Start-up vortex flow past an accelerated flat plate

Abstract

Viscous flow past a finite flat plate accelerating in the direction normal to itself is studied numerically. The plate moves with nondimensional speed tp, where p = 0, 1/2, 1, 2. The work focuses on resolving the flow at early to moderately large times and determining the dependence on the acceleration parameter p. Three stages in the vortex evolution are identified and quantified. The first stage, referred to as the Rayleigh stage [Luchini and Tognaccini, “The start-up vortex issuing from a semi-infinite flat plate,” J. Fluid Mech. 455, 175–193 (2002)], consists of a vortical boundary layer of roughly uniform thickness surrounding the plate and its tip, without any separating streamlines. This stage is present only for p > 0, for a time-interval that scales like p3, as p → 0. The second stage is one of self-similar growth. The vortex trajectory and circulation satisfy inviscid scaling laws, the boundary layer thickness satisfies viscous laws. The self-similar trajectory starts immediately after the Rayleigh stage ends and lasts until the plate has moved a distance d = 0.5 to 1 times its length. Finally, in the third stage, the image vorticity due to the finite plate length becomes relevant and the flow departs from self-similar growth. The onset of an instability in the outer spiral vortex turns is also observed, however, at least for the zero-thickness plate considered here, it is shown to be easily triggered numerically by underresolution. The present numerical results are compared with experimental results of Pullin and Perry [“Some flow visualization experiments on the starting vortex,” J. Fluid Mech. 97, 239–255 (1980)], and numerical results of Koumoutsakos and Shiels [“Simulations of the viscous flow normal to an impulsively started and uniformly accelerated flat plate,” J. Fluid Mech. 328, 177–227 (1996)].