Flows of Reactive Fluids

FLUID MECHANICS AND ITS APPLICATIONSVolume 94

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1 C

Roger Prud’homme

Flows of Reactive Fluids

Roger Prud’hommeInstitut Jean Le Rond d’AlembertUniversité Pierre et Marie Curie and CNRS75252 Paris Cedex 05Franceroger.prud_homme@upmc.fr

ISSN 0926-5112ISBN 978-0-8176-4518-2 e-ISBN 978-0-8176-4659-2DOI 10.1007/978-0-8176-4659-2Springer New York Dordrecht Heidelberg London

Library of Congress Control Number: 2010930226

© Springer Science+Business Media, LLC 2010All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connec-tion with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden.The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.

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Mathematics Subject Classification (2010): 76Fxx, 76G25, 76H05, 76J20, 76L05, 76Nxx, 76Txx, 76Vxx, 80Axx, 80A25, 80A32

Springer is part of Springer Science+Business Media (www.springer.com)

To Hayat.

In memory of Marcel and Simone Barrere.

Preface

Reactive fluids are present in many situations of great importance, such as incombustion chambers or around spacecraft re-entering the atmosphere. An-alyzing the flow properties of such fluids represents one of the most difficultchallenges to current technology. Indeed, all of the most difficult aspects offluid mechanics appear to be grouped together in this research field! Suchfluids are complex mixtures with compositions that vary rapidly in time andspace. They are not usually at thermodynamic equilibrium, since the reactiontimes of the chemical reactions involved may not be negligible in comparisonwith the transit time of the fluid. However, the author of this book limits itsscope to typical phenomena that are not very far from local equilibrium butcan nevertheless exhibit the most important types of irreversible processes.The production of entropy is highly dependent on the chemical reaction path-way, which is difficult to simplify. Also, most of the classical problems thatcharacterize fluid mechanics—such as turbulence, the presence of thin bound-ary layers or shear layers, and the propagation of acoustic waves and shockwaves—are also present, and are much more difficult to analyze and describethan they are for homogeneous fluids, because reactive mixtures interact withthese phenomena. For example, density is highly dependent on the chemicalpathway since it is determined by the local and instantaneous production ofchemical species, and so its value affects many other quantities through theequation of state and the balances of mass, momentum, and energy.

This book is a remarkable and quite pedagogical synthesis. It presents allof these problems in a logical and systematic way, step by step in the differ-ent chapters, without going into the complexities of the very many particularclasses of them. Indeed, the author pays more attention to general conceptsand to guiding ideas (thus justifying the formulation of a very general the-ory) than to combining them for particular applications. To be able to followthe text, the reader should understand mathematics to the level requiredfor most graduate courses in fluid mechanics. This involves a knowledge ofclassical techniques such as multiple-scale analysis and matched asymptoticexpansions, but without the need to dwell on their mathematical justification.

VIII Preface

Dimensional analysis is proposed as a systematic and powerful tool for reduc-ing the general set of equations to the relevant formulation for a given classof phenomena, and a list of the most important nondimensional numbers isgiven.

It was a true pleasure to look through this text, noting its good organiza-tion and reading some of the chapters and paragraphs in depth. I am convincedthat readers of the book—it is aimed at graduate students as well as experi-enced scientists or engineers—will find an abundance of useful resources, andsome will definitely keep it on their desk.

Rene MoreauProfesseur Emerite a Grenoble INP

Acknowledgments

I am indebted to all the people who have, in several ways, contributed tothe final realization of this book: Professors Marcel Barrere, Paul Germainand Luigi Napolitano, who greatly contributed to my training in continuummechanics and reactive fluids; Professor Nicola Bellomo for having proposedthis book for publication by Springer/Birkhauser; Tom Grasso, Editor atBirkhauser Boston, who managed efficiently the realization of this book; andProfessor Rene Moreau, who wrote the preface for this book and suggestedpublication in the Fluid Mechanics and its Applications (FMIA) series.

Early versions of the manuscript were critically read by Jean-SylvestreDarrozes, Renee Gatignol, Paul Kuentzmann, Gerard Maugin and PierreSagaut, who made pertinent suggestions. I am also indebted to colleagues fromthe Universite Pierre et Marie Curie (UPMC), the National Center for Scien-tific Research (CNRS), and the French Aerospace Lab (ONERA), includingNicolas Bertier, Francis Dupoirieux, Guillaume Legros, Lionel Matuszewski,Yves Mauriot, and Angela Vincenti, who kindly agreed to read some of thechapters in order to improve the English scientific terminology, as well asCedric Croizet, Daniel Fruman and Denis Sebart, who helped in other ways.

