼ ; 2
a2 ¼ 0;
1 a3 ¼ ; 2
a1 ¼ a2 ¼ a3 ¼ 0;
1 0 a1 ¼ ; 4
1 0 a2 ¼ ; 2
a3 ¼
1 0 a1 ¼ ; 4
1 0 a2 ¼ ; 2
a3 ¼
0
1 4
0
(7.158)
1 4
Then we obtain Pu ¼
u ð x g Þ
1=4
; 1=2
P1u ¼
1=4
yg 1=4
x n1=2
;
Py ¼ #;
P1y ¼
yg 1=4
x a1=2
(7.159)
According to the expressions (7.159), (7.156) and (7.157) take the the following form u ¼ ðx g Þ1=2 ’ðÞ
(7.160)
pffiffiffiffiffi y ¼ #ð PrÞ
(7.161) 1=4
1=4
where Pr ¼ n=a is the Prandtl number, and ¼ yg =x n1=2 : Ð Bearing in mind that the stream function is defined as c ¼ udy and v ¼ @c=@x, we find 1=4 1=2 ’ðÞ c ¼ x1=4 g n
(7.162)
n o e 0 1=4 1=2 v ¼ x1=4 3’ðÞ ’ ðÞ g n 4
(7.163)
Substitution of the expressions for u; v and y into (7.150) and (7.151) yields the pffiffiffiffi ffi following system of ODEs for the unknown functions ’ðÞ and #ð PrÞ (Pohlhausen, 1921) 000
00
02
’ þ 3’’ 2’ þ # ¼ 0 00
0
# þ 3 Pr ’# ¼ 0
(7.164) (7.165)
7.6 Heat and Mass Transfer in Natural Convection
193
where subscript corresponds to differentiation by : Note, that imposing the conditions that the equations for the velocity and temperature do not incorporate an arbitrary constant e, it should be taken as e ¼ 4. The boundary conditions for (7.164) and (7.165) are 0
0
¼ 0; ’ ¼ ’ ¼ 0 # ¼ 1; ! 1; ’ ¼ 0 # ! 0
(7.166)
The local heat flux qðxÞ is expressed as qðxÞ ¼ k
@T @# ¼ kcx1=4 ðTw T1 Þ @y y¼0 @ ¼0
(7.167)
1=4
where k is thermal conductivity, c ¼ ðg =4n2 Þ : The value of ð@#=@Þ¼0 depends on the Prandtl number value. It can be found only by solving (7.164) and (7.165). For example, ð@#=@Þ¼0 ¼ 0:508 for Pr ¼ 0:773(Schlichting 1979). The total heat flux from a plate of length l to the surrounding fluid is ðl
4 Q ¼ b qðxÞdx ¼ 0:508 bl3=4 ckðTw T1 Þ 3
(7.168)
0
where b is the plate width. The average Nusselt number based on Q is expressed as Nu ¼ 0:478ðGrÞ1=4
(7.169)
where Gr ¼ gl3 ðTw T1 Þ=n2 T1 is the Grashof number.
7.6.5
Mass Transfer to a Vertical Reactive Plate Under the Conditions of Natural Convection
Consider mass transfer to a vertical reactive plate in contact with a liquid reactant solution. Assume that at the surface of this plate (but not in the solution bulk) takes place a chemical reaction the rate of which is much large then the rate of reactant diffusion. In this case the reactant concentration will be equal zero at the wall. As a result, a not-uniform reactant concentration field arises near the plate. Since the density of solution depends on reactant concentration, a non-uniform density field arises as well. The latter results in natural convection that determines the intensity of mass transfer to the reactive wall. In the framework of the boundary layer approximation the momentum and species balance equations of the problem have the form (Levich 1962)
194
7 Heat and Mass Transfer
u
@u @u @2u þv ¼ n 2 þ gc c @x @y @y
(7.170)
@c @c @ 2 c þv ¼D 2 @x @y @y
(7.171)
u
where gc ¼ ðgc1 =rÞð@r=@cÞc¼c1 ; c ¼ ðc1 cÞ=c1 ; g is gravity acceleration due, r ¼ rðcÞ; D is the reactant diffusivity coefficient, and subscript 1 corresponds to reactant concentration in the undisturbed solution. The boundary conditions for (7.170) and (7.171) read u ¼ v ¼ 0 c ¼ 1 at y ¼ 0; u ¼ v ¼ 0 c ! 1 at y ! 1
(7.172)
Equations 7.170 and 7.171 and conditions (7.172) formally coincide with (7.150) and (7.151) and conditions (7.152). Accordingly, the expressions for the components of velocity and reactant concentration can be recast as follows u ¼ ðx gc Þ1=2 ’ðÞ n o 1 0 1=4 1=2 g D 3’ðÞ ’ ðÞ v ¼ e x1=4 c 4 pffiffiffiffiffiffi c ¼ #c ð SmÞ
(7.173) (7.174) (7.175)
1=4
where Sc ¼ n=D is the Schmidt number, ¼ ðg =4n2 Þ y=x1=4 . Substitution of the expressions (7.173) and (7.175) into (7.170)pand ffiffiffiffiffi (7.171) leads (at e ¼ 4Þ to the system of ODEs for the functions ’ðÞ and #ð ScÞ 000
00
02
’ þ 3’’ 2’ þ #c ¼ 0 00
0
#c þ 3Sc#c ¼ 0
(7.176) (7.177)
where primes ðÞ0 correspond to differentiation by the dimensionless variable . The mass flux of reactant at the wall is found by calculating the derivative ð@c =@yÞy¼0 ¼ ð@c =@Þ¼0 @=@y and using Fick’s law. The expressions that determine the local j and total J mass flux at the vertical wall of height h and width b reads (Levich 1962) (
j ¼ 0:7Sc
1=4
)1=4 gc1 @r c1 D 2 4n r @c c¼c1 x1=4
( J ¼ 0:9Sc
1=4
)1=4 gc1 @r bh3=4 c1 D 4n2 r @c c¼c1
(7.178)
(7.179)
7.7 Heat Transfer From a Flat Plate in a Uniform Stream
7.7
195
Heat Transfer From a Flat Plate in a Uniform Stream of Viscous, High Speed Gas
Consider laminar boundary layer over a wall subjected to parallel uniform stream of viscous, high velocity, perfect gas. It is easy to see that pressure along the wall is constant in this particular case. Then, the system of equations that correspond to this problem reads (Loitsyanskii 1966) ru
@u @u @ @u þ rv ¼ m @x @y @y @y @ru @rv þ ¼0 @x @y
ru
2 @h @h k @ @h @u þm þ rv ¼ m @x @y mcP @y @y @y g1 h rh; m ¼ m0 f P¼ g h1
(7.180)
(7.181)
(7.182)
(7.183)
where r; u; v; P and h are the density, longitudinal and lateral velocity components, pressure and enthalpy, respectively; m; k and cp are the viscosity, thermal conductivity and specific heat of fluid, respectively; g ¼ cP =cV is the adiabatic index (the ratio of specific heats at constant pressure or constant volume).The boundary conditions for (7.180)–(7.182) are y ¼ 0 : u ¼ 0; v ¼ 0 h ¼ hw ; y ! 1 : u ! u1 ; h ! h1 ; P ! P1
(7.184)
for the plate with a given constant temperature, and y ¼ 0 : u ¼ 0; v ¼ 0;
@h ¼ 0; y ! 1 : u ! u1 ; h ! 1; P ! 1 @y
(7.185)
for thermally insolated wall (subscripts w and 1 refer to the wall and undisturbed flow, respectively). The density of the ambient gas (far away from the wall) determines (at a given of the enthalpy, h1 Þ the value of pressure, which follows from the first (7.183). Taking into account this circumstance, it is possible to write the following functional equations for the longitudinal velocity component and enthalpy u ¼ fu ðx; y; r1 ; u1 ; h1 ; hw ; P1 ; m1 ; k1 ; cP1 Þ
(7.186)
h ¼ hh ðx; y; r1 ; u1 ; h1 ; hw ; P1 ; m1 ; k1 ; cP Þ
(7.187)
196
7 Heat and Mass Transfer
It shows that u and h are the function of ten dimensional parameters: ½ x ¼ L; ½ y ¼ L; ½r1 ¼ L3 M; ½u1 ¼ LT 1 ; ½P ¼ L1 MT 2 ; ½h1 ¼ JM1 ; ½hw ¼ JM1 ; ½k1 ¼ JT 1 L1 y1 ; ½cP ¼ JM1 y1 and ½m ¼ L1 MT 1 . Taking into account that (7.182) includes the ratio ½k=mcP ¼ 1 as a single complex, it is possible to diminish the number of the governing parameters in (7.186) and (7.187) and present them in the following form u ¼ fu ðx; y; r1 ; u1 ; h1 ; hw ; P1 ; m1 ;
k Þ mcP
(7.188)
h ¼ fh ðx; y; r1 ; u1 ; h1 ; hw ; P1 ; m1 ;
k Þ mcP
(7.189)
Equations 7.188 and 7.189 contain nine governing parameters, four of which have independent dimensions. Then, according with the Pi-theorem, (7.188) and (7.189) reduce to the two following dimensionless equations Pu ¼ fu ðP1 ; P2 ; P3 ; P4 ; P5 Þ
(7.190)
Ph ¼ fh ðP1 ; P2 ; P3 ; P4 ; P5 Þ
(7.191)
where Pu and Ph are the dimensionless velocity and enthalpy, respectively; the dimensionless groups Pi (i¼1,. . .5) are given by the following expressions: 0
a
0
0
a
a
0
00
00
a
a
00
a
00
000
a
000
a
000
a
000
P1 ¼ y=xa1 r12 u13 h14 ; P2 ¼ hw =xa1 r12 u13 h14 ; P3 ¼ P1 =xa1 r12 u13 h14 ; P4 ¼ 1V
a1V a1V a1V
V
a V aV aV
m1 =xa1 r12 u13 h14 and P5 ¼ ðk1 =m1 cP1 Þ=xa1 r12 u13 h14 Bearing in mind that Pu ¼ u=xa1 ra12 ua13 ha14 and Ph ¼ h=xa1 ra12 ua13 ha14 , we can rewrite (7.190) and (7.191) in the following form u y hw P m1 k1 ¼ ’u ; ; ; ; u1 x h1 r1 u21 xr1 u1 m1 cP
(7.192)
h y hw P 1 m1 k1 ¼ ’h ; ; ; ; h1 x h1 r1 u21 xr1 u1 m1 cP
(7.193)
It is seen that the dimensionless velocity and enthalpy are determined by the functions of two dimensionless variables y=x and m1 =xr1 u1 and three constants. That shows that (7.180)–(7.182) cannot be reduce ODEs. In order to solve this problem it is necessary to use a special transformation (7.180)–(7.182). First of all, we transform (7.180)–(7.182), with the boundary conditions (7.183) and the correlations (7.183) to the dimensionless form normalizing the parameters by their values in the undisturbed flow (u ¼ u=u1 ; h ¼ h=h1 , etc.) and the variables x and y by some length scaleL (x ¼ x=L and y ¼ y=LÞ. Omiting for brevity bars
7.7 Heat Transfer From a Flat Plate in a Uniform Stream
197
over the dimensionless parameters and assumed that viscosity is a linear function of temperature, we rewrite (7.180)–(7.183) as follows @u @u @ @u (7.194) ¼ m ru þ rv @x @y @y @y @ru @rv þ ¼0 @x @y 2 @h @h @u 1 @ @u 2 ru þ rv þ ¼ ðg 1ÞM1 m m @x @y @y Pr @y @y 1 r ¼ ; m ¼ hn h
(7.195)
(7.196)
(7.197)
Planar compressible problems described by (7.194)–(7.197) are further simplified by employing the Dorodnitsyn (1942), Illingworth (1949), Stewartson (1949) transformation (Loitsyanskii 1966; Schlichting 1979) ðy x ¼ x; ¼ rdy
(7.198)
0
The transformation reduces the compressible (7.194)–(7.197) to the icompressible boundary layer equations, which take the following dimensionless form @u @u @ 2 u þ ve ¼ 2 @x @ @
(7.199)
@u @e v þ ¼0 @x @y
(7.200)
2 @h @h 1 @2h @u 2 þ ve ¼ þ ðg 1ÞM 1 @x @ Pr @h2 @
(7.201)
u
u
with the boundary conditions ¼ 0 : u ¼ 0; h ¼ hw ; ! 1 : u ! 1; h ! 1
(7.202)
for plates with a constant temperature given, and ¼ 0; u ¼ 0;
@h ¼ 0; ! 1 : u ! 1; h ! 1 @
(7.203)
198
7 Heat and Mass Transfer
In (7.199)–(7.201) ve ¼ u
@ þ rv @x
(7.204)
Equations (7.199) and (7.201) and the boundary conditions (7.202) and (7.203) show that the dynamic problem formulated in new variables and coordinates becomes autonomous and the dimensionless velocity u depends on two dimensionless variables x and as u ¼ f ð; xÞ
(7.205)
Any combination of these variables is also dimensionless. Similarly to flows of incompressible fluid, one can assume that u is self-similar in a sense that f in (7.205) is a function of one dimensionless variable w¼a
xa
(7.206)
where a and a are some constants. Introducing function ’w ðwÞ instead of f in such a way that u is expressed as 0
u ¼ a’w ðwÞ
(7.207)
we find the expressions for the stream function c and a transformed lateral velocity component ve as ð c ¼ udy ¼ xa ’ðwÞ ve ¼
@c 0 ¼ axa1 ða’w x ’Þ @x
(7.208)
(7.209)
where prime denotes derivatives in w. Substitution of the expressions for u and ve into (7.194) yields a 000 a 00 ’w þ ’w ’ ¼ 0 x x2a
(7.210)
Requiring that (7.210) become an ODE in w, while variable x cancels, one finds that a ¼ a ¼ 1=2. Then (7.210) takes the form 000
00
’ w þ ’w ’ ¼ 0
(7.211)
7.8 Heat Transfer Related to Phase Change
199
The boundary conditions for (7.211) are 0
0
w ¼ 0; ’ ¼ 0 ’w ¼ 0; w ! 1; ’ ! 2
(7.212)
Correspondingly, the energy equation 7.203 reduces to the following ODE Pr 2 00 ’w ¼ 0 ðg 1ÞM1 4
(7.213)
w ¼ 0 : h ¼ hw ; w ! 1 : h ! 1
(7.214)
00
0
hw þ Pr ’hw þ with the boundary conditions
for a constant temperature wall, and w¼0:
@h ¼ 0; w ! 1 : h ! 1 @w
(7.215)
for the thermally-insulated one. The problem (7.211) and (7.212) formally coincides with the problem on the incompressible boundary layer near a plane wall (the Blasius problem). Accordingly, the drag coefficient at m ¼ hn and n ¼ 1 is expressed as cf ¼
1 1 00 pffiffiffiffiffiffiffiffi ’w ð0Þ 2 Rex
(7.216)
00
where Rex ¼ u1 x=n1 and ’w ð0Þ ¼ 1:328. The solution of (7.213) leads to the following dependence for the the average Nusselt number for a plate of length L Nu ¼
pffiffiffiffiffiffiffiffiffi Tw T s f ðPrÞ Re1 Tw T1
(7.217)
where Ts is the stagnation temperature, Re1 ¼ uL=n.
7.8 7.8.1
Heat Transfer Related to Phase Change Heat Transfer Due to Condensation of Saturated Vapor on a Vertical Wall
Consider the heat transfer process during condensation of saturated vapor on a cold vertical wall. As a result of condensation, a thin liquid film forms at the wall surface. This film flows downward due to gravity. Latent heat that is released due
200
7 Heat and Mass Transfer
to vapor condensation is transferred from film to the wall by conduction and convection. In a thin liquid film flowing with low velocity (Re 1Þthe convective component of the heat flux qconv is negligible in comparison with the conductive one qcond, which is supported by the following estimates qcond
@T kL kL ðTs Tw Þ d @y y¼0
(7.218)
qconv rucP dðTs Tw Þ
(7.219)
In Eqs. (7.218) and (7.219) d is the liquid film thickness; rL ; kL and cPL are the density, thermal conductivity and specific heat at constant pressure of the liquid, Ts and Tw are the vapor saturated corresponding to a given pressure and the wall temperatures, respectively. The estimations (7.218) and (7.219) show that qconv
Re Pr 1 at Re 1 and Pr 1 qcond
(7.220)
The assumptions adopted here were first used by Nusselt (1916) in a simplified theory of heat transfer during film condensation of saturated vapor on a cold vertical wall. Below we consider this problem using the Pi-theorem. First of all, consider the parameters which can affect the heat transfer from a saturated vapor to a cold vertical wall. The set of the governing parameters should inevitably include the liquid density rL and viscosity mL as well as gravity acceleration g: It is necessary to account for the latent heat of condensation Dhfv ; which is the origin of heat flux due to vapor condensation q. The set of the governing parameters should also include the vapor density rV (in fact, the difference rL rV Þ: However, since rL >>rV ; it is possible to omit rV from the set of the governing parameters. Accordingly, we can write the functional equation for the liquid film thickness at the wall as follows d ¼ f ðr; m; Dhfv ; g; q; xÞ
(7.221)
where x is the longitudinal coordinate reckoned down the wall. In order to decrease the number of the governing parameters, it is naturally to assume that the parameters g and mL group with the liquid density rL as the specific weight grL and kinematic viscosity mL =rL ¼ nL : Then (7.221) transforms into d ¼ f ðgrL ; nL ; Dhfv ; q; xÞ
(7.222)
In the framework of the boundary layer approximation one can introduce three different scales of length Lx , Ly and Lz for the longitudinal and two lateral directions
7.8 Heat Transfer Related to Phase Change
201
x, y and z, and use the system of units Lx Ly Lz MTJ: Then the dimensions of the governing parameters in (7.222) are expressed as
1 2 2 1 ½rL g ¼ L1 y Lx MT ; ½nL ¼ Ly M; Dhfv ¼ JM ; ½q 1 ¼ JT 1 L1 x Lz ; ½ x ¼ L
(7.223)
It is seen that all the governing parameters have independent dimensions, so that (7.223) takes the form d ¼ cðrL gÞa1 ðnL Þa2 ðDhfv Þa3 ðqÞa4 ðxÞa5
(7.224)
where c is a constant. Taking into account the dimension of ½d ¼ Ly , we find the values of the exponents ai as a1 ¼ 1=3; a2 ¼ 1=3; a3 ¼ 1=3; a4 ¼ 1=3 and a5 ¼ 1=3. Then (7.224) takes the form
nL qx d¼c ðrL gÞDhfv
1=3 (7.225)
Assuming that q ¼ kL ð@T=@yÞy¼0 kL ðTs Tw Þ=d, we obtain d ¼ c1
kL mL ðTs Tw Þ 1=4 x r2L gDhfv
(7.226)
where c1 ¼ c3/4 Under the assumption that the temperature distribution within the liquid film is linear, the local heat transfer coefficient is found as 2 1=4 rL gDhfv kL3 1 kL a ¼ ¼ c2 d mL ðTs Tw Þ x
(7.227)
where c2 ¼ c1 1 is a constant. The comparison of the predictions of Nusselt’s theory with the experimental data on film condensation shows that there is a significant deviation of the theoretical results from the experimental results (about 25%). The latter stems from significant oversimplification of the complex phenomenon in Nusselt’s theory which does not account for a number of factors that affect the heat transfer intensity. These factors are: wave formation on the film surface, temperature dependence of the physical properties of fluid, etc. (Baehr and Stephan 1998). A detailed analysis of steady laminar film condensation (condensation of stagnant and flowing vapor, the self-similar solution in the cases of free and forced convection, etc.) can be found in the monographs by Fujii (1991) and Stephan (1992).
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7 Heat and Mass Transfer
7.8.2
Freezing of a Pure Liquid (The Stefan Problem)
Let the half-space 0bxb1 be filled with a pure liquid (say, water) at rest with an initial temperature #21 >0(# ¼ T Tf ; where Tf is the freezing temperature) (Fig. 7.5). On the left side of the water-filled domain a refrigerator is put at the initial time moment t¼0. Its temperature #10 is kept constant below the freezing temperature #f : #10 <#f ¼ 0: Because of the conductive heat transfer to the refrigerator, liquid temperature decreases in the negative x direction. As a result of liquid cooling, near the contact with the refrigerator forms an ice layer and its thickness increases in time. The interface that separates the ice and liquid domains propagates into liquid (in the positive x direction). The main aim is in determining the rate of growth of the ice layer. From the physical point of view it is plausible to assume that the ice layer location of the freezing interface x is determined by the physical properties of liquid and its ice (cP1 ; cP2 ; k1 and k2 ; with subscript 1 being used for ice and subscript 2-for liquid), latent heat of solidification qs ; the initial temperature of liquid far away from the refrigerator #20 , the refrigerator temperature #10 , densities of the ice and liquid r1 and r2 , as well as time t x ¼ f ðcP1 ; cP2 ; k1 ; k2 ; #10 ; #20 ; r1 ; r2 ; qs ; tÞ
(7.228)
The dimensions of the governing parameters are as follows ½cP ¼ JM1 y1 ; ½k ¼ JT 1 L1 y1 ; ½#i ¼ y; ½qs ¼ JM1 ; ½t ¼ T
(7.229)
Equations 7.228 and 7.229 show that the present problem contains ten dimensional parameters, five of them possessing independent dimensions. Choosing as the parameters with the independent dimensions r2 ; cP2 ; k2 ; #21 and t, we trabsform (7.228) to the following form P ¼ ’ðP1 ; P2 ; P3 ; P4 Þ
(7.230)
where the dimensionless groups are expressed as P¼
x ra2 cbP2 k2g #e20 to
;
Pi ¼
wi ai bi gi ei oi r2 cP2 k2 #20 t
;
(7.231)
J1 = J2 = Jf = 0
J1.– <Jf
uf
ice
0
Fig. 7.5 Freezing of pure liquid
Water
J2, +>Jf
7.8 Heat Transfer Related to Phase Change
203
with wi being one of the following five parameters r1 ; cP1 ; k1 ; #10 and qs , and i ¼ 1; 2; 3; 4 and 5, respectively. Determining the values of the exponents a; b; g; e; o; and ai ; bi ; gi ; ei ,oi , we arrive at the following expression x ¼ ða2 tÞ1=2 ’ðr12 ; cP12 ; k12 ; #12 ; StÞ
(7.232)
where a2 is the thermal diffusivity of liquid, r12 ¼ r1 =r2 ; cP12 ¼ cP1 =cp2 ; k12 ¼ k1 =k2 ; #12 ¼ #10 =#20 , and St ¼ qs =cP2 #20 being the Stanton number. The speed of the freezing front is Vs ¼
1 a2 1=2 ’ðr12 ; cP12 ; k12 ; #12 ; StÞ 2 t
(7.233)
Equations 7.232 and 7.233 show that the freezing front propagates as t1=2 , whereas its speed decreases as t1=2 . In the framework of the dimensional analysis it is impossible to establish the exact form of the dependence ’ðr12 ; cP12 ; k12 ; #12 ; StÞ and calculate the exact values of x and Vs . In order to find the exact form of the solution, the corresponding equations should be solved exactly. The system of the governing equations of the Stefan problem reads @#1 @ 2 #1 ¼ a1 2 @t @t
(7.234)
for the ice domain 1<x<x, and @#2 @ 2 #2 ¼ a2 @t @x2
(7.235)
for the liquid domain x<x<1. The corresponding boundary conditions are x ¼ x; # ¼ #1 ¼ #2 ¼ #f ¼ 0
(7.236)
x ¼ 1; #1 ¼ #10 ; x ¼ 1; #2 ¼ #20
(7.237)
In addition, the system of Eqs (7.234) and (7.235) is subjected to the following boundary condition expressing the thermal balance at the freezing front @#1 @#2 dx k1 k2 ¼ qs r @x x¼x @x x¼xþ dt
(7.238)
This additional condition allows finding the coordinate of the freezing front x as a function of time. It is emphasized that the problem allows determining x only up
204
7 Heat and Mass Transfer
to an additive constant, which corresponds to the initiation of freezing from any cross-section x (in particular, from x ¼ 1). Equation 7.234 and 7.235 with the boundary conditions (7.236) and (7.237) admit solutions in the form of the following functional equations for the temperature fields in the ice and liquid domains #1 ¼ f1 ða1 ; #10 ; x; tÞ
(7.239)
#2 ¼ f2 ða2 ; #20 ; x; tÞ
(7.240)
Since the governing parameters in (7.239) as well as in Eq (7.240) contains three parameters with independent dimensions, these equations can be transformed to the following self-similar forms #1 ¼ ’1 ð1 Þ
(7.241)
#2 ¼ ’2 ð2 Þ
(7.242)
pffiffiffiffiffiffi pffiffiffiffiffiffi #1 ¼ #1 =#10 and #2 ¼ #2 =#20 with 1 ¼ x= a1 t and 2 ¼ x= a2 t, where respectively. The conditions (7.236) and (7.237) take the form x ¼ x; ’1 ð1x Þ ¼ ’2 ð2x Þ ¼ 0
(7.243)
x ¼ 1; ’1 ð1Þ ¼ 1; x ¼ 1; ’2 ð1Þ ¼ 1
(7.244)
pffiffiffiffiffiffi pffiffiffiffiffiffi where 1x ¼ x= a1 t and 2x ¼ x= a2 t. Accordingly, (7.238) shows that the only possible law of propagation of the freezing front (up to an additive constant) is given by pffi x¼b t
(7.245)
where b is a constant (an eigenvalue of the problem) which is also detrmined by (7.238). In particular, substituting (7.241)–(7.245) into (7.234)–(7.238), one finds the self-similar solutions as ’1 ð1 Þ ¼ A1 þ B1 erf ð1 =2Þ
(7.246)
’2 ð2 Þ ¼ A2 þ B2 efrð2 =2Þ
(7.247)
pffiffiffiffiffi erf b=2 a2 1 A1 ¼ 1; B1 ¼ pffiffiffiffiffi ; A2 ¼ pffiffiffiffiffi ; B2 erf b=2 a1 1 erf b=2 a2 ¼
1 pffiffiffiffiffi 1 erf b=2 a2
(7.248)
Problems
205
and the following equation determining b pffiffiffi k1 exp b=4a21 k2 exp b=4a22 p þ ¼ qs r2 b 2 a1 erf ðb=2a1 Þ a2 ½1 erf ðb=2a2 Þ
(7.249)
Problems P.7.1. Show that the evolution of the thermal field driven by radiative heat flux in a uniform medium after an instantaneous pointwise energy release by a powerful source corresponds to a self-similar solution. The equation that determines temperature field driven by radiative heat flux reads @T 1 @ @T ¼ 2 wr 2 @t r @r @r
(P.7.1)
where w ¼ aT n is the radiant thermal diffusivity coefficient, and it is assumed that the initial energy release happens at r¼0. Integrating (P.7.1) from r ¼ 0 to r ¼ 1 yields the folloving integral invariant 1 ð
Tr 2 dr ¼ Q
4p
(P.7.2)
0
Equations P.7.1 and P.7.2 show that the problem under consideration contains four governing parameters: two dimensional constants, a and Q, and two independent variables r and t. Accordingly, the functional equation for the temperature field reads T ¼ f ða; Q; r; tÞ
(P.7.3)
The dimensions of the governing parameters are
a L2 T 1 yn ;
Q yL3 ;
r ½ L;
t½T
(P.7.4)
Since three of the four governing parameters have independent dimensions, (P.7.3) can be reduced to the following dimensionless equation P ¼ ’ðP1 Þ 0
0
0
where P ¼ T=aa1 Qa2 ta3 and P1 ¼ r=aa1 Qa2 ta3 :
(P.7.5)
206
7 Heat and Mass Transfer
Bearing in mind the dimensions of T; a; Q; r and t, we arrive at the following 0 set of algebraical equations determining the exponents ai and ai 0
0
2a1 þ 3a2 ¼ 0; 2a1 þ 3a2 ¼ 1 0
0
a1 þ a3 ¼ 0; a1 þ a3 ¼ 0 0
(P.7.6)
0
a1 n þ a2 ¼ 1; a1 n þ a2 ¼ 0 Equations P.7.6 yield the following values a1 ¼
1 ; n þ 2=3
a2 ¼
2 1 ; 3 n þ 2=3
a3 ¼
1 ; n þ 2=3
n 1 1 0 0 ; a2 ¼ ; a3 ¼ a1 ¼ 3n þ 2 3n þ 2 3n þ 2
(P.7.7)
0
Then (P.7.5) takes the following form T¼
Q2 ðatÞ3
!1=ð3nþ2Þ ( ’
r
)
ðaQn tÞ1=ð3nþ2Þ
(P.7.8)
Substitution of the expression (P.7.8) into (P.7.1) leads to the following ODE determining the function ’ðxÞ 00 0 2 1 3 0 ð’nþ1 Þx þ ð’nþ1 Þx þ x’ þ ’¼0 x 3n þ 2 x 3n þ 2 1=ð3nþ2Þ
(P.7.9)
, and subscript x denotes differentiation by x. where x ¼ r=ðaQn tÞ P.7.2. Determine mass flux to a pipe wall in the case of fully developed laminar flow of liquid reactant solution. The rate of the heterogeneous reaction at the wall is assumed to be infinite, and accordingly, the reactant concentration at the wall equals zero. A sketch of a fully developed (hydrodynamically) laminar flow is shown in Fig. 5.1. In this case the flow field does not change with x (along the pipe). On the contrary, the reactant concentration field, and in particular, the thickness of the diffusion boundary layer near the pipe wall increases in flow direction. The reactant concentration changes in a thin boundary layer since the kinematic viscosity is typically much larger than the diffusion coefficient, n D. Accordingly, the flow structure and the reactant concentration distribution can be characterized by the maximum velocity u0 at the pipe axis and the reactant concentration c0 in the core of the flow. Thus, as the governing parameters that determine the diffusion flux of the reactant toward the wall one can choose u0 ; c0 , pipe radius R, the diffusion coefficient D; as well as the longitudinal coordinate x as the parameter responsible
