A conservative finite volume method for incompressible two-phase flows on unstructured meshes

References

• S. Osher and J. A. Sethian, “Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations,” J. Comput. Phys., vol. 79, no. 1, pp. 12–49, 1988. Nov. DOI: 10.1016/0021-9991(88)90002-2.
   Web of Science ®Google Scholar
• F. Gibou, R. Fedkiw and S. Osher, “A review of level-set methods and some recent applications,” J. Comput. Phys., vol. 353, pp. 82–109, 2018. DOI: 10.1016/j.jcp.2017.10.006.
   Web of Science ®Google Scholar
• M. Sussman and E. Fatemi, “An efficient, interface-preserving level set redistancing algorithm and its application to interfacial incompressible fluid flow,” SIAM J. Sci. Comput., vol. 20, no. 4, pp. 1165–1191, 1999. DOI: 10.1137/S1064827596298245.
   Web of Science ®Google Scholar
• Z. Gu, H. Wen, C. Yu and T. W. Sheu, “Interface-preserving level set method for simulating dam-break flows,” J. Comput. Phys., vol. 374, pp. 249–280, 2018. DOI: 10.1016/j.jcp.2018.07.057.
   Web of Science ®Google Scholar
• E. Olsson and G. Kreiss, “A conservative level set method for two phase flow,” J. Comput. Phys., vol. 210, no. 1, pp. 225–246, 2005. DOI: 10.1016/j.jcp.2005.04.007.
   Web of Science ®Google Scholar
• T. Wacławczyk, “A consistent solution of the re-initialization equation in the conservative level-set method,” J. Comput. Phys., vol. 299, no. Supplement C, pp. 487–525, 2015. DOI: 10.1016/j.jcp.2015.06.029.
   Google Scholar
• R. Chiodi and O. Desjardins, “A reformulation of the conservative level set reinitialization equation for accurate and robust simulation of complex multiphase flows,” J. Comput. Phys., vol. 343, pp. 186–200, 2017. DOI: 10.1016/j.jcp.2017.04.053.
   Web of Science ®Google Scholar
• T. Wacławczyk, “On a relation between the volume of fluid, level-set and phase field interface models,” Int. J. Multiphase Flow, vol. 97, pp. 60–77, 2017. DOI: 10.1016/j.ijmultiphaseflow.2017.08.003.
   Web of Science ®Google Scholar
• W. G. Price and Y. G. Chen, “A simulation of free surface waves for incompressible two-phase flows using a curvilinear level set formulation,” Int. J. Numer. Meth. Fluids, vol. 51, no. 3, pp. 305–330, 2006. DOI: 10.1002/fld.1126.
   Web of Science ®Google Scholar
• Y. Aiming, C. Sukun, Y. Liu and Y. Xiaoquan, “An upwind finite volume method for incompressible inviscid free surface flows,” Comput. Fluids, vol. 101, pp. 170–182, 2014.
   Web of Science ®Google Scholar
• S. Bhat and J. C. Mandal, “Contact preserving Riemann solver for incompressible two-phase flows,” J. Comput. Phys., vol. 379, pp. 173–191, 2019. DOI: 10.1016/j.jcp.2018.10.039.
   Web of Science ®Google Scholar
• B. Vermeire, N. Loppi and P. Vincent, “Optimal runge-kutta schemes for pseudo time-stepping with high-order unstructured methods,” J. Comput. Phys., vol. 383, pp. 55–71, 2019. DOI: 10.1016/j.jcp.2019.01.003.
   Web of Science ®Google Scholar
• B. R. Hodges, “An artificial compressibility method for 1d simulation of open-channel and pressurized-pipe flow,” Water, vol. 12, no. 6, pp. 1727, 2020. DOI: 10.3390/w12061727.
   Web of Science ®Google Scholar
• S. Parameswaran and J. C. Mandal, “A novel Roe solver for incompressible two-phase flow problems,” J. Comput. Phys., vol. 390, pp. 405–424, 2019. DOI: 10.1016/j.jcp.2019.04.012.
   Web of Science ®Google Scholar
• E. Olsson, G. Kreiss and S. Zahedi, “A conservative level set method for two phase flow II,” J. Comput. Phys., vol. 225, no. 1, pp. 785–807, 2007. DOI: 10.1016/j.jcp.2006.12.027.
   Web of Science ®Google Scholar
• N. Balcázar, L. Jofre, O. Lehmkuhl, J. Castro and J. Rigola, “A finite-volume/level-set method for simulating two-phase flows on unstructured grids,” Int. J. Multiphase Flow, vol. 64, pp. 55–72, 2014. DOI: 10.1016/j.ijmultiphaseflow.2014.04.008.
   Web of Science ®Google Scholar
• J. C. Mandal and J. Subramanian, “On the link between weighted least-squares and limiters used in higher-order reconstructions for finite volume computations of hyperbolic equations,” Appl. Numer. Math., vol. 58, no. 5, pp. 705–725, 2008. DOI: 10.1016/j.apnum.2007.02.003.
