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

Abstract

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 Mt2/(Ms2−1), where Mt is a turbulent Mach number and Ms 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, [Mt2/(Ms2−1)]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 Mt/(Ms − 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 Mt2/(Ms2−1).

Thermochemical effects on hypersonic shock waves interacting with weak turbulence

“…The amplitudes of the acoustic modes of the streamwise and transverse velocity fluctuations in (48) are proportional to the amplitude of the acoustic pressure, U a = s = a =! a and V a = s = 1=!…”

“…where the acoustic and vortical modes of the dimensionless velocity fluctuations in the far field are given in (48) and (56). The relations between the modes of the streamwise and transverse velocity fluctuations are provided by the irrotationality condition v a = a u a for the acoustic mode, and by the solenoidal condition k y v r = Rk x u r for the vortical mode.…”

“…which satisfies the normalization R 1 0 P( )d = 1. In addition, the velocity amplitudes U a , U r , V a and V r are obtained using the long-time far-field asymptotic expressions (48) and (57). The lower integration limit of K a is = 1 since the acoustic mode decays exponentially with distance downstream of the shock in the long-wave regime < 1.…”

“…Most early work on the interaction of shock waves with turbulence have been limited to calorically perfect gases in boundary layers [21][22][23][24][25][26][27][28][29][30] , and isotropic free streams [31][32][33][34][35] . Large-scale numerical simulations, including DNS [36][37][38][39][40][41][42][43][44][45][46][47][48][49][50] , LES [51][52][53] , and RANS 54,55 , have been the pacing item for those investigations. Nonetheless, the rapid progress in large-scale numerical simulations during the last decades has not abated the fundamental role that theoretical analyses have played in understanding shock/turbulence interactions by providing closed-form solutions.…”

Compressibility effects on Reynolds stress amplification and shock structure in shock–isotropic turbulence interactions

“…(2018), Tanaka et al. (2018, 2020), Chen & Donzis (2019) and Gao et al. (2020) cover a wide range of Mach numbers and turbulence intensities .…”