Theoretical analysis of liquid wave motion in rotating cylinder depending on liquid depth ratio

Abstract

The fluid force caused by the wave motion of liquid that partially fills a hollow rotor was theoretically investigated, with a focus on the motion's relationship with the radial liquid depth. The fluid force causes a self-excited whirl of the rotor. To obtain the excitation force as a function of the liquid depth, a nonlinear analysis method that does not use shallow water approximation is presented. Gravity is negligible and the liquid motion is assumed to be axially uniform. The linear eigenmodes of the two-dimensional flow in the radial and circumferential directions are nonlinearly coupled through the Galerkin method. The periodic solution of the wave response to the whirl motion is obtained and is compared with the conventional shallow water approximation. In the present theory, the excitation force reaches its maximum at a specific liquid depth that is consistent with previous experimental studies, while the maximum does not appear in the shallow water approximation. Furthermore, the present theory indicates that the response of the first eigenmode that causes the excitation force is suppressed by nonlinearity when the liquid is shallow; in contrast, when the liquid is deep, the first eigenmode self-balances with disturbance of the whirl motion without the help of nonlinearity. The decrease in the helping role of nonlinearity in a deep liquid comes from the ratio of the radial flow velocity to the circumferential flow velocity in the eigenfunction becoming large as the liquid depth ratio increases.