Optimization of nonlinear error for weighted essentially non-oscillatory methods in direct numerical simulations of compressible turbulence

Weighted essentially non-oscillatory (WENO) methods have been developed to simultaneously provide robust shock-capturing in compressible fluid flow and avoid excessive damping of fine-scale flow features such as turbulence. Under certain conditions in compressible turbulence, however, numerical dissipation remains unacceptably high even after optimization of the linear component that dominates in smooth regions. We therefore construct and evaluate WENO schemes that also reduce dissipation due to one source of nonlinear error: the smoothness measurement that governs the application of stencil adaptation away from the linear optimal stencil. Direct numerical simulations (DNS) include a one-dimensional Euler solution and three-dimensional compressible isotropic turbulence. We find that the smoothness measurement modifications that we call the “relative smoothness limiter” and the “relative total variation limiter” each significantly enhance thez grid-convergence properties of WENO schemes while generating, respectively, small and moderate additional computational expense. Moreover, we observe these techniques to be broadly effective regardless of flow configuration.