Turbulence amplification by a shock wave and rapid distortion theory

Amplification of turbulent kinetic energy in an axial compression is examined in the frame of homogeneous rapid distortion theory (RDT) by using the Craya-Herring formalism. By separating the turbulent field into solenoidal and dilatational modes (Helmholtz decomposition), one can show the dilatational mode is mediated by the parameter Δm₀=D₀/a ₀k₀, which corresponds to the initial ratio between the acoustic time scale (a₀k₀)^-1 and the compression time scale D₀^-1, with D₀ the compression rate. It is shown here that amplification of total kinetic energy is then limited by two analytical solutions obtained for Δm₀=0 (purely solenoidal-acoustical regime) and for Δm₀≫1 ("pressure released" regime), respectively. The results of the theory are first compared to results of direct numerical simulations (DNS) on homogeneous axial compression. The applicability of this homogeneous approach to the shock wave turbulence interaction, is then discussed. Considering a shock-induced compression at given Mach number, it is shown that the corresponding amplification factors predicted by homogeneous RDT largely differs from that obtained from Ribner's linear interaction analysis and DNS on the shock problem.