{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,13]],"date-time":"2026-06-13T18:59:17Z","timestamp":1781377157311,"version":"3.54.1"},"reference-count":49,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2025,9,26]],"date-time":"2025-09-26T00:00:00Z","timestamp":1758844800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100012190","name":"Ministry of Science and Higher Education of the Russian Federation","doi-asserted-by":"crossref","award":["FSWE-2024-0001"],"award-info":[{"award-number":["FSWE-2024-0001"]}],"id":[{"id":"10.13039\/501100012190","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["www.mdpi.com"],"crossmark-restriction":true},"short-container-title":["Algorithms"],"abstract":"<jats:p>This article presents the results of a study on the possibility of using a single-speed multiphase model with free surface allowance for simulating boiling and condensation processes. The simulation is based on the VOF method, which allows the position of the interphase boundary to be tracked. To increase the stability of the iterative procedure for numerically solving volume fraction transfer equations using a finite volume discretization method on arbitrary unstructured grids, the basic VOF method is been modified by writing these equations in a semi-divergent form. The models of Tanasawa, Lee, and Rohsenow are considered models of interphase mass transfer, in which the evaporated or condensed mass linearly depends on the difference between the local temperature and the saturation temperature with accuracy in empirical parameters. This paper calibrates these empirical parameters for each mass transfer model. The results of our study of the influence of the values of the empirical parameters of models on the intensity of boiling and evaporation, as well as on the dynamics of the interphase boundary, are presented. This research is based on Stefan\u2019s problem of the movement of the interphase boundary due to the evaporation of a liquid and the problem of condensation of vapor bubbles water columns. As a result of a series of numerical experiments, it is shown that the average error in the position of the interfacial boundary for the Tanasawa and Lee models does not exceed 3\u20136%. For the Rohsenow model, the result is somewhat worse, since the interfacial boundary moves faster than it should move according to calculations based on analytical formulas. To investigate the possibility of condensation modeling, the results of a numerical solution of the problem of an emerging condensing vapor bubble are considered. A numerical assessment of its position in space and the shape and dynamics of changes in its diameter over time is carried out using the VOF method, taking into account the free surface. It is shown herein that the Tanasawa model has the highest accuracy for modeling the condensation process using a VOF method taking into account the free surface, while the Rohsenow model is most unstable and prone to deformation of the bubble shape. At the same time, the dynamics of bubble ascent are modeled by all three models. The results obtained confirm the fundamental possibility of using a VOF method to simulate the processes of boiling and condensation and taking into account the dynamics of the free surface. At the same time, the problem of the studied models of phase transitions is revealed, which consists of the need for individual selection of optimal values of empirical parameters for each specific task.<\/jats:p>","DOI":"10.3390\/a18100604","type":"journal-article","created":{"date-parts":[[2025,9,26]],"date-time":"2025-09-26T12:33:03Z","timestamp":1758889983000},"page":"604","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Application Features of a VOF Method for Simulating Boiling and Condensation Processes"],"prefix":"10.3390","volume":"18","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3247-0835","authenticated-orcid":false,"given":"Andrey","family":"Kozelkov","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics, Nizhny Novgorod State Technical University n.a. R.E. Alekseev, 603155 Nizhny Novgorod, Russia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3828-6406","authenticated-orcid":false,"given":"Andrey","family":"Kurkin","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Nizhny Novgorod State Technical University n.a. R.E. Alekseev, 603155 Nizhny Novgorod, Russia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Andrey","family":"Puzan","sequence":"additional","affiliation":[{"name":"National Center for Physics and Mathematics, 607328 