Turbulent Combustion

Turbulent Combustion

The combustion of fossil fuels remains a key technology for the foreseeable future. It is therefore important that we understand the mechanisms of combus-tion and, in particular, the role of turbulence within this process. Combuscombus-tion always takes place within a turbulent flow field for two reasons: Turbulence increases the mixing process and enhances combustion, but at the same time combustion releases heat, which generates flow instability through gas expan-sion and buoyancy, thus enhancing the transition to turbulence.

The four chapters of this book present a thorough introduction to the field of turbulent combustion. After an overview of modeling approaches, the three remaining chapters consider the three distinct cases of premixed, nonpremixed, and partially premixed combustion, respectively.

This book will be of value to researchers and students of engineering and applied mathematics by demonstrating the current theories of turbulent com-bustion within a unified presentation of the field.

Norbert Peters has been a professor in mechanics at RWTH Aachen in Germany since 1976. Currently he is deputy editor of Combustion and Flame and editor of Flow, Turbulence and Combustion. In 1990 he received the Leibniz-Prize of the Deutsche Forschungsgemeinschaft, and in 1994 he received an honorary doctorate from the Universit´e L´ıbre de Bruxelles.

FOUNDING EDITOR

G. K. Batchelor

GENERAL EDITORS

S. Davis

Walter P. Murphy Professor

Applied Mathematics and Mechanical Engineering Northwestern University

L. B. Freund

Henry Ledyard Goddard University Professor Division of Engineering

Brown University

S. Leibovich

Sibley School of Mechanical & Aerospace Engineering Cornell University

V. Tvergaard

Department of Solid Mechanics The Technical University of Denmark

T U R B U L E N T C O M B U S T I O N

N O R B E R T P E T E R S

The Pitt Building, Trumpington Street, Cambridge, United Kingdom

  

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First published in printed format

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Contents

Preface page xi

1 Turbulent combustion: The state of the art 1

1.1 What is specific about turbulence with combustion? 1 1.2 Statistical description of turbulent flows 5 1.3 Navier–Stokes equations and turbulence models 10 1.4 Two-point velocity correlations and turbulent scales 13 1.5 Balance equations for reactive scalars 18 1.6 Chemical reaction rates and multistep asymptotics 22

1.7 Moment methods for reactive scalars 29

1.8 Dissipation and scalar transport of nonreacting and

linearly reacting scalars 30

1.9 The eddy-break-up and the eddy dissipation

models 33

1.10 The pdf transport equation model 35

1.11 The laminar flamelet concept 42

1.12 The concept of conditional moment closure 53

1.13 The linear eddy model 55

1.14 Combustion models used in large eddy simulation 57 1.15 Summary of turbulent combustion models 63

2 Premixed turbulent combustion 66

2.1 Introduction 66

2.2 Laminar and turbulent burning velocities 69 2.3 Regimes in premixed turbulent combustion 78 2.4 The Bray–Moss–Libby model and the Coherent

Flame model 87

2.5 The level set approach for the corrugated flamelets

regime 91

2.6 The level set approach for the thin reaction zones

regime 104

2.7 A common level set equation for both regimes 107 2.8 Modeling premixed turbulent combustion based on

the level set approach 109

2.9 Equations for the mean and the variance of G 114

2.10 The turbulent burning velocity 119

2.11 A model equation for the flame surface area ratio 127 2.12 Effects of gas expansion on the turbulent burning

velocity 137

2.13 Laminar flamelet equations for premixed

combustion 146

2.14 Flamelet equations in premixed turbulent

combustion 152

2.15 The presumed shape pdf approach 156

2.16 Numerical calculations of one-dimensional and

multidimensional premixed turbulent flames 157 2.17 A numerical example using the presumed shape pdf

approach 162

2.18 Concluding remarks 168

3 Nonpremixed turbulent combustion 170

3.1 Introduction 170

3.2 The mixture fraction variable 172

3.3 The Burke–Schumann and the equilibrium solutions 176

3.4 Nonequilibrium flames 178

3.5 Numerical and asymptotic solutions of counterflow

diffusion flames 186

3.6 Regimes in nonpremixed turbulent combustion 190 3.7 Modeling nonpremixed turbulent combustion 194

3.8 The presumed shape pdf approach 196

3.9 Turbulent jet diffusion flames 198

3.10 Experimental data from turbulent jet diffusion

flames 203

3.11 Laminar flamelet equations for nonpremixed

combustion 207

3.12 Flamelet equations in nonpremixed turbulent

Contents ix 3.13 Steady versus unsteady flamelet modeling 219 3.14 Predictions of reactive scalar fields and pollutant

formation in turbulent jet diffusion flames 222 3.15 Combustion modeling of gas turbines, burners, and

direct injection diesel engines 229

3.16 Concluding remarks 235

4 Partially premixed turbulent combustion 237

4.1 Introduction 237

4.2 Lifted turbulent jet diffusion flames 238 4.3 Triple flames as a key element of partially premixed

combustion 245

4.4 Modeling turbulent flame propagation in partially

premixed systems 251

4.5 Numerical simulation of lift-off heights in turbulent

jet flames 255

4.6 Scaling of the lift-off height 258

4.7 Concluding remarks 261 Epilogue 263 Glossary 265 Bibliography 267 Author Index 295 Subject Index 302

Preface

Fossil fuels remain the main source of energy for domestic heating, power generation, and transportation. Other energy sources such as solar and wind energy or nuclear energy still account for less than 20% of total energy con-sumption. Therefore combustion of fossil fuels, being humanity’s oldest tech-nology, remains a key technology today and for the foreseeable future. It is well known that combustion not only generates heat, which can be converted into power, but also produces pollutants such as oxides of nitrogen (NOx), soot, and unburnt hydrocarbons (HC). Ever more stringent regulations are forcing manufacturers of automotives and power plants to reduce pollutant emissions, for the sake of our environment. In addition, unavoidable emissions of CO2 are believed to contribute to global warming. These emissions will be reduced by improving the efficiency of the combustion process, thereby increasing fuel economy.

In technical processes, combustion nearly always takes place within a tur-bulent rather than a laminar flow field. The reason for this is twofold: First, turbulence increases the mixing processes and thereby enhances combustion. Second, combustion releases heat and thereby generates flow instability by buoyancy and gas expansion, which then enhances the transition to turbulence. This book addresses gaseous turbulent flows only. Although two-phase tur-bulent flows such as fuel sprays are also of much practical interest, they are omitted here, because their fundamentals are even less well understood than those of turbulent combustion. We also restrict ourselves to low Mach num-ber flows, because high speed turbulent combustion is an area of its own, with practical applications in supersonic and hypersonic aviation only.

Technical processes in gaseous turbulent combustion can be subdivided in terms of mixing: premixed, nonpremixed, or partially premixed turbulent combustion. For example, combustion in homogeneous charge spark-ignition

engines or in lean-burn gas turbines occurs under premixed conditions. In con-trast, combustion in a diesel engine or in furnaces essentially takes place under nonpremixed or partially premixed conditions.