Problems
207
for the growth of the diffusion boundary layer thickness in flow direction. Then the functional equation for the reactant diffusion flux reads qm ¼ f ðu0 ; c0 ; R; D; xÞ
(P.7.10)
The dimensions of the diffusion flux and the governing parameters are ½qm ¼ ML2 T 1 ; ½u0 ¼ LT 1 ; ½c0 ¼ L3 M; ½ R ¼ L; ½ D ¼ L2 T 1 ; ½ x ¼L (P.7.11) Three of the five governing parameters in (P.7.10) have independent dimensions. Therefore, according to the Pi-theorem, (P.7.10) can be transformed to the following form P ¼ ’ðP1 ; P2 Þ 0 a1
0 a2
0 a3
(P.7.12) 00 a1
00 a2
00 a3
where P ¼ qm =ca01 Da2 Ra3 ; P1 ¼ x=c0 D R and P2 ¼ u0 =c0 D R : 0 00 Determining the values of the exponents ai ; ai and ai , we find a1 ¼ 1; a2 ¼ 1; 0 0 0 00 00 00 a3 ¼ 1; a1 ¼ 0; a2 ¼ 0; a3 ¼ 1; a1 ¼ 0; a2 ¼ 1 and a3 ¼ 1. Then, we obtain Sh ¼ ’ðx; Ped Þ
(P.7.13)
where Sh ¼ qm R=co D and Ped ¼ u0 R=D are the Sherwood and the diffusional Peclet numbers, and x ¼ x=R: Equation P.7.13 shows that the dimensionless mass transfer coefficient is determined by two dimensionless groups x and Ped . In the framework of the dimensional analysis it is impossible to specify this dependence further more.. However, this problem can be solved in the framework of the dimensional analysis if the diffusion equation is simplified in the boundary layer approximation, which allows another set of the governing parameters to be chosen. For this aim consider the rigorous mathematical formulation the problem (but not its exact solution). The reactant diffusion equation in the thin diffusion layer near the pipe wall reads (Levich 1962) 2u0 @c @2c y ¼D 2 R @x @y
(P.7.14)
c ! c0 at y ! 1; c ¼ 0 at y ¼ 0
(P.7.15)
where y ¼ R r; and r is the radial coordinate in the cylindrical system reckoned from the pipe axis. of coordinates. Introducing the new variable y ¼ yðu0 =DRÞ1=3 , transform (P.7.14) to the following form @c 1 1 @ 2 c ¼ @x 2 y @y2
(P.7.16)
208
7 Heat and Mass Transfer
Equation P.7.16 with the boundary conditions (P.7.15) show that concentration c derends on three dimensional parameters c ¼ f ðc0 ; x; yÞ
(P.7.17)
Two governing parameters in (P.7.17) have independent dimensions, so that this equation can be transformed to the following dimensionless form P ¼ ’ðP1 Þ a
0
(P.7.18)
0
where P ¼ c=ca01 xa2 and P1 ¼ y=c01 xa2 : 0 The corresponding values of the exponents ai and ai are found as: 0 0 a1 ¼ 1; a2 ¼ 0; a1 ¼ 0 and a2 ¼ 1=3. Then, (P.7.18) takes the form c ¼ c0 ’ðÞ
(P.7.19)
where ¼ yðu0 =DRxÞ1=3 . The reactant flux to the pipe wall is u 1=3 1 @c @c 0 qm ¼ D ¼ D c0 @y y¼0 @ ¼0 DR x1=3
(P.7.20)
or Sh ¼
@c @
Ped 1=3 x1=3
(P.7.21)
¼0
Using the expression (P.7.20), we find the cumulative mass flux of reactant to the wall in a pipe section of length x as 2 1=3 ð u0 x I ¼ 2pR qm dx ¼ Ac0 DR DR
(P.7.22)
where A ¼ ð2p=3Þð@c=@Þ¼0 The exact analytical solution of the problem reads (Levich 1962) 2 1=3 u0 x I ¼ 2:01pc0 DR DR
(P.7.23)
References
209
References Acrivos A, Taylor TD (1962) Heat and mass transfer from single spheres in Stokes flow. Phys Fluids 5:378–394 Baehr HD, Stephan K (1998) Heat and mass transfer. Springer, Heidelberg Banks WHH (1965) The thermal laminar boundary layer on a rotating sphere. J Appl Math Phys 16:780–788 Bayley FJ, Owen JN, Turner AB (1972) Heat transfer. Nelson, London Brigman PW (1922) Dimensional analysis. Yale University Press, New Haven Chao BT, Greif R (1974) Laminar forced convection over rotating bodies. Trans ASME J Heat Trans 100:497–502 Dorfman LA (1967) Hydrodynamic resistence and the heat loss of rotating solids. Prentice Hall, Englewood Cliffs Dorodnitsyn AA (1942) Boundary layer in compressible gas. Appl Math Mech 6:449–485 Eastop TD (1973) The influence of rotation on the heat transfer from a sphere to an air stream. Int J Heat Mass Transf 16:1954–1957 Frankel NA, Acrivos A (1968) Heat and mass transfer from small spheres and cylinders freely suspended in shear flow. Phys Fluids 11:1913–1918 Fujii T (1991) Theory of laminar film condensation. Springer, Heidelberg Hussaini MY, Sastry MS (1976) The laminar compressible boundary layer and rotating sphere with heat transfer. Trans ASME J Heat Transf 98:533–535 Illingworth CR (1949) Steady flow in the laminar boundary layer of a gas. Proc Roy Soc A 199:533–558 Kaviany M (1994) Principles of convective heat transfer. Springer, Heidelberg Kays WM (1975) Convective heat and mass transfer. McGraw-Hill, New York Kays WM, Crawford ME (1980) Convective heat and mass transfer, 2nd edn. McGraw-Hill, New York Kestin J (1966) The effect of free-stream turbulence on heat transfer rates. In: Irvin TE, Hartnett JP (eds). Advances in Heat Transf. 3: 1–32 Kreith F (1968) Convective heat transfer in rotating systems. In: Irvin TE, Hartnett JP (eds). Advances in Heat Transf. 5: 129–251 Kutateladze SS (1963) Fundamentals of heat transfer. Academic, New York Landau LD, Lifshitz EM (1987) Fluid mechanics, 2nd edn. Pergamon, New York Lavender WJ, Pei DCT (1967) The effect of fluid turbulence on the rate of heat transfer from spheres. Int J Heat Mass Transf 10:529–539 Lee MH, Jeng DR, De Witt KJ (1978) Laminar boundary layer transfer over rotating bodies in forced flow. Trans ASME J Heat Transf 100:497–502 Levich VG (1962) Physicochemical hydrodynamics. Prentice-Hill, Englewood Cliffs Loitsyanskii LG (1966) Mechanics of liquid and gases. Pergamon, Oxford Nusselt W (1916) Die oberflachen Kondensation des wasserdampes. Zeitsehrift des Vereines Deutscher Ingenieure 60(2):541–546 Pohlhausen E (1921) Der warmeaustausch zwischen festen Korpern and Flussigkeiten mit kleiner Reibung and kleiner Warmeleitung. ZAMM 1:115 Raithby CA, Hollands KGT (1998) Natural convection. In: Rohsenow WW, Hartnett JP, Cho YI (eds) Handbook of heat transfer, 3rd edn. McGraw-Hill, New York Rayleigh L (1915) The principle of similitude. Nature 95:66–68 Riabochinsky D (1915) The principle of similitude-letter to the editor. Nature 95:591 Rohsenow WM, Hartnett JP, Cho YI (1998) Handbook of heat transfer, 3rd edn. McGraw-Hill, New York Schlichting H (1979) Boundary layer theory, 7th edn. McGraw-Hill, New York Sedov LI (1993) Similarity and dimensional methods in mechanics, 10th edn. CRC Press, Boca Raton Soo SL (1990) Multiphase fluid dynamics. Science Press and Gower Technical, Beijing
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Spalding DB (1963) Convective mass transfer. Edward Arnold, London Stephan K (1992) Heat transfer in condensation and boiling. Springer, Heidelberg Stewartson K (1949) Correlated compressible and incompressible boundary layers. Proc Roy Soc A 200:84–100 Tieng SM, Yan AC (1993) Experimental investigation on convective heat transfer of heated spinning sphere. Int J Heat Mass Transf 36:599–610 White FM (1988) Heat and mass transfer. Addison-Wesley, New York Zel’dovich Ya.B, Kompaneyets AS (1970) Towards a theory of heat propagation with conductivity depending on temperature. In: Collective works in commemoration of A. F. Ioffe. The USSR Acad Sci 61–71 (In Russian) Zel’dovich YaB, Raizer YP (2002) Physics of shock waves and high-temperature hydrodynamic phenomena. Dover, New York
Chapter 8
Turbulence
8.1
Introductory Remarks
The turbulence represents itself a very complicated hydrodynamic phenomenon characterized by irregular unsteady fluid motion. It emerges in liquid and gas flows at sufficiently high Reynolds numbers when laminar flow regime becomes unstable and strongly perturbed. This process is accompanied by arising turbulent eddies of different sizes which are, in their turn, sources of velocity disturbances at each point of the flow field. The amplitudes and frequencies of such disturbances depend on the Reynolds number value. The scales of these disturbances decrease as Re increases, whereas their frequency is proportional to the Reynolds number. An exceptional complexity of turbulence impedes the theoretical analysis of this phenomenon. In this situation a number of important results (mostly qualitative) may be obtained using the methods of the similarity theory and dimensional analysis (Kolmogorov, 1941a, b; Obukhov 1941). Following these works we address briefly some problems related to the uniform isotropic turbulence, as the examples illustrating applications of the dimensional considerations to study turbulent flows. The actual velocity vector v at any point of developed turbulent flows can be presented as a sum of the mean velocity v (obtained by averaging the actual velocity 0 over a long time interval) and fluctuation velocity v . The existence of turbulent fluctuation velocity and other fluctuating hydrodynamic and thermal characteristics significantly affect the flow structure, as well as the intensity of processes of momentum, heat and mass transfer. Turbulent flows can be represented schematically as a conglomerate of turbulent eddies of different sizes that generate fluctuations of velocity, temperature, etc. The influence of large and small eddies on flow field is essentially different. The large eddies are the bearers of kinetic energy of turbulent flow, which is transferred though a cascade of smaller and smaller eddies to the smallest ones. On the other hand, the smallest eddies are subjected to significant viscous forces which dissipate the transferred kinetic energy into heat. The energy transfer from large to small eddies occurs practically without dissipation, so that the energy flux from large L.P. Yarin, The Pi-Theorem, Experimental Fluid Mechanics, DOI 10.1007/978-3-642-19565-5_8, # Springer-Verlag Berlin Heidelberg 2012
211
212
8 Turbulence
eddies equals energy dissipation in the smallest ones. That allows the assumption that in developed turbulent flows a continuous energy transfer from large to small eddies takes place. Such peculiarity of turbulent flows has a principal meaning for the evaluation of a number of important characteristics of developed turbulent flows. Indeed, using the concept of the energy transfer from large to the smallest eddies, it is possible to evaluate the dissipation of kinetic energy by analyzing solely the large scale motions. Taking into account the above-mentioned peculiarities of the energy transfer, it is possible to neglect the influence of viscosity on flow characteristics within the domain of large eddies. In this case the energy dissipation is determined by only three parameters, namely, the fluid density r, the characteristic size of large eddies l which is of the same order as the external flow scale, and the mean velocity change Du over the length l. Accordingly, the functional equation for the energy dissipation per unit time E takes the form E ¼ f ðr; Du; lÞ
(8.1)
All the governing parameters in (8.1) have independent dimensions. Therefore, according to the Pi-theorem, (8.1) can be written as E ¼ cra1 Dua2 la3
(8.2)
where c is a dimensionless constant. Taking into account the dimension ½E ¼ L1 MT 3 , we find the values of the exponents ai as a1 ¼ 1; a2 ¼ 3 and a3 ¼ 1, which allows transformation of (8.2) to the following form e¼c
Du3 l
(8.3)
where e ¼ E=r is the energy dissipation per unit time and unit mass. In saddition, the functional equation for the turbulent (eddy) viscosity associated with the turbulent Reynolds stresses can be also expressed through the same set of the governing parameters mT ¼ f ðr; Du; lÞ
(8.4)
Using the Pi-theorem, we arrive at the following expression nT ¼ c1 lDu
(8.5)
where nT ¼ mT =r is the turbulent kinematic viscosity and c1 is a dimensionless constant.
8.1 Introductory Remarks
213
Then the relation between the turbulent and physical (molecular) viscosities can be expressed as nT ¼ c1 Re n
(8.6)
where Re ¼ lDu=n. Equation (8.6) shows that the ratio nT =n increases with the Reynolds number. Consider now some local properties of turbulence that characterize its features at the scales of the order l much smaller than the characteristic scale of large eddies l but much larger than the scale of the smallest dissipative eddies l0 . It is possible to assume that turbulence is isotropic in flow regions far away from solid surfaces. Using this assumption, we evaluate first the change of velocity of turbulent motion vl over a distance of the order of l. Choosing as the governing parameters the energy dissipation e and length l, we present the functional equation for vl as follows vl ¼ f ðe; lÞ
(8.7)
Bearing in mind the dimensions ½e ¼ L2 T 3 ; ½l ¼ L and ½nl ¼ LT 1 , we arrive at the scaling law vl ðelÞ1=3
(8.8)
It expresses the Kolmogorov-Obukhov law: the change of velocity over a small distance l is scales as l1=3 . The local Reynolds number for the smallest dissipative eddies corresponds to the borderline where molecular viscosity begins to play role. Therefore, this Reynolds number should be of the order of one. On the other hand, it is determined by velocity vl corresponding to l ¼ l0 , the scale of the smallest eddies responsible for energy dissipation l0 , and fluid viscosity n. Accordingly, we arrive at the following estimate of l0 l0
l Re3=4
(8.9)
In the framework of the dimensional analysis it is possible to evaluate velocity change over the smallest dissipative eddy scale l0 [which is given by (8.8) and (8.9)], the frequency of velocity pulsations, as well as the turbulent pulsations of temperature at l >> l0 . In order to estimate the temperature fluctuations in a nonuniformly heated fluid over a distance l, consider the dependence for the temperature difference Tl (temperature fluctuations) following Obukhov (1949). For this aim, we use the expression for the energy dissipation due to thermal conductivity
214
8 Turbulence
ET k
ðrT Þ2 ð Tl = l Þ k T T
(8.10)
where ET is the energy dissipation due to thermal conductivity of fluid ð½ET ¼ L1 MT 3 Þ, k is the termal conductivity (½k ¼ LMT 3 y1 ), Tis the actual temperature. Accordingly, the rate of dissipative changing of temperature may be estimated as follows 0
ET wt
ð Tl = l Þ 2 T
(8.11)
0 0 where ET is the rate of dissipative changing temperature ( ET ¼ yT 1 ), and wt is the turbulent thermal diffusivity (½wt ¼ L2 T 1 ). 0 Assuming that ET ¼ ’=T , where ’ is a certain factor that determine the local features of turbulence in a non-uniformly heated fluid, we can postulate the following relation wt
ð Tl = l Þ 2 ’ ¼ T T
(8.12)
At small intensity of temperature fluctuations (Tl =T <<1) it is possible to replace the actual temperature T by the mean temperature of fluid and assume that ’ ¼ wt
2 Tl l
(8.13)
Thus, the factor ’ depends on three dimensional parameters, namely, (1) turbulent thermal diffusivity wt , (2) temperature fluctuations Tl , and (3) the distance l ’ ¼ f ðwt ; Tl ; lÞ
(8.14)
Since all the governing parameters in the functional (8.14) have independent dimensions, according with the Pi-theorem, it takes the following form ’ ¼ wat 1 Tla2 la3
(8.15)
where c is a dimensionless constant. Bearing in mind the dimension of ’ (½’¼y2 T 1 ), we find using the principle of dimensional homogeneity that the values of the exponents ai are: a1 ¼1;a2 ¼2; anda3 ¼1. Then (8.15) reads
8.2 Decay of Isotropic Turbulence
215
’ ¼ cwt
2 Tl l
(8.16)
Accounting that wt nt lvl , and vl ¼ ðelÞ1=3 , we arrive at the following estimate for the temperature fluctuations Tl2 ’e1=3 l2=3
(8.17)
It is seen that temperature fluctuations like the velocity fluctuations are proportional to l1=3 on the scale l>>l0 . Below we consider in detail the applications of the Pi-theorem to a number of important types of turbulent flows, in particular, turbulent near-wall flows, flows in smooth and rough pipes and channels, as well as various kinds of turbulent jets.
8.2
Decay of Isotropic Turbulence
The behavior of isotropic turbulence is described by the von Karman and Howarth (1938) [see also Pope (2000)] @bdd 1 @ 1 @ 4 4 @bdd þ 4 ¼ 2n 4 r ðr bddd Þ @t @r r @r r @r
(8.18)
where bdd and bddd are the components of the second and third correlation tensor, respectively. At very the smallest-scale dissipative eddies the corresponding small velocity fluctuations are dominated by the viscous effects and the second term on the righthand side in (8.18) can be omitted. Then the von Karman and Howarth equation takes the form @bdd 1 @ @bdd r4 ¼ 2n 4 r @r @t @r
(8.19)
To find the solution of (8.19), it is necessary to impose the initial distribution of the second correlation bdd ðr; tÞ, i.e. bdd ðr; 0Þ. Equation (8.19) admits the invariant (the Loitsyanskii invariant L0 ) which can be determined through the initial condition bdd ðr; 0Þ (Loitsyanskii 1939) 1 ð
L0 ¼
r 4 bdd dr 0
(8.20)
216
8 Turbulence
The value of the invariant does not change in time and is valid for all time moments t > 0. Consider the problem (8.19) and (8.20) assuming that the value of L0 is non-zero and finite, 0 < L0 < 1, as the most plausible assumption (Landau and Lifshitz 1987). Introducing the new variables bdd ¼ bdd =L0 and t ¼ 2t, we transform (8.19) and (8.20) to the following form @bdd 1 @ 4 @bdd ¼n 4 r @t @r r @r
(8.21)
1 ð
r4 bdd dr ¼ 1
(8.22)
0
From (8.21) and (8.22) it follows that bdd depends on three governing parameters bdd ¼ f ðt ; n; rÞ
(8.23)
½n ¼ L2 T 1 and ½r ¼ L, These parameters have the following dimensions½t ¼ T; 5 whereas the dimension of the unknown quantity is bdd ¼ L . It is seen that two of the three governing parameters have independent dimensions. Then, in accordance with the Pi-theorem, the dimensionless form of (8.23) reads P ¼ ’ðP1 Þ 0 a1
(8.24)
0
where P ¼ bdd =ta1 na2 and P1 ¼ r=t na2 . Using the principle of the dimensional homogeneity, we find the value of the 0 0 0 exponents ai and ai as a1 ¼ a2 ¼ 5=2 and a1 ¼ a2 ¼ 1=2. Then, (8.24) transforms to the following expression for bdd bdd
r pffiffiffiffiffiffi ¼ 5 ’ nt ðnt Þ2 1
(8.25)
To find an exact expression for the function ’, one should express the derivatives in (8.21) using (8.25)
n
@bdd 1 1 d’ ¼ 5’ þ x @t 2 ðnt Þ5=2 t dx
(8.26)
1 @ 1 4 d’ d 2 ’ 4 @bdd r þ ¼ 2 r 4 @r @r ðnt Þ5=2 t x dx dx
(8.27)
8.3 Turbulent Near-Wall Flows
217
where ’ ¼ ’ðxÞ and x ¼ r=ðnt Þ1=2 . Substituting (8.26) and (8.27) into (8.21), we arrive at the following ODE for the function ’ ¼ ’ðxÞ d2 ’ 4 x d’ þ þ þ 2’ ¼ 0 x 2 dx dx2
(8.28)
The Loitsyanskii invariant takes the form 1 ð
x4 ’ðxÞdx ¼ 1
(8.29)
0
The solution of (8.28) and (8.29) and a detailed analysis of the problem on decay of isotropic turbulence can be found in the monographs of Sedov (1993) and Barenblatt (1996).
8.3 8.3.1
Turbulent Near-Wall Flows Plane-Parallel Flows
Turbulent flows over smooth and rough walls were a subject of a large number of experimental and theoretical investigations. The most important results of these works are combined in several well-known research monographs by Hinze (1975), Rotta (1962), Schlichting (1979) and Monin and Yaglom (1965-Part 1, 1967-Part 2), as well as in numerous text- and reference books on hydrodynamics. In the present sub-section we focus our attention on the application of the Pi-theorem to reveal some fundamental features of turbulent near-wall flows, while addressing an interested reader to find the discussion of the other aspects of such flows in the above-mentioned monographs. We begin with the general form of the distribution of mean velocity in planeparallel flows over a plate. Let vbe the mean velocity vector in a plane-parallel flow with the components u ¼ uðyÞ; v ¼ 0 and w ¼ 0 in the x, y and z directions, respectively. It is seen that flow characteristics, in particular, velocity u, in such flows depend only on the transversal coordinate y normal to the wall. It is plausible to assume that local velocity uðyÞ is determined by the following four parameters: the fluid density ½r ¼ L3 M and viscosity ½m ¼ L1 MT 1 , the shear stress at the wall ½t ¼ L1 MT 2 and the distance from the wall ½ y ¼ L u ¼ f ðr; m; t ; yÞ
(8.30)
218
8 Turbulence
Three of the four governing parameters in (8.30) have independent dimensions, so that according to the Pi-theorem, this equation transforms to the following dimensionless form P ¼ ’ðP1 Þ 0
0
(8.31)
0 a3
where P ¼ u=ra1 ma2 ta3 and P1 ¼ y=ra1 ma2 t . 0 The values of the exponents ai and ai are found as a1 ¼ 1=2 , a2 ¼ 0 , 0 0 0 a3 ¼ 1=2; a1 ¼ 1=2; a2 ¼ 1 and a3 ¼ 1=2, which allows us to present (8.31) as follows yu u ¼’ n u
(8.32)
uþ ¼ ’ðyþ Þ
(8.33)
or
=n are In (8.33) uþ ¼ u=u and yþ ¼ yu pffiffiffiffiffiffiffiffiffi ffi the dimensionless velocity and distance from the wall, respectively, u ¼ t =r is the friction velocity. Equations (8.32) or (8.33) expresses the Prandtl law-of-the wall (Prandtl, 1925b). The exact form of the function ’ðyþ Þcan be determined in two limiting cases corresponding to small or large values of yþ . At small yþ (in the viscous sublayer) velocity is so low rgat it should not dependend on density, since the inertia effect is negligible. Then the functional equation for the velocity in the viscous sublayer reads u ¼ f ðm; t ; yÞ
(8.34)
Since all the governing parameters in (8.34) have independent dimensions, it takes the form u ¼ cma1 ta2 ya3
(8.35)
where c is a constant. Determining the values of the exponents ai as a1 ¼ 1; a2 ¼ 1 and a3 ¼ 1, we obtain u yu ¼c n u
(8.36)
Equation (8.36) shows that fluid velocity increases linearly in y within viscous sublayer. This conclusion agrees fairly well with the experimental data in the 0 < yþ < 5.
8.3 Turbulent Near-Wall Flows
219
Consider now the flow far enough from the wall. At large distances from the wall, turbulent transfer plays the dominant role. That allows one to omit molecular viscosity in the set of the governing parameters, and assume that flow characteristics are determined by three parameters: r; t and y. It is easy to see that in this case it is impossible to assume m ¼ 0 in (8.30) and write the functional equation for velocity in the form u ¼ f ðr; t ; yÞ. That would result in the unrealistic outcome that velocity does not depend on y. Indeed, since all the governing parameters in this equation have independent dimensions, it takes the following form u ¼ cra1 ta2 ya3 , where c is a dimensionless constant, and a1 ¼ 1=2;a2 ¼ 1=2 and a3 ¼ 0. At the same time, this set of governing parameters determines the velocity gradient far from the wall, so that functional equation for the problem can be written in the form du ¼ f ðr; t ; yÞ dy
(8.37)
It is emphasized that (8.37) implies a dependence of velocity gradient on r; t and y due to the fact that at large Reynolds numbers, when the influence of molecular viscosity is negligible, the value of the velocity gradient at each point must be determined by only two parameters r; t and the distance y (Landau and Lifshitz 1979). All governing parameters in (8.37) have independent dimensions. In this case the Pi-theorem determines the following form of (8.37) du ¼ w ra1 ya2 ta3 dy
(8.38)
where w ¼ x1 is a constant. Also, a1 ¼ 1=2; a2 ¼ 1 and a3 ¼ 1=2. Then (8.38) takes the form du u ¼ dy wy
(8.39)
u ðln y þ cÞ w
(8.40)
Integrating (8.39) yields u¼
The constant c in (8.40) is found from the matching condition at the outer boundary layer of the viscous sublayer y0 n=u . There the velocity is close to the friction velocity: u u . The latter results in the logarithmic velocity profile u¼
u yu ln w n
(8.41)
220
8 Turbulence
The accuracy of the law-of-the wall (8.41) can be improved by including an empirical constant in addition to the logarithmic term. Then, the comparison with the experimental data shows that the von Karman constant w ¼ 0.4 and the updated equation (with the additional constant) becomes u ¼ u ½2:5 lnðyu =nÞ þ 5:1 ¼ 2:5u lnðyu =0:13nÞ as suggested by Coles (1955).
