A. Y. Klimenko

Conditional moment closure for turbulent combustion

This paper reviews the fundamentals of conditional moment closure (CMC) methods for the prediction of turbulent reacting flows, with particular emphasis on combustion. It also surveys several of the applications that have been made. CMC methods predict the conditional averages and higher moments of quantities such as species mass fractions and enthalpy, conditional on the mixture fraction or reaction progress variable having a particular value. A brief introduction is given to generalized functions and probability density function (pdf) methods. This is followed by an exposition on the various methods of derivation for the CMC equation and the general characteristics of this equation and its boundary conditions. Simplifications that can be made in slender layer flows such as jets and plumes are outlined and examples of application of the technique to such flows are given. The method allows the definition of a new class of simplified reactors related to the well known perfectly stirred reactor and plug flow reactor: these are outlined. CMC predictions are compared to experiment and direct numerical simulations for flows with homogeneous turbulence. Derivation and modeling of the equations for conditional variances and covariances are outlined and their use in second-order CMC illustrated. Brief review is made of progress on application of the method to problems involving differential diffusion, multiple conditioning, sprays and premixed combustion.

The modeling of turbulent reactive flows based on multiple mapping conditioning

A new modeling approach—multiple mapping conditioning (MMC)—is introduced to treat mixing and reaction in turbulent flows. The model combines the advantages of the probability density function and the conditional moment closure methods and is based on a certain generalization of the mapping closure concept. An equivalent stochastic formulation of the MMC model is given. The validity of the closuring hypothesis of the model is demonstrated by a comparison with direct numerical simulation results for the three-stream mixing problem.

Testing multiple mapping conditioning mixing for Monte Carlo probability density function simulations

Mitarai [Phys. Fluids 17, 047101 (2005)] compared turbulent combustion models against homogeneous direct numerical simulations with extinction/recognition phenomena. The recently suggested multiple mapping conditioning (MMC) was not considered and is simulated here for the same case with favorable results. Implementation issues crucial for successful MMC simulations are also discussed.

A unified model of flames as gasdynamic discontinuities

Viewed on a hydrodynamic scale, flames in experiments are often thin so that they may be described as gasdynamic discontinuities separating the dense cold fresh mixture from the light hot burned products. The original model of a flame as a gasdynamic discontinuity was due to Darrieus and to Landau. In addition to the fluid dynamical equations, the model consists of a flame speed relation describing the evolution of the discontinuity surface, and jump conditions across the surface which relate the fluid variables on the two sides of the surface. The Darrieus-Landau model predicts, in contrast to observations, that a uniformly propagating planar flame is absolutely unstable and that the strength of the instability grows with increasing perturbation wavenumber so that there is no high-wavenumber cutoff of the instability. The model was modified by Markstein to exhibit a high-wavenumber cutoff if a phenomenological constant in the model has an appropriate sign. Both models are postulated, rather than derived from first principles, and both ignore the flame structure, which depends on chemical kinetics and transport processes within the flame. At present, there are two models which have been derived, rather than postulated, and which are valid in two non-overlapping regions of parameter space. Sivashinsky derived a generalization of the Darrieus-Landau model which is valid for Lewis numbers (ratio of thermal diffusivity to mass diffusivity of the deficient reaction component) bounded away from unity. Matalon & Matkowsky derived a model valid for Lewis numbers close to unity. Each model has its own advantages and disadvantages. Under appropriate conditions the Matalon-Matkowsky model exhibits a high-wavenumber cutoff of the Darrieus-Landau instability. However, since the Lewis numbers considered lie too close to unity, the Matalon-Matkowsky model does not capture the pulsating instability. The Sivashinsky model does capture the pulsating instability, but does not exhibit its high-wavenumber cutoff. In this paper, we derive a model consisting of a new flame speed relation and new jump conditions, which is valid for arbitrary Lewis numbers. It captures the pulsating instability and exhibits the high-wavenumber cutoff of all instabilities. The flame speed relation includes the effect of short wavelengths, not previously considered, which leads to stabilizing transverse surface diffusion terms.

On simulating scalar transport by mixing between Lagrangian particles

Lagrangian particles with mixing can be used as direct numerical simulations (DNS), large eddy simulations (LES), or filtered density function (FDF) methods depending on conditions of the simulations. We estimate major parameters associated with the DNS, LES, and FDF regimes and demonstrate that, under certain conditions specified in the paper, simulations using different mixing models approach the DNS limit.