In the spark-ignition engine, fuel and oxidizer are mixed by turbulence for a sufficiently long period of time before the electrical spark ignites the mixture. The deposition of electrical energy from the spark ionizes the gas and heats it to several thousand degrees Kelvin. At temperatures above one thousand degrees Kelvin chemical reactions are initiated. These generate a flame kernel that grows at first by laminar, then by turbulent flame propagation. The turbulent burning velocity and its prediction from first principles will play a central role in the chapter on premixed turbulent combustion.

In stationary lean-burn gas turbines that have recently been developed for power generation, the fuel is prevaporized and premixed with air before entering into the combustion chamber. To homogenize the temperature field and thereby reduce emissions, in particular those of NOx, turbulent mixing is very strong and dominates the combustion process. In the postflame region mixing with secondary air reduces the temperature to levels that are low enough not to destroy the turbine blades downstream of the exhaust. The reduction in temperature also freezes the NOxformation in the postflame region.

In the rich-to-lean-burn gas turbines used in aircraft engines, liquid fuel is injected into the combustion chamber where it evaporates and burns in a turbulent diffusion flame, which is stabilized by swirl and recirculation. As in lean-burn gas turbines secondary air is mixed into the product stream of the flame to oxidize the remaining hydrocarbons, CO, and soot and to reduce the temperature at the exhaust.

In the diesel engine, several liquid fuel sprays are injected into hot com-pressed air; the fuel evaporates and mixes partially with the air before auto-ignition occurs. The auto-auto-ignition process happens quite randomly and inde-pendently at several locations within the sprays. These ignition kernels will then initiate the burnout of the partially premixed fuel–air mixture. The final burnout occurs essentially under nonpremixed conditions.

In furnaces, finally, gaseous, liquid, or solid fuels are injected separately from the air into the combustion chamber. The air may be preheated or partially diluted by hot exhaust gases. Once the mixture is ignited, the flame propa-gates toward the nozzle until it stabilizes at a distance, called the lift-off height, downstream of the nozzle. Partial premixing takes place in the region between the nozzle and the lift-off height and determines the stabilization of the turbu-lent flame. Further downstream, combustion again occurs under nonpremixed conditions.

Preface xiii It is clear that in addition to premixed and nonpremixed combustion, par-tially premixed combustion plays, at least locally, an important role in practical applications. Another example for partially premixed combustion is the mod-ern direct-injection spark-ignition engine where the charge is stratified such that the flame initiated at the spark propagates through a partially premixed inhomogeneous mixture.

A second criterion for the subdivision of turbulent combustion relates to the ratio of turbulent to chemical time scales. Because of chain-branching reactions, hydrocarbon oxidation occurs if the temperature is above a certain crossover temperature, but it ceases if the temperature falls below that temperature. The crossover temperature is defined as the temperature where the effects of chain-branching reactions just balance that of chain-breaking reactions. At ambient pressure this temperature lies between 1,300 K and 1,500 K for hydrocarbon flames but it increases with pressure. At temperatures lower than the crossover temperature, extinction occurs, which must in general be avoided in practical applications. Therefore, combustion engines typically operate safely apart from extinction at conditions where temperatures are high enough and chemical reactions occur rapidly. This is often referred to as fast chemistry.

Slow chemistry does not very often occur in practical processes. Although NOxformation is a relatively slow process, it is nearly always linked to the fast chemistry of the main combustion reactions, since it depends on the presence of radicals. There are a few applications of slow oxidation chemistry just above the crossover temperature between chain-branching and chain-breaking reac-tions, for example low NOxburners, which operate at temperatures where both oxidation and NOxchemistry are relatively slow. This process has been called MILD combustion. An example is the MILD combustion mode, which will be discussed in Chapter 3.

The four chapters in this book are organized as follows:

In Chapter 1, after a general introduction, the more prominent current mod-eling approaches for turbulent flows with combustion will be presented. At the end of that chapter an overview of the existing models in terms of mixing and in terms of infinitely fast or finite rate chemistry will be presented.

In Chapter 2, which deals with premixed turbulent combustion, emphasis will be placed on a combustion model that uses the level set approach to de-termine the location of the premixed flame surface. Damk¨ohler’s regimes of large scale and small scale turbulence will be associated with the corrugated flamelets regime and the thin reaction zones regime, respectively. Finally, based on an equation for the flame surface area ratio, an expression will be derived for the turbulent burning velocity, valid in both regimes. Other models and

experimental data will be discussed in the light of this formulation. Although the mathematics behind the level set approach is rather demanding, the author believes that it is necessary in order to capture the physics of a propagating premixed turbulent flame surface.

In Chapter 3, models for nonpremixed combustion will be presented. Em-phasis is placed on models that are based on the mixture fraction as independent variable. The definition and the role of the mixture fraction will be discussed in detail. The derivation of the flamelet equations will be based on two-scale asymptotic arguments. Filtering of the mixture fraction field and that of the reactive scalars outside of the thin reaction zone will justify the use of flamelet equations in the entire available mixture fraction range. The flamelet model will be compared to experiments and other models used for nonpremixed tur-bulent combustion. Scaling laws for round turtur-bulent jet flames including NOx formation and the influence of buoyancy will be presented.

Chapter 4 is concerned with partially premixed combustion. The classical problem in partially premixed combustion is flame stabilization at the lift-off height in a turbulent jet diffusion flame, which will be discussed in detail. The key element of the instantaneous flame front is the triple flame structure. The level set formulation for premixed combustion and the mixture fraction formu-lation for nonpremixed combustion will be combined to obtain an expression for the turbulent burning velocity in partially premixed systems. This expres-sion is validated with respect to its capability to predict lift-off heights of lifted turbulent diffusion flames.

Finally, in the epilogue the principles behind turbulent combustion modeling are emphasized.

The turbulence models used in the four chapters of this book rely on mod-eling procedures that were developed for nonreacting constant density flows. They are highly disputed even for those applications because they rely on em-piricism and some kind of intuition supplemented by physical arguments. It will become clear that with combustion, empiricism and the number of neces-sary simplifications increase. This is reflected by the large variety of different combustion models that have been formulated and that are pursued and con-tinuously improved by different groups in the combustion community. In some cases competing models are based on the same physical concepts, with differ-ent modeling strategies leading to differdiffer-ent formulations. In other cases, the concepts are fundamentally different. For instance, Lagrangian models such as those used in the pdf transport equation model differ in their physical and mathematical formulation from classical flamelet models formulated in the Eulerian framework. As a consequence, if one needs to solve a combustion problem numerically, an a priori choice has to be made as to which model

Preface xv to use. This is regretable, since no model is infallible and certainly no single approach will provide definite answers to the large variety of problems in tur-bulent combustion.

