Statistical analysis of deformation of a shock wave propagating in a local turbulent region

Direct numerical simulation is performed for analyzing the interaction between a normal shock wave and turbulence. The shock wave is initially located in a quiescent fluid and propagates into a local turbulent region. This flow setup allows investigation of the initial transition and statistically steady stages of the interaction. Shock deformation is quantified using the local shock wave position. The root-mean-square (rms) fluctuation in the shock wave position increases during the initial stage of the interaction, for which the time interval divided by the integral time scale increases with M_t²/(M_s²-1 ) , where M_t is a turbulent Mach number and M_s is a shock Mach number. In late time, the rms fluctuation in the shock wave position hardly depends on the propagation time and follows a power law, [M_t^{2/(M_s2-1 ) ] 0.46}, whose exponent is similar to the power law exponent of the rms pressure-jump fluctuation reported in experimental studies. Fluctuations in the shock wave position have a Gaussian probability density function. The spectral analysis confirms that the length scale that characterizes shock wave deformation is the integral length scale of turbulence. The fluctuating shock wave position is correlated with dilatation of the shock wave, where the correlation coefficient increases with M_t/(M_s - 1). In addition, the shock wave that deforms backward tends to be stronger than average and vice versa. Mean pressure jumps across the shock wave are different between areas with forward and backward deformations. This difference increases with the rms fluctuation in the shock wave position and is well-represented as a function of M_t²/(M_s²-1 ) .