Comparison of Surface Tension Models for the Volume of Fluid Method
Abstract
:1. Introduction
- two-dimensional capillary rise.
2. Numerical Model
2.1. Governing Equations
2.2. Surface Tension Models
2.2.1. The Continuum Surface Force (CSF) Model
2.2.2. The Smoothed CSF Model
2.2.3. The Sharp Surface Force (SSF) Model
3. Solver Settings
4. Validation: Benchmark Test Cases
4.1. Two Dimensional Rising Bubbles
4.2. Two-Dimensional Capillary Rise
5. Analysis: Spurious Current
5.1. Stagnant Bubble of Few Millimetres
5.2. Effect of Time Step
5.3. Effect of Under-Relaxation Factor
6. Conclusions
- For a stationary bubble with a 2.5 mm radius, CSF and SSF models generate the most and least amount of spurious currents, respectively. For the finest mesh used, Smoothed CSF and SSF models reduce spurious currents by nearly one-tenth and one-twentieth of the CSF model (when no under-relaxation factor is used), respectively. When using a lower under-relaxation factor (for the finest mesh), Smoothed CSF and SSF models reduce the spurious currents by approximately one-fourth of the CSF model.
- The time step constraints proposed by Brackbill et al. [66] and Deshpande et al. [63] show that spurious currents generated by the CSF is significantly reduced while using a lower under-relaxation factor. In Smoothed CSF and SSF models, when using the same under-relaxation factor, the time step constraint slightly reduces the spurious currents by 6–7%. Interestingly, when no under-relaxation is used, the CSF model generates marginally larger (nearly 0.1%) spurious currents with the time step constraint proposed by Deshpande et al. [63], but other models show a reduction in spurious current by less than 10%.
- The Laplace pressure in the bubbles predicted by Smoothed CSF and SSF is more accurate with an error of 7–9% for the higher mesh densities than CSF model with negligible imbalance in mass of the phases.
- Although using a lower under-relaxation factor reduces the spurious currents and predicts the Laplace pressure in the stationary bubble (for all the surface tension models considered) quite reasonably, it can adversely effect the accuracy of dynamic cases such as rising bubbles by underestimating the flow variables.
- Using a higher mesh density results in larger spurious currents for CSF model but they are reduced for both Smoothed CSF and SSF models for the static case considered.
- The effect of mesh resolution was studied only for the stationary bubble in this work. For the case of rising bubbles, previous works [36,55], using the CSF model, reported challenges in achieving a mesh independent solution. Similarly, for capillary rise using the CSF model, Yamamoto et al. [36] reported an increasing error when using a finer mesh. The meshes used in this paper correspond to the finest grid (used in FreeLIFE solver) implemented by Hysing et al. [54] and the grid that provided a most accurate model for capillary rise in the work by Yamamoto et al. [36]. We expect similar effects of mesh resolution for both Smoothed CSF and SSF models for dynamic cases, as they are variants of the same formulation. The quantification of these errors will be treated in future work.
- Modelling the capillary rise with SSF was shown to provide a more accurate representation than the CSF model. Interestingly, the Smoothed CSF could not reliably simulate capillary rise due to spurious currents.
Supplementary Materials
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
- Basaran, O.A.; Gao, H.; Bhat, P.P. Nonstandard inkjets. Annu. Rev. Fluid Mech. 2013, 45, 85–113. [Google Scholar] [CrossRef]
- Clark, I.C.; Abate, A.R. Microfluidic bead encapsulation above 20 kHz with triggered drop formation. Lab Chip 2018, 18, 3598–3605. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Jomy Vachaparambil, K. An Analytical and Numerical Study of Droplet Formation and Break-off for Jetting of Dense Suspensions. Master’s Thesis, KTH, School of Engineering Sciences (SCI), Mechanics, Stockholm, Sweden, 2016. [Google Scholar]
