#### Abstract

This paper extends a recent theoretical study that was previously presented in the form of a brief communication (Zimont, C&F, 192, 2018, 221-223), in which we proposed a simple splitting method for the derivation of two-fluid conditionally averaged equations of turbulent premixed combustion in the flamelet regime, formulated more conveniently for applications involving unclosed equations without surface-averaged unknowns. This two-fluid conditional averaging paradigm avoids the challenge in the Favre averaging paradigm of modeling the countergradient scalar transport phenomenon and the unusually large velocity fluctuations in a turbulent premixed flame. It is a more suitable conceptual framework that is likely to be more convenient in the long run than the traditional Favre averaging method. In this article, we further develop this paradigm and pay particular attention to the problem of modeling turbulent premixed combustion in the context of a two-fluid approach. We formulate and analyze the unclosed differential equations in terms of the conditions of the Reynolds stresses , and the mean chemical source , which are the only modeling unknowns required in our alternative conditionally averaged equations. These equations are necessary for the development of model differential equations for the Reynolds stresses and the chemical source in the advanced modeling and simulation of turbulent premixed combustion. We propose a simpler approach to modeling the conditional Reynolds stresses based on the use of the two-fluid conditional equations of the standard “” turbulence model, which we formulate using the splitting method. The main problem arising here is the appearance in these equations of unknown terms describing the exchange of the turbulent energy and dissipation rate in the unburned and burned gases. We propose an approximate way to avoid this problem. We formulate a simple algebraic expression for the mean chemical source that follows from our previous theoretical analysis of the transient turbulent premixed flame in the intermediate asymptotic stage, in which small-scale wrinkles in the instantaneous flame surface reach statistical equilibrium, while the large-scale wrinkles remain in statistical nonequilibrium.

#### 1. Introduction

We use the term ‘paradigm’ in the title of this article to emphasize that the two-fluid approach is a conceptual framework for analyzing and modeling turbulent premixed combustion in the flamelet regime. Paradigms in turbulent combustion research, with their basic conceptual viewpoints and principles of theory and modeling, have been discussed in the literature (see, for example, the paper entitled “Paradigm in Turbulent Combustion Research” by Bilger et al. [1], which contains relevant citations). In this paper, we primarily analyze turbulent premixed combustion in the context of “Damköhler’s paradigm” [1], which implies that instantaneous combustion takes place in a strongly wrinkled laminar flame, and we touch only briefly upon the thickened flamelet regime. Bray, Moss, and Libby proposed the progress variable for use in the description of premixed combustion of a single quantity and developed BML formalism, which forms the basis of the BML model developed using the Favre ensemble averaging method [2]. This more elaborate Damköhler’s laminar flamelet paradigm can be referred to as the “BML Favre averaging paradigm”. We remind readers that this paradigm leads to the emergence of a fundamental difficulty in the theory and modeling of turbulent premixed combustion involving the adequate prediction of the unknown mean flux of the progress variable and stress tensor , which appear in the unclosed Favre-averaged combustion and hydrodynamics equations of the problem. The point is that the scalar flux is predominantly (but not always) countergradient (sometimes called ‘countergradient turbulent diffusion’), and the stress tensor strongly depends on abnormally large velocity fluctuations observed in the turbulent premixed flame. The gas dynamic (nonturbulent) nature of these phenomena was clearly explained by Pope in [3]:

“As well as looking at the detail structure of a turbulent premixed flame, we can examine mean quantities. Here too, in comparison to other turbulent flows, there are some unusual observations, the most striking being countergradient diffusion. Within the flame there is a mean flux of reactants due to the fluctuating component of the velocity field. Contrary to normal expectations and observations in other flows, it is found that this flux transports reactants up the mean-reactants gradient, away from the products (hence countergradient diffusion). A second notable phenomenon is the large production of turbulent energy within the flame: Behind the flame the velocity variance can be 20 times its upstream value. Both these phenomena result from the large density difference between reactants and products and from the pressure field due to volume expansion […] From the Euler equations it is readily seen that a given pressure gradient accelerates the light products more than the heavier reactants. This mechanism is responsible both for countergradient diffusion and for turbulent energy production”.

Many articles have been devoted to theoretical and experimental studies of these phenomena. More advanced theoretical and modeling approaches in the context of the BML Favre averaging paradigm use the Favre-averaged unclosed equations in terms of the components of the scalar flux and stress tensor , and further approximation of the unknowns appears within turbulence theories (see, for example, [4]). To justify the need for our study, we emphasize that the countergradient scalar flux and abnormally large velocity fluctuations cannot be described adequately in the context of the BML Favre averaging paradigm (as we will illustrate below using as an example the results obtained in [4]). The reason for this is that these phenomena are caused by the large difference in the conditional mean velocities and , due to the different pressure-driven acceleration of the heavier unburned and lighter burned gases. These velocities, which cannot be predicted by the Favre-averaged equations, are at the same time described directly by the two-fluid conditional equations. It would be appropriate here to quote a statement made by Forman Williams in his Hottel lecture entitled “The Role of Theory in Combustion Science” [5]:

“It is relevant to distinguish between the science and the technology of the subject. The march of technology has never hesitated. It uses science whenever possible but often, especially in combustion, forges ahead by trial and error, or fortuitously by application of scientific misconception, but without scientific understanding”.

In this context, the use of Favre averaging in the modeling of turbulent premixed combustion in the flamelet regime is, strictly speaking, the “application of scientific misconception”, in contrast to two-fluid conditional averaging, which is conceptually adequate. Hence, two fundamental modeling problems arise in the turbulent combustion theory developed in the framework of the Favre averaging paradigm: the issue of adequate prediction in the premixed flame of both phenomena (“countergradient turbulent diffusion” and “unusually large generation of turbulence”) disappears in the framework of the two-fluid conditional average paradigm.

