Sub-grid scale characterization and asymptotic behavior of multi-dimensional upwind schemes for the vorticity transport equations

Abstract

We study the sub-grid scale characteristics of a vorticity-transport-based approach for large-eddy simulations. In particular, we consider a multi-dimensional upwind scheme for the vorticity transport equations and establish its properties in the underresolved regime. The asymptotic behavior of key turbulence statistics of velocity gradients, vorticity, and invariants is studied in detail. Modified equation analysis indicates that dissipation can be controlled locally via non-linear limiting of the gradient employed for the vorticity reconstruction on the cell face such that low numerical diffusion is obtained in well-resolved regimes and high numerical diffusion is realized in under-resolved regions. The enstrophy budget highlights the remarkable ability of the truncation terms to mimic the true sub-grid scale dissipation and diffusion. The modified equation also reveals diffusive terms that are similar to several commonly employed sub-grid scale models including tensor-gradient and hyper-viscosity models. Investigations on several canonical turbulence flow cases show that large-scale features are adequately represented and remain consistent in terms of spectral energy over a range of grid resolutions. Numerical dissipation in under-resolved simulations is consistent and can be characterized by diffusion terms discovered in the modified equation analysis. A minimum state of scale separation necessary to obtain asymptotic behavior is characterized using metrics such as effective Reynolds number and effective grid spacing. Temporally-evolving jet simulations, characterized by large-scale vortical structures, demonstrate that high Reynolds number vortex-dominated flows are captured when criteria is met and necessitate diffusive non-linear limiting of vorticity reconstruction be employed to realize accuracy in under-resolved simulations.