Minimal seed in supersonic boundary layer at $M=3$

TL;DR

The paper develops a nonlinear non-modal optimization framework to identify the minimal seed for laminar-to-turbulent transition in a supersonic boundary layer at $M=3.0$ and $Re=300$. It reveals that finite-amplitude nonlinear effects reshape the optimal perturbation, producing flattened outer-layer structures and near-wall streamwise vortices that drastically accelerate transition compared to linear predictions. Although the late-stage evolution aligns with oblique breakdown and $ ext{Λ}$-vortices, the nonlinear path reduces the disturbance energy required for transition, and this mechanism persists under wall cooling ($T_w=0.6T_{ad}$) where the generalized inflection point is suppressed. The study highlights a robust nonlinear amplification pathway that leverages the interaction between near-wall planar waves and outer-layer vortex patterns to form streaks via vortex self-induction, offering a more efficient transition route than traditional linear analyses.

Abstract

This study investigates the minimal seed for laminar-to-turbulent transition in a supersonic boundary layer at $M=3.0$ and $Re=300$ using adjoint-based nonlinear non-modal analysis. While linear theory identifies oblique waves as the optimal disturbances for transient growth, we demonstrate that nonlinear effects fundamentally alter the optimal perturbation structure as the initial amplitude exceeds a critical threshold. Our analysis reveals that the nonlinear optimal perturbation exhibits a distinctive spatial distribution characterized by flattened structures in the outer layer and streamwise vortices near the wall, leading to a more rapid transition compared to the linear counterpart. A key finding is that this nonlinear amplification mechanism remains robust even under wall-cooled conditions ($T_w = 0.6 T_{ad}$), where the disappearance of the generalized inflection point (GIP) suppresses linear instabilities of Mack's first mode. This rapid growth is driven by the nonlinear interaction between two-dimensional planar waves near the wall and staggered vortex patterns in the outer layer, facilitating streak formation through vortex self-induction. Although the late-stage evolution eventually converges to the classical oblique breakdown scenario as characterized by the formation of $Λ$-vortices, the identified nonlinear path significantly reduces the disturbance energy required for transition.

Figures (20)

• Figure 1: Comparison of growth rate of nonlinear non-modal analysis at incompressible limit ($M=0.4$) with the literature values.
• Figure 2: Comparison of wall-normal profiles at the inflow boundary for $T_w = T_{ad}, 0.6 T_{ad}$: numerical calculation (open circles) versus self-similar solution (lines).
• Figure 3: Convergence of optimization process in the nonlinear non-modal analysis at $\delta = 1.0, t_f=150, T_w =0.6 T_{ad}$.
• Figure 4: $[60, 80]$
• Figure 5: $[80, 100]$