Abstract
A numerical method is given for determining the transient flow past a sphere which is impulsively started from rest with constant velocity in a viscous fluid. One of the features of the method is that the calculation of the flow at early times is performed using boundary-layer variables, which leads to very accurate solutions. The problem is formulated in terms of the stream function and the vorticity. The method used is the semi-analytical one of series truncation in which the stream function and vorticity are expanded in a series of Legendre functions with argument z = cosθ, where θ is the polar angle. The governing equations are thus reduced to sets of time-dependent differential equations in the radial variable. In theory the number of these equations is infinite but in practice only a finite number can be solved to give an approximation to the flow. They are solved numerically by the Crank-Nicolson procedure. Numerical solutions are given for the cases R = 20, 40, 100, 200, 500, 1000, and ∞, where R is the Reynolds number based on the diameter of the sphere. For R = 20, 40, and 100 the equations are integrated to times at which the solutions may be considered close to steady motion. The developments of various physical properties of the flow are calculated and compared, where possible, with the results of other calculations.
Published Version
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