A minimization principle for the description of modes associated with finite-time instabilities
We introduce a minimization formulation for the determination of a finite-dimensional, time-dependent, orthonormal basis that captures directions of the phase space associated with transient instabilities. While these instabilities have finite lifetime, they can play a crucial role either by alterin...
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| Format: | Article |
| Language: | en_US |
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Royal Society, The
2017
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| Online Access: | http://hdl.handle.net/1721.1/109178 https://orcid.org/0000-0002-6318-2265 https://orcid.org/0000-0003-0302-0691 |
| _version_ | 1811074834928500736 |
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| author | Babaee, Hessameddin Sapsis, Themistoklis P. |
| author2 | Massachusetts Institute of Technology. Department of Mechanical Engineering |
| author_facet | Massachusetts Institute of Technology. Department of Mechanical Engineering Babaee, Hessameddin Sapsis, Themistoklis P. |
| author_sort | Babaee, Hessameddin |
| collection | MIT |
| description | We introduce a minimization formulation for the determination of a finite-dimensional, time-dependent, orthonormal basis that captures directions of the phase space associated with transient instabilities. While these instabilities have finite lifetime, they can play a crucial role either by altering the system dynamics through the activation of other instabilities or by creating sudden nonlinear energy transfers that lead to extreme responses. However, their essentially transient character makes their description a particularly challenging task. We develop a minimization framework that focuses on the optimal approximation of the system dynamics in the neighbourhood of the system state. This minimization formulation results in differential equations that evolve a time-dependent basis so that it optimally approximates the most unstable directions. We demonstrate the capability of the method for two families of problems: (i) linear systems, including the advection–diffusion operator in a strongly non-normal regime as well as the Orr–Sommerfeld/Squire operator, and (ii) nonlinear problems, including a low-dimensional system with transient instabilities and the vertical jet in cross-flow. We demonstrate that the time-dependent subspace captures the strongly transient non-normal energy growth (in the short-time regime), while for longer times the modes capture the expected asymptotic behaviour. |
| first_indexed | 2024-09-23T09:56:11Z |
| format | Article |
| id | mit-1721.1/109178 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T09:56:11Z |
| publishDate | 2017 |
| publisher | Royal Society, The |
| record_format | dspace |
| spelling | mit-1721.1/1091782022-09-30T17:51:22Z A minimization principle for the description of modes associated with finite-time instabilities Babaee, Hessameddin Sapsis, Themistoklis P. Massachusetts Institute of Technology. Department of Mechanical Engineering Babaee, Hessameddin Sapsis, Themistoklis P. We introduce a minimization formulation for the determination of a finite-dimensional, time-dependent, orthonormal basis that captures directions of the phase space associated with transient instabilities. While these instabilities have finite lifetime, they can play a crucial role either by altering the system dynamics through the activation of other instabilities or by creating sudden nonlinear energy transfers that lead to extreme responses. However, their essentially transient character makes their description a particularly challenging task. We develop a minimization framework that focuses on the optimal approximation of the system dynamics in the neighbourhood of the system state. This minimization formulation results in differential equations that evolve a time-dependent basis so that it optimally approximates the most unstable directions. We demonstrate the capability of the method for two families of problems: (i) linear systems, including the advection–diffusion operator in a strongly non-normal regime as well as the Orr–Sommerfeld/Squire operator, and (ii) nonlinear problems, including a low-dimensional system with transient instabilities and the vertical jet in cross-flow. We demonstrate that the time-dependent subspace captures the strongly transient non-normal energy growth (in the short-time regime), while for longer times the modes capture the expected asymptotic behaviour. United States. Army Research Office (66710-EG-YIP) United States. Office of Naval Research (ONR N00014-14-1-0520) United States. Defense Advanced Research Projects Agency (HR0011-14-1-0060) 2017-05-18T19:12:17Z 2017-05-18T19:12:17Z 2016-02 2015-11 Article http://purl.org/eprint/type/JournalArticle 1364-5021 1471-2946 http://hdl.handle.net/1721.1/109178 Babaee, H. and Sapsis, T. P. “A Minimization Principle for the Description of Modes Associated with Finite-Time Instabilities.” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science 472, no. 2186 (February 2016): 20150779. https://orcid.org/0000-0002-6318-2265 https://orcid.org/0000-0003-0302-0691 en_US http://dx.doi.org/10.1098/rspa.2015.0779 Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf Royal Society, The arXiv |
| spellingShingle | Babaee, Hessameddin Sapsis, Themistoklis P. A minimization principle for the description of modes associated with finite-time instabilities |
| title | A minimization principle for the description of modes associated with finite-time instabilities |
| title_full | A minimization principle for the description of modes associated with finite-time instabilities |
| title_fullStr | A minimization principle for the description of modes associated with finite-time instabilities |
| title_full_unstemmed | A minimization principle for the description of modes associated with finite-time instabilities |
| title_short | A minimization principle for the description of modes associated with finite-time instabilities |
| title_sort | minimization principle for the description of modes associated with finite time instabilities |
| url | http://hdl.handle.net/1721.1/109178 https://orcid.org/0000-0002-6318-2265 https://orcid.org/0000-0003-0302-0691 |
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