   Web of Science ®Google Scholar
• J. K. Patel and G. Natarajan, “A novel consistent and well-balanced algorithm for simulations of multiphase flows on unstructured grids,” J. Comput. Phys., vol. 350, pp. 207–236, 2017. DOI: 10.1016/j.jcp.2017.08.047.
   Web of Science ®Google Scholar
• J. Brackbill, D. Kothe and C. Zemach, “A continuum method for modeling surface tension,” J. Comput. Phys., vol. 100, no. 2, pp. 335–354, 1992. DOI: 10.1016/0021-9991(92)90240-Y.
   Web of Science ®Google Scholar
• A. J. Chorin, “A numerical method for solving incompressible viscous flow problems,” J. Comput. Phys., vol. 2, no. 1, pp. 12–26, 1967. DOI: 10.1016/0021-9991(67)90037-X.
   Google Scholar
• J. L. Chang and D. Kwak, “On the method of pseudo compressibility for numerically solving incompressible flows,” AIAA 22nd Aerospace Sciences Meeting, 1984. in cited By 11. DOI: 10.2514/6.1984-252.
   Google Scholar
• E. Turkel, “Preconditioned methods for solving the incompressible and low speed compressible equations,” J. Comput. Phys., vol. 72, no. 2, pp. 277–298, 1987. DOI: 10.1016/0021-9991(87)90084-2.
   Web of Science ®Google Scholar
• N. Hakimi, “Preconditioning Methods for Time Dependent Navier-Stokes Equations,” Phd thesis, Dept. of Fluid Mechanics, Vrije Universiteit Brussel, December, 1997.
   Google Scholar
• P. Nithiarasu, “An efficient artificial compressibility (ac) scheme based on the characteristic based split (cbs) method for incompressible flows,” Int. J. Numer. Meth. Eng., vol. 56, no. 13, pp. 1815–1845, 2003. DOI: 10.1002/nme.712.
   Web of Science ®Google Scholar
• A. Rizzi and L.-E. Eriksson, “Computation of inviscid incompressible flow with rotation,” J. Fluid Mech., vol. 153, no. 1, pp. 275–312, 1985. DOI: 10.1017/S0022112085001264.
   Google Scholar
• A. G. Malan, R. W. Lewis and P. Nithiarasu, “An improved unsteady, unstructured, artificial compressibility, finite volume scheme for viscous incompressible flows: part I. theory and implementation,” Int. J. Numer. Meth. Eng., vol. 54, no. 5, pp. 695–714, 2002. DOI: 10.1002/nme.447.
   Web of Science ®Google Scholar
• A. G. Malan, R. W. Lewis and P. Nithiarasu, “An improved unsteady, unstructured, artificial compressibility, finite volume scheme for viscous incompressible flows: part II. application,” Int. J. Numer. Meth. Eng., vol. 54, no. 5, pp. 715–729, 2002. DOI: 10.1002/nme.443.
   Web of Science ®Google Scholar
• A. L. Gaitonde, “A dual-time method for two-dimensional unsteady incompressible flow calculations,” Int. J. Numer. Meth. Eng., vol. 41, no. 6, pp. 1153–1166, 1998. DOI: 10.1002/(SICI)1097-0207(19980330)41:6<1153::AID-NME334>3.0.CO;2-9.
   Web of Science ®Google Scholar
• S. Gottlieb, “On high order strong stability preserving runge-kutta and multi step time discretizations,” J. Sci. Comput., vol. 25, no. 1, pp. 105–128, 2005. Oct. DOI: 10.1007/s10915-004-4635-5.
   Web of Science ®Google Scholar
• R. S. Chandrakant, “Incompressible Flow Computations: HLLC-AC based Numerical Approach,” Phd thesis, Dept. of Aerospace Engineering, Indian Institute of Technology, July, 2016., Bombay
   Google Scholar
• S. Parameswaran and J. Mandal, “A stable interface-preserving reinitialization equation for conservative level set method,” Eur. J. Mech. - B/Fluids, vol. 98, pp. 40–63, 2023. DOI: 10.1016/j.euromechflu.2022.11.001.
   Web of Science ®Google Scholar
• I. Chakraborty, G. Biswas and P. Ghoshdastidar, “A coupled level-set and volume-of-fluid method for the buoyant rise of gas bubbles in liquids,” Int. J. Heat Mass Transfer, vol. 58, no. 1-2, pp. 240–259, 2013. DOI: 10.1016/j.ijheatmasstransfer.2012.11.027.
   Web of Science ®Google Scholar
• J. C. Mandal, S. Rao and J. Subramanian, “High-resolution finite volume computations using a novel weighted least-squares formulation,” Int. J. Numer. Meth. Fluids, vol. 56, no. 8, pp. 1425–1431, 2008. DOI: 10.1002/fld.1707.
