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Reynolds stress anisotropy in shock/isotropic turbulence interactions

Published online by Cambridge University Press:  26 February 2021

Nathan E. Grube  and
M. Pino Martín Open the ORCID record for M. Pino Martín [Opens in a new window]
Nathan E. Grube
Affiliation:
Department of Aerospace Engineering, University of Maryland, Glenn L. Martin Hall, 4298 Campus Drive, College Park, MD20742, USA
M. Pino Martín*
Affiliation:
Department of Aerospace Engineering, University of Maryland, Glenn L. Martin Hall, 4298 Campus Drive, College Park, MD20742, USA
*
†Email address for correspondence: mpmartin@umd.edu
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Abstract

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.

JFM classification

Turbulent Flows: Isotropic turbulence Compressible Flows: Shock waves Turbulent Flows: Turbulence modelling
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 913 , 25 April 2021 , A19
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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