On the Reynolds number scaling of vorticity production at no-slip walls during vortex-wall collisions

Abstract

Recently, numerical studies revealed two different scaling regimes of the peak enstrophy Z and palinstrophy P during the collision of a dipole with a no-slip wall [Clercx and van Heijst, Phys. Rev. E 65, 066305, 2002]: Z ∝ Re 0.8 and P ∝ Re 2.25 for 5 × 10 2 ≤ Re ≤ 2 × 10 4 and Z ∝ Re 0.5 and P ∝ Re 1.5 for Re ≥ 2 × 10 4 (with Re based on the velocity and size of the dipole). A critical Reynolds number Re c (here, Re c ≈ 2 × 10 4 ) is identified below which the interaction time of the dipole with the boundary layer depends on the kinematic viscosity ν. The oscillating plate as a boundary-layer problem can then be used to mimick the vortex-wall interaction and the following scaling relations are obtained: Z ∝ Re 3/4 , P ∝ Re 9/4 , and dP/dt ∝ Re 11/4 in agreement with the numerically obtained scaling laws. For Re ≥ Re c the interaction time of the dipole with the boundary layer becomes independent of the kinematic viscosity and, applying flat-plate boundary-layer theory, this yields: Z ∝ Re 1/2 and P ∝ Re 3/2 .

A numerical study of a vortex ring impacting a permeable wall

“…Apart from vortex ring, a vortex dipole impacting on a solid wall has been studied in the past. [3][4][5] This interaction shares many of the salient flow features encountered in vortex ring/wall interaction. Earlier experimental and numerical studies have uncovered much about these interactions, and the physics behind it are fairly well understood.…”

“…Earlier experimental and numerical studies have uncovered much about these interactions, and the physics behind it are fairly well understood. [2][3][4][5][6][7][8][9][10][11][12] In the case of a vortex ring, replacing a solid wall with a permeable wall can lead to significant changes in the flow field, and the final outcome of the interaction is strongly influenced by surface permeability, wall structure dimension, wall thickness, and the Reynolds number. Under certain favourably conditions, the primary vortex ring can even pass through the permeable wall and continues as a modified vortex ring in its lee.…”

The effect of slip length on vortex rebound from a rigid boundary

“…Nguyen van yen, Farge, and Schneider 1 found that Z ∝ Re and hence an energy dissipation rate independent of viscosity, and energy dissipation therefore approaches a constant in the vanishing viscosity limit. Keetels et al 17 used an oscillating plate model to predict the vorticity, enstrophy, and palinstrophy production from a dipole colliding with a no-slip boundary. In particular, they find a scaling of Z ∝ Re 0.75 for Reynolds number less than some critical value, and Z ∝ Re 0.5 for Reynolds number greater than the critical value.…”

Impact of a vortex dipole with a semi-infinite rigid plate

“…18 Collocated meshes are known to be nonconservative for these secondary conservation variables. 32,33 To assess, and ultimately minimize, the impact of the solution procedure on non-physical energy dissipation, the motion of a dipole in an inviscid fluid (ν = 0 in Eq.…”

“…Scaling laws for the vorticity generation on no-slip walls via a vortex/wall interaction are presented in Ref. 18 Herein, we report on the impact of a vortex dipole with the tip of a semi-infinite rigid plate. Specifically, we numerically investigate the dynamics of a Lamb dipole 28 in an otherwise quiescent incompressible Newtonian fluid, interacting with the vorticity induced along and shed from a semiinfinite rigid plate.…”