- Valette, R.; Hachem, E.; Khalloufi, M.; Pereira, A.; Mackley, M.; Butler, S. The effect of viscosity, yield stress, and surface tension on the deformation and breakup profiles of fluid filaments stretched at very high velocities. J. Non-Newton. Fluid Mech. 2019, 263, 130–139. [Google Scholar] [CrossRef]
- Høst-Madsen, J.; Jensen, K.H. Laboratory and numerical investigations of immiscible multiphase flow in soil. J. Hydrol. 1992, 135, 13–52. [Google Scholar] [CrossRef]
- Muggeridge, A.; Cockin, A.; Webb, K.; Frampton, H.; Collins, I.; Moulds, T.; Salino, P. Recovery rates, enhanced oil recovery and technological limits. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 2014, 372, 20120320. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Osborn, J.L.; Lutz, B.; Fu, E.; Kauffman, P.; Stevens, D.Y.; Yager, P. Microfluidics without pumps: reinventing the T-sensor and H-filter in paper networks. Lab Chip 2010, 10, 2659–2665. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Brussieux, C.; Viers, P.; Roustan, H.; Rakib, M. Controlled electrochemical gas bubble release from electrodes entirely and partially covered with hydrophobic materials. Electrochimica Acta 2011, 56, 7194–7201. [Google Scholar] [CrossRef] [Green Version]
- Guan, P.; Jia, L.; Yin, L.; Tan, Z. Effect of bubble contact diameter on bubble departure size in flow boiling. Exp. Heat Transf. 2016, 29, 37–52. [Google Scholar] [CrossRef]
- Clift, R.; Grace, J.; Weber, M. Bubbles, Drops, and Particles; Dover Civil and Mechanical Engineering Series; Dover Publications: Mineola, NY, USA, 2005. [Google Scholar]
- Vuorinen, V.A.; Hillamo, H.; Kaario, O.; Nuutinen, M.; Larmi, M.; Fuchs, L. Effect of droplet size and atomization on spray formation: A priori study using large-eddy simulation. Flow Turbul. Combust. 2011, 86, 533–561. [Google Scholar] [CrossRef]
- Hilz, E.; Vermeer, A.W. Spray drift review: The extent to which a formulation can contribute to spray drift reduction. Crop Prot. 2013, 44, 75–83. [Google Scholar] [CrossRef]
- Bouffard, J.; Kaster, M.; Dumont, H. Influence of process variable and physicochemical properties on the granulation mMechanism of mannitol in a fluid bed top spray granulator. Drug Dev. Ind. Pharm. 2005, 31, 923–933. [Google Scholar] [CrossRef] [PubMed]
- Gallino, G.; Gallaire, F.; Lauga, E.; Michelin, S. Physics of Bubble-Propelled Microrockets. Adv. Funct. Mater. 2018, 28, 1800686. [Google Scholar] [CrossRef]
- Vachaparambil, K.J.; Einarsrud, K.E. Analysis of bubble nucleation mechanisms in supersaturated solutions: A macroscopic perspective. Meet. Abstr. 2018, MA2018-01, 1366. [Google Scholar]
- Vachaparambil, K.J.; Einarsrud, K.E. Explanation of bubble nucleation mechanisms: A gradient theory approach. J. Electrochem. Soc. 2018, 165, E504–E512. [Google Scholar] [CrossRef]
- Xu, W.; Lan, Z.; Peng, B.; Wen, R.; Ma, X. Heterogeneous nucleation capability of conical microstructures for water droplets. RSC Adv. 2015, 5, 812–818. [Google Scholar] [CrossRef]
- Alexiadis, A.; Dudukovic, M.; Ramachandran, P.; Cornell, A.; Wanngård, J.; Bokkers, A. Liquid–gas flow patterns in a narrow electrochemical channel. Chem. Eng. Sci. 2011, 66, 2252–2260. [Google Scholar] [CrossRef]
- Gupta, A.; Roy, S. Euler–Euler simulation of bubbly flow in a rectangular bubble column: Experimental validation with Radioactive Particle Tracking. Chem. Eng. J. 2013, 225, 818–836. [Google Scholar] [CrossRef]
- Shams, E.; Finn, J.; Apte, S.V. A numerical scheme for Euler–Lagrange simulation of bubbly flows in complex systems. Int. J. Numer. Methods Fluids 2011, 67, 1865–1898. [Google Scholar] [CrossRef]
- Legendre, D.; Zevenhoven, R. A numerical Euler–Lagrange method for bubble tower CO2 dissolution modeling. Chem. Eng. Res. Des. 2016, 111, 49–62. [Google Scholar] [CrossRef]
- Tuković, Ž.; Jasak, H. A moving mesh finite volume interface tracking method for surface tension dominated interfacial fluid flow. Comput. Fluids 2012, 55, 70–84. [Google Scholar] [CrossRef] [Green Version]
- Charin, A.; Tuković, Ž.; Jasak, H.; Silva, L.; Lage, P. A moving mesh interface tracking method for simulation of liquid–liquid systems. J. Comput. Phys. 2017, 334, 419–441. [Google Scholar] [CrossRef]
