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Generalized Algebraic Kernels and Multipole Expansions for massively parallel Vortex Methods

Parallel vortex particle methods are an efficient technique for large-scale simulations of turbulent fluid flows. One of their big advantages is the intrinsic adaptivity of vortex particles, since computational elements exist only where the vorticity field is non-zero. To overcome O(N2)-complexity of the corresponding N-body problem, multipole-based fast summation methods can be used here as well, which reduce the computational costs to at least O(NlogN).

Evolution of a spherical vortex sheetEvolution of a spherical vortex sheet. The initial conditions are the solution to the problem of flow past a sphere with unit free-stream velocity along the z-axis. While moving downwards in z-direction, the sphere collapses from the top and wraps into its own interior, forming a large vortex ring in the inside.

Since vortex particle methods are dominated by high order non-Coulombian interaction kernels, the parallel Barnes-Hut tree code PEPC and its underlying theory have been extended to handle generalized algebraic smoothing kernels of arbitrary order. However, a stable implementation using particles for discretizing the vorticity field must provide a scheme for treating the overlap condition, which is required for convergent regularized vortex particle methods. Therefore, the code also includes an implementation of the concept of remeshing to account for long-term accuracy.

Development of side-by-side collision of two viscous vortex ringsDevelopment of side-by-side collision of two viscous vortex rings. Starting from the (x,z)-plane, the rings travel in the positive y-direction and move towards the x = 0 plane, being attracted by mutual induction. Induced by the collision of the inner cores, vortex bridges are formed and the inner core strengths decrease while the circulation inside the bridges increases rapidly.