A multienvironment conditional probability density function model for turbulent reacting flows

Abstract

The multienvironment conditional probability density function (MECPDF) model was first proposed by Fox [Computational Models for Turbulent Reacting Flows (Cambridge University Press, Cambridge, 2003)] as a simple extension of multienvironment probability density function models for turbulent reacting flows. Like the conditional moment closure (CMC) and the laminar flamelet model (LFM), the MECPDF model describes the reacting scalars conditioned on the value of the mixture fraction. However, unlike CMC and LFM, the new model provides a consistent description of conditional fluctuations in both the scalar dissipation rate and the reacting scalars, and hence can be used to model partial extinction and reignition in homogeneous turbulent reacting flows. In this work, a general derivation of the MECPDF model is presented for a single reaction-progress variable using the direct quadrature method of moments. Extensions of the model to multiple reaction-progress variables and conditioning on the mixture-fraction vector are also discussed. After deriving the model, the closure assumptions are validated using direct simulations for pure diffusion of two randomly distributed, initially correlated scalar fields. Two homogeneous applications are then considered: nonreactive mixing starting from nontrivial initial conditions, and reactive mixing with partial extinction and reignition. The multienvironment conditional probability density function (MECPDF) model was first proposed by Fox [Computational Models for Turbulent Reacting Flows (Cambridge University Press, Cambridge, 2003)] as a simple extension of multienvironment probability density function models for turbulent reacting flows. Like the conditional moment closure (CMC) and the laminar flamelet model (LFM), the MECPDF model describes the reacting scalars conditioned on the value of the mixture fraction. However, unlike CMC and LFM, the new model provides a consistent description of conditional fluctuations in both the scalar dissipation rate and the reacting scalars, and hence can be used to model partial extinction and reignition in homogeneous turbulent reacting flows. In this work, a general derivation of the MECPDF model is presented for a single reaction-progress variable using the direct quadrature method of moments. Extensions of the model to multiple reaction-progress variables and conditioning on the mixture-fraction vector are also discussed. After deriving the model, the closure assumptions are validated using direct simulations for pure diffusion of two randomly distributed, initially correlated scalar fields. Two homogeneous applications are then considered: nonreactive mixing starting from nontrivial initial conditions, and reactive mixing with partial extinction and reignition. Keywords Center for Turbulence Research Disciplines Chemical Engineering Comments This article is from Physics of Fluids

A term-by-term direct numerical simulation validation study of the multi-environment conditional probability-density-function model for turbulent reacting flows

“…This numerical approximation is sufficient to close the conditional reaction source term 2,10 and no physical modeling is required, similar to the situation in Lagrangian PDF methods. The remaining correlations of the conditional dissipation terms are approximated as…”

“…2 The use of Gaussian quadrature in the current derivation offers perspective on how the weights and abscissas can be determined from the conditional moments. Most readers will be familiar with a simplified form of Gaussian quadrature where the weighting function, f Y͉ , is uniform over the interval ͑−1,1͒, for which the abscissas are given as the roots of the Legendre polynomial of order N E .…”

“…2 However, the "conservation correction terms" are neglected because conservation is obtained by enforcing Eq. ͑18͒ ͓or equivalently Eq.…”

Direct Quadrature Conditional Moment Closure for Modelling of Turbulent Combustion

“…The DQCMC method is derived starting from the following transport equation for the spatially homogeneous equation for the joint PDF [2,11] as:…”

“…The present method is closely related to the direct quadrature method of moments (DQMOM) by Fox and co-workers [11][12][13][14]. However, there are some distinct differences which will be outlined below.…”