As far as we know, the pioneers of the theoretical two-fluid approach to turbulent premixed combustion were Spalding [6] and Weller [7]. In [6], Spalding also analyzed some previous papers related to the two-fluid approach. In [7], Weller presented conditionally averaged two-fluid equations that in fact were model equations. The exact, unclosed, two-fluid conditionally averaged mass and momentum equations, which served as a starting point for our study, were obtained in [8]. This paper extends our recent theoretical development of the two-fluid theory of turbulent premixed combustion described in [9] in the format of a brief communication, in which we proposed a simple and physically clear splitting method for the derivation of the unclosed two-fluid conditional equations of turbulent premixed combustion. This method made it possible to rederive the conditional equations obtained in [8] and to understand the reason for the appearance of surface-averaged variables in these, the impossibility of the well-founded modeling of which is one of the obstacles to using this approach in applications; in addition, this allowed us to indicate a way of excluding their appearance. The only required modeling unknowns appearing in the original alternative two-fluid conditional equations formulated in [9] are the conditional Reynolds stress tensors and in the unburned and burned gases, and the mean chemical source .

This paper makes the following contributions:(i)We describe in more detail the splitting method and the derivation of the unclosed two-fluid conditionally averaged equations, both those described in the literature and the alternative versions.(ii)We obtain the unclosed equations in terms of the conditional moments and , by splitting the known unclosed equations in terms of the Favre average , and present the unclosed equation for the mean chemical source , which can be used for advanced modeling of the unknowns in the two-fluid conditionally averaged equations.(iii)We formulate simple models for estimating the unknown chemical source in the laminar and thickened flamelet combustion regimes, based on our previous theoretical studies of transient turbulent premixed flames in the intermediate asymptotic stage, which were published elsewhere.(iv)We reformulate the equations for the classical “” turbulence model in terms of the conditional turbulent energy and dissipation rate using the slipping method and estimate the potentialities and problems arising from the use of the obtained two-fluid “” model to model turbulent characteristics and the conditional Reynolds stresses and .

We consider the two-fluid approach to the theory and modeling of turbulent premixed combustion in the flamelet regime as a promising alternative in the long run to traditional methods based on Favre ensemble averaging. The two-fluid conditionally averaged equations adequately describe the hydrodynamic (due to the different pressure-driven acceleration of the unburned and burned gases) and turbulent (using the corresponding two-fluid unclosed equation and corresponding two-fluid turbulence model) effects and their interaction in the turbulent premixed flame. This allows us to eliminate the necessity of modeling the mean scalar flux and stress tensor, which presents a challenge in the context of the Favre averaging framework.

A strong argument in favor of the two-fluid approach is provided by a review [10] of many studies of the structure of the instantaneous combustion zone in turbulent mixed flames, which showed that combustion occurs in thin, strongly wrinkled flamelet sheets: “Thin flamelets are found to occur even when the Karlovitz number greatly exceeds unity. The preheat zone average thickness is no larger than the laminar value in many studies, while in some cases it is 2–4 times larger.” The two-fluid approach is applicable for modeling not only premixed combustion in the laminar flamelet regime, but also flames in the thickened flamelet regime. We draw the reader’s attention to the paper [11] quoted in [10], where the local flame structure of a premixed swirl-stabilized gas turbine burner has been investigated. We consider the result obtained in [11] that the flamelet thermal thickness in the investigated highly turbulent lean premixed flames is close to the thermal thickness of the laminar flame as a significant argument in favor of the use of the two-fluid approach.

The two-fluid conditionally averaged unclosed equations directly describe the conditional mean velocities and , the conditional mean pressures and , the mean progress variable , and hence the probabilities of unburned gas and burned gas . These can be used to calculate the mean density and pressure , the Favre-averaged progress variable , the Reynolds- and Favre-averaged velocities and , and the mean scalar flux . In conjunction with the corresponding combustion and turbulence models, these equations also describe the mean chemical source , the conditional Reynolds stresses and , the Favre-averaged stresses , and some other turbulent characteristics.

In our study, together with the conditional mean variables, we also use the Reynolds- and Favre-averaged ones, which are useful in the interpretation of the results of numerical simulations and comparison with experimental data. However, the chemical source , which depends on curtain statistical characteristics of randomly wrinkled instantaneous flame, is not a conditional mean characteristic. The source term represents the production rate per unit volume of the product. The rate per unit mass is a Favre-averaged characteristic . The unclosed equation for the chemical source, which is considered in this paper, was formulated in [12] in terms of . We express the variables and appearing in this equation using , , and , which are directly described by the two-fluid conditional equations.

The paper is organized as follows:

Section 2: The two-fluid mathematical model and the conditions of its applicability to premixed combustion in the laminar and thickened flamelet regimes are described.

Section 3: The potentialities and limitations of the Favre and two-fluid conditional averaging frameworks are discussed, and the aim of the study is formulated.

Section 4: A splitting method for the derivation of unclosed two-fluid conditional equations is presented; it explains the origin of surface-averaged unknowns and demonstrates how to avoid them.

Section 5: Two statistical concepts of premixed combustion in the flamelet regime are described in the form of a three-stage and a global one-stage process, resulting in equations with and without surface-averaged unknowns, respectively.

Section 6: An alternative system is presented for the two-fluid unclosed equations, in which the only unknowns that need to be modeled are the conditional Reynolds stresses and the chemical source.

Section 7: Unclosed equations are obtained for the conditional Reynolds stress using the splitting method.

Section 8: Unclosed equations for the mean chemical source are presented and analyzed.

Section 9: Equations for the conditional mean turbulent energy and dissipation rate are obtained by splitting the Favre-averaged equations of the classical “” turbulence model.

Section 10: The chemical source is modeled for transient flames in the laminar and thickened regimes.

Section 11: A summary of this work and a conclusion are presented.

Appendix: The hydrodynamic/hydraulic analytical theory of the countergradient scalar flux is described in detail, and its applications for impinging and Bunsen flames and criteria for transition of the scalar flux are discussed.

The preliminary results of the current paper have been presented at three conferences (Zimont, V.L., Gas Dynamics in Turbulent Premixed Combustion: Conditionally Averaged Unclosed Equations and Analytical Formulation of the Problem, the International Colloquium on the Dynamics of Explosions and Reactive System, University of California, Irvine, USA, July 24-29, 2011, pp. 1-6. Zimont, V.L., An Alternative Approach to Modeling Turbulent Premixed Combustion, the Seventh Mediterranean Combustion Symposium, Chia Laguna, Cagliari, Sardinia, Italy, September 11-15, 2011, pp. 1-11. Zimont, V.L., An Approach in Turbulent Premixed Combustion Research Based on Conditional Averaging, Joint Meeting of the British and Scandinavian-Nordic Section of the Combustion Institute, Cambridge, UK, March 27-28, 2014, pp. 67-68.).

