# American Institute of Mathematical Sciences

December  2020, 7(2): 291-312. doi: 10.3934/jcd.2020012

## An energy dissipative semi-discrete finite-difference method on staggered meshes for the 3D compressible isothermal Navier–Stokes–Cahn–Hilliard equations

 1 Keldysh Institute of Applied Mathematics, Miusskaya sqr., 4, 125047 Moscow, Russia 2 National Research University Higher School of Economics, Pokrovskii bd. 11, 109028 Moscow, Russia, Keldysh Institute of Applied Mathematics, Miusskaya sqr., 4, 125047 Moscow, Russia

* Corresponding author

Received  September 2019 Published  July 2020

Fund Project: The study was supported by the Russian Science Foundation, project no. 19-11-00169

We consider the initial-boundary value problem for the 3D regularized compressible isothermal Navier–Stokes–Cahn–Hilliard equations describing flows of a two-component two-phase mixture taking into account capillary effects. We construct a new numerical semi-discrete finite-difference method using staggered meshes for the main unknown functions. The method allows one to improve qualitatively the computational flow dynamics by eliminating the so-called parasitic currents and keeping the component concentration inside the physically reasonable range $(0,1)$. This is achieved, first, by discretizing the non-divergent potential form of terms responsible for the capillary effects and establishing the dissipativity of the discrete full energy. Second, a logarithmic (or the Flory–Huggins potential) form for the non-convex bulk free energy is used. The regularization of equations is accomplished to increase essentially the time step of the explicit discretization in time. We include 3D numerical results for two typical problems that confirm the theoretical predictions.