In view of these difficulties one may be tempted to give up trying to model turbulent combustion and revert to direct numerical simulations (DNS). How-ever, as pointed out by Bray (1996), “DNS are an extremely valuable research tool. . . from which much can be learned. . . . However, DNS cannot and will not meet the pressing need for improved predictive methods to aid. . . the design of practical high Reynolds number combustion systems.” The reason for this lies in the large range of scales that would have to be resolved if a full numerical simulation were to be performed. Since combustion requires molecular mixing on the smallest scale of turbulence, the Kolmogorov scale, this scale and the even smaller scales of the thin reaction layers would have to be resolved. The inertial subrange between the integral scale and the Kolmogorov scale η ex-tends typically over two orders of magnitude in engineering applications with combustion. The size of the combustion chamber L is estimated as at least one order of magnitude larger than the integral scale. Therefore there are at least three orders of magnitude between L and the mesh size needed to resolve processes occuring at the Kolmogorov scale by a direct numerical simulation. With constant mesh sizes and a three-dimensional (3D) mesh this would result in at least 109_{grid points, which makes such computations prohibitive for many} years to come.

However, quite elaborate numerical codes using adaptive gridding to account for complex geometries have been developed in recent years and are commer-cially available. These codes use RANS (Reynolds averaged Navier–Stokes equations) models and resolve the flow by using mesh sizes of the order of the integral length scale. Industrial users of these codes naturally demand that they are also applicable to flows with combustion. This sets a framework for turbulent combustion models that is quite restrictive and may require compro-mises.

A way to improve the predictive capability of these models, in particular with respect to the flow field, is to employ large eddy simulations (LES) for flows with combustion. LES models employ a larger number of grid points to resolve the energy containing scales within the inertial subrange. It will be argued at the end of Chapter 1 that some RANS models, in particular flamelet models combined with the presumed shape pdf approach, can easily be extended to LES. I am indebted to many friends and colleagues in the combustion commu-nity for sharing with me over the past twenty years their stimulating ideas on turbulent combustion. Their papers are referenced in this book to the best of my knowledge. The book could also not have been written without the thesis

work of many of my students and collaborators who have amply contributed to many of the sections in Chapters 2 to 4. That work has been funded generously by the Deutsche Forschungsgemeinschaft, which is gratefully acknowledged. Early versions of the manuscript have received critical reading by Bill Ashurst, Bernd Binninger, Philip de Goey, Johannes Janicka, Alain Kerstein, Rupert Klein, Moshe Matalon, Heinz Pitsch, Martin Oberlack, Luc Vervisch, and many others. This has considerably improved the text and is also gratefully acknow-ledged. Last but not least, I want to thank Beate Dieckhoff for her considerate and careful preparation of the manuscript.

1

Turbulent combustion: The state of the art

1.1 What Is Specific about Turbulence with Combustion?

In recent years, nothing seems to have inspired researchers in the combustion community so much as the unresolved problems in turbulent combustion. Tur-bulence in itself is far from being fully understood; it is probably the most significant unresolved problem in classical physics. Since the flow is turbulent in nearly all engineering applications, the urgent need to resolve engineer-ing problems has led to preliminary solutions called turbulence models. These models use systematic mathematical derivations based on the Navier–Stokes equations up to a certain point, but then they introduce closure hypotheses that rely on dimensional arguments and require empirical input. This semiempirical nature of turbulence models puts them into the category of an art rather than a science.

For high Reynolds number flows the so-called eddy cascade hypothesis forms the basis for closure of turbulence models. Large eddies break up into smaller eddies, which in turn break up into even smaller ones, until the smallest eddies disappear due to viscous forces. This leads to scale invariance of energy transfer in the inertial subrange of turbulence. We will denote this as inertial range invariance in this book. It is the most important hypothesis for large Reynolds number turbulent flows and has been built into all classical turbulence models, which thereby satisfy the requirement of Reynolds number independence in the large Reynolds number limit. Viscous effects are of importance in the vicinity of solid walls only, a region of minor importance for combustion.

The apparent success of turbulence models in solving engineering problems has encouraged similar approaches for turbulent combustion, which conse-quently led to the formulation of turbulent combustion models. This is, however, where problems arise.

Combustion requires that fuel and oxidizer be mixed at the molecular level. How this takes place in turbulent combustion depends on the turbulent mixing process. The general view is that once a range of different size eddies has deve-loped, strain and shear at the interface between the eddies enhance the mixing. During the eddy break-up process and the formation of smaller eddies, strain and shear will increase and thereby steepen the concentration gradients at the interface between reactants, which in turn enhances their molecular interdiffu-sion. Molecular mixing of fuel and oxidizer, as a prerequisite of combustion, therefore takes place at the interface between small eddies. Similar considera-tions apply, once a flame has developed, to the conduction of heat and the diffusion of radicals out of the reaction zone at the interface.

While this picture follows standard ideas about turbulent mixing, it is less clear how combustion modifies these processes. Chemical reactions consume the fuel and the oxidizer at the interface and will thereby steepen their gradients even further. To what extent this will modify the interfacial diffusion process still needs to be understood.

This could lead to the conclusion that the interaction between turbulence and combustion invalidates classical scaling laws known from nonreacting tur-bulent flows, such as the Reynolds number independence of free shear flows in the large Reynolds number limit. To complicate the picture further, one has to realize that combustion involves a large number of elementary chemical reactions that occur on different time scales. If all these scales would inter-act with all the time scales within the inertial range, no simple scaling laws could be found. Important empirical evidence, however, does not confirm such pessimism:

• The difference between the turbulent and the laminar burning velocity, nor-malized by the turbulence intensity, is independent of the Reynolds number. It is Damk¨ohler number independent for large scale turbulence, but it be-comes proportional to the square root of the Damk¨ohler number for small scale turbulence (cf. Section 2.10).

• The flame length of a nonbuoyant turbulent jet diffusion flame, for instance, is Reynolds number and Damk¨ohler number independent (cf. Section 3.9). • The NO emission index of hydrogen–air diffusion flames is independent of the Reynolds number but proportional to the square root of the Damk¨ohler number (cf. Section 3.14).

• The lift-off height in lifted jet diffusion flames is independent of the noz-zle diameter and increases nearly linearly with the jet exit velocity (cf. Section 4.6).

1.1 What is specific about turbulence with combustion? 3 Power law Damk¨ohler number scaling laws may be the exception rather than the rule, but they indicate that there are circumstances where only a few chemical and turbulent time scales are involved. As far as Reynolds number in-dependence is concerned, it should be noted that the Reynolds number in many laboratory experiments is not large enough to approach the large Reynolds number limit. A remaining Reynolds number dependence of the turbulent mix-ing process would then show up in the combustion data. Apart from these experimental limitations (which become more serious owing to the increase of viscosity with temperature) it is not plausible that there would be a Reynolds number dependence introduced by combustion, because chemical reactions in-troduce additional time scales but no viscous effects. Even if chemical time scales interact with turbulent time scales in the inertial subrange of turbulence, these interactions cannot introduce the viscosity as a parameter for dimensional scaling, because it has disappeared as a parameter in that range. This does not preclude that ratios of molecular transport properties, Prandtl or Lewis num-bers, for instance, would not appear in scaling laws in combustion. As we have restricted the content of this book to low speed combustion, the Mach number will not appear in the analysis.