#### 2. The Two-Fluid Mathematical Model and Its Applicability to Premixed Combustion

In this section, we consider the conditions for applicability of the two-fluid mathematical model (in which it is postulated that instantaneous combustion takes place in a wrinkled flame of zero thickness) to combustion in the flamelet regime (where instantaneous combustion takes place in a strongly wrinkled laminar flame) and to combustion in the thickened (microturbulent) flamelet regime, where eddies smaller than the laminar flame thickness penetrate and thicken the preheat zone of the instantaneous flame (Figure 1).

##### 2.1. The Two-Fluid Mathematical Model

The two-fluid mathematical model corresponds to the limiting case in which the width of the instantaneous laminar flame is zero; i.e., instantaneous combustion takes place in a strongly wrinkled flame surface that propagates in a direction normal to itself with speed . Thus, the turbulent flame in the “flame surface regime” consists of two fluids: entirely unburned and completely burned gases.

This model corresponds to a limiting case of the laminar flame, in which the molecular transfer coefficient and chemical time in the known expressions for the speed and width tend to zero, , , while their ratio tends to a constant . These conditions are as follows: where is the turbulent Reynolds number (in which the kinematical viscosity coefficient and molecular transfer coefficient are of the same order, ), is the Damköhler number (where is the turbulent time), and is the Kolmogorov microscale (the size of the minimal eddies). Equation (1b) shows that close to the limit, the size of the minimal eddies remains much larger than the width of the flame . This indicates that the influence of the stretch and curvature of the flame on its structure and speed caused by turbulence is negligible; i.e., in this limiting case, the speed is not an assumption, but an exact result.

In this case, the probability density function (PDF) of the progress variable is bimodal: , where and are the probabilities of unburned and burned gases, respectively. The symbol denotes the Dirac generalized function, and the values of the progress variable are for unburned and for burned gases.

##### 2.2. The Two-Fluid Mathematical Model and Premixed Combustion in the Laminar Flamelet Regime

The laminar flamelet regime arises in the case of developed turbulence (), when the thickness of the laminar flame is much less than the size of the smallest eddies, . Since in this case, instantaneous combustion occurs in a highly wrinkled, thin laminar flame, and the application of the two-fluid mathematical model is justified. We rewrite the inequality using the expressions for , and the definition of the Damköhler number, as follows:

This inequality shows that in order for the instantaneous flame to be laminar, the Damköhler number must be very large, even in the case of smaller Reynolds numbers. For example, for values of and the LHS inequalities become and . At smaller Damköhler numbers, the laminar flamelet regime does not exist, and the question of the applicability of the two-fluid approach and the corresponding criteria need special consideration.

##### 2.3. The Two-Fluid Mathematical Model and Combustion in the Microturbulent Flamelet Regime

In the case of smaller Damköhler numbers for which the width of the laminar flame becomes larger than the sized of minimal turbulent eddies, , the smaller eddies penetrate the flame, intensifying the transport processes and thickening the flame. If this thickening is not very large, so that the instantaneous microturbulent flame remains strongly curved and the probability of the intermediate states is small, a two-fluid mathematical model will be applicable to this case. We now obtain the necessary conditions using our theoretical results for the parameters of the thickened (microturbulent) flame, as reported in [16]. These results served as the basis for the TFC turbulent premixed combustion model [17], which we use in this paper to model the mean chemical source appearing in the two-model conditionally averaged equations.

The speed , width , and microturbulent transfer coefficient of the thickened flamelet are described by the following expressions (Eq. in [16] and Eq. in [17]): which were obtained on the assumption that the microturbulent transport inside the thickened flame is controlled by turbulence at statistical equilibrium with a “” spectrum.

The speed of the turbulent flame is , where is the dimensionless area of the thickened flamelet sheet. Since (meaning that the flamelet sheet is strongly wrinkled) and (the speed of the developed turbulent premixed flame is of an equal order of magnitude to the characteristic velocity fluctuation), it follows that . From this and the inequality, we can derive a criterion for the existence of a combustion regime in which the microturbulent flamelet sheet remains relatively thin () and strongly wrinkled () (Eqs. and in [16] and Eq. in [17]): We now modernize this criterion slightly, keeping in mind that thickening of the instantaneous flame takes place when , i.e., when , where is the Karlovitz number. When the condition for strong wrinkling of the thickened flamelet sheet by large-scale eddies, , is added, the altered criterion is as follows:

These two criteria for the existence of a strongly wrinkled thickened flamelet sheet are similar in meaning and are satisfied at large Reynolds and moderately large Damköhler numbers.

*Example.* For , , (4) and (5) become and ; for , , (4) and (5) become and .

We now estimate the probability of the intermediate states, . We assume that the speed and width of the turbulent flame are and , and hence . The ratio of the volume per unit area of the flamelet sheet and the turbulent flame is and , giving a small characteristic value of the probability of finding a flamelet sheet in the flame: .

We see that the two-fluid approach is approximately applicable to the thickened flamelet combustion regime when the values of the Damköhler number are sufficiently large, . For smaller Damköhler numbers, , at which the preheat zone becomes thicker and less wrinkled, and especially for small numbers, , direct application of the two-fluid approach becomes less suitable.

In [16], a physical explanation was given for why the thickened flamelet, which occurs when , remains thin, meaning that there is no successive involvement of turbulent eddies of all sizes in the instantaneous flame and a distributed combustion regime is achieved. When its structure reaches statistical equilibrium, flamelet thickening reaches a limit in which “the heat fluxes in the front because of heat conduction and convection, and heat liberation because of chemical reaction have one order of magnitude” (see Eq. in [16]).

Even in the case of premixed flames subjected to extreme levels of turbulence studied in a recent paper [18], the instantaneous reaction layer remains thin, continuous, and strongly wrinkled:

“Unlike the preheat zones, which grew exponentially with increasing values of (up to 10 times their laminar value), the reaction layers in all 28 cases study here remained relatively thin, not exceeding 2 times their respective measured laminar values, even though the turbulence level () increased by a factor of 60 […] The reaction layers are also observed to remain continuous; that is, local extinction events are rarely observed”.

