Buoyancy-driven algebraic (localised) boundary-layer disturbances

Abstract

We show that a new class of steady linear eigenmodes exist in the Falkner–Skan boundary layer, associated with an algebraically developing, thermally coupled three-dimensional perturbation that remains localised in the spanwise direction. The dominant mode has a weak temperature difference that decays (algebraically) downstream, but remains sufficient (for favourable pressure gradients that are below a critical level) to drive an algebraically growing disturbance in the velocity field. We determine the critical Prandtl number and pressure gradient parameter required for downstream algebraic growth. We also march the nonlinear boundary-region equations downstream, to demonstrate that growth of these modes eventually gives rise to streak-like structures of order-one aspect ratio in the cross-sectional plane. Furthermore, this downstream flow can ultimately become unstable to a two-dimensional Rayleigh instability at finite amplitudes.