On the motion of laminar wing wakes in a stratified fluid

We present numerical solutions for two-dimensional laminar symmetric vortex systems descending in a stably stratified fluid, within the Boussinesq approximation. Three types of flows are considered: (I) tight vortices; (II) those deriving from an elliptical wing lift distribution; and (III) those deriving from a 'high-lift' distribution, with a part-span flap on the wing. The non-dimensional stratification ranges from zero to moderate, as it does for airliners. For Types I and II, with high Reynolds numbers and weak stratification, the solutions confirm the theory of Scorer & Davenport (1970) (their article lacks a crucial. link which we provide, equivalent to one of Crow (1974)). Contrary to common conceptions and observations in small-scale experiments, the descent velocity increases exponentially with time, as the distance between vortices decreases and the circulation of the vortices proper is conserved. With moderate stratification, wakes with sufficient energy also attain the accelerating regime, until the vortex cores make contact. However, they first experience a rebound, which is both of practical importance and out of reach of simple formulas. Type III wakes produce two durable vortex pairs which tumble, and mitigate the buoyancy effect by exchanging fluid with the surroundings. These phenomena are obscured by low wing aspect ratios, Reynolds numbers below about 10(5), or appreciable surrounding turbulence; this may explain why neither a clear rebound nor an acceleration can be reconciled with experiments to date. We argue that airliner wakes have very little inherent diffusion, and that a rapid end to the wake's descent must reveal effects other than simple buoyancy. In particular, stratification promotes the Crow instability.