Boundary layer transition over a lifting body of 1.6 m length at $2^\circ$ angle of attack has been simulated at Mach 6 and a unit Reynolds number $1.0 \times 10^7$ m $^{-1}$ . The model geometry is the same as the Hypersonic Transition Research Vehicle designed by the China Aerodynamics Research and Development Center. Four distinct transitional regions are identified, i.e. windward vortex region, shoulder vortex region, windward cross-flow region and shoulder cross-flow region. Multi-dimensional linear stability analyses by solving the two-dimensional eigenvalue problem (spatial BiGlobal approach) and the plane-marching parabolized stability equations (PSE3D approach) are further carried out to uncover the dominant instabilities in the last three regions as well as the shoulder attachment-line region. The shoulder vortex is conducive to both inner and outer modes of shear-layer instability, of which the latter most likely trigger the vortex breakdown. A novel method is presented to substantially reduce the computational cost of BiGlobal and PSE3D in resolving the cross-flow instabilities in cross-flow regions. The peak frequencies of cross-flow modes lie between 15 and 45 kHz. Whereas oblique second Mack modes are marginally unstable in the windward cross-flow region, they could be strong enough to compete with the cross-flow modes in the shoulder cross-flow region. In the shoulder attachment-line region, there exists only one unstable mode of Mack instability, differing from previous studies that show a hierarchy of modes in the context of symmetrical attachment-line flows. Results of the numerical simulation and multi-dimensional stability analyses are compared when possible, showing a fair agreement between the two approaches and highlighting the necessity of considering non-parallel effects.

Hypersonic boundary layer transitions caused by unsteady blowing and suction are investigated with linear stability analyses and direct numerical simulations (DNS). Three blowing-suction frequencies, i.e. 15 kHz, 30 kHz and 45 kHz, are separately utilized to excite a pair of unsteady Görtler instability waves (the first two cases) or first-mode instability waves (the last case). These two primary instabilities respectively induce diamond-shaped and Λ-shaped structures through self interactions. These structures are highly susceptible to high-frequency secondary instabilities, as is demonstrated by global Floquet analyses which take in account both temporal unsteadiness and spanwise spatial variations of the base flow. The secondary instability manifests as hairpin packets riding on the downstream end of the diamond-shaped structures or resides in the outward sides of the two legs of the Λ-shaped structures. The theoretical results quantitatively agree with the DNS results. Energy analyses further reveal that the wall-normal productions dominate the energy transfer for the secondary instability of the unsteady Görtler vortices, while the spanwise productions are crucial to the secondary instabilities in the first-mode oblique breakdown. Quasi-steady analyses based on the "frozen" base flow are also performed, whose results compare favorably with those from Floquet analyses in the lowest-frequency case.

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We studied the stability of leeward streamwise vortices over a hypersonic inclined blunt cone under a flight condition by solving the two-dimensional spatial eigenvalue problem (BiGlobal) and plane-marching parabolized stability equations (PSE3D). The stability analyses were performed based on the laminar flow obtained by direct numerical simulation. Due to the azimuthal pressure gradient and a large bluntness, a pair of large-scale inward vortices dominate in the vicinity of the leeward ray. Such vortical structures are different from previous studies, where outward vortices are most prominent. Two types of unstable modes are identified, namely `inner mode' with low phase velocities and `outer mode' with high phase velocities. The inner modes are unstable in a wide frequency range that comparable to the outer modes, differing from previous studies where inner modes are deemed to lie in a much lower frequency range compared to the outer modes. Moreover, the inner modes, with supersonic phase velocities, are found to be able to radiate weak acoustics outside the vortices. Mack mode, belonging to the outer-mode instability, is most amplified that it is likely responsible for the breakdown of the vortices. BiGlobal and PSE3D results show a good agreement regarding the downstream developments of the Mack mode and inner modes, and yet exhibit remarkable discrepancies in tracing the evolution of the outer modes, highlighting the necessity of considering the non-parallel effects when studying the stability of streamwise vortices.

A systematic parametric study is presented on the modal global linear instability of Mach 6 boundary-layer flow over an inclined cone at 6 degree angle of attack under typical wind-tunnel conditions. The analysis is performed using the spatial BiGlobal theory. Results reveal two classes of global instabilities in the outboard region away from the leeward plane: high-frequency (hundreds of kHz) second mode instabilities and low-frequency (tens of kHz) crossflow instabilities. In the global eigenvalue spectra of the second mode instability, two distinct branches of significantly unstable modes are identified, namely an approximately continuous spectrum (branch S) and a few discrete points (branch D). The branch D modes peak at the windward centerline while branch S modes lie at a certain distance away from the windward meridian. The global stability characteristics of the crossflow instability over a conical configuration are uncovered for the first time. In the global eigenvalue spectra of the crossflow instability, many unstable modes emerge and form a complex pattern, with the most amplified frequency of around 30 kHz. The unstable crossflow modes are mainly distributed on the leeward side and wave angles obviously increase towards the leeward centerline. The peak growth rates are slightly smaller than those of second modes, and yet the most amplified frequency varies rather mildly along the streamwise direction so that they may achieve larger integrated growth rates than second modes do.