There remains, however, the issue of to what extent we can expect an in-teraction between chemical and turbulent scales in the inertial subrange. Here, we must realize that combustion differs from isothermal mixing in chemically reacting flows by two specific features:

• heat release by combustion induces an increase of temperature, which in turn

• accelerates combustion chemistry. Because of the competition between chain branching and chain breaking reactions this process is very sensitive to temperature changes.

Heat release combined with temperature sensitive chemistry leads to typical combustion phenomena, such as ignition and extinction. This is illustrated in Figure 1.1 where the maximum temperature in a homogeneous flow combustor is plotted as a function of the Damk¨ohler number, which here represents the ratio of the residence time to the chemical time. This is called the S-shaped curve in the combustion literature. The lower branch of this curve corresponds to a slowly reacting state of the combustor prior to ignition, where the short residence times prevent a thermal runaway. If the residence time is increased by lowering the flow velocity, for example, the Damk¨ohler number increases until the ignition point I is reached. For values larger than DaI thermal runaway

Tmax

Q

DaQ DaI

I

Da

Figure 1.1. The S-shaped curve showing the maximum temperature in a well-stirred

reactor as a function of the Damk¨ohler number.

leads to a rapid unsteady transition to the upper close-to-equilibrium branch. If one starts on that branch and decreases the Damk¨ohler number, thereby moving to the left in Figure 1.1, one reaches the point Q where extinction occurs. This is equivalent to a rapid transition to the lower branch. The middle branch between the point I and Q is unstable.

In the range of Damk¨ohler numbers between DaQand DaI, where two sta-ble branches exist, any initial state with a temperature in the range between the lower and the upper branch is rapidly driven to either one of them. Owing to the temperature sensitivity of combustion reactions the two stable branches repre-sent strong attractors. Therefore, only regions close to chemical equilibrium or close to the nonreacting state are frequently accessed. In an analytic study of stochastic Damk¨ohler number variations Oberlack et al. (2000) have recently shown that the probability of finding realizations apart from these two steady state solutions is indeed very small.

Chemical reactions that take place at the high temperatures on the upper branch of Figure 1.1 are nearly always fast compared to all turbulent time scales and, with the support of molecular diffusion, they concentrate in thin layers of a width that is typically smaller than the Kolmogorov scale. Except for density changes these layers cannot exert a feedback on the flow. Therefore they cannot influence the inertial range scaling. If these layers extinguish as the result of excessive heat loss, the temperature decreases such that chemistry becomes very slow and mixing can also be described by classical inertial range scaling. In both situations, fast and slow chemistry, time and length scales of com-bustion are separated from those of turbulence in the inertial subrange. This scale separation is a specific feature of most practical applications of turbulent

1.2 Statistical description of turbulent flows 5 combustion.†It makes the mixing process in the inertial range independent of chemistry and simplifies modeling significantly. Almost all turbulent combus-tion models explicitly or implicitly assume scale separacombus-tion.

As a general theme of this chapter, we will investigate whether the turbulence models to be discussed are based on the postulate of scale separation between turbulent and chemical time scales. In addition, it will be pointed out if a com-bustion model does not satisfy the postulate of Reynolds number independence in the large Reynolds number limit.

1.2 Statistical Description of Turbulent Flows

The aim of stochastic methods in turbulence is to describe the fluctuating ve-locity and scalar fields in terms of their statistical distributions. A convenient starting point for this description is the distribution function of a single variable, the velocity component u, for instance. The distribution function Fu(U ) of u is defined by the probability p of finding a value of u< U:

Fu(U )= p(u < U), (1.1)

where U is the so-called sample space variable associated with the random stochastic variable u. The sample space of the random stochastic variable u consists of all possible realizations of u. The probability of finding a value of u in a certain interval U−≤ u < U+is given by

p(U₋≤ u < U₊)= Fu(U+)− Fu(U−). (1.2) The probability density function (pdf ) of u is now defined as

Pu(U )=d Fu(U )

dU . (1.3)

It follows that Pu(U )dU is the probability of finding u in the range U ≤ u < U+ dU. If the possible realizations of u range from −∞ to +∞, it follows that

_+∞

−∞ Pu(U ) dU = 1, (1.4)

which states that the probability of finding the value u between−∞ and +∞ is certain (i.e., it has the probability unity). It also serves as a normalizing condition for Pu.

†_{A potential exception is the situation prior to ignition, where chemistry is neither slow enough} nor fast enough to be separated from the turbulent time scales. We will discuss this situation in detail in Chapter 3, Section 3.12.

In turbulent flows the pdf of any stochastic variable depends, in principle, on the position x and on time t. These functional dependencies are expressed by the following notation:

Pu(U ; x, t). (1.5)

The semicolon used here indicates that Pu is a probability density in U -space and is a function of x and t. In stationary turbulent flows it does not depend on t and in homogeneous turbulent fields it does not depend on x. In the following, for simplicity of notation, we will not distinguish between the random stochastic variable u and the sample space variable U , dropping the index and writing the pdf as

P(u; x, t). (1.6)

Once the pdf of a variable is known one may define its moments by u(x, t)n =

_+∞ −∞ u

n_{P(u; x, t) du.}

(1.7) Here the overbar denotes the average or mean value, sometimes also called expectation, of un_{. The first moment (n= 1) is called the mean of u:}

¯u(x, t) =
_+∞

−∞ u P(u; x, t) du. (1.8)

Similarly, the mean value of a function g(u) can be calculated from ¯g(x, t) =

_+∞

−∞ g(u)P(u; x, t) du. (1.9)

Central moments are defined by [u(x, t) − u(x, t)]n =

_+∞ −∞ (u− ¯u)

n_{P(u; x, t) du,}

(1.10) where the second central moment

[u(x, t) − u(x, t)]2 =
_+∞

−∞ (u− ¯u)

2_{P(u; x, t) du (1.11)} is called the variance. If we split the random variable u into its mean and the fluctuations uas

u(x, t) = ¯u(x, t) + u_{(x, t), (1.12)} where u= 0 by definition, the variance is found to be related to the first and second moment by

1.2 Statistical description of turbulent flows 7 Models for turbulent flows traditionally start from the Navier–Stokes equa-tions to derive equaequa-tions for the first and the second moments of the flow variables using (1.12). Since the three velocity components and the pressure depend on each other through the solutions of the Navier–Stokes equations they are correlated. To quantify these correlations it is convenient to introduce the joint probability density function of the random variables. For instance, the joint pdf of the velocity components u andv is written as

P(u, v; x, t).

The pdf of u, for instance, may be obtained from the joint pdf by integration over all possible realizations ofv,

P(u)=
_+∞

−∞ P(u, v) dv, (1.14)

and is called the marginal pdf of u in this context. The correlation between u andv is given by

uv=
_+∞

−∞
_+∞

−∞ (u− ¯u)(v − ¯v)P(u, v) dudv. (1.15) This can be illustrated by a so-called scatter plot (cf. Figure 1.2). If a series of instantaneous realizations of u andv are plotted as points in a graph of u andv, these points will scatter within a certain range. The means ¯u and ¯v are the average positions of the points in u andv directions, respectively. The correlation uv/(u2v2₎1/2_{is proportional to the slope of the average straight} line through the data points.

