Stabilization of hypersonic boundary layers by linear and nonlinear optimal perturbations

Abstract

The effect of stationary, finite-amplitude, linear and nonlinear optimal perturbations on the modal disturbance growth in a Mach 6 axisymmetric flow over a 7 • half-angle cone with 0.126 mm nose radius and 0.305 m length is investigated. The freestream parameters (M = 6, Re = 18×10 6 /m) are selected to match the flow conditions of a previous experiment in the VKI H3 hypersonic tunnel. Plane-marching parabolized stability equations are used in conjunction with a partial-differential equation based planar eigenvalue analysis to characterize the boundary layer instability in the presence of azimuthally periodic streaks. The streaks are observed to stabilize nominally planar Mack mode instabilities, although oblique Mack mode and first-mode disturbances are destabilized. Experimentally measured transition onset in the absence of any streaks correlates with an amplification factor of N = 6 for the planar Mack modes. For high enough streak amplitudes, the transition threshold of N = 6 is not reached by the Mack mode instabilities within the length of the cone; however, subharmonic first-mode instabilities, which are destabilized by the presence of the streaks, do reach N = 6 near the end of the cone. The highest stabilization is observed at streak amplitudes of approximately 20 percent of the freestream velocity. Because the use of initial disturbance profiles based on linear optimal growth theory may yield suboptimal control in the context of nonlinear streaks, the computational predictions are extended to nonlinear optimal growth theory. Results show that by using nonlinearly optimal perturbation leads to slightly enhanced stabilization of plane Mack mode disturbances as well as reduced destabilization of subharmonic first-mode disturbances.

Nonlinear Transient Growth and Boundary Layer Transition

“…Therefore, the mode shapes for the present case are omitted from this section. Further results for a wider range of streak amplitudes and instability frequencies, including the effect of the finite-amplitude streaks on the second mode instability, can be found in Paredes et al 32,49 N -factor calculations are presented next for the S and SS instability modes by using the linear form of the plane-marching PSE. As shown by Paredes et al, 32 the N -factor curves corresponding to S and SS modes show a similar trend as that in Fig.…”

From primary instabilities to secondary instabilities in Görtler vortex flows

“…On the other hand, synchronization R1 might not be qualitatively affected since it is between the planar and oblique components of second-mode instability. As a result, the planar second-mode components might still merge with a continuous spectrum, while the oblique second-mode components would be moderately enhanced like the case studied by [27] as it travels downstream.…”

Reused LU factorization as a preconditioner for efficient solution of the Parabolized Stability Equations

“…In the case of swept wings, stationary crossflow vortices can be attenuated with discrete roughness elements [44,45]. In hypersonic flow, appropriately tuned streamwise streaks can stabilize Mack-modes (Mode-2 instabilities) both on flat plates [46] and cones [47]; a control technique that can be achieved by suitable vortex generators [48,49]. The PSE method could be used to improve the Technology Readiness Level of any of the above methods, with sufficient improvements in the computational cost.…”

“…Although explicit treatment of the nonlinear terms is used in almost all studies (since this way, the equation of each mode can be solved separately), implicit discretization of the nonlinear terms may improve the convergence of NLPSE [5]. However, this is a computationally demanding approach and is seldom used except for a few studies study of streak evolution in hypersonic boundary layers [46,47,61,62]. Alternatively, implicit discretization of the mean flow distortion (MFD) in the explicit formulation may also improve convergence [4,30].…”