Revisiting Implicit Large Eddy Simulation for Discontinuous High Order Methods

Implicit Large Eddy Simulation (ILES) has been widely used in the CFD community for scale-resolving simulations with dissipative numerical methods, such as high-resolution methods with limiters, filtered compact difference, and discontinuous high-order methods, including discontinuous Galerkin, spectral difference, and flux reconstruction methods. In ILES, the embedded numerical dissipation serves as the subgrid-scale stress (SGS) model to dissipate unresolved high-wave-number modes that may contaminate the resolved scales. The SGS stress is known to be a second-order term, as verified by both mathematical analysis and numerical test in the present study. Thus, all high-order schemes with order of accuracy higher than 2nd order do not have sufficient dissipation to completely drain the energy from the unresolved turbulent scales in the limit of mesh refinement. As a result, aliasing errors arising from the unresolved scales make the flow more energetic near the highest resolvable wave number, producing a visible pileup in the kinetic energy spectrum. In a test with under-resolved turbulent channel flow, we observe that the energy pileup improved the skin-friction prediction for several high-order schemes with solution polynomial orders up to 4, i.e., P2 - P4. We believe that the energy pileup compensated for the energy loss due to the LES filtering, generating results closer to the Direct Numerical Simulation (DNS) data. In addition, we also observed that the sixth-order P5 scheme overcompensated for the energy loss due to filtering and produced a skin-friction larger than the DNS result. In this case, an explicit energy-dissipating SGS model, such as the Vreman model, can be used to improve the skin friction predictions on coarse meshes, which agree very well with DNS data. In other words, P5 simulations coupled with an SGS model can obtain DNS-like accuracy on much coarser meshes.