8.3.2
Pipe Flows
Consider fully developed turbulent flows in straight pipes with circular crosssection of radius R. The local velocity in such flows is determined by the distance from wall y ¼ R r (r is the corresponding distance from the pipe axis), fluid density r and viscosity m and friction velocity u (Monin and Yaglom, 1971) u ¼ f ðR; y; r; m; u Þ
(8.42)
The transformation of (8.42) to thee dimensionless form by with the help of the Pi-theorem yields P ¼ ’ðP1 ; P2 Þ 0 a1
0
0
(8.43) 00 a1
00
00 a3
where P ¼ u=ua1 Ra2 ra3 ; P1 ¼ y=u Ra2 ra3 and P2 ¼ m=u Ra2 r . Allying the principle of the dimensional homogeneity and find the values of the 0 00 0 0 0 exponents ai ; ai and ai as a1 ¼ 1; a2 ¼ 0; a3 ¼ 0; a1 ¼ 0; a2 ¼ 1; a3 ¼ 0; 00 00 00 a1 ¼ 1; a2 ¼ 1 and a3 ¼ 1, we transform (8.43) to the form of a dependence of the dimensionless velocity u=u on two dimensionless groups y=R and u R=n u y u R ¼’ ; u R n
(8.44)
There are two important cases when the function ’ðy=R; u R=nÞ can be reduced to a function of one dimensionless variable. The first of them corresponds to small values of the ratio y=R 1, at which the dependence of the local velocity on radius of the pipe becomes unimportant. In this case the functional equation for local velocity u reduces to u ¼ f ðy; r; m; u Þ
(8.45)
Transforming (8.45) to dimensionless form by using the Pi-theorem, we arrive at u y u ¼’ u n
(8.46)
8.3 Turbulent Near-Wall Flows
221
The second limiting case corresponds to flow in the so-called ‘turbulent core’, i.e. the region surrounding the pipe axis. In this region the turbulent shear stress is larger than the viscous one. Correspondingly, the functional equation for the velocity gradient du=dy should be written as du ¼ f ðy; R; u Þ dy
(8.47)
Applying the Pi-theorem to (8.47), we reduces it to the form du u y ¼ ’ dy R R
(8.48)
Assuming that ’ðy=RÞ ¼ Rdcðy=RÞ=dy with cð0Þ ¼ 0, and integrating (8.48) from y to R, we arrive at the following equation for the defect of velocity that was found by von Karman (1930) y u0 u ¼c u R
(8.49)
where u0 is the velocity at the boundary of the turbulent core. Equation (8.49) expresses the law of the defect of velocity. It is valid within the turbulent core of pipe flows where 1 < < 1, with ¼ y=R and 1 and 1 being the dimensionless coordinates of the boundaries of the turbulent core.
8.3.3
Turbulent Boundary Layer
The flow in the turbulent boundary layer over a flat plate is significantly different from plan-parallel and pipe flows. The differences stem from the different conditions at the outer boundary of the boundary layer and a pipe axis, as well as from the dependence flow characteristics on coordinates. In particular, in planeparallel and pipe flows velocity depends only on the normal-to-wall coordinate y, whereas in turbulent boundary layers velocity depends on the normal y and longitudinal x coordinates. Accordingly, in turbulent boundary layers the friction velocity is not constant but changes with x. Assume that the defect of velocity within the outer part of the turbulent boundary layer u1 u depends on local parameters of the flow corresponding to a fixed x, e.g. on the thickness of the boundary layer dðxÞ, the friction velocity u , as well as on the velocity of the undisturbed fluid u1 , and distance from the wall y. Then we can write the following functional equation
222
8 Turbulence
ðu1 uÞ ¼ f ðu1 ; d; y; u Þ
(8.50)
Applying the Pi-theorem, we transform (8.50) to the following dimensionless form P ¼ ’ðP1 ; P2 Þ
(8.51)
where P ¼ ðu1 uÞ=u ; P1 ¼ y=d and P2 ¼ u1 =u . The experimental data on the velocity distribution in turbulent boundary layers show that the dependence of the dimensionless group P on P1 is very weak (Clauser 1956). That allows simplification of (8.51) to the form similar to the one for pipe flows [(8.49)] y u1 u ¼’ u d
(8.52)
On the other hand, the flow in a thin viscous sublayer adjacent to the wall is described by (8.36).
8.4 8.4.1
Friction in Pipes and Ducts Friction in Smooth Pipes
The application of the Pi-theorem to study friction in smooth pipes of different geometry in the case of laminar flow of incompressible fluid was considered in Chap. 4. In the present subsection we briefly address this problem in relation to friction in fully developed turbulent flows in smooth pipes. It is quite plausible to assume that friction factor, and thus pressure gradient ½dP=dx ¼ L2 MT 2 , in such flows is determined by four dimensional parameters: (1) the mean velocity ½u ¼ LT 1 , pipe diameter ½d ¼ L, as well as fluid density ½r ¼ L3 M and viscosity ½m ¼ L1 MT 1 dP ¼ f ðu; d; r; mÞ dx
(8.53)
It is emphasized that there is a principal distinction between the functional equations for friction or pressure gradient in pipes with fully developed turbulent and laminar flows. In the case of turbulent pipe flows, (8.53) contains fluid density as one of the governing parameters. The latter shows that the inertial forces play an important role in fully developed turbulent flows in pipes. In contrast, in steadystate, fully developed laminar flow, the inertial forces are immaterial and the fluid density is absent from the corresponding (8.53).
8.4 Friction in Pipes and Ducts
223
In (8.53) three of the four governing parameters have independent dimensions, which allows transformation of (8.53) to the following dimensionless form P ¼ ’ðP1 Þ 0
(8.54) 0
0
where P ¼ ðdP=dxÞ=d a1 ra2 ma3 and P1 ¼ u=da1 ra2 ma3 . Taking into account the dimensions of the pressure gradient dP=dx and the governing parameters with independent dimensions, we find the values of the 0 0 0 0 exponents ai and ai as a1 ¼ 3; a2 ¼ 1; a3 ¼ 2; a1 ¼ 1; a2 ¼ 1 and a3 ¼ 1. Then, (8.54) takes the form l¼
’ðReÞ ¼ cðReÞ Re2
(8.55)
where l ¼ 2ðdP=dxÞd=ru2 is the friction factor and Re ¼ ud=n: The friction factor l can be described by different empirical and semi-empirical correlations for the function cðReÞproposed for smooth pipes, for example by the Blasius correlation cðReÞ ¼ 0:3164Re1=4
(8.56)
(valid for Re < 105 ) or the Prandtl universal friction law for smooth pipes pffiffiffi 1 pffiffiffi ¼ 2:01 lg Re l 0:8 l
(8.57)
which is valid up to Re ¼ 3.4 106 (Schlichting 1979).
8.4.2
Friction in Rough Pipes
Numerous experimental investigations performed during the last century show that roughness significantly affects friction, and thus the pressure gradient, in turbulent flows in rough pipes. The functional equation for the pressure gradient dP=dx in this case can be written as1
1 Strictly speaking, (8.58) should be written as: dP=dx ¼ f ðu; def ; m; r; ks ; a1 ; a2 ai Þ; where a1 ; a2 ai are the parameters characterizing the shape and distribution of rough elements on the surface.
224
8 Turbulence
dP ¼ f ðu; def ; m; r; kÞ dx
(8.58)
with def and k being the effective diameter and characteristic roughness amplitude (corresponding to sand grains glued to the inner wall) of such pipes (the choice of the characteristic scales of rough pipes is discussed in Herwig et al. 2008). Transforming (8.58) to the dimensionless form with the help of the Pi-theorem, we arrive at the following equation for the friction factor l l ¼ ’ðRe; kÞ
(8.59)
where l ¼ 2ðdP=dxÞd=ru2 and the relative sand roughness is k ¼ k=def . Equation (8.59) shows that the friction factor of rough pipes depends on two dimensionless parameters: the Reynolds number based on the effective diameter of a pipe and the relative sand roughness ks . Typically, three different regimes of turbulent flows in rough pipes are distinguished: (1) the hydraulically smooth region corresponds to u k=n < 5, (2) the transition region-to 5 < u k=n < 70; and (3) the completely rough region with u k=n>>70. At small enough values of the dimensionless ratio u k=n, rough pipes are practically hydraulically smooth since their friction factor depends only on the Reynolds number [as for example, in the Blasius correlation (8.56)]. At large values of ks , the riction factor of rough pipes depends practically only on the relative roughness. In such flows the quadratic resistance law in which l depends only on ks rather than on Re (i.e. l is proportional to ru2 ) is valid.
8.5 8.5.1
Turbulent Jets Eddy Viscosity and Thermal Conductivity
In the aero- and hydromechanics of submerged turbulent jets the main physical feature to be accounted for is the fact that the intensity of molecular momentum, heat and mass transfer in these flows is negligibly small than the intensity of turbulent transfer due to fluctuation motion of fluid associated with eddies. The latter allows one to simplify the system of the turbulent Reynolds equations describing turbulent jets of incompressible fluid
@u @u 1 dP @ @u j ¼ þ y ðn þ nT Þ u þv @x @y r dx @y @y
(8.60)
@uyj @vyj þ ¼0 @x @y
(8.61)
8.5 Turbulent Jets
225
where u, v and P are the longitudinal and lateral velocity components and pressure, respectively (with bars denoting the averaged parameters), n and nT are the molecular and eddy kinematic viscosities of fluid and subscripts j ¼ 0 and 1 corresponds to the plane and axisymmetric jets, respectively. The physically plausible assumption that nT >> n allows us to omit n in (8.60) and reduce the transfer term in this equation to a simpler form
@ @u @ @u ¼ yj ðn þ nT Þ yj nT @y @y @y @y
(8.62)
As usual with the turbulent Reynolds equations, the system of (8.60–8.61) should be supplemented by a semi-empirical correlation that determine the dependence of the eddy viscosity on the characteristics of the mean velocity field. A number of such correlations were suggested by Prandtl (1925a, 1942), von Karman (1930) and Taylor (1932) in the framework of the semi-empirical theories of turbulence. There are two different approaches to express of the eddy viscosity: (1) differential approach, and (2) the integral one. The former implies that turbulent transfer in shear flow (such as submerged jets) is determined by local features of the mean velocity field in the vicinity of a considered point. In particular, as the governing parameters that determine turbulent transfer one can take the local velocity gradient of the mean flow du=dy, the fluid density r, as well as some characteristic length l that account for the displacement of fluid elements in the lateral y-direction and is termed the mixing length. Accordingly, the functional equation for the eddy viscosity in one-dimensional shear flow with zero pressure gradient dP=dx ¼ 0(including the submerged jet flows) reads mT ¼ f1
du r; l; dy
(8.63)
where mT is the turbulent eddy viscosity. Since all the governing parameters in (8.63) have independent dimensions, namely ½r ¼ L3 M, ½l ¼ L and ½du=dy ¼ T 1 , it reduces to m T ¼ c 1 ra 1 la 2
du dy
a3 (8.64)
where c1 is a constant. Taking into account the dimension of the eddy viscosity ½mT ¼ L1 MT 1 , we find the values of the exponents ai as a1 ¼ 1, a2 ¼ 2 and a3 ¼ 1, and obtain the following relations for mT and nT mT ¼ c1 rl2
du dy
(8.65)
226
8 Turbulence
n T ¼ c1 l 2
du dy
(8.66)
In the case when we take into account both the first and second derivatives, the functional equation for the eddy viscosity reads du d 2 u mT ¼ f2 r; l; ; 2 dy dy
(8.67)
Equation (8.67) contains four governing parameters, three of them have independent dimensions. Then (8.67) reduces to P ¼ ’ðP1 Þ
(8.68) 0 a 0 a2 2 2 a3 a1 2 2 a3 1 where P ¼ mT = r ðdu=dyÞ ðd u=dy Þ and P1 ¼ l= r ðdu=dyÞ ðd u=dy Þ .
0
0 a2
0
Finding the values of the exponents ai and ai as a1 ¼ 1, a2 ¼ 3, a3 ¼ 2; a1 ¼ 0 0 0 a2 ¼ 1 and a3 ¼ 1, we arrive at the following correlation
l mT ¼ r ’ ðdu=dyÞ=ðd 2 u=dy2 Þ ðd2 u=dy2 Þ2 ðdu=dyÞ3
(8.69)
The expression for the eddy viscosity can be significantly simplified by using the von Karman similarity hypothesis (von Karman 1930) which implies that the mixing length l is determined by the local characteristics of the mean velocity field in the vicinity of a point of consideration, in particular, by the values of the derivatives du=dy and d 2 u=dy2 as l ¼ cl
du d 2 u ; 2 dy dy
(8.70)
Bearing in mind (8.70), we can exclude l from (8.67) and transform this equation to the following form mT ¼ f2
du d2 u r; ; 2 dy dy
(8.71)
All the governing parameters in (8.71) have independent dimensions. Therefore, in accordance with the Pi-theorem, we obtain mT ¼ c 2 r
a1
du dy
a2
d2 u dy2
a3 (8.72)
8.5 Turbulent Jets
227
where c2 is a constant. Taking into account the dimensions of mT , r, du=dy, and d 2 u=dy2 , we find the values of the exponents ai as a1 ¼ 1, a2 ¼ 3 and a3 ¼ 2. After that, (8.72) takes the following form mT ¼ c2 r
ðdu=dyÞ3 ðd 2 u=dy2 Þ2
(8.73)
The integral approach to the eddy viscosity implies that turbulent transfer in shear flows is determined by the integral flow characteristics (in particluar, for the far field of a turbulent submerged jet by its total momentum flux Jx ), as well as fluid density and local coordinates x and y mT ¼ f ðr; Jx ; x; yÞ
(8.74)
where the governing parameters have the following dimensions ½r ¼ L3 M; ½Jx ¼ Lj MT 2 ; ½ x ¼ L; ½ y ¼ L
(8.75)
Since in the framework of the Pi-theorem the difference n k ¼ 1, (8.74) takes the form P ¼ ’1 ðP1 Þ 0
0
a
(8.76)
0
where P ¼ mT =ra1 Jxa2 xa3 and P1 ¼ y=ra1 Jx 2 xa3 . 0 For the axisymmetric jets (j ¼ 1) the exponents ai and ai are found as a1 ¼ 1=2, 0 0 0 a2 ¼ 1=2, a3 ¼ 0; a1 ¼ 0, a2 ¼ 0 and a3 ¼ 1 which yields y mT ¼ ðrJx Þ1=2 ’ x
(8.77)
Within the far field of turbulent axisymmetric jets, the distribution of the longitudinal velocity component u can be presented as y u ¼c um d
(8.78)
where um is the axial velocity and d is the half-width of the jet. Indeed, since the velocity profile in the far field of the jet is determined by three dimensional parameters, namely the velocity at the jet axis um , the jet thickness d, and a coordinate y of a point under consideration, the functional equation u ¼ f ðum ; d; yÞ is valid. It is easy to see that two of the three governing parameters in this equation possess independent dimensions. Therefore, applying the Pi-theorem results in (8.78).
228
8 Turbulence
Beating in mind (8.78), and the integral invariant of the axisymmetric jets Ðy Jx ¼ ru2 ydy ¼ const, we obtain 0
91=2 81 sffiffiffiffi = <ð Jx cðÞd ¼ um d ; : r
(8.79)
0
where ¼ y=d. Substituting (8.79) into (8.77), we obtain the following expression for the kinematic eddy viscosity 91=2 81 = <ð y nT ¼ ðum dÞ cðÞd ’ ; : x
(8.80)
0
In the particular case of ’ðy=dÞ being a week function ofy=x, it is possible to assume that ’1 is a constant. Then, (8.80) becomes identical Prandtl’s relation for the eddy viscosity in submerged turbulent jets (Prandtl 1942) nT ¼ const ðum dÞ
(8.81)
According to (8.81), the kinematic eddy viscosity is constant in the axisymmetric turbulent jets, whereas in plane turbulent jets nT ¼ nT ðxÞ (Vilis and Kashkarov 1965). Corrsin and Uberoi (1950), Antonia et al. (1975), Chevray and Tutu (1978), Chua and Antonia (1990) showed that in reality the eddy viscosity changes across the axisymmetric turbulent jets. It appears to be constant in the inner part of the jet flow and decreases rapidly at toward the external boundary of the mixing layer at the jet edge. The thermal analog of nT is the eddy thermal diffusivity aT : The experimental data show that there is a certain difference between the values of nT and aT , so that the turbulent Prandtl number PrT ¼ nT =aT is not equal to one (Table 8.1, Mayer and Divoky 1966). The experimental data also show that nT and aT , as well as the turbulent Prandtl number change over the cross-section of submerged turbulent jets (Fig. 8.1). The mean values of PrT in the far field of submerged jets are about 0.75–0.8 (Vulis and Kashkarov, 1965; Chua and Antonia, 1990). Note, that the values of the turbulent Prandtl and Schmidt numbers practically do not depend on the physical properties of fluid, i.e. on the molecular values of the Prandtl and Schmidt numbers. [The Schmidt number represents itself the ration of the kinematic viscosity to diffusivity and in mass transfer processes plays a similar role to that of the Prandtl number in the heat transfer processes.] The experimental studies of the velocity, temperature or concentration fields in turbulent jets of different fluids (mercury: Pr 102 ; oil: Pr 103 )
8.5 Turbulent Jets
229
Table 8.1 Turbulent Prandtl number in turbulent jet flows Flow field Type of flow PrT Planar Wake of a heated cylinder 0.54 Planar Heated jet 0.54 Planar Heated jet 0.42–0.59 Axisymmetric Heated jet with tracers 0.74 Axisymmetric Hydrogen jet 0.72 Axisymmetric Heated jet 0.7 Axisymmetric Heated jet 0.71 Axisymmetric Submerged water jet 0.72–0.83
Authors Fage and Falkner Reichardt Van der Hegge Zijnen Van der Hegge Zijnen Keary and Weller Corrsin Forstall Forstall and Gaylo-rd
0.9 0.8 0.7
PrT
0.6 0.5 0.4 0.3 0.2 0.1 0
0
0.5
1
η
1.5
2
2.5
Fig. 8.1 Turbulent Prandtl number in the axisymmetric turbulent jet. ■-Corrsin and Uberoi (1950), ~-Chevray and Tutu (1978), ●-Chua and Antonia (1990) (measurements 90 X- probe), ¼ y=y1=2 is the dimensionless lateral coordinate, y1=2 is the half-velocity coordinate where u ¼ um =2, u and um are the local longitudinal velocity and the axial velocity, respectively. Reprinted from Chua and Antonia (1990) with permission
and aqueous solt solutions (Sc 103 ) reveal that the turbulent Prandtl and Schmidt numbers are always about 0.75–0.8 (Sakipov 1961; Sakipov and Temirbaev 1962; Forstall and Gaylord 1955). These data confirm that turbulent transfer of momentum, heat and species is dominant in submerged turbulent jets. A comprehensive analysis of a number of semi-empirical correlations for the kinematic viscosity and thermal diffusivity can be found in the monograph by Hinze (1975)
8.5.2
Plane and Axisymmetric Turbulent Jets
Consider velocity distribution in the far field of a plane or axisymmetric turbulent jet. The governing parameters in this case are selected as: (1) the fluid density r, (2) the total momentum flux Jx which is constant along the jet, (3) longitudinal and
230
8 Turbulence
lateral coordinates x and y determining the location of a point of consideration. The eddy viscosity mT , of course, also affects the flow field, however, it is fully determined by Jx ; x and y and thus, is not an independent parameter. Therefore, the functional equation for the longitudinal mean velocity u reads u ¼ f ðr; Jx ; x; yÞ
(8.82)
Here and hereinafter bars above mean flow characteristics are omitted for brevity. The dimensions of the governing parameters in (8.82) are ½r ¼ L3 M; Jx ¼ MT 2 ; ½ x ¼ L; ½ y ¼ L
(8.83)
It is seen that three governing parameters possess independent dimensions, so that the difference n k ¼ 1. Then (8.82) reduces to the following form P ¼ ’1 ðP1 Þ 0
0 a2
(8.84)
0
where P ¼ u=ra1 Jxa2 xa3 and P1 ¼ y=ra1 Jx xa3 . Taking into account the dimensions of the parameters involved in the expressions for P and P1 and applying the principle of the dimensional homoge0 neity, we find the values of the exponents ai and ai as a1 ¼ 1=2, a2 ¼ 1=2, 0 0 0 a3 ¼ 1=2; a1 ¼ 0, a2 ¼ 0, and a3 ¼ 1. Accordingly, we obtain sffiffiffiffi Jx y 1=2 ’ x u¼ r 1 x
(8.85)
The functional equation for the axial velocity um and the jet thickness d are um ¼ fu ðr; Jx ; xÞ
(8.86)
d ¼ fd ðr; Jx ; xÞ
(8.87)
Applying the Pi-theorem, we transform (8.86) and (8.87) to the following form um ¼ c1 rb1 Jxb2 xb3
(8.88)
d ¼ c2 rg1 Jxg2 xg3
(8.89)
where c1 and c2 are constants, and the exponents bi and gi are equal to: b1 ¼ 1=2, b2 ¼ 1=2, b3 ¼ 1=2, g1 ¼ 0, g2 ¼ 0, and g3 ¼ 1. As a result, we arrive at the following equations
8.5 Turbulent Jets
231
sffiffiffiffi Jx 1=2 um ¼ c1 x r d ¼ c2 x
(8.90) (8.91)
Bearing in mind (8.89) and (8.90), we can rewrite (8.85) as follows u ¼ ’1 ðÞ um
(8.92)
where ¼ y=d. The axial velocity um and the jet thickness di are defined by the profile of the i-th characteristic being used as the characteristic scale, which allows us to present the profiles of different flow characteristics in the form similar to that of (8.92). Indeed, the functional equation for any of the flow characteristic Ni reads Ni ¼ fi ðum ; y; di Þ
(8.93)
pffiffiffiffiffiffi pffiffiffiffiffi where Ni ¼ ui ; u02 ; v02 ; u0 v0 . . . which represent any of the mean velocity components, velocity pulsations, the shear stress, etc. Applying the Pi-theorem, we arrive at Pi ¼ ’i ðP Þ
(8.94)
where Pi ¼ Ni =uami , P ¼ y=di ¼ , etc., as well as a1 ¼ a2 ¼ a3 ¼ 1 and a4 ¼ 2 for i ¼ 1,. . .4. Accordingly, we arrive at the following equations pffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffi u u02 v02 u0 v0 ¼ ’1 ðÞ; ¼ ’2 ðÞ; ¼ ’3 ðÞ; ¼ ’4 ðÞ um u2m um um
(8.95)
Choosing as the characteristic scales of temperature and species concentrations their values at the jet flow axis, as well as the thicknesses of the thermal and diffusion layers dT and dc , respectively, we express the functional equations for DT and DCj as follows DT ¼ fT ðDTm ; dT ; yÞ
(8.96)
DCj ¼ fC DCj ; dc ; y
(8.97)
where DT ¼ T T1 , DTm ¼ Tm T1 , DCj ¼ Cj Cj1 , and T1 and Cj1 are the temperature and concentration of the j th species in in the undisturbed fluid outside the jet.
232
8 Turbulence
The application of the Pi-theorem to (8.96) and (8.97) leads to DT ¼ ’T ðT Þ DTm
(8.98)
DC ¼ ’C ðC Þ DCm
(8.99)
where T ¼ y=dT and C ¼ y=dC . Similar functional equations for fluctuations of temperature and species concentrations read pffiffiffiffiffiffi T 02 ¼ ’T ðT Þ DTm pffiffiffiffiffiffiffi C02 ¼ ’C ðC Þ DCm
(8.100)
(8.101)
The assumption that the axial velocity (temperature or species concentrations) and the thickness of the dynamic (thermal or diffusional) layer can serve as the characteristic scales for submerjed jets is based only on the symmetry of these flows and does not imply whether they are plane or axisymmetric. Therefore, (8.95–8.101) are valid as for both plane and axisymmetric jets. These equations can be used for generalizing of the experimental data for the mean and pulsation velocity, temperature and concentration distributions in the far field of any axisymmetric jets. The corresponding generalized experimental results for the velocity, temperature and concentration in plane and axisymmetric turbulent jets are presented in Figs. 8.2–8.6. It is seen that experimental data obtained at different jet cross-sections collapse at single curves corresponding to the self-similar behavior uncovered in the present subsection.
