Low Reynolds number Navier-Stokes simulation for rotating liquid-filled containers

Abstract

Three-dimensional, steady, laminar, incompressible Navier-Stokes equations are solved using an implicit finite-difference scheme based on successive over-relaxation. These low Reynolds number numerical simulations are used to predict the behavior of liquids undergoing steady spin and steady precession at a fixed precession angle. Low Reynolds number simulations are applicable to slowly spinning vehicles or highly viscous liquid payloads. The liquid is contained in a fully filled axisymmetric container; the particular geometry examined in this paper is a right-circular cylinder with flat or rounded endcaps. Numerical predictions of viscous and pressure moments due to the liquid fill at low Reynolds number are reported. These moments tend to increase the precession angle and reduce the spin rate of the container. Liquid-induced roll and side (yaw) moments are computed as functions of endcap height to cylinder radius (0 < e/a < 1), cylinder half-height to radius (0.5 < c/ a < 4), Reynolds number (20 < Re < 3 X 102), ratio of precession to spin rate (0.05 < r < 0.95), and precession angle (ac = 2 deg). For a given cylinder, rounded endcaps can decrease the resonant liquid-induced moment by about 20% and shift the resonance to a smaller Reynolds number (for fixed precessional frequency). Thus, rounded endcaps can produce flight stability for some projectiles with highly viscous liquid payloads.