The Interaction of Turbulence with Shock Waves

The interaction of turbulence and shock waves is considered self-consistently so that the back-reaction of the turbulence and its associated reaction on the turbulence is addressed. This approach differs from previous studies that considered the interaction of linear modes with a shock. The most basic model of hypersonic flow, described by the inviscid form of Burgers' equation, is used. An energy-containing model, which couples the turbulent energy density and correlation length of the flow and the mean flow, is developed. Upstream turbulence interacting with a shock wave is found to mediate the shock by 1) increasing the mean shock speed, and 2) decreasing the efficiency of turbulence amplification at the shock as the upstream turbulence energy density is increased. The implication of these results is that the energy in upstream turbulent fluctuations, while being amplified at the shock, is also being converted into mean flow energy downstream. The variance in both the shock speed and position is computed, leading to the suggestion that, in an ensemble-averaged sense, the turbulence-mediated shock will acquire a characteristic thickness given by the standard deviation of the shock position. Lax's geometric entropy condition is used to show that as the upstream turbulent energy density increases, the shock is eventually destabilized, and may emit one or more shocks to produce a system of multiple shock waves. Finally, turbulence downstream of the shock is shown to decay in time according to -2/3 law.