Generalized theory for numerical instability of the Gaussian-filtered Navier-Stokes equations as a model system for large eddy simulation of turbulence

Abstract

The Gaussian-filtered Navier-Stokes equations are examined theoretically and a generalized theory of their numerical stability is proposed. Using the exact expansion series of subfilter-scale stresses or integration by parts, the terms describing the interaction between the mean and fluctuation portions in a statistically steady state are theoretically rewritten into a closed form in terms of the known filtered quantities. This process involves high-order derivatives with time-independent coefficients. Detailed stability analyses of the closed formulas are presented for determining whether a filtered system is numerically stable when finite difference schemes or others are used to solve it. It is shown that by the Gaussian filtering operation, second and higher even-order derivatives are derived that always exhibit numerical instability in a fixed range of directions; hence, if the filter widths are unsuitably large, the filtered Navier-Stokes equations can in certain cases be unconditionally unstable even though there is no error in modeling the subfilter-scale stress terms. As is proved by a simple example, the essence of the present discussion can be applied to any other smooth filters; that is, such a numerical instability problem can arise whenever the dependent variables are smoothed out by a filter.