A study on the numerical dissipation of the Spectral Difference method for freely decaying and wall-bounded turbulence

This paper aims at understanding the numerical dissipation mechanisms related to the Spectral Difference (SD) method in the context of three-dimensional (3D) turbulence. The numerical dissipation stemming from the discretization of the convective terms is studied by performing inviscid computations of the transitional Taylor-Green vortex and isotropic turbulence configurations. The Taylor-Green vortex computations show that the increase in the order of accuracy restricts the numerical dissipation to smaller scales which, in turn, leads to a better representation of transitional mechanisms. However, isotropic turbulence computations using a fifth-order accuracy or above show obvious manifestations of under-resolution (such as the onset of oscillations and numerical noise), which suggests that the high-order numerical dissipation alone is unable to mimic the dissipation originating from sub-grid scales in the case freely decaying turbulence. Computations of the channel flow configuration at Reτ=1000 at typical large-eddy simulation resolutions show that under-resolved SD discretizations using a high order of accuracy (fifth and sixth) lead to an excellent prediction of the wall-friction, the velocity profiles, the turbulent structures near the wall and the energy spectra, while lower order discretizations lead to an underestimation of the wall-friction and globally a poor representation of wall-bounded turbulence. The present study emphasizes the benefit of using high-order SD discretizations for an accurate representation of turbulent phenomena (namely, transitional and wall-bounded turbulence) but also the necessity of combining this approach with dynamic large-eddy simulation models or appropriate regularization techniques which would activate only where needed to recover physically consistent results, e.g., in regions where fully developed turbulence is present.