{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,2]],"date-time":"2025-11-02T06:25:05Z","timestamp":1762064705346,"version":"build-2065373602"},"reference-count":62,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2022,8,27]],"date-time":"2022-08-27T00:00:00Z","timestamp":1661558400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Numerical Wind Tunnel Project","doi-asserted-by":"publisher","award":["11902266","11902264","12072283","B17037"],"award-info":[{"award-number":["11902266","11902264","12072283","B17037"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"111 Project of China","award":["11902266","11902264","12072283","B17037"],"award-info":[{"award-number":["11902266","11902264","12072283","B17037"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The multiphase model based on free-energy theory has been experiencing long-term prosperity for its solid foundation and succinct implementation. To identify the main hindrance to developing a free-energy-based discrete unified gas-kinetic scheme (DUGKS), we introduced the classical lattice Boltzmann free-energy model into the DUGKS implemented with different flux reconstruction schemes. It is found that the force imbalance amplified by the reconstruction errors prevents the direct application of the free-energy model to the DUGKS. By coupling the well-balanced free-energy model with the DUGKS, the influences of the amplified force imbalance are entirely removed. Comparative results demonstrated a consistent performance of the well-balanced DUGKS despite the reconstruction schemes utilized. The capability of the DUGKS coupled with the well-balanced free-energy model was quantitatively validated by the coexisting density curves and Laplace\u2019s law. In the quiescent droplet test, the magnitude of spurious currents is reduced to a machine accuracy of 10\u221215. Aside from the excellent performance of the well-balanced DUGKS in predicting steady-state multiphase flows, the spinodal decomposition test and the droplet coalescence test revealed its stability problems in dealing with transient flows. Further improvements are required on this point.<\/jats:p>","DOI":"10.3390\/e24091202","type":"journal-article","created":{"date-parts":[[2022,8,29]],"date-time":"2022-08-29T01:39:19Z","timestamp":1661737159000},"page":"1202","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Free-Energy-Based Discrete Unified Gas Kinetic Scheme for van der Waals Fluid"],"prefix":"10.3390","volume":"24","author":[{"given":"Zeren","family":"Yang","sequence":"first","affiliation":[{"name":"School of Aeronautics, Northwestern Polytechnical University, Xi\u2019an 710072, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sha","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Aeronautics, Northwestern Polytechnical University, Xi\u2019an 710072, China"},{"name":"National Key Laboratory of Science and Technology on Aerodynamic Design and Research, Northwestern Polytechnical University, Xi\u2019an 710072, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Congshan","family":"Zhuo","sequence":"additional","affiliation":[{"name":"School of Aeronautics, Northwestern Polytechnical University, Xi\u2019an 710072, China"},{"name":"National Key Laboratory of Science and Technology on Aerodynamic Design and Research, Northwestern Polytechnical University, Xi\u2019an 710072, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7664-437X","authenticated-orcid":false,"given":"Chengwen","family":"Zhong","sequence":"additional","affiliation":[{"name":"School of Aeronautics, Northwestern Polytechnical University, Xi\u2019an 710072, China"},{"name":"National Key Laboratory of Science and Technology on Aerodynamic Design and Research, Northwestern Polytechnical University, Xi\u2019an 710072, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,8,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1016\/0021-9991(81)90145-5","article-title":"Volume of fluid (VOF) method for the dynamics of free boundaries","volume":"39","author":"Hirt","year":"1981","journal-title":"J. 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