Abstract
In this article, we describe some aspects of the diffuse interface modelling of incompressible flows, composed of three immiscible components, without phase change. In the diffuse interface methods, system evolution is driven by the minimisation of a free energy. The originality of our approach, derived from the Cahn–Hilliard model, comes from the particular form of energy we proposed in Boyer and Lapuerta (M2AN Math Model Numer Anal, 40:653–987,2006), which, among other interesting properties, ensures consistency with the two-phase model. The modelling of three-phase flows is further completed by coupling the Cahn–Hilliard system and the Navier–Stokes equations where surface tensions are taken into account through volume capillary forces. These equations are discretized in time and space paying attention to the fact that most of the main properties of the original model (volume conservation and energy estimate) have to be maintained at the discrete level. An adaptive refinement method is finally used to obtain an accurate resolution of very thin moving internal layers, while limiting the total number of cells in the grids all along the simulation. Different numerical results are given, from the validation case of the lens spreading between two phases (contact angles and pressure jumps), to the study of mass transfer through a liquid/liquid interface crossed by a single rising gas bubble. The numerical applications are performed with large ratio between densities and viscosities and three different surface tensions.
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References
Anderson D.M., McFadden G.B., Wheeler A.A.: Diffuse-interface methods in fluid mechanics. Annu. Rev. Fluid Mech. 30, 139–165 (1998)
Bonometti T., Magnaudet J.: An interface-capturing method for incompressible two-phase flows. Validation and application to bubble dynamics. Int. J. Multiph. Flow 33, 109–133 (2007)
Boyer F.: A theoretical and numerical model for the study of incompressible mixture flows. Comput. Fluids 31, 41–68 (2002)
Boyer F., Lapuerta C.: Study of a three component Cahn-Hilliard flow model. M2AN, Math. Model. Numer. Anal. 40, 653–687 (2006)
Boyer, F., Minjeaud, S.: Numerical schemes for a three component Cahn-Hilliard model, in preparation (2008)
Boyer, F., Lapuerta, C., Minjeaud, S., Piar, B.: A multigrid method with local adaptive refinement, application to a ternary Cahn-Hilliard model. ESAIM: Proc. (2009)
Fichot F., Meekunnasombat P., Belloni J., Duval F., Garcia A., Quintard M.: Two-phase flows in porous media: Prediction of pressure drops using a diffuse interface mathematical description. Nucl. Eng. Des. 237, 1887–1898 (2007)
Greene G.A., Chen J.C., Conlin M.T.: Onset of entrainment between immiscible liquid layers due to rising gas bubbles. Int. J. Heat Mass Transf. 31, 1309–1317 (1988)
Greene G.A., Chen J.C., Conlin M.T.: Bubble induced entrainment between stratified liquid layers. Int. J. Heat Mass Transf. 34, 149–157 (1991)
Guermond J.-L., Quartapelle L.: A projection FEM for variable density incompressible flows. J. Comput. Phys. 165, 167–188 (2000)
Jacqmin D.: Calculation of two-phase Navier-Stokes flows using phase-field modelling. J. Comput. Phys. 155, 96–127 (1999)
Jamet D., Torres D., Brackbill J.U.: On the theory and computation of surface tension: the elimination of parasitic currents through energy conservation in the second-gradient method. J. Comput. Phys. 182, 262–276 (2002)
Keller A.A., Chen M.: Effect of spreading coefficient on three-phase relative permeability of NAPL. Water Resour. Res. 39, 1288 (2003)
Kim J.: A continuous surface tension force formulation for diffuse-interface models. J. Comput. Phys. 204, 784–804 (2005)
Kim J., Lowengrub J.: Phase field modeling and simulation of three-phase flows. Int. Free Bound. 7, 435–466 (2005)
Kim J.: Phase field computations for ternary fluid flows. Comput. Methods Appl. Mech. Eng. 196, 45–48 (2007)
Kim J., Kang K., Lowengrub J.: Conservative multigrid methods for ternary Cahn-Hilliard systems. Commun. Math. Sci. 2, 53–77 (2004)
Krysl P., Grinspun E., Schröder P.: Natural hierarchical refinement for finite element methods. Int. J. Numer. Meth. Eng. 56, 1109–1124 (2003)
Lowengrub J., Truskinovsky L.: Quasi-incompressible Cahn-Hilliard fluids and topological transitions. Proc. R. Soc. Lond. A 454, 2617–2654 (1998)
Mani V., Mohanty K.K.: Effect of the spreading coefficient on three-phase flow in porous media. J. Colloid Interf. Sci. 187, 45–56 (1997)
PELICANS, Collaborative Development Environment: https://gforge.irsn.fr/gf/project/pelicans/
Rowlinson J.S., Widom B.: Molecular Theory of Capillarity. Clarendon Press, Oxford (1982)
Yue P., Feng J.J., Liu C., Shen J.: A Diffuse-interface method for simulating two-phase flows of complex fluids. J. Fluid Mech. 515, 293–317 (2004)
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Boyer, F., Lapuerta, C., Minjeaud, S. et al. Cahn–Hilliard/Navier–Stokes Model for the Simulation of Three-Phase Flows. Transp Porous Med 82, 463–483 (2010). https://doi.org/10.1007/s11242-009-9408-z
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DOI: https://doi.org/10.1007/s11242-009-9408-z