Drag force and torque of a stationary sphere in a uniform laminar open channel flow

An analytical solution for a stationary sphere immersed in a steady and uniform laminar open channel flow with the surrounding fluid slowly passing by is presented. The solution satisfies the open channel boundary conditions by using a non-slip bottom wall and a free-slip top, and the non-slip condition on the sphere surface is achieved by a converged collocation method. The hydrodynamic drags and torques of the sphere located at different positions in the flow field are calculated. The results show that compared to Stokes' drag law, the drag force tends to be larger when the sphere is close to the bottom wall and smaller when closer to the slip top. The torque gradually changes from positive to negative as the distance from the bottom increases in an open channel flow. When the top boundary is set far enough from the sphere, the data results are consistent with those of a stationary sphere in a linear shear flow near a single wall. Approximate expressions for the drag force and torque with respect to the correlated length scales are fitted using the calculated results of the present solution, and the expressions can be simplified to single variable functions of the ratio between the sphere-bottom distance and the sphere radius h/alpha with sufficient accuracy when the water depth is considered to be infinite. Published by AIP Publishing.