Start-up vortex flow past an accelerated flat plate

Abstract

Viscous flow past a finite flat plate moving in direction normal to itself is studied numerically.The plate moves with velocity $at^p$, where $p=0,0.5,1,2$. We present the evolution of vorticity profiles, streaklines and streamlines, and study the dependence on the acceleration parameter $p$. Four stages in the vortex evolution, as proposed by Luchini & Tognaccini (2002), are clearly identified. The initial stage, in which the vorticity consists solely of a Rayleigh boundary layer, is shown to last for a time-interval whose length shrinks to zero like $p^3$, as $p \to 0$. In the second stage, a center of rotation develops near the tip of the plate, well before a vorticity maximum within the vortex core develops. Once the vorticity maximum develops, its position oscillates and differs from the center of rotation. The difference between the two increases with increasing $p$, and decreases in time. In the third stage, the center of rotation and the shed circulation closely satisfy self-similar scaling laws for inviscid flow. Finally, in the fourth stage, the finite plate length becomes relevant and the flow begins to depart from the self-similar behaviour. While the core trajectory and circulation closely satisfy inviscid scaling laws, the vorticity maximum and the boundary layer thickness follow viscous scaling laws. The results are compared with experimental results of Pullin & Perry (1980), and Taneda & Honji (1971), where available

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“…All our observations are consistent with the ones made by Xu and Nitsche 37 . However, we also observed the formation of secondary vortex structures emanating from the edges, seen in Figure 22(a4, c4), which was not reported in their study.…”

“…The computational setup for this flow is shown in Figure 9 along with the boundary conditions. For this flow setup, Reynolds number is defined as , where l is the plate height, ν is the kinematic viscosity of the fluid, and T is a characteristic timescale defined by for some constant a 37 …”

Drag force on an accelerating submerged plate

“…However, the first three stages already occur within a small motion of the plate, i.e. within a travelled distance of 0.5-1 times the plate height (Xu & Nitsche 2015). Each of our experiments runs far into the fourth phase, which has not been investigated in great detail.…”

“…Previously, when this behaviour was observed during experiments by Lian & Huang (1989), the same observed flow behaviour was attributed to experimental defects. However, this behaviour being intrinsic to the flow was later disputed once again by Xu & Nitsche (2015), as they showed that by increasing the simulation resolution the instabilities disappear. However, Schneider et al (2014) report that the instabilities are affected by the shape of the plate tip which suggests that they are intrinsic to the flow.…”

“…The dimensionless time t * is essentially the number of plate heights travelled by the plate. A similar notation was adopted by Xu & Nitsche (2015) who reported that it was more suitable to compare results with different accelerations at the same distances travelled than identical times travelled. Gharib, Rambod & Shariff (1998) called this dimensionless time the formation time where it proved to be a universal time scale for the generation of a vortex ring by a piston.…”