Stochastic forcing to a linearized Navier-Stokes based model for laminar compressible boundary layers

Abstract

We consider a linearized Navier–Stokes based model for compressible laminar boundary layers and study the response of these equations to stochastic white-in-time forcing. In particular we look at the different components of the forcing and the response of this linear model with the aim of understanding how the different mechanisms captured by the model change with increasing compressibility effects. We therefore analyze the response of the linear operator to individual components of forcing, i.e. the forcing to each of the momentum equations, the continuity and the energy equations of the linear operator. We also analyze the response obtained in the three components of velocity, in density and in temperature individually. For a fixed Reynolds number of Re=400, we consider Mach numbers ranging between Ma=0.05 and Ma=10 and different wall-cooling ratios. We find that, for all the Mach numbers considered here, the most amplified structures are the streamwise streaks forced by streamwise vortices. Previous studies have shown that these modes are highly amplified in the incompressible regime as well. However, as the Mach number increases, the contribution of the streamwise velocity to these streaks decrease, and the contribution of density and temperature to the streaks increase. Finally, we briefly look at the resolvent operator of the flow, and find that all the components of the forcing are important for the amplification of the Mach waves of the flow, and these modes are not captured by the stochastically forced model.