These experimental data show that a distributed reaction zone is not observed and that accounting for pressure-driven effects can be important for a proper description of the flame even in cases of very strong turbulence. The reaction layers in all 28 cases studied here remained relatively thin, not exceeding two times their respective measured laminar values, even though the turbulence level () increased by a factor of “60”. We notice that the preheat zones were investigated in [18] using laser induced fluorescence of formaldehyde. Obtained images of the preheat zones do not allow us to find the density gradient that is the relevant quantity for the estimation of a characteristic width of the instantaneous flame. The characteristic width can be significantly smaller than what the images show keeping in mind the very small thickness of the instantaneous reaction zone. The applicability of the two-fluid approximation to the turbulent premixed flames subjected to extreme levels of turbulence remains an open question.

#### 3. Unclosed Favre-Averaged and Two-Fluid Conditional Equations: Formulation of the Problem

In this section, we consider two systems of unclosed equations for a turbulent premixed flame in the context of the Favre averaging and two-fluid conditional averaging frameworks; compare their potentialities and limitations; and formulate problems arising in the context of the two-fluid framework.

##### 3.1. Unclosed Favre-Averaged Equations and the Challenge of Countergradient Turbulent Diffusion

The Favre-averaged combustion and hydrodynamics equations for premixed combustion are as follows:

The Favre-averaged value of a variable is and its instantaneous value is , where the notation identifies Reynolds averaging, the mean density and , and are the probabilities of the unburned and burned gases . The term in the combustion equation (6a) is a scalar flux for which the component is . In the momentum hydrodynamics equations (6c), is a tensor for which the component is . In (6b) for the mean density, and are the densities of the unburned and burned gases.

The variables defined for the system in (6a), (6b), (6c), and (6d) are the mean density and pressure , the Favre-averaged progress variable , and speed . The unknowns in the modeling are the scalar flux (the mean flux of the progress variable) , the stress tensor , and the mean chemical source (the mass consumption rate of the unburned gas per unit volume), which in the case of the flame surface regime is , where is the speed of the instantaneous flame relative to the unburned gas and is the flame surface density (the mean area of the instantaneous flame per unit volume). The mass equation (6d) does not contain unknown terms.

“The use of Favre averaging was initially introduced to reduce the system of transport equations to a form equivalent to the case of constant density flows and then using the same turbulent closures, for example, gradient transport with eddy diffusivity modeling of second order transport correlation”. The challenge of modeling the scalar flux and stress tensor in the turbulent premixed flame arises from the fact that the former is unusual in the turbulent diffusion countergradient direction, and the latter is abnormally large for the turbulent flow velocity fluctuations described by the diagonal terms of the tensor. As mentioned in the introduction, the gas dynamics nature of these phenomena was clearly explained by Pope in [3].

It is unlikely that the phenomena of the countergradient scalar flux and abnormally large velocity fluctuations in the turbulent premixed flame, caused by the different pressure-driven acceleration of the unburned and burned gases, can be modeled adequately in the context of the Favre averaging paradigm, even using an advanced approach based on attracting the unclosed differential equations for the scalar flux and stresses, and even though these equations involving and implicitly contain this hydrodynamic mechanism. The point is that closure approximations for these equations in the context of the Favre averaging framework, where the unknowns are expressed in terms of the Favre-averaged parameters defined by the equations of the problem, do not provide adequate modeling of the hydrodynamic pressure-driven effect. This type of closure is performed in the same manner as that commonly used in turbulence theory; the scalar flux and stresses are presented in the model equations as turbulent phenomena. As an illustration of this shortcoming, we will refer to the results of calculations of the scalar flux on the axis of the impinging premixed flame presented in [19], in which a model theory of this type was developed for the mean fluxes and stresses. The results of the numerical simulations performed in [4] (Figures 5 and 6 in [4]) show a qualitatively different behavior of the pressure profiles on the axis of the flame. In the flame close to the wall (Figure 5 in [4]) the pressure increases slightly across the flame due to the strong influence of the wall on the pressure profile, meaning that in this case, the scalar flux cannot be countergradient. In the free-standing impinging flame, where influence of the wall is small, the strong pressure drops across the flame where (Figure 6 in [4]) can yield a countergradient scalar flux. At the same time, the numerically simulated scalar fluxes presented in these figures are in both cases countergradient and close in value, suggesting that the Favre averaging paradigm does not provide a theoretically adequate description of the effects caused by the different pressure-driven acceleration of the unburned and burned gases.

The reason for this drawback is that the closure of these equations is performed in the context of the Favre averaging framework, in which the unknowns are expressed in terms of the Favre-averaged parameters described by the equations of the problem, and does not adequately model the hydrodynamic pressure-driven effect described without modeling in the context of a conditional averaging paradigm. This type of closure is performed in the same way as that commonly used in the turbulence theory; that is, the scalar flux is interpreted as a turbulent phenomenon. The closure used in [4] allowed the authors to show agreement between their results (using a corresponding empirical constant) and known experimental data for the free-standing flame where the scalar flux was countergradient (Figure 4 in [4]). However, the results of numerical modeling with these empirical constants for the impinging flame close to the wall were not physically feasible, even in a qualitative sense.

The results of numerical simulations of the impinging flame presented in [19] show a gradient scalar flux in the flame close to the wall, and a predominantly countergradient flux in the free-standing flame. In these simulations, the mean scalar flux was estimated as a sum of the terms that describe the countergradient hydrodynamic contribution due to the differences in pressure-driven acceleration of the heavier unburned and lighter burned gases, and the gradient contribution due to turbulent diffusion. We discuss these results in the appendix.