- Menon, S.; Mooney, K.G.; Stapf, K.; Schmidt, D.P. Parallel adaptive simplical re-meshing for deforming domain CFD computations. J. Comput. Phys. 2015, 298, 62–78. [Google Scholar] [CrossRef] [Green Version]
- Hirt, C.; Nichols, B. Volume of fluid (VOF) method for the dynamics of free boundaries. J. Comput. Phys. 1981, 39, 201–225. [Google Scholar] [CrossRef]
- Chang, Y.; Hou, T.; Merriman, B.; Osher, S. A Level Set Formulation of Eulerian Interface Capturing Methods for Incompressible Fluid Flows. J. Comput. Phys. 1996, 124, 449–464. [Google Scholar] [CrossRef] [Green Version]
- Jacqmin, D. Calculation of Two-Phase Navier–Stokes Flows Using Phase-Field Modeling. J. Comput. Phys. 1999, 155, 96–127. [Google Scholar] [CrossRef]
- Peskin, C.S. Flow patterns around heart valves: A numerical method. J. Comput. Phys. 1972, 10, 252–271. [Google Scholar] [CrossRef]
- Pivello, M.; Villar, M.; Serfaty, R.; Roma, A.; Silveira-Neto, A. A fully adaptive front tracking method for the simulation of two phase flows. Int. J. Multiph. Flow 2014, 58, 72–82. [Google Scholar] [CrossRef]
- Cubos-Ramírez, J.; Ramírez-Cruz, J.; Salinas-Vázquez, M.; Vicente-Rodríguez, W.; Martinez-Espinosa, E.; Lagarza-Cortes, C. Efficient two-phase mass-conserving level set method for simulation of incompressible turbulent free surface flows with large density ratio. Comput. Fluids 2016, 136, 212–227. [Google Scholar] [CrossRef]
- Yue, P.; Zhou, C.; Feng, J.J. Spontaneous shrinkage of drops and mass conservation in phase-field simulations. J. Comput. Phys. 2007, 223, 1–9. [Google Scholar] [CrossRef] [Green Version]
- Samkhaniani, N.; Ansari, M. Numerical simulation of bubble condensation using CF-VOF. Prog. Nucl. Energy 2016, 89, 120–131. [Google Scholar] [CrossRef]
- Baltussen, M.; Kuipers, J.; Deen, N. A critical comparison of surface tension models for the volume of fluid method. Chem. Eng. Sci. 2014, 109, 65–74. [Google Scholar] [CrossRef]
- Patel, H.; Kuipers, J.; Peters, E. Computing interface curvature from volume fractions: A hybrid approach. Comput. Fluids 2018, 161, 74–88. [Google Scholar] [CrossRef] [Green Version]
- Jafari, A.; Shirani, E.; Ashgriz, N. An improved three-dimensional model for interface pressure calculations in free-surface flows. Int. J. Comput. Fluid Dyn. 2007, 21, 87–97. [Google Scholar] [CrossRef]
- Yamamoto, T.; Okano, Y.; Dost, S. Validation of the S-CLSVOF method with the density-scaled balanced continuum surface force model in multiphase systems coupled with thermocapillary flows. Int. J. Numer. Methods Fluids 2017, 83, 223–244. [Google Scholar] [CrossRef]
- Gerlach, D.; Tomar, G.; Biswas, G.; Durst, F. Comparison of volume-of-fluid methods for surface tension-dominant two-phase flows. Int. J. Heat Mass Transf. 2006, 49, 740–754. [Google Scholar] [CrossRef]
- Yokoi, K. A practical numerical framework for free surface flows based on CLSVOF method, multi-moment methods and density-scaled CSF model: Numerical simulations of droplet splashing. J. Comput. Phys. 2013, 232, 252–271. [Google Scholar] [CrossRef] [Green Version]
- Yokoi, K. A density-scaled continuum surface force model within a balanced force formulation. J. Comput. Phys. 2014, 278, 221–228. [Google Scholar] [CrossRef]
- Ningegowda, B.; Premachandran, B. A coupled level set and volume of fluid method with multi-directional advection algorithms for two-phase flows with and without phase change. Int. J. Heat Mass Transf. 2014, 79, 532–550. [Google Scholar] [CrossRef]
- Albert, C.; Raach, H.; Bothe, D. Influence of surface tension models on the hydrodynamics of wavy laminar falling films in volume of fluid-simulations. Int. J. Multiph. Flow 2012, 43, 66–71. [Google Scholar] [CrossRef]
- Pavuluri, S.; Maes, J.; Doster, F. Spontaneous imbibition in a microchannel: analytical solution and assessment of volume of fluid formulations. Microfluid. Nanofluid. 2018, 22, 90. [Google Scholar] [CrossRef] [Green Version]
- Gumulya, M.; Utikar, R.P.; Pareek, V.; Mead-Hunter, R.; Mitra, S.; Evans, G.M. Evaporation of a droplet on a heated spherical particle. Chem. Eng. J. 2015, 278, 309–319. [Google Scholar] [CrossRef]