I extend my appreciation to people who have granted me permission toreprint illustrations and to Christian Peak, for his copyediting work and Eng-lish revisions to the final edition. My thanks also go to the directors of theInstitute Jean le Rond d’Alembert (CNRS/UPMC) and ONERA/DEFA fortheir encouragement.

Roger Prud’homme

List of Symbols

Latin Symbols

a, b: van der Waals coefficientsaj: activity of species jA: chemical affinityA: adiabatic transformationAl: chemical elementB: Spalding parameter, or Arrhenius coefficientB, C, C′: virial coefficientsBkj : virial coefficients of a mixtureBo: Bond numberc: velocity of sound, or molecular velocityC: total number of moles per unit volumeCf : friction coefficientCj : molar concentration per unit volumeCp, Cv: specific heat at constant pressure or volume, respectivelyCr: crispation numberd: droplet diameter, molecular diameter, or distanceD: diffusion coefficientD: strain rate tensorDa: first Damkohler parameterDT : thermal diffusion coefficiente: internal energy per unit massE: internal energyE : equilibrium stateEa: activation energyE(k): energy spectrum of turbulence

X List of Symbols

E(te): distribution of residence timesEj: chemical speciesf : undefined quantity, Helmholtz free energy per unit mass, or Blasius

functionf : force acting on a unit massf j : force acting on a unit mass of species jF : Helmholtz free energy, or generalized forceF: forceFr: Froude numberg: Gibbs free enthalpy per unit mass, or the common value of λ/cpf and

ρD used in the study of droplet combustion when the Lewis number Le isequal to one

g(ξ, ζ): function of space variables in the Emmons problemg(U): function of the reduced velocity in the study of the planar detonation

wave structureg: shear-force vectorG: Gibbs free enthalpy, reduced gradient of stagnation temperature in a

detonation wave, filter function for wavenumbers, or production rate of entitiesper unit volume

GC : concentration gradientGr: Grashov numberGT : temperature gradienth: enthalpy per unit massH : enthalpyHi Hickman number1: unit tensorJ : generalized fluxJ: total mass fluxJj : flux of species jJ Dj : diffusion flux of species jk: Boltzmann constant, bulk viscosity, wavenumber, or kinetic energy per

unit massK: kinetic energy, compressibility, number of chemical reactions in a mix-

ture, or wavenumberkc: cut-off wavenumberk(T ): specific reaction rateKa, KC , Kp: equilibrium constants for activity, concentration, partial

pressure, respectivelyKF : coefficient of turbulent exchange for quantity F

List of Symbols XI

l: latent heat per unit mass, or mean free path�: integral scale of turbulence�f : laminar premixed flame thickness�K : microscale of Kolmogorov�δ: reaction thickness in a laminar premixed flameL: length, liquid, molar latent heat, number of chemical elements in a

mixture, or crystal sizeLe: Lewis number, or phenomenological coefficientsLp: Prandtl mixing lengthm: total massM : molecular massM: mean molar massM: bending stress tensormj: mass of species jMj: molar mass of species jm: mass flow rateMa: Benard–Marangoni numbern: total number of molesnj: number of moles of species jN : number of species, or number of molecules per unit volumeN: unit normal vector to an interfaceNu: Nusselt numberp: thermodynamic pressure, or probability densityP: pressure tensorP : mechanical powerPr: Prandtl numberPe: Peclet numberq: parameter, or heat fluxq: heat flux vectorq′: heat flux vector due to temperature gradientq: volume flow rateQ: heat quantityQ: caloric power(q0

f )j : enthalpy of formation per unit mass of species j(Q0

f )j = (H00 )j : molar enthalpy of formation of species j

r: perfect gas constant per unit mass, radius, or caloric power received perunit volume

R: universal molar gas constant, radius, or number of independent speciesin a mixture