A joint pdf of two variables can always be written as a product of a conditional pdf of one variable times the marginal pdf of the other, for

v

v

u u

Figure 1.2. A scatter plot of two velocity components u andv illustrating the correlation

example

P(u, v; x, t) = P(u | v; x, t)P(v; x, t). (1.16) This is called Bayes’ theorem. In this example the conditional pdf P(u| v; x, t) describes the probability density of u, conditioned at a fixed value ofv. If u andv are not correlated they are called statistically independent. In that case the joint pdf is equal to the product of the marginal pdfs:

P(u, v; x, t) = P(u; x, t)P(v; x, t). (1.17) By using this in (1.15) and integrating, we easily see that uvvanishes, if u andv are statistically independent. In turbulent shear flows uvis interpreted as a Reynolds shear stress, which is nonzero in general. The conditional pdf P(u| v; x, t) can be used to define conditional moments. For example, the conditional mean of u, conditioned at a fixed value ofv, is given by

u | v =
_+∞

−∞ u P(u| v) du. (1.18)

In the following we will use angular brackets for conditional means only. As a consequence of the nonlinearity of the Navier–Stokes equations sev-eral closure problems arise. These are not only related to correlations between velocity components among each other and the pressure, but also to correla-tions between velocity gradients and correlacorrela-tions between velocity gradients and pressure fluctuations. These appear in the equations for the second mo-ments as dissipation terms and pressure–strain correlations, respectively. The statistical description of gradients requires information from adjacent points in physical space. Very important aspects in the statistical description of turbulent flows are therefore related to two-point correlations, which we will introduce in Section 1.4.

For flows with large density changes as occur in combustion, it is often convenient to introduce a density-weighted average ˜u, called the Favre average, by splitting u(x, t) into ˜u(x, t) and u(x, t) as

u(x, t) = ˜u(x, t) + u_{(x, t). (1.19)} This averaging procedure is defined by requiring that the average of the product of uwith the densityρ (rather than uitself) vanishes:

ρu_{= 0. (1.20)}

The definition for ˜u may then be derived by multiplying (1.19) by the density ρ and averaging:

1.2 Statistical description of turbulent flows 9 Here the average of the productρ ˜u is equal to the product of the averages ¯ρ and ˜u, since ˜u is already an average defined by

˜u= ρu/ ¯ρ. (1.22)

This density-weighted average can be calculated, if simultaneous measurements ofρ and u are available. Then, by taking the average of the product ρu and dividing it by the average ofρ one obtains ˜u. While such measurements are often difficult to obtain, Favre averaging has considerable advantages in simplifying the formulation of the averaged Navier–Stokes equations in variable density flows. In the momentum equations, but also in the balance equations for the temperature and the chemical species, the convective terms are dominant in high Reynolds number flows. Since these contain products of the dependent variables and the density, Favre averaging is the method of choice. For instance, the average of the product of the densityρ with the velocity components u and v would lead with conventional averages to four terms,

ρuv = ¯ρ ¯u ¯v + ¯ρu_v_{+ ρ}_u_{v + ρ¯} _v_{¯u+ ρ}_u_v_{. (1.23)} Using Favre averages one writes

ρuv = ρ(˜u + u_{)( ˜v + v}₎

= ρ ˜u ˜v + ρu_{v + ρv˜} _{˜u+ ρu}_v_{. (1.24)} Here fluctuations of the density do not appear. Taking the average leads to two terms only,

ρuv = ¯ρ ˜u ˜v + ¯ρ
uv. (1.25) This expression is much simpler than (1.23) and has formally the same structure as the conventional average of uv for constant density flows:

uv = ¯u ¯v + uv. (1.26)

Difficulties arising with Favre averaging in the viscous and diffusive transport terms are of less importance since these terms are usually neglected in high Reynolds number turbulence.

The introduction of density-weighted averages requires the knowledge of the correlation between the density and the other variable of interest. A Favre pdf of u can be derived from the joint pdf P(ρ, u) as

¯ ρ ˜P(u) =
_ρmax ρmin ρ P(ρ, u) dρ =
_ρmax ρmin ρ P(ρ | u)P(u) dρ = ρ | uP(u). (1.27)

Multiplying both sides with u and integrating yields ¯ ρ
_+∞ −∞ u ˜P(u) du=
_+∞

−∞ ρ | uu P(u) du, (1.28) which is equivalent to ¯ρ ˜u = ρu. The Favre mean value of u therefore is defined as

˜u=
_+∞

−∞ u ˜P(u) du. (1.29)

1.3 Navier–Stokes Equations and Turbulence Models

In the following we will first describe the classical approach to model turbulent flows. It is based on single point averages of the Navier–Stokes equations. These are commonly called Reynolds averaged Navier–Stokes equations (RANS). We will formally extend this formulation to nonconstant density by introducing Favre averages. In addition we will present the most simple model for turbulent flows, the k–ε model. Even though it certainly is the best compromise for engineering design using RANS, the predictive power of the k–ε model is, except for simple shear flows, often found to be disappointing. We will present it here, mainly to help us define turbulent length and time scales.

For nonconstant density flows the Navier–Stokes equations are written in conservative form: Continuity ∂ρ ∂t + ∇ · (ρv) = 0, (1.30) Momentum ∂ρv ∂t + ∇ · (ρvv) = −∇ p + ∇ · τ + ρg. (1.31) In (1.31) the two terms on the left-hand side (l.h.s.) represent the local rate of change and convection of momentum, respectively, while the first term on the right-hand side (r.h.s.) is the pressure gradient and the second term on the r.h.s. represents molecular transport due to viscosity. Hereτ is the viscous stress tensor τ = µ  2 S−2 3δ∇ · v  (1.32) and S= 1 2(∇v + ∇v T ) (1.33)

1.3 Navier–Stokes equations and turbulence models 11 is the rate of strain tensor, where∇vT is the transpose of the velocity gradient andµ is the dynamic viscosity. It is related to the kinematic viscosity ν as µ = ρν. The last term in (1.31) represents forces due to buoyancy.

Using Favre averaging on (1.30) and (1.31) one obtains ∂ ¯ρ

∂t + ∇ · ( ¯ρ ˜v) = 0, (1.34)

∂ ¯ρ ˜v

∂t + ∇ · ( ¯ρ ˜v ˜v) = −∇ ¯p + ∇ · ¯τ − ∇ · ( ¯ρ
vv)+ ¯ρg. (1.35) This equation is similar to (1.31) except for the third term on the l.h.s. containing the correlation− ¯ρ
vv, which is called the Reynolds stress tensor.