##### 3.2. The Two-Fluid Equations and Modeling Challenges

The conservation equations for the unclosed two-fluid mass (7a), (7b), momentum (7c), (7d), and progress variable were formulated in [8] (Eqs. (23)-(26a) and (26b)) and (17), as follows: where the indexes “” and “” refer to the unburdened and burned gases, respectively; and are the displacement speeds of the instantaneous flame surface relative to the unburned and burned gases ( using a common notation); is the flame surface density (FSD), the mean area of the instantaneous flame per unit volume; and is the unit normal vector on the flame surface toward the unburned side. The mean progress variable is equal to the probability of the burned gas , where the values of the progress variable are and , i.e.,

In our analysis, we will use the mean chemical source rather than the FSD , where is the mass flux per unit area of the instantaneous flame. We can therefore designate the RHS terms in (7a) and (7b) as and , the second RHS terms in (7c) and (7d) as as and , and the final RHS term in (7e) as .

The system in (3a), (3b), and (3c) describes the variables , , , The unknowns that need to be modeled are the conditional tensors and (or the conditional Reynolds stress tensors and ), the mean chemical source , and the surface-averaged variables , , , , and defined on the surfaces in the unburned and burned gases adjacent to the instantaneous flame, respectively.

In contrast to the Favre averaging paradigm, the mean scalar flux does not need to be modeled, as it is described by the following closed expression: where and , in terms of the variable , are as follows: We can see that the phenomenon of the countergradient scalar flux in the turbulent premixed flame does need to be modeled, since , , and are described by the unclosed system in (7a), (7b), (7c), (7d), and (7e).

The components of the tensor are described by the following expression: Eq. (10) shows that the “abnormally large velocity fluctuations in the turbulent premixed flame” mentioned above, which are described by the diagonal terms of the tensor , are controlled by the differences in the mean conditional velocities described directly by the unclosed system in (7a), (7b), (7c), (7d), and (7e).

We proceed on the premise that the conditional averaging framework is likely to be more convenient in the long run than the Favre averaging framework, since the two-fluid equations more adequately describe the pressure-driven hydrodynamic processes in the premixed flame caused by a large difference in the densities of unburned and burned gases. Given the possibility of practical use of the two-fluid approach, we identified a critical obstacle in the fundamental difficulty of modeling of the surface mean variables, i.e., their expression in terms of the variables described by the system. At the same time, we proceeded based on the possibility of theoretical modeling of the unknown conditional Reynolds stresses and mean chemical source by modifying existing modeling approaches in the turbulence and combustion theories for our case.

Our formulation of the problem for this theoretical study follows from these considerations.

##### 3.3. Formulation of the Problem

When we began to address this problem, we noticed a connection between the structures of the Favre-averaged mass and moment equations (6d) and (6c), and the two-fluid conditionally averaged mass and moment equations (7a), (7b) and (7c), (7d), which influenced the formulation of the problem. By comparing the mass equations (6d) and (7a), (7b), we can see that the LHS of each conditional mass equation for the unburned (7a) and burned (7b) gases has the structure of the Favre-averaged equation (6d), but is expressed in terms of the conditional variables. The LHS of each conditional equation is weighted by the probabilities of the unburned () and burned ( gas, respectively. Each equation has sink and source terms on the RHS, which are denoted as and , where the chemical source is the same as in the Favre-averaged equation (6a). A similar connection exists between the structures of the momentum equations (6c) and (7c), (7d), where the sink and source represented by the two last terms in (7c), (7d) express the impulse exchange between the unburned and burned gases.

In the derivation proposed by Lee and Huh [8], generalized functions were used in intermediate, rather cumbersome mathematical manipulations and disappeared in the final conditional equations. The authors used zone averaging (as reflected in the title of the article) and did not actually rely on the two-fluid mathematical model. Since cumbersome calculations using an intermediate value of the progress variable that is nonexistent in the two-fluid mathematical model, , led to the physically obvious result mentioned above, a simple method for obtaining these equations by splitting the Favre-averaged mass and momentum equations must therefore exist.

We therefore formulate the research problems considered in this paper as follows:(1)A simple and physically insightful slipping method for the derivation of the two-fluid conditional mean equations, which allows us to formulate alternative unclosed two-fluid conditionally averaged momentum equations (using sound physical reasoning) that do not contain the surface-averaged unknowns; to formulate unclosed two-fluid equations for the conditional Reynolds stresses; and to reformulate the Favre-averaged equations of the “” turbulence model in terms of the mean conditional variables.(2)A derivation using an analysis of the splitting method of the two-fluid conditional equation in terms of the conditional Reynolds stresses and their analysis.(3)An unclosed equation for the mean chemical source in the context of the two-fluid approach.(4)Practical approaches to modeling turbulence in the unburned and burned gases in the context of a two-fluid version of the standard “” turbulence model, and the mean chemical source using a hypothesis of statistical equilibrium of the small-scale structures.

In preparation for analyzing and resolving the listed tasks in further sections, we aim to do the following:(i)To remind readers of the two-fluid one-dimensional analytical theory of the countergradient phenomenon developed by the present authors in [15] and to formulate a criterion for the transition from countergradient to gradient scalar flux in the premixed flame.(ii)To illustrate these results using the examples of impinging and Bunsen flames considered in [19] and [15], respectively.

Consideration of these results yields strong arguments in the favor of the two-fluid conditional averaging framework. We present these materials in the appendix, since although they are not required in order to read the subsequent sections of this article, they not only can help the reader to understand the problem more deeply, but also may be of interest in their own right.

#### 4. A Splitting Method and Alternative Derivation of the Known Conditional Equations

In this section, we consider the original derivation splitting method proposed in [9] and demonstrate its effectiveness by rederiving the known zone conditionally averaged mass and momentum equations reported in [8]. We notice that the derivation in [8] is rather cumbersome due to the use of generalized functions and a particular value of the progress variable in the intermediate mathematical manipulations, which disappear from the final unclosed conditionally averaged equations. Our derivation is more direct and simple.

In the next section, we derive alternative conditionally averaged momentum equations using the splitting method, in which the only unknowns are , , and .

##### 4.1. The Original Splitting Method

The central idea of this approach is simple: we split the Favre-ensemble-averaged mass (6d) and momentum (6c) equations into averaged two-fluid mass and momentum equations for the unburned and burned gases, respectively, using the obvious identity (where may be a constant, scalar, vector, or tensor variable). We then use the conservation laws for the instantaneous flame to determine the sink and source terms appearing in the two-fluid conditional mass and momentum equations.

