Numerical simulation of the fluid dynamics of 2D rigid body motion with the vortex particle method

A viscous vortex particle method is presented for computing the fluid dynamics of two-dimensional rigid bodies in motion. The Navier–Stokes equations are solved using a fractional step procedure. Smooth particles carry vorticity and exchange strength to account for convection and viscous diffusion. The spurious slip resulting from this half-step is identified with a surface vortex sheet, and the slip is eliminated by diffusing the sheet to adjacent particles. Particles are remeshed every few time steps to a Cartesian grid with a ‘body-ignorant’ interpolation using simple symmetric stencils. Kelvin’s circulation theorem remains enforced by accounting for the circulation leaked into the body during this procedure, and redistributing it to the particles in the subsequent sheet diffusion. The stability and convergence with respect to numerical parameters are explored in detail, with particular focus on the residual slip velocity. The method is applied to two problems that demonstrate its utility for investigating biological locomotion: a flapping elliptical wing with hovering insect kinematics, with good agreement of forces with previous simulations and experiments; and a three-linkage ‘fish’ undergoing undulatory mechanics.