Instability wave–streak interactions in a high Mach number boundary layer at flight conditions

Abstract

The interaction of stationary streaks undergoing non-modal growth with modally unstable instability waves in a high Mach number boundary-layer flow is studied using numerical computations. The geometry and flow conditions are selected to match a relevant trajectory location from the ascent phase of the HIFiRE-1 flight experiment; namely, a $7^{\circ }$ half-angle, circular cone with $2.5$ mm nose radius, free-stream Mach number equal to $5.30$, unit Reynolds number equal to $13.42~\text{m}^{-1}$ and wall-to-adiabatic temperature ratio of approximately $0.35$ over most of the vehicle. This paper investigates the nonlinear evolution of initially linear optimal disturbances that evolve into finite-amplitude streaks, followed by an analysis of the modal instability characteristics of the perturbed, streaky boundary-layer flow. The investigation is performed with a stationary, full Navier–Stokes equations solver and the plane-marching parabolized stability equations (PSE), in conjunction with partial-differential-equation-based planar eigenvalue analysis. The overall effect of streaks is to reduce the peak amplification factors of instability waves, indicating a possible downstream shift in the onset of laminar–turbulent transition. The present study confirms previous findings that the mean-flow distortion of the nonlinear streak perturbation reduces the amplification rates of the Mack-mode instability. More importantly, however, the present results demonstrate that the spanwise varying component of the streak can produce a larger effect on the Mack-mode amplification. The analysis of planar and oblique Mack-mode waves modulated by the presence of the streaks shows that the planar Mack mode still dominates the instability characteristics of the flow. The study with selected azimuthal wavenumbers for the stationary streaks reveals that a wavenumber of approximately $1.4$ times larger than the optimal wavenumber is more effective in stabilizing the planar Mack-mode instabilities. In the absence of unstable first-mode waves for the present cold-wall condition, transition onset is expected to be delayed until the peak streak amplitude increases to nearly 35 % of the free-stream velocity, when intrinsic instabilities of the boundary-layer streaks begin to dominate the transition process. For streak amplitudes below that limit a significant net stabilization is achieved, yielding a potential transition delay that can exceed 100 % of the length of the laminar region in the uncontrolled case.