- Gumulya, M.; Joshi, J.B.; Utikar, R.P.; Evans, G.M.; Pareek, V. Bubbles in viscous liquids: Time dependent behaviour and wake characteristics. Chem. Eng. Sci. 2016, 144, 298–309. [Google Scholar] [CrossRef] [Green Version]
- Li, D.; Duan, X. Numerical analysis of droplet impact and heat transfer on an inclined wet surface. Int. J. Heat Mass Transf. 2019, 128, 459–468. [Google Scholar] [CrossRef]
- Sontti, S.G.; Atta, A. Formation characteristics of Taylor bubbles in power-law liquids flowing through a microfluidic co-flow device. J. Ind. Eng. Chem. 2018, 65, 82–94. [Google Scholar] [CrossRef]
- Wang, Z.; Yang, J.; Koo, B.; Stern, F. A coupled level set and volume-of-fluid method for sharp interface simulation of plunging breaking waves. Int. J. Multiph. Flow 2009, 35, 227–246. [Google Scholar] [CrossRef]
- Khismatullin, D.; Renardy, Y.; Renardy, M. Development and implementation of VOF-PROST for 3D viscoelastic liquid–liquid simulations. J. Non-Newton. Fluid Mech. 2006, 140, 120–131. [Google Scholar] [CrossRef]
- Guo, Z.; Fletcher, D.F.; Haynes, B.S. Implementation of a height function method to alleviate spurious currents in CFD modelling of annular flow in microchannels. Appl. Math. Model. 2015, 39, 4665–4686. [Google Scholar] [CrossRef]
- Fuster, D.; Agbaglah, G.; Josserand, C.; Popinet, S.; Zaleski, S. Numerical simulation of droplets, bubbles and waves: State of the art. Fluid Dyn. Res. 2009, 41, 065001. [Google Scholar] [CrossRef]
- Aboukhedr, M.; Georgoulas, A.; Marengo, M.; Gavaises, M.; Vogiatzaki, K. Simulation of micro-flow dynamics at low capillary numbers using adaptive interface compression. Comput. Fluids 2018, 165, 13–32. [Google Scholar] [CrossRef] [Green Version]
- Raeini, A.Q.; Blunt, M.J.; Bijeljic, B. Direct simulations of two-phase flow on micro-CT images of porous media and upscaling of pore-scale forces. Adv. Water Resour. 2014, 74, 116–126. [Google Scholar] [CrossRef]
- Liu, Q.; Palm, B. Numerical study of bubbles rising and merging during convective boiling in micro-channels. Appl. Therm. Eng. 2016, 99, 1141–1151. [Google Scholar] [CrossRef]
- Hysing, S.; Turek, S.; Kuzmin, D.; Parolini, N.; Burman, E.; Ganesan, S.; Tobiska, L. Quantitative benchmark computations of two-dimensional bubble dynamics. Int. J. Numer. Methods Fluids 2009, 60, 1259–1288. [Google Scholar] [CrossRef]
- Klostermann, J.; Schaake, K.; Schwarze, R. Numerical simulation of a single rising bubble by VOF with surface compression. Int. J. Numer. Methods Fluids 2013, 71, 960–982. [Google Scholar] [CrossRef]
- Hoang, D.A.; van Steijn, V.; Portela, L.M.; Kreutzer, M.T.; Kleijn, C.R. Benchmark numerical simulations of segmented two-phase flows in microchannels using the Volume of Fluid method. Comput. Fluids 2013, 86, 28–36. [Google Scholar] [CrossRef]
- Hardt, S.; Wondra, F. Evaporation model for interfacial flows based on a continuum-field representation of the source terms. J. Comput. Phys. 2008, 227, 5871–5895. [Google Scholar] [CrossRef]
- Rusche, H. Computational Fluid Dynamics of Dispersed Two-Phase Flows at High Phase Fractions. Ph.D. Thesis, Imperial College London (University of London), London, UK, 2003. [Google Scholar]
- Roenby, J.; Bredmose, H.; Jasak, H. A computational method for sharp interface advection. Royal Soc. Open Sci. 2016, 3, 160405. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Scheufler, H.; Roenby, J. Accurate and efficient surface reconstruction from volume fraction data on general meshes. J. Comput. Phys. 2019, 383, 1–23. [Google Scholar] [CrossRef] [Green Version]
- Cifani, P.; Michalek, W.; Priems, G.; Kuerten, J.; van der Geld, C.; Geurts, B. A comparison between the surface compression method and an interface reconstruction method for the VOF approach. Comput. Fluids 2016, 136, 421–435. [Google Scholar] [CrossRef] [Green Version]
- Harvie, D.; Davidson, M.; Rudman, M. An analysis of parasitic current generation in volume of fluid simulations. Appl. Math. Model. 2006, 30, 1056–1066. [Google Scholar] [CrossRef]
- Deshpande, S.S.; Anumolu, L.; Trujillo, M.F. Evaluating the performance of the two-phase flow solver interFoam. Comput. Sci. Discov. 2012, 5, 014016. [Google Scholar] [CrossRef]
