On the construction of discrete filters for symmetry-preserving regularization models

Since direct numerical simulations cannot be computed at high Reynolds numbers, a dynamically less complex formulation is sought. In the quest for such a formulation, we consider regularizations of the convective term that preserve the symmetry and conservation properties exactly. This requirement yielded a novel class of regularizations [Verstappen R. On restraining the production of small scales of motion in a turbulent channel flow. Comput Fluids 2008;37:887–97.] that restrains the convective production of smaller and smaller scales of motion in an unconditionally stable manner, meaning that the velocity cannot blow up in the energy-norm (in 2D also: enstrophy-norm). The numerical algorithm used to solve the governing equations must preserve the symmetry and conservation properties too. To do so, one of the most critical issues is the discrete filtering. The method requires a list of properties that, in general, is not preserved by classical filters for LES unless they are imposed a posteriori. In the present paper, we propose a novel class of discrete filters that preserves such properties per se  . They are based on polynomial functions of the discrete diffusive operator, D∼, with the general form F=I+∑m=1MdmD∼m. Then, the coefficients, dm, follow from the requirement that, at the smallest grid scale kc, the amount by which the interactions between the wavevector-triples (kc, kc − q, q) are damped must become virtually independent of the qth Fourier-mode. This allows an optimal control of the subtle balance between convection and diffusion at the smallest grid scale to stop the vortex-stretching. Finally, the resulting filters are successfully tested for the Burgers’ equation.