Vortex methods are a powerful tool for the simulation of incompressible flows at high Reynolds number. They rely on a discrete Lagrangian representation of the vorticity field to approximately satisfy the Kelvin and Helmholtz theorems which govern the dynamics of vorticity for inviscid flows. A time-splitting technique can be used to include viscous effects. The diffusion equation is considered separately after convecting the particles with an inviscid vortex method. In our work, the viscous effects are represented by the so-called deterministic method. The approach was extended to problems where a flux of vorticity is used to enforce the no-slip boundary condition.
In order to accurately compute the viscous transport of vorticity, gradients need to be well resolved. As the Reynolds number is increased, these gradients get steeper and more particles are required to achieve the requisite resolution. In practice, the computing cost associated with the convection step dictates the number of vortex particles and puts an upper bound on the Reynolds number that can be simulated with confidence. That threshold can be increased by reducing the asymptotic time complexity of the convection step from to . The nearfield of every vortex particle is identified. Within that region, the velocity is computed by considering the pairwise interaction of vortices. The speedup is achieved by approximating the influence of the rest of the domain, the farfield. In that context, the interaction of two vortex particles is treated differently depending on their spatial relation. The resulting computer code does not lend itself to vectorization but has been successfully implemented on concurrent computers.