Inertial and viscous forces on a rigid sphere in straining flows at moderate Reynolds numbers

The focus of this paper is the effect of spatial non-uniformity in the ambient flow on the forces acting on a rigid sphere when the sphere Reynolds number, Re, is in the range 10 to 300. Direct numerical simulations (DNS) based on a pseudospectral methodology are carried out to solve for the unsteady three-dimensional flow field around a sphere which is either held stationary or allowed to translate freely under the hydrodynamic forces. The various components of the total force, namely the inertial, steady viscous, and history forces, are systematically estimated in the context of linearly varying straining flows. The inertial forces are isolated by computing the rapid changes in the drag and lift forces in response to a rapid acceleration of the ambient flow. It is shown that the inertial forces arising due to convective acceleration at moderate Reynolds numbers follow the inviscid flow result. While the effect of temporal acceleration depends only on the sign and magnitude of the acceleration, the effect of convective acceleration is shown to depend also on the initial state of the ambient flow. A simple theoretical argument is presented to support the numerical observations. It is also shown that the effect of convective acceleration on the steady viscous force can be realized on a slower time scale. The results show that the history kernels currently available in the literature are not adequate to represent the effect of non-uniformity on the history force. We isolate the steady viscous force by considering the simulation results for a stationary sphere subjected to steady straining flows. It is shown that the steady viscous forces under such non-uniform ambient conditions cannot be adequately represented by Schiller-Neumann-type drag laws. A generalized representation for the steady viscous force on a sphere subjected to straining flows at moderate Re is presented. The strain-induced corrections to the steady viscous force, under some situations, are shown to be significant and of at least the same order as the inertial forces. In order to further estimate the importance of different forces, we consider direct numerical simulations of the unsteady free translation of a sphere in straining flows. The predictions based on the Schiller-Neumann drag significantly misrepresent the exact force obtained from DNS. The inclusion of the inertial forces improves the prediction when the sphere moves within the same plane of strain, and worsens when the sphere moves away from the plane of strain. The DNS results can be predicted well when the strain-induced corrections to the viscous drag are included. Analysis of the different components of the total force suggests that the Schiller-Neumann drag, the inertial forces due to convective acceleration, and the strain-induced viscous corrections are the dominant components. The contributions from the acceleration of the sphere and the history force are consistently small.