Reynolds-averaged Navier–Stokes model predictions of linear instability. I: Buoyancy- and shear-driven flows

Abstract

Reynolds-averaged Navier–Stokes (RANS) models based on turbulent kinetic energy and dissipation rate or length scale transport equations are commonly used for fully developed turbulence arising from buoyancy, shear, or shocks. It is shown here that RANS models also describe the linear instability phase for buoyancy instabilities (specifically, the Rayleigh–Taylor instability) and shear instabilities (specifically, the Kelvin–Helmholtz instability and stratified shear flow). During the initial phases of the evolution of these models, turbulent diffusion is negligible, the fluctuation energy grows exponentially with negligible coupling to the mean flow, and the model coefficients can be chosen such that the growth rate is equivalent to the physical growth rate of short-wavelength perturbations. The reduced equations obtained by neglecting turbulent diffusion correspond to the unclosed RANS equations under the assumptions of linear theory (i.e., retaining only second-order correlations). The correspondence with linear theory requires interpreting the model length scale as a Lagrangian fluid displacement rather than a quantity derived from the turbulent kinetic energy dissipation rate. The analytical solutions obtained for both the K-ε and K-ℓ models in the absence of diffusion 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 discontinuous mean flows. Paper II [B.M. Johnson and O. Schilling, Reynolds-averaged Navier–Stokes model predictions of linear instability. II: Shock- driven flows, J. Turbul. 12(33) (2011), pp. 1–31] applies the analysis developed herein to shock–turbulence interaction and the Richtmyer–Meshkov instability.