Reynolds stress anisotropy in shock/isotropic turbulence interactions

Linear interaction analysis (LIA) predicts that isotropic vortical turbulence interacting with a normal shock wave at Mach 2 or greater produces a post-shock Reynolds stress tensor whose transverse components R-22 and R-33 are greater than the streamwise component R-11. However, computational and experimental studies have consistently reported Reynolds stress tensors with R-11 > R-22 in this regime. This discrepancy is primarily due to nonlinear terms which have been neglected in the linear analysis. However, if the effect of the nonlinear terms is to move the anisotropy from a state with R-22 > R-11, through isotropy, to a state of qualitatively opposite anisotropy, this would seem contrary to the usual return-to-isotropy effect. Here, we explain the redistribution of energy between the transverse and streamwise Reynolds stresses by considering the wavelengths of the vortical waves emitted by the interaction. Our study shows that the transverse Reynolds stresses are concentrated at smaller length scales than the streamwise stress. In turn, the evolution of the modes associated with R-22 occurs on a faster time scale, and their energy is more quickly transferred away, allowing R-11 to dominate. The overall effect is that the global anisotropy quickly attains a state where R-11 > R-22. Furthermore, a quantitative model is developed based on this understanding of the flow physics. The model uses LIA along with an eddy viscosity approximation for the nonlinear interactions in order to quantify the redistribution of energy between the transverse and streamwise Reynolds stresses. The model predicts anisotropy levels approximately matching those observed in direct numerical simulation data.

Automated summary

The study explores the discrepancy between theoretical predictions and computational/experimental results regarding the Reynolds stress tensor in vortical turbulence interacting with a normal shock wave at Mach 2 or greater. It is found that the redistribution of energy between transverse and streamwise Reynolds stresses is influenced by the wavelengths of the vortical waves emitted during the interaction, leading to the dominance of R-11. A quantitative model based on this understanding of flow physics accurately predicts anisotropy levels observed in direct numerical simulation data.