8.5.3
Inhomogeneous Turbulent Jets
The inhomogeneous turbulent jets emerge when turbulent mixing of gas streams of different densities is encountered. The existence of non-uniform density field significantly affects the aerodynamics of inhomogeneous jets, in particular, the decay of the mean velocity, velocity pulsations, and concentration along the jet axis, as well as distributions of these parameters in jet cross-sections, the jet ejection features, etc. (Abramovich,1974; Panchapakesan and Lumley 1993). The flow field in turbulent inhomogeneous jets is described by the continuity, momentum and species balance equations for turbulent motion of the inhomogeneous mixtures of variable density (Shin et al. 1982). Consider a plane or axisymmetric
8.5 Turbulent Jets 1.2
233
a
1
u um
0.8 0.6 0.4 0.2 0 0.5
0
1
2
1.5
y y1
2.5
2
0.35
b
0.3
u'2 um
0.25 0.2 0.15 0.1 0.05 0 0
0.5
1
1.5 y y1
2
2.5
3
2
Fig. 8.2 Distribution of the mean (a) and (b) velocity pulsations in the cross- sections of turbulent jet. (a)x=d: ♦118, ■108, ~103, 76, *65; (b) x=d: ♦143, ■129, ~118, 106, *95. Reprinted from Gutmark and Wygnanski (1976) with permission
turbulent jet of species 1 injected into space submerged by a mixture of species 1 and 2. After a number of simplifications of the mean momentum balance and species balance equations, the integration of these equations across the jet yields 1 ð
ru2 yj dy ¼ Jx 0
(8.102)
234
8 Turbulence
0.03 0.025
u2m
u′ v′
0.02 0.015 0.01 0.005 0 0
0.5
1 y y1
1.5
2
2
Fig. 8.3 The distribution of the turbulent shear stress in cross-sections of planar turbulent jet. x=d: ♦143, ■129, ~118, 106, +95. Reprinted from Gutmark and Wygnanski (1976) with permission 1.2 1
ΔT ΔTm
0.8 0.6 0.4 0.2 0
0
0.5
1
y y1
1.5
2
2.5
2
Fig. 8.4 Radial profiles of the mean temperature DT=DT m ¼ T T1 = T m T1 in the crosssections of an axisymmetric turbulent jet x=d: ◊7.0, □10, ~15, 20.21, +25.47. Reprinted from Lockwood and Moneib (1980) with permission 1 ð
ruDc1 yj dy ¼ Gx
(8.103)
0 1
where r ¼ ½ðr1 1 r2 1 Þc1 þ r2 1 is the mean local density of the gas mixture, c1 is the mean local concentration of the injected gas, r1 and r2 are the mean densiy of the injected (1) and ambient (2) species, Dc1 ¼ c1 c11 is the excess concentration of the injected gas, subscript 1 corresponds to the ambient conditions far away from the jet axis, j ¼ 0 or 1 for plane and axisymmetric jets, respectively. As the governing parameters determining the velocity and concentration fields in the far field of submerged inhomogeneous turbulent jets it is natural to choose the
8.5 Turbulent Jets
235
1.6 1.4 1.2
eT
eT.m
1 0.8 0.6 0.4 0.2 0 0
0.5
1
1.5 y y1
2
2.5
3
2
Fig. 8.5 Radial distributions of the root-mean-square temperature fluctuations in the crosssections of an axisymmetric turbulent jet x=d: ◊7.0, □10.0, D 15:0, 20.21, +25.47, ~30.75, ♦36.66; eT -rms of temperature fluctuations. Reprinted from Lockwood and Moneib (1980) with permission
local density of the gas mixture ½r ¼ L3 M, the total momentum flux along the jet Jx ¼ Lj MT 2 , the total mass flux of the injected gas ½Gx ¼ Lj1 MT 1 ; as well as the coordinates ½ x ¼ L and½ y ¼ L. (Note that Jx and Gx do not change along the jet and, thus are considered as given invariant constants). Accordingly, we can state the following functional equations for the thickness of a turbulent inhomogeneous jet d, as well as for the axial velocity um and concentration Dc1m in it d ¼ f1 ðr; Jx ; Gx ; xÞ
(8.104)
um ¼ f2 ðr; Jx ; Gx ; xÞ
(8.105)
Dcm1 ¼ f3 ðr; Jx ; Gx ; xÞ
(8.106)
Applying the Pi-theorem to (8.104–8.106), we arrive at the following dimensionless equations Pk ¼ ’k ðP Þ
(8.107) 0
0 a2
0 a3
00
a
00
a
00
where k ¼ 1; 2; 3; P1 ¼ d=ra1 Jxa2 Gax 3 , P2 ¼ um =ra1 Jx Gx , P3 ¼ Dcm =ra1 Jx 2 Gx 3 a a and P ¼ x=ra1 Jx 2 Gx 3 . Bearing in mind the dimensions of d, um , and Dcm , and applying the principle of the dimensional homogeneity, we find the values of the exponents involved in 0 0 0 00 (8.107) as: a1 ¼ 1=2; a2 ¼ 1=2, a3 ¼ 1; a1 ¼ 0, a2 ¼ 1, a3 ¼ 1; a1 ¼ 1=2, 00 00 a2 ¼ 1=2, a3 ¼ 1; a1 ¼ 1=2, a2 ¼ 1=2, a3 ¼ 1. Accordingly, we obtain the following equations for d; um , and Dcm
236
8 Turbulence
a
1.2
cC(c, h) k
1 0.8 0.6 0.4 0.2 0 0
0.05
0.1
0.15
0.2
0.25
η
b
0.3
cC¢rms(c, h) k
0.25 0.2 0.15 0.1 0.05 0 0
0.05
0.15
0.1
0.2
0.25
η
Fig. 8.6 Concentration distribution in cross-sections of turbulent axisymmetric jet. (a) mean concentration, (b)concentration pulsations. x=d : □20, ◊40, D60, 80; Reprinted from Doweling and Dimotakis (1990) with permission
d
(
) x
1=2 ¼ ’1 1=2 G2x =rJx G2x =rJx ( 1=2 ) um G2x ¼ ’2 x= ðJx =Gx Þ rJx Dcm1
( 1=2 ) G2x ¼ ’3 x= rJx
(8.108)
(8.109)
(8.110)
Taking into account that the invariants Jx and Gx are constants, we find that Jx ¼ r1 u210 djþ1 , Gx ¼ r1 u10 Dc10 d jþ1 , and Dc10 ¼ c10 ¼ 1 for issuing of pure injected gas into space filled by another gas, where u10 and c10 are the initial velocity and concentration of the injected gas and d is the nozzle diameter. Then, we obtain
8.5 Turbulent Jets
237
Jx ¼ u10 ; Gx
G2x rJx
1=2 ¼
rffiffiffiffiffi r1 ðjþ1Þ=2 d r
(8.111)
The gas mixture density in the far field of the jet is of close to the ambient one, and it is possible to assume that r r2 . Then (8.108–8.110) take the form
pffiffiffiffi d ¼ ’1 x o
(8.112)
pffiffiffiffi um ¼ ’ 2 x o
(8.113)
pffiffiffiffi Dcm1 ¼ ’3 x o
(8.114)
pffiffiffiffi where d ¼ d=d o, um ¼ um =u10 , Dcm1 ¼ Dcm1 =Dc10 ¼ Dcm1 , ðDcm1 ¼ cm1 when c11 ¼ 0Þ, x ¼ x=d and o ¼ r2 =r1 . Thus, the variation of the longitudinal velocity and concentration of the issued species along the axis of anpinhomogeneous turbulent gas jet is determined by a ffiffiffiffi single dimensionless group x o. The experimental data on the dependence Dcm1 pffiffiffiffi ðx oÞ are presented in Fig. 8.7. The figure also includes the experimental data on the axial decay of the excess of enthalpy in high temperature turbulent jets which is expected to be indentical to the decay of the excess concentration. It is seen that all the experimental data corresponding to the different inhomogeneous ð0:27 o 8:2Þand pffiffiffiffi high temperature ðo > 15Þ turbulent jets groip around a single curve cm ðx oÞ. (Abramovich et al. 1974) Choosing as the characteristic scales of velocity, concentration and the corresponding fluctuations their values at the jet axis, we present distribution of pffiffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffiffiffi u, u02 , v02 , and c1 and c1 02 as follows
pffiffiffiffi pffiffiffiffi Fig. 8.7 Turbulent mixing, axsisymmetric gas jets; the dependence cm1 ðx oÞ, with x o: □0.27, ◊1.3, D8.2, 1.5, ~21, ♦26
238
8 Turbulence
Ni ¼ f1 um; y; d
(8.115)
Mk ¼ fk ðcm1 ; y; dÞ
(8.116)
pffiffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffiffiffi where Ni ¼ u, u02 or v02 and i ¼ 1,2 and 3; also, Mk ¼ c1 or c1 02 and k ¼ 4 and 5. Applying the Pi-theorem to (8.115) and (8.116), we arrive at the following equations pffiffiffiffiffiffi pffiffiffiffiffi y y y c y u u02 v02 1 ¼ f1 ¼ f2 ¼ f3 ¼ ’4 ; ; ; ; um um um d d d cm1 d pffiffiffiffiffiffiffi y c1 02 ¼ ’5 cm1 d
(8.117)
Using a similar approach, we can find the following dimensionless equations for the turbulent correlations u0 v0 , v0 c0 , etc. in jet cross-sections y v0 c 0 y u0 v 0 1 ¼ f ¼ f7 ; 6 2 um d um cm1 d
(8.118)
Equations (8.117) and (8.118) show that the profiles of the dimensional characteristics corresponding to different cross-sections of the inhomogeneous jet can be presented in the parametric planes Ni , and Mi (where ¼ y=dÞ in the form of universal dependences Ni ðÞ and Mi ðÞ (cf. Figs. 8.8–8.10)
8.5.4
Co-flowing Jets
Co-flowing turbulent jet is formed when a turbulent jet is issued into a uniform fluid flow in an infinite domain. Detailed experimental data for the mean and characteristics and turbulent pulsations in co-flowing turbulent jets were obtained by Maczynski, 1962; Bradbury and Riley 1967; Antonia and Bigler 1973; Everitt and Robins 1978; Abramovich et al. 1984; Nickels and Perry 1996. Some general considerations of the features of co-flowing turbulent jets can be found in the monograph by Townsend (1956) who analyzed the conditions corresponding to self-similar flows in such jets. He also dealt with the effect of viscosity on flow characteristics, e.g. the independence of the decay of the mean and pulsation velocities along the jet axis on the Reynolds number, etc. For the additional details, a reader can consult the above-mentioned works, while we restrict our discussion to the applications of the Pi-theorem to study the aerodynamics of co-flowing turbulent jets, and especially to generalize the experimental data for the far field in these flows.
8.5 Turbulent Jets
a
239
1.2 1
u um
0.8 0.6 0.4 0.2 0
b
0
0.05
0.1
0.15 y x
0.2
0.25
0.3
0
0.05
0.1
0.15 y x
0.2
0.25
0.3
1.2 1
c1 cm1
0.8 0.6 0.4 0.2 0
Fig. 8.8 The mean velocity (a) and mean helium concentration (b) distributions in the axisymmetric turbulent jet. x=d: □90, ◊100, D110, 120. Reprinted from Panchapakesan and Lumley (1993) with permission
A sketch of co-flowing turbulent jet with a uniform velocity profile at the orifice exit x ¼ 0 is depicted in Fig. 8.11. At some distance from the jet orifice, the initialvelocity profile transforms into a smooth profile with a characteristic maximum at the jet axis, which gradually decreases toward the outer edge of the jet where the longitudinal velocity u matches the free stream velocity u1 . In the framework of the boundary layer theory the flow in co-flowing turbulent jets is described by the momentum balance and continuity equations subjected to the boundary conditions that account for the existence of the outer flow y ¼ 0;
@u ¼ 0; v ¼ 0 y ! 1 u ! u1 @y
(8.119)
240
8 Turbulence
0.4
a
0.35 0.3
u′ um
0.25 0.2 0.15 0.1 0.05 0 0
0.5
1
y y1
1.5
2
2.5
0.25
0.3
2
0.4
b
0.35 0.3
c′1
cm1
0.25 0.2 0.15 0.1 0.05 0 0
0.05
0.1
0.15 y y1
0.2
2
Fig. 8.9 The distribution of the intensities of fluctuations, a – the root – mean-square of velocity (u0 =um ) and b – concentration (c1 0 =cm1 )in cross-sections of the axisymmetric turbulent jet x=d: □50, ■60, D 70, 80, ~90, ♦100, + 110. Reprinted from Panchapakesan and Lumley (1993) with permission
where subscript 1 refers to the undisturbed co-flow. The integration of the momentum balance equation across the jet accounting for the boundary conditions (8.119) yields the following integral invariant of coflowing jets 1 ð
Jx ¼
ruðu u1 Þyj dy 0
(8.120)
8.5 Turbulent Jets 0.3
241
a
0.25
u′v′ um2
0.2 0.15 0.1 0.05 0 0
0.5
1
y y1
1.5
2
2.5
2
0.035
b
0.03 0.025
c1′v′ cm1um
0.02 0.015 0.01 0.005 0 0
0.05
0.1
0.15 y x
0.2
0.25
0.3
Fig. 8.10 Variation of the Reynolds stress (u0 v0 =u2m )a and radial scalar flux (c1 0 v0 =cm1 um )b across the axisymmetric turbulent jet. x=d: D 50, □60, ◊70, 90, ■100, ~110. Reprinted from Panchapakesan and Lumley (1993) with permission
which is the momentum flux along the jet and where j ¼ 0 or 1correspond to plane or axisymmetric jets, respectively. Far from the orifice where the jet ‘forgets’ the initial conditions, the longitudinal velocity is determined by fluid density ½r ¼ L3 M, the total momentum flux ½Jx ¼ Lj MT 2 , the free stream velocity ½u1 ¼ LT 1 and the longitudinal and lateral coordinates ½ x ¼ L and ½ y ¼ L u ¼ f1 ðr; Jx ; u1 ; x; yÞ
(8.121)
242
8 Turbulence
Fig. 8.11 Sketch of coflowing turbulent jet with a step-wise initial velocity distribution
Beating in mind the dimensions of the governing parameters we, find that in the present case the difference n k ¼ 2 and, accordingly, (8.121) reduces to the dimensionless equation P ¼ ’1 ðP1 ; P2 Þ 1=2
(8.122) 1=2
where P ¼ u=u1 , P1 ¼ x=ðJx =rxjþ1 Þ and P2 ¼ y=ðJx =rxjþ1 Þ . Thus, the dimensionless velocity in co-flowing jets is a function of two dimensionless groups. That means that the present problem has no self-similar solution. Note that excluding u1 from the boundary conditions (8.119) by introducing the excess of velocity u u1 does not allow one to decrease the number of the governing parameters and thereby obtain a self-similar solution, since such transformation introduces u1 into the momentum balance equation. Within the far field of co-flowing turbulent jets, velocity change across the jet is small enough compared to the free stream velocity. This allows one to recover an approximate self-similarity of co-flowing turbulent jets (Townsend 1956). Indeed, the functional equations for the mean flow characteristics of plane co-flowing turbulent jets can be written now as U ¼ f2 ðr; Jx ; x; yÞ
(8.123)
Um ¼ f3 ðr; Jx xÞ
(8.124)
d ¼ f4 ðr; Jx ; xÞ
(8.125)
where U ¼ u u1 is the excess of velocity, subscript m refers to the jet axis and d denotes the jet thickness. Applying the Pi-theorem to (8.123–8.125), we arrive at the following equations y u u1 pffiffiffiffiffiffiffiffiffiffiffiffi ¼ ’2 x Jx =rx
(8.126)
8.5 Turbulent Jets
243
um u1 pffiffiffiffiffiffiffiffiffiffiffiffi ¼ C1 Jx =rx
(8.127)
d ¼ C2 x
(8.128)
where C1 ¼ ’2 ð0Þ ¼ const. Equations (8.127) and (8.128) show that the excess of axial velocity in the far field of plane co-folwing turbulent jets is inversely proportional to x1=2 , whereas the jet thickness is directly proportional to x. These results agree fairly well with the experimental data of Bradbury and Riley (1967) and Everitt and Robins (1978) for plane co-flowing turbulent jets (cf. Figs. 8.12 and 8.13). It is seen that d x (with x ¼ x=y) and Dum x1=2 at x > 40 (d ¼ d=y, þ1 Ð x ¼ x=y, Dum ¼ Dum =u1 , Dum ¼ um u1 , y ¼ u=u1 ðu=u1 1Þdy. It is 1
emphasized that all the data corresponding to different values of the co-flow parameter m(the ratio of the exit to the free stream velocity) collapse near single curves dðxÞ and Dum ðxÞ in the parametric planes d x and Dum x. Using (8.126–8.128), we express the velocity distribution across the co-flowing jet as y u u1 ¼ c3 ’2 um u 1 d
(8.129)
where c3 ¼ c1 1 .
2.5
d , Δum u∞ q
2
1.5
1
0.5
0 0
20
40
60 x q
80
100
120
Fig. 8.12 Variation of the axial velocity and thickness of co-flowing jet along the flow axis. Reprinted from Bradbury and Riley (1967) with permission
244
8 Turbulence 3 2.5
d q
2 1.5 1 0.5 0 0
20
40
60
80
100
120
140
x q
Fig. 8.13 Variation of the thickness of a plane turbulent co-flowing jet along its axis. y is the momentum thickness of the jet . u=u1 : D2.60, ◊3.03, □3.24, 3.29, ~3.78, ■6.72, +17.08. Reprinted from Everitt and Robins (1978) with permission 1.2 1
U–U1 U0
0.8 0.6 0.4 0.2 0 -3
-2
-1
0 y y1
1
2
3
2
Fig. 8.14 Velocity distribution in cross-sections of plane turbulent jet issuing into a parallel moving air stream. U0 =U1 : ◊6.64, □1.93, D0,41, 20.2, ~ Reprinted from Bradbury and Riley (1967) with permission
The experimental data on the velocity distribution in different cross-sections of co-flowing plane turbulent jet are presented in Fig. 8.14. The data related to the distributions of turbulent pulsations across co-flowing turbulent ‘strong’ and ‘weak’
8.5 Turbulent Jets
245
0.1 0.09 0.08 0.07 u'2 U02
0.06 0.05 0.04 0.03 0.02 0.01 0 0
0.5
1
1.5 y y1
2
2.5
3
2
Fig. 8.15 Distribution of the velocity fluctuations in cross-sections of plane turbulent jets. D-strong jet, □-weak jet, ▪▪▪-wake. Reprinted from Everitt and Robins (1978) with permission
jets (i.e. the jet flows where the excess of center-line velocity is much larger or much smaller than the free stream velocity) are shown in Fig. 8.15 It is seen that in both cases the experimental data can be generalized in the form of universal dependences in the parametric planes Ni where Ni ¼ Du=Dum and u02 =ðDuÞ2 .
8.5.5
Turbulent Jets in Crossflow
The aerodynamics of turbulent jets in crossflow was a subject of a number of experimental works. They contain detailed data on the jet trajectory, decay of centerline velocity, distributions of the mean and pulsation characteristics, as well as the vorticity field in such jets (Chassaing et al. 1974; Moussa et al. 1977; Andreopoulos and Rodi 1985; Fric and Roshko 1994; Kelso et al. 1996; Smith and Mungal 1998). In addition, a number of the empirical and semi-empirical correlations for predicting the trajectory of turbulent jets in crossflow and variation of the flow characteristics along the curved jet axis was proposed (Keffer and Baines 1963; Abramovich 1963). A self-consistent analysis of the aerodynamics of turbulent jets in crossflow based on the similarity theory was recently proposed by Hasselbrink and Mungal (2001). Following the ideas of the latter work, we outline below the application of the dimensional analysis to study characteristics of turbulent jets in crossflow. Our aim is to describe the flow field in an incompressible turbulent jet in crossflow with a known trajectory yc ðxc Þfor a given blowing ratio
246
8 Turbulence
Fig. 8.16 Sketch of a jet in crossflow. The origin of the coordinates x andyis at the center of the nozzle issuing the jet. Adenotes a jet crosssection with a center at x ¼ xc and y ¼ yc . Subscripts j; 1and c correspond to the nozzle, the undisturbed crossflow and the centerline of the jet, respectively. Reprinted from Hasselbrink and Mungal (2001) with permission
ðrj u2j r1 v21 Þ1=2 (Fig. 8.16). We will proceed from the assumption that the local values of any of the flow characteristics Ni ¼ r; u; v are fully determined by the axial value of a given characteristic in the same cross-section Nic , the jet thickness there d, the position along the centerline xc , and coordinates x; y of a point under consideration. It is emphasized that the velocity vector v in the general case has three components u; v and w, but following Hasselbrink and Mungal (2001), we consider the flow to be planar and thus, are dealing with only u and v. Then we can write the functional equation for Ni as follows Ni ¼ f ðNic ; x; y; xc ; dÞ
(8.130)
where Ni ¼ r; u; v. The dimensions of Nic are La M b T g , whereas the dimensions of the other governing parameters in (8.130) are all ½ L. Therefore, in the case under consideration the total number of the governing parameters is five, whereas the number of the governing parameters with independent dimensions equals two, so that n k ¼ 3. Then, according to the Pi-theorem, (8.130) can be transformed to the following dimensionless equation Ni ¼ Nic ’ð; g; xc Þ
(8.131)
where , g and xc are the coordinates x,y and xc normalized by d. The density r and velocity component distributions can be expressed as follows r ¼ rc ðxc Þf1 ð; g; xc Þ
(8.132)
u ¼ uc ðxc Þf2 ð; g; xc Þ
(8.133)
½v1 vð; g; xc Þ ¼ ½v1 vc ðxc Þf3 ð; g; xc Þ
(8.134)
8.5 Turbulent Jets
247
where the functions f1 , f2 , and f3 are assumed to be universal within each of the characteristic flow domains of jets in crossflow: (1) potential core, (2) the near-field region, and (3) the far-field region. In order to determine the value of the flow characteristics at the jet axis, it is necessary to employ some additional relations that follow from the mass, momentum and momentum excess (based on Dv ¼ v1 vc ) balance equations. In particular, they have the form m I 1 rc u c d 2
(8.135)
J I2 rc u2c d2
(8.136)
y I3 rc d2 ðv1 vc Þuc
(8.137)
for the near-field of the jet, and m I4 rc v1 d2
(8.138)
J I5 rc v1 uc d2
(8.139)
y I6 rc v1 ðv1 vc Þd2
(8.140)
for the far-field of the jet where m ¼ mj þ m1 is the mass flux,Jand y are the momentum and momentum excess fluxes, respectively,Ðthrough a cross-section at a Ð given distance from the nozzle. In addition, I1 ¼ f1 f2 ddg, I2 ¼ f1 f22 ddg A A Ð Ð Ð Ð I3 ¼ f1 f2 f3 ddg, I4 ¼ f1 ddg, I5 ¼ f1 f2 ddg and I6 ¼ f1 f3 ddg, where A A
A
A
A
denotes jet cross-section. Neglecting the effect of the molecular viscosity, we assume that the functional equations for the axial velocity and turbulent jet thickness in the near field region of the jet are uc ¼ c1 ðJ; r1 ; xÞ
(8.141)
d ¼ c2 ðJ; r1 ; xÞ
(8.142)
Taking into account the dimensions of J, r1 and x, we find uc ¼ c1
J r1
1=2
d ¼ c2 x where c1 and c2 are constants.
x1
(8.143) (8.144)
248
8 Turbulence
The dependence of vm on x found using (8.140) is ðv1 vc Þ
y 1=2 r1 J 1=2
x1
(8.145)
For the far-field region of the jet flow we find the following expressions for uc ; d and vc uc
J x2 r1 v1
(8.146)
dx
(8.147)
ðv1 vc Þ
J x2 r1 v 1
(8.148)
The relations (8.143–8.145) and (8.146–8.148) allow us to evaluate the change in the velocity and thickness withing the near and far regions of of the jet. It is seen that the centerline velocity fades according to the different laws at small and large distances from the nozzle. In particular, at small distance from the nozzle the centerline velocity fades as x1 , wereas at large x it fades as x2 .In both regions the jet thickness is proportional to x.
8.5.6
Turbulent Wall Jets
Consider a turbulent wall jet outflowing from a two-dimensional slot in contact with a wall into a fluid at rest. The velocity profile in a cross-section of the wall jet is sketched in Fig. 8.17. It results from the interection of the jet with the solid surface,
Fig. 8.17 Sketch of turbulent wall jet
8.5 Turbulent Jets
249
as well as with the surrounding fluid. In the far-field region of turbulent wall jets the profile of the longitudinal mean velocity component has a maximum located at some distance y ¼ ym from the wall which depends on x. On both sides from the maximum at y ¼ ym velocity gradually decreases to zero at y ¼ 0 (at the wall, due to the no-slip condition) and y ! 1 (in the surrounding fluid at rest). The measurements show that turbulent wall jets have a very complicated structure because of the interplay of the molecular viscousity and the inertial effects (Launder and Rodi 1981; 1983). The influence of various factors on flow characteristics is different in different domains of turbulent wall jets. Within the near-wall region the molecular viscous effects are dominant, whereas at large distance from the wall fluid inertia plays an important role. Such structure of turbulent wall jets allows one to select two characteristic domains: (1) the inner one close to the wall (with dominant viscous effects), and (2) the outer one in which the inertial effects are dominant. In the two domains the velocity distribution is affected by different parameters responsible for specific flow features in the inner and outer parts of turbulent wall jets. In the framework of such model the characteristic velocity and length can be chosen as the friction velocity u ¼ ðtw =rÞ1=2 and length d ¼ n=u in the inner domain, and the maximum longitudinal velocity um and the jet thickness d (e.g. corresponding to the velocity value u ¼ um =2Þ for the outer one. At the Reynolds number being infinite, there exists a self-similar solution of the problem (George et al. 2000; Bergstrom and Tachie 2001), and the functional equation for the longitudinal velocity in the inner and outer domains can be written as uin ¼ fin ðu ; y; vÞ
(8.149)
uout ¼ fout ðum ; d; yÞ
(8.150)
Applying the Pi-theorem to (8.149) and (8.150), we transform them to the following dimensionless form Pin ¼ ’in ðP1in Þ
(8.151)
Pout ¼ ’out ðP1out Þ
(8.152)
where the dimensionless groups read Pin ¼
u yu uout y ; P1in ¼ ; P1out ¼ ; Pout ¼ n um u d
(8.153)
Equations (8.151) and (8.152) show that the dimensionless velocity in the inner and outer domains is a function of a single dimensionless group. The latter makes it possible to collapse the experimental data for turbulent wall jets as a single curve Uþ ¼ u=u versus yþ ¼ yu =n for the inner layer, and another single curve
250
8 Turbulence
u=u versus y=d for the outer one .The experimental data for the mean velocity profiles in the outer (y y1=2 ) and inner (yþ ) coordinates, and the variation of the half-width with the streamwise distance in turbulent plane jet are shown in Figs. 8.18, 8.19 and 8.20.
1.2 1
u um
0.8 0.6 0.4 0.2 0 0
0.5
1
y y1
1.5
2
2.5
2
Fig. 8.18 The mean velocity profiles in the wall jet near a smooth wall. ◊Re0 ¼ 12500; x=h ¼ 50; □Re0 ¼ 12500; x=h ¼ 80; ■Re0 ¼ 9100; x=h ¼ 50; Re0 ¼ 6100; x=h ¼ 30; ~Re0 ¼ 10000 (data by Karlsson et al., 1993). Reprinted from Tachie et al. (2004) with permission
1.00E+10
n2
y1 / 2I0
1.00E+09
1.00E+08
1.00E+07 1.00E+08
1.00E+09
1.00E+10 xI0
1.00E+11
n2
Fig. 8.19 Variation of the half-width of turbulent plane wall jet in the streamwise direction. Re0 : 1240, ◊11900, ~10300, □9100, 6100, ♦12500, ■5900. Reprinted from Tachie et al. (2004) with permission
8.5 Turbulent Jets
251
U+
1.00E+02
1.00E+01
1.00E+00 1.00E+00
1.00E+01
1.00E+02
1.00E+03
y+
Fig. 8.20 The mean velocity profile in the inner coordinates in a turbulent plane wall jet. Re: ◊12500, D9100, □6100. Reprinted from Tachie et al. (2004) with permission
In order to determine the character of variation of um and d along the turbulent wall jet some addition considerations of the governing parameters that determine flow in the outer domain should be involved. Narasimha et al. suggested that the set of parameters should also include the momentum flux at the jet origin divided by density ½I0 ¼ L3 T 2 and molecular kinematic viscosity ½n ¼ L2 T 1 (Narasimha et al. 1973). Accordingly, one can present the functional equations for um and d as follows um ¼ f1 ðn; I0 ; xÞ
(8.154)
d ¼ f2 ðn; I0 ; xÞ
(8.155)
Then, applying the Pi-theorem, we arrive at um n xI0 ¼ ’1 2 n I0 dI0 xI0 ¼ ’2 2 2 n n
(8.156)
(8.157)
252
8.5.7
8 Turbulence
Impinging Turbulent Jet
A sketch of the turbulent jet impinging normally onto an infinite solid wall is shown in Fig. 8.21. The impinging jets can be subdivided into three characteristic regions. In the first one (1 in Fig. 8.21) free turbulent mixing of the jet with the surrounding medium is dominant. In region 2 direct interaction of the jet with the solid wall is accompanied by an abrupt deceleration of the jet and its deflection in the direction normal to its initial path. In region 3 the jet-wall interaction resembles that one for a walljet. The centerline velocity um gradually changes from its initial value at the nozzle exit u0 to zero at the stagnation point at the wall surface. The distribution of the centerline velocity corresponding to different distances from the nozzle exit down to the solid surface is shown in Fig. 8.22 where u0 and d0 are the jet velocity at the nozzle axis and the nozzle diameter, respectively. It is seen that the jet-wall interaction affects significantly the longitudinal velocity um ðxÞ. In order to gain an insight into the peculiarities of the velocity distribution we apply -theorem. In this case the governing parameters determining the centerline P the (for a ½Jx ¼ MT 2 the total momentum flux ½r ¼ L3 M, velocity are: the fluid density and the ½h ¼ L plane jet), the distance between the solid surface and the nozzle exit. Accordingly, the functional equation for the centerline velocity reads ½x ¼ L coordinate um ¼ f ðr; Jx ; h; xÞ
Fig. 8.21 Sketch of the impinging turbulent jet
(8.158)
8.5 Turbulent Jets
253
Fig. 8.22 Distribution of the centerline velocity at several cross-sections. Curve 1 corresponds to a free jet, h ¼ h1 ¼ 1. Curves 2–4 correspond to different distances from the nozzle: h2 > h3 > h4
Since three of the four governing parameters in (8.158) have independent dimensions, this equation reduces to the following dimensionless equation P ¼ ’ðP1 Þ 0
0 a2
(8.159)
0
where P ¼ um =ra1 Jxa2 ha3 and P1 ¼ x=ra1 Jx ha3 . Taking into account the dimensions of um , r, Jx , h and x, we find the values of the 0 0 0 0 exponents ai and ai as a1 ¼ 1=2, a2 ¼ 1=2, a3 ¼ 1=2; a1 ¼ a2 ¼ 0, and a3 ¼ 1. Then (8.159) takes the form x um h 1=2 ¼’ u0 d0 h
(8.160)
In (8.160) it is accounted for the fact that Jx ru20 d0 . Equation (8.160) shows that dimensionless centerline velocity ue ¼ ðum =u0 Þ ðh=d0 Þ1=2 is a universal function of a single dimensionless variable x ¼ x=h. The experimental data by Gutmark et al. (1978) on the centerline velocity distribution in plane impinging turbulent jet are presented in Fig. 8.23. They correspond to different valuesof the ratio h=d0 and the Reynolds number. Figure 8.23 shows that all experi mental data collapse around a single curve u ðx Þ according to the results of the dimensional analysis. It is worth noting that near the wall there is a narrow layer where the centerline velocity is proportional to the distance from the wall. This linear relation corresponds to the inviscid stagnation flow. A significant deflection of the dimensionless velocity u from the one corresponding to a free jet takes place in the region 0 < x 0:2.
