A new kinetic-energy-preserving method based on the convective rotational form

Abstract

We present a new conservative discretization of the Navier-Stokes convective term that is also kinetic energy preserving (KEP). For divergence-free velocity fields, the specialized splitting recovers the traditional convective rotational form and therefore exhibits enhanced discrete secondary consistency with respect to vortical dynamics. The proposed method is first developed in the incompressible setting and is then extended to compressible flows via a square-root density weighting. It furthermore forms the basis for identifying a novel one-parameter family of KEP methods that leverages small-scale corrections to the momentum pressure gradient. Matrix-vector analysis associated with diagonal-norm finite difference summation-by-parts operators is used to straightforwardly study the preservation properties of different convective term formulations, including the new method, per the limitations of a discrete product rule. Preliminary evaluations on canonical test cases (i.e., isentropic and Taylor Green vortex simulations) are used to demonstrate the robustness of the current splitting and its performance with respect to representing secondary quantities.