XII List of Symbols

rN : rate of nucleationrT,p: energy of a chemical reactionRa: Rayleigh numberRe: Reynolds numberRi: Richardson numberRj : mass of j produced by a chemical reactionRr: chemical reactions: entropy per unit mass, or Arrhenius exponentS: entropyS: surfaceSc: Schmidt numberSh: Sherwood numberSr: Strouhal numberSt: Stanton numbert: timeT : absolute temperaturete: residence timeTa: Taylor numberT (k): turbulent transfer functionu, v, w: components of the velocity v with respect to x, y, z

u, v: coefficients in the equation of state for a real gasU, U∞: reference velocityv: barycentric velocity vector

∑Nj=1 vj

vj : velocity vector of species jV : velocity, or force potentialV: undefined vector, or composite velocity vector defined for the interface

V = V// + wNV : volume(V): manifold of equilibrium states in the thermodynamic spaceV i: surface viscosity numberVj : diffusion velocity of species = vj − vvr, vθ, vz: components of v in cylindrical coordinatesw: normal velocity of a surface, or velocity vector in phase spaceW : workW: local velocity vector of a discontinuity, or velocity vector of a fictitious

motionWe: Weber numberWF : rate of production of quantity F

x, y, z: Cartesian coordinatesx: position vector

List of Symbols XIII

X, Y, Z: coordinates in a relative frameXj: molar fraction of species jYj : mass fraction of species jZ: transfer function

Greek Symbols

α: coefficient of heat exchange, coefficient of dilatation, universal exponentfor Cv, or constant in the linearized theory of droplet vaporization−β: temperature gradient in a fluid layerβj : reduced concentrationβlj : number of atoms of l in the jth moleculeβT : reduced temperatureχ: compressibility coefficientδ: boundary layer thickness, infinitesimal difference, or universal exponent

along critical isothermΔ: difference, or Laplacianε: small parameter, or turbulent dissipation rateφ: velocity potential, or pre-exponential factor of ψφ: linear density of torquesϕ: surface density of torquesϕj : partial molar quantity associated with the quantity ϕ

Δϕm: mixture quantity associated with ϕ

γ: isentropic coefficient cp/cv, or universal exponent for isothermal com-pressibility

Γ : circulation of the velocity vectorη: partial bulk viscosity, or reduced coordinateκ: thermal diffusivity λ/ρcp, or mean curvature of flame surfaceλ: coefficient of thermal conductivity, or scale factorΛ: coefficient of load loss, or heat exchange coefficientμ: coefficient of shear viscosity, or Gibbs free energy per moleν: kinematic viscosity, or universal critical exponent for the correlation

lengthνc: collision frequencyνj : algebraic stoichiometric coefficient νj = ν′′j − ν′jν′j : stoichiometric coefficient of the direct reactionν′′j : stoichiometric coefficient of the reverse reactionπ: average normal pressureΠi: dimensionless ratio of physical quantities

XIV List of Symbols

θ: temperature, or angular coordinateϑ: volume per unit mass (inverse of density)Θ: reduced stagnation temperature in a detonation waveρ: densityρF : density of property F

ρj: partial densityσ: surface tensionσ: surface tension tensor, or membrane stress tensorΣ: surface areaΣ: stress tensorσT : temperature derivative of surface tensionτ : characteristic time, reduced enthalpy of a chemical reaction, or residence

timeτT : thermal diffusion timeτv: lifetime of a vaporizing dropletω: rotation velocity, or pulsation of an oscillatory waveω: rotation vectorΩ: rotation velocityξ: progress variable per unit mass, reduced coordinate, or correlation

lengthψ: stream function, or entity number in phase spaceζ: progress variable per unit volume, or reduced variableζ: position vector in phase spaceζ: production rate for a chemical reaction

Subscripts, Superscripts, and Other Symbols

a: interfacec: concentrationC : critical pointchem: chemicalD: direct, dissociation, or diffusivee: equilibrium flow, external, or exit sectionf : frozen flow, or flameG: gasi, j : speciesi: internal, or irreversibleL: localNL: nonlocalm: mixture, or mass

List of Symbols XV

mech: mechanico: opening sectionp: at constant pressureP : partition wallr: chemical reaction, reference, or chemical reactorR: reverse, or recombinations: steady state, or surfaceS : surfacet: turbulentT : temperature, or at constant temperatureth: thermalV : vapor

//: parallel to a surface⊥: normal to a surfaceg: gasl: liquidS : interfaceS : scalarV : vectorT : second-order tensor0: reference quantity•: pure simple substance(): per unit time, or for a production rate(): thermodynamic quantity per unit mole, or average quantity()′: perturbation from average quantity

