Generalized Smagorinsky model in physical space

A simple expression for the Smagorinsky length scale is derived using arguments in physical space. This expression is sensitive to the choice of the basis functions employed in a numerical method and can be used with the finite volume or the finite element method on fully unstructured grids and meshes. It is useful in implementing both the constant parameter and dynamic versions of the Smagorinsky model. For a polynomial basis, the dependence of the Smagorinsky length scale on the order of completeness of the basis functions and the degree of anisotropy of the grid is examined. It is found that the length scale decreases with increasing polynomial order and that it scales with the smallest dimension of the grid. Comparisons are drawn between the new physical space approach and the traditional wavenumber space approach and the extension of the new approach to the finite difference method is discussed. In addition, the effect of a finite inertial range and contribution from molecular viscosity on the Smagorinsky length scale is considered.