The Reynolds stress tensor is unknown and represents the first closure prob-lem for turbulence modeling. It is possible to derive equations for the six com-ponents of the Reynolds stress tensor. In these equations several terms appear that again are unclosed. Those so-called Reynolds stress models have been pre-sented for nonconstant density flows, for example, by Jones (1994) and Jones and Kakhi (1996).

Although Reynolds stress models contain a more complete description of the physics, they are not yet widely used in turbulent combustion. Many industrial codes still rely on the k–ε model, which, by using an eddy viscosity, introduces the assumption of isotropy. It is known that turbulence becomes isotropic at the small scales, but this does not necessarily apply to the large scales at which the averaged quantities are defined. The k–ε model is based on equations where the turbulent transport is diffusive and therefore is more easily handled by numerical methods than the Reynolds stress equations. This is probably the most important reason for its wide use in many industrial codes.

An important simplification is obtained by introducing the eddy viscosity νt, which leads to the following expression for the Reynolds stress tensor:

− ¯ρ
v_v_{= ¯ρν} t  2 ˜S−2 3δ∇ · ˜v  −2 3δ ¯ρ ˜k. (1.36)

Hereδ is the tensorial Kronecker symbol δi j (δi j = 1 for i = j and δi j = 0 for i = j) and νt is the kinematic eddy viscosity, which is related to the Favre average turbulent kinetic energy

˜k= 1 2v

_{· v} _(1.37)

and its dissipation ˜ε by

νt = cµ ˜k 2 ˜

The introduction of the Favre averaged variables ˜k and ˜ε requires that mod-eled equations are available for these quantities. These equations are given here in their most simple form:

Turbulent kinetic energy ¯ ρ∂ ˜k ∂t + ¯ρ ˜v · ∇ ˜k = ∇ ·  ¯ ρνt σk ∇ ˜k  − ¯ρ
v_v_{:∇ ˜v − ¯ρ ˜ε, (1.39)} Turbulent dissipation ¯ ρ∂ ˜ε ∂t + ¯ρ ˜v · ∇ ˜ε = ∇ ·  ¯ ρνt σε∇ ˜ε  − cε1ρ¯ε˜_˜k
vv:∇ ˜v − cε2ρ¯ε˜ 2 ˜k . (1.40) In these equations the two terms on the l.h.s. represent the local rate of change and convection, respectively. The first term on the r.h.s. represents the turbu-lent transport, the second one turbuturbu-lent production, and the third one turbuturbu-lent dissipation. As in the standard k–ε model, the constants σk = 1.0, σ_ε = 1.3, cε1 = 1.44, and cε2= 1.92 are generally used. A more detailed discussion con-cerning additional terms in the Favre averaged turbulent kinetic energy equation may be found in Libby and Williams (1994).

It should be noted that for constant density flows the k-equation can be derived with few modeling assumptions quite systematically from the Navier– Stokes equations. From this derivation follows the definition of the viscous dissipation as

ε = ν[∇v_{+ ∇v}T

] :∇v. (1.41)

Theε-equation, however, cannot be derived in a systematic manner. The basis for the modeling of that equation are the equations for two-point correlations. Rotta (1972) has shown that by integrating the two-point correlation equa-tions over the correlation coordinate r one can derive an equation for the integral length scale , which will be defined below. This leads to a k–-model. The -equation has been applied, for example, by Rodi and Spald-ing (1970) to turbulent jet flows. It is easily shown that from this model and from the algebraic relation between, k, and ε a balance equation for ε can be derived. A similar approach has recently been used by Oberlack (1997) to derive an equation for the dissipation tensor that is needed in Reynolds stress models.

The dissipationε plays a fundamental role in turbulence theory, as will be shown in the next section. The eddy cascade hypothesis states that it is equal to the energy transfer rate from the large eddies to the smaller eddies and therefore is invariant within the inertial subrange of turbulence. By using this property for ε in the k-equation and by determining ε from an equation like (1.40) rather than

1.4 Two-point velocity correlations and turbulent scales 13 from its definition (1.41) one obtains Reynolds number independent solutions for free shear flows where, owing to the absence of walls, the viscous stress tensor can be neglected compared to the Reynolds stress tensor. This is how inertial range invariance is built into turbulence models.

It may be counterintuitive to model dissipation, which is active at the small scales, by an equation that contains only quantities that are defined at the large integral scales (cf. Figure 1.5 below). However, it is only because inertial range invariance has been built into turbulence models that they reproduce the scaling laws that are experimentally observed. Based on the postulate formulated at the end of Section 1.1 the same must be claimed for turbulent combustion models in the large Reynolds number limit. Since combustion takes place at the small scales, inertial range invariant quantities must relate properties defined at the small scales to those defined at the large scales, at which the models are formulated.

1.4 Two-Point Velocity Correlations and Turbulent Scales

A characteristic feature of turbulent flows is the occurrence of eddies of dif-ferent length scales. If a turbulent jet shown in Figure 1.3 enters with a high velocity into initially quiescent surroundings, the large velocity difference be-tween the jet and the surroundings generates a shear layer instability, which, after a transition, becomes turbulent further downstream from the nozzle exit. The two shear layers merge into a fully developed turbulent jet. In order to characterize the distribution of eddy length scales at any position within the jet, one measures at point x and time t the axial velocity u(x, t), and simultaneously at a second point (x+ r, t) with distance r apart from the first one, the velocity u(x+ r, t). Then the correlation between these two velocities is defined by the

* * air air fuel unstable shear layer transition to turbulence fully developed turbulent jet (x) (x+ r) r

Figure 1.3. Schematic presentation of two-point correlation measurements in a

Figure 1.4. The normalized two-point velocity correlation for homogeneous isotropic

turbulence as a function of the distance r between the two points.

average

R(x, r, t) = u_{(x, t)u}_{(x+ r, t). (1.42)} For homogeneous isotropic turbulence the location x is arbitrary and r may be replaced by its absolute value r= | r|. For this case the normalized correlation f (r, t) = R(r, t)/u2(t) (1.43) is plotted schematically in Figure 1.4. It approaches unity for r→ 0 and decays slowly when the two points are only a very small distance r apart. With increasing distance it decreases continuously and may even take negative values. Very large eddies corresponding to large distances between the two points are rather seldom and therefore do not contribute much to the correlation.

Kolmogorov’s 1941 theory for homogeneous isotropic turbulence assumes that there is a steady transfer of kinetic energy from the large scales to the small scales and that this energy is being consumed at the small scales by viscous dissipation. This is the eddy cascade hypothesis. By equating the energy transfer rate (kinetic energy per eddy turnover time) with the dissipationε it follows that this quantity is independent of the size of the eddies within the inertial range. For the inertial subrange, extending from the integral scale to the Kolmogorov scaleη, ε is the only dimensional quantity apart from the correlation coordinate r that is available for the scaling of f (r, t). Since ε has the dimension [m2_/s3_], the second-order structure function defined by

F2(r, t) = (u(x, t) − u(x+ r, t))2= 2 u2(t)(1− f (r, t)) (1.44) with the dimension [m2_/s2_{] must therefore scale as}

1.4 Two-point velocity correlations and turbulent scales 15 where C is a universal constant called the Kolmogorov constant. In the case of homogeneous isotropic turbulence the velocity fluctuations in the three coordi-nate directions are equal to each other. The turbulent kinetic energy

k=1

2v· v (1.46)

is then equal to k= 3u2/2. Using this one obtains from (1.44) and (1.45) f (r, t) = 1 − 3

4 C k(ε r)

2/3_{, (1.47)}

which is also plotted in Figure 1.4.

