The start-up vortex issuing from a semi-infinite flat plate

Abstract

The subject of the present work is the start-up vortex issuing from a sharp trailing edge accelerated from rest in still air. A numerical simulation of the flow has been performed in the case of a semi-infinite flat plate by solving the Navier-Stokes equations in the psi-omega formulation. The numerical algorithm is based on a fast multigrid implicit integration of the difference equations in an unstructured mesh that is dynamically built to minimize the computational costs. A local refinement of the mesh near the edge of the plate increases the accuracy of the simulation. The results show that the asymptotic stage of the vortex evolution is self-similar in the mean, but the appearance of instabilities produces a time-dependent flow which is not instantaneously self-similar

Scaling behaviour in impulsively started viscous flow past a finite flat plate

“…It is noteworthy that in our computations, the recirculation region and associated negative vorticity is visible already at time t = O( t), within the first 10 timesteps, indicating that it forms immediately after the start of the motion. Luchini & Tognaccini (2002) suggest that the initial stage in viscous separation, referred to as the Rayleigh stage, is one in which the flow is potential everywhere except for a thin viscous boundary layer of constant thickness around all of the plate, including the plate tip. Our results indicate that the Rayleigh stage is absent for impulsively started flow.…”

“…This implies that one can recover the solution at time t with one value of Re from the solution with another Re = Re/τ 1/3 at another time t = τ t. Consequently, knowing the solution for one Re at all times yields the full solution for any other Re. This is equivalent to the fact shown by Luchini & Tognaccini (2002), that under appropriate rescaling the Navier-Stokes equations for flow past a semi-infinite plate reduce to Re-independent equations. We will show that the finite plate flow studied here satisfies the scaling (3.3) at early times.…”

“…Related numerical results include the simulations of Wang (2000) and Eldredge (2007) for viscous flow past thin rounded plates, those of Hudson & Dennis (1985), Dennis et al (1993) and Koumoutsakos & Shiels (1996) for flow past finite plates of zero thickness, and Luchini & Tognaccini (2002), who studied flow past a semi-infinite plate of zero thickness. These works report vortex fields, vortex core trajectories and induced forces at intermediate to relatively large times.…”

“…Here, we use a split method in time in which advection is treated using a semi-Lagrangian scheme, diffusion is treated with a three-level Crank-Nicholson method, and all finite difference and interpolation approximations are of fourth order. The method uses ideas from several previous works, including Staniforth & Côté (1991), E & Liu (1996), Johnston & Krasny (2002), Luchini & Tognaccini (2002), Seaid (2002) and Nitsche, Taylor & Krasny (2003). It is of second order for the present highly singular flow.…”

“…Here, we identify three types of scaling laws. (i) For viscous flow past a semi-infinite plate, one can eliminate the dependence on Re by appropriately rescaling length and time, see for example Luchini & Tognaccini (2002). As a result, the evolution in time with fixed Re equals the solution with variable Re at a fixed time, up to scale.…”