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

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 << 1 and beta >> 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, A(K), mapped out in the K-beta space, shows that Lambda(K) < 1.05 for K <= similar to 0.8 and 1 <= beta <= 10(6), which contradicts its counterpart based on experimental results. The large Lambda(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 Lambda(K) = 1.1 with beta = 200-20 950. The flow separation actually occurs over the parameter space where Lambda(K)( )approximate to 1.0. It is the spatio-temporal extent of flow separation rather than separation itself that causes large Lambda(K) values. The proposed measure for the spatio-temporal extent, which is more sensitive to K than beta, correlates extremely well with Lambda(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 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 < similar to 2.0.