- Abadie, T.; Aubin, J.; Legendre, D. On the combined effects of surface tension force calculation and interface advection on spurious currents within Volume of Fluid and Level Set frameworks. J. Comput. Phys. 2015, 297, 611–636. [Google Scholar] [CrossRef]
- Francois, M.M.; Cummins, S.J.; Dendy, E.D.; Kothe, D.B.; Sicilian, J.M.; Williams, M.W. A balanced-force algorithm for continuous and sharp interfacial surface tension models within a volume tracking framework. J. Comput. Phys. 2006, 213, 141–173. [Google Scholar] [CrossRef]
- Brackbill, J.; Kothe, D.; Zemach, C. A continuum method for modeling surface tension. J. Comput. Phys. 1992, 100, 335–354. [Google Scholar] [CrossRef]
- Ubbink, O. Numerical Prediction of Two Fluid Systems with Sharp Interfaces. Ph.D Thesis, Imperial College London (University of London), London, UK, 1997. [Google Scholar]
- Heyns, J.A.; Oxtoby, O.F. Modelling surface tension dominated multiphase flows using the VOF approach. In Proceedings of the 6th European Conference on Computational Fluid Dynamics, Barcelona, Spain, 20–25 July 2014. [Google Scholar]
- Sussman, M.; Puckett, E.G. A coupled level set and volume-of-fluid method for computing 3D and axisymmetric incompressible two-phase flows. J. Comput. Phys. 2000, 162, 301–337. [Google Scholar] [CrossRef]
- Albadawi, A.; Donoghue, D.; Robinson, A.; Murray, D.; Delauré, Y. Influence of surface tension implementation in volume of fluid and coupled volume of fluid with level set methods for bubble growth and detachment. Int. J. Multiph. Flow 2013, 53, 11–28. [Google Scholar] [CrossRef]
- Raessi, M.; Mostaghimi, J.; Bussmann, M. A volume-of-fluid interfacial flow solver with advected normals. Comput. Fluids 2010, 39, 1401–1410. [Google Scholar] [CrossRef]
- Renardy, Y.; Renardy, M. PROST: A Parabolic Reconstruction of Surface Tension for the Volume-of-Fluid Method. J. Comput. Phys. 2002, 183, 400–421. [Google Scholar] [CrossRef] [Green Version]
- Youngs, D. Time-dependent multi-material flow with large fluid distortion. In Numerical Methods for Fluid Dynamics; Morton, K.W., Baines, M.J., Eds.; Academic Press: New York, NY, USA, 1982; pp. 273–285. [Google Scholar]
- Pilliod, J.E.; Puckett, E.G. Second-order accurate volume-of-fluid algorithms for tracking material interfaces. J. Comput. Phys. 2004, 199, 465–502. [Google Scholar] [CrossRef] [Green Version]
- Popinet, S. An accurate adaptive solver for surface-tension-driven interfacial flows. J. Comput. Phys. 2009, 228, 5838–5866. [Google Scholar] [CrossRef] [Green Version]
- Raeini, A.Q.; Blunt, M.J.; Bijeljic, B. Modelling two-phase flow in porous media at the pore scale using the volume-of-fluid method. J. Comput. Phys. 2012, 231, 5653–5668. [Google Scholar] [CrossRef]
- Denner, F.; Evrard, F.; Serfaty, R.; van Wachem, B.G. Artificial viscosity model to mitigate numerical artefacts at fluid interfaces with surface tension. Comput. Fluids 2017, 143, 59–72. [Google Scholar] [CrossRef] [Green Version]
- Lafaurie, B.; Nardone, C.; Scardovelli, R.; Zaleski, S.; Zanetti, G. Modelling merging and fragmentation in multiphase flows with SURFER. J. Comput. Phys. 1994, 113, 134–147. [Google Scholar] [CrossRef]
- Galusinski, C.; Vigneaux, P. On stability condition for bifluid flows with surface tension: Application to microfluidics. J. Comput. Phys. 2008, 227, 6140–6164. [Google Scholar] [CrossRef] [Green Version]
- Popinet, S. Numerical models of surface tension. Annu. Rev. Fluid Mech. 2018, 50, 49–75. [Google Scholar] [CrossRef]
- Wang, Y.; Dandekar, R.; Bustos, N.; Poulain, S.; Bourouiba, L. Universal rim thickness in unsteady sheet fragmentation. Phys. Rev. Lett. 2018, 120, 204503. [Google Scholar] [CrossRef] [PubMed]
- Saha, A.A.; Mitra, S.K. Effect of dynamic contact angle in a volume of fluid (VOF) model for a microfluidic capillary flow. J. Colloid Interface Sci. 2009, 339, 461–480. [Google Scholar] [CrossRef] [PubMed]
- Issa, R. Solution of the implicitly discretised fluid flow equations by operator-splitting. J. Comput. Phys. 1986, 62, 40–65. [Google Scholar] [CrossRef]
- Greenshields, C.J. OpenFOAM User Guide; Version 6; The OpenFOAM Foundation: London, UK.
