The Pi-Theorem: Applications to Fluid Mechanics and Heat and Mass Transfer

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, 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 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 L
1 (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 L
1 y T , the dimension of viscosity is expressed as
1
1 ½m ¼¼ L
1 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 L
1 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 L
1 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 L
1 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 x
1 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 xA
1=5 t
3ðaþ1Þ=5 ; and c ¼ 1=c. Substitution of the expression (3.126) into (3.129) yields 0

h ¼ c A
1=5 tð2a
1Þ=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 L
3 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> 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 Re0

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)

162

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)

164

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 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.

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260

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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 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 cj1 ; 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 > 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