Sarov, Russia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Vadim","family":"Kurulin","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Nizhny Novgorod State Technical University n.a. R.E. Alekseev, 603155 Nizhny Novgorod, Russia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Natalya","family":"Tarasova","sequence":"additional","affiliation":[{"name":"Independent Researcher, 607188 Sarov, Russia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Vitaliy","family":"Gerasimov","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Nizhny Novgorod State Technical University n.a. R.E. Alekseev, 603155 Nizhny Novgorod, Russia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,9,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"106840","DOI":"10.1016\/j.compchemeng.2020.106840","article-title":"Numerical simulation of natural convection and boil-off in a small size pressurized LNG storage tank","volume":"138","year":"2020","journal-title":"Comput. Chem. Eng."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"3133","DOI":"10.1007\/s12206-012-0820-x","article-title":"Numerical study of natural convection in a liquefied natural gas tank","volume":"26","author":"Roh","year":"2012","journal-title":"J. Mech. Sci. Technol."},{"key":"ref_3","first-page":"106","article-title":"Evaluation of numerical diffusion of the finite volume method when modelling surface waves","volume":"24","author":"Tyatyushkina","year":"2019","journal-title":"Comput. Technol."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Kozelkov, A., Kurulin, V., Kurkin, A., Taranov, A., Plygunova, K., Krutyakova, O., and Korotkov, A. (2023). Numerical Approach Based on Solving 3D Navier\u2013Stokes Equations for Simulation of the Marine Propeller Flow Problems. Fluids, 8.","DOI":"10.3390\/fluids8030104"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Korotkov, A.V., Kozelkov, A.S., Kurkin, A.A., Giniyatullin, R.R., and Lashkin, S.V. (2023). Numerical Simulation of the Conjugate Heat Transfer of a \u00abFluid\u2013Solid Body\u00bb System on an Unmatched Grid Interface. Fluids, 8.","DOI":"10.3390\/fluids8100266"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Kurul, N., and Podowski, M. (1990, January 19\u201324). Multidimensional effects in forced convection subcooled boiling. Proceedings of the Ninth International Heat Transfer Conference, Jerusalem, Israel.","DOI":"10.1615\/IHTC9.40"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1016\/j.ijheatmasstransfer.2013.11.017","article-title":"A fully coupled numerical simulation of sessile droplet evaporation using Arbitrary Lagrangian-Eulerian formulation","volume":"70","author":"Yang","year":"2014","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"3015","DOI":"10.1016\/j.ijheatmasstransfer.2011.02.047","article-title":"Numerical investigation of heat and mass transfer from an evaporating meniscus in a heated open groove","volume":"54","author":"Wang","year":"2011","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1016\/0021-9991(81)90145-5","article-title":"Volume of fluid (VOF) method for dynamics of free boundaries","volume":"39","author":"Hirt","year":"1981","journal-title":"J. Comput. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1016\/0021-9991(88)90002-2","article-title":"Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations","volume":"79","author":"Osher","year":"1988","journal-title":"J. Comput. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"776","DOI":"10.1006\/jcph.2001.6810","article-title":"Coupling of the Interface Tracking and the Two-Fluid Models for the Simulation of Incompressible Two-Phase Flow","volume":"171","author":"Cerne","year":"2001","journal-title":"J. Comput. Phys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"106446","DOI":"10.1016\/j.ultsonch.2023.106446","article-title":"Investigation of cavitation noise using Eulerian-Lagrangian multiscale modeling","volume":"97","author":"Li","year":"2023","journal-title":"Ultrason. Sonochem."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Alkhezzi, A., and Fan, Y. (2023). Improved Two-Fluid Model for Segregated Flow and Integrated Multiphase Flow Modeling for Downhole Pressure Predictions. Energies, 16.","DOI":"10.3390\/en16247923"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"108699","DOI":"10.1016\/j.mineng.2024.108699","article-title":"Experimental and numerical investigation of turbulent multiphase jets","volume":"211","author":"Kamble","year":"2024","journal-title":"Miner. Eng."