Numerical errors in subfilter scalar variance models for large eddy simulation of turbulent combustion
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Subfilter scalar variance is a key quantity for scalar mixing at the small scales of a turbulent flow and thus plays a crucial role in large eddy simulation (LES) of combustion. While prior studies have mainly focused on the physical aspects of modeling subfilter variance, the current work discusses variance models in conjunction with numerical errors due to their implementation using finite difference methods. Because of the prevalence of grid-based filtering in practical LES, the smallest filtered scales are generally under-resolved. These scales, however, are often important in determining the values of subfilter models. A priori tests on data from direct numerical simulation (DNS) of homogenous isotropic turbulence are performed to evaluate the numerical implications of specific model forms in the context of practical LES evaluated with finite differences. As with other subfilter quantities, such as kinetic energy, subfilter variance can be modeled according to one of two general methodologies. In the first of these, an algebraic equation relating the variance to gradients of the filtered scalar field is coupled with a dynamic procedure for coefficient estimation. Although finite difference methods substantially underpredict the gradient of the filtered scalar field, the dynamic method is shown to mitigate this error through overestimation of the model coefficient. The second group of models utilizes a transport equation for the subfilter variance itself or for the second moment of the scalar. Here, it is shown that the model formulation based on the variance transport equation is consistently biased toward underprediction of the subfilter variance. The numerical issues stem from making discrete approximations to the chain rule manipulations used to derive convective and diffusive terms in the variance transport equation associated with the square of the filtered scalar. This set of approximations can be avoided by solving the equation for the second moment of the scalar, suggesting that model's numerical superiority.