An unconditionally energy stable method for binary incompressible heat conductive fluids based on the phase-field model. (English) Zbl 1524.76143
Summary: This paper proposes an unconditionally energy stable method for incompressible heat conductive fluids under the phase-field framework. We combine the complicated system by the Navier-Stokes equation, Cahn-Hilliard equation, and heat transfer equation. A Crank-Nicolson type scheme is employed to discretize the governing equation with the second-order temporal accuracy. The unconditional energy stability of the proposed scheme is proved, which means that a significantly larger time step can be used. The Crank-Nicolson type discrete framework is applied to obtain the second-order temporal accuracy. We perform the biconjugate gradient method and Fourier transform method to solve the discrete system. Several computational tests are performed to show the efficiency and robustness of the proposed method.
MSC:
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 35Q35 | PDEs in connection with fluid mechanics |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 76T06 | Liquid-liquid two component flows |
Keywords:
unconditionally energy stable; two-phase thermodynamic flow; phase-field model; Navier-Stokes equationReferences:
| [1] | Discacciati, M.; Quarteroni, A., Navier-Stokes/Darcy coupling: modeling, analysis, and numerical approximation, Rev. Mat. Complut., 22, 315-426 (2009) · Zbl 1172.76050 |
| [2] | Li, Y.; Wang, K.; Yu, Q.; Xia, Q.; Kim, J., Unconditionally energy stable schemes for fluid-based topology optimization, Commun. Nonlinear Sci., 111, Article 106433 pp. (2022) · Zbl 07526842 |
| [3] | Thomas, S., Enhanced oil recovery-an overview, Oil Gas Sci. Technol., 63, 9-19 (2008) |
| [4] | You, B.; Xia, Q., Continuous data assimilation algorithm for the two dimensional Cahn-Hilliard-Navier-Stokes system, Appl. Math. Optim., 85, 1-19 (2022) · Zbl 1487.35331 |
| [5] | Dam, J.; Feddes, R. A., Numerical simulation of infiltration, evaporation and shallow groundwater levels with the Richards equation, J. Hydrol., 233, 72-85 (2000) |
| [6] | Taylor, C.; Stefan, H., Shallow groundwater temperature response to climate change and urbanization, J. Hydrol., 375, 601-612 (2009) |
| [7] | Li, Y.; Liu, R.; Xia, Q.; He, C.; Li, Z., First- and second-order unconditionally stable direct discretization methods for multi-component Cahn-Hilliard system on surfaces, J. Comput. Appl. Math., 401, Article 113778 pp. (2022) · Zbl 1503.65178 |
| [8] | Freistuhler, H.; Kotschote, M., Phase-field and Korteweg-type models for the time-dependent flow of compressible two-phase fluids, Arch. Ration. Mech. Anal., 224, 1-20 (2017) · Zbl 1366.35130 |
| [9] | Wang, Z.; Zheng, X.; Chryssostomidis, C.; Karniadakis, G. E., A phase-field method for boiling heat transfer, J. Comput. Phys., 435, Article 110239 pp. (2021) · Zbl 07503725 |
| [10] | Chen, W.; Han, D.; Wang, X.; Zhang, Y., Uniquely solvable and energy stable decoupled numerical schemes for the Cahn-Hilliard-Navier-Stokes-Darcy- Boussinesq system, J. Sci. Comput., 85, 45 (2020) · Zbl 1458.35323 |
| [11] | Salimi, M. R.; Taeibi-Rahni, M.; Jam, F., Heat transfer analysis of a porously covered heated square cylinder, using a hybrid Navier-Stokes lattice Boltzmann numerical method, Int. J. Therm. Sci., 91, 59-75 (2015) |
| [12] | Zheng, X.; Babaee, H.; Dong, S.; Chryssostomidis, C.; Karniadakis, G. E., A phase-field method for 3D simulation of two-phase heat transfer, Int. J. Heat Mass Transf., 82, 282-298 (2015) |
| [13] | Hu, H. H.; Patankar, N. A.; Zhu, M. Y., Direct numerical simulations of fluid-solid systems using the arbitrary Lagrangian-Eulerian technique, J. Comput. Phys., 169, 427-462 (2001) · Zbl 1047.76571 |
| [14] | Chen, J.; Sun, S.; Wang, X.-P., A numerical method for a model of two-phase flow in a coupled free flow and porous media system, J. Comput. Phys., 268, 1-16 (2014) · Zbl 1349.76187 |
