Fluid Motion in a Spinning, Coning Cylinder via Spatial Eigenfunction Expansion

摘要

The first attempts to explain the motion of a liquid-filled projectile were confined to the limit Reynolds Number = Re approaches infinity and linear theory. Recently, the need became apparent for the limit Re approaches 0 for which the spatial eigenvalue method was developed; it is not restricted in Re, however. The eigenvalue problem is defined by ordinary differential equations in the radial direction. The eigenvalues are determined by an iterative process for which sufficiently accurate initial estimates are required. The flow variables are expanded in a eigenfunction series with coefficients determined by satisfying the boundary conditions; a least squares method and collocation method are used for this purpose. The pressure and shear stress so determined give the pressure coefficient and overturning moment. The accuracy of the calculation is discussed. Results are given over a range of Re, aspect ratio, and nutational frequency. The CPU time required on the VAX 8600 varies from 10 seconds at Re = 10 to 30 minutes at Re = 1,000. Results are compared with experimental measurements. Comparisons are also made with results from the large scale finite difference program of Strikwerda.