Exact expressions for the unresolved stress in a finite-volume based large-eddy simulation

In this article we propose new discretization-informed expressions for the residual stress tensor (RST) in a finite-volume based large-eddy simulation (LES-FVM). In addition to the classical RST $\overline{u u} - \bar{u} \bar{u}$ resulting from the non-commutation between filtering and the nonlinear stress, our RST also contains contributions from the numerical flux, discrete divergence, and pressure terms. Unlike the classical RST, our proposed RST is non-symmetric and non-local. The proposed form of the RST is important for generating appropriate reference data for LES closure modeling. Based on DNS results of the 1D Burgers and 3D incompressible Navier-Stokes equations, we show that the discretization-induced parts of the RST play an important role in the LES-FVM equation for common LES filter widths. When the discrete contribution is included, our RST expression gives zero a-posteriori error in LES, while existing RST expressions give errors that increase over time. For a Smagorinsky model, we show that the Smagorinsky coefficient is higher when fitted to our new RST than when fitted to the classical RST and gives improved results.