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Computation of multiphase systems with phase field models. (English) Zbl 1076.76517
Summary: Phase field models offer a systematic physical approach for investigating complex multiphase systems behaviors such as near-critical interfacial phenomena, phase separation under shear, and microstructure evolution during solidification. However, because interfaces are replaced by thin transition regions (diffuse interfaces), phase field simulations require resolution of very thin layers to capture the physics of the problems studied. This demands robust numerical methods that can efficiently achieve high resolution and accuracy, especially in three dimensions. We present here an accurate and efficient numerical method to solve the coupled Cahn-Hilliard/Navier-Stokes system, known as Model H, that constitutes a phase field model for density-matched binary fluids with variable mobility and viscosity. The numerical method is a time-split scheme that combines a novel semi-implicit discretization for the convective Cahn-Hilliard equation with an innovative application of high-resolution schemes employed for direct numerical simulations of turbulence. This new semi-implicit discretization is simple but effective since it removes the stability constraint due to the nonlinearity of the Cahn-Hilliard equation at the same cost as that of an explicit scheme. It is derived from a discretization used for diffusive problems that we further enhance to efficiently solve flow problems with variable mobility and viscosity. Moreover, we solve the Navier-Stokes equations with a robust time-discretization of the projection method that guarantees better stability properties than those for Crank-Nicolson-based projection methods. For channel geometries, the method uses a spectral discretization in the streamwise and spanwise directions and a combination of spectral and high order compact finite difference discretizations in the wall normal direction. The capabilities of the method are demonstrated with several examples including phase separation with, and without, shear in two and three dimensions. The method effectively resolves interfacial layers of as few as three mesh points. The numerical examples show agreement with analytical solutions and scaling laws, where available, and the 3D simulations, in the presence of shear, reveal rich and complex structures, including strings.

MSC:
76D05 Navier-Stokes equations for incompressible viscous fluids
76M25 Other numerical methods (fluid mechanics) (MSC2010)
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35Q30 Navier-Stokes equations
Software:
FFTW
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[1] Cahn, J.W.; Hilliard, J.E., Free energy of a nonuniform system I, J. chem. phys., 28, 258, (1958)
[2] Cahn, J.W.; Hilliard, J.E., Free energy of a nonuniform system III, J. chem. phys., 31, 688, (1959)
[3] Hohenberg, P.C.; Halperin, B.I., Theory of dynamic critical phenomena, Rev. mod. phys., 49, 3, 435, (1977)
[4] Lowengrub, J.; Truskinovsky, L., Quasi-incompressible cahn – hilliard fluids and topological transitions, Proc. R. soc. lond. A, 454, 2617, (1998) · Zbl 0927.76007
[5] Lele, S.K., Compact finite difference schemes with spectral-like resolution, J. comput. phys., 103, 16, (1992) · Zbl 0759.65006
[6] Chella, R.; Viñals, V., Mixing of a two-phase fluid by a cavity flow, Phys. rev. E, 53, 3832, (1996)
[7] Jacqmin, D., Calculation of two phase Navier Stokes flows using phase-field modeling, J. comput. phys., 115, 96, (1999) · Zbl 0966.76060
[8] Kendon, V.M.; Cates, M.E.; Barraga, I.P.; Desplat, J.-C.; Blandon, P., Inertial effects in three-dimensional spinodal decomposition of a symmetric binary fluid mixture: a lattice Boltzmann study, J. fluid mech., 440, 147, (2001) · Zbl 1049.76052
[9] Wu, Y.; Skrdla, H.; Lookman, T.; Chen, S., Spinodal decomposition in binary fluids under shear flow, Physica A, 239, 428-436, (1997)
[10] Penrose, O.; Fife, P., Thermodynamically consistent models of phase-field type for the kinetics of phase transitions, Physica D, 43, 44, (1990) · Zbl 0709.76001
[11] Bates, P.W.; Fife, P.C., The dynamics of nucleation for the cahn – hilliard equation, SIAM J. appl. math., 53, 990, (1993) · Zbl 0788.35061
