History force on a sphere due to a step change in the free-stream velocity

Abstract Finite-difference solutions for unsteady flows over a stationary sphere due to a step change in the free-stream velocity from U 1 to U 2 (0 U 1 UZ ) are obtained, from which the unsteady drag is evaluated, for Reynolds numbers, Re (based on the diameter of the sphere and the free-stream velocity U 2 ), ranging from 0.1 to 100 over a large range of time. The history force on the sphere is determined by subtracting the quasi-steady drag from the computed total drag. The numerical result shows a complicated behavior of the history force at finite Re for both U 1 = 0 and U 1 > 0. It decays as t − 1 2 for small time; it then decays as t − n ( n ⩾ 2 with n = 2 for small Re) for an intermediate range of time; and it decays exponentially at large time. The numerical results are used to assess a recently developed expression for the history force for finite Re. Good overall agreement is observed for the history force between the analytical prediction and the finite-difference solution for small and intermediate time for the Re values tested.