A full numerical simulation based on spectral methods is used to investigate linearly accelerating and decelerating flows past a rigid sphere. Although flow separation does not occur at Reynolds numbers below 20 for a steady flow, in the linearly decelerating flow separation is observed at much lower Reynolds numbers with complete detachment of vorticity possible in certain cases. The existence of a large recirculation region contributes to the result that a negative viscous force on the sphere is possible. The contribution of the pressure to the force includes a component that is well described by the inviscid added-mass term in both the accelerating and decelerating cases. The force on the sphere is found in general to initially decay in a power law manner after acceleration or deceleration ends followed by rapid convergence at later times to the steady state. For the cases examined this convergence is found to be exponential except for those in which the sphere is brought to rest in which case the convergence remains algebraic. This includes the special case of an infinite acceleration or deceleration where the free stream velocity is impulsively changed.

A direct numerical simulation, based on spectral methods, has been used to compute the time-dependent, axisymmetric viscous flow past a rigid sphere. An investigation has been made for oscillatory flow about a zero mean for different Reynolds numbers and frequencies. The simulation has been verified for steady flow conditions, and for unsteady flow there is excellent agreement with Stokes flow theory at very low Reynolds numbers. At moderate Reynolds numbers, around 20, there is good general agreement with available experimental data for oscillatory motion. Under steady flow conditions no separation occurs at Reynolds number below 20; however in an oscillatory flow a separation bubble forms on the decelerating portion of each cycle at Reynolds numbers well below this. As the flow accelerates again the bubble detaches and decays, while the formation of a new bubble is inhibited till the flow again decelerates. Steady streaming, observed for high frequencies, is also observed at low frequencies due to the flow separation. The contribution of the pressure to the resultant force on the sphere includes a component that is well described by the usual added-mass term even when there is separation. In a companion paper the flow characteristics for constant acceleration or deceleration are reported.