254
8 Turbulence 1.6 1.4 1.2
h d0
0.8
um u0
1 2
1
0.6 0.4 0.2 0 0
0.2
0.4
0.6 x
0.8
1
1.2
h
Fig. 8.23 The distribution of the normalized mean longitudinal velocity component um =u0 ðh=d0 Þ1=2 along the jet center line. DRe ¼ 3104 ; h=d ¼ 100, □Re ¼ 4:3104 ; h=d ¼ 40, ◊Re¼ 5:6103 ; h=d ¼ 31,Re¼ 5:6103 ; h=d ¼ 43:6, +Re¼5:6103 ; h=d ¼67:5. Reprinted from Gutmark et al. (1978) with permission
Problems P.8.1. Establish the functional dependence of the dimensionless centerline velocity in the axisymmetric impinging turbulent jet on the dimensionless distance from the wall. The governing parameters which determine the centerline velocity in such a jet are: ½r ¼ L3 M, ½Jx ¼ LMT 2 , ½h ¼ L and ½ x ¼ L. Accordingly, the functional equation for the centerline velocity reads u ¼ f ðr; Jx h; xÞ
(P.8.1)
Applying the Pi-theorem to (P.8.1), we obtain P ¼ ’ðP1 Þ 0
a
0
(P.8.2)
0
where P ¼ um =ra1 Jxa2 ha3 and P1 ¼ x=ra1 Jx 2 ha3 . Taking into account the dimensions of um ; r, Jx , h and x, we find the values of 0 0 0 0 the exponents ai and ai as a1 ¼ 1=2, a2 ¼ 1=2, a1 ¼ 0,a2 ¼ 0, a3 ¼ 1. Then we obtain from (P.8.2)
um u0
x h ¼’ d0 h
(P.8.3)
Problems
255
Fig. 8.24 A cavity formed at the surface of a liquid pool by an impinging axisymmetric gas jet
u0 h
Note that in (P.8.3) accounts for the fact that Jx ru20 d02 , where d0 is the diameter of the nozzle. P.8.2. Consider cavity formation at the free surface of a liquid pull by an impinging axisymmetric turbulent gas jet. (1) Determine the dimensionless groups of the flow, and (2) present the experimental data for the cavity depth and diameter in an appropriate dimensionless form. The flow under consideration is depicted in Fig. 8.24. An axisymmetric turbulent gas jet is issued from a nozzle with the exit diameter dj . The jet is directed normally to the unperturbed free surface of the pool. The distance between the nozzle exit and the unperturbed surface in the liquid pool is h. The impinging jet causes a depression of the liquid surface. A sufficiently strong jet creates a visible cavity at the free surface. Gas penetration into liquid is accompanied by the jet deceleration and formation of the annular reverse gas flow. The liquid surface is deformed due to the action of of the dynamic pressure and friction from the gas side as well as the cavity shape is affected by liquid surface tension. In some cases the cavity surface is unstable and gas bubble entrainment can take place there, the phenomenon which is disregarded here. Thus, the cavity formation depends on several competing factors: the physical properties of the gas and liquid phases, the initial jet diameter and velocity of the jet, etc.
256
8 Turbulence
Assuming that the gas velocity distribution at the nozzle exit is uniform, we list the governing parameters of the flow rG L3 M ; rL L3 M ; mG L1 MT 1 ; mL L1 MT 1 ;
(P.8.4)
s½MT 2 , g½LT 2 , uj ½LT 1 , dj ½ L, h½ L where r and m denote density and viscosity, respectively, s is the surface tension, g is the gravity acceleration, uj is the jet velocity at the nozzle exit, and subscripts G and L refer to the gas and liquid, respectively. Three of the nine governing parameters in (P.8.4) possess independent dimensions. According to the Pi-theorem, the number of the dimensions groups that determine the dimensionless characteristics of the cavity equals to six. These are the following rGL ; h; ReG ; ReL ; Fr; We
(P.8.5)
where rGL ¼ rG =rL , H ¼ h=dj . Also, ReG ¼ uj dj =nG , ReL ¼ uj dj =nL , Fr ¼ u2j =gdj and We ¼ dj rG u2j =s are the two Reynolds numbers, the Froude and Weber numbers, respectively, with n being the kinematic viscosity. In the particular case where the Reynolds and Weber numbers are sufficiently large and the viscous and surface tension effects are negligible, the number of the dimensionless groups can be reduced significantly. At a large distance between the nozzle exit and the unperturbed liquid surface (h >> dj Þ the characteristics of the gas jet are mostly determined by its total momentum flux Jx ¼ rG u2j dj2 p=4, whereas the effect of the gas pressure at the cavity surface (which is determined by the weight of the liquid displaced from the cavity) can be related to the specific weight of the liquid g ¼ rL g½L2 MT 2 . Then, the number of the governing parameters reduces to three, namely, Jx , g and h, so that the functional equation for the cavity depth hc becomes hc ¼ f1 ðJx ; g; hÞ
(P.8.6)
In (P.8.6) the number of the governing parameters with independent dimensions equals two. Accordingly, (P.8.6) reduces to the following dimensionless equation P ¼ ’ðP1 Þ 0 a1
(P.8.7)
0
where P ¼ hc =Jxa1 ga2 and P1 ¼ h=Jx ga2 . 0 Taking into account the dimensions of hc , h, g and Jx , we find that a1 ¼ a1 ¼ 1=2 0 and a2 ¼ a2 ¼ 1=3. Then (P.8.7) takes the form ( hc ðJx =gÞ1=3
¼’
) h ðJx =gÞ1=3
(P.8.8)
Problems 3.5
257
a
3
dc[in]
2.5 2 1.5 1 0.5 0 0
0.01
0.02
0.03
0.04
0.05
Ix[lb] 1.6
b
1.4 1.2
hc[in]
1 0.8 0.6 0.4 0.2 0 0
0.1
0.2
0.3 Jx[lb]
0.4
0.5
0.6
Fig. 8.25 Variation of the cavity diameter (a) and depth (b) with the distance between the nozzle and the unperturbed free surface (h) and the momentum flux of the jet. h½in: ~2.0, +3.0, 4.0, D5.0, □5.0, ◊7.0. Reprinted from Cheslak et al. (1969) with permission
or hc ¼ h
( ) 1=3 Jx 1 h Jx ’1 ¼ F 1 g gh3 h ðJx =gÞ1=3
(P.8.9)
258
8 Turbulence
Since the cavity diameter dc is related with its depth, it is possible to state the functional equation for dc as follows dc ¼ f2 ðJx ; g; hc Þ
(P.8.10)
Applying the Pi-theorem to (P.8.10), we arrive at dc Jx ¼ F2 hc gh3c
(P.8.11)
The experimental data on the cavity diameter and depth are presented in Figs. 8.25a and 8.25b. It is seen that an increase in h is accompanied by the growth of the cavity diameter and a decrease in its depth. An increase in the total momentum flux of the jet leads to an increase of the cavity depth and diameter. Equations (P.8.9) and (P.8.11) show that the dimensionless cavity depth hc =h and diameter dc =hc are functions of the dimensionless group Jx =gh3 and Jx =gh3c , respectively. Accordingly, one can expect that all the data points corresponding to different experimental
conditions should collapse at single curves ðhc =hÞðJx =gh3 Þ and ðdc =hc Þ Jx =gh3c in the parametric planes hc =h versus Jx =gh3 and dc =hc versus Jx =gh3c . This result, indeed, agrees with the data by Banks and Chandrasekhara (1963), as well as with several other measurements.
References Abramovich GN (1963) Theory of turbulent jets. MTI Press, Boston Abramovich GN, Krasheninnikov SYu, Sekundov AN, Smirnova IP (1974) Turbulent mixing of Gas jets. Nauka, Moscow (in Russian) Abramovich GN, Girshovich TA, Krasheninnikov SYu, Sekundov AN, Smirnova IP (1984) Theory of turbulent jets. Nauka, Moscow (in Russian) Andreopoulos J, Rodi W (1985) On the structure of jets in crossflow. J Fluid Mech 138:93–127 Antonia RA, Prabhu A, Stephenson SE (1975) Conditionally sampled measurements in a heated turbulent jet. J Fluid Mech 72:455–480 Antonia RA, Bigler RW (1973) An experimental investigation of an axisymmetric jet in coflowing air stream. J Fluid Mech 61:805–822 Banks RB, Chandrasekhara DV (1963) Experimental investigation of the penetration of a highvelocity gas jet through a liquid surface. J Fluid Mech 15:13–34 Barenblatt GI (1996) Similarity, self-similarity, and intermediate asymptotics. Cambridge University Press, Cambridge Bergstrom DJ, Tachie MF (2001) Application of power laws to low Reynolds number boundary layers on smooth and rough surfaces. Phys Fluids 13:3277–3284 Bradbury LJS, Riley J (1967) The spread of turbulent plane jet issuing into a parallel moving airstream. J Fluid Mech 27:381–394 Chassaing P, George J, Claria A, Sananes F (1974) Physical characteristics of subsonic jets in a cross-stream. J Fluid Mech 62:41–64
References
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Cheslak FR, Nicholles JA, Sichel M (1969) Cavities formed on liquid surfaces by impinging gas jets. J Fluid Mech 36:55–63 Chevray R, Tutu NK (1978) Intermittency and preferential transport of heat in a round jet. J Fluid Mech 88:133–160 Chua LP, Antonia RA (1990) Turbulent Prandtl number in a circular jet. Int J Heat Mass Transf 33:331–339 Clauser FH (1956) The turbulent boundary layer. Adv Appl Mech 56:1–51 Coles D (1955) The law of the wall in turbulent shear flow, 50 jahre grenzschicht-forschung. Vieweg, Braunschweig, pp 153–163 Corrsin S, Uberoi MS (1950) Further experiments on the flow and heat transfer in a heated turbulent air jet. NACA Report 998, NACA - TN - 1865 Doweling DR, Dimotakis PE (1990) Similarity of the concentration field of gas-phase turbulent jet. J Fluid Mech 218:109–141 Everitt KM, Robins AG (1978) The development and structure of turbulent plane jets. J Fluid Mech 88:563–583 Fric TF, Roshko A (1994) Vortical structure in the wake of a transverse jet. J Fluid Mech 279:1–47 Forstall W, Gaylord EW (1955) Momentum and mass transfer in submerged water jets. J Appl Mech 22:161–171 George WK, Abrahamsson H, Eriksson J, Karlsson RI, Lofdahl L, Wosnik M (2000) A similarity theory for the turbulent plane wall jet without external stream. J Fluid Mech 425:367–411 Gutmark E, Wygnanski I (1976) The planar turbulent jet. J Fluid Mech 73:465–495 Gutmark E, Wolfshtein M, Wygnanski I (1978) The plane turbulent impinging jet. J Fluid Mech 88:737–756 Hasselbrink EF, Mungal MG (2001) Transverse jet and jet features. Part 1. Scaling laws for strong transverse jets. J Fluid Mech 443:1–25 Herwig H, Gloss D, Wenterodt T (2008) A new approach to understanding and modeling the influence of wall roughness on friction factors for pipe and channel flows. J Fluid Mech 613:35–53 Hinze JO (1975) Turbulence, 2nd edn. McGraw Hill, New York Karlsson RI, Eriksson JE, Persson J (1993) LDV measurements in a plane wall jet in large enclosure. In: proceeding of the 6th International symposium on applications of laser techniques to fluid mechanics, 20–23 July. Lisabon, Portugal, paper 1:5 von Karman Th (1930) Mechanische Ahnlichkeit und Turbulenz. Nachr Ges Wiss Gottingen Math Phys Klasse 58:271–286 von Karman Th, Howarth L (1938) On the statistical theory of isotropic turbulence. Proc Roy Soc A 164:192–215 Keffer JF, Baines WD (1963) The round turbulent jet in a cross wind. J Fluid Mech 15:481–496 Kelso RM, Lim TT, Perry AE (1996) An experimental study of round jets in cross-flow. J Fluid Mech 306:111–144 Kolmogorov AN (1941a) Local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. DAN SSSR 30(4):299–303, in Russian Kolmogorov AN (1941b) Disperse energy at local isotropic turbulence. DAN SSSR 32(1):19–21 Landau LD, Lifshitz EM (1979) Fluid mechanics, 2nd edn. Pergamon, London Launder BE, Rodi W (1981) The turbulent wall jet. Prog Aerospace Sci 19:81–128 Launder BE, Rodi W (1983) The turbulent wall jet-measurement and modeling. Annu Rev Fluid Mech 15:429–459 Lockwood FC, Moneib HA (1980) Fluctuating temperature measurements in a heated round free jet. Comb Sci Tech 22:63–81 Loitsyanskii LG (1939) Some fundamental laws of isotropic turbulent flow. Trans TZAGI 440:3–23 Maczynski JFJ (1962) A round jet in an ambient co-axial stream. J Fluid Mech 13:597–608 Mayer E, Divoky D (1966) Correlation of intermittency with preferential transport of heat and chemical species in turbulent shear flows. AIAA J 4:1995–2000
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Moussa ZM, Trischka JW, Eskinazi S (1977) The near field in the mixing of a round jet with a cross-stream. J Fluid Mech 80:49–80 Monin AS, Yaglom AM (1965-Part 1, 1967-Part 2) Statistical fluid dynamics (in Russian). Nauka. Moscow (English Translation, 1971, MIT Press, Boston) Narasimha R, Narayan KY, Parthasarathy SP (1973) Parametric analysis of turbulent wall jets in still air. Aeronautical J 77:335–359 Nickels TB, Perry AE (1996) An experimental and theoretical study of the turbulent co-flowing jet. J Fluid Mech 309:157–182 Obukhov AM (1941) On energy distribution in the spectrum of turbulent flow. Izv AN SSSR Ser Geogr Geoph 5(4–5):453–466, in Russian Obukhov AM (1949) Structure of the temperature field in a turbulent flow. Izv AN SSSR Ser Geogr Geoph 13:58–69 (in Russian) Panchapakesan NR, Lumley JL (1993) Turbulence measurements in axisymmetric jets of air and helium. Part 2. Helium jet. J Fluid Mech 246:225–247 Pope SB (2000) Turbulent flows. Cambridge University Press, Cambridge Prandtl L (1925a) Uber die ausgeloildete Turbulenz. ZAMM 5:136–139 Prandtl L (1942) Bemerkungen zur Theorie der freien Turbulenz. ZAMM 22:241–243 Prandtl L (1925b) Bericht uber Untersuchungen zur ausgebildeten Turbulenz. ZAMM 5:136–139 Rotta JC (1962) Turbulent boundary layers in incompressible flow. In: Ferri A, Kuchemann D, Sterne LHG (eds) Progress in aeronautical sciences vol 2. pp 1–219, Pergamon Press Sakipov ZB (1961) On the ratio of the coefficients of turbulent exchange of momentum and heat in free turbulent jet. Izv AN Kaz, SSR, 19 Sakipov ZB, Temirbaev DZ (1962) On the ratio of the coefficient of turbulent exchange of momentum and heat in free turbulent jet of mercury. Izv AN Kaz, SSR, 22 Schlichting H (1979) Boundary layer theory, 7th edn. McGraw-Hill, New York Sedov LI (1993) Similarity and dimensional methods in mechanics 10th edn CRC Press, Boca Raton Shin T-H, Lumley JL, Jonicka J (1982) Second-order modeling of a variable-density mixing layer. J Fluid Mech 180:93–116 Smith SH, Mungal MG (1998) Mixing structure and scaling of the jet in cross-flow. J Fluid Mech 357:83–122 Tachie MF, Balachander R, Bergstrom DJ (2004) Roughness effects on turbulent plane wall jets in an open channel. Exp Fluids 37(2):281–292 Taylor GI (1932) The transport of vorticity and heat through fluids in turbulent motion. Proc Roy Soc London A 135:685–705 Townsend AA (1956) The structure of turbulent shear flow. Cambridge University Press, Cambridge Vilis LA, Kashkarov VP (1965) The theory of viscous fluid jets. Nauka, Moscow (in Russian)
Chapter 9
Combustion Processes
9.1
Introductory Remarks
Combustion presents itself complicated physicochemical process which proceeds due to progressively self-accelerating exothermal chemical oxidation reactions sustained by an intensive heat release. A strong dependence of the chemical reaction rate on temperature according to the Arrhenius law determines a very high sensitivity of combustion processes to small disturbances of the governing parameters. It also determines an almost abrupt transition of reactive systems from a low temperature state to a high temperature state which is associated with ignition. The existence of a critical state corresponding to ignition, as well as the ability of combustion oxidation reactions to sustain a self-propagating flame front over reactive media represent themselves main features of combustion process. Combustion of continuous gaseous media is described by the system of equations including the Navier–Stokes, continuity, energy and species balance equations @r þ rðv rÞv ¼ rP þ rðmrvÞ @t
(9.1)
@r þ r ðrvÞ ¼ 0 @t
(9.2)
@h þ rðv rÞh ¼ rðkrTÞ þ qW @t
(9.3)
@cj þ rðv rÞcj ¼ rðrDrcj Þ Wj @t
(9.4)
r
r
where r, v, P, T, h and cj are the density, velocity vector, pressure, temperature, enthalpy, and species concentrations, respectively, q is the heat release of the combustion oxidation reaction (it is assumed that the whole complicated chemical L.P. Yarin, The Pi-Theorem, Experimental Fluid Mechanics, DOI 10.1007/978-3-642-19565-5_9, # Springer-Verlag Berlin Heidelberg 2012
261
262
9 Combustion Processes
process can be reduced to a single equivalent reaction), W is the rate of the chemical reaction, Wj is the rate of conversion of j th species (positive for reagents, which are consumed and negative for the reaction products which are produced), m; k and D are the viscosity, thermal conductivity and diffusivity, respectively. It is emphasized that for simplicity all the transport coefficients and physical properties of the parameters are assumed to be identical for all species involved. The system of (9.1–9.4) should be supplemented by an equation of state of the gas, a microkinetic law for the rate of chemical reaction and the correlations determining the dependence of the physical properties on temperature. Solving the highly nonlinear (9.1), (9.3) and (9.4) is extremely difficult. Therefore, the analytical models of the combustion processes, as a rule, involve various simplifications and approximations. Consider briefly some of them. First, we discuss simplifications related to the chemical reaction rate W. For this aim, we assume that: (1) the reactive mixture represents itself perfect gases, (2) the thermal and species diffusivities are equal, and thermal conductivity and specific heat cP are invariable, so that the enthalpy is expressed as h ¼ cP T, (3) a simple equivalent single-step chemical reaction with the rate which can be factorized as a product of two functions depending solely either on temperature or concentration Wðc; TÞ ¼ ’ðcÞcðTÞ
(9.5)
Moreover, we assume that there is only one limiting species in the system (fuel) and, as a result, only one species (fuel) balance equation is to be considered. Its concentration is denoted as c. Then, the energy and fuel balance equations read @h þ ðv rÞh ¼ ar2 h þ qWðc; TÞ @t
(9.6)
@c þ ðv rÞc ¼ Dr2 c Wðc; TÞ @t
(9.7)
where a and D are the thermal and mass diffusivity coefficients. Assume that a ¼ D, i.e. the Lewis number Le ¼ a=D ¼ 1. Then, eliminating the source terms from (9.6) and (9.7), we obtain the equation for the total (physical and chemical) enthalpy @H þ ðv rÞH ¼ ar2 H @t
(9.8)
where H ¼ h þ qc: For an isolated system ð@H=@nÞS ¼ 0; with S being the system envelop and n is the normal to it. In this case (9.8) is integrated as H ¼ const
(9.9)
9.1 Introductory Remarks
263
The constant in (9.9) is found from the condition that the maximum temperature Tm corresponds to complete fuel consumption, i.e. to c ¼ 0: Accordingly, that the constant is equal to cP Tm , and thus the current fuel concentration is related to the current temperature as c¼
cp ðTm TÞ q
(9.10)
Therefore, the function ’ðcÞin (9.5) can be presented as cP ðTm TÞ ’ðcÞ ¼ ’ ¼ ’ ðTÞ q
(9.11)
Then the reaction rate becomes Wðc; TÞ ¼ ’ ðTÞcðTÞ ¼ WðTÞ
(9.12)
Assuming the nth order reaction depending on temperature via the Arrhenius law, i.e. Wðc; TÞ ¼ zcn expðE=RT Þ, we arrive at the following explicit expression for WðTÞ E WðTÞ ¼ zðTm TÞn exp RT
(9.13)
where z is pre-exponent, E is the activation energy and R is the universal gas constant. It is emphasized that (9.12) and (9.13) are valid only when the Lewis number Le ¼ 1. In the general case when Le 6¼ 1; the distribution of the total enthalpy within a reactive medium is not given by a constant according to (9.9) but is more complicated. In combustion of gaseous mixtures, which is a particular case of homogeneous combustion), H has extrema in the vicinity of the flame front. It was shown in Zel’dovich et al. (1985) that this is the result of the energy redistribution due to different rates of heat and mass transfer at Le 6¼ 1: An additional simplification that is widely use in the theory of thermal explosion and ignition is related to the Frank-Kamenetskii transformation of the exponent in the Arrhenius law. Following Frank-Kamenetskii (1969), we present the ratio E=RTas E E E 1 E E ¼ DT ¼ RT RðT þ DTÞ RT ð1 þ DT=T Þ RT RT2
(9.14)
where as the temperature T is close to the temperature at which the chemical reaction proceeds: to the initial temperature T0 in the problem of self-ignition, or to the maximum temperature Tm in the problem of combustion wave propagation; D ¼ T T :
264
9 Combustion Processes
According to (9.14), the exponent in the Arrhenius law takes the form eRT eRT ey E
E
(9.15)
where y ¼ E=RT2 ðT T Þ. The expression (9.15) is an accurate approximation of the Arrhenius law at temperatures near to T , albeit without linearization. The latter feature is important, since such phenomena as ignition, extinction or thermal explosions are basically nonlinear and to be able to address them, nonlinear (even though simplified) nature of the problem should be preserved (Frank-Kamenetskii 1969; Zel’dovich et al. 1985). An additional significant simplification of the problems related to combustion of non-premixed gases can be achieved by means of the Schvab–Zel’dovich transformation (Schvab 1948; Zel’dovich 1948). It allows removal of the non-linear term WðTÞ from the all but one species balance equations and transforms them to the form identical to the form for inert gases. In order to illustrate this approach, we consider application of the Schvab–Zel’dovich transformation to the species equations in the case of reaction between a single fuels and a single oxidizer r
@ca þ rðv rÞca ¼ rDr2 ca Wa @t
(9.16)
r
@cb þ rðv rÞcb ¼ rDr2 cb Wb @t
(9.17)
where subscripts aand b refer to fuel and oxidizer, respectively. Taking into account that the reaction rates Wa and Wb are related by the stoichiometric relation dictated by the corresponding chemical reaction Wa ¼
Wb O
(9.18)
with O being the stoichiometric coefficient, we transform the system of equations (9.16) and (9.17) in the following way. We divide (9.17) by O and subtract (9.17) from (9.16). As a result, we arrive at the following equation r
@b þ rðv rÞb ¼ rDr2 b @t
(9.19)
where b ¼ ca cb =O. Equation (9.19) has the same form as the species balance equation the inert flows without any chemical reactions. This equation admits solutions which differ up to a 0 constant, i.e. b ¼ b þ const: The latter allows one to reduce the boundary conditions for axisymmetric reactive flows (submerged torches) to the form
9.2 Thermal Explosion
265
y¼0
@b0 ¼ 0; @y
y ! 1 b0 ! 0
(9.20)
where is the radial coordinate.
9.2
Thermal Explosion
Thermal explosions corresponds to unstable states of reactive system at which any initial temperature distribution leads to a disruption of the thermal equilibrium resulting from a runaway rate of the exothermal chemical reaction with heat release higher than the rate of heat removal to the environment. The latter leads to progressive temperature growth and an abrupt transition from the initial low temperature state to the final stable high temperature state. In order to illustrate the nature of thermal explosions, we consider the thermal regime of combustion of an ideally stirred reactor (for example, a jet-stirred reactor, JSR) which represents itself a closed volume filled with a homogeneous reactive mixture of fuel and oxidizer. The wall temperature of JSR is assumed to be fixed and equal to a low temperature T0 imposed by the surrounding medium. The temperature field inside JSR is assumed to be uniform, which corresponds to an infinite rate of mixing. Changes in the reactant concentrations in time are assumed to be negligible during the extremely short period of time preceding thermal explosion. Then, the specific rate of heat release by chemical reaction QI and the rate of heat removal to the environment QII (both divided by the reactor volume V) read E QI ¼ qW ¼ const exp RT
(9.21)
S QII ¼ h ðT T0 Þ V
(9.22)
where q is heat of reaction, W ¼ zcea csb expðE=RT Þ is the rate of chemical reaction (without simplification by means of the Frank-Kamenetskii transformation), z is the pre-exponent, ca and cb are reactant concentrations (fuel and oxidizer, respectively), e and s are constant (they denote the reaction orders in fuel and oxidizer, respectively), h is the heat transfer coefficient at the outer wall of the reactor, S and Vare the surface area and volume of JSR. The curves of heat release QI and heat removal QII are plotted versus temperature T in Fig. 9.1. In the general case there are three intersection points which correspond to the low (1), intermediate (2) and high temperature (3) states. It is easy to see that intermediate state (2) is unstable whereas the states (1) and (3) are stable. Indeed, any perturbation increasing the mixture temperature relative to that corresponding to point 2 (T > T2 Þis accompanied by an excess of the heat release
266
9 Combustion Processes
rate over the intensity of the heat removal rate. As a result, temperature T should keep increasing and thus, the intermediate point 2 is unstable. Also, if due to a perturbation the temperature decreases (T < T2 ), the heat removal rate exceeds the heat release rate. As a result, temperature T keeps decreasing and once more it is seen that point 2 is unstable. On the other hand, similar arguments show that points 1 and 3 are stable. Among the possible mutual locations of curves QI ðTÞ and QII ðTÞthere are two particular ones where they are tangent to each other. These two cases signify the two critical states: (1) the transition from the low- to hightemperature state, which corresponds to the mixture ignition-point I in Fig. 9.1, and (2) the transition from the high- to low-temperature state, which corresponds the mixture extinction-point E in Fig. 9.1. Therefore, the thermal explosion corresponds to the ignition condition, at which a low-temperature stationary state of a reactive system becomes impossible. This is an outline of the non-stationary theory of thermal explosion elaborated by Semenov (1935). On the other hand, the stationary theory of thermal explosion developed by Frank-Kamenetskii (1969) treats thermal explosion as the situation at which no stationary solution of the thermal balance equation can be found. A detailed exposition of both theories of thermal explosion can be found in the monograph by Zel’dovich et al. (1985); see also Frank-Kamenetskii (1969) and Vulis (1961). Referring the interested readers to these monographs, we restrict our consideration to the applications of the Pi-theorem to the stationary problem of thermal explosion. Consider stationary temperature distribution in a symmetric reactor with characteristic size r0 and wall temperature T0 : Assuming as before that changes in reactant concentrations are negligible, we determine the governing parameters of the
Q 3
E
QI
2
Fig. 9.1 The curves corresponding to heat release QI, and heat removal QII . The lower, intermediate and upper intersection points 1, 2 and 3, respectively, correspond to the stable lower temperature state, intermediate (unstable) state, and the stable high temperature state
QII I 1
0
To
T
9.2 Thermal Explosion
267
problem. From the physical point of view the local temperature inside a reactor depends on the thermal conductivity of gas mixture ½k ¼ JL1 T 1 y1 , kinetic factors in the Arrhenius law z expðE=RT Þ; namely on ½z ¼ T1 ; ½E ¼ Jmol1 and ½R ¼ Jy1 mol1 ; the heat of reaction ½q ¼ JL3 ; the reactor size ½r0 ¼ L; the wall temperature ½T0 ¼ y and the coordinate r ½ L of the point under consideration T ¼ f ðr0 ; r; T0 ; k; z; E; R; qÞ
(9.23)
The temperature distribution defined by (9.23) satisfies the following conditions T ¼ T0
at
dT ¼0 dr
r ¼ r0 ;
at
r¼0
(9.24)
Equation (9.23) and the boundary conditions (9.24) contain eight dimensional parameters including five parameters with independent dimensions. Then, according to the Pi-theorem, (9.23) reduces to the following dimensionless equation b ¼ ’ðx; g; b0 Þ
(9.25)
where b ¼ RT=E; b0 ¼ RT0 =E; g ¼ qzRr02 =kE and x ¼ r=r0 . Equation (9.25) shows that using the Pi-theorem, it was possible to decrease the number of the governing parameters from eight to three. However, even in this case the study of the critical state which corresponds to thermal explosion is highly complicated. In this situation, similarly to the analytical solutions of the combustion theory discussed above, it is useful to employ a physically-based simplification, namely the Frank-Kamenetskii transformation. Since the ignition (or thermal explosion) process occurs at temperatures close to the wall temperature T0 , similarly to (9.15) we have RTE
zeRT ze E
0
E
2 ðTT0 Þ
eRT0
(9.26)
Then, (9.23) can be replaced by the following equation
z; T0 ; qÞ DT ¼ f ðr; r0 ; k; e
(9.27)
where e z ¼ z expðE=RT0 Þ; T0 ¼ RT02 =E and DT ¼ T T0 : Applying the Pi-theorem to the simplified (9.27), we arrive at the following simpler dimensionless equation # ¼ cðx; dÞ
(9.28)
where # ¼ EðT T0 Þ=RT02 and d ¼ qEzr02 =kRT02 expðE=RT Þ is the FrankKamenetskii parameter, which is the sole dimensionless constant on the righthand side in (9.28).
