{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T10:01:23Z","timestamp":1777456883568,"version":"3.51.4"},"reference-count":31,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2024,2,1]],"date-time":"2024-02-01T00:00:00Z","timestamp":1706745600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12372214"],"award-info":[{"award-number":["12372214"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12202452"],"award-info":[{"award-number":["12202452"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["P2022-C-III-001-001"],"award-info":[{"award-number":["P2022-C-III-001-001"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Science Center for Gas Turbine Project","award":["12372214"],"award-info":[{"award-number":["12372214"]}]},{"name":"Science Center for Gas Turbine Project","award":["12202452"],"award-info":[{"award-number":["12202452"]}]},{"name":"Science Center for Gas Turbine Project","award":["P2022-C-III-001-001"],"award-info":[{"award-number":["P2022-C-III-001-001"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In isotropic turbulence, symmetry of different directions can reduce the number of independent components for velocity gradient tensors. In three-dimensional isotropic turbulence, the independent components under either incompressible or compressible conditions have already been analyzed in the literature. However, for two-dimensional isotropic turbulence, they are still unclear. We derive rigorously the independent components for velocity gradient tensors of two-dimensional isotropic turbulence and give physical explanations. These theoretical results are validated using high-resolution direct numerical simulations (DNSs) of two-dimensional compressible turbulence. Results show that the present DNS setup is still not sufficient to capture the isotropy of third-order moments, suggesting that more investigations on determining the smallest scale and improving the numerical schemes for two-dimensional compressible turbulence are required.<\/jats:p>","DOI":"10.3390\/sym16020175","type":"journal-article","created":{"date-parts":[[2024,2,1]],"date-time":"2024-02-01T08:44:08Z","timestamp":1706777048000},"page":"175","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Low-Order Moments of Velocity Gradient Tensors in Two-Dimensional Isotropic Turbulence"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0009-0004-9990-3177","authenticated-orcid":false,"given":"Chensheng","family":"Luo","sequence":"first","affiliation":[{"name":"Laboratory of Complex System, Ecole Centrale de P\u00e9kin\/School of General Engineering, Beihang University, Beijing 100191, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1949-500X","authenticated-orcid":false,"given":"Ping-Fan","family":"Yang","sequence":"additional","affiliation":[{"name":"The State Key Laboratory of Nonlinear Mechanics, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, China"},{"name":"School of Engineering Sciences, University of Chinese Academy of Sciences, Beijing 100049, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5756-7102","authenticated-orcid":false,"given":"Le","family":"Fang","sequence":"additional","affiliation":[{"name":"Laboratory of Complex System, Ecole Centrale de P\u00e9kin\/School of General Engineering, Beihang University, Beijing 100191, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,2,1]]},"reference":[{"key":"ref_1","first-page":"16","article-title":"Dissipation of energy in locally isotropic turbulence","volume":"32","author":"Kolmogorov","year":"1941","journal-title":"Dokl. 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