## Abstract

We study the convergence characteristics of two algebraic kernels used in vortex calculations: the Rosenhead-Moore kernel, which is a low-order kernel, and the Winckelmans-Leonard kernel, which is a high-order kernel. To facilitate the study, a method of evaluating particle-cluster interactions is introduced for the Winckelmans-Leonard kernel. The method is based on Taylor series expansion in Cartesian coordinates, as initially proposed by Lindsay and Krasny [J. Com-put. Phys., 172 (2001), pp. 879-907] for the Rosenhead-Moore kernel. A recurrence relation for the Taylor coefficients of the Winckelmans-Leonard kernel is derived by separating the kernel into two parts, and an error estimate is obtained to ensure adaptive error control. The recurrence relation is incorporated into a tree-code to evaluate vorticity-induced velocity. Next, comparison of convergence is made while utilizing the tree-code. Both algebraic kernels lead to convergence, but the Winckelmans-Leonard kernel exhibits a superior convergence rate. The combined desingularization and discretization error from the Winckelmans-Leonard kernel is an order of magnitude smaller than that from the Rosenhead-Moore kernel at a typical resolution. Simulations of vortex rings are performed using the two algebraic kernels in order to compare their performance in a practical setting. In particular, numerical simulations of the side-by-side collision of two identical vortex rings suggest that the three-dimensional evolution of vorticity at finite resolution can be greatly affected by the choice of the kernel. We find that the Winckelmans-Leonard kernel is able to perform the same task with a much smaller number of vortex elements than the Rosenhead-Moore kernel, greatly reducing the overall computational cost.

Original language | English |
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Pages (from-to) | 2510-2527 |

Number of pages | 18 |

Journal | SIAM Journal on Scientific Computing |

Volume | 31 |

Issue number | 4 |

DOIs | |

State | Published - 2009 |

## Keywords

- Computational particle methods
- Convergence
- Hierarchical methods
- Key words. numerical simulation
- TV-body problems
- Tree-code
- Vortex methods