- Van Leer, B. Towards the ultimate conservative difference scheme. II. Monotonicity and conservation combined in a second-order scheme. J. Comput. Phys. 1974, 14, 361–370. [Google Scholar] [CrossRef]
- Maki, K. Ship Resistance Simulations with OpenFOAM. In Proceedings of the 6th OpenFOAM Workshop: The Pennsylvania State University, State College, PA, USA, 13–16 June 2011. [Google Scholar]
- Greenshields, C. OpenFOAM 2.3.0: Multiphase Modelling. 2014. Available online: https://openfoam.org/release/2-3-0/multiphase/ (accessed on 12 August 2019).
- Song, B.; Liu, G.R.; Lam, K.Y.; Amano, R.S. On a higher-order bounded discretization scheme. Int. J. Numer. Methods Fluids 2000, 32, 881–897. [Google Scholar] [CrossRef]
- Min, T.; Yoo, J.Y.; Choi, H. Effect of spatial discretization schemes on numerical solutions of viscoelastic fluid flows. J. Non-Newton. Fluid Mech. 2001, 100, 27–47. [Google Scholar] [CrossRef]
- Bullard, J.W.; Garboczi, E.J. Capillary rise between planar surfaces. Phys. Rev. E 2009, 79, 011604. [Google Scholar] [CrossRef] [PubMed] [Green Version]
Publication | Remarks |
---|---|
Brackbill et al. [66] | Introduced the Continuum Surface Force (CSF) and density scaled CSF models. These methods are very common due to its relatively straightforward implementation in a VOF framework. |
Ubbink [67] | Proposed using a smoothed volume fraction to calculate curvature (referred to as “Smoothed CSF” in this paper). Using a smoothed volume fraction to compute the curvature instead of non-smoothed volume fraction in CSF model reduced spurious current up to one order of magnitude [56]. This method has been used in modelling condensing bubbles [32] and droplets in microfluidic devices [56]. A similar smoothening of was proposed by Heyns and Oxtoby [68]. |
Sussman and Puckett [69] | Developed a fully coupled level-set VOF (CLSVOF) method which combines the mass conservativeness of VOF method with smoothness of the level-set function to reduce spurious currents. The CLSVOF method has been used to applications such as splashing droplets [45], flow through microfluidic devices [46], wave breaking [47] and droplet evaporation [43]. Another variant of coupled level set approach is the simple coupled level-set VOF (S-CLSVOF) proposed by Albadawi et al. [70]. |
Raessi et al. [71] | Proposed a method to calculate based on advected normals. The spurious currents were lower than CSF (within the same order) while modelling cases such as stationary bubble, rising bubble and Rayleigh–Taylor instability [71]. |
Renardy and Renardy [72] | Introduced parabolic reconstruction of surface tension (PROST) algorithm which uses a least-squares fit of the interface to a quadratic surface. The spurious current produced by the algorithm is lower by two orders of magnitude compared to CSF [72]. The model was used to simulate droplet deformation including breakup [48,72]. |
Cifani et al. [61] | Implemented piecewise linear interface calculation (PLIC) algorithm (proposed by Youngs [73]) to reconstruct the interface in interFoam and managed to reduce spurious currents. |
Pilliod and Puckett [74] | Developed an efficient least squares volume-of-fluid interface reconstruction algorithm (ELIVRA) which reconstructs the interface using a least square method to fit the interface to a linear surface. |
Popinet [75] | Proposed calculating curvature using height functions. Use of height functions have reduced spurious current (slightly outperformed PROST algorithm [75]) and has been shown to model flow in microchannels [49], rising bubble [34,44] and other multiphase flows [50]. |