() : transposed matrix, or Favre average()′′: perturbation from Favre averaged: reversible infinitesimal transformation(): pre-exponential factor

()0

T : standard thermodynamic function×: vector product⊗: tensor product·: scalar product (contracted tensor product): : dyadic product (doubly contracted tensor product)∧: exterior product∗: sonic conditions, or reference state<>: ensemble average∇: nabla (gradient operator)[ ]+−: jump in a quantity across an interface

XVI List of Symbols

d(): differentialδ(): small variationd/dt: material derivative∂/∂t: partial time derivativedW/dt: material derivative associated with the velocity W of a fictitious

motion

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII

List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Equations of State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1 Defining the State Variables of a Mixture . . . . . . . . . . . . . . . . . . . 8

2.1.1 Classical Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.2 Chemical Progress Variable . . . . . . . . . . . . . . . . . . . . . . . . . 92.1.3 Definitions Required to Describe a Reactive Fluid . . . . . 102.1.4 The Case for Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Thermodynamic Functions and Equation of State for SimpleFluids and Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.1 Thermodynamics Reminder . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.2 Properties of Simple Fluids at Equilibrium . . . . . . . . . . . 142.2.3 Examples of Laws of State . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2.4 EOS for Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3 Properties of Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.3.1 General Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.3.2 Partial Molar Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.3.3 Ideal Mixture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.3.4 Mixture Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.3.5 Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3.6 Ideal Mixture of Perfect Gases . . . . . . . . . . . . . . . . . . . . . . 292.3.7 A Mixture of Real Gases That Obeys the Virial Relation 312.3.8 Liquid Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.3.9 Mixture of Real Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.4 Reactive Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.4.1 Enthalpy of a Chemical Reaction . . . . . . . . . . . . . . . . . . . . 352.4.2 Entropy Production in a Homogeneous Reactive Mixture 36

XVIII Contents

2.4.3 Chemical Reaction at Equilibrium for a Mixture ofPerfect Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.4.4 Mixture of Perfect Gases in Chemical Equilibrium . . . . . 382.4.5 Unspecified Multireactive Mixtures . . . . . . . . . . . . . . . . . . 402.4.6 Reactive Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.4.7 Extension to Nonequilibrium Mixtures . . . . . . . . . . . . . . . 41

2.5 Thermodynamic Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.5.1 The Stability Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.5.2 Case of a Simple Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.5.3 Case of a Mixture with One Degree of Chemical Freedom 462.5.4 Case of a Mixture with Several Degrees of Chemical

Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.6 Surface Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.6.1 One-Component Fluid–Fluid Interfaces . . . . . . . . . . . . . . . 472.6.2 Multicomponent Fluid–Fluid Interfaces . . . . . . . . . . . . . . 50

3 Transfer Phenomena and Chemical Kinetics . . . . . . . . . . . . . . . 513.1 General Information on Irreversible Phenomena . . . . . . . . . . . . . 52

3.1.1 A Chemical Reaction Near Equilibrium . . . . . . . . . . . . . . 523.1.2 Thermal Exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.2 Presenting the Coefficients of Transfer via theThermodynamics of Irreversible Processes . . . . . . . . . . . . . . . . . . 613.2.1 Basic Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.2.2 Species Diffusion, Heat Conduction, and Viscosity . . . . . 62

3.3 Other Ways of Presenting the Transfer Coefficients . . . . . . . . . . 633.3.1 Presenting the Transfer Coefficients via the Simplified

Kinetic Theory of Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.3.2 More Precise Estimation of the Transfer Coefficients . . . 663.3.3 Liquids and Dense Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.4 Elements of Chemical Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4 Balance Equations for Reactive Flows . . . . . . . . . . . . . . . . . . . . . 734.1 Passage to the Continuum: Example of Thermal Transfer in

a Continuous Medium at Rest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.2 Reminder of the Concepts of the Material Derivative and

Strain in a Simple Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.3 Mass Balance of Species j and Total Mass Balance in a

Composite Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.4 General Balance Equation for a Property F . . . . . . . . . . . . . . . . . 80

4.4.1 Balance Equation Based on the Mean Material Motionv(x, t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.4.2 Balance Equation for a Property F Based on anArbitrary Continuous Motion W(x, t) . . . . . . . . . . . . . . . 82

