Energy Dissipating Structures Produced by Walls in Two-Dimensional Flows at Vanishing Viscosity

Abstract

We perform numerical experiments of a dipole crashing into a wall, a generic event in two-dimensional incompressible flows with solid boundaries. The Reynolds number (Re) is varied from 985 to 7880, and no-slip boundary conditions are approximated by Navier boundary conditions with a slip length proportional to Re(-1). Energy dissipation is shown to first set up within a vorticity sheet of thickness proportional to Re(-1) in the neighborhood of the wall, and to continue as this sheet rolls up into a spiral and detaches from the wall. The energy dissipation rate integrated over these regions appears to converge towards Re-independent values, indicating the existence of energy dissipating structures that persist in the vanishing viscosity limit.

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

“…The critical Reynolds number was found to be around Re = 20 000, higher than considered here and higher than considered by Nguyen van yen, Farge, and Schneider. 1 The predicted scaling agrees well with observed peak enstrophy in their simulations. Importantly these results suggest that the enstrophy growth, and hence energy dissipation, will slow as the Reynolds number becomes increasingly large.…”

“…The slip length relation s L = 4/Re is motivated by the finding of Nguyen van yen, Farge, and Schneider 1 for the boundary condition of the penalisation method and similarly s L = 0.003 is the slip length given by this approximation for Re ≈ 1300, near the lower Reynolds numbers considered in Nguyen van yen, Farge, and Schneider. 1 As discussed earlier, the slip length s L = 0.0001 should approximate the no-slip results. The parameters chosen in the calculations are shown in Table I.…”

“…Multiplying both sides of the equation by s L and taking the limit s L → 0 recovers the influence matrix for the no-slip boundary conditions. The determinant of the influence matrix can be simplified using u 1 2 which is never zero and hence the matrix is invertible.…”

“…The crucial difference between these three cases occurs in the enstrophy growth results, in particular the rate at which enstrophy growth varies with Reynolds number. The no-slip, s L = 0.0001, and s L = 4/Re cases, are shown in Figure 16 1 For lower Reynolds numbers the enstrophy growth is much more rapid, and we have omitted Re < 1700 from Figure 17 for clarity.…”

“…However, we do find unusual behaviour if the slip length is allowed to be inversely proportional to Reynolds number, with our results closely paralleling those of Nguyen van yen, Farge, and Schneider. 1 …”