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1016\/j.ijmst.2022.09.016","article-title":"Particulate flow modelling in a spiral separator by using the Eulerian multi-fluid VOF approach","volume":"33","author":"Meng","year":"2023","journal-title":"Int. J. Min. Sci. Technol."},{"key":"ref_16","unstructured":"Ubbink, O. (1997). Numerical Prediction of Two Fluid Systems with Sharp Interfaces. [Ph.D. Thesis, University of London]."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1007\/s42401-020-00082-7","article-title":"Application of mathematical modeling to solve the emergency water landing task in the interests of passenger aircraft certification","volume":"4","author":"Kozelkov","year":"2021","journal-title":"Aerosp. Syst."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"709","DOI":"10.1134\/S0015462816060016","article-title":"Numerical Modeling of the Free Rise of an Air Bubble","volume":"51","author":"Kozelkov","year":"2016","journal-title":"Fluid Dyn."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Kozelkov, A., Galanov, N., Semenov, I., Zhuchkov, R., and Strelets, D. (2023). Computational Investigation of the Water Droplet Effects on Shapes of Ice on Airfoils. Aerospace, 10.","DOI":"10.3390\/aerospace10100906"},{"key":"ref_20","unstructured":"Knudsen, M. (1934). The Kinetic Theory of Gases. Some Modern Aspects, Methuen and Co., Ltd."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Schrage, R.W. (1953). A Theoretical Study of Interphase Mass Transfer, Columbia University Press.","DOI":"10.7312\/schr90162"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1164","DOI":"10.1016\/j.ijheatmasstransfer.2016.12.065","article-title":"Review of computational studies on boiling and condensation","volume":"108","author":"Kharangate","year":"2017","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_23","unstructured":"Hartnett, J.P., and Irvine, T.F. (1991). Advances in condensation heat transfer. Advances in Heat Transfer, Academic Press."},{"key":"ref_24","unstructured":"Lee, W.H. (1980). A Pressure Iteration Scheme for Two-Phase Flow Modeling. Multiphase Transport Fundamentals, Reactor Safety, Applications, Hemisphere."},{"key":"ref_25","first-page":"969","article-title":"A method of correlating heat transfer data for surface boiling of liquids","volume":"74","author":"Rohsenow","year":"1952","journal-title":"Trans. ASME"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"111553","DOI":"10.1016\/j.anucene.2025.111553","article-title":"Numerical study of effects of system variables on rising vapour bubble in convective boiling conditions","volume":"222","author":"Sharif","year":"2025","journal-title":"Ann. Nucl. Energy"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"127456","DOI":"10.1016\/j.ijheatmasstransfer.2025.127456","article-title":"Numerical simulations of flow boiling in constricted microchannels: Bubble dynamics and heat transfer characteristics","volume":"253","author":"Darshan","year":"2025","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"100065","DOI":"10.1016\/j.scca.2025.100065","article-title":"Numerical investigation of drop spreading on a heated surface","volume":"6","author":"Vishal","year":"2025","journal-title":"Sustain. Chem. Clim. Action"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Li, Q., Ge, H., Pan, R., Zhang, Z., and Chen, R. (2021). Numerical Study on Flow and Release Characteristics of Gas Extinguishing Agent under Different Filling Pressure and Amount Conditions. Processes, 9.","DOI":"10.3390\/pr9091683"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"105289","DOI":"10.1016\/j.csite.2024.105289","article-title":"Unsteady numerical calculation of flow boiling heat transfer in microchannels with Kagome structures","volume":"63","author":"Wan","year":"2024","journal-title":"Case Stud. Therm. Eng."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"125607","DOI":"10.1016\/j.ijheatmasstransfer.2024.125607","article-title":"Simulation of a two-phase loop thermosyphon using a new interface-resolved phase change model","volume":"228","author":"Caner","year":"2024","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"105044","DOI":"10.1016\/j.csite.2024.105044","article-title":"Analysis of pool boiling heat transfer characteristics on copperbased structured surfaces modified with superhydrophilic, hydrophobic, superhydrophobic and hybrid biphilic properties","volume":"61","author":"Liu","year":"2024","journal-title":"Case Stud. Therm. Eng."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1016\/0021-9991(92)90240-Y","article-title":"A continuum method for modeling surface tension","volume":"100","author":"Brackbill","year":"1992","journal-title":"J. Comput. Phys."