Generalised Langevin Model in Correspondence with a Chosen Standard Scalar-Flux Second-Moment Closure

Abstract

In the context of transported joint velocity-scalar probability density function methods, the correspondence between Generalised Langevin Models (GLM) for Lagrangian particle velocity evolution and Eulerian Reynolds-stress turbulence models has been established in the 1990's by S.B. Pope. It was shown that the GLM representation of a given Reynolds stress model is not unique. It was also shown that a given GLM together with a given mixing model for particle composition evolution implies a differential scalar-flux model. In this paper, we study how extra constraints can be applied on the choice of the GLM coefficients in order to imply a chosen scalar-flux model. This correspondence between GLM-implied and standard scalar-flux models is based on the linear relaxation term and on the mean velocity gradient contributions in the rapid term. In general, GLM-implied models possibly involve more terms (including anisotropy effects in the scalar-flux decay rate and some high-order terms in the rapid-pressure-scrambling term). The proposed form of the GLM supposes a non-constant value for the diffusion coefficient C0, originally identified as a Kolmogorov constant. Here, the value of C0 is determined in order to yield the Monin model for linear relaxation of the scalar-flux, and the constant in the rapid-pressure contribution is related to the choice of the parameter β in the GLM. We finally show how GLM-implied scalar-flux models are in general dependent on the choice of the mixing model and how the proposed GLM can reduce this dependency. These developments are illustrated by results obtained from calculations of the Sydney bluff-body stabilised flame HM1.Response to Reviewers: First of all, we would like to thank the reviewers for their comments which helped to improve the paper. #1We included all the suggestions of Reviewer #1. About remark 5: we used the more general formalism introduced by Wouters [REF. 4] that allows to deal with variable density flows, and with Reynoldsstress models that include cubic terms (we added a remark in the text after Equation (27) at the beginning of Section 4.1). #2The remarks of Reviewer #2 allowed us to distinguish more clearly in the text two aspects: 1. setting the C_0 value depending on the Monin linear relaxation term and 2. removing the effect of the mixing model as much as possible. This lead to some modifications of the text. The modification of β in order to be in agreement with Launder's model for the rapid-pressure-scrambling term is indeed pragmatic. We added some clarifications on this aspect on page 12 in the paragraph "Rapid-pressure-scrambling term". About the "philosophical questions" raised by the reviewer, here are some remarks on the modification of C_0 in order to lead the Monin linear return to isotropy (the main contribution of this paper). As mentioned by the reviewer "the scalar field is dynamically passive". With Taylor's hypothesis, we see the relation between the Monin constant and the GLM coefficient C_0. As discussed in the paper, there is no re...

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