Convergence Characteristics and Computational Cost of Two Algebraic Kernels in Vortex Methods with a Tree-Code Algorithm
Name
Wee-2009-CONVERGENCE CHARACTE.pdf
Size
2.02 MB
Format
Adobe PDF
Checksum (MD5)
3e5761c9e3b654684430b1cc26057942
Author(s)
Marzouk, Youssef M.
Schlegel, Fabrice
Ghoniem, Ahmed F.
Date Issued
June 2009
Journal
SIAM Journal on Scientific Computing
Publisher
Society for Industrial and Applied Mathematics
Citation
Wee, D. et al. “Convergence Characteristics and Computational Cost of Two Algebraic Kernels in Vortex Methods with a Tree-Code Algorithm.” SIAM Journal on Scientific Computing 31.4 (2009): 2510-2527. ©2009 Society for Industrial and Applied Mathematics
Version
Final published version
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. Comput. 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.
MIT Department
Massachusetts Institute of Technology. Department of Aeronautics and Astronautics
Massachusetts Institute of Technology. Department of Mechanical Engineering
Terms of Use
Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
Persistent DSpace Link
DOI of Published Version
http://dx.doi.org/10.1137/080726872