4.5 Momentum Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.6 Energy Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Contents XIX

4.7 Flux and Entropy Production in a Discrete System . . . . . . . . . . 854.8 Entropy Balance in a Continuous Medium . . . . . . . . . . . . . . . . . . 894.9 Balance Laws for Discontinuities in Continuous Media . . . . . . . 904.10 Other Methodologies for Balance Laws . . . . . . . . . . . . . . . . . . . . . 92

4.10.1 Total Deterministic Balance . . . . . . . . . . . . . . . . . . . . . . . . 924.10.2 Probabilistic Population Balance . . . . . . . . . . . . . . . . . . . . 93

5 Dimensionless Numbers and Similarity . . . . . . . . . . . . . . . . . . . . 975.1 Elements of Dimensional Analysis: Πi Ratios . . . . . . . . . . . . . . . 98

5.1.1 Basic Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 985.1.2 Vashi–Buckingham or Π Theorem . . . . . . . . . . . . . . . . . . . 995.1.3 Practical Utility of Dimensional Analysis . . . . . . . . . . . . . 995.1.4 Example: Head Loss in a Cylindrical Pipe . . . . . . . . . . . . 100

5.2 Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.2.2 Example: Similarity of a Flexible Balloon Subjected

to a Wind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1025.3 Analytical Search for the Solutions to a Heat Transfer

Problem (Self-Similar Solution) . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.4 A Few Dimensionless Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6 Chemical Reactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096.1 Ideal and Real Reactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096.2 Homogeneous Perfectly Stirred Chemical Reactors . . . . . . . . . . . 111

6.2.1 Basic Equations of Perfectly Stirred Reactors . . . . . . . . . 1116.2.2 Steady Regimes of Perfectly Stirred Reactors . . . . . . . . . 1146.2.3 Stability Analysis of a Perfectly Stirred Reactor . . . . . . . 116

6.3 Residence Time Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1216.3.1 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1216.3.2 Homogeneous Well-Stirred Reactor in Steady Mode . . . . 1226.3.3 Piston Reactor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1226.3.4 Poiseuille Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1226.3.5 Real Reactor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

7 Coupled Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1277.1 General Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

7.1.1 Types of Coupled Phenomena . . . . . . . . . . . . . . . . . . . . . . . 1287.1.2 Incompressible Nonviscous Fluid . . . . . . . . . . . . . . . . . . . . 1297.1.3 Incompressible Viscous Fluid . . . . . . . . . . . . . . . . . . . . . . . 1317.1.4 Reactive Incompressible Viscous Flow . . . . . . . . . . . . . . . . 132

7.2 Coupling Between Chemical Kinetics and NondissipativeFlow: Compressible Reactive Fluid . . . . . . . . . . . . . . . . . . . . . . . . 132

7.3 Thermal Transfer and Mass Diffusion . . . . . . . . . . . . . . . . . . . . . . 1407.4 Shvab–Zel’dovich Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 1427.5 Phase Change of a Pure Constituent in a Gaseous Mixture . . . 144

XX Contents

7.6 Thermal Osmosis: Minimum Entropy Production . . . . . . . . . . . . 1467.7 Coupling Between Chemical Kinetics and Dissipative Flow:

Laminar Flames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1497.7.1 Example of a Laminar Premixed Flame: The Bunsen

Thin Flame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1507.7.2 An Example of a Laminar Diffusion Flame . . . . . . . . . . . 154

7.8 Coupling Between Heat and Momentum Transfer in thePresence of Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1587.8.1 Historical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . 1587.8.2 Rayleigh–Benard Instability . . . . . . . . . . . . . . . . . . . . . . . . 159

7.9 Surface Tension and Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1617.9.1 Marangoni Effect in a Highly Conducting Fluid Layer . . 1617.9.2 Benard–Marangoni Instability . . . . . . . . . . . . . . . . . . . . . . . 165

8 Turbulent Flow Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1698.1 Experimental Evidence for Turbulence . . . . . . . . . . . . . . . . . . . . . 1708.2 Turbulence Onset and Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

8.2.1 Turbulence Onset Mechanisms: Laminar FlowInstabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

8.2.2 Turbulence Decay Mechanisms: Vortex Dissipation . . . . 1858.3 Classical Turbulence Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

8.3.1 Turbulent Transfer Coefficients and Chemical Kinetics(Simplified Statistical Theory, Incompressible Case) . . . . 189

