Most-Critical Transient Disturbances in an Incompressible Flat-Plate Boundary Layer

Abstract

Transient growth is a linear disturbance growth mechanism that plays a key role in roughness-induced boundary-layer transition. It occurs when superposed stable, non-orthogonal continuous spectrum modes experience algebraic disturbance growth followed by exponential decay. Algebraic disturbance growth can modify the basic state making it susceptible to secondary instabilities rapidly leading to transition. Optimal disturbance theory was developed to model the most-dangerous disturbances. However, evidence suggests roughness-induced transient growth is sub-optimal yet leads to transition earlier than optimal theory suggests. This research computes initial disturbances most unstable to secondary instabilities to further develop the applicability of transient growth theory to surface roughness.

The main approach is using nonlinear adjoint optimization with solutions of the parabolized Navier–Stokes and BiGlobal stability equations. Three objective functions were considered: disturbance kinetic energy growth, sinuous instability growth rate, and Tollmien–Schlichting (TS) wave growth rate. The first objective function was used as validation of the optimization method. Counter-rotating streamwise vortices located low in the boundary layer maximize the sinuous instability growth rate. Sinuous instabilities were observed at disturbance amplitudes as low as 2:5% spanwise root-mean-square. The near wake of the initial disturbance is potentially much less stable than the far field. TS wave stabilization was achieved for all parameters considered and becomes more effective at higher frequencies.