Comparison of gradient approximation methods in schemes designed for scale-resolving simulations

Various methods for improved accuracy approximation of the gradients entering the diffusion fluxes are considered. Linear combinations of 2$^{\mathrm{nd}}$ order difference schemes for a non-uniform grid that transform into 4$^{\mathrm{th}}$ order schemes in the uniform case were investigated. We also considered 3$^{\mathrm{rd}}$ and 4$^{\mathrm{th}}$ order schemes for approximating gradients on a non-uniform grid in the normal and tangent directions to the cell face, respectively, based on Lagrange polynomials. The initial testing was carried out on one-dimensional functions: a smooth Gauss function and a piecewise linear function. Next, the schemes were applied in direct numerical simulation of the Taylor–Green vortex.