Secondary global instabilities in shock-wave–boundary-layer interactions: direct numerical simulations and three-dimensional global stability analysis

Secondary instabilities developing from the stationary primary global mode are investigated for an incident shock flow at Mach 2.15. Direct numerical simulations (DNS) and three-dimensional global stability analysis (3-D GSA) are combined to elucidate the evolution and characteristics of the associated secondary branches. Two spanwise domain widths are considered, lamda Subscript m λ m $\\lambda _m$ and 2 lamda Subscript m 2 λ m $2\\lambda _m$ , where lamda Subscript m λ m $\\lambda _m$ denotes the spanwise wavelength of the primary global mode. In the lamda Subscript m λ m $\\lambda _m$ domain, a time-periodic unsteadiness emerges as a secondary instability after saturation of the primary global mode, which is identified as a supercritical Hopf bifurcation. In the 2 lamda Subscript m 2 λ m $2\\lambda _m$ domain, an additional secondary growth accompanied by wavelength doubling is observed following the onset of unsteadiness. A corresponding stationary secondary instability with wavelength 2 lamda Subscript m 2 λ m $2\\lambda _m$ is determined to be the mechanism underlying this secondary growth, which is a subcritical pitchfork bifurcation. Overall, DNS and 3-D GSA provide a consistent interpretation of the secondary-instability scenario, and highlight the sensitivity of the observed dynamics to the spanwise domain width.