Sub-Filter Scale Models for Scalar Transport in Large Eddy Simulations

Large eddy simulation (LES) of turbulent heat transfer in an infinite channel has been used to compare the performance of several promising sub-filter-scale models for modelling the transport of a passive scalar. The dynamic mixed model and the dynamic reconstruction model (a higher order version of the mixed model) have been reported in the literature to perform very well in LES of turbulent flow. Here these models are tested to determine the model’s suitability for modelling transport of a passive scalar. These models together with the dynamic Smagorinsky model and a no-model case, are tested at a Prandtl number of 0.71 and Reynolds number of 180 based on wall friction velocity and channel half width. Both the dynamic reconstruction model and the dynamic mixed model perform very well showing clear improvement in the prediction of the mean flow and other turbulent statistics compared to the no-model case. The standard dynamic Smagorinsky model without the additional reconstruction terms performs quite poorly. Introduction In a large eddy simulation, a low pass filter is applied to the governing equations, separating the large resolved scales from the unresolved sub-filter-scales (SFS). In most LES simulations the computational grid and the discretisation of the equations provide the implicit filter, where the filter width is taken as being proportional to the grid dimensions. There are several difficulties however with the implicit nature of the filter in these simulations. Firstly, with low order accuracy finite difference schemes, the implicit filtering is smooth, meaning it removes energy from the large resolved scales as well as the small scales [3]. The energy removed from the large scales then needs to be reconstructed by the SFS model. When only an implicit filter is used, the shape of the filter is unknown making this reconstruction difficult. Secondly, unless high order finite differencing schemes are used, the numerical error in the small resolved scales is significant. It has been long known that using a grid size smaller than an explicitly applied filter would provide a means of reducing the numerical error in the smallest resolved scales. Recent work has suggested revisiting these ideas [1, 2]. Carati et al. [4] illustrate how combined discretisation — implicit filtering (denoted by an operator G̃) and explicit filtering (denoted by the operator Ḡ) — affects the decomposition of the velocity field. The authors re-write the governing equations to distinguish between the explicit filtering and discretisation operations as follows, ∂ũi ∂xi = 0, (1) ∂ũi ∂t + ∂ ̃ (ũiũ j) ∂x j =− ∂ p̃ ∂xi +ν ∂ũi ∂x j∂x j − ∂τ̃i j ∂x j , (2) where τi j = (uiu j)− (ũiũ j). Carati et al. [4] proposed that the residual stresses from equation (2) could be decomposed to τi j = τRSFS + τSGS where τSGS = (uiu j − ũiũ j) and τRSFS = (ũiũ j− ̄̃ ui ̄̃ u j). τSGS is the sub grid scale (SGS) stress that cannot be captured by the grid or implicit filter. τRSFS represents the interactions of the resolved scales ( ̄̃ u) and SFS motions (ũ− ̄̃ u), which are the filtered scales that are still supported by the grid. These ideas have been successfully applied to several LES simulations of turbulent flow. Winckelmans et al. [6] formulated a dynamic reconstruction model (DRM) based on the explicit filtering framework they introduced. Gullbrand and Chow [7] implemented a higher order version of the reconstruction model and found improved performance over the dynamic mixed model (DMM) of Zang et al. [8] and the dynamic Smagorinsky model (DSM) of Germano et al. [9] in a turbulent channel flow simulation. Gullbrand and Chow formulated the SFS stress as follows, τRSFS = (ũi ũ ? j )− ( ̄̃ ui ̄̃ u j) and τSGS = −2cs(∆̃)| ̄̃ S| ̄̃ Si j , where ũi is an approximation of ũi found using, ũi ≈ ũi = N ∑ n=0 (I− Ḡ)ũi. (3) In this way, the model is simply a higher order version of the dynamic mixed model of Zang et al. [8], where ũ in the τRSFS term is approximated by ̄̃ u instead of u?. Chow et al. [5] applied the DRM to an atmospheric boundary layer simulation and found improved performance compared with DSM and DMM models. In this study the interest is in determining how the DRM model performs when applied to turbulent transport of a passive scalar. Much of the development in SFS heat flux models has followed directly from models of the residual stress tensor in the momentum equations. The dynamic heat flux model proposed by Moin et al. [10] is based on the dynamic Smagorinsky model of Germano et al. [9]. The SFS heat flux (γ j) is modelled using, γ j =−cθ∆̃|S̃| ∂θ̃ ∂x j , where the model coefficient cθ is calculated dynamically. Following this work a number of researchers have proposed non-linear models for the SFS heat flux term, which removes the assumption of alignment with the resolved temperature gradient. Salvetti and Banerjee [11] developed a dynamic two parameter model (DTM) which is similar to DMM of Zhang et al. [8]. In a priori tests the authors found both DMM and DTM had a high degree of correlation with DNS data for both heat flux and SFS stresses, while DSM was less satisfactory. Jiménez et al. [12] tested DMM, DTM and DSM in a mixing layer and found that the eddy diffusivity model works well, provided the resolved velocity field is captured well. In a posteriori tests, the authors found comparable results when DSM was used for modelling γ j and DMM used for modelling τi j and when DMM was used for both γ j and τi j. The results were not as good when DSM was used for modelling both γ j and τi j. Peng and Davidson [13] developed a tensor diffusivity model which formulates γ j ∝ −Si j∂θ/∂x j . Yin et al. [14] applied this model in a simulation of turbulent channel flow with buoyancy.