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. Results from three promising shock-capturing schemes a) high order WENO, b) nonlinear artificial diffusivity with compact finite differences, and c) a hybrid approach combining high-order central differencing with WENO near the shocks are compared using the Taylor-Green problem and compressible isotropic turbulence with eddy-shocklets. The performance of each scheme is characterized in terms of an effective bandwidth. The comparison highlights the damaging effect of numerical dissipation when the WENO scheme is applied everywhere. The hybrid approach is found to be best suited for studying shock-turbulence interactions. Results from previous DNS and LES studies of the canonical shock-turbulence interaction problem, i.e. the interaction of isotropic turbulence with a (nominally) normal shock and comparison with available theory and experimental data are recalled. The principal physical effects include the amplification of turbulent kinetic energy across the shock and its anisotropy, change in turbulence length scales across the shock, departure from the common assumption of strong Reynolds analogy (used in modeling turbulence in high-speed flows), and the distortion of the shock due to its interaction with the incident turbulence. In this context new results from our SciDAC sponsored project are shown and open issues are mentioned. New DNS results achieve a significantly larger turbulence Reynolds number and allow an exploration of the nonlinear effects. While the linear interaction theory of Ribner provides useful estimates of the amplification of vorticity fluctuations across the shock, it misses the strong nonlinear dynamics of the energized and highly anisotropic vorticity downstream of the shock. It is found that previous DNS studies also underestimated this effect. The simulations show that turbulent self-stretching and tilting mechanisms bring about a relatively rapid return to isotropy in the turbulent vorticity field. The turbulent velocity field, however, does not show any appreciable tendency towards isotropy. It is further 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. Estimates for the computational resources necessary for studying this fundamental shock-turbulence interaction problem at higher Reynolds number on peta-scale computing systems are given.
