Efficient numerical method for the direct numerical simulation of the flow past a single light moving spherical body in transitional regimes

An efficient numerical method for solving the coupled system of Navier-Stokes equations and equations of motion of a single spherical particle is presented. It combines a spectral-spectral-element spatial discretization in a cylindrical domain moving translationally with the free particle and particle equations of motion involving hydrodynamic forces integrated on the particle surface. The time discretization is semi-implicit with a third-order accurate explicit treatment of the advective terms and a fully implicit treatment of the remaining linear problem consisting in Stokes-like equations coupled with the particle equations of motion. It is shown that the fully implicit approach is the only way to account for very light spheres. Moreover, no reduction of the time step is necessary. The particle equations of motion are re-formulated as a simple system of six linear equations for six unknowns using the fact that the six components combining the hydrodynamic forces and torques depend linearly on particle translation and angular velocities. They are solved directly and are thus exactly satisfied at each time step. Numerical tests show that the increase of computing costs needed to account for the free sphere degrees of freedom remains within about 20% per time step. The accuracy and resolution independence of the solution are tested at the primary instability threshold and for a strongly supercritical zigzagging trajectory. A partial validation using available experimental results is also presented. Very satisfactory accuracy is shown to be obtained with only a very limited number of azimuthal modes.