Combining the vortex-in-cell and parallel fast multipole methods for efficient domain decomposition simulations

Abstract

A new combination of vortex-in-cell and parallel fast multipole methods is presented which allows to efficiently simulate, in parallel, unbounded and half-unbounded vortical flows (flows with one flat wall). In the classical vortex-in-cell (VIC) method, the grid used to solve the Poisson equation is typically taken much larger than the vorticity field region, so as to be able to impose suitable far-field boundary conditions and thus approximate the truly unbounded (or half-unbounded) flow; an alternative is to assume periodicity. This approach leads to a solution that depends on the global grid size and, for large problems, to unmanageable memory and CPU requirements. The idea exploited here is to work on a domain that contains tightly the vorticity field and that can be decomposed in several subdomains on which the exact boundary conditions are obtained using the parallel fast multipole (PFM) method. This amounts to solving a 3-D Poisson equation without requiring any iteration between the subdomains (e.g., no Schwarz iteration required): this is so because the PFM method has a global view of the entire vorticity field and satisfies the far-field condition. The solution obtained by this VIC–PFM combination then corresponds to the simulation of a truly unbounded (or half-unbounded) flow. It requires far less memory and leads to a far better computational efficiency compared to simulations done using either (1) the VIC method alone, or (2) the vortex particle method with PFM solver alone. 3-D unbounded flow validation results are presented: instability, non-linear evolution and decay of a vortex ring (first at a moderate Reynolds number using the sequential version of the method, then at a high Reynolds number using the parallel version); instability and non-linear evolution of a two vortex system in ground effect. Finally, a space-developing simulation of an aircraft vortex wake in ground effect is also presented.