Hydrodynamic damping of an oscillating cylinder at small Keulegan–Carpenter numbers

Abstract

Direct numerical simulations (DNS) of oscillatory flow around a cylinder show that the Stokes–Wang (S–W) solution agrees exceptionally well with DNS results over a much larger parameter space than the constraints of $\beta K^2\ll 1$ and $\beta \gg 1$ specified by the S–W solution, where $K$ is the Keulegan–Carpenter number and $\beta$ is the Stokes number. The ratio of drag coefficients predicted by DNS and the S–W solution, $\varLambda _K$ , mapped out in the $K\text {--}\beta$ space, shows that $\varLambda _K < 1.05$ for $K\leq {\sim }0.8$ and $1 \leq \beta \leq 10^6$ , which contradicts its counterpart based on experimental results. The large $\varLambda _K$ values are primarily induced by the flow separation on the cylinder surface, rather than the development of three-dimensional (Honji) instabilities. The difference between two-dimensional and three-dimensional DNS results is less than 2 % for $K$ smaller than the corresponding $K$ values on the iso-line of $\varLambda _K = 1.1$ with $\beta = 200\text {--}20\,950$ . The flow separation actually occurs over the parameter space where $\varLambda _K\approx 1.0$ . It is the spatio-temporal extent of flow separation rather than separation itself that causes large $\varLambda _K$ values. The proposed measure for the spatio-temporal extent, which is more sensitive to $K$ than $\beta$ , correlates extremely well with $\varLambda _K$ . The conventional Morison equation with a quadratic drag component is fundamentally incorrect at small $K$ where the drag component is linearly proportional to the incoming velocity with a phase difference of ${\rm \pi} /4$ . A general form of the Morison equation is proposed by considering both viscous and form drag components and demonstrated to be superior to the conventional equation for $K < {\sim }2.0$ .