##### 4.2. Conditionally Averaged Mass Equations

To transform (6d) using (11), we use and . This results in the following equation: where the expressions in the brackets and refer to reactants and products, respectively. We can easily check whether the expression in the second braces is equal to . To do this, we must eliminate from (1a) using (6a) and the obvious expressions:Using (12) then gives , and the conditional mass equations are as follows: (14a), (14b) are identical to the two-fluid mass equations and in [8].

##### 4.3. Conditionally Averaged Momentum Equations

In a similar way to the analysis in the previous subsection, we present the Favre-averaged momentum equations (6c) as a sum of two groups of terms, which contain conditionally averaged parameters referring to the unburned and burned gases. To obtain these, we insert into (6c) the expressions yielded by (11) with and . This results in the following equation: where and are conditionally averaged moments and pressures. Splitting the equation in (15) yields the following two-fluid momentum equations: where and are equal in value and oppositely directed () terms that express the impulse sink and source due to the mass exchange between the unburned and burned gases, and the inequality of the pressure on the different sides of the instantaneous flame surface. The equation expresses the conservation of the averaged impulse on the instantaneous flame surface. The equation follows from the instantaneous impulse conservation law where is the unit vector normal to the instantaneous flame surface. As we shall see below, the left-hand side and right-hand side of the equation (18) are equal, respectively, to and .

We represent the RHS of (6c) as follows: where the expression in the brackets is equal to zero in accordance with (17). After splitting the two-fluid momentum equations, (6c) with the RHS presented by (19) yields which are identical to the conditional averaged momentum equations and in [8] that were derived using generalized functions (see (7c) and (7d) above).

A comparison of (16a), (16b), (20a), and (20b) shows that the sink and source terms in (16a) and (16b) are as follows: To avoid misunderstanding, we should emphasize that the splitting method uniquely determines the conditionally averaged mass equations (14a) and (14b) and momentum equations (20a) and (20b).

The alternative conditional momentum equations without the surface-averaged unknowns, which we formulate in the next section, are based on an original concept of a one-step statistical interpretation of instantaneous combustion and an alternative averaged impulse conservation law corresponding to this concept.

#### 5. Original Statistical Concept of Premixed Combustion in the Flame Surface Regime

The surface-averaged terms in (21a) and (21b) describe the momentum exchange between the unburned and burned gases. To avoid the use of the surface-averaged variable, we must express the momentum exchange between the gases in terms of the conditional mean variables. We do this within the framework of the original statistical concept of an instantaneous flame, and we consider this below.

##### 5.1. Statistical Concepts of the One- and Three-Step Processes in Flame Surface Combustion

We consider two statistical concepts of the combustion process in the turbulent premixed flame in the flame surface regime. In accordance with the first concept, the combustion is considered as a global one-step process in which unburned gas with parameters transforms into burned gas with parameters , with a rate of transformation equal to . This global one-step process can be represented schematically as follows:

In accordance with the second concept, the global process is split into three subprocesses; these correspond to three successive stages and are represented as follows:

The second stage corresponds to the stepwise transformation of the unburned gas, located on the surface adjacent (infinitely close) to the instantaneous flame, to burned gas, appearing on the infinitely close adjacent surface on the other side of the flame. In the first and third stages, the change in the averaged variables takes place in the unburned and burned gases, respectively.

It is important to keep in mind that the averaged parameters and are defined at every point of the turbulent premixed flame, while the instantaneous parameters and , are defined in the unburned and burned gases, respectively, and the instantaneous parameters and are defined on the corresponding surfaces adjacent to the instantaneous flame, as shown in Figure 2.

In the general case, the conditional and surface-averaged variables in the unburned and burned gases are not equal, i.e., and . The surface-averaged velocity of the products directly generated by the instantaneous flame is not necessarily equal to the conditional mean velocity of the burned gas . Thus, the interaction of these “newborn” volumes of products with the main flow of the products generates the pressure gradient in the rear zone adjacent to the instantaneous flame surface, giving rise to the equalization of the velocities in the third stage. Similarly, the velocities and are not necessarily equal due to the pressure gradient generated by combustion in the front zone, which yields the transition .

The flamelet sheet separates the gases with conditional mean parameters and ; that is, this sheet includes the instantaneous flame surface and several adjacent layers of unburned and burned gases in which the transitions and take place. In our mathematical analysis, the width of this flamelet sheet is assumed to be zero.

We draw the reader’s attention to an analogy between the two statistical concepts of the flamelet combustion described above and two classical theoretical concepts of a detonation wave:(i)In the* Zel’dovich*-Neumann-Doering (ZND) theory, a detonation wave is a shock wave followed by a combustion front, i.e., a global process of transformation of cold reactants into hot products that involves two stages: compression of the unburned gas in the shock wave and its chemical transformation into burned gas at the combustion front. Thus, the intermediate parameters of the compressed reactants appear in the equations of the ZND theory.(ii)In the Chapman-Jouguet (C-J) theory, a detonation wave is considered to be a surface that divides the unburned and burned gases. This surface includes the shock wave and combustion front. Hence, the intermediate parameters of the gas do not appear in the equations of the C-J theory, which include only the initial parameters of the reactants and the final parameters of the products.

Using this analogy, we can say that the momentum equations containing intermediate surface-averaged variables (derived by Lee and Huh in [8]) correspond conceptually to the ZND theory, whereas our alternative momentum equations, which do not contain intermediate surface-averaged variables (see (27a) and (27b) below), correspond to the C-J theory.

##### 5.2. Impulse Conservation Law Corresponding to the Two Concepts of the Flame Sheet

The impulse conservation law (17) that was used for formulation of the sink and source terms on the LHS of (20a) and (20b) corresponds to the second concept of the flamelet sheet. It refers to the second stage in (23), in which the unburned gas flow crossing the instantaneous flame becomes the burned gas. The problem of estimating the unknown surface-averaged parameters appearing in this case is reduced to one of modeling the effects of the hydrodynamic mechanisms in the unburned and burned gases that control the transformations of the gas parameters and in the first and third stages.

