Modal and nonmodal stability analysis of shock-wave/boundary-layer interactions

Abstract

This dissertation is about the modal and nonmodal stability of an oblique shock wave impinging on a Mach 5.92 laminar boundary layer at a transitional Reynolds number. The adverse pressure gradient of the oblique shock wave causes the laminar boundary layer to separate from the wall, resulting in the formation of a recirculation bubble. For sufficiently large oblique shock angles, the recirculation bubble is unstable to three-dimensional perturbations, and the flow bifurcates from its original laminar state. We use direct numerical simulation (DNS) and global stability analysis (GSA) to show that this first occurs at a critical oblique shock angle of 12.9 degrees. The least-stable global mode is stationary at bifurcation, and it takes place at a nondimensional spanwise wavenumber of 0.25, in good agreement with the DNS results. Examination of the critical global mode reveals that it originates from an interaction between small spanwise corrugations at the incident shock base, streamwise vortices inside the recirculation bubble, and spanwise modulation of the bubble strength. Furthermore, the global mode drives the formation of long streamwise streaks downstream of the bubble. This stationary three-dimensional instability is similar to other mechanisms observed in laminar recirculation bubbles. We show that centrifugal instability plays no role in the self-sustaining mechanism of the stationary global mode. Further, we employ an adjoint solver to corroborate our physical interpretations by showing that the critical global mode is most sensitive to base flow modifications that are entirely contained inside the recirculation bubble. We also perform a parametric study to determine the effect of freestream Mach number on shock-wave/boundary-layer interaction (SWBLI) instability. Along with DNS and GSA, we investigate the physical mechanisms responsible for transient growth in an SWBLI using a power iteration method. This approach lets perturbations propagate upstream and downstream, which allows us to capture the complex physics associated with the recirculation bubble and understand how it amplifies fluctuations. For a Mach 5.92 boundary layer with no oblique shock wave, we demonstrate that the transient response arises from the inviscid Orr mechanism, the Landahl lift-up effect, and first-mode instability. The optimal transient growth for this spatially-developing boundary layer with a nondimensional streamwise domain length of 235 is G=1.69x10^3 and occurs at a spanwise wavenumber of 0.6. This corresponds to an amplification of 4.11x10^1, which is similar to that seen in a variety of parallel boundary layer flows. We compute the optimal transient growth of an SWBLI at the exact same conditions as the spatially-developing boundary layer. The presence of an oblique shock wave changes the optimal transient response such that G=1.36x10^7 at a spanwise wavenumber of 0.6. Hence, the transient growth in an SWBLI is four orders of magnitude larger than the transient growth in a spatially-developing boundary layer. The nondimensional spanwise wavenumber of the optimal transient response also increases from 0.6 to 2.6. Moreover, the corresponding optimal spanwise wavelength for the SWBLI is on the order of twice the boundary-layer thickness, agreeing with SWBLI experiments. These changes are attributed to the sudden change in the streamline curvature in the upstream region of the flow field. Furthermore, the optimal initial condition for the SWBLI consists of elongated streaks in the upstream boundary layer. As this initial condition evolves to its final state, we observe the formation of streamwise streaks in the recirculation bubble (that are further amplified in the downstream boundary layer) along with a large perturbation that comes off of the bubble apex and convects downstream. Our results demonstrate large transient growth in a Mach 5.92 SWBLI and suggest that inevitable imperfections in a hypersonic wind tunnel would play an important role in the early stages of transition to turbulence.