Predicting the boundary-layer transition process is a hot research issue in aerodynamics as well as an engineering demand for aircraft design. The e^N method based on linear stability theory is widely regarded as the most scientific method for transition prediction, which has been applied in engineering practices. However, this method has theoretical shortcomings as it traditionally ignores the receptivity of the boundary layer to external disturbances, i.e., the process of exciting boundary-layer unstable waves by external disturbances. The influence factors of receptivity are numerous, but at present, their influence laws are relatively less studied and not systematically understood yet. In this study, the high-speed cone boundary layer is taken as the object to investigate the influence of key parameters such as nose bluntness, wall temperature, unit Reynolds number, and frequency on the receptivity of boundary layers to freestream slow acoustic waves. The used methods include numerical simulations of receptivity and linear stability theory. It is found that, overall, the receptivity coefficient decreases as the nose bluntness increases, and increases as the wall temperature decreases, but is almost unaffected by the unit Reynolds number. For freestream disturbances with different frequencies, the unstable mode excited in the boundary layer under adiabatic or heated wall temperature conditions include the first and second modes, among which the receptivity coefficient of the first mode is 1-2 orders of magnitude higher than that of the second mode. In addition, the receptivity coefficient decreases with increasing frequency in the first-mode excited frequency range, while it is mainly affected by the nose bluntness in the second-mode excited frequency range. The receptivity coefficient decreases first then increases with the frequency for small bluntness, decreases monotonically with the frequency for medium bluntness, and increases first then decreases for large bluntness.

An inclined cone with exists on a azimuthal pressure gradient on its surface, which drives the flow from the high-pressure region on the windward side to the low-pressure region on the leeward side. This results in the formation of large-scale flow vortex structures near the leeward centerline, which further destabilizing and triggering transitions. Linear global stability analyses were conducted by solving the two-dimensional eigenvalue problem (BiGlobal) and using plane-marching parabolized stability equations (PSE3D) to reveal the instabilities of streamwise vortices on the leeward side of the cones. Results showed that the Mack mode instability prevails in the upstream region for all cases. As the leeward base flow begins to distort significantly, the Mack mode weakens, while the intrinsic instability modes emerge and eventually take over. The downstream development of N factors for the unstable modes is obtained by PSE3D. According to the measured transition locations in experiments, the transition N factors are estimated to be less than 3 for the cases with a 2-deg angle of attack and approximately 6 for the case with a 4-deg angle of attack. The low N-factor at the transition location of the case with a 2-deg angle of attack implies that non-modal growth of disturbances may play a role in the transition process.