Numerical Experiments with Shock-Turbulence Interaction

Many applications in engineering and physical sciences involve turbulent flows interacting with shock waves. High-speed flows around aerodynamic bodies and through propulsion systems for high-speed flight abound with interactions of shear driven turbulence with complex shock waves. Supernova explosions and implosion of a cryogenic fuel pellet for inertial confinement fusion also involve the interaction of shockwaves with turbulence and strong density variations. Numerical simulations of such physical phenomena impose conflicting demands on the numerical algorithms. Capturing broadband spatial and temporal variations in a turbulent flow suggests the use of high-bandwidth schemes with minimal dissipation and dispersion, while capturing the flow discontinuity at a shock wave requires numerical dissipation. We summarize results from a series of benchmark test problems for assessing the ability of three different approaches to shock capturing: high order WENO, nonlinear artificial diffusivity with compact finite differences, and a hybrid approach combining high-order central differencing with WENO near the shocks. These test problems allow an assessment of the tradeoff needed between preserving non-dissipation of small-scale flow fluctuations and avoiding significant Gibbs' oscillation near a shock. Numerical experiments on Taylor-Green problem and compressible isotropic turbulence are used to evaluate the performance of these schemes on flows with broadband fluctuations. The compressible turbulence test case also contains local eddy-shocklets. The performance of each scheme is characterized in terms of an effective bandwidth. Finally some results on a canonical shock-turbulence interaction problem, i.e. the interaction of isotropic turbulence with a (nominally) normal shock, are discussed. These results achieve a turbulence Reynolds number which is significantly larger than previous DNS studies of this problem. It is observed that when the turbulence interacting with the shock is sufficiently energetic the instantaneous shock structure is significantly modified; local regions of significant over-compression are found as well as regions where the mean shock compression is nearly isentropic. Damaging effect of numerical dissipation were also noted for high-order WENO scheme. Some estimates are given at the end for the computational resources needed for studying this fundamental shock-turbulence interaction problem on peta-scale computing systems.