A fully three-dimensional linear stability analysis is carried out to investigate the unstable bifurcations of a compressible viscous fluid past a sphere. A time-stepper technique is used to compute both equilibrium states and leading eigenmodes. In agreement with previous studies, the numerical results reveal a regular bifurcation under the action of a steady mode and a supercritical Hopf bifurcation that causes the onset of unsteadiness but also illustrate the limitations of previous linear approaches, based on parallel and axisymmetric base flow assumptions, or weakly nonlinear theories. The evolution of the unstable bifurcations is investigated up to low-supersonic speeds. For increasing Mach numbers, the thresholds move towards higher Reynolds numbers. The unsteady fluctuations are weakened and an axisymmetrization of the base flow occurs. For a sufficiently high Reynolds number, the regular bifurcation disappears and the flow directly passes from an unsteady planar-symmetric solution to a stationary axisymmetric stable one when the Mach number is increased. A stability map is drawn by tracking the bifurcation boundaries for different Reynolds and Mach numbers. When supersonic conditions are reached, the flow becomes globally stable and switches to a noise-amplifier system. A continuous Gaussian white noise forcing is applied in front of the shock to examine the convective nature of the flow. A Fourier analysis and a dynamic mode decomposition show a modal response that recalls that of the incompressible unsteady cases. Although transition in the wake does not occur for the chosen Reynolds number and forcing amplitude, this suggests a link between subsonic and supersonic dynamics.

The source of unsteadiness in shock-wave/boundary-layer interactions is currently disputed. This paper considers a two-dimensional separation bubble induced by an oblique shock wave interacting with a laminar boundary layer at a free-stream Mach number of 1.5. The global response of the separated region to white noise forcing is analyzed for different interaction strengths, which generate small and large separation bubbles. Forcing location and amplitude effects have been examined. For both interaction strengths and for forcing both upstream and inside the bubble, the wall-pressure spectra downstream of the separation show a high-frequency peak that is demonstrated to be a Kelvin-Helmholtz instability. A low-frequency response at the separation point is also found when the separation bubble is only forced internally, therefore with a disturbance-free upstream boundary layer. For low-amplitude internal forcing, the low-frequency response at the separation point and downstream of the bubble is linear. However, when forced upstream the low-frequency unsteadiness of the large separation bubble is found to be driven by nonlinearities coming from the downstream shedding. The same nonlinear behavior is found when the separation bubble is internally forced over a narrow band around the shedding frequency, without low-frequency disturbances. This analysis for a laminar interaction is used to interpret the low-frequency unsteadiness found at the foot of the shock of turbulent interactions. Here, the low-frequency unsteadiness occurs in the absence of upstream disturbances and a linear relationship is found between the internal forcing and the response near the separation point. When low-frequencies are not present in the forcing they are generated from weak nonlinearities of the shear-layer instability modes.

Three-dimensional direct numerical simulations (DNS) of a shock-induced laminar separation bubble are carried out to investigate the flow instability and origin of any low-frequency unsteadiness. A laminar boundary layer interacting with an oblique shock wave at $M=1.5$ is forced at the inlet with a pair of monochromatic oblique unstable modes, selected according to local linear stability theory (LST) performed within the separation bubble. Linear stability analysis is applied to cases with marginal and large separation, and compared to DNS. While the parabolized stability equations approach accurately reproduces the growth of unstable modes, LST performs less well for strong interactions. When the modes predicted by LST are used to force the separated boundary layer, transition to deterministic turbulence occurs near the reattachment point via an oblique-mode breakdown. Despite the clean upstream condition, broadband low-frequency unsteadiness is found near the separation point with a peak at a Strouhal number of $0.04$, based on the separation bubble length. The appearance of the low-frequency unsteadiness is found to be due to the breakdown of the deterministic turbulence, filling up the spectrum and leading to broadband disturbances that travel upstream in the subsonic region of the boundary layer, with a strong response near the separation point. The existence of the unsteadiness is supported by sensitivity studies on grid resolution and domain size that also identify the region of deterministic breakdown as the source of white noise disturbances. The present contribution confirms the presence of low-frequency response for laminar flows, similarly to that found in fully turbulent interactions.

A computational study of the interaction between an oblique shock-wave and a laminar boundary-layer on a flat plate is carried out to investigate the unsteady character of a two-dimensional (2D) separation bubble at a free-stream Mach number of 1.5. In order to validate the code, a review of the case is presented, highlighting some significant differences compared to previous works that were not run sufficiently far in time. A steady base flow is forced with white noise and a global analysis is carried out to investigate the response of the shockinduced separation bubble. An amplitude sensitivity study is carried out for two different forcing locations, one upstream and one within the separation bubble. Independently of the forcing location all the spectra show a high frequency peak, which is demonstrated to be a Kelvin-Helmholtz instability by the linear e N -method, and an unsteady low-frequency behaviour. For low amplitude forcing applied within the bubble the low-frequency response is linear. For upstream forcing the low-frequency response is driven by the non-linearity of the downstream shedding. The same phenomenon occurs when downstream forcing is applied over a narrow band that does not include low frequencies. The occurrence of low-frequency unsteadiness in a 2D laminar problem shows the possibility to link recent works on turbulent interaction back to simpler laminar problems. In particular it is shown that the low-frequency occurs in the absence of upstream disturbances and that it can be effectively driven by broadband disturbances within the separation bubble, for example arising from weakly non-linear interaction of shear-layer modes instability.