[12] Elliot, C.M., The cahn – hilliard model for the kinetics of phase separation, (), 35-72
[13] Gurtin, M.E.; Polignone, D.; Viñals, J., Two-phase binary fluids and immiscible fluids described by an order parameter, Math. models meth. appl. sci., 6, 6, 815, (1996) · Zbl 0857.76008
[14] Bray, A.J., Theory of phase-ordering kinetics, Adv. phys., 43, 3, 357-459, (1994)
[15] van der Waals, J.D., The thermodynamic theory of capillarity flow under the hypothesis of a continuous variation of density (in Dutch), Verhandel/konink. akad. weten., 1, 8, (1879)
[16] Langer, J.S.; Baron, M.; Miller, H., New computational method in theory of spinodal decomposition, Phys. rev. A, 11, 4, 1417, (1975)
[17] Cook, A.W.; Dimotakis, P.E., Transition stages of rayleigh – taylor instability between miscible fluids, J. fluid mech., 443, 69, (2001) · Zbl 1015.76037
[18] Buell, J.C., A hybrid numerical-method for 3-dimensional spatially-developing free shear flows, J. comput. phys., 95, 2, 313-338, (1991) · Zbl 0725.76072
[19] Douglas, J.J.; Dupont, T., Alternating-direction Galerkin methods on rectangles, (), 133-213
[20] Gottlieb, D.; Orszag, S.A., ()
[21] Ceniceros, H.D., A semi-implicit moving mesh method for the focusing Schrödinger equation, Commun. pure appl. anal., 1, 1, (2002) · Zbl 1010.35098
[22] Ascher, U.M.; Ruuth, S.J.; Wetton, B.T.R., Implicit – explicit methods for time dependent partial differential equations, SIAM J. numer. anal., 32, 797, (1995) · Zbl 0841.65081
[23] D.J. Eyre, An unconditionally stable one-step scheme for gradient systems (preprint)
[24] P. Smereka, Semi-implicit level set methods for motion by mean curvature and surface diffusion (preprint) · Zbl 1035.65098
[25] Zhu, J.; Chen, L.-Q.; Shen, J.; Tikare, V., Coarsening kinetics from a variable-mobility cahn – hilliard equation: application of a semi-implicit Fourier spectral method, Phys. rev. E, 60, 4, 3564-3572, (1999)
[26] Karniadakis, G.E.; Israeli, M.; Orszag, S.A., High-order splitting methods for the incompressible navier – stokes equations, J. comput. phys., 97, 414-443, (1991) · Zbl 0738.76050
[27] Sussman, M.; Smereka, P.; Osher, S., A level set approach for computing solutions to incompressible two-phase flow, J. comput. phys., 114, 146-159, (1994) · Zbl 0808.76077
[28] Brackbill, J.U.; Kothe, D.B.; Zemach, C., A continuum method for modeling surface tension, J. comput. phys., 100, 335, (1992) · Zbl 0775.76110
[29] Taylor, G.I., The formation of emulsions in definable fields of flows, Proc. R. soc. lond. A, 146, 501-523, (1934)
[30] Shapira, M.; Haber, S., Low Reynolds number motion of a droplet in shear flow including wall effects, Int. J. multiphase flow, 16, 2, 305-321, (1990) · Zbl 1134.76668
[31] Rallison, J.M., The deformation of small viscous drops and bubbles in shear flow, Annu. rev. fluid mech., 16, 45-66, (1984) · Zbl 0604.76079
[32] Roths, T.; Friedrich, C.; Marth, M.; Honerkamp, J., Dynamics and rheology of the morphology of immiscible polymer blends—on modeling and simulation, Rheologica acta, 41, 3, 211-222, (2002)
[33] Puckett, E.G.; Almgren, J.B.B.A.S.; Marcus, D.L.; Rider, W.J., A high-order projection method for tracking fluid interfaces in variable density incompressible flows, J. comput. phys., 130, 2, 269-282, (1997) · Zbl 0872.76065
[34] Copetti, M.; Elliot, C., Kinetics of phase decomposition process: numerical solutions to the cahn – hilliard equation, Mater. sci. technol., 6, 273, (1990)
[35] Chakrabarti, A.; Toral, R.; Gunton, J.D., Late-stage coarsening for off-critical quenches: scaling functions and the growth law, Phys. rev. E, 47, 3025-3038, (1993)
[36] Onuki, A., Phase transitions of fluids in shear flow, J. phys.: condes. matter, 9, 6119, (1997)
[37] Frischknecht, A., Stability of cylindrical domains in phase-separating binary fluids in shear flow, Phys. rev. E, 58, 3, 3495-3514, (1998)
[38] Hashimoto, T.; Matsuzaka, K.; Moses, E.; Onuki, A., String phase in phase-separating fluids under shear flow, Phys. rev. lett., 74, 1, 126-129, (1995)
[39] Migler, K.B., String formation in sheared polymer blends: coalescence, breakup, and finite size effects, Phys. rev. lett., 86, 6, 1023-1026, (2001)
[40] Frigo, M.; Johnson, S.G., FFTW: an adaptive software architecture for the fft, ICASSP conf. proc., 3, 1381-1384, (1998)
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