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

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 ∿ Re0.8 and P ∿ Re2.25 for 5 × 102 ≤ Re ≤ 2 × 104 and Z ∿ Re0.5 and P ∿ Re1.5 for Re ≥ 2 × 104 (with Re based on the velocity and size of the dipole). A critical Reynolds number Rec(here, Rec ≈ 2 × 104) 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 ∿ Re11/4 in agreement with the numerically obtained scaling laws. For Re ≥ Rec 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 ∿ Re1/2 and P ∿ Re3/2.