Reynolds-averaged Navier–Stokes model predictions of linear instability. II. Shock-driven flows

Abstract

Applying the formalism for buoyancy- and shear-driven instabilities developed in Paper I [B.M. Johnson and O. Schilling, J. Turbul. 12(36) (2011), pp. 1–38], it is shown here that Reynolds-averaged Navier–Stokes (RANS) models describe the early phase of shock-driven instabilities: the amplification of velocity fluctuations due to the passage of a shock for shock–turbulence interaction and the initial deposition of solenoidal velocity at an interface by a shock for the Richtmyer–Meshkov instability. The correspondence between the closed RANS equations in the absence of turbulent diffusion and the unclosed RANS equations under the assumptions of linear theory discussed in Paper I is shown here to extend to shock-driven flows. In order to compare the models with linear theory, a form of linear interaction analysis appropriate for short-wavelength perturbations is developed and applied to a shock interacting with an ambient velocity field. It is shown that relaxing Morkovin's hypothesis can result in arbitrarily large amplification of ambient velocity fluctuations due to the passage of a shock. The classical Richtmyer–Meshkov instability corresponds to the case in which the amplification is formally infinite as the initial velocity fluctuation is zero. The solutions obtained here for both the K-ε and K-ℓ models provide insight into the choice of appropriate initial conditions for RANS models, impose various constraints on the model coefficients and can be used to verify numerical discretizations of the model equations. The analysis also quantifies and addresses issues regarding convergence under grid refinement in the presence of shocks: it is shown that convergence in a shock-driven flow can be improved by scaling the initial length scale with the grid spacing.

Keywords:

• shock–turbulence interaction
• Richtmyer–Meshkov instability
• two-equation turbulence models

Acknowledgements

This work was performed under the auspices of the US Department of Energy by Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344.

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