| [15] | Pilliod, J. E.; Puckett, E. G., Second-order accurate volume-of-fluid algorithms for tracking material interfaces, J. Comput. Phys., 199, 465-502 (2004) · Zbl 1126.76347 |
| [16] | Welch, S. W.J.; Wilson, J., A volume of fluid based method for fluid flows with phase change, J. Comput. Phys., 160, 662-682 (2000) · Zbl 0962.76068 |
| [17] | Osher, S.; Fedkiw, R. P., Level set methods: an overview and some recent results, J. Comput. Phys., 169, 475-502 (2001) · Zbl 0988.65093 |
| [18] | Adalsteinsson, D.; Sethian, J. A., A fast level set method for propagating interfaces, J. Comput. Phys., 118, 269-277 (1995) · Zbl 0823.65137 |
| [19] | Xia, Q.; Kim, J.; Li, Y., Modeling and simulation of multi-component immiscible flows based on a modified Cahn-Hilliard equation, Eur. J. Mech. B, Fluids, 95, 194-204 (2022) · Zbl 1495.76123 |
| [20] | Xia, Q.; Yu, Q.; Li, Y., A second-order accurate, unconditionally energy stable numerical scheme for binary fluid flows on arbitrarily curved surfaces, Comput. Methods Appl. Mech. Eng., 384, Article 113987 pp. (2021) · Zbl 1506.76116 |
| [21] | Anderson, D. M.; Mcfadden, G. B.; Wheeler, A. A., Diffuse-interface methods in fluid mechanics, Annu. Rev. Fluid Mech., 30, 139-165 (1998) · Zbl 1398.76051 |
| [22] | Li, Y.; Xia, Q.; Lee, C.; Kim, S.; Kim, J., A robust and efficient fingerprint image restoration method based on a phase-field model, Pattern Recognit., 123, Article 108405 pp. (2022) |
| [23] | Li, Y.; Xia, Q.; Yoon, S.; Lee, C.; Lu, B.; Kim, J., Simple and efficient volume merging method for triply periodic minimal structures, Comput. Phys. Commun., 264, Article 107956 pp. (2021) |
| [24] | Wise, S. M.; Wang, C.; Lowengrub, J. S., An energy-stable and convergent finite-difference scheme for the phase field crystal equation, SIAM J. Numer. Anal., 47, 2269-2288 (2009) · Zbl 1201.35027 |
| [25] | Shen, J.; Wang, C.; Wang, X.; Wise, S. M., Second-order convex splitting schemes for gradient flows with Ehrlich-Schwoebel type energy: application to thin film epitaxy, SIAM J. Numer. Anal., 50, 105-125 (2012) · Zbl 1247.65088 |
| [26] | Shen, J.; Yang, X., Numerical approximations of Allen-Cahn and Cahn-Hilliard equations, Discrete Contin. Dyn. Syst., 28, 1669-1691 (2010) · Zbl 1201.65184 |
| [27] | Shen, J.; Xu, J.; Yang, J., The scalar auxiliary variable (SAV) approach for gradient flows, J. Comput. Phys., 353, 407-416 (2018) · Zbl 1380.65181 |
| [28] | Yang, X., Linear, first and second-order, unconditionally energy stable numerical schemes for the phase field model of homopolymer blends, J. Comput. Phys., 327, 294-316 (2016) · Zbl 1373.82106 |
| [29] | Yue, P.; Feng, J.; Liu, C.; Shen, J., A diffuse-interface method for simulating two-phase flows of complex fluids, J. Fluid Mech., 515, 293-317 (2004) · Zbl 1130.76437 |
| [30] | Sayevand, K.; Tenreiro Machado, J.; Moradi, V., A new non-standard finite difference method for analyzing the fractional Navier-Stokes equations, Comput. Math. Appl., 78, 1681-1694 (2019) · Zbl 1442.65177 |
| [31] | Nikan, O.; Avazzadeh, Z.; Tenreiro Machado, J. A., Numerical treatment of microscale heat transfer processes arising in thin films of metals, Int. J. Heat Mass Transf., 132, Article 105892 pp. (2022) |
| [32] | Huang, Z.; Lin, G.; Ardekani, A. M., A consistent and conservative Phase-Field model for thermo-gas-liquid-solid flows including liquid-solid phase change, J. Comput. Phys., 449, Article 110795 pp. (2022) · Zbl 07524791 |
| [33] | Gong, Y.; Liu, X.; Wang, Q., Fully discretized energy stable schemes for hydrodynamic equations governing two phase viscous fluid flows, J. Sci. Comput., 69, 921-945 (2016) · Zbl 1397.76093 |
| [34] | Han, D.; Wang, X., A second order in time, uniquely solvable, unconditionally stable numerical scheme for Cahn-Hilliard-Navier-Stokes equation, J. Comput. Phys., 290, 139-156 (2015) · Zbl 1349.76213 |