268
9 Combustion Processes
The critical value of dcorresponding to the ignition (or thermal explosion) can be found only by solving the thermal balance equation r2x # ¼ d expð#Þ
(9.29)
subjected to the boundary conditions # ¼ 0 at
x ¼ 1;
d# ¼0 dx
at
x¼0
(9.30)
The corresponding critical condition found from (9.29) and (9.30) when the stationary solution becomes impossible reads d ¼ dcr ¼ const
(9.31)
where dcr ¼ 0:88 for plane reactors and 0.33 for the spherical ones.
9.3
Combustion Waves
The present section is devoted to a simple estimate of the speed of combustion wave that propagate in homogeneous infinite reactive media. The analysis is based on the approach of the thermal theory of combustion that imply that combustion wave propagation is a result of heat transfer from a high temperature reaction zone to a relatively cold fresh mixture of fuel and oxidizer due to thermal conductivity. This theory was developed by Zel’dovich (cf. Zel’dovich et al. 1985) and FrankKamenetskii (1969). The thermal theory of combustion accounts for the main features of the process, namely, the sharp dependence of the chemical reaction rate on temperature, the intensive heat release within a thin reaction front, as well as for the heat and mass transfer due to molecular thermal conductivity and molecular diffusion. According to this theory, the mechanism of combustion wave propagation in homogeneous mixtures is the following. An instantaneous heating of a thin layer of a preliminarily cold reactive mixture by an external source triggers chemical reaction within the heated layer (Fig. 9.2). The heat released by the exothermal chemical reaction, in its turn, leads to a further heating of the mixture in this layer and its ignition under certain conditions. Heat transfer from the high temperature zone to the cold mixture ensures heating and ignition of the neighboring layers, i.e. propagation of a self-sustained chemical reaction zone (the flame front, or combustion wave) over reactive medium. At the transient stage of the combustion wave propagation, the process develops under the conditions of a continuous variation of the temperature and concentration fields. Also, the speed of the combustion wave varies until its value will not approach the one corresponding to the stationary regime of combustion.
9.3 Combustion Waves
269
Fig. 9.2 The structure of a combustion wave at a certain moment of time. The wave is propagating from right to left. I-heating zone, II-reaction zone, III-high temperature zone
In the framework of the thermal theory there are two main factors that determine the speed of combustion wave in homogeneous reactive mixtures: (1) the exothermal chemical reaction accompanied by an intense heat release, and (2) the heat transfer from the high-temperature reaction zone to the cold fresh mixture by thermal conductivity. Under these conditions the governing parameters of the process are as follows: the mixture density ½r ¼ L3 M; thermal conductivity ½k ¼ LMT 3 y1 ; specific heat ½cP ¼ L2 T 2 y1 , and the characteristic time of chemical reaction ½tm ¼ T determined by the maximal temperature. In addition, it is assumed here that the Lewis number Le ¼ 1. Then, the thermal diffusivity a ¼ k/ rcP ¼ D, and thus, the diffusion coefficient D should not be included separately in the set of the governing parameters. Accordingly, the functional equation for the speed of combustion wave uf ¼ LT 1 has the form uf ¼ f ðr; k; cP ; tm Þ
(9.32)
All the governing parameters in (9.32) have independent dimensions. Then, according to the Pi-theorem, (9.32) transforms to uf ¼ cra1 cap2 ka3 ta4
(9.33)
where c is a dimensionless constant. Using the principle of the dimensional homogeneity and accounting for the dimensions of uf ; r; cP ; k and tm , we arrive at the following system of equations for the exponents ai 3a1 þ 2a2 þ a3 1 ¼ 0 a1 þ a2 ¼ 0 2a1 3a3 þ a4 þ 1 ¼ 0 a2 þ a3 ¼ 0
(9.34)
270
9 Combustion Processes
Equations (9.34) yield 1 a1 ¼ ; 2
1 a2 ¼ ; 2
1 a3 ¼ ; 2
a4 ¼
1 2
(9.35)
and thus uf ¼ c
rffiffiffiffiffi a tm
(9.36)
The value of the dimensional constant c depends on the other dimensionless groups of the problem. They are E=RT0 and E=RTm (Frank-Kamenetskii 1969). On the other hand, when the Frank-Kamenetskii transformation (9.15) is employed to simplify the Arrhenius law, the constant c depends a single dimensionless group ym ¼ EðTm T0 Þ=RTm2 combining the two previously mentioned groups. Using (9.36) it is possible to estimate the effect of pressure on the speed of combustion wave propagation. Assuming that c isp a ffiffiffiffiffiffiffiffiffiffi weak function of the dimensionless groups E=RT0 and E=RTm , we see that uf a=tm : The thermal diffusivity of gases is inversely proportional to pressure, a P1 . Based on the chemical kinetics n n data, one can also expect that t1 m r expðE=RTm Þ P expðE=RTm Þ; were n is the reaction order. As a result, we obtain the flame speed as uf P
n1 2
E exp 2RTm
(9.37)
It is seen that the flame speed uf does not depend on pressure in the case of a first order reaction. On the other hand, in the case of a second order reaction the flame speed uf is proportional toP1=2 : As was noted before, a detailed form of the dependence of the combustion wave speed on the physicochemical and kinetic parameters can be found by solving the energy and diffusion equations. At Le ¼ 1when the profiles of fuel concentration and the normalized temperature are similar to each other, the problem reduces to the integration of the energy equation. In the frame of reference associated with the moving combustion front (flame) the equation reads dT d dT þ qWðTÞ ¼ k ruf cP dx dx dx
(9.38)
In an infinite premixed mixture of fuel and oxidizer the boundary conditions for (9.38) have the form x ¼ 1;
T ¼ T0 ;
x ¼ þ1;
T ¼ Tm
(9.39)
9.4 Combustion of Non-premixed Gases
271
It is easy to see that if T(x) is a solution of the problem (9.38) and (9.39), then T(x þ c) (with c being an arbitrary constant) is also a solution. The latter means that the constant c is undetermined in principle, and one of the boundary conditions (9.39) becomes redundant. However, (9.38) contains a still unknown flame speed uf, which shows that a seemingly redundant boundary condition should be used to find uf, i.e. the flame speed uf represents itself an eigenvalue of the problem (9.38) and (9.39). A comprehensive discussion of the analytical solutions for the speed of combustion waves in homogeneous mixtures can be found in the following monographs and surveys: Frank-Kamenetskii (1969), Williams (1985), Zel’dovich et al. (1985), Merzhanov and Khaikin (1992). Numerical solutions of this problem were discussed in Spalding (1953), Zel’dovich et al. (1985) and Merzhanov et al. (1969).
9.4
Combustion of Non-premixed Gases
Consider combustion of non-premixed gases in an adiabatic cylindrical chamber (Fig. 9.3). The gaseous reactants are supplied through a core tube of cross-sectional radius r1 (fuel) and an annular gap of thickness r2 r1 (oxidizer). It is assumed that the velocity distribution at any cross-section of the combustion chamber (burner) is uniform, i.e. fuel and oxidizer are issued with the same speed and the effect of viscous friction at the wall is negligible. The mass flux does not change downstream in the chamber. In addition, it is assumed that the diffusion transfer in radial direction is much large than in the longitudinal one. Regarding the rate of chemical reaction at the flame front, it is assumed that it is infinite, and therefore, concentrations of fuel and oxidizer at the flame front are zero. The assumptions made follow those in the seminal work of Burke and Schumann (1928), as well as the detailed analysis of the corresponding problem is covered in the monographs by Vulis (1961), Williams (1985), Zel’dovich (1948) and Zel’dovich et al. (1985). Below we discuss briefly the formulation of this problem and concentrate of the application of the dimensional analysis to in this particular case.
r
Oxidizer Fuel
2
r2 r1
x 1
Fig. 9.3 Sketch of a nonpremixed gas burner
Oxidizer
272
9 Combustion Processes
The above assumptions and the Schvab-Zel’dovich transformation allow us to reduce the species balance equations to the following single equation we write the governing equation in the form ru
@b 1 @ @b ¼ rD r @x r @r @r
(9.40)
[cf. (9.19)] where b ¼ ca cb =O; ca and cb are the concentration of fuel and oxidizer, respectively, O is the stoichiometric oxidizer-to-fuel mass ratio, ru is the mass flow rate and rD the product of density and diffusion coefficient; both ru and rD are constant in the present case. The boundary conditions for (9.40) read x¼0:
0 r r1 x>0:
b ¼ ca0 ; r¼0
r1 < r < r2
@b ¼ 0; @r
r ¼ r2
b¼
cb0 O
@b ¼0 @r
(9.41)
The conditions at x ¼ 0 determine the uniform distribution of fuel and oxidizer at the burner inlet; the boundary conditions at x > 0 determine the flow symmetry and correspond to the absence of the chemical reaction at the wall. The solution of (9.40) with the boundary conditions (9.41) is (Zel’dovich et al. 1985) bðr; xÞ ¼ b
1 X
CI J0 ðr’i =r2 Þ expðSI xÞ
(9.42)
i¼1
The following notation is used in (9.42): b ¼ ca0 ½ðca0 O þ cb0nÞ=Oðr1 =r2 Þo2 ;
Si ¼ ðrD=ruÞð’2i =r22 Þ,
Ci ¼ 2ðcb0 =OÞð1 þ wÞðr1 =r2 ÞJ1 ðr1 ’i =r2 Þ= ’i ½J0 ð’i Þ2 ;
w ¼ ðca0 O=cb0 Þ; J0 ð Þ and J1 ð Þ are the Bessel functions of the first kind of zero and first orders, and ’i are the roots of the equation J1 ð’Þ ¼ 0: Concentrations ca0 and cb0 correspond to fuel and gas at the burner entrance at x ¼ 0: The corresponding approximate expression for the flame length xf ¼ lf is (Zel’dovich et al. 1985) lf ¼
ur22 2ð1 þ wÞðr1 =r2 Þ2 J1 ðr1 ’1 =r2 Þ h i ln 2 D’1 ’ ½J0 ð’ Þ2 w ð1 þ wÞðr1 =r2 Þ2 1 1
(9.43)
where ’1 ¼ 3:83 and J0 ð’1 Þ ¼ 0:4: The expression (9.43) corresponds to combustion at the excess of oxidizer and the whole fuel is consumed at a finite length xf which corresponds to the tip of curve 1 in Fig. 9.3. In the opposite case when combustion proceeds at the lack of oxidizer, the term ½J0 ð’1 Þ2 in the denominator of (9.43) should be replaced by ½J0 ð’1 Þ.
9.4 Combustion of Non-premixed Gases
273
Then, (9.43) describes the distance at which all oxidizer will be fully consumed, which corresponds to the right-hand side end of curve 2 in Fig. 9.3. Consider the Burke-Schumann problem in the framework of the dimensional analysis. The problem formulation reveals that at Le ¼ 1 when the thermal and mass diffusivities are equal to each other, the field of the compound concentration b in coaxial burner is determined by nine parameters b ¼ f ðu; D; r; x; r1 ; r2 ; ca0 ; cb0 ; OÞ
(9.44)
These governing parameters have the following dimensions ½u ¼ Lx T 1 ; ½ D ¼ L2y T 1 ; ½r ¼ Ly ; ½ x ¼ Lx ; ½r1 ¼ Ly ; ½r2 ¼ Ly ; ½ca0 ¼ 1; ½cb0 ¼ 1; ½O ¼ 1
(9.45)
Among of the six dimensional parameters in (9.45), three parameters have independent dimensions. Therefore, it is possible to form three dimensionless groups r1 ; r2
r ; r2
xD ur22
(9.46)
Then (9.44) reduces to the following dimensionless form r xD r1 ; ; ; ca0 ; cb0 ; O b¼’ r2 ur22 r2
(9.47)
As it was mentioned above, the concentrations of reactants at the combustion front at xf ¼ xf ðrf Þ are equal to zero, so that b ¼ 0 there. Then, (9.47) yields ’
r f xf D r 1 ; ; c ; c ; O ¼0 a0 b0 r2 ur22 r2
(9.48)
Solving (9.48) relative to the dimensionless group xf D=ur22 , we obtain xf D rf r1 ¼c ; ; ca0 ; cb0 ; O r2 r2 ur22
(9.49)
Equation (9.49) determines geometry of the diffusion flame of non-premixed reagents. It contains four constants r1 =r2 ; ca0 ; cb0 and O that account for the burner geometry, as well as the characteristics of reactive system. The dependence xf D=ur22 ¼ c rf =r2 found from the exact solution (9.43) is shown in Fig. 9.4. As discussed, the shape of hthe diffusion iflame depends on a relation between w ¼ ðca0 O=cb0 Þ and ðr1 =r2 Þ= 1 ðr1 =r2 Þ2 which determines where the system has
274
9 Combustion Processes
Fig. 9.4 Configurations of the diffusion flame of nonpremixed gases. 1: The case of the excess of oxidizer. 2: The case of the excess of fuel
xfD
ur 22
1
0.1
2
0
rf r2
0.5
either an excess of oxidizer or fuel. In the first case the flame tip is located at the flow axis, whereas in the second one at the wall of the burner. Assuming in (9.49) rf ¼ 0, we obtain the following expression for the diffusion flame length xf ¼ lf r1 lf ¼ Pe c ; ca0 ; cb0 ; O r2
(9.50)
where Pe ¼ ur2 =D is the Peclet number and lf ¼ l=r2 . Equation (9.50) shows that with r1 =r2 ; ca0 ; cb0 and O being constant, the flame length lf ¼ lf =r2 Pe i.e. lf ur22 =D. The volumetric flow rate of the gaseous phase is Gv ur2 e for the planar flame, and Gv ur22 for the axisymmetric flame (e ¼ 1is the unit of length). Then, we arrive at the conclusion that lf ;planar Gv r2 =D and lf ;axisymm Gv =D, i.e. the length of the axisymmetric flame does not depend on the radius of combustion chamber, whereas the length of a planar flame is directly proportional to r2 when Gv ¼ const. Note, that in the planar case the difference r2 h is equal to the semi-height of the channel (Vulis 1961).
9.5
Diffusion Flame in the Mixing Layer of Parallel Streams of Gaseous Fuel and Oxidizer
The flow and flame structure under consideration are sketched in Fig. 9.5. Two uniform streams of gaseous non-premixed reactants moving over both sides of a semi-infinite plate which ends at x ¼ 0 come in contact to each other. The fuel
9.5 Diffusion Flame in the Mixing Layer of Parallel Streams
275
Fig. 9.5 Diffusion flame in the mixing layer of parallel streams of gaseous and oxidizer
stream is supplied at y < 0, whereas the oxidizer-at y > 0. The ignition takes place at the line x ¼ 0; y ¼ 0: When the rate of chemical reaction is large enough, conversion of reactants into combustion products occurs within a thin reaction zone that can be considered practically infinitesimally thin and viewed as the flame front (Zel’dovich et al. 1985); cf. Fig. 9.5. Then, domain I in Fig. 9.5 is filled with the oxidizer and fully converted combustion products, whereas domain II is filled with fuel and combustion products. Chemical reaction takes place neither in domain I nor in domain II but solely at the flame front. On the other hand, pure mixing takes place in domains I and II. In the framework of this model, and assuming the low Mach number (M << 1) and h ¼ cp T(cP ¼ const), the velocity, temperature and fuel and oxidizer concentration fields are determined by the following equations @u @u @ @u ¼ m ru þ rv @x @y @y @y
(9.51)
@ru @rv þ ¼0 @x @y
(9.52)
@T @T @ @T þ rvcP ¼ k rucP @x @y @y @y
(9.53)
276
9 Combustion Processes
@cj @cj @ @cj þ rv ¼ rDj ru @x @y @y @y
(9.54)
where subscript j corresponds to the j th species (j ¼ a for fuel, and j ¼ b for oxidizer). The system of (9.51–9.54) is supplemented by the equation of state (9.55) accounting for the fact that pressure is constant in the mixing layer, as well as the dependences of the physical parameters on temperature (9.56) rT ¼ const mðTÞ;
kðTÞ;
(9.55)
rDj ðTÞ
(9.56)
The boundary conditions at x > 0 for (9.51–9.54) read y ! þ1;
u ! uþ1 ;
T ! Tþ1 ;
ca ! caþ1
y ! 1;
u ! u1 ;
T ! T1 ;
cb ! cb1
(9.57)
At the flame front y ¼ yf ðxÞ the reactant concentrations are zero, since the reaction rate is practically infinite, whereas the diffusion fluxes of fuel and oxidizer are in stoichiometric ratio T ¼ Tf ;
ca ¼ cb ¼ 0
Db ð@cb =@nÞf Da ð@ca =@nÞf
¼O
(9.58) (9.59)
where Tf is the flame temperature which is equal to the adiabatic temperature of combustion of non-premixed fuel and oxidizer, O is the stoichiometric coefficient and @=@n is the derivative along the normal to the flame front. It is emphasized that the boundary condition (9.59) allows one to determine the location of the flame front. The problem we are dealing with in the present section represents itself a compressible flow. In such cases the Dorodnitsyn–Illingworth–Stewartson transformation discussed previously in Sect. 7.7 of Chap. 7 allows one to reduce compressible problems to the corresponding incompressible ones. In particular, Ry introducing new the variables x ¼ x and ¼ rdy and assuming that the dependences mðTÞ; kðTÞ and rDðTÞ are linear, it is0 possible to reduce (9.51–9.54) to the form identical to the incompressible equations corresponding to the same flow geometry but with r ¼ const: The dimensional analysis of these system of equations shows that there exists the self-similar solution of the dynamic, thermal and species balance equations in the following form u ¼ F0 ð’Þ;
DT ¼ yð’Þ;
cj ¼ #ð’Þ
(9.60)
9.5 Diffusion Flame in the Mixing Layer of Parallel Streams
277
with u ¼ 2u=ðuþ1 þ u1 Þ; DT a ¼ T Tþ1 = Tf Tþ1 ; DT b ¼ ðT T1 Þ= pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi RY g Tf T1 ; ’ ¼ x ; ¼ rdy; ¼ x; y ¼ ðy=l Þ ½Reð1 þ mÞ=2; x ¼ 0
x=l ; ra ¼ ra =raþ1 ; rb ¼ rb =rb1 ; m ¼ u1 =uþ1 ; l is an arbitrary length scale, x and are the Dorodnitsy n variables, and g is a constant. 0 The functions Fj ð’Þ; yj ð’Þ and #j ð’Þ are determined by the following ODEs 1 00 000 Fj þ Fj Fj ¼ 0 2
(9.61)
00
Pr 0 yj Fj ¼ 0 2
(9.62)
00
Sc 0 #j Fj ¼ 0 2
(9.63)
yj þ #j þ
[subscript j ¼ a or b], with the boundary conditions 2 ; ya ¼ 0; #a ¼ 1 at 1þm m 0 Fb ¼ ; yb ¼ 0; #b ¼ 1 at 1þm ya;b ¼ 1; #a;b ¼ 0 at ’ ¼ ’f 0
Fa ¼
’ ! þ1 ’ ! 1
(9.64)
The integration of (9.61–9.63) with the boundary conditions (9.64) leads to the following expressions for the velocity, temperature and concentration distributions in the mixing layer u 1 ¼ fð1 þ mÞ þ ð1 mÞerf ð’Þg uþ1 2 pffiffiffiffiffi 1 erf ð’ PrÞ pffiffiffiffiffi ; ya ¼ 1 erf ð’f PrÞ
(9.65)
pffiffiffiffiffi 1 erf ð’ ScÞ pffiffiffiffiffi #a ¼ 1 1 erf ð’f ScÞ
(9.66)
pffiffiffiffiffi 1 þ erf ð’ ScÞ pffiffiffiffiffiffiffi 1 þ erf ð’f ScÞ
(9.67)
for ’f < ’ < þ 1; and yb ¼
pffiffiffiffiffi 1 þ erf ð’ PrÞ pffiffiffiffiffi ; 1 þ erf ð’f PrÞ
#b ¼ 1
for 1 < ’ < ’f : To find the location of the flame front, we take into account the following evaluations valid for the boundary layer in which the flame front slope relative to the x-axis is small enough, so that cosðn; xÞ 0 and cosðn; yÞ 1: For example, at
278
9 Combustion Processes
O ¼ 15; Pr ¼ 1 and Rex ¼ 100 1000; cosðn; yÞ 0:99 0:998. Under these conditions @=@n ¼ @=@x cosðn; xÞ þ @=@u cosðn; yÞ d=dy. Calculating the derivatives ð@ca =@nÞf and ð@cb =@nÞf by using the expressions (9.66) and (9.67) and substituting them into (9.59), we arrive at the following expression for the coordinate of the flame front ’f pffiffiffiffiffi 1e erf ð’f ScÞ ¼ 1þe
(9.68)
where e ¼ ðcaþ1 =cb1 ÞðDa =Db ÞO1 : The results presented above are related to the aerodynamics of non-premixed diffusion flames at small speed of fluid and oxidizer. Now we consider some features of the non-premixed diffusion flames in high velocity flows where the energy dissipation significantly affects the flame characteristics, such as, for example, the flame front temperature. We will consider diffusion combustion of non-premixed gases in the boundary layer formed when a high speed uniform semiinfinite gaseous fuel flow comes in contact with a semi-space filled with gaseous oxidizer at rest. Mixing of the gaseous fuel with gaseous oxidizer begins at crosssection x ¼ 0: The ignition of the reactive mixture occurs by an external source located at point x ¼ 0; y ¼ 0: As a result of the ignition, the reactive mixture in the boundary layer forms a thin reaction zone that can be presented as an infinitely thin flame front. The system of the governing equations describing velocity, enthalpy and species concentration distribution in diffusion combustion of non-premixed gases in high speed flows takes the following form (in the boundary layer approximation after the Dorodnitsyn–Illingworth–Stewartson transformation has been applied; cf. Sect. 7.7, Chap. 7) @u @v 1 @ 2u þ ve ¼ @ Re @2 @x
(9.69)
2 @h @h 1 @2h @u 2 þ ve ¼ þ ð g 1 ÞM þ1 2 @ Pr @ @ @x
(9.70)
@cj @cj 1 @cf ¼ þ ve @ Sc @2 @x
(9.71)
@u @e v þ ¼0 @x @
(9.72)
u
u
u
where u and v are component of the velocity, cj is the concentration pffiffiffiffiffi u ¼ u=uþ1 ; v ¼ ðv=uþ1 Þ Pr; ve ¼ rv þ u@=@x; h ¼ h=hþ1 ; h ¼ cP T; ðcP ¼ constÞ r ¼ r=rþ1 ; x ¼ x=l ; ¼ =l ; x and are the Dorodnitsy n variables, Re and Mþ1 are the Reynolds and Mach numbers, respectively.
9.5 Diffusion Flame in the Mixing Layer of Parallel Streams
279
The boundary conditions corresponding to the case of gaseous fuel issuing into the oxidizer at rest (u1 ¼ 0) are posed at the both edges of the mixing layer and the flame front. They are as following @u ¼ 0 at ! þ1 @ u ¼ 0; h ¼ h1 ; cb ¼ 1 at ¼ 1 ca ¼ 0; cb ¼ 0 at ¼ f u ¼ 1;
h ¼ 1;
ca ¼ 1;
(9.73)
The conditions (9.73) should be also added to the boundary conditions (9.59) to determine the location of the flame front. Also, the boundary condition describing the thermal balance at the flame front is needed. The latter is necessary to determine the combustion temperature, since in high velocity flow it depends not only on the heat of reaction but also on the heating due to the energy dissipation. The boundary condition which expresses the thermal balance at the flame front reads qrf Daf
@ca @n
þ kf
f
@T @n
¼ kf f
@T @n f
(9.74)
where q is the heat reaction. The dimensional analysis shows that there exists a self-similar solution of (9.69–9.72) subjected to all above-mentioned boundary conditions. The self-similar solution allows us to reduce the system of the partial differential equations of the problem to the corresponding system of ODEs for the functions Fð’Þ), yð’Þ and #ð’Þ 000
00
Fj þ 2Fj Fj ¼ 0
(9.75)
00 2 00 0 2 Fj ¼0 yj þ 2 Pr Fj yj þ Prðg 1ÞMþ1
(9.76)
00
0
#j þ 2ScFj #j ¼ 0
(9.77)
The boundary conditions for (9.75–9.77) read 0
00
Fa ¼ 1; ya ¼ 1; #a ¼ 1; Fa ¼ 0 0
at
c ! þ1
Fb ¼ 0; yb ¼ h1 ; #b ¼ 1 at c ! 1 y ¼ yf ; #a ¼ #b ¼ 0 at c ¼ cf
(9.78)
pffiffiffi 0 where Fj ¼ uj =uþ1 ; yj ¼ hj =hþ1 ; #j ¼ cj cj 1 ; c ¼ 2 x; x and are the Dorodnitsyn variables, and Mþ1 is the Mach number of the undisturbed flow. The boundary conditions (9.78) should be supplemented by the balance relations (9.59) and (9.74) that determine the position of the flame front, as well as its
280
9 Combustion Processes
temperature. The first of the latter is determined (as in the flow with small velocity) by (9.68). However, in high velocity flows the temperature of the flame front depends not only on the physicochemical properties of the reactants but also on the velocity of the undisturbed flow. The transformation of (9.74) leads to the following relation for the flame temperature Tf qcaþ1 1 g1 2 e ¼1þ þ Mþ1 Tþ1 cP Tþ1 1 þ e 2 ð1 þ eÞ2
(9.79)
where g ¼ cp =cv is the ratio of the specific heats at constant pressure and volume, respectively. The solution of (9.75–9.77) with the boundary conditions (9.58) that determine the velocity, enthalpy and reactant and combustion products concentration fields, as well as the configuration of the flame front and its temperature was found by Vulis et al. (1968). A similar approach can be also used to study combustion of liquid fuel in a stream of gaseous oxidizer which blows over its surface, for example, the combustion of large oil spots (Yarin and Sukhov 1987). Solution of the latter problem is very similar to the one described above, albeit it involves additional thermal and mass balance conditions, which are required for calculation of the temperature and vapor concentration at the free surface.
9.6
Gas Torches
Gas torches represent themselves submerged jets in which the intensive exothermal chemical oxidation reaction (combustion) proceeds. The conversion of the initial reactants into combustion products occurs in such jets within a thin high temperature zone that is identified with the flame front. The thickness of this zone can be estimated by the dimension consideration. Since the combustion process is determined by the two general factors, (1) kinetics of chemical reactions and (2) diffusion, we can assume that the thickness of the reaction zone depends on the rate constant ½Z ¼ T 1 of the chemical reaction and diffusivity ½ D ¼ L2 T 1 d ¼ f ðZ; DÞ
(9.80)
where Z ¼ Z0 expðE=RT Þ is the Arrhenius factor, k0 is the pre-exponential, E and R are the activation energy and the universal gas constant, respectively. According to the Pi-theorem, because k and D have independent dimensions, (9.80) takes the form d ¼ c Z a1 D a2 where c is a constant.