There are eddies of a characteristic size containing most of the kinetic energy. At these eddies there remains a relatively large correlation f (r, t) before it decays to zero. The length scale of these eddies is called the integral length scale and is defined by

(t) =
_∞

0

f (r, t) dr. (1.48)

The integral length scale is also shown in Figure 1.4.

We denote the root-mean-square (r.m.s.) velocity fluctuation by

v₌_{2 k/3, (1.49)}

which represents the turnover velocity of integral scale eddies. The turnover time/vof these eddies is then proportional to the integral time scale

τ = k

ε. (1.50)

For very small values of r only very small eddies fit into the distance between

x and x+ r. The motion of these small eddies is influenced by viscosity, which

provides an additional dimensional quantity for scaling. Dimensional analysis then yields the Kolmogorov length scale

η =  ν3 ε 1/4 , (1.51)

which is also shown in Figure 1.4.

The range of length scales between the integral scale and the Kolmogorov scale is called the inertial range. In addition toη a Kolmogorov time and a velocity scale may be defined as

t_η=ν ε

1/2

The Taylor length scaleλ is an intermediate scale between the integral and the Kolmogorov scale. It is defined by replacing the average gradient in the definition of the dissipation (1.41) byv/λ. This leads to the definition

ε = 15νv2

λ2. (1.53)

Here the factor 15 originates from considerations for isotropic homogeneous turbulence. Using (1.52) we see that λ is proportional to the product of the turnover velocity of the integral scale eddies and the Kolmogorov time:

λ = (15ν v2_/ε)1/2_{∼ v}_t

η. (1.54)

Thereforeλ may be interpreted as the distance that a large eddy convects a Kolmogorov eddy during its turnover time t_η. As a somewhat artificially defined intermediate scale it has no direct physical significance in turbulence or in turbulent combustion. We will see, however, that similar Taylor scales may be defined for nonreactive scalar fields, which are useful for the interpretation of mixing processes.

According to Kolmogorov’s 1941 theory the energy transfer from the large eddies of size is equal to the dissipation of energy at the Kolmogorov scale η. Therefore we will relateε directly to the turnover velocity and the length scale of the integral scale eddies,

ε ∼ v3

 . (1.55)

We now define a discrete sequence of eddies within the inertial subrange by n = 

2n ≥ η, n= 1, 2, . . . . (1.56) Sinceε is constant within the inertial subrange, dimensional analysis relates the turnover time tn and the velocity differencevn across the eddyn toε in that range as ε ∼ v2n tn ∼ v3 n n ∼ 2 n t3 n . (1.57)

This relation includes the integral scales and also holds for the Kolmogorov scales as ε = v 2 η tη = v3 η η. (1.58)

A Fourier transform of the isotropic two-point correlation function leads to a definition of the kinetic energy spectrum E(k), which is the density of kinetic

1.4 Two-point velocity correlations and turbulent scales 17 energy per unit wavenumber k. Here, rather than presenting a formal derivation, we relate the wavenumber k to the inverse of the eddy sizenas

k= −1n . (1.59)

The kinetic energyv2

nat scalenis then v2

n ∼ (ε n)2/3= ε2/3k−2/3 (1.60) and its density in wavenumber space is proportional to

E(k)= dv 2 n dk ∼ ε

2/3_k−5/3_{. (1.61)}

This is the well-known k−5/3law for the kinetic energy spectrum in the inertial subrange.

If the energy spectrum is measured in the entire wavenumber range one obtains the behavior shown schematically in a log–log plot in Figure 1.5. For small wavenumbers corresponding to large scale eddies the energy per unit wavenumber increases with a power law between k2 _{and k}4_{. This range is} not universal and is determined by large scale instabilities, which depend on the boundary conditions of the flow. The spectrum attains a maximum at a wavenumber that corresponds to the integral scale, since eddies of that scale contain most of the kinetic energy. For larger wavenumbers corresponding to the inertial subrange the energy spectrum decreases following the k−5/3law. There is a cutoff at the Kolmogorov scaleη. Beyond this cutoff, in the range called

Figure 1.5. Schematic representation of the turbulent kinetic energy spectrum as a

the viscous subrange, the energy per unit wavenumber decreases exponentially owing to viscous effects.

In one-point averages the energy containing eddies at the integral length scale contribute the most to the kinetic energy. Therefore RANS averaged mean quantities essentially represent averages over regions in physical space that are of the order of the integral scale. This was meant by the statement at the end of Section 1.3 that RANS averages are defined at the large scales. In Large Eddy Simulations (LES), to be discussed in Section 1.14, filtering over smaller regions than the integral length scale leads to different mean values and, in particular, to smaller variances.

1.5 Balance Equations for Reactive Scalars

Combustion is the conversion of chemical bond energy contained in fossil fuels into heat by chemical reactions. The basis for any combustion model is the continuum formulation of the balance equations for energy and the chemical species. We will not derive these equations here but refer to Williams (1985a) for more details. We consider a mixture of n chemically reacting species and start with the balance equations for the mass fraction of species i ,

ρ∂Yi

∂t + ρv · ∇Yi = −∇ · ji+ ωi, (1.62) where i = 1, 2, . . . , n. In these equations the terms on the l.h.s. represent the local rate of change and convection. The diffusive flux in the first term on the r.h.s. is denoted by jiand the last termωiis the chemical source term.

The molecular transport processes that cause the diffusive fluxes are quite complicated. A full description may be found in Williams (1985a). Since in models of turbulent combustion molecular transport is less important than tur-bulent transport, it is useful to consider simplified versions of the diffusive fluxes; the most elementary is the binary flux approximation

ji = −ρ Di∇Yi, (1.63)

where Diis the binary diffusion coefficient, or mass diffusivity, of species i with respect to an abundant species, for instance N2. It should be noted, however, that in a multicomponent system this approximation violates mass conservation, if nonequal diffusivities Diare used, since the sum of all n fluxes has to vanish and the sum of all mass fractions is unity. Equation (1.63) is introduced here mainly for the ease of notation, but it must not be used in laminar flame calculations.

1.5 Balance equations for reactive scalars 19 For simplicity it will also be assumed that all mass diffusivities Di are proportional to the thermal diffusivity denoted by

D= λ/ρ cp (1.64)

such that the Lewis numbers

Lei = λ/(ρ cpDi)= D/Di (1.65) are constant. In these equationsλ is the thermal conductivity and cpis the heat capacity at constant pressure of the mixture.

