Secondary Instabilities Of G?rtler Vortices In High-speed Boundary Layers And Control On Flow Transition

G?rtler vortices as well as the stabilization/destabilization of 2-D disturbances in high-speed boundary layer flows are numerically investigated.In the framework of linear stability theory,evolution of the discrete spectrum in a M a = 4.5 boundary layer is studied first.Both two-dimensional(2-D)and three-dimensional(3-D)disturbances are considered with streamwise curvature effects.The concave curvature shows a destabilizing effect on the 2-D second/third mode when the fast mode(mode F(1),mode F(2)...)synchronizes with the slow mode(mode S).The spectrum branching in the synchronization between the mode F(2)and mode S is also observed.The increase in the spanwise wavenumber(3-D disturbances),on the other hand,suppresses the synchronization between mode F and mode S and reduces the growth rate of the unstable mode.With regard to the 3-D disturbances subjecting to the concave curvature,the mode S originating from the slow acoustic wave amounts to the unsteady G?rtler mode while the quasi-steady G?rtler mode emanates from the continuous spectrum of the vorticity/entropy wave.The secondary instabilities of G?rtler vortices in high-speed boundary layer flows are then investigated.To uncover the compressibility effects,five Ma numbers,covering incompressible to hypersonic flows,at Ma = 0.015,1.5,3.0,4.5and 6.0 are specified.G?rtler vortices in subsonic and moderate supersonic flows(Ma = 0.015,1.5,3.0)are governed by the conventional wall-layer mode(mode W).In hypersonic flows(Ma = 4.5,6.0),the trapped-layer mode(mode T)becomes dominant.This difference maintains and intensifies downstream leading to different scenarios of secondary instabilities.In fact,when Re is large enough(Re is based on local boundary layer thickness),competition between mode W and mode T occurs in hypersonic cases.The linear and nonlinear development of G?rtler vortices which are governed by dominant modal disturbances are investigated with direct marching of the nonlinear parabolic equations.The secondary instabilities of G?rtler vortices set in when the resulting streaks are adequately developed.They are studied with Floquet theory at multiple streamwise locations.The secondary perturbations become unstable downstream following the sequence of sinuous mode type I,varicose mode and sinuous mode type II indicating an increasing threshold amplitude.Onset conditions are determined for these modes.The above three modes each can have the largest growth rate under proper conditions.In hypersonic cases,the threshold amplitude A(u)is dramatically reduced showing a great impact of the thermal streaks.To investigate the parametric effect of the spanwise wavenumber,three global wavenumbers(B = 0.5,1.0 and2.0 × 10-3)are specified.The relationship between the dominant mode(sinuous or varicose)and the spanwise wavenumber of G?rtler vortices found in incompressible flows is shown not fully applicable in high-speed cases.The sinuous mode becomes the most dangerous regardless of the spanwise wavelength when Ma > 3.0.The subharmonic type can be the most dangerous mode while the detuned type can be neglected although some of the sub-dominant secondary modes reach their peak growth rates under detuned states.We make use of the streaks developed from Klebanoff(optimal perturbations)modes to stabilize the flow.The boundary-layer flows at Mach numbers 4.5 and6.0 are studied in which both the first-and second modes are supported.When the streak amplitude is in an appropriate range,i.e.large enough to modulate the laminar boundary layer but low enough to not trigger secondary instability,both the first-and second modes can effectively be suppressed.On the other hand,G?rtler modes are utilized to develop a criterion on the geometry/arrangement of roughness elements to achieve flow transition.