Citation: Vladislav Balashov, Alexander Zlotnik. An energy dissipative semi-discrete finite-difference method on staggered meshes for the 3D compressible isothermal Navier–Stokes–Cahn–Hilliard equations. Journal of Computational Dynamics, 2020, 7 (2) : 291-312. doi: 10.3934/jcd.2020012
##### References:
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Blunt, Multiphase Flow in Permeable Media: A Pore-Scale Perspective, Cambridge University Press, 2017.  doi: 10.1017/9781316145098.  Google Scholar [10] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.   Google Scholar [11] B. N. Chetverushkin, Kinetic Schemes and Quasi-Gasdynamic System of Equations, CIMNE, Barcelona, 2008. Google Scholar [12] K. Connington and T. Lee, A review of spurious currents in the lattice Boltzmann method for multiphase flows, J. Mech. Sci. Technology, 26 (2012), 3857-3863.  doi: 10.1007/s12206-012-1011-5.  Google Scholar [13] M. I. M. Copetti and C. M. Elliott, Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy, Numer. Math., 63 (1992), 39-65.  doi: 10.1007/BF01385847.  Google Scholar [14] A. Debussche and L. 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Comm., 219 (2017), 20-34.  doi: 10.1016/j.cpc.2017.05.002.  Google Scholar [19] L. Goudenège, D. Martin and G. Vial, High order finite element calculations for the Cahn-Hilliard equation, J. Sci. Comput., 52 (2012), 294-321.  doi: 10.1007/s10915-011-9546-7.  Google Scholar [20] J.-L. Guermond and B. Popov, Viscous regularization of the Euler equations and entropy principles, SIAM J. Appl. Math., 74 (2014), 284-305.  doi: 10.1137/120903312.  Google Scholar [21] Z. Guo, P. Lin, J. Lowengrub and S. M. Wise, Mass conservative and energy stable finite difference methods for the quasi-incompressible Navier–Stokes–Cahn–Hilliard system: Primitive variable and projection-type schemes, Comput. Meth. Appl. Mech. Eng., 326 (2017), 144-174.  doi: 10.1016/j.cma.2017.08.011.  Google Scholar [22] D. J. E. Harvie, M. R. Davidson and M. Rudman, An analysis of parasitic current generation in volume of fluid simulations, Appl. Math. Model., 30 (2006), 1056-1066.  doi: 10.1016/j.apm.2005.08.015.  Google Scholar [23] P. C. Hiemenz and T. P. Lodge, Polymer Chemistry, 2$^{nd}$ edition, CRC Press, 2007.   Google Scholar [24] D. Jacqmin, Calculation of two-phase Navier–Stokes flows using phase-field modeling, J. Comput. Phys., 155 (1999), 96-127.  doi: 10.1006/jcph.1999.6332.  Google Scholar [25] D. Jamet, D. Torres and J. U. Brackbill, 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 (2002), 262-276.  doi: 10.1006/jcph.2002.7165.  Google Scholar [26] J. Liu, Thermodynamically Consistent Modeling and Simulation of Multiphase Flows, PhD dissertation, the University of Texas at Austin, 2014. Google Scholar [27] J. Lowengrub and L. Truskinovsky, Quasi-incompressible Cahn-Hilliard fluids and topological transitions, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 454 (1978), 2617-2654.  doi: 10.1098/rspa.1998.0273.  Google Scholar [28] S. Minjeaud, An adaptive pressure correction method without spurious velocities for diffuse-interface models of incompressible flows, J. Comput. Phys., 236 (2013), 143-156.  doi: 10.1016/j.jcp.2012.11.022.  Google Scholar [29] N. Provatas and K. Elde, Phase-Field Methods in Material Science and Engineering, Willey-VCH, Weinheim, 2010. doi: 10.1002/9783527631520.  Google Scholar [30] Yu. V. Sheretov, Continuum Dynamics under Spatiotemporal Averaging, RKhD, Moscow-Izhevsk, 2009 [in Russian]. Google Scholar [31] M. Svärd, A new Eulerian model for viscous and heat conducting compressible flows, Phys. A, 506 (2018), 350-375.  doi: 10.1016/j.physa.2018.03.097.  Google Scholar [32] G. Tierra and F. Guillén-González, Numerical methods for solving the Cahn-Hilliard equation and its applicability to related energy-based models, Arch. Comput. Meth. Eng., 22 (2015), 269-289.  doi: 10.1007/s11831-014-9112-1.  Google Scholar [33] P. Yue, C. Zhou and J. J. Feng, Spontaneous shrinkage of drops and mass conservation in phase-field simulations, J. Comput. Phys., 223 (2007), 1-9.  doi: 10.1016/j.jcp.2006.11.020.  Google Scholar [34] I. Zacharov et al., "Zhores" – Petaflops supercomputer for data-driven modeling, machine learning and artificial intelligence installed in Skolkovo Institute of Science and Technology, preprint, arXiv: 1902.07490. Google Scholar [35] A. Zlotnik, On the energy dissipative spatial discretization of the barotropic quasi-gasdynamic and compressible Navier–Stokes systems of equations in polar coordinates, Russ. J. Numer. Anal. Math. Model., 33 (2018), 199-210.  doi: 10.1515/rnam-2018-0017.  Google Scholar [36] A. A. Zlotnik, On conservative spatial discretizations of the barotropic quasi-gasdynamic system of equations with a potential body force, Comput. Math. Math. Phys., 56 (2016), 303-319.  doi: 10.1134/S0965542516020160.  Google Scholar [37] A. A. Zlotnik and T. A. Lomonosov, Conditions for $L^2$-dissipativity of linearized explicit difference schemes with regularization for 1D barotropic gas dynamics equations, Comput. Math. Math. Phys., 59 (2019), 452-464.  doi: 10.1134/S0965542519030151.  Google Scholar [38] A. A. Zlotnik and T. A. Lomonosov, On $L^2$-dissipativity of a linearized explicit finite-difference scheme with quasi-gasdynamic regularization for the barotropic gas dynamics system of equations, Dokl. Math., 101 (2020) (in press). Google Scholar

