Simulation of particles in fluid: a two-dimensional benchmark for a cylinder settling in a wall-bounded box

This paper presents numerical experiments inspired by the theoretical work of Faxén for predicting the terminal velocity of a cylinder, settling halfway between two parallel walls at low Reynolds numbers. It is demonstrated that unexpected correlations exist between Faxén's results and the relaxation of a rigid disk initially suspended in a wall-bounded square box. To this end, the 1-Fluid (1F) method is used within a frame of Direct Numerical Simulation (DNS). In first place, the assessment of 1F method in two dimensions is presented. Simulations are in good agreement with Faxén's approach in half-bounded domains, and with simulation data from literature as well. Numerical experiments are then designed in order to investigate the transient behavior of a circular disk in a wall-bounded square box. Significant ranges of particle-to-wall containment ratios, density ratios and Galileo numbers were used in simulations. In the case where the aspect ratio belongs to the range [0.005,0.4] and the Galileo number is smaller than 1, it is found that the wall correction factor based on the maximum settling velocity could be correlated directly with the Faxén's correction factor based on the terminal settling velocity. For extreme values of containment, Faxén's theory gives irrelevant predictions, and alternative approaches based on 1F simulations are suggested. Finally, an original benchmark is designed as an efficient and inexpensive tool for validating numerical approaches to fluid/particle systems.