Is No-Slip Necessary for Vorticity Generation and Shedding over a Circular Cylinder?

Abstract

It is traditionally believed that the no-slip boundary condition is necessary for vorticity generation, as might be implied by the Lighthill vorticity generation mechanism. However, the investigations of Morton (1984) and Terrington et al. (2020) assert that the vorticity-generation mechanism is independent of the no-slip boundary condition. To investigate this hypothesis, we simulate viscous flow over a circular cylinder in the laminar periodic von K\'arm\'an vortex-shedding regime at Re = 170, without enforcing the no-slip condition at the wall. Instead, we close the wall tangential velocity and its normal derivative using one-sided finite-difference approximations. Accordingly, the wall tangential velocity is dynamically determined from the evolving interior solution rather than prescribed. The resulting flows exhibit qualitatively similar patterns of periodic vortex shedding across all three wall treatments (no-slip, second- and third-order one-sided approximations), although the mean drag coefficient, r.m.s. lift coefficient, and Strouhal number differ from those of the standard no-slip simulation. More significantly, the velocity distributions just outside the boundary layer closely match those of the standard no-slip simulation, regardless of the approximation order. In particular, the circulation evaluated along a contour at the edge of the boundary layer, which corresponds to the total vorticity contained within the layer in the no-slip case, is found to be remarkably insensitive to the wall treatment. These results suggest that the no-slip condition is not necessary for vorticity generation or periodic shedding. However, it may still be required for accurate quantitative prediction of the integral flow quantities and shedding dynamics.