Effect of cavity-induced perturbation interactions on the transitional flow after the trailing edge

TL;DR

The paper addresses how perturbations generated by a subsonic open cavity amplify and interact to trigger flow transition downstream. It combines DNS data at $\mathrm{Ma}=0.6$, $\mathrm{Re}_D=1500$ with classical resolvent analysis about the mean flow and harmonic resolvent analysis about a time-periodic base flow at the Rossiter frequency to identify modal and non-modal amplification mechanisms. Key findings show that the mean flow supports modal Rossiter modes and a 3D centrifugal instability, while non-modal lift-up yields downstream streaks; when the unsteady base flow is included, cross-frequency interactions further amplify 3D perturbations via oblique Tollmien–Schlichting waves, forming a feedback-driven pathway to transition. These results illuminate multi-mode cavity dynamics and offer a quantitative framework for predicting and potentially mitigating downstream flow modifications caused by cavity perturbations.

Abstract

We investigate the modal and non-modal linear amplification mechanisms in the flow over a subsonic open cavity and their subsequent interactions to identify optimal flow perturbations that propagate downstream the cavity and trigger flow transitions. Using both the stationary and time-varying base flows from a Direct Numerical Simulation of a cavity flow at Mach 0.6, we employ classical and harmonic resolvent analyses to explain the role of the cavity-generated perturbations in destabilizing the flow downstream. Our analysis of perturbation amplification about the mean flow identifies structures that resemble Tollmien-Schlichting (T-S) waves at the Rossiter frequency in the attached boundary layer region after the cavity. A low-frequency centrifugal instability dominates inside the cavity. The mean flow also amplifies stationary streaks via a lift-up mechanism that extends throughout the boundary layer region downstream of the cavity. The harmonic resolvent analysis (HRA) reveals the amplification of additional perturbations by the unsteady Rossiter base flow. By restricting the input and output at the same frequency in the HRA, we find the amplification of the stationary perturbation to be the most dominant 3D instability mechanism. This amplification is driven by the interaction of the 3D streaks with the unsteady Rossiter base flow, which generates internal forcing in the form of oblique T-S waves, thereby further amplifying the streaks. The interaction between the centrifugal perturbation and the unsteady flow also generates streamwise elongated structures in the boundary layer region after the cavity. Together, the centrifugal-Rossiter and streak-Rossiter interactions synergistically amplify perturbations downstream of the cavity.

Figures (10)

• Figure 1: (a) Computational setup for the DNS of the flow over a rectangular cavity at $\hbox{\it Ma}_{\infty}=0.6$ (drawing is not to scale). Instantaneous iso-surface of Q-criterion at level $QD^2/\tilde{U}_{\infty}^2=0.03$ is colored by streamwise velocity, and iso-surface of fluctuating streamwise velocity is shown at levels $u'/\tilde{U}_{\infty}=\pm0.05$ (Red: positive, Blue: negative). (b) Power spectral density of the fluctuating vertical velocity ($v'$) sampled at ($x,y,z$)=(1.0,0.0,0.0).
• Figure 2: (a) Eigenspectrum of the operator $\mathsfbi{A}(\overline{\boldsymbol{Q}},\beta)$. (b) The optimal energy gain $\sigma_1$ over ranges of spanwise wavenumbers ($\beta D$) and frequencies ($St$) about the spanwise and time-averaged base flow.
• Figure 3: (a,b) Streamwise variation of the disturbance energy of the optimal forcing and response modes and (c,d,e) iso-surface of the real component of the optimal forcing and response modes obtained from CRA at ($\beta D, St$)=$(0.0,0.36),(2\pi,0.025),(2\pi,0.0)$. The gray solid region in (a,b) indicates the extent of the cavity. The forcing modes are plotted at levels $\pm0.2\max(|\mathcal{R}\{\phi_{u}\}|)$ (Yellow: positive, Black: negative), and the response modes are plotted at levels $\pm0.3\max(|\mathcal{R}\{\psi_{u}\}|)$ (Red: positive, Blue: negative) in (c,d,e).
• Figure 4: (a) Streamwise variation of the disturbance energy in three velocity components of the optimal classical resolvent response mode at $(\beta D,St)=(2\pi,0.0)$. (b) Planar ($y$--$z$) view of the spatial structures of the optimal streamwise velocity response (contour plots) and streamwise vorticity forcing (contour lines, solid-positive, dashed-negative) corresponding to $(\beta D,St)=(2\pi,0.0)$.
• Figure 5: (a) Normalized Floquet exponents $s$ of the operator $\mathsfbi{T}(\beta)$ in the complex plane for $s_iD/2\pi \tilde{U}_{\infty}\in [0,St_{II}/2]$. (b) A zoomed plot of Floquet multipliers $\alpha=\mathrm{e}^{sT_0}$ showing the growth of perturbations over the period $T_0$.