Eigenfunction expansion of the flow in a spinning and nutating cylinder

Abstract

Spin-stabilized projectiles with liquid payloads may experience different types of flight instabilities caused by the fluid motion in the payload cylinder. The first type is known to occur in low viscosity fluids, i.e., at high Reynolds numbers owing to resonance with inertial waves at critical frequencies. The second type originates from a forced secondary flow at arbitrary frequency and is most pronounced for fluids of high viscosity, i.e., relatively low Reynolds numbers. In this paper a method is developed that permits unified analysis of both types of instability since it can be used for flows at high Reynolds numbers inaccessible to other approaches. The method is based on a spatial eigenfunction expansion of the axial velocity component and reduces the three-dimensional problem to one dimension. The eigenfunctions are given in closed form and the eigenvalues are determined numerically by solving a system of nonlinear equations.