Reynolds stress anisotropy in shock/isotropic turbulence interactions

• Published in: Journal of fluid mechanics 2021-02, Vol.913, Article A19
• Main Authors: Grube, Nathan E., Martín, M. Pino
• Format: Article
• Language: English
• Subjects: Agreements; Anisotropy; Approximation; Computational fluid dynamics; Computer applications; Direct numerical simulation; Eddy viscosity; Energy; Experiments; Fluid mechanics; Geometry; Isotropic turbulence; Isotropy; JFM Papers; Laboratories; Linear analysis; Mathematical analysis; Mathematical models; Normal shock waves; Physics; Reynolds number; Reynolds stress; Reynolds stresses; Shock waves; Simulation; Stress tensors; Tensors; Turbulence; Turbulence models; Velocity; Viscosity; Wavelengths
• ISSN: 0022-1120; 1469-7645
• DOI: 10.1017/jfm.2020.1070

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.