Modification of the Hugoniot adiabat due to turbulence generated in shocked deuterium-filled CH foams

Direct-drive laser targets are often designed with DT-filled CH foam ablator. Accurate modeling of these targets requires understanding of shock propagation in such non-uniform media. The interaction of the shock front with the preshock random density non-uniformities generates a random motion (turbulence) in the postshock flow. The energy coupled into the postshock turbulent motion, in turn, modifies the shock adiabat. As first detected in simulations by G. Hazak \textit{et al}., Phys. Plasmas \textbf{5}, 4357 (1998), shock compression and shock velocity in a deuterium-filled foam would be, respectively, less and greater than those predicted for the uniform medium of the same average density. We report an exact analytical theory of this ``shock undercompression'' effect and present explicit formulas for the shock adiabat modification. We discuss the contributions of post-shock Reynolds stresses, acoustic energy flux emitted downstream and correlations between vortical and entropy perturbations and highlight the difference between the cases of 2D and 3D turbulence.