Hydrodynamic Interaction between Two Cylinders with Rotation

Abstract

The hydrodynamic interaction between two cylinders with rotational motion through an inviscid and incompressible fluid is investigated theoretically. The dynamical behaviors of an elliptic cylinder moving around a fixed circular cylinder are described based on the dynamical equations of motion in the plane of motion. In a relative coordinate system moving with the stream, the kinetic energy of the fluid is expressed as a function of fifteen generalized added masses due to the planar motion of the two cylinders. By means of the generalized added masses, the planar motion of an elliptic cylinder around a fixed circular cylinder can be computed without considering the flow field. In order to proceed the problem analytically, a set of transformations of harmonics between two corresponding spaces are obtained. These transformations are applied to derive the complete complex potentials by using the successive potential procedure, which is an extension of the circle theorem in two dimensions. These results are utilized to predict trajectories of an elliptic cylinder around a fixed circular cylinder in planar motion and to estimate the effects of non-circularity, initial position and initial velocity on the interaction between two cylinders. The numerical results show explicitly that the dynamical behaviors of the moving bodies with rotational motion appear nonlinear. Their moving properties exhibit significant difference from those in the particle dynamics.