Before going into the definition of the chemical source termωi to be pre-sented in the next section, we want to consider the energy balance in a chemi-cally reacting system. The enthalpy h is the mass-weighted sum of the specific enthalpies hi of species i : h= n i=1 Yihi. (1.66)

For an ideal gas hi depends only on the temperature T : hi = hi_,ref+

T Tref

c_{p i}(T ) d T. (1.67) Here cpi is the specific heat capacity of species i at constant pressure and T is the temperature in Kelvins. The chemical bond energy is essentially contained in the reference enthalpies hi,ref. Reference enthalpies of H2, O2, N2, and solid carbon are in general chosen as zero, while those of combustion products such as CO2 and H2O are negative. These values as well as polynomial fits for the temperature dependence of c_{p i} are documented, for instance, for many species used in combustion calculations in Burcat (1984). Finally the specific heat capacity at constant pressure of the mixture is

cp= n

i=1

Yic_{p i}. (1.68)

A balance equation for the enthalpy can be derived from the first law of ther-modynamics as (cf. Williams, 1985a)

ρ∂h_∂t + ρv · ∇h = ∂p_∂t + v · ∇ p − ∇ · jq+ qR. (1.69) Here the terms on the l.h.s. represent the local rate of change and convection of enthalpy. We have neglected the term that describes frictional heating because it is small for low speed flows. The local and convective change of pressure is important for acoustic interactions and pressure waves. We will not consider the termv · ∇ p any further since we are interested in the small Mach number limit

only. The transient pressure term∂p/∂t must be retained in applications for reciprocating engines but can be neglected in open flames where the pressure is approximately constant and equal to the static pressure. The heat flux jq includes the effect of enthalpy transport by the diffusive fluxes ji:

jq= −λ∇T + n

i=1

hiji. (1.70)

Finally, the last term in (1.69) represents heat transfer due to radiation and must be retained in furnace combustion and whenever strongly temperature dependent processes, such as NOxformation, are to be considered.

The static pressure is obtained from the thermal equation of state for a mixture of ideal gases

p = ρRT

W . (1.71)

Here R is the universal gas constant and W is the mean molecular weight given by W = _n i₌₁ Yi Wi −1 . (1.72)

The molecular weight of species i is denoted by Wi. For completeness we note that mole fractions Xican be converted into mass fractions Yi via

Yi = Wi

W Xi. (1.73)

We now want to simplify the enthalpy equation. Differentiating (1.66) one obtains dh= cpd T + n i₌₁ hidYi, (1.74)

where (1.67) and (1.68) have been used. If (1.70), (1.74), and (1.63) are inserted into the enthalpy equation (1.69) with the termv · ∇ p removed, it takes the form ρ∂h_∂t + ρv · ∇h = ∂p_∂t + ∇ · _λ cp∇h  + qR − n i=1 hi∇ · _λ cp − ρ Di  ∇Yi  . (1.75)

It is immediately seen that the last term disappears, if all Lewis numbers are assumed equal to unity. If, in addition, unsteady pressure changes and radiation heat transfer can be neglected, the enthalpy equation contains no source terms.

1.5 Balance equations for reactive scalars 21 It then has a form similar to the equation for a conserved scalar such as the mixture fraction that will be introduced in Chapter 3.

Another form of the energy equation can be written in terms of the temper-ature. Inserting (1.62), (1.70), and (1.74) into (1.69) one obtains

ρcp∂T_∂t + ρcpv· ∇T = ∂p ∂t + ∇ · (λ∇T ) − n i=1 c_{p i}ji· ∇T − n i=1 hiωi+ qR. (1.76) If, for simplicity, the specific heat capacities cp,i are all assumed equal and constant, the pressure is constant, and the heat transfer due to radiation is neglected, the temperature equation becomes

ρ∂T

∂t + ρv · ∇T = ∇ · (ρ D∇T ) + ωT. (1.77) Here (1.64) was used and the heat release due to chemical reactions is written as ωT = −1 cp n i=1 hiωi. (1.78)

This form of the temperature equation resembles that for the mass fractions of species i , which becomes with the binary diffusion approximation (1.63)

ρ∂Yi_∂t + ρv · ∇Yi = ∇ · (ρ Di∇Yi)+ ωi. (1.79) If, in addition, a one-step reaction and equal diffusivities (Di = D) are assumed, coupling relations between the temperature and the species mass fractions can be derived (cf. Williams, 1985a). These assumptions are often used in mathe-matical analyses of combustion problems.

In the following we will use the term “reactive scalars” for the mass fraction of all chemical species and temperature and introduce the vector

ψ = (Y1, Y2, . . . , Yn, T ). (1.80) Here n is the number of reactive species. For simplicity of notation, the balance equation for the reactive scalarψiwill be written

ρ∂ψi

∂t + ρv · ∇ψi = ∇ · (ρ Di∇ψi)+ ωi, (1.81) where i= 1, 2, . . . , n + 1. The diffusivity Di(i= 1, 2, . . . , n) is the mass dif-fusivity for the species and the thermal difdif-fusivity D is Dn₊₁. Similarly,ωn₊₁ is defined asωT. The chemical source term will also be written as

1.6 Chemical Reaction Rates and Multistep Asymptotics

Over the past thirty years, knowledge about gas phase reactions relevant to com-bustion has continuously improved. Most of the rate data for pure hydrocarbon fuels up to n-heptane and iso-octane and some aromatic fuels such as toluol are sufficiently accurate to predict ignition delay times, laminar burning velocities, and extinction strain rates (cf. the recent review by Lindstedt, 1998). Many data collections are available in the combustion literature (cf., for instance, Baulch et al., 1994). Many authors have selected reactions and their rate data and have combined them into tables called elementary reaction mechanisms. For example, a mechanism for methane oxidation comprising 277 reactions among 49 chemical species, called GRI-Mech 2.11, has been compiled by the Gas Research Institute (cf. Bowman et al., 1999). Kinetic data for C1–C3 hydro-carbons have been evaluated by Leung and Lindstedt (1995) for the purpose of calculating laminar diffusion flames. A comprehensive mechanism for hydro-carbon and methanol combustion may also be found in Warnatz et al. (1996).

The rate of reaction k in a mechanism containing r chemical reactions is wk= kf k n  j₌₁  ρYj Wj _ν j k − kbk n  j₌₁  ρYj Wj _ν j k . (1.83)

Here kf kand kbkare the rate coefficients of the forward and backward reaction, respectively. In general they are temperature dependent and may also depend on pressure. The exponentsνj kandνj k are the stoichiometric coefficients of reaction k in forward and backward direction, respectively.

The chemical source termωi, which is the mass of species i produced per unit volume and unit time, is the sum over all reactions in the mechanism:

ωi = Wi r k₌₁

νi kwk, (1.84)

whereνi k= νi k − νi k . The sum over all source terms vanishes: n

i=1

ωi = 0. (1.85)

Using (1.84) the heat release rate in (1.76) may be transformed to − n i₌₁ hiωi= − r k₌₁ n i₌₁ νi kWihiwk= r k₌₁ Qkwk, (1.86) where Qk= − n i₌₁ νi kWihi (1.87)