A detailed quantitative analysis of sparse-Lagrangian filtered density function simulations in constant and variable density reacting jet flows

Sparse-Lagrangian filtered density function (FDF) simulations using a generalized multiple mapping conditioning mixing model and density coupling via a conditional form of the equivalent enthalpy method are performed for both constant density and variable density turbulent jet diffusion flames. The consistency between the sparse-Lagrangian FDF for the reactive species and the Eulerian large eddy simulation (LES) for velocity along with the accuracy of the reactive species predictions relative to the exact equilibrium solution are presented in detail. The sensitivity to the number of particles used in the simulations, the mixing localization structure, chemistry and numerical time step are all investigated. The analysis shows that consistency between the FDF and LES fields is relatively insensitive to the sparseness of the particle distributions and other model parameters but that the reactive species are strongly dependent on the degree of mixing localization in the LES mixture fraction space. An algorith...

Stability of planar flames as gasdynamic discontinuities

The stability of a steadily propagating planar premixed flame has been the subject of numerous studies since Darrieus and Landau showed that in their model flames are unstable to perturbations of any wavelength. Moreover, the instability was shown to persist even for very small wavelengths, i.e. there was no high-wavenumber cutoff of the instability. In addition to the Darrieus-Landau instability, which results from thermal expansion, analysis of the diffusional thermal model indicates that premixed flames may exhibit cellular and pulsating instabilities as a consequence of preferential diffusion. However, no previous theory captured all the instabilities including a high-wavenumber cutoff for each. In Class, Matkowsky & Klimenko (2003) a unified theory is proposed which, in appropriate limits and under appropriate assumptions, recovers all the relevant previous theories. It also includes additional new terms, not present in previous theories. In the present paper we consider the stability of a uniformly propagating planar flame as a solution of the unified model. The results are then compared to those based on the models of Darrieus-Landau, Sivashinsky and Matalon-Matkowsky. In particular, it is shown that the unified model is the only model to capture the Darrieus-Landau, cellular and pulsating instabilities including a high-wavenumber cutoff for each.

On the relation between the conditional moment closure and unsteady flamelets

We consider the relation between the conditional moment closure (CMC) and the unsteady flamelet model (FM). The CMC equations were originally constructed as global equations, while FM was derived asymptotically for a thin reaction zone. The recent tendency is to use FM-type equations as global equations. We investigate the possible consequences and suggest a new version of FM: coordinate-invariant FM (CIFM). Unlike FM, CIFM complies with conditional properties of the exact transport equations which are used effectively in CMC. We analyse the assumptions needed to obtain another global version of FM: representative interactive flamelets (RIF), from original FM and demonstrate that, in homogeneous turbulence, one of these assumptions is equivalent to the main CMC hypothesis.

Matching the conditional variance as a criterion for selecting parameters in the simplest multiple mapping conditioning models

The simplest model within the multiple mapping conditioning (MMC) approach, that involves a single mixture-fraction-like reference variable, is considered in the Brief Communication. An important parameter—the minor dissipation time—remains unknown in the probabilistic version of the model. The present work demonstrates by the specially developed asymptotic analysis that the simplest MMC possesses an ability (although somewhat limited) to match the physical intensity of the conditional fluctuations and this match represents the criterion for proper selection of the minor dissipation time.

Lagrangian particles with mixing. II. Sparse-Lagrangian methods in application for turbulent reacting flows

Both parts of this work present a more detailed and specific analysis of ideas introduced in the previously published letter [Phys. Fluids 19, 031702 (2007)]. In Paper I [Phys. Fluids 19, 065101 (2009)], we show that the continuous scalar transport and diffusion can be accurately specified by means of mixing between randomly walking Lagrangian particles with scalar properties. Here, in Paper II, we deal with the situation where the number of particles is not sufficient to resolve all scales in turbulent flows and Lagrangian particles with mixing become an approximate model rather than a stochastic framework for solving exact scalar transport equations. We consider sparse-Lagrangian methods that use relatively small numbers of particles compared to the number of Eulerian grid points and discuss similarities and differences with conventional large eddy simulations—filtered density function (LES-FDF) methods. Special attention is paid to multiple mapping conditioning (MMC) formulation for sparse-Lagrangian s...