Nonparallel stability analysis of three-dimensional disturbance wave in boundary layers

In this paper, the nonparallel stability of 3D TS waves for the spatial mode in the boundary layers is analyzed using parabolized stability equations (PSE), which are new effective approach to study the evolution of disturbance waves in the weakly nonparallel flows such as boundary layers. The governing equations are derived directly from full Navier-Stokes equations in primitive-variables form. Numerical computation is exact by using fourth-order compact scheme in normal-wise. The PSE is solved stably by using a nonsingularly mapped domain with modified outer boundary conditions, and overcome the step-size limitation and the transient phase introduced by the parallel initial solution by larger step size at beginning. The solver of single element band matrix with complement zero block is presented, which is quite efficient. The results presented clearly show the evolutions of the 3D TS waves and the effects of the nonparalellism on the stability of 3D waves in incompressible boundary layers. Paticularly, the nonparalell effects, which is found in the amplitude curves of TS waves with different spanwise wavenumber, can change stability characteristics of 3D disturbance wave from stability with parallel flow to instability with nonparallel flow under some conditions. The role of pressure gradient is also investigated. The results computated are quite consistent with available data.