8.3.2 Some Definitions Relating to Turbulence . . . . . . . . . . . . . 1938.3.3 k–ε Modeling (Closing the Transfer Terms) . . . . . . . . . . . 1968.3.4 Spectral Analysis and Kolmogorov’s Theory . . . . . . . . . . 199

8.4 Turbulent Combustion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2028.4.1 Averaged Balance Equation for Turbulent Combustion . 2038.4.2 Turbulent Regimes for Premixed Combustion . . . . . . . . . 2058.4.3 Turbulent Regimes for Nonpremixed Combustion . . . . . . 2088.4.4 Combustion Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

8.5 Concepts of Large Eddy Simulation . . . . . . . . . . . . . . . . . . . . . . . . 2198.5.1 Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2208.5.2 Filtered Balance Equations . . . . . . . . . . . . . . . . . . . . . . . . . 2228.5.3 Closure Relations for Filtered Balance Equations . . . . . . 2238.5.4 Spectral Analysis and LES . . . . . . . . . . . . . . . . . . . . . . . . . 226

8.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

9 Boundary Layers and Fluid Layers . . . . . . . . . . . . . . . . . . . . . . . . . 2319.1 Unsteady Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

9.1.1 Viscous Boundary Layer in the Laminar Flow of anIncompressible Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

9.1.2 Diffusional Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . 2349.2 Steady Flow of a Viscous Incompressible Fluid Between Two

Coaxial Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

Contents XXI

9.2.1 Laminar Couette Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2359.2.2 Blowing and Aspiration at the Walls . . . . . . . . . . . . . . . . . 2379.2.3 Taylor–Couette Instability . . . . . . . . . . . . . . . . . . . . . . . . . . 240

9.3 Steady Incompressible Laminar Boundary Layer Above aFlat Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2439.3.1 Basic Equations of the Boundary Layer . . . . . . . . . . . . . . 2439.3.2 Self-Similar Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2479.3.3 Blowing and Aspiration at the Wall . . . . . . . . . . . . . . . . . . 249

9.4 Steady Laminar Boundary Layers with Chemical ReactionsAbove a Flat Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2519.4.1 Boundary Layers with Diffusion . . . . . . . . . . . . . . . . . . . . . 2519.4.2 The Emmons Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

9.5 The Rotating Disc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2609.5.1 Viscous Boundary Layer in Laminar Flow . . . . . . . . . . . . 2609.5.2 Diffusion in the Vicinity of a Rotating Disc . . . . . . . . . . . 2649.5.3 A Rotating Disc in Turbulent Flow . . . . . . . . . . . . . . . . . . 266

9.6 Turbulent Boundary Layer and Dimensional Analysis . . . . . . . . 2679.6.1 Turbulent Boundary Layer on a Flat Plate . . . . . . . . . . . 2679.6.2 Method of Multiple Scales and Dimensional Analysis . . . 2709.6.3 Turbulent Diffusion and First-Order Chemical

Reaction in the Vicinity of a Wall . . . . . . . . . . . . . . . . . . . 273

10 Reactive and Nonreactive Waves . . . . . . . . . . . . . . . . . . . . . . . . . . 27710.1 Continuous and Discontinuous One-Dimensional Waves in a

Barotropic Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27810.1.1 Basic Equations of One-Dimensional Waves . . . . . . . . . . . 27810.1.2 Piston Moving in an Infinite Cylinder . . . . . . . . . . . . . . . . 28010.1.3 Speed of a Normal Shock Wave Generated by a Piston . 283

10.2 Small Motions of a Fluid in Linearized Theory . . . . . . . . . . . . . . 28710.2.1 Case of a Nonreactive Fluid . . . . . . . . . . . . . . . . . . . . . . . . 28710.2.2 Case of a Monoreactive Fluid . . . . . . . . . . . . . . . . . . . . . . . 289

10.3 The Case of Small Stationary Disturbances . . . . . . . . . . . . . . . . . 29410.3.1 Linearized Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29410.3.2 Small Singular Disturbances and Transonic Flow . . . . . . 296

10.4 The Rankine–Hugoniot Relations . . . . . . . . . . . . . . . . . . . . . . . . . . 30210.5 Deflagration Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

10.5.1 Steady Propagation of an Adiabatic Planar Flame . . . . . 30910.5.2 Curved Nonadiabatic Flames . . . . . . . . . . . . . . . . . . . . . . . 313