Raeini et al. [76] | Introduced a sharp surface force formulation to calculate the capillary force, which is then filtered to reduce the spurious currents (known as FSF model). Neglecting the filtering terms in the FSF model provides a sharp surface formulation of surface tension known as SSF, which is described in [42]. The SSF has been reported to be reduce the spurious currents by two to three orders in comparison to CSF [42]. The FSF model has been reported to provide periodic bursts in the velocity fields but lower spurious current than SSF [42]. The approach has been used to model bubbles in microfluidic devices [51] and flow through porous media [52]. |
Denner et al. [77] | Proposed the use of artificial viscosity model, which applies artificial shear stress in the tangential direction to interface, to dampen the spurious currents. The model has been used to model rising bubble and capillary instability of a water jet [77]. |
Lafaurie et al. [78] | Proposed an alternative to the CSF model, known as the Continuum Surface Stress (CSS) model, determines surface tension as divergence of stress tensor without relying on complex curvature calculations. Due to imbalance in the surface tension and pressure, CSS model can also produce spurious currents [35] which has reported to be in the same order as CSF [72]. CSS model has been used to model static droplets and rising bubbles [35], but it does not provide reliable results for falling films [41]. |
Modeling Term | Keyword | Scheme | Remarks |
---|---|---|---|
Time derivatives | ddtSchemes | Euler | First order implicit method (see [84]) |
Divergence term | vanLeerV vanLeer interfaceCompression | Modified vanLeer for vector fields (see [84]) See [85] See [63] | |
Gradient term | gradSchemes | linear | Operator with ∇ (see [84]) |
Laplacian term | laplacianSchemes | linear corrected | Operator with (see [84]) |
Others | snGradSchemes interpolationSchemes | corrected linear | Surface normal gradients (see [84]) Interpolates values (see [84]) |
Equation | Linear Solver | Smoother/Preconditioner | Tolerance |
---|---|---|---|
Pressure correction equation | PCG | DIC | 10 (based on [63]) |
Momentum equation | smoothSolver | symGaussSeidel | 10 |
Volume fraction equation | smoothSolver | symGaussSeidel | 10 |
Parameter | Value | Notes |
---|---|---|
nAlphaCorr | 2 | Number of correction [55]; typically set equal to 1 or 2 for time-dependent flows [86]. |
nAlphaSubCycles | 1 | Represents the number of sub-cycles within equation [84]. |
cAlpha () | 1 | Used for interface compression in Equation (9). |
MULESCorr | yes | Switches on semi-implicit MULES [87]. |
nLimiterIter | 3 | Number of MULES iterations over the limiter [87]. |
momentumPredictor | no | Controls solving of the momentum predictor; typically set to ’no’ for multiphase and low Reynolds number flows [84]. |
minIter | 1 | Minimum number of iterations used in momentum calculation. |
nOuterCorrectors | 1 | PISO algorithm is selected by setting this parameter equal to unity (in PIMPLE algorithm) [84]. |
nCorrectors | 3 | The number of times the PISO algorithm solves the pressure and momentum equation in each step; usually set to 2 or 3 [84]. |
nNonOrthogonalCorrectors | 0 | Used when meshes are non-orthogonal [84]. |
Cases | (kg/m) | (kg/m) | (m/s) | (m/s) | (N/m) | (m/s) * | |||
---|---|---|---|---|---|---|---|---|---|
