Resonance phenomena in viscous fluids inside partially filled spinning and nutating cylinders

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. For cylinders completely filled with a single fluid, these instabilities were analyzed by eigenfunction expansion developed by Selmi, Li, and Herbert [Phys. Fluids A 4, 1998 (1992)]. The method permits unified analysis of both types of instability, since it can be used for flows at moderate as well as high Reynolds numbers. Often in practice, cylinders are made to include a central rod to alter resonance properties or are partially filled during production, to ensure safety as the liquid payload may expand under different conditions. In this paper, the eigenfunction approach is extended to analyze the moments caused by the flow in a spinning and nutating cylinder, containing a partial fill or a central rod. The analysis shows that the fill ratio (defined as the ratio of the volume of the fluid to the volume of the cylinder) affects resonance with inertial waves. The inviscid flow equations are solved analytically to provide criteria for the onset of resonance in the two configurations.