(9.81)
9.6 Gas Torches
281
Taking into account the dimensions of d; Z and D and applying the principle of dimensional homogeneity, we find the values of the exponents ai as a1 ¼ 1=2; a2 ¼ 1=2 which transforms (9.81) as follows rffiffiffiffi D d¼c Z
(9.82)
Equation (9.82) shows that the characteristic size of the reaction zone in torches of non-premixed gases (the diffusion flame) is the order of the flame front thickness in homogeneous mixtures. Indeed, d
rffiffiffiffi rffiffiffi D a uf Z Z Z
(9.83)
pffiffiffiffiffiffi where a is the thermal diffusivity, uf aZ is the speed of combustion wave in homogeneous mixtures. The estimate (9.83) reflects the physical similarity of the processes that occur in combustion of homogeneous reactive mixtures and in reaction zones of torches of non-premixed gases. Assuming that the characteristic size of the mixing zone in a gas torch is l (for submerged torches l is on the order of the boundary layer thickness), we arrive at the following estimate of the relative thickness of the reaction zone in diffusion flames rffiffiffiffiffiffiffi rffiffiffiffiffi D tk E d exp ¼ tD l2 Z 2RT
(9.84)
where d ¼ d=l; and tk ¼ Z01 and tD l2 =D are the characteristic kinetic and diffusion time, respectively. It is emphasized that the estimate (9.84) is valid not only in laminar torches but also in turbulent ones. In the latter case (9.84) implies not the molecular diffusion and thermal conductivity D and k but rather their turbulent analogs. Equation (9.84) shows that the relative thickness of reaction zone d depends essentially on the values of the kinetic and diffusion times. If the rate of chemical reaction is large enough so that tk << tD ; the relative thickness of the reaction zone is small enough, since as tk =tD ! 0; the value of d ! 0: Combustion temperature also affects significantly the thickness of the reaction zone. An increase in T (due to combustion of high caloric fuels) is accompanied by decreasing d: That allows one to assume that the chemical reaction of combustion proceeds within a very thin (in the limiting case of an extremely small ratio tk =tD , in an infinitesimally thin) flame front at a temperature close to the maximum one. Outside of flame front only the inert transfer of mass, momentum and energy take plays in submerged laminar and turbulent torches. The latter makes it possible to apply the method of the theory of the gas jets in studying gas torches (Abramovich 1963; Vulis et al. 1968; Vulis and Yarin 1978).
282
9 Combustion Processes
One can distinguish two main types of gas torches: (1) torches of premixed homogeneous mixtures, and (2) torches of non-premixed gaseous fuel and oxidizer. In the first type of torches, a premixed reactive mixture is supplied directly into the flame front, whereas in the second one a separate supply of reactants into the reaction zone occurs. Accordingly, in homogeneous torches combustion process is determined by the rate of chemical reaction, whereas in torches of non-premixed gases it is determined predominantly by the mixing rates and to some extent by the reaction rate. Moreover, in the non-premixed torches, as a rule, the mixing is the limiting process. Therefore, torches of non-premixed gases are longer and less intense than the homogeneous ones. In homogeneous torches combustion is fully completed within the entrance section of the jet, at a distance of about five nozzle calibers. In this case the flame front is located near the boundary of the potential core of the jet. Under such conditions the geometry of a homogeneous torch is determined by the velocity distribution at the nozzle exit, as well as the speed of combustion wave in homogeneous mixture. Thus, the functional equation for the length of homogeneous torch lf is lf ¼ f ðu0 ; uf ; dÞ
(9.85)
where u0 ¼ u0 ðyÞ is the velocity of the reactive mixture at the nozzle exit, with y being transversal coordinate. Equation (9.85) also incorporates the speed of combustion wave uf which represents the physicochemical characteristics of the reactive mixture and can be calculated using the well-known methods of the combustion theory (Zel’dovich et al. 1985; Williams 1985), and the nozzle diameter d. According to the Pi-theorem, (9.85) reduces to the following dimensionless form lf u0 ¼’ d uf
(9.86)
It is seen that the relative lengths of homogeneous torches depend only on the ratio of the issue velocity to the speed of combustion wave. At a known uf , the length and shape of homogeneous torches are found by integrating the following equation dx ¼ dy
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 u0 1 uf
(9.87)
Integration of (9.87) in the case of a parabolic velocity profile at the nozzle exit results in the following expression (Khitrin 1957)
1 k2 1þk lf xf ¼ N sin ’ cos ’D’ Fð’; KÞ þ 2 Eð’; kÞ k k2
(9.88)
9.6 Gas Torches
283
Fig. 9.6 Sketch of a homogeneous mixture torch
x u0
q uf y
pffiffiffiffiffiffiffiffiffiffiffi where sin2 ’ ¼ yf ðu 1Þ; k2 ¼ ðu 1Þ=ðu þ 1Þ; N ¼ ðu 1Þ u þ 1; D’ ¼ 1=2 ; u ¼ u0 =uf ; y ¼ yf =r0 ; x ¼ xf =r0 ; lf ¼ lf =r0 ; r0 ¼ d=2 is the 1 k2 sin2 ’ nozzle radius, Fð’; kÞ and Eð’; kÞ are the elliptic integral the first and second kind, respectively; cf. Fig. 9.6. When the velocity distribution at the nozzle exit is uniform, i.e. u0 ¼ const, and the outflow velocity of reactive mixture significantly exceeds the velocity of combustion wave, the length of homogeneous torch is given by lf u0 ¼ d uf
(9.89)
This relation shows that the length of laminar homogeneous torches is proportional to the outflow velocity. This result agrees well with the experiments on combustion in laminar torches of premixed gases, whereas the experimental data for the turbulent torches show that their length weakly depends on u0 , since ufT un0 ; with ufT being the speed of turbulent flame and n 0:7 0:8: Consider now vertical torches of non-premixed gases. First, we reveal the effect of flow parameters on the length of such torches. For this aim, we begin with the simplest example, namely a torch which forms when gaseous fuel is issued from a cylindrical nozzle into a gaseous medium containing oxidizer, which is at rest far
284
9 Combustion Processes
away in the transverse direction from the nozzle. Assuming that the velocity of the fuel jet is large enough, we neglect the any possible effect of buoyancy forces, which might arise from the disparity of fuel and oxidizer densities. We also neglect the effect of density change and assume that r is constant. At fixed initial concentrations of fuel in the gas jet and oxidizer in the ambient medium, we state the functional equation for the torch length lf as follows lf ¼ f ðu0 ; D; dÞ
(9.90)
Bearing in mind that gas torches features stem from those of submerged jets, we define the dimensions of the governing parameters in the system of units Lx Ly MT ½u0 ¼ Lx T 1 ;
½ D ¼ L2y T 1 ;
½d ¼ Ly
(9.91)
Since all the governing parameters possess independent dimensions, (9.91) reduces to lf ¼ cua01 Da2 d a3
(9.92)
where c is a dimensionless constant that depend on concentrations of the fuel in the gas jet cg and oxygen in the ambientcox , as well as on stoichiometric fuel-tooxygen mass ratio O. The latter follows directly from the functional equation for the torch length written with the account of the dimensional (½u0 ¼ Lx T 1 ; 2 1 ½ D ¼ Ly T ; ½d ¼ Ly ) and dimensionless (½O ¼ 1; cg ¼ 1; ½cox ¼ 1) charac teristics. Since the dimension of lf ¼ Lx can be constructed as a combination of the dimensions of the parameters u0 ; D and d, such an equation acquires the form of lf ¼ ua01 Da2 d a3 f ðO; cg ; cox Þ, i.e. the form of (9.92) where the constant c ¼ f ðO; cg ; cox Þ. Taking into account that the dimension of the torch length lf ¼ Lx and applying the principle of the dimensional homogeneity, we find the values of the exponents ai as a1 ¼ 1; a2 ¼ 1 and a3 ¼ 2: Accordingly, we obtain that in the case of n ¼ D lf ¼ cRe
(9.93)
where lf ¼ lf =d and Re ¼ u0 d=n is the Reynolds number. It is seen that the length of laminar axisymmetric torches of non-premixed gases (with n ¼ const) is proportional to the Reynolds number that is defined by the outflow velocity and the nozzle diameter, whereas the length of turbulent torches (with nT u0 d) lf does not depend on Re as follows from (9.93). (Hottel, Hawthorne, 1949) The experimental data shown in Fig. 9.7 agree with these predictions. The length of non-premixed gases torches is large enough to assume lf >> 1: Then, the characteristics of non-premixed diffusion flames practically do not depend on the outflow conditions at the nozzle exit and are determined by the
9.6 Gas Torches
285
f
II
I
III
0
Re
Fig. 9.7 The dependence lf ðReÞof the length of non-premixed torches on the Reynolds number. I-laminar torches, II-transitional torches and III-turbulent torches
integral parameters of the flow: the total momentum flux and mass fluxes of species. In order to determine these characteristics, we use the continuity, momentum and species balance equations in the boundary layer form @ruyk @rvyk þ ¼0 @x @y
(9.94)
ru
@u @u 1 @ k þ rv ¼ yt @x @y yk @y
(9.95)
ru
@b @b 1 @ k þ rv ¼ yg @x @y yk @y
(9.96)
where t and g are the shear stress and the compound species flux, respectively, the compound concentration b ¼ ca cb þ 1; with c1 and c2 being the fuel and oxygen concentration, respectively, ca ¼ ca O=cb1 ; cb ¼ cb =cb1 : Also, O is the stoichiometric fuel-to-oxygen mass ratio, subscript 1 refers to the ambient conditions, k ¼ 0 and 1 for the plane and axisymmetric torches, respectively. It is emphasized that (9.96) for the compound concentration b was obtained from the species balance equations by using the Schvab–Zel’dovich method, which allows us to exclude the source terms associated with the rate of chemical reaction. The boundary conditions for (9.95) and (9.96) read
286
9 Combustion Processes
u!0 b!0
at
y ! 1;
@u @b ¼ ¼ 0 at @y @y
y¼0
(9.97)
for submerged torches, and @b ¼ 0 at y ¼ 0 (9.98) @y for the wall torches issued from a slit parallel to the adjacent wall (in the case k ¼ 0 only). By integrating this system of equations with the boundary conditions (9.97), we arrive (at r ¼ const) the following integral invariants for the free torches u¼0
b ¼ 0 at
y ! 1;
1 ð
u¼0
u2 yk dy ¼ Jex
(9.99)
ex ubyk dy ¼ G
(9.100)
0 1 ð
0
ex are constant determined by the conditions at the nozzle exit and, where Jex and G thus, are given. It is emphasized that the invariant (9.100) stems from the fact that the compound concentration b is described by (9.96) which does not involve any term related to the rate of chemical reaction. Assuming that the longitudinal convective species fluxes within the far field of ex , we formulate the non-premixed torches are determined by the invariants Jex and G functional equation for (ubÞm as follows ex Þ ðubÞm ¼ f ðJex ; G
(9.101)
where subscript m refers to the torch axis. h i 2 Taking into account that the dimensions of Jex ¼ L2x L1þk and y T h i 1þk 1 e Gx ¼ Lx L T are independent, we arrive at the simpler form of (9.101) y
u m bm ¼ c
Jex fx G
(9.102)
where c is a constant. The dimensional analysis of the flow in the far field of submerged jets shows (Chap. 6) that the problem under consideration based on (9.95) and (9.96) has the following self-similar solutions u ¼ ’ðÞ; um
b ¼ cðÞ; bm
um ¼ Axa
(9.103)
9.6 Gas Torches
287
where ¼ y=d; d is the thickness of the boundary layer of the jet, and A and a are constants, the functions ’ðÞ and cðÞ are determined by a system of the ordinary differential equations that can be found transforming (9.94–9.96) using the Pitheorem. The substitution of the expression (9.103) for the axial velocity um into (9.102) yields xa ¼ bm Ac1
fm G Jf m
(9.104)
where c1 ¼ c1 is the constant. Taking in (9.104) x ¼ lf and accordingly bm ¼ 1 which corresponds to the complete combustion at the flame tip, we obtain the following expression for the length of non-premixed gas torch la f ¼ Ac1
fx G Jex
(9.105)
Using the known expressions for the coefficients A and a (see Chap. 6) and fx ¼ pk u0 b0 lkþ1 (where l0 is the nozzle characteristic accounting for the fact that G size and b0 is the given compound concentration value at the nozzle exit), we arrive at the following relations for the dimensionless length of torches of non-premixed gases of different types (Table 9.1). It is seen from Table 9.1 that the length of laminar non-premixed torches depends on the issue velocity (through the Reynolds number Re) and parameter e ¼ 1 þ ðca0 =cb0 ÞO accounting for the reactant concentrations and the stoichiometric fuel-to-oxygen mass ratio. On the other hand, the length of turbulent torches depends only on the ratio of reactant concentrations and the stoichiometric fuel-to-oxygen mass ratio. The constants Ci in the expressions for lf in Table 9.1 depend on the type of the jet flow (free or wall flow), its geometry (plane or axisymmetric), as well as on the flow regime (laminar or turbulent). It is emphasized that the dependence of lf on the reactant concentrations is different for different types of non-premixed torches. For example, in plane laminar torches lf e3 , whereas in the axisymmetric ones lf e. That results from the different mixing intensity in various types of submerged torches. Table 9.1 Length of diffusion torches of different geometry
Type of flow Plane laminar torch Plane laminar wall-torch over an adiabatic plate Axisymmetric laminar torch Plane turbulent torch Axisymmetric turbulent torch
lf C1 Ree3 C2 Ree4 C3 Ree C4 e2 C5 e
288
9.7
9 Combustion Processes
Immersed Flames
Consider an immersed diffusion flame formed by a jet of gaseous oxidizer issuing into liquid reagent (fuel); cf. Fig. 9.8. As a result of the gas–liquid interaction, namely its breakup into bubbles and atomization of liquid in the form of liquid droplets, a two-phase jet-like flow forms in the liquid medium. The general characteristics of combustion process in this situation, such as the intensity of heat release, completeness of combustion, etc., are determined to a considerable extent by the volumetric content of the gaseous phase mv : Depending on the value of mv , the immersed jet flow acquires a form of either bubbly or gasdroplet jets (Abramovich et al. 1984). At a high initial temperature of reactants (or due to the presence of an external igniter, chemical reaction between fuel and oxidizer begins. In consequence of heating and vaporization of liquid reagent due to the chemical reaction between fuel vapor and the oxidizer supplied as the gas jet, recognizable domains filled with vapor–oxidizer mixture containing droplets of reactive liquid (fuel) or bubbles filled by the oxidizer/fuel vapor mixture are formed in the liquid. A high temperature zone-a diffusion flame is formed in the two-phase mixture. In the case when combustion products are gaseous, the diffusion flame is located in an open cavity (Fig. 9.8a). On the other hand, when combustion products represent an immiscible liquid, the diffusion flame is located in a closed gas cavity (Fig. 9.8b). Its size depends not only on the physicochemical properties of reactants and the intensity of their mixing but also on the boiling temperature of combustion products that determine the position of the condensation zone. Sukhov and Yarin (1981, 1983) developed a theory of laminar immersed flames in the case when combustion products are gaseous. In this case, assuming that the rate of chemical reaction is infinite, the thermal conductivity and diffusion coefficients are constant and the effect of buoyancy force is negligible, the problem reduces to the following system of equations
a
b
x
x
1
f
1
f
2
2
0
y
0
y
Fig. 9.8 Sketch of the immersed flame. 1: Gaseous phase (a mixture of vapor of reactive liquid, gaseous oxidizer and combustion products). 2: Reactive liquid
9.7 Immersed Flames
289
@ui @ui k @ k @ui þ vi ¼ ni y y ui @x @y @y @y
(9.106)
@ @ ðui yk Þ þ ðvi yk Þ ¼ 0 @x @y
(9.107)
@Ti @Ti @ @Ti þ vi ¼ ai y k yk @x @y @y @y
(9.108)
@cj @cj k @ k @cj þ vi ¼ Dj y y ui @x @y @y @y
(9.109)
ui
X
cj ¼ 1
(9.110)
j
with the boundary conditions @ui @ca @cb @T1 ¼ 0; vi ¼ 0; ¼ ¼ 0; ca ¼ 0; ¼0 @y @y @y @y y ¼ yf ðxÞ ca ¼ cb ¼ 0; cc ¼ 1; T ¼ Tf @u1 @u2 ¼ r21 n21 ; cb ¼ 0; T1 ¼ T2 ¼ T y ¼ y ðxÞ u1 ¼ u2 ¼ u ; @y @y y ! 1 u2 ¼ 0; T2 ! T21
y¼0
(9.111) where x and y are longitudinal and transverse coordinates, u and v are the longitudinal and transverse velocity components, T and cj are the temperature and concentration, n; a and Dj are the kinematic viscosity, thermal and species diffusivities, respectively, k ¼ 0 or 1 for the plane and axisymmetric flows, subscripts f and correspond to the combustion front and the liquid–gas interface, subscript 1 and 2 refer to the gaseous and liquid phases, and j ¼ a; b and c correspond to the fuel, oxidizer and combustion products, respectively, r21 ¼ r2 =r1 and n21 ¼ n2 =n1 : The conditions (9.111) should be also supplemented with the Clausius– Clapeyron equation determining the equilibrium concentration of vapor at the liquid–gas interface, as well as by the mass and thermal balances at the flame and the interface. The Clausius–Clapeyron equation reads qe ca ¼ w exp Ra T
(9.112)
where ca is the vapor concentration at the interface, w is the pre-exponential factor, qe is the heat of evaporation.
290
9 Combustion Processes
In the framework of the boundary layer theory, the slope of the flame front to the flow axis is sufficiently small, so that ð@=@nÞf ð@=@yÞf ; with n being the normal to the flame front. Then assuming the Lewis number Le ¼ 1; we obtain the balance conditions at the flame front and the liquid–gas interface in the following form
@cb @y
þO f
@ca @y
¼0
@T @T q @ca þ ¼0 @y af @y bf rcP @y f v1 cc þ D k1
@T1 @y
@cc @y
(9.113)
f
(9.114)
¼0
@T2 @ca k2 þ r1 D ¼0 @y @y
(9.115)
r2 v2 r1 v1 ca þ r1 D
@ca ¼0 @y
(9.116)
(9.117)
where k and cP are the thermal conductivity and specific heat, respectively, and q is the heat of reaction. The conditions (9.111) should be supplemented with the integral invariants that are needed to obtain a non-trivial solution. Using the Schvab–Zel’dovich variable and integrating (9.106) and (9.109) across the immersed jet, we arrive at the following invariants 1 ð
yð
u21 yk dy
þ r21
u22 yk dy ¼ Ix ¼ const
(9.118)
y
0 yð
u1 byk dy ¼ Gx ¼ const
(9.119)
0
where b ¼ 1 þ Dc; Dc ¼ cb =O ca and r21 ¼ r2 =r1 . Introducing the dimensionless variables ui ¼ ui =u0 and y ¼ y=y0 , we transform (9.118) and (9.119) as follows 1 ð
yð
u2i yk dy 0
þ r21
u22 yk dy ¼ 1 y
(9.120)
9.7 Immersed Flames
291
Zy u1 byk dy ¼ 1
(9.121)
0
where u0 and y0 are the scales of velocity and length defined as u0 ¼ ðIx =Gx Þ and 1=ðkþ1Þ y0 ¼ G2x =Ix . The totality of the boundary conditions (9.111), balance equations (9.113–9.117) and the integral relations (9.120) and (9.121) fully determines the problem on the immersed diffusion torch. It allows finding the profiles of all characteristic parameters, values of temperature at the flame front, the temperature and concentrations at the interface surface, as well as the lengths and configurations of the flame front and gaseous cavity (Yarin and Sukhov 1987). However, instead of describing the theoretical solution of (9.106–9.110), following the approach of the present book, we focus our attention at the calculation of the length and configuration of the immersed diffusion flame using the dimensional analysis. For this aim, we use the approach developed in Vulis et al. (1968) and Vulis and Yarin (1978) for the investigation of the aerodynamics of diffusion flames. It consists in the calculation of the axial velocity and concentration and then determining the flow parameters corresponding to the flame front. In the framework of the model of an infinitely thin flame front (corresponding to the assumption on an infinitely large rate of chemical reaction), the length of the immersed flame is found from the following conditions: ca ¼ cb ¼ 0 at x ¼ lf , with subscript f corresponding to the tip of the torch. In order to determine the velocity in the immersed flame, we use the approach developed in Chap. 6 for the free laminar jets. We write the following functional equations for the local velocity u; the axial velocity um , and the thickness of gaseous cavity y ui ¼ fi ðum ; y; y Þ
(9.122)
um ¼ fm ðIx ; n1 ; xÞ
(9.123)
y ¼ f ðIx ; n1 ; xÞ
(9.124)
where subscripts i ¼ 1 and 2 correspond to the gaseous and liquid phases, respectively. Note that the kinematic viscosity of liquid n2 ; as well as densities of both phases r1 and r2 are not included in the sets of the governing parameters in (9.122–9.124) because they are accounted for in the expression for the kinematic momentum Ix : To transform (9.122–9.124) to the dimensionless form, we use the Lx Ly Lz MT system of units. The dimensions of the parameters involved in (9.122–9.124) in this system of units are as follows ½ui ¼ Lx T 1 ; ½um ¼ Lx T 1 ; ½Ix ¼ L2x Ly T 2 ; ½n1 ¼ L2y T 1
½ y ¼ Ly ; ½ x ¼ Lx (9.125)
292
9 Combustion Processes
It is seen that two governing parameters in (9.122) have independent dimensions, whereas the dimensions of the governing parameters in (9.123–9.124) are independent. Then, according to the Pi-theorem, (9.122–9.124) take the form ui ¼ um ci ð’Þ
(9.126)
um ¼ Ai xe
(9.127)
y ¼ Bxg
(9.128)
2 kþ1 kþ1 1=ð3kÞ 1=ð3kÞ where Ai ¼ ci u1 ; B ¼ c n2 d02 =Ix ; e ¼ ðk þ 1Þ= 0 Ix d0 =n1 ð3 kÞ, g ¼ 2=ð3 kÞ; ui ¼ u=u0 ; um ¼ um =u0 ; y ¼ y=y0 ; ’ ¼ y=y ; ci and c are constants. Equations (9.126–9.128) show that the system of PDEs determining the velocity distribution in the immersed laminar torch can be reduced to a system of ODEs. Therefore, the velocity can be expressed as a function cð’Þ of a single variable 0
F ð’Þ ci ð’Þ ¼ i k ’
(9.129)
where the function Fð’Þ should to satisfy the following conditions
0
F1 ’k
0
F1 ¼ 1; F1 ¼ 0 ’k 0 0 0 0 F1 F 0 0 ¼ r21 n21 2k ’ ¼ 1 F1 ¼ n21 F2 ; k ’ ’ 0 0 0 F2 F2 ’!1 ! 0; !0 k ’ ’k ’¼0
¼ 0;
(9.130)
Here primes denote differentiation with respect to ’; and n21 ¼ ðn2 =n1 Þ. Determine the concentration distribution. Since the physically realistic selfsimilar solution of (9.109) is absent under the condition ca ¼ const, we use the integral method of calculation distribution of b: Approximate the actual concentration profile by the series b¼
1 X
an ðxÞy
(9.131)
n¼0
where the coefficients an are found from the conditions at the flow axis and the interface surface
9.7 Immersed Flames
y¼0
293
db=dy ¼ 0;
b ¼ bm ;
y ¼ y ðxÞ b ¼ b
(9.132)
Taking into account three terms of the series (9.131), we arrive at the expression b ¼ bm ð1 ’2 Þ þ b ’2
(9.133)
Using (9.121) and (9.133), we find the dependence bm ðxÞ as n kþ1 o 1 bm ¼ x3k ðA1 Bkþ1 Þ b I1 ½F1 ð1Þ I1 1
(9.134)
Ð1 0 where I1 ¼ ’2 F1 ð’Þd’. 0
Bearing in mind that at the flame front bf ¼ 1, and that at the tip of the diffusion flame bm ¼ 1, we arrive at the following equations for the shape and length of the immersed torch yf ¼ Bx2=ð3kÞ
bm 1 bm b
1=2 (9.135)
n oð1kÞ=ð1þkÞ 1 lf ¼ ðA1 Bkþ1 Þ b I1 ½F1 ð1Þ I1 ð1 b Þðk3Þ=ðkþ1Þ
(9.136)
where lf is the length of the immersed flame, and subscript f corresponds to the flame front. The correlations (9.135) and (9.136) are qualitative, since they contain factors A and B that incorporate the unknown constants ci and c , as well as the vapor concentration at the interface surface b : The latter can be determined from the Clapeyron–Clausius equation for a given temperature at the interface T : The actual values of the factors Aand B are found from (9.106) and (9.107) after substituting into these equations the expressions (9.127–9.129) Ai ¼ ðI2 þ
r21 n221 I3 Þ2=ð3kÞ
Re 6 5k
B ¼ ðI2 þ r21 n221 I3 Þ1=ð3kÞ
where I2 ¼
R1 0
ð1þkÞ=ð3kÞ
Re1 6 5k
; A2 ¼ n21 A1
2=ð3kÞ (9.137)
R1 0 2 k 0 2 F1 =’k d’; I3 ¼ F2 =’ d’; and Re1 ¼ u0 y0 =n1 . 1
Accordingly, the expression (9.136) takes the form lf ¼
Re1 ðI2 þ r21 n221 I3 Þð1kÞ=ð1þkÞ ½F1 ð1Þ I1 ð1 b Þðk3Þ=kþ1Þ 6 5k
(9.138)
294
9 Combustion Processes
Fig. 9.9 The effect of an inert admixture of the length of plane (k ¼ 0) and axisymmetric (k ¼ 1) immersion flames
The effect of various parameters on the characteristics of the immersed flames is clearly visible through the dependence of its length on the reactants temperatures, the issue velocity, composition of gaseous oxidizer, etc. In particular, an increase in the reactants temperatures is accompanied by shortening of the immersed torch length. This results from an increase in vapor concentration the interface (a decrease in b Þ and an increase of the term I3 ð1 b Þ in (9.138). An opposite effect takes place at increasing the latent heat of evaporation: an increase in the value of qe leads to a significant growth of the torch length lf : As with the other types of diffusion flames, the characteristics of the immersed flames depend on the flow geometry. For example, the length of the plane (k ¼ 0) and axisymmetric (k ¼ 1) flame is inversely proportional, respectively, to the third and first powers of the factor ½F1 ð1Þ I3 ð1 b Þ which accounts for the effect of vapor concentration at the interface surface. It is emphasized that in both cases the length of the immersed flames is directly proportional to Re: A possible presence of an inert admixture in the gas jet also affects the flame characteristics (Sukhov and Yarin 1983, 1987). An increase in the content of an inert admixture in the gaseous oxidizer jet is accompanied by a significant decrease in the length of the axisymmetric and plane flames (cf. Fig. 9.9).
Problems P.9.1. Evaluate the burning time of a liquid fuel droplet at its combustion in a stagnant atmosphere that contains gaseous oxidizer. The process of droplet burning involves a number of simultaneously happening physical processes such as liquid vaporization, mixing of gaseous reagents and combustion of the resulting vapor–oxidizer mixture. These processes are also accompanied by heat and mass transfer, as well as by the diminishment of the droplet size and surface and the corresponding displacement of the reaction zone. Accordingly, a theoretical description of droplet combustion implies solving the coupled non-steady equations governing the mass, momentum, energy and species transfer in the liquid and gaseous phases (Yarin and Hetsroni 2004).The non-linear terms accounting for the heat release and species consumption involved in these equations make the theoretical analysis of the problem extremely difficult.