TC1 | 1000 | 100 | 10 | 10 | 24.5 | (0 −0.98 0) | 35 | 10 | 0.286 |
TC2 | 1000 | 1 | 10 | 10 | 1.96 | (0 −0.98 0) | 35 | 125 | 3.571 |
Parameter | CSF | Smoothed CSF | SSF | [55] | Benchmark ([54]) |
---|---|---|---|---|---|
0.2366 | 0.2375 | 0.2386 | 0.2365 | 0.2419 | |
0.9632 | 0.9491 | 0.9104 | 0.9219 | 0.9270 |
Parameter | CSF | Smoothed CSF | SSF | [55] | Benchmark ([54]) |
---|---|---|---|---|---|
0.2434 | 0.2429 | 0.2427 | 0.2431 | 0.2514 | |
0.7663 | 0.7637 | 0.7502 | 0.7250 | 0.7281 | |
0.2302 | 0.2290 | 0.2260 | 0.2302 | 0.2440 | |
1.9721 | 1.9700 | 1.9729 | 1.9594 | 1.9844 |
(kg/m) | (kg/m) | (m/s) | (m/s) | (N/m) | (m/s) * | |
---|---|---|---|---|---|---|
1000 | 1 | 10 | 1.48 × 10 | 0.07 | (0 −10 0) | 0.0014 |
Surface Tension Model | (mm) | |
---|---|---|
CSF | 9.16 | −0.076 |
Smoothed CSF | Capillary height did not stabilise during simulations (see Figure 7) | |
SSF | 9.26 | −0.065 |
(kg/m) | (kg/m) | (m/s) | (m/s) | (N/m) | (m/s) |
---|---|---|---|---|---|
1000 | 1 | 10 | 1.48 × 10 | 0.07 | (0 0 0) |
Mesh | Mesh Resolution (mm) | Total Number of Cells | * | Maximum Time Step (s) |
---|---|---|---|---|
M0 | 0.5 × 0.5 | 400 | 5 | 9 × 10 |
M1 | 0.25 × 0.25 | 1600 | 10 | 3 × 10 |
M2 | 0.125 × 0.125 | 6400 | 20 | 1 × 10 |
M3 | 0.083 × 0.083 | 14400 | 30 | 6 × 10 |
Surface Tension Model | Mesh | Mass Imbalance | ||||
---|---|---|---|---|---|---|
CSF | M0 | 0.133 | 0.002 | 22.29 | −0.20 | 0 |
M1 | 0.171 | 0.002 | 23.03 | −0.18 | 0 | |
M2 | 0.174 | 0.002 | 24.06 | −0.14 | 0 | |
M3 | 0.189 | 0.003 | 24.77 | −0.12 | 0 | |
Smoothed CSF | M0 | 0.096 | 0.001 | 24.12 | −0.14 | 0 |
M1 | 0.088 | 0.001 | 25.14 | −0.10 | 0 | |
M2 | 0.062 | 0.001 | 25.19 | −0.10 | 0 | |
M3 | 0.049 | 0.001 | 26.09 | −0.07 | 0 | |
SSF | M0 | 0.045 | 0.001 | 23.95 | −0.14 | 0 |
M1 | 0.087 | 0.001 | 25.12 | −0.10 | 0 | |
M2 | 0.036 | 0.001 | 25.88 | −0.08 | 0 | |
M3 | 0.041 | 0.001 | 25.55 | −0.09 | 0 |
Surface Tension Model | based on Brackbill et al. [66] | based on Deshpande et al. [63] |
---|---|---|
CSF | 0.395950 | 0.189170 |
Smoothed CSF | 0.052188 | 0.048619 |
SSF | 0.038550 | 0.040984 |
Surface Tension Model | Mesh | Mass Imbalance | ||||
---|---|---|---|---|---|---|
CSF | M0 | 0.158 | 0.002 | 22.27 | −0.20 | 0 |
M1 | 0.279 | 0.004 | 23.09 | −0.18 | 0 | |
M2 | 0.510 | 0.007 | 24.34 | −0.13 | 0 | |
M3 | 0.723 | 0.010 | 24.47 | −0.13 | 0 | |
Smoothed CSF | M0 | 0.154 | 0.002 | 24.02 | −0.14 | 0 |
M1 | 0.122 | 0.002 | 24.97 | −0.11 | 0 | |
M2 | 0.104 | 0.001 | 25.21 | −0.10 | 0 | |
M3 | 0.075 | 0.001 | 26.03 | −0.07 | 0 | |
SSF | M0 | 0.042 | 0.001 | 24.07 | −0.14 | 0 |
M1 | 0.065 | 0.001 | 24.86 | −0.11 | 0 | |
M2 | 0.033 | 0.000 | 26.04 | −0.07 | 0 | |
M3 | 0.036 | 0.001 | 25.64 | −0.08 | 0 |
Surface Tension Model | based on Brackbill et al. [66] | based on Deshpande et al. [63] |
---|---|---|
CSF | 0.722930 | 0.723390 |
Smoothed CSF | 0.082800 | 0.075458 |
SSF | 0.040016 | 0.035903 |
© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Vachaparambil, K.J.; Einarsrud, K.E. Comparison of Surface Tension Models for the Volume of Fluid Method. Processes 2019, 7, 542. https://doi.org/10.3390/pr7080542
Vachaparambil KJ, Einarsrud KE. Comparison of Surface Tension Models for the Volume of Fluid Method. Processes. 2019; 7(8):542. https://doi.org/10.3390/pr7080542
Chicago/Turabian StyleVachaparambil, Kurian J., and Kristian Etienne Einarsrud. 2019. "Comparison of Surface Tension Models for the Volume of Fluid Method" Processes 7, no. 8: 542. https://doi.org/10.3390/pr7080542