Further Development of the Navier-Stokes Equations-Based Mean Flow Perturbation Technique

Abstract

The evolution of disturbances in a mean basic state can be accomplished in several different ways. A relatively recent approach proposed by Touber and Sandham1 for shock-turbulent boundary layer interaction (STBLI) performs this task by advancing the Navier-Stokes equations and factoring out the changes associated with the mean basic state. In a recent effort,2 we have shown that this Navier-Stokes based Mean Flow Perturbation (NS-MFP) method reverts to the Parabolized, Linear and Global stability methods under appropriate conditions. NS-MFP solves an initial boundary value problem and is in principle capable of tracking the complete spatio-temporal evolution of disturbances in any 3-D mean flow. Here, we further analyze NS-MFP in the presence of (a) strong-gradient/discontinuities and (b) viscous stresses with the ultimate aim of using it for tracking disturbance evolution in 3-D swept STBLI. To analyze the effects of strong gradient and/or discontinuities, we investigate the interaction of an entropic spot with a Mach 2 normal shock using NS-MFP. The results obtained are quantitatively compared with corresponding theoretical linear interaction analysis (LIA) presented in the literature. To illustrate that NS-MFP captures the viscous terms accurately and is applicable to wall bounded flows; we examine the evolution of pressure perturbations on a Mach 2.0 laminar flat plate boundary layer. Finally, we apply the method to a 2-D STBLI problem, where an impinging shock with 9◦ wedge angle interacts with a Mach 2.3 turbulent boundary layer (Reynolds number 17, 500). Our results successfully capture the biglobal mode reported by other researchers, along with the intermediate frequencies characteristic of Kelvin-Helmholtz shedding in the mixing layer. Finally, we establish the suitability of the approach in analyzing a Reynolds Averaged Navier Stokes (RANS) generated basic state as an initial exercise to extend the technique to fully 3-D swept STBLI, for which LES are expensive.