Convergence Characteristics and Computational Cost of Two Algebraic Kernels in Vortex Methods with a Tree-Code Algorithm
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 i...
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| Format: | Article |
| Language: | en_US |
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Society for Industrial and Applied Mathematics
2010
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| Online Access: | http://hdl.handle.net/1721.1/52343 https://orcid.org/0000-0001-8242-3290 https://orcid.org/0000-0001-8730-272X |
| _version_ | 1826203687586889728 |
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| author | Marzouk, Youssef M. Schlegel, Fabrice Ghoniem, Ahmed F. |
| author2 | Massachusetts Institute of Technology. Department of Aeronautics and Astronautics |
| author_facet | Massachusetts Institute of Technology. Department of Aeronautics and Astronautics Marzouk, Youssef M. Schlegel, Fabrice Ghoniem, Ahmed F. |
| author_sort | Marzouk, Youssef M. |
| collection | MIT |
| description | 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. |
| first_indexed | 2024-09-23T12:41:27Z |
| format | Article |
| id | mit-1721.1/52343 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T12:41:27Z |
| publishDate | 2010 |
| publisher | Society for Industrial and Applied Mathematics |
| record_format | dspace |
| spelling | mit-1721.1/523432022-09-28T09:26:04Z Convergence Characteristics and Computational Cost of Two Algebraic Kernels in Vortex Methods with a Tree-Code Algorithm Marzouk, Youssef M. Schlegel, Fabrice Ghoniem, Ahmed F. Massachusetts Institute of Technology. Department of Aeronautics and Astronautics Massachusetts Institute of Technology. Department of Mechanical Engineering Marzouk, Youssef M. Marzouk, Youssef M. Schlegel, Fabrice Ghoniem, Ahmed F. 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. U.S. Department of Energy, Mathematical, Information, and Computational Sciences (MICS) program (DE-FG02-98ER25355) 2010-03-05T16:42:45Z 2010-03-05T16:42:45Z 2009-06 2009-03 Article http://purl.org/eprint/type/JournalArticle 1064-8275 http://hdl.handle.net/1721.1/52343 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 https://orcid.org/0000-0001-8242-3290 https://orcid.org/0000-0001-8730-272X en_US http://dx.doi.org/10.1137/080726872 SIAM Journal on Scientific Computing 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. application/pdf Society for Industrial and Applied Mathematics SIAM |
| spellingShingle | Marzouk, Youssef M. Schlegel, Fabrice Ghoniem, Ahmed F. Convergence Characteristics and Computational Cost of Two Algebraic Kernels in Vortex Methods with a Tree-Code Algorithm |
| title | Convergence Characteristics and Computational Cost of Two Algebraic Kernels in Vortex Methods with a Tree-Code Algorithm |
| title_full | Convergence Characteristics and Computational Cost of Two Algebraic Kernels in Vortex Methods with a Tree-Code Algorithm |
| title_fullStr | Convergence Characteristics and Computational Cost of Two Algebraic Kernels in Vortex Methods with a Tree-Code Algorithm |
| title_full_unstemmed | Convergence Characteristics and Computational Cost of Two Algebraic Kernels in Vortex Methods with a Tree-Code Algorithm |
| title_short | Convergence Characteristics and Computational Cost of Two Algebraic Kernels in Vortex Methods with a Tree-Code Algorithm |
| title_sort | convergence characteristics and computational cost of two algebraic kernels in vortex methods with a tree code algorithm |
| url | http://hdl.handle.net/1721.1/52343 https://orcid.org/0000-0001-8242-3290 https://orcid.org/0000-0001-8730-272X |
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