show all references

##### References:
 [1] M. O. Abu-Al-Saud, S. Popinet and H. A. Tchelepi, A conservative and well-balanced surface tension model, J. Comput. Phys., 371 (2018), 896-913.  doi: 10.1016/j.jcp.2018.02.022.  Google Scholar [2] D. M. Anderson, G. B. McFadden and A. A. Wheeler, Diffuse-interface methods in fluid mechanics, Ann. Rev. Fluid Mech., 30 (1998), 139-165.  doi: 10.1146/annurev.fluid.30.1.139.  Google Scholar [3] A. Arakawa and V. R. Lamb, Computational design of the basic dynamical processes of the UCLA general circulation model, Methods Comput. Phys., 17 (1977), 173-265.   Google Scholar [4] V. A. Balashov and E. B. Savenkov, Quasi-hydrodynamic model of multiphase fluid flows taking into account phase interaction, J. Appl. Mech. Tech. Phys., 59 (2018), 434-444.  doi: 10.1134/S0021894418030069.  Google Scholar [5] V. A. Balashov and E. B. Savenkov, Thermodynamically consistent spatial discretization of the one-dimensional regularized system of the Navier–Stokes–Cahn–Hilliard equations, J. Comput. Appl. Math., 372 (2020), 112743, 16 pp. doi: 10.1016/j.cam.2020.112743.  Google Scholar [6] V. Balashov, E. Savenkov and A. Zlotnik, Numerical method for 3D two-component isothermal compressible flows with application to digital rock physics, Russian J. Numer. Anal. Math. Modelling, 34 (2019), 1-13.  doi: 10.1515/rnam-2019-0001.  Google Scholar [7] V. Balashov and A. Zlotnik, An energy dissipative spatial discretization for the regularized compressible Navier–Stokes–Cahn–Hilliard system of equations, Math. Model. Anal., 25 (2020), 110-129.  doi: 10.3846/mma.2020.10577.  Google Scholar [8] V. Balashov, A. Zlotnik and E. Savenkov, Analysis of a regularized model for the isothermal two-component mixture with the diffuse interface, Russian J. Numer. Anal. Math. Modelling, 32 (2017), 347-358.  doi: 10.1515/rnam-2017-0033.  Google Scholar [9] M. J. Blunt, Multiphase Flow in Permeable Media: A Pore-Scale Perspective, Cambridge University Press, 2017.  doi: 10.1017/9781316145098.  Google Scholar [10] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.   Google Scholar [11] B. N. Chetverushkin, Kinetic Schemes and Quasi-Gasdynamic System of Equations, CIMNE, Barcelona, 2008. Google Scholar [12] K. Connington and T. Lee, A review of spurious currents in the lattice Boltzmann method for multiphase flows, J. Mech. Sci. Technology, 26 (2012), 3857-3863.  doi: 10.1007/s12206-012-1011-5.  Google Scholar [13] M. I. M. Copetti and C. M. Elliott, Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy, Numer. Math., 63 (1992), 39-65.  doi: 10.1007/BF01385847.  Google Scholar [14] A. Debussche and L. Dettori, On the Cahn-Hilliard equation with a logarithmic free energy, Nonlinear Anal., 24 (1995), 1491-1514.  doi: 10.1016/0362-546X(94)00205-V.  Google Scholar [15] A. Yu. Demianov, O. Yu. Dinariev and N. V. Evseev, Introduction to the Density Functional Method in Hydrodynamics, Fizmatlit, Moscow, 2014. Google Scholar [16] T. G. Elizarova, Quasi-Gas Dynamic Equations, Computational Fluid and Solid Mechanics. Springer, Dordrecht, 2009. doi: 10.1007/978-3-642-00292-2.  Google Scholar [17] F. Frank, C. Liu, F. O. Alpak and B. Riviere, A finite volume/discontinuous Galerkin method for the advective Cahn-Hilliard equation with degenerate mobility on porous domains stemming from micro-CT imaging, Comput. Geosci., 22 (2018), 543-563.  doi: 10.1007/s10596-017-9709-1.  Google Scholar [18] Y. Gong, J. Zhao and Q. Wang, An energy stable algorithm for a quasi-incompressible hydrodynamic phase-field model of viscous fluid mixtures with variable densities and viscosities, Comput. Phys. Comm., 219 (2017), 20-34.  doi: 10.1016/j.cpc.2017.05.002.  Google Scholar [19] L. Goudenège, D. Martin and G. Vial, High order finite element calculations for the Cahn-Hilliard equation, J. Sci. Comput., 52 (2012), 294-321.  doi: 10.1007/s10915-011-9546-7.  Google Scholar [20] J.-L. Guermond and B. Popov, Viscous regularization of the Euler equations and entropy principles, SIAM J. Appl. Math., 74 (2014), 284-305.  doi: 10.1137/120903312.  Google Scholar [21] Z. Guo, P. Lin, J. Lowengrub and S. M. Wise, Mass conservative and energy stable finite difference methods for the quasi-incompressible Navier–Stokes–Cahn–Hilliard system: Primitive variable and projection-type schemes, Comput. Meