This work investigates the origination of the secondary instability in Görtler vortices using the linear stability theory, BiGlobal analysis, three-dimensional linear parabolized stability equations (3DLPSEs), and direct numerical simulation (DNS). The flow over a concave wall suffering from the Görtler instability and first/second Mack mode instability is selected. Furthermore, this work simulates the evolution of infinitesimal Mack mode disturbance in a flow perturbed by finite-amplitude Görtler vortices by using DNS and 3DLPSE methods. The 3DLPSE approach accurately predicts the process of Mack mode disturbance evolving into the secondary instability of Görtler vortices, and a perfect agreement with results by DNS is obtained. The results indicate that the secondary instability of stationary Görtler vortices can originate from the first/second Mack mode. The evolutions of first/second Mack mode with different spanwise wavenumbers are performed based on 3DLPSE and compared against the BiGlobal method. The results show that the shape functions and growth rates of disturbances always tend to the results of dominant modes obtained by the BiGlobal method. Because the dominant mode might shift from one to another, the overall evolution cannot be predicted only by the BiGlobal method based on a fixed mode. According to our computations, it is deduced that the Mack modes with the same frequency and symmetric characteristics would finally develop into the secondary instability with similar shapes.

An investigation into the linear instability of streamwise vortices over a yawed blunt cone with a half-angle of 7° at 6° angle of attack, a free-stream Mach number of 6, and a Reynolds number of 1.0 × 10⁷/m was performed. The steady laminar base flow obtained using direct numerical simulation (DNS) has a mushroom structure. Three classes of unstable modes (inner mode, outer mode, and Mack mode modified by the streamwise vortices) were identified using a two-dimensional spatial global stability analysis (bi-global). Among them, the inner mode and the modified Mack mode were discovered for the first time for such a configuration. Notably, the inner mode appears only over a short distance upstream of the cone, and its energy is mainly in the vicinity of the symmetry plane of the mushroom structure. By comparing with DNS results, it was shown that multiple instability modes can coexist physically during the transition in the leeward plane. In addition, the eN method based on global stability theory was used to predict the location of the transition along the centerline on the leeward side of the cone. It was found that the inner modes grow rapidly in the upstream region, while the outer mode near the stem of the mushroom structure dominates the transition in the downstream region. Further analysis suggests that the inner modes dominate the transition only in a noisy environment, whereas the traditional centerline modes dominate the transition in quiet free-stream conditions.

The problem of interaction between disturbances and shock waves was solved by a theoretical approach called linear interaction analysis in the mid-twentieth century. More recently, great progress has been made in analysing shock–turbulence interactions by direct numerical simulation. However, an unsolved theoretical problem remains: What happens when no acoustic waves are stimulated behind the shock wave? The concept of a damped wave is introduced, which is a type of excited plane wave. Based on this, the dispersion and amplitude relationships between any incident plane wave and resulting stimulated waves are constructed analytically, systematically and comprehensively. The physical essence of damped waves and the existence of critical angles are clarified. It is demonstrated that a damped wave is a complex number space solution to the acoustic dispersion relationship under certain conditions. It acts as a bridge connecting fast and slow acoustic waves at the position where the $x$ component of the group velocity is zero. There are two critical angles that can excite fast and slow acoustic waves, which determine the conditions that stimulate a damped wave. Our results show good agreement with theoretical and simulation results. The contribution of each excited wave to the transmission coefficient is evaluated, the distribution of the transmission coefficient is analysed and application to an engineering wedge model is performed.

Three-dimensional (3D) boundary layers are common flows in aircraft but present many problems to instability analysis and transition prediction. The difficulties of 3D boundary layers are reviewed and a method is proposed to predict the linear evolution of infinitesimal perturbations in 3D boundary layers, named as RTPSE, in which the line-marching parabolized stability equation (PSE) is improved by applying the ray tracing (RT) theory. Two major improvements are achieved. One is that the marching line is predefined along the direction of group velocity, which is related to both the characteristic line of local dispersion relation and the direction of energy propagation. Another is that the variation of the real part of the spanwise wave number is predicted by RT theory, while its imaginary part is determined based on the conservation relation of generalized growth rate. The implementation of RTPSE for 3D boundary layers is given in detail and involves linear stability theory, the PSE, and RT theory. Both the tracing ray and spanwise wave number are calculated in the real number space, only leading to a second-order error. Direct numerical simulation is performed to verify and validate the prediction by RTPSE in a 3D supersonic boundary layer on a blunt cone with a half angle of $7^{\circ }$ and an angle of attack of $9^{\circ }$. Results show that RTPSE can accurately predict the variation of spanwise wave number and linear evolution of disturbances for the whole wave packet, for stationary crossflow waves and for traveling crossflow waves, while the traditional PSE cannot. The application condition of RT theory is investigated numerically, and the caustic does not occur for unstable disturbances, implying that RTPSE is fully applicable to predict the linear evolution of disturbances in 3D boundary layers.

This paper describes the design of a class of three-dimensional (3D) smooth humps mounted on a yawed cone model for a high-precision compressible Navier–Stokes solver. The influence of isolated 3D humps with different heights on the linear stability of streamwise vortices over a yawed cone with a $7°$ half-angle at a $6°$ angle-of-attack, free-stream Mach number of 6, and unit Reynolds number of $1.0\times {10}^7$ /m is investigated. The results show that smooth humps induce stationary streaks inside the streamwise vortices over a yawed cone, effectively increasing the energy of the inward vortex and inhibiting the evolution and development of the outward vortex. Stability analysis shows that the induced streaks enhance the inner mode instability, suppress the outer mode instability, and reduce the dominant frequency of the outer mode. The stability properties of the hump-induced streaks are studied, and the physical process of the transformation between the new streak instability mode and the original boundary layer mode is revealed. The eN method based on global stability theory is used to predict the transition location along the centerline on the leeward side of the cone. It is found that configurations with hump heights of 0.153 and 0.267 times the local boundary layer thickness result in a transition delay of ∼3% and ∼11%, respectively.

Zero- or low-frequency crossflow instabilities play a dominant role in the hypersonic boundary layer over a swept blunt plate. However, high-frequency second-mode disturbances dominate and lead to transition due to the wall cooling effect. The nonlinear interaction of high-frequency disturbances generates zero- or low-frequency disturbances. This paper investigates the relationship between the generated disturbances and the crossflow instability disturbances through nonlinear parabolized stability equations (NPSE) and direct numerical simulations. It is found that a special zero-frequency disturbance is generated by two second-mode disturbances with the same frequency but different spanwise wavenumbers, defined as a forced solution in this paper. This differs from the so-called eigenvalue solution obtained by linear stability theory, with the growth rate of the former being much larger than that of the latter. Additionally, the profile of the forced solution is similar to that of the second-mode disturbances, but not the crossflow-mode disturbances. The evolution of the forced solution is determined by the initial amplitude, spanwise wavenumber, and frequency of the second-mode disturbances, and is independent of the initial amplitude of crossflow-mode disturbances. If the second-mode disturbances begin to attenuate, the forced solution cannot maintain itself and transitions to the eigenvalue solution. The amplitude of the forced solution can be predicted using magnitude analysis through the amplitude of the second-mode disturbances, and the theoretical predictions are consistent with those of the NPSE when the second-mode disturbances are unstable. The results demonstrate that the forced solution is a mechanism for the second-mode disturbances to promote transition.