Simulation of vortex ring interaction

A numerical solution of the Navier-Stokes equations for incompressible flows is used to simulate the interaction of isolated vortex rings in unbounded domains. The algorithm is based on a pressure correction scheme, in which the Poisson equation for the pressure is solved with a conjugate gradient method with a preconditioning based on an incomplete lower-upper decomposition. The integration in time is carried out with an explicit Adams-Bashforth scheme on a non-staggered grid. The algorithm is efficiently implemented on vector-parallel computers. Hill's solution of the Euler equations for vortex rings is used as an initial condition of the velocity field. The time development of two vortex rings approaching each other under an angle of 40-90 degrees is simulated on a grid, moving with the propagation velocity of the vortex rings. The generation and connection of vortex rings in a bounded domain is investigated with a solution of the Navier-Stokes equations for compressible hows in a cylinder of a piston engine. In this case an explicit time-stepping scheme with centrally discretized convective and viscous terms is applied on general curvilinear coordinates in block-structured moving grids. Two vortex rings are generated by the flow through the open intake valves, which connect to a single vortex at a crank angle of about 150 degrees. Vortex lines are integrated to visualize the flow field of the time dependent solution.