   Web of Science ®Google Scholar
• J. Blazek, “Chapter 5 - Unstructured Finite-Volume Schemes,” in Computational Fluid Dynamics: Principles and Applications (Third Edition) (J. Blazek, ed.), pp. 121–166. Oxford: butterworth-Heinemann, third edition ed., 2015,
   Google Scholar
• W. J. Coirier, “An adaptively-refined, Cartesian, cell-based scheme for the Euler and Navier-Stokes equations,” PhD thesis, The University of Michigan, 1994.
   Google Scholar
• S. Veluri, C. Roy, A. Choudhary and E. Luke, “Finite volume diffusion operators for compressible CFD on unstructured grids,” in 19th AIAA Computational Fluid Dynamics, 2009. DOI: 10.2514/6.2009-4141.
   Google Scholar
• E. Sozer, C. Brehm and C. C. Kiris, “Gradient calculation methods on arbitrary polyhedral unstructured meshes for cell-centered CFD solvers,” 52nd Aerospace Sciences Meeting, 2014. in DOI: 10.2514/6.2014-1440.
   Google Scholar
• T. W. H. Sheu and C. H. Yu, “Numerical simulation of free surface by an area-preserving level set method,” Commun. Comput. phys, vol. 11, no. 4, pp. 1347–1371, 2012. DOI: 10.4208/cicp.120510.150511s.
   Web of Science ®Google Scholar
• T. W. H. Sheu, C. H. Yu and P. H. Chiu, “Development of level set method with good area preservation to predict interface in two-phase flows,” Int. J. Numer. Meth. Fluids, vol. 67, no. 1, pp. 109–134, 2011. DOI: 10.1002/fld.2344.
   Web of Science ®Google Scholar
• S. T. Zalesak, “Fully multidimensional flux-corrected transport algorithms for fluids,” J. Comput. Phys., vol. 31, no. 3, pp. 335–362, 1979. DOI: 10.1016/0021-9991(79)90051-2.
   Web of Science ®Google Scholar
• I. Tadjbakhsh and J. B. Keller, “Standing surface waves of finite amplitude,” J. Fluid Mech., vol. 8, no. 03, pp. 442–451, 07 1960. DOI: 10.1017/S0022112060000724.
   Google Scholar
• S. Kang and F. Sotiropoulos, “Numerical modeling of 3d turbulent free surface flow in natural waterways,” Adv. Water Resour., vol. 40, pp. 23–36, 2012. DOI: 10.1016/j.advwatres.2012.01.012.
   Web of Science ®Google Scholar
• S. P. Bhat and J. Mandal, “An improved HLLC-type solver for incompressible two-phase fluid flows,” Comput. Fluids, vol. 244, pp. 105570, 2022. DOI: 10.1016/j.compfluid.2022.105570.
   Web of Science ®Google Scholar
• J. C. Martin and W. J. Moyce, “Part IV. an experimental study of the collapse of liquid columns on a rigid horizontal plane,” Philos. Trans. R. Soc. Lond. A, vol. 244, no. 882, pp. 312–324, 1952.
   Web of Science ®Google Scholar
• Y. Zhang, Q. Zou and D. Greaves, “Numerical simulation of free-surface flow using the level-set method with global mass correction,” Int. J. Numer. Meth. Fluids, vol. 63, no. June 2009, pp. n/a–n/a, 2009. DOI: 10.1002/fld.2090.
   Google Scholar
• S. Hysing, “Mixed element FEM level set method for numerical simulation of immiscible fluids,” J. Comput. Phys., vol. 231, no. 6, pp. 2449–2465, 2012. DOI: 10.1016/j.jcp.2011.11.035.
   Web of Science ®Google Scholar
• N. Balcázar, O. Lehmkuhl, L. Jofre, J. Rigola and A. Oliva, “A coupled volume-of-fluid/level-set method for simulation of two-phase flows on unstructured meshes,” Comput. Fluids, vol. 124, pp. 12–29, 2016. Special Issue for ICMMES-2014. DOI: 10.1016/j.compfluid.2015.10.005.
   Web of Science ®Google Scholar
• S. Hysing, et al., “Quantitative benchmark computations of two-dimensional bubble dynamics,” Int. J. Numer. Meth. Fluids, vol. 60, no. 11, pp. 1259–1288, 2009. DOI: 10.1002/fld.1934.
   Web of Science ®Google Scholar
• J. H. Duncan, “The breaking and non-breaking wave resistance of a two-dimensional hydrofoil,” J. Fluid Mech, vol. 126, pp. 507–520, 1983. DOI: 10.1017/S0022112083000294.
   Web of Science ®Google Scholar
• A. D. Mascio, R. Broglia and R. Muscari, “On the application of the single-phase level set method to naval hydrodynamic flows,” Comput. Fluids, vol. 36, no. 5, pp. 868–886, 2007. DOI: 10.1016/j.compfluid.2006.08.001.
   Web of Science ®Google Scholar