Optimal transitional mechanisms of incompressible separated shear layers subject to external disturbances

TL;DR

This paper investigates the optimal transition mechanisms of incompressible shear layers over separation bubbles, highlighting how pressure gradients influence instability modes and using advanced analysis to identify disturbances that maximize skin friction drag.

Contribution

It introduces a combined linear resolvent and nonlinear Harmonic-Balanced Navier-Stokes framework to analyze and optimize disturbance growth in separated shear layers with pressure gradients.

Findings

Pressure gradient shifts instability modes from T-S waves to K-H and centrifugal instabilities.

Centrifugal instability generates streamwise vortices leading to streaks and breakdown.

Optimal disturbances that maximize skin friction are identified through adjoint optimization.

Abstract

Optimal transitional mechanisms are analysed for an incompressible shear layer developing over a short, pressure gradient-induced laminar separation bubble (LSB) with peak reversed flow of 2%. Although the bubble remains globally stable, the shear layer destabilises due to the amplification of external time- and spanwise-periodic disturbances. Using linear resolvent analysis (RA), we demonstrate that the pressure gradient modifies boundary layer receptivity, shifting from Tollmien-Schlichting (T-S) waves and streaks in a zero pressure gradient (ZPG) environment to Kelvin-Helmholtz (K-H) and centrifugal instabilities in the presence of the LSB. To characterise the non-linear evolution of these disturbances, we employ the Harmonic-Balanced Navier-Stokes (HBNS) framework, solving the Navier-Stokes equations in spectral space with a finite number of Fourier harmonics. Additionally, adjoint optimisation is incorporated to identify forcing disturbances that maximise the mean skin friction drag, conveniently chosen as the cost function for the optimisation problem since it is commonly observed to increase in the transitional stage. Compared to attached boundary layers, this transition scenario exhibits both similarities and differences. While oblique T-S instability is replaced by oblique K-H instability, both induce streamwise rotational forcing through the quadratic non-linearity of the N-S equations. However, in separated boundary layers, centrifugal instability first generates strong streamwise vortices due to multiple centrifugal resolvent modes, which then develop into streaks via lift-up. Finally, we show that the progressive distortion and disintegration of K-H rollers, driven by streamwise vortices, lead to the breakdown of large coherent structures.