| [35] | Guo, Z.; Lin, P.; Lowengrub, J. S., A numerical method for the quasi-incompressible Cahn-Hilliard-Navier-Stokes equations for variable density flows with a discrete energy law, J. Comput. Phys., 276, 486-507 (2014) · Zbl 1349.76057 |
| [36] | Han, D.; Brylev, A.; Yang, X.; Tan, Z., Numerical analysis of second order, fully discrete energy stable schemes for phase field models of two phase incompressible flows, J. Sci. Comput., 70, 965-989 (2017) · Zbl 1397.76070 |
| [37] | Shen, J.; Yang, X., Decoupled energy stable schemes for phase-field models of two-phase complex fluids, SIAM J. Sci. Comput., 36, B122-B145 (2014) · Zbl 1288.76057 |
| [38] | Shen, J.; Yang, X., An efficient moving mesh spectral method for the phase field model of two phase flows, J. Comput. Phys., 228, 2978-2992 (2009) · Zbl 1159.76032 |
| [39] | Shen, J.; Yang, X., A phase-field model and its numerical approximation for two-phase incompressible flows with different densities and viscosities, SIAM J. Sci. Comput., 32, 1159-1179 (2010) · Zbl 1410.76464 |
| [40] | Chen, W.; Han, D.; Wang, X.; Zhang, Y., Conservative unconditionally stable decoupled numerical schemes for the Cahn-Hilliard-Navier-Stokes-Darcy-Boussinesq system, Numer. Methods Partial Differ. Equ. (2021) |
| [41] | Guo, Z.; Lin, P.; Lowengrub, J. S.; Wise, S. M., Mass conservative and energy stable finite difference methods for the quasi-incompressible Navier-Stokes-Cahn-Hilliard system: primitive variable and projection-type schemes, Comput. Methods Appl. Mech. Eng., 326, 144-174 (2017) · Zbl 1439.76121 |
| [42] | Chen, W.; Han, D.; Wang, X., Uniquely solvable and energy stable decoupled numerical schemes for the Cahn-Hilliard-Stokes-Darcy system for two-phase flows in karstic geometry, Numer. Math., 137, 229-255 (2017) · Zbl 1476.76089 |
| [43] | Li, Y.; Choi, J. I.; Kim, J., Multi-component Cahn-Hilliard system with different boundary conditions in complex domains, J. Comput. Phys., 323, 1-16 (2016) · Zbl 1415.65219 |
| [44] | Chorin, A. J., A numerical method for solving incompressible viscous flow problems, J. Comput. Phys., 2, 12-26 (1967) · Zbl 0149.44802 |
| [45] | Cahn, J. W., On spinodal decomposition, Acta Metall., 9, 795 (1961) |
| [46] | Li, Y.; Jeong, D.; Shin, J.; Kim, J., A conservative numerical method for the Cahn-Hilliard equation with Dirichlet boundary conditions in complex domain, Comput. Math. Appl., 65, 102-115 (2013) · Zbl 1268.65114 |
| [47] | Dodd, M. S.; Ferrante, A., A fast pressure-correction method for incompressible two-fluid flows, J. Comput. Phys., 273, 416-434 (2014) · Zbl 1351.76161 |
| [48] | Liu, J.; Dedè, L.; Evans, J. A.; Borden, M. J.; Hughes, T. J.R., Isogeometric analysis of the advective Cahn-Hilliard equation: spinodal decomposition under shear flow, J. Comput. Phys., 242, 321-350 (2013) · Zbl 1311.76069 |
| [49] | Kim, J.; Lowengrub, J., Phase field modeling and simulation of three-phase flows, Interfaces Free Bound., 7, 435-466 (2005) · Zbl 1100.35088 |
| [50] | Yang, J.; Kim, J., A phase-field model and its efficient numerical method for two-phase flows on arbitrarily curved surfaces in 3D space, Comput. Methods Appl. Mech. Eng., 372, Article 113382 pp. (2020) · Zbl 1506.76104 |
| [51] | Wallis, G.; Makkenchery, S., The hanging film phenomenon in vertical annular two-phase flow, J. Fluids Eng., 96, 297-298 (1974) |
| [52] | Ranocha, H.; Dalcin, L.; Parsani, M., Fully discrete explicit locally entropy-stable schemes for the compressible Euler and Navier-Stokes equations, Comput. Math. Appl., 80, 1343-1359 (2020) · Zbl 1524.65677 |
| [53] | Témam, R., Sur l’approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires II, Arch. Ration. Mech. Anal., 33, 377-385 (1969) · Zbl 0207.16904 |
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