Stability Studies Of Compressible Boundary Layer Flows Based On PSE

In this article, the new theory and methods of PSE(Parabolized Stability Equations) are used to research the stability of the compressible boundary layer flows. The principal idea of PSE is that the disturbance flow field can be resolved into the sinusoidal variation in space and time of the traveling Tollmien-Schlichting (T-S) wave component, and the residual field with a certain velocity profile and a characteristic wavelength and growth rate for cases of general convective instability. For a weakly nonparallel basic flow, the growth rate of the remaining field in the disturbance varies slowly in the streamwise direction. The stability equations can be parabolized by neglecting the smaller second and higher derivatives of the slowly varying functions with respect to the streamwise direction, resulting in the parabolized stability equations. Because the parallel approximation and the limitation of the initial amplitude are not introduced, PSE approach can be utilized to analyze simultaneously the non-parallel and nonlinear stability problem. The computed results of PSE in this article agree well with the relevant data in references.The stabilities of unparallel compressible boundary layer flows are studied. Parabolized stability equations are derived from fully compressible Navier-Stokes equations. Developed accuracy methods, including high order differential methods and effective algebraic transformation, have improved the accuracy of the numerical solution and the velocity of convergence. According the feature of PSE, as predict-correction and spatial marching procedures are implemented, and the critical normalization condition of limiting the variation of disturbance mass is satisfied, the stability of numerical computa-tion can be guaranteed. The influences of various parameters, such as frequencies, spanwise wave numbers of disturbances, wall temperature and Mack number of free flows, on the stability are ana-lyzed.Recently, the supersonic and hypersonic boundary layer stabilities are the important topic of dis-cussion, and are emphasized in this article. According to the characteristic of the stability equations for high Mach number boundary layer flows, the saw tooth integral path on the complex plane and the expansion of the basic flow onto the complex plane by using Taylor series are adopted to deal with the singularity in the numerical integral process. Multiple unstable modes are obtained. Moreover, the spatial marching of the first modes and second modes are carried out by using PSE method.Next, we study nonlinear stability of the compressible boundary layer flows. Nonlinear parabo-lized stability equations(NPSE) are derived by the decomposition of the disturbance waves into basic modes and high order modes using fast Fourier transformation (FFT). The nonlinear stability among the T-S wave, the three-dimensional subharmonic wave and the produced high order waves in super- sonic boundary layer flows are researched. The nonlinear interaction of harmonic waves and the effect of different initial amplitudes of T-S wave and subharmonic wave on the stability are analyzed in de-tail. The nonlinear evolution and physic properties of the streamwise and spanwise vortexes and so on are clearly exhibited. The computational costs of NPSE method are much smaller than that of DNS method, which indicates that NPSE method is a powerful tool for the studies of the nonlinear stability.The boundary layer stabilities of the swept wing, which are extremely complicated, are also re-searched. Firstly, boundary layer equations in the curvilinear coordinates are derived and solved ex-actly, and the basic flow parameters are attained. Secondly, the stability equations in the non-orthogonal curvilinear coordinates are derived, and solved by using accurate numerical methods. The stability problems including crossflow instability are investigated. The influences of various pa-rameters, such as frequencies, wave angles, and the streamwise and spanwise positions, on the stabil-ity are further analyzed. The boundary layer stabilities on different wing surface regions are compared, which provide the reliable basis for engineering application of the transition prediction and the boundary layer controls.In conclusion, a systematic study of the compressible boundary layer flow stabilities is con-ducted. The stabilities of extensive velocity ranges, including subsonic, supersonic and hypersonic boundary layer flows, are researched. The multiple stability modes, specially the second modes which are most important for high Mach number flows, are used in the computation. The nonparallel and nonlinear stability problems, especially the nonlinear evolution of the disturbance waves and the complex vortices, are studied by using NPSE, which have important theoretical sense. The stability researches of the boundary layer for three dimensional objects, such as swept wing, are very valuable for engineering application like aircraft design.