Differential formulation of the viscous history force on a particle for efficient and accurate computation

It is well known that the computation of the Basset-like history force is very demanding in terms of CPU and memory requirements, since it requires the evaluation of a history integral. We use the recent rational theory of Beylkin & Monzon (Appl. Comput. Harmon. Anal., vol. 19, 2005, pp. 17-48) to approximate the history kernel in the form of exponential sums to reformulate the viscous history force in a differential form. This theory allows us to approximate the history kernel in terms of exponential sums to any desired order of accuracy. This removes the need for long-time storage of the acceleration histories of the particle and the fluid. The proposed differential form approximation is applied to compute the history force on a spherical particle in a synthetic turbulent flow and a wall-bounded turbulent channel flow. Particles of various diameters are considered, and results obtained using the present technique are in reasonable agreement with those achieved using the full history integral.