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"662","DOI":"10.1006\/jcph.2000.6481","article-title":"A volume of fluid based method for fluid flows with phase change","volume":"160","author":"Welch","year":"2000","journal-title":"J. Comput. Phys."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1080\/08916159008946385","article-title":"Condensing phenomena of a single vapor bubble into subcooled water","volume":"3","author":"Kamei","year":"1990","journal-title":"Exp. Heat Transf. Int. J."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Ferziger, J.H., and Peric, M. (2002). Computational Method for Fluid Dynamics, Springer.","DOI":"10.1007\/978-3-642-56026-2"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Vachaparambil, K.J., and Einarsrud, K.E. (2019). Comparison of Surface Tension Models for the Volume of Fluid Method. Processes, 7.","DOI":"10.3390\/pr7080542"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"865","DOI":"10.1016\/j.ijheatmasstransfer.2015.02.037","article-title":"Experimental and computational investigation of vertical downflow condensation","volume":"85","author":"Lee","year":"2015","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1720","DOI":"10.1134\/S0965542517100086","article-title":"Method for taking into account gravity in free-surface flow simulation","volume":"57","author":"Efremov","year":"2017","journal-title":"Comput. Math. Math. Phys."},{"key":"ref_40","unstructured":"Landau, L.D., and Lifshitz, V.M. (1988). Hydrodynamics, Nauka."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1242","DOI":"10.1134\/S0021894417070069","article-title":"Efficiency Analysis of the Parallel Implementation of the SIMPLE Algorithm on Multiprocessor Computers","volume":"58","author":"Lashkin","year":"2017","journal-title":"J. Appl. Mech. Tech. Phys."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Sarazov, A.V., Kozelkov, A.S., Strelets, D.Y., and Zhuchkov, R.N. (2023). Modeling Object Motion on Arbitrary Unstructured Grids Using an Invariant Principle of Computational Domain Topology: Key Features. Symmetry, 15.","DOI":"10.3390\/sym15112081"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"087102","DOI":"10.1063\/5.0058987","article-title":"A method for simulating interfacial mass transfer on arbitrary meshes","volume":"33","author":"Giustini","year":"2021","journal-title":"Phys. Fluids"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"451","DOI":"10.1016\/j.ijheatmasstransfer.2012.12.010","article-title":"Numerical investigation of hydrodynamics and heat transfer of elongated bubbles during flow boiling in a microchannel","volume":"59","author":"Magnini","year":"2013","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_45","first-page":"1","article-title":"Study on condensation of a single vapor bubble into subcooled water\u2014Part 2; Experimental analysis","volume":"19","author":"Kamei","year":"1990","journal-title":"Heat Trans.\u2013Jpn. Res."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"5871","DOI":"10.1016\/j.jcp.2008.02.020","article-title":"Evaporation model for interfacial flows based on a continuum-field representation of the source terms","volume":"227","author":"Hardt","year":"2008","journal-title":"J. Comput. Phys."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1016\/S0017-9310(00)00086-7","article-title":"Analysis of the evaporation coefficient and the condensation coefficient of water","volume":"44","author":"Marek","year":"2001","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"336","DOI":"10.1016\/j.pnucene.2015.04.011","article-title":"Numerical simulation of single bubble condensation in subcooled flow using OpenFOAM","volume":"83","author":"Zeng","year":"2015","journal-title":"Prog. Nucl. Energy"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"120","DOI":"10.1016\/j.pnucene.2016.02.004","article-title":"Numerical simulation of bubble condensation using CF-VOF","volume":"89","author":"Samkhaniani","year":"2016","journal-title":"Prog. Nucl. Energy"}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/18\/10\/604\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,28]],"date-time":"2025-09-28T04:37:39Z","timestamp":1759034259000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/18\/10\/604"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,26]]},"references-count":49,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2025,10]]}},"alternative-id":["a18100604"],"URL":"https:\/\/doi.org\/10.3390\/a18100604","relation":{},"ISSN":["1999-4893"],"issn-type":[{"value":"1999-4893","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,9,26]]}}}