Assessment of high-resolution methods for numerical simulations of compressible turbulence with shock waves

Abstract

a b s t r a c tFlows in which shock waves and turbulence are present and interact dynamically occur in a wide range of applications, including inertial confinement fusion, supernovae explosion, and scramjet propulsion. Accurate simulations of such problems are challenging because of the contradictory requirements of numerical methods used to simulate turbulence, which must minimize any numerical dissipation that would otherwise overwhelm the small scales, and shock-capturing schemes, which introduce numerical dissipation to stabilize the solution. The objective of the present work is to evaluate the performance of several numerical methods capable of simultaneously handling turbulence and shock waves. A comprehensive range of high-resolution methods (WENO, hybrid WENO/central difference, artificial diffusivity, adaptive characteristic-based filter, and shock fitting) and suite of test cases (Taylor-Green vortex, Shu-Osher problem, shock-vorticity/entropy wave interaction, Noh problem, compressible isotropic turbulence) relevant to problems with shocks and turbulence are considered. The results indicate that the WENO methods provide sharp shock profiles, but overwhelm the physical dissipation. The hybrid method is minimally dissipative and leads to sharp shocks and well-resolved broadband turbulence, but relies on an appropriate shock sensor. Artificial diffusivity methods in which the artificial bulk viscosity is based on the magnitude of the strain-rate tensor resolve vortical structures well but damp dilatational modes in compressible turbulence; dilatation-based artificial bulk viscosity methods significantly improve this behavior. For well-defined shocks, the shock fitting approach yields good results.

Studies of Inviscid Flux Schemes for Acoustics and Turbulence Problems

“…Similar conclusions were reached in a comprehensive study by Johnsen et al 13 The recent review by Pirozzoli 14 summarizes much of the work to date.…”

“…(9)- (13). The divergence form was tested in the course of this study to confirm the well-known instability characteristics of the method.…”

On the performance of relaxation filtering for large-eddy simulation

“…It is one of the simplest environments to study the breakdown process of large-scale vortices into successively smaller ones, and the resulting homogeneous isotropic turbulence [24,25]. In the past decade, the Taylor-Green vortex has become a popular reference case, used in a series of studies on LES methods, e.g., by Fauconnier et al [25], Drikakis et al [26], Chandy and Frankel [27], Johnsen et al [28], Adams [29], and Gassner and Beck [30]. We select the Reynolds number Re = 3000, which is large enough so that natural transition into small-scale homogeneous isotropic decaying turbulence occurs.…”

Subgrid-scale backscatter after the shock-turbulence interaction

“…Thus, turbulence simulations re-quire the minimization of numerical dissipation for small scale representation, while the shocks require increased local dissipation to regularize the algorithm [4]. Explicit subgrid models also need to account for the presence of the shock.…”