Resonance phenomena in viscous fluid configurations inside a spinning and coning cylinder

The moments exerted by viscous fluids in cylindrical payloads can cause severe flight instabilities of liquid-filled projectiles. This study is concerned with the calculations of these moments for a spinning and coning payload having the following configurations: (1) A single fluid in a completely filled cylinder. (2) A single fluid in a cylinder containing a coaxial rod. (3) A single fluid in a partially filled cylinder. Small percentages of air inclusion during production essentially lead to this case. (4) Two immiscible fluids of different density and viscosity in a completely filled cylinder. Since a negative roll moment is caused by shear stresses at the cylinder wall, a heavy low-viscosity additive can reduce this moment. The viscous fluid motions, and consequently the moments, are analyzed with numerical methods based on spatial eigenfunction expansions (for the linearized problems) or spectral techniques (for the linear and nonlinear problems). For completely filled cylinders, it is found that the yaw and roll moments acquire maxima at critical aspect ratios (i.e. ratio of length to diameter of the cylinder) as the Reynolds number is increased. Similar results are found for the other configurations, but now at critical fill ratios (i.e. ratio of the volume of one fluid to the volume of the cylinder) for a given aspect ratio. These maxima are due to resonant inertial waves. The inviscid fluid motions for the various configurations are also analyzed by eigenfunction expansions for the linearized equations and their solutions, in analytical form, provide criteria for the onset of resonance. Spectral techniques are used to investigate the effect of nonlinearities on the moments, and to calculate the pitch moment accurately since unlike the roll and yaw moments it is only partially predicted by linear theory. Comparisons of the linear and nonlinear results reveal that nonlinear effects are negligible for all practical applications.