The interaction of turbulence with shock waves: a basic model

Analytic work addressing the interaction of turbulence with a shock wave has concentrated on investigating the generation of sound by jet engines. Hydrodynamic modes (acoustic, vortical, and entropic) can experience considerable amplification on passage through a shock [Moore, 1954; Ribner, 1954, 1987; McKenzie & Westphal, 1968; Mahesh et al., 1995], while downstream of a shock there exists a critical angle for incident acoustic modes where the reflection coefficient is 1. The analytic studies cited assume that the incident, radiated and resulting perturbations, including the shock front distortion, is of small amplitude, allowing a linearized description. Not addressed in this approach is the nonlinear response of the shock wave to the incident turbulence, nor the conversion of mean shock wave momentum and energy to the amplification of the incident turbulence. However, upstream turbulence can affect significantly the mean flow variables of a shock wave. Instead, the classical Rankine-Hugoniot (R-H) conditions, which relate the upstream and downstream states of a shock wave, are no longer exact for the mean flow. The mean R-H conditions are modified by contributions to the mass, momentum, and energy fluxes from turbulent fluctuations. To properly understand the interaction of turbulence with shocks, one must incorporate the back reaction of the shock-turbulence interaction. One must therefore incorporate statistically variation in ram pressure, energy, and mass flux in the mean R-H conditions and then determine the mean position, velocity, and strength of the shock. This is an entirely different perspective from that developed in previous analytic studies, although captured in principle by numerical studies. We present initial results for a highly simplified system that goes some way towards addressing the non-linear interaction of fully developed turbulence with shock waves.