We formulate the impulse conservation law for this sheet as follows: where is the averaged unit vector normal to the instantaneous flame surface. The terms and describe the sink and source of the impulse per unit volume caused by transformation of the unburned gas (with mean speed ) into the burned gas (with mean speed ), while the terms and describe the sink and source due to differences between the mean pressures and in the unburned and burned gases. Eq. (24) does not contain a term describing the correlation between the instantaneous pressures and and the unite vector , since these pressures refer to the unburned and burned gases, while the unit vector shows the local orientation of the instantaneous flame surface. We remind the reader that this correlation is significant in (17), since the pressures and are defined on adjacent surfaces that are infinitely close to the instantaneous flame with local orientations described by the unit vector .

In essence, the originality of our theoretical analysis consists of using a novel concept of a one-step conversion of unburned gas with conditional mean parameters into burned gas with parameters . This allows us to avoid the appearance of intermediate unknown surface-averaged parameters in the conditionally averaged momentum equations.

It seems that it is impossible to exclude surface-averaged parameters from the exact equation (17), meaning that there is no mathematical equivalence between (17) in terms of surface-averaged parameters and (24) in terms of conditional mean parameters. We therefore consider (24) as an approximation till a possible more rigorous proof. It should be noted that (24) retains original physical meaning also in the case of the thickened flamelet regime, in contrast to (17), where the surface-averaged variables that are defined on the surfaces adjacent to the instantaneous flame surface may cease to be strictly defined. Below we shall validate this equation against a result, known in the literature, from direct numerical simulation (DNS) of the one-dimensional premixed flame. In answer to possible objections from purists, we note that full mathematical strictness of the unclosed equations is not obligatory, since any turbulence and combustion model that could be used for estimation of the unknown conditional Reynolds stresses and mean chemical source appearing in the unclosed equations would inevitably be approximate.

In order to better understand the relationship between these equations, we use an analogy based on the result from turbulence theory that gives an exact formula for the mean dissipation rate in terms of the velocity derivatives. This result is controlled by small-scale turbulence; the formula for the dissipation rate in strong turbulence expressed in terms of the large-scale turbulent parameters (where is an empirical constant) does not follow from the previous formula, but is a consequence of Kolmogorov’s hypothesis of statistical equilibrium in the small-scale structure of turbulence. Although the exact formula and its approximate treatment cannot be mathematically equivalent, the scientific community consider Kolmogorov’s expression for the dissipation rate to be a theoretical result of turbulence mechanics.

Analogously, we consider the approximation in (24), which fortunately does not contain an empirical constant, as an approximate impulse conservation law for the instantaneous flame in the turbulent region at high Reynolds numbers (i.e., developed turbulence), large Damköhler numbers (i.e., fast chemistry), and large density ratios (i.e., a strong pressure-driven effect). In contrast to the exact equation in (17), this equation is expressed in terms of the conditionally averaged parameters that are described by the conditional mass equations (14a) and (14b) and formulated in the alternative momentum equations (27a) and (27b) below.

##### 5.3. Accuracy of the Alternative Impulse Conservation Law

We have not attempted to estimate the accuracy of (24) with respect to the global parameters of the turbulent flame. We do not expect high accuracy at constant density, , where both sides of (17) are identical. At the same time, the conditional mean speeds in (24) are not equal in the general case: and (the former difference controls the scalar flux in (7a) and affects the stresses in (7b)). Absent DNS results would indicate the extent to which the difference in the velocities in the case of constant density is compensated for by the inequality of the conditional mean pressures, . The DNS results obtained in [20] for a steady-state, planar premixed flame with density ratio 7.53 and Damköhler number 18.1 demonstrate that, in this case, the exact and approximate equations are fairly close. In order to estimate the terms in the exact impulse equation (17), the pressure fluctuation correlations and are ignored, and the terms containing pressure are presented in the forms and . The authors calculated these expressions, which using our notation become and . These formulas approximately express the differences in the RHS and LHS terms of (17) and (24), respectively. The graph for the former expression, given in Figure 4 of [20] for the range , shows that the value of this expression is close to zero over the whole range. The graph in Figure 6 of [20] for the latter expression presented for the range shows that the value of this expression is nearly zero in the range , becomes negative and decreases relatively slowly in the range , and then increases to zero in the range . In general, these results support the conclusion that the exact and approximate equations (17) and (23) are close.

We note that the authors of [20] proposed the use of approximation described above for the terms and in modeling conditionally averaged momentum equations that do not contain unknown surface-averaged terms. The pressure-related terms in these equations differ from the corresponding terms in the momentum equations that are obtained in the next section, using a statistical one-step model for transformation in the premixed flame of the unburned gas (with parameters , , ) into burned gas (with the parameters , , ).

It should be mentioned that this DNS has been performed for the turbulent premixed flame with a relatively low Reynolds number. We have no numerical data for the validation of (24) in the case of much higher Reynolds number where the interaction between hot and cold volumes can be more significant and the difference in the conditional mean velocities lower.

#### 6. Alternative Momentum Equations and Complete System of Unclosed Equations

In this section, we obtain conditionally averaged momentum equations without unknown surface-averaged terms and formulate the complete system of unclosed equations as a basis for modeling, as used in further sections.

##### 6.1. Formulation of the Alternative Conditional Momentum Equations

To find and in (16a) and (16b), which yield the alternative momentum equations, we represent the RHS of (6a) as follows: where the expression in the braces is equal to zero in accordance with (24). After splitting the expressions for and , (25) yields Hence, the desired alternative momentum equations are as follows:

We eliminate the variable using the expression , where . In order to eliminate the unit vector from (26a) and (26b), we use (24) to obtain ; i.e., the averaged unit vector is described by (28a): Combining (28a) with (24) and using the expression result in (28b), which we will use below. It follows from (28b) that the pressure difference becomes equal to the pressure drop across the laminar flame, , at the points of the turbulent flame where the vectors , and are collinear.

The reader needs to keep two points in mind: firstly, we regard the unburned and burned gases as incompressible fluids; secondly, the structures of the Favre-averaged equation and corresponding Reynolds-averaged equation of the same problem (which is associated with the constant density approximation) are identical. In our case, these are the momentum equation (6c) and the same equation in which the density is assumed to be constant.