10.6 Structure of the Planar Detonation Wave . . . . . . . . . . . . . . . . . . . 31610.7 Spherical Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

10.7.1 Case of Small Movements . . . . . . . . . . . . . . . . . . . . . . . . . . 32310.7.2 Spherical Flames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32410.7.3 Blast Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326

XXII Contents

11 Interface Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33311.1 General Information About Interfaces . . . . . . . . . . . . . . . . . . . . . . 334

11.1.1 Interfaces and Interfacial Layers . . . . . . . . . . . . . . . . . . . . . 33411.1.2 Types of Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33411.1.3 Thermodynamic Definitions . . . . . . . . . . . . . . . . . . . . . . . . 336

11.2 General Form of Interface Balance Laws . . . . . . . . . . . . . . . . . . . . 33811.2.1 Appropriate Form of Bulk Equations . . . . . . . . . . . . . . . . . 33811.2.2 General Balance Equation of an Interface . . . . . . . . . . . . . 341

11.3 Interface Balance Laws When Surface Variables ObeyClassical Thermodynamic Relations . . . . . . . . . . . . . . . . . . . . . . . . 34311.3.1 Classical Thermodynamic Relations . . . . . . . . . . . . . . . . . 34311.3.2 Interface Balance Laws for Species, Mass, Momentum

and Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34411.3.3 Interfacial Entropy Production . . . . . . . . . . . . . . . . . . . . . . 344

11.4 Constitutive Relations of Interfaces . . . . . . . . . . . . . . . . . . . . . . . . 34511.4.1 Constitutive Relations Deduced Directly from Linear

Irreversible Thermodynamics for Two-DimensionalInterfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

11.4.2 Constitutive Relations for Interfaces Deducedby Applying Irreversible Thermodynamics toThree-Dimensional Interfacial Layers . . . . . . . . . . . . . . . . . 352

11.5 Interfaces with Resistance to Wrinkling . . . . . . . . . . . . . . . . . . . . 35811.6 Concepts of Second-Gradient Theory . . . . . . . . . . . . . . . . . . . . . . . 36311.7 Conclusions Regarding Interface Equations . . . . . . . . . . . . . . . . . 364

12 Multiphase Flow Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36512.1 Formation of a Two-Phase Flow: Droplet Generation . . . . . . . . . 36712.2 Simplified Model of a Flow with Particles . . . . . . . . . . . . . . . . . . . 373

12.2.1 Variables That Characterize Two-Phase Flow . . . . . . . . . 37312.2.2 Balance Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37512.2.3 Application to the Study of Small Disturbances . . . . . . . 37812.2.4 Application to the Study of a Vortex in a Dilute

Suspension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38112.3 Flow with Evaporating Droplets . . . . . . . . . . . . . . . . . . . . . . . . . . . 390

12.3.1 Flow Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39012.3.2 Balance Equations for the Flow . . . . . . . . . . . . . . . . . . . . . 39112.3.3 Application to the Study of Spray Flame Propagation . 395

12.4 Problems at the Particle Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39912.4.1 Force Exerted by a Fluid on a Spherical Particle . . . . . . 40012.4.2 Heat Exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40812.4.3 Steady Combustion of a Fuel Drop in a Combustive

Atmosphere at Rest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41012.4.4 Transient Vaporization of a Droplet . . . . . . . . . . . . . . . . . . 42012.4.5 Other Cases of Droplet Vaporization and Combustion . . 427

Contents XXIII

A Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429A.1 Tensor Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429A.2 Motion and Field of Deformation of an Interfacial Layer . . . . . . 431A.3 Geometry of Interfaces and Interfacial Layers in Curvilinear

Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432A.4 Kinematics of Interfaces and Interfacial Layers . . . . . . . . . . . . . . 439A.5 Supercritical Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442

A.5.1 Thermodynamic Properties of Supercritical Fluids . . . . . 442A.5.2 Supercritical Fluid Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . 446

A.6 More on Transfer Coefficient Determination . . . . . . . . . . . . . . . . . 449A.6.1 Collision Integrals and Cross-sections . . . . . . . . . . . . . . . . 449A.6.2 Transfer Coefficients for Pure Gases . . . . . . . . . . . . . . . . . 452A.6.3 Transfer Coefficients for Gas Mixtures . . . . . . . . . . . . . . . 454

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471