Transient State Analysis of Separated Flow Around a Sphere

Transient state solutions of the Navier-Stokes equations were obtained for incompressible flow around a sphere accelerating from zero initial velocity to its terminal free falling velocity. By assuming rotational symmetry about the axis in the direction of motion, the Navier-Stokes equations and the continuity equation were simplified in terms of vorticity and stream function. The instantaneous acceleration of the falling sphere was calculated by considering the difference between the gravitational force and the drag force in a transient state. A set of implicit finite difference equations was developed. In order to obtain accurate information around the body, an exponential transformation along the radial direction was used to provide finer meshes in the vicinity of the surface of the sphere. The vorticity equation was solved by an alternating direction implicit (ADI) method while the stream function equation was solved by a successive over-relaxation (SOR) method. Simultaneous solutions were obtained. Transient state solutions were compared with steady state solutions for Reynolds numbers up to 300. Separations first occurred at a Reynolds number 20 for steady state flows and at Reynolds numbers 22·46 and 28·24 for transient state flows with terminal Reynolds numbers of 100 and 300, respectively. Separation angles, sizes of separation regions, and drag coefficents were calculated for both steady and unsteady states. Good agreement was obtained with existing experimental data in the steady state