Hence, the constant density equation (29) also can be used for the formulation of two-fluid conditional equations, leading to (16a) and (16b). We use this consideration in Section 7 when deriving the unclosed two-fluid conditional equations (33a) and (33b) for and , proceeding from the constant density equation (32) for . This trick can be useful when deriving other two-fluid conditional equations.

##### 6.2. Complete System of Unclosed Equations for Turbulent Premixed Combustion

In this section, we summarize the previous results and represent the complete system of unclosed equations, where the unknowns that need to be modeled are the conditionally averaged Reynolds stresses and the mean chemical source. (14a), (14b), (27a), (27b), (28a), and (28b) give the following system of unclosed equations: The system includes three scalar equations, (30a), (30b), and (30e), and two vector equations, (30c) and (30d). These describe three scalar variables, , and , and two vector variables, and . The unknowns are the conditional stress tensors and , with components and , and the chemical source .

In the context of the two-fluid approach, the terms that describe the scalar flux and stress tensor do not appear in the equations; this means that the challenge of modeling the phenomena of the countergradient scalar flux and abnormal velocity fluctuations in the turbulent flame, which are artifacts of Favre ensemble averaging, does not arise, as mentioned above. At the same time, a quantitative estimate of the mean scalar flux and stresses, and the Reynolds- and Favre-averaged parameters, may be necessary for physical interpretation of the results of numerical simulations and comparison with experimental data. These variables are described by the following exact expressions:

(31a), (31b), (31c), (31d), (31e), (31f), and (31g) are not an inherent part of the system in (30a), (30b), (30c), (30d), and (30e), as they are not required for a solution of (30a), (30b), (30c), (30d), and (30e) for given values of , and . (31a), (31b), (31c), (31d), (31e), (31f), and (31g) are needed to calculate , , , , , and using the values of the variables , , , and described by the system in (30a), (30b), (30c), (30d), and (30e), which can be used in the models for the unknown , and and for the interpretation of numerical simulations. We refer to the system that includes (30a), (30b), (30c), (30d), (30e), (31a), (31b), (31c), (31d), (31e), (31f), and (31g) as the complete equation system, as it describes the main Reynolds-, Favre-, and conditionally averaged variables used in applications. We can add equations, for example, formulas for , , , and , if the temperatures and species concentrations in the unburned and burned gases are known.

We call attention to the absence from the system in (30a), (30b), (30c), (30d), and (30e) of a specific combustion balance equation that is similar to (6a). Eqs. (30a)–(30d) are hydrodynamic mass and momentum equations containing the source , meaning that the general problem cannot be broken down into combustion and hydrodynamics subproblems. This does not contradict the initial formulation of the problem, in which (6a), (6b) and (6c), (6d) correspond to the combustion and hydrodynamic subproblems, respectively: the basic Favre-averaged combustion equation (6a) follows from the conditionally averaged mass equations (30a), (30b) and the expression (31f).

In the following two sections, we present unclosed equations for the unknown conditional Reynolds stresses and the mean chemical source, which can form the basis for advanced modeling approaches. In these sections, we propose practical approaches for the estimation of these unknowns using a two-fluid version of the standard “” turbulence model and simple theoretical expressions describing the mean chemical source in the transient turbulent premixed flame observed in the experiments.

The obtained system of the two-fluid equations can be numerically simulated using as a model called “Eulerian Multifluid Modeling” that is embedded in the commercial package Ansys Fluent 6.3 (https://www.sharcnet.ca/Software/Fluent6/html/ug/node900.htm#sec-multiphase-eulerian). Fluent’s Eulerian multiphase model does not distinguish between fluid-fluid and fluid-solid (granular) multiphase flows. The Fluent solution can be summarized as follows: (i) momentum and continuity equations are solved for each phase (in our case, for the unburned and burned gases); (ii) a single pressure is shared by all phases. The latter condition means that we must set in the two-fluid momentum equations (30c) and (30d) and omit (30e), which is superfluous in this case. This assumption seems permissible from a physical point of view.

It is necessary to stress that we set only due to peculiarities of this commercial package; that is, this assumption is not caused by the essence of the two-fluid approach. Note that assuming that conditional mean pressures are equal, strictly speaking, contradicts the basic idea of modeling the momentum interaction between the two fluids. A consequence of this assumption is that the conditional equations (30c) and (30d) lose accuracy (although the Favre-average equation (6d) remains exact while being the sum of these inaccurate equations) and therefore the conditional mean speeds as described by those equations are not quite accurate. However, such inaccuracies may be insignificant—as may be indicated indirectly by a comparison of the results, which follow from the simple hydraulic two-fluid theory of the countergradient phenomenon—using the assumption of equality of conditionally averaged pressures, with known experimental measurements of the countergradient scalar flux in the turbulent premixed flame; see Appendix A.1.

#### 7. Unclosed Equations in Terms of Moments

In this section, we obtain the two-fluid unclosed equations for the unknown conditional moments and . We use the splitting method to formulate the exact equations, which inevitably contain surface-averaged unknowns, and approximate alternative unclosed differential equations that do not contain these unknowns (this derivation is analogous to the use of the splitting method for the two-fluid conditional momentum equations).

We start with the constant density unclosed equations for that are as follows [21] (molecular viscous terms are omitted): Based on the remark at the end of Section 6.1, (32) leads to unclosed conditionally averaged equations for and as follows: Here and express the sink and source, .

##### 7.1. Exact Equations Containing Surface-Averaged Unknowns

We first derive exact expressions for and in terms of the surface-averaged parameters, which need to be modeled, and then avoid modeling by the formulation of approximate expressions for the sink and source in terms of conditional mean parameters.

We begin the derivation of the exact expressions for and by formulating the following conservation law for the and components of the impulse, which connect the parameters defined on the surfaces adjacent to the instantaneous flame: which follow from the instantaneous and averaged impulse equations for the component and similar equations for the component. By multiplying the RHS and RHS of (35a) and (35b) and averaging, we obtain the conservation law for the surface-averaged conditional second moments and as follows: When (i.e., there is no combustion), (34a) and (34b) become the identity , as all of the mean parameters defined on both adjacent surfaces are equal. For the case , we divide (36) by and then multiply it by ; remembering that , we obtain the following equation: Hence, the exact sink and source terms in (33a) and (33b) are as follows: