Improvement of the parabolized stability equation to predict the linear evolution of disturbances in three-dimensional boundary layers based on ray tracing theory

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 degrees and an angle of attack of 9 degrees. 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.