Applications Of PSE To Evolution Of Disturbances In Compressible Boundary Layers And To Secondary Instability In Supersonic Boundary Layers

In this paper, parabolized stability equations (PSE) approach is used to investigate the evolution of disturbances in compressible boundary layers on flat plates. Firstly, the linear and nonlinear parabolized stability equations are derived. On this basis, the linear PSE is used to study the evolution of small amplitude disturbances in compressible boundary layers and the nonparallel effects of the basic flow to the neutral curves of compressible boundary layers. Then, the nonlinear PSE is employed to study the evolution of finite amplitude disturbances in compressible boundary layers, and the results are compared with those obtained by DNS so that its reliability is confirmed. Finally, the nonlinear PSE is applied as a tool to investigate the problem of secondary instability in supersonic boundary layers. The following conclusions are drawn:(1) For three typical small amplitude disturbances, i.e. the T-S wave in subsonic boundary layers and the first mode and second mode T-S waves in supersonic boundary layers, their linear evolutions computed by linear PSE agree almost perfectly with those obtained by linear stability theory for both parallel and nonparallel basic flows, provided the Reynolds number is large.(2) Linear PSE is applied to search the neutral curves of 2-D small amplitude disturbances in compressible boundary layers, and the results are compared with those obtained by LST so that the nonparallel effects to neutral curves are found. No matter the boundary layer is subsonic or supersonic, nonparallel effects are obvious at the critical Reynolds number, such that the critical Reynolds number is reduced and the frequency range of unstable disturbances is broadened at the critical Reynolds number. At large Reynolds number, as the basic flow is closer to parallel flow, the nonparallel effect to boundary layer stability becomes insignificant.(3) For three typical disturbances, i.e. the finite amplitude T-S wave in subsonic boundary layers and the finite amplitude first mode and second mode T-S waves in supersonic boundary layers, their evolutions obtained by using nonlinear PSE agree reasonably well with those obtained by spatial mode direct numerical simulations, including the amplitudes and shapes of the mean flow distortion, the fundamental disturbance and higher harmonics. Consequently, nonlinear PSE can be applied to investigate the evolution of finite amplitude disturbances in compressible boundary layers.(4) The nonlinear PSE method is applied to study the secondary instability mechanism in a supersonic boundary layer with Mach number 4.5. The result shows that the mechanism of secondary instability does work, no matter the fundamental wave is first mode or second mode T-S wave. The variation of the growth rates of the 3-D sub-harmonic wave against its span-wise wave number and the amplitude of the 2-D fundamental wave is found to be similar to those found in incompressible boundary layers. But even as the amplitude of the 2-D fundamental wave is as large as at the order of 2%, the maximum growth rate of the 3-D sub-harmonic is still much smaller than the growth rate of the most unstable second mode 2-D T-S wave. Consequently, secondary instability is unlikely the main cause leading to transition in supersonic boundary layers.