Problems
295
Therefore, as a rule, droplet burning is studied using models based on a number of simplifying assumptions: (1) the chemical reaction rate is infinite, (2) the droplet temperature is uniform, (3) the effects of buoyancy and radiant heat transfer are negligible, (4) the Lewis number equals one, (5) the physical properties of the liquid and gaseous phases are constant. The assumption (1) allows one to consider the reaction zone as an infinitesimally thin flame front which separates the flow field into two domains: the inner one (near the droplet surface) filled with a mixture of the fuel vapor and combustion products, and the outer one filled with a mixture of the oxidizer and combustion products. Also, in this case the vapor and oxidizer concentrations at the flame front are equal to zero. The other assumptions make it possible to use the lumped capacitance heat transfer model for the droplet, as well as to consider a spherically-symmetric flame (for the burning in a stagnant atmosphere). The flame temperature equals the adiabatic combustion temperature in this case. Moreover, the droplet surface temperature changes only slightly during the combustion process and can be taken as a constant equal to the boiling temperature of liquid fuel. Based on the above-mentioned simplifications, assume that during the combustion process the droplet diameter d depends on: (1) densities of liquid fuel and gaseous oxidizer, (2) species diffusivity (assumed being identical for all the components), (3) total enthalpy of fuel, (4) latent heat of evaporation, (5) droplet initial diameter, and (6) time d ¼ f ðr1 ; r2 ; D; qt ; qe ; d0 ; tÞ
(P.9.1)
Here r1 and r2 are the density of gaseous and liquid phases, respectively, D is the diffusivity, qt ¼ c01 q=O þ cP ðT1 Ts Þis the total enthalpy of the fuel with q being the heat of reaction, c01 the oxidizer concentration in the surrounding medium, O the stoichiometric oxidizer-to-fuel mass ratio, qe the latent heat of evaporation, cP the specific heat of gaseous phase, T1 and Ts being the ambient and saturated temperature, respectively; d0 is the initial droplet diameter, and t is time. The dimensions of the governing parameters involved are ½r1 ¼ L3 M;
½r2 ¼ L3 M;
½qe ¼ JM1 ;
½d0 ¼ L;
½D ¼ L2 T 1 ; ½t ¼ T
½qt ¼ JM1 ; (P.9.2)
Five of the seven governing parameters possess independent dimensions. Then, according to the Pi-theorem, the number of dimensionless groups of the present problem is equal two, and (P.9.1) reduces to the following dimensionless equation P ¼ ’ðP1 ; P2 Þ
(P.9.3)
where P ¼ d=d0 ; P1 ¼ ðr1 =r2 Þ Dt=d02 and P2 ¼ qt =qe ¼ B (B is the Spalding transfer number.
296
9 Combustion Processes
Droplet is burnt completely at the moment t ¼ tb when d ¼ 0. At that moment (P.9.3) yields ’ðP1b ; P2 Þ ¼ 0; where P1b ¼ ðr1 =r2 Þ Dtb =d02 : Solving the latter equation for P1b , we obtain the following expression for the droplet burning time tb ¼
r2 d02 cðBÞ r1 D
(P.9.4)
It is emphasized that the analytical solution of the problem yields the following expression for tb tb ¼
r2 d02 ½lnð1 þ BÞ1 r1 8D
(P.9.5)
References Abramovich GN (1963) The theory of turbulent jets. MTI Press, Cambridge Abramovich GN, Girshovich TA, Krasheninnikov SY, Sekundov AN, Smirnova IP (1984) Theory of turbulent jets. Nauka, Moscow (in Russian) Burke SP, Schumann TE (1928) Diffusion flames. Ind Eng Chem 20:998–1004 Frank-Kamenetskii DA (1969) Diffusion and heat transfer in chemical kinetics, 2nd edn. Plenum Press, New York Hottel HC, Hawthorne WR (1949) Diffusion in laminar flame jets. In: Proceeding of third symposium on combustion and flame and explosion phenomena, Williams & Wilkins, Baltimore, pp 254–266 Khitrin LN (1957) The physics of combustion. Moscow University, Moscow (in Russian) Merzhanov AG, Khaikin BI (1992) Theory of combustion waves in homogeneous media. AN SSSR, Chernogolovka (in Russian) Merzhanov AG, Khaikin BI, Shkadinskii KG (1969) The establishment of a steady-state regime of flame propagation after gas ignition by an overheated surface. Prikl Mech Tech Phys 5:42–48 Schvab BA (1948). A relation between temperature and velocity fields in gaseous flame. In: The investigation of the process of fossil fuel combustion, Gosenergoizdat, Moscow-Leningrad, pp 231–248 (in Russian) Semenov NN (1935) Chemical kinetics and chain reactions. Oxford University Press, Oxford Spalding DB (1953) Theoretical aspects of flame stabilization: an approximate graphical method for the flame speed of mixed gases. Aircraft Eng 25:264–276 Sukhov GS, Yarin LP (1981) Combustion of a jet of immiscible fluids. Combust. Explos. Shock. Waves 17:146–151 Sukhov GS, Yarin LP (1983) Calculating the characteristics of immersion burning. Combust. Explos. Shock waves 19:155–158 Vulis LA (1961) Thermal regime of combustion. McGraw-Hill, New York Vulis LA, Yarin LP (1978) Aerodynamics of a torch. Energia, Leningrad (in Russian) Vulis LA, Ershin SA, Yarin LP (1968) Foundations of the theory of gas torches. Energia, Leningrad (in Russian) Williams FA (1985) Combustion theory, 2nd edn. Benjamin-Cummings, Menlo Park Yarin LP, Hetsroni G (2004) Combustion of two-phase reactive media. Springer, Berlin Yarin LP, Sukhov GS (1987) Foundations of combustion theory of two-phase media. Energoatomizdat, Leningrad (in Russian) Zel’dovich YB (1948) Toward a theory of non-premixed gas combustion. J Tech Phys 19:199–210 Zel’dovich YB, Barenblatt GI, Librovich VB, Makhviladze GM (1985) Mathematical theory of combustion and explosion. Plenum Press, New York
Author Index
A Abrahamsson, H., 259 Abramovich, G.N., 131, 156, 232, 237, 238, 245, 258, 281, 288, 296 Acrivos, A., 169, 175, 209 Adamson, T.C., 102 Adler, M., 119, 120, 129 Adrian, R.J., 102 Akatnov, N.I., 146, 148, 156 Alhama, F., 7, 38 Andrade, E.N., 143, 156 Andreopoulos, J., 245, 258 Antonia, R.A., 228, 229, 238, 258, 259 Anton, T.R., 80. 101 Armstrong, R.C., 129 Astarita, G., 113, 129
B Baehr, H.D., 39, 69, 201, 209 Bagananoff, D., 157 Bahrami, M., 113, 129 Baines, W.D., 154, 156, 245, 259 Balachander, R., 260 Banks, R.B., 258 Banks, W.H.H., 170, 209 Barenblatt, G.I., 7, 23, 37, 217, 258, 296 Barua, S.N., 119, 129 Basset, A.B., 20, 101 Batchelor, G.K., 54, 69, 71, 101, 149, 153, 156 Bayazitoglu, Y., 139 Bayley, F.J., 179, 209 Bearman, P.W., 76, 81, 101 Berger, S.A., 118, 129 Bergstorm, D.J., 249, 258, 260 Berlemont, A., 80, 101 Bernulli, D., 10
Bigler, R.W., 238, 258 Bird, R.B., 113, 129 Blackman, D.R., 4, 37 Blasius, H., 47, 69 Bloonfield, L.J., 154, 156 Boothroyt, R.G., 84, 101 Boussinesq, J., 80, 101 Bradbury, L.I.S., 238, 243, 244, 258 Brenner, H., 71, 102 Bridgmen, P.W., 11, 23, 37, 87, 101, 166, 167, 209 Britter, R.E., 63, 69 Buchanan, H.J., 75, 102 Buckingham, E., 23, 37 Burke, S.P., 271, 296
C Campbelle, I.H., 156 Carpenter, L.H., 81, 102 Celata, G.P., 112, 129 Champagne, F.H., 147, 149, 157 Chandrasekhara, D.V., 258 Chang, E.J., 80, 101 Chao, B.T., 170. 209 Chassaing, P., 245, 258 Chen, A.M.L., 113, 114, 116, 130 Chen, C.S., 105, 129 Cheslak, F.R., 257, 259 Chevray, R., 228, 229, 259 Cho, Y.I., 209 Chua, L.P., 226, 229, 259 Claria, A., 258 Clauser, F.H., 222, 259 Clift, R., 71, 78, 101 Coles, D., 220, 259 Corrsin, S., 228, 229, 259
L.P. Yarin, The Pi-Theorem, Experimental Fluid Mechanics, DOI 10.1007/978-3-642-19565-5, # Springer-Verlag Berlin Heidelberg 2012
297
298 Crawford, M.E., 39, 69, 183, 209 Culham, J., 129
D Dandy, D.S., 80, 101 Dean, W.R., 118, 129 Derjagin, B.M., 96, 101 Desjanquers, P., 101 De Witt, K.J., 209 Desjonquers, P., 101 Didden, N., 63.69 Dimotakis, P.E., 236, 259 Divoky, D., 228, 259 Dodson, D.S., 102 Dorfman, L.A., 170, 209 Dorodnizin, A.A., 197, 209 Douglas, J.F., 7, 38, 45, 69 Doweling, D.R., 236, 259 Dowirie, M.J., 101 Du, D.X., 130 Dunkan, A.B., 112, 129 Dwyer, H.A., 80, 82, 101, 102
E Eastor, T.D., 171, 209 Emery, A.E., 105, 129 Eriksson, J., 259 Ershin, S.A., 296 Eskinazi, S., 259 Everitt, K.M., 238, 243–245, 259
F Fargie, D., 105, 129 Fendell, F.E., 82, 102 Flakner, V.M., 68, 69 Forstall, W., 229, 259 Frankel, N.A., 175, 209 Frank-Kamenetskii, D.A., 263, 264, 266, 268, 270, 271, 296 Fric, T.F., 245, 259 Friedman, M., 105, 129 Fujii, T., 201, 209
G Gad-el-Hak, M., 112, 129 Garimella, S., 112, 129 Gaylord, E.M., 229, 259 George, J., 249, 258 George, W.K., 249, 259
Author Index Germano, M., 120, 129 Gilis, J., 129 Girshovich, T.A., 258, 296 Glauert, M.B., 146, 148, 157 Gloss, D., 130, 259 Couesbet, G., 101 Grace, J.R., 101 Graham, J.M.R., 76, 81, 101, 102 Greif, R., 170, 209 Gua, Z.Y., 130 Gutmark, E., 133, 157, 233, 234, 253, 254, 259
H Hadamard, J.S., 85, 102 Hagen, G., 103, 129 Hamel, G., 43, 69 Hamilton, W.S., 80, 102 Happel, J., 71, 102 Hartnett, J.P., 209 Hassager, O., 129 Hasselbrink, E.F., 245, 246, 259 Hausner, O., 112, 130 Hawkis, G.A., 30, 38 Hawthorne, W.R., 284, 296 Herwig, H., 110, 112, 129, 130, 224, 259 Hetsroni, G., 71, 102, 112, 113, 130, 294, 296 He, Y.-L., 130 Hinze, J.O., 28, 38, 131, 157, 217, 229, 259 Ho, C.-M., 112, 130, 133, 157 Hollands, K.G.T., 187, 209 Hottel, H.C., 284, 296 Hoult, D.P., 60, 69 Howarth, L., 215, 259 Huntley, H.E., 7, 38, 45, 69, 88, 102 Huppert, H.E., 60, 61, 63, 69 Hussain, F., 133, 157 Hussain, H.S., 133, 157 Hussaini, M.Y., 171, 209
I Illigworth, C.R., 197, 209 Incorpera. F.P., 112, 130 Ipsen. D.C., 4, 38 Ito. H., 119, 120, 130
J Jaluria, Y., 149, 157 Jeng, D.R., 209 Jones, J.B., 30, 38 Jonicka, J., 260
Author Index K Kakas, S., 112, 130 Kandlicar, S.G., 110, 130 Karamcheti, K., 157 Karanfilian, S.K., 80, 102 Karlsson, R.I., 250, 259 Karman, Th., 55, 69, 215, 221, 225, 226, 259 Karthpalli, A., 133, 157 Kashkarov, V.P., 54, 70, 131, 133, 147, 157, 228, 260 Kassoy, D.R., 82, 102 Kaviany, M., 159, 209 Kays, W.M., 39, 69, 159, 183, 209 Keffer, J.F., 245, 259 Kelso, R.M., 245, 259 Kenlegan, G.H., 81, 102 Kerr, R.C., 154, 156 Kestin, J., 9, 30, 38, 172, 209 Khaikin, B.I., 271, 296 Khitrin, L.N., 282, 296 Kolmogorov, A.N., 211, 259 Kompaneyets, A.S., 162, 210 Konsovinous, N.S., 136, 157 Kotas, T.J., 80, 102 Krashenninikov, S.Yu., 258, 296 Kreith, F., 169, 170, 209 Kuta teladse, S.S., 18, 38, 158, 209
L Lamb, H., 83, 90, 102 Landau, L.D., 37, 38, 52, 53, 54, 69, 71, 94, 95, 98, 102, 103, 130, 133, 157, 159, 173, 209, 216, 219, 259 Launder, B.F., 249, 259 Lavender, W.J., 172, 209 Lawerence, C.J., 102 Lee, M.H., 170, 209 Levich, V.G., 29, 38, 65, 69. 94, 95, 102, 159, 177, 189, 193, 194, 207–209 Levi, S.M., 96, 101 Librovich, V.B., 296 Li, D., 130 Lifshitz, E.M., 37, 38, 54, 69, 71, 98, 102, 103, 130, 133, 157, 159, 173, 209, 216, 219, 259 Lim, T.T., 259 Liron, N., 129 List, E.J., 154, 157 Li, Z., 113, 130 Li, Z.X., 113, 139 Lockwood, F.C., 234, 235, 259
299 Lofdahl, L., 259 Loitsyanskii, L.G., 22, 28, 38, 42, 43, 69, 75, 101–103, 105, 112, 120. 122, 123, 130, 195,,197, 209, 269 London, A.L., 111, 130 Lumley, J.L., 232, 239–241, 260 Lykov, A.M., 18, 38
M Maczynski, J.F.J., 238, 259 Madrid, C.N., 7, 38 Ma, H.B., 111, 130 Makhviladze, G.M., 296 Mala, G.M., 130 Marrucci, G., 113, 129 Martin, B.W., 105, 129 Maxey, H.R., 80, 101, 102 Maxworthy, T., 63, 69 Mayer, E., 228, 259 Mc Laughlin, J.B., 80, 102 Mei, R., 80, 102 Merzhanov, A.G., 271, 296 Messiter, L.F., 102 Mikhailov Yu,A., 18, 38 Maneib, H.A., 234, 235, 259 Monin, A.S., 217, 220, 260 Moody, L.F., 111, 130 Mori, Y., 119, 130 Morton, B.R., 149, 153, 154, 157 Mosyak, A., 130 Moussa, Z.M., 245, 259 Mungal, M.G., 245, 246, 259, 260
N Nakayama, W., 119, 130 Narasimha, R., 251, 260 Narasyan, K.Y., 260 Nicholles, J.A., 259 Nickels, T.B., 238, 260 Nikuradse, J., 109, 129, 130 Nusselt, W., 200, 209
O Obasaju, E.D., 101 Obukhov, A.M., 211, 213, 260 Odar, F., 80, 102 Oseen, C.W., 80, 102 Owen, J.N., 209
300 P Pai, S.I., 131, 157 Panchapakesan, N.R., 232, 239–241, 260 Papanicolaou, P.N., 154, 157 Parthasarthy, S.R., 260 Pei, D.C.T., 172, 209 Perry, A.E., 238, 259, 260 Persson, J., 259 Peterson, G.P., 111, 112, 129, 130 Pfund, D., 113, 130 Plam, B., 112, 130 Pogrebnyak, E., 130 Pohlhausen, E., 63, 70, 192 Poiseuille, J., 103, 130 Pope, S.P., 215, 260 Prabhu, A., 258 Prandtl, L., 218, 225, 228. 260
Q Qu, W., 113, 130
R Raithby, C.A., 187, 209 Raizer, G.P., 162, 164, 209 Ramaswamy, G.S. 4, 38 Rao, V.V.L., 4, 38 Rayleigh, L., 165, 209 Raynolds, O., 35, 38 Rector, D., 130 Riabouchinsky, D., 166, 209 Riley, J.J., 80, 102, 238, 243, 244, 258 Robins, A.G., 238, 243–245, 259 Rodi, W., 245, 249, 258, 259 Rohsenow, W.M., 159, 269 Rosenhead, L., 43, 70 Roshko, A., 245, 259 Rotta, J.C., 217, 245, 260 Rybczynskii, W., 85, 102
S Saffman, P.S., 79, 93, 102 Sakipov, Z.B., 229, 260 Sananes, F., 258 Sasty, M.S., 171, 209 Schiller, L., 104, 129, 130 Schlichting, H., 39, 52, 58, 64, 67, 70, 75, 102–104, 107, 110, 111, 131, 157, 159, 193, 197, 209, 217, 223, 260
Author Index Schneider, W., 136, 157 Schumann, T.E., 271, 296 Schvab, B.A., 264, 296 Sedov, L.I., 6, 10, 15, 23, 38, 43, 54, 55, 70, 71, 83, 87, 102, 166, 209, 217 Sekundov, A.N., 258, 296 Semenov, N.N., 266, 296 Sforzat, P., 133, 157 Shah, R.K., 111, 130 Sherman, F.S., 55, 70 Shekarriz, A., 130 Shih, C.C., 75, 102, 232 Shin, T.-H., 232, 260 Shkadinskii, K.G., 296 Sichel, M., 259 Simpson, J.E., 60, 70 Skan, S.W., 68, 69 Smirnova, I.P., 258, 296 Smith, S.H., 245, 260 Sobhan, C., 112, 129 Soo, S.L., 71, 102, 169, 209 Spolding, D.B., 159, 209, 271, 296 Sprankle, M.L., 102 Spurk, J.H., 23, 38 Stephan, K., 39, 69, 201, 209 Stephenson, S.E., 258 Stewardson, K., 197, 209 Stokes, G.C., 44, 70, 87, 102 Sukhov, G.S., 280, 288, 291, 294, 296
T Tabol, L., 129 Tachie, M.F., 249–251, 258, 260 Tai, Y.-C., 112, 130 Tang, G.-H., 130 Tao, W.-Q., 130 Taylor, G.I., 157, 225, 260 Taylor, T.D., 169, 209 Temirbaev, D.Z., 229, 260 Tieng, S.M., 189, 210 Towendsend, A.A., 131, 157, 238, 242, 260 Trentacoste, N., 133, 157 Trischka, J.W., 259 Turner, J.S., 149, 154, 156, 157 Turner, A.B., 209 Tutu, N.K., 228, 229, 259
U Uberoi, M.S., 228, 229, 259
Author Index V Vasiliev, L.L., 130 Vulis, L.A., 54, 70, 131, 133, 147, 157, 228, 260, 266, 271, 274, 280, 281, 291, 296 Van Dyke, M., 120, 129
W Wang, H.L., 113, 130 Wang, Y., 113, 130 Ward-Smith, A.C., 103, 111, 125, 130 Weast, R.C., 38 Weber, M.E., 101 Weiss, D.A., 58, 59, 70 Wenterodt, T., 130, 259 White, C.M., 119, 120, 130 White, F.M., 111, 130, 159, 210 Wilkinson, W.L., 113, 114, 116, 130 Williams, F.A., 271, 282, 296 Williams, W., 6, 38
301 Wolfshtein, M., 259 Wosnik, M., 259 Wygnanscki, I., 147, 149, 157, 233, 234, 259
Y Yaglom, A.M., 217, 220, 260 Yalin, M.S., 90, 92, 93, 102 Yan, A.S., 189, 210 Yao, L.-S., 129 Yarin, A.L., 58, 59, 70 Yarin, L.P., 71, 102, 110, 112, 113, 129, 130, 280, 281, 288, 291, 294, 296 Yenter, Y., 130 Yavanovich, M.M., 129
Z Zel’dovich Ya, B., 149, 153, 157, 162, 164, 209, 263, 264, 266, 268, 271, 272, 275, 282, 296
.
Subject Index
A Acceleration effect, 80–81 Activation energy, 263, 280 Applying the II-theorem to transform PDE into ODE, 63 Archimedes number, 19 Arrhenius law, 261, 263–264, 267, 270
B Bessel function, 123, 272 Biot number, 19 Bond number, 19 Brinkman number, 14, 19 Buoyant jet, 149–154
C Capillary number, 19, 95 Capillary waves in liquid lamella after a weak drop impact onto a thin liquid film, 58–60 Clausius–Clapeyron equation, 289, 293 Co-flowing turbulent jets, 238–239, 242–244 Combustion of non-premixed gases, 264, 271–274, 278 Combustion waves, 268–271 Continuity equation, 7, 13, 31–34, 49, 52, 56, 67, 68, 71, 77, 85, 104, 134, 168, 239 Convective heat and mass transfer, 2, 160, 166 Couette flow, 178–179 Critical conditions, 268
D Damkohler number, 17, 19, 21 Darcy number, 19
Dean number, 19, 118–120 Deborah number, 19 Delta function, 160 Diffusion boundary layer over a flat reactive plate, 65–67 Diffusion flame, 273–281, 284, 288, 291, 293, 294 Dimensional and dimensionless parameters, 3–7 Dimensionless groups, 1, 3, 9, 13, 18–22, 26, 29, 40, 45, 48, 51, 53, 73, 81, 84, 87, 90–91, 110, 111, 115, 117, 118, 120, 122, 124, 139–141, 167, 179, 188, 189, 196, 202, 207, 220, 242, 249, 255, 256, 270, 273, 295 Dorodnitsyn–Illingworth–Stewardson transformation, 197, 276, 278 Drag of a body partially dipped in liquid, 87 of a deformable particle, 84–86 on a flat plate, 73–76 force, 1, 4, 7, 23–28, 71–101, 145 of an irregular particle, 82–84 on a solid particle, 76–82 of a spherical particle at low, moderate and high Reynolds number, 1, 76–79
E Eckert number, 19, 179 Eddy viscosity and thermal conductivity, 224–229 Effect of energy dissipation, 112 of the free-stream turbulence, 1, 81, 171–173 of particle acceleration, 80
L.P. Yarin, The Pi-Theorem, Experimental Fluid Mechanics, DOI 10.1007/978-3-642-19565-5, # Springer-Verlag Berlin Heidelberg 2012
303
304 Effect (cont.) of particle-fluid temperature difference, 1, 82 of particle rotation, 1, 79, 169–171 of velocity gradient, 174–175 Ekman number, 19 Enthalpy, 5, 16, 17, 134, 135, 184, 195, 196, 237, 261–263, 278, 280, 295 Entrance flow regime, 106–108 region of pipe, 106, 107, 180–181 Euler number, 14, 17, 19, 22
F Flow in curved pipes, 116–120 in irregular pipes, 111–112 over a plane wall which has instantaneously started moving from rest, 44–47 in straight rough pipes, 1 Fourier number, 122 Frank-Kamenetskii approximation, 263, 264 parameter, 266, 267 Freezing of a pure liquid, 202–205 Froude number, 14, 17, 73, 86, 87, 256 Fully developed flow in rough pipes, 109–111 in smooth pipes, 109
G Gas torches, 2, 280–287 Grashof number, 19, 170, 186, 188–190, 193
H Hadamard–Rybczinskii formula, 85 Heat of reaction, 265, 267, 279, 290, 295 release, 19, 21, 66, 162, 261, 265, 266, 268, 269, 288, 294 Heat transfer accompanying condensation of saturated vapor on a vertical wall, 199–201 in channel and pipe flows, 2, 178–183 under the conditions of phase change, 160 from a flat plate in a uniform stream of viscous high-speed perfect gas, 195–199 in forced convection, 165–178
Subject Index from a hot particle immersed in fluid flow, 2, 165–169 in mixed convection, 2, 187–188 in natural convection, 186–194 from a spherical particle, 186–187 from a spinning particle, 170, 187–188 from a vertical hot wall, 190–193
I Ideally stirred reactor, 265 Immersed flame, 288–294 Impinging turbulent jet, 252–254 Inhomogeneous turbulent jet, 232–238
J Jacob number, 19 Jet flow, 43, 51, 131–156, 160, 225, 228, 229, 231, 245, 248, 287, 288
K Knudsen number, 19, 112, 113 Kutateladze number, 19
L Laminar boundary layer over a flat plate, 14, 47–51 flow near a rotating disk, 55–58 flows in channels and pipes, 103–129 jets issuing from a thin pipe, 134 submerged jet, 51–54, 131, 141–143, 154–156, 185 wake of a solid body, 143–146 Lewis number, 18, 19, 262, 263, 269, 290, 295
M Mach number, 19, 275, 278, 279 Mass diffusivity coefficient, 262 flux, 23, 190, 194, 206, 208, 235, 247, 271, 285 transfer from a spherical particle in natural and mixed convection, 189–190 transfer in forced convection, 165–178 transfer to a vertical reactive plate in natural convection, 193–194 transfer to solid particles and drops immersed in fluid flow, 176–178
Subject Index Micro-channel flows, 112–113 Microkinetic law, 262 Mixing length, 153, 225, 226
N Navier–Stokes equations, 7, 13, 27, 28, 31, 32, 34, 39, 52, 56, 67, 71, 77, 96, 104, 105, 112, 118, 120, 124, 136, 168, 261 Newton’s law, 28, 79 Nondimensionalization of the governing equations, 1, 16 Non-Newtonian fluid flows, 1, 18, 113–116 Nusselt number, 15, 20, 166, 168–172, 175, 179, 181–183, 186–189, 193, 199
O Oscillatory motion, 75–76
P Peclet number, 14, 17, 20, 21, 166, 168, 169, 175–178, 190, 207, 274 Plane jet, 137, 147, 183, 250, 252 Prandtl number, 18, 20, 65, 82, 151, 168–170, 172–174, 179, 180, 182, 183, 186–188, 192, 193, 225, 228–229 Pre-exponential, 280, 289 Propagation of viscous-gravity currents over a solid horizontal surface, 60–63
R Rate of conversion, 262 Rayleigh number, 20, 187 Reynolds number, 1, 2, 14, 17, 18, 20–22, 25, 27, 28, 34, 72–84, 86, 88, 89, 91, 96, 97, 103, 105, 107, 110–111, 113, 115, 117, 118, 120, 124, 128, 142, 143, 153, 155, 168–170, 172–176, 180, 182, 188, 189, 211, 213, 219 Richardson number, 20 Rossby number, 20
S Schmidt number, 18, 20, 30, 67, 194, 228, 229 Schvab–Zel’dovich transformation, 264, 272 Sedimentation, 1, 90–93 Self-similar solution, 39, 41–43, 58, 63, 67, 149, 152, 191, 201, 204, 205, 242, 249, 276, 279, 286
305 Semenov number, Shear stress, 33, 36, 45, 50, 74, 77, 78, 88, 113, 114, 217, 221, 231, 234, 285 Sherwood number, 20, 176–178, 190, 207 Similarity, 1, 3, 11, 21–22, 211, 226, 245, 281 Single-one-step chemical reaction, 262 Spalding transfer number, 296 Stoichiometric coefficient, 264, 276 Stokes equations, 97 Strouhal number, 14, 17, 20, 75
T Taylor number, 20 Temperature field, 150, 151, 153, 160–165, 183, 184, 191, 204, 205, 265 induced by a plane instantaneous thermal source, 160–161 induced by a pointwise instantaneous thermal source, 161–162 Terminal velocity of small heavy spherical particle in viscous liquid, 87–90 Thermal boundary layer over a flat plate, 63–65 characteristics of laminar jets, 183–185 diffusivity coefficient, 162, 163, 205, 262 explosion, 2, 263–268 particle in viscous liquid, 87–90 Thin liquid film on a plane withdrawn from a pool filled with viscous liquid, 93–96 Total enthalpy, 263, 295 Total momentum flux, 52, 54, 133, 135, 136, 143, 146, 227, 229, 235, 241, 252, 256, 258, 285 Transfer in turbulent jet, 224 Turbulent jet, 2, 135, 147, 153, 215, 224–254 wall jet, 248–251 Two-phase flow, 19, 71, 93
U Unsteady flows in straight pipes, 120–123
V Van-der Waals equation, 30–31
W Wall jet over plane and curved surfaces, 146–149 Weber number, 20, 84, 100, 256
