Numerical study of viscous starting flow past a flat plate

TL;DR

This study numerically investigates the early viscous flow past a flat plate impulsively started normal to itself, analyzing vorticity generation, circulation shedding, and flow scaling laws across different Reynolds numbers.

Contribution

It provides detailed numerical analysis of vorticity dynamics and circulation shedding in viscous flow past a flat plate, including new scaling laws and benchmark results.

Findings

Vorticity remains attached to the plate at early times.

Viscous diffusion significantly influences circulation shedding.

Flow quantities follow inviscid self-similar scaling laws.

Abstract

Viscous flow past a finite plate which is impulsively started in direction normal to itself is studied numerically using a high order mixed finite difference and semi-Lagrangian scheme. The goal is to resolve details of the vorticity generation at early times, and to determine the effect of viscosity on flow quantities such as the core trajectory and vorticity, and the shed circulation. Vorticity contours, streaklines and streamlines are presented for a range of Reynolds numbers $Re \in [250, 2000]$ and a range of times $t \in[0. 0002, 5]$. At early times, most of the vorticity is attached to the plate. The paper proposes a definition for the shed circulation at early as well as late times, and shows that it indeed represents vorticity that separates from the plate without reattaching. The contribution of viscous diffusion to the circulation shedding rate is found to be significant, but, interestingly, to depend only slightly on the value of the Reynolds number. The shed circulation and the vortex core trajectories follow scaling laws for inviscid self-similar flow over several decades in time. Scaling laws describing the core vorticity, core dissipation, boundary layer thickness, drag and lift forces in time and Reynolds number are also presented. The simulations provide benchmark results to evaluate, for example, simpler separation models such as point vortex and vortex sheet models.