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

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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, we demonstrate that the pressure gradient modifies boundary layer receptivity, shifting from Tollmien–Schlichting (T-S) waves and streaks in a zero-pressure-gradient environment to Kelvin–Helmholtz (K-H) and centrifugal instabilities in the presence of the LSB. To characterise the nonlinear evolution of these disturbances, we employ the harmonic-balanced Navier–Stokes (N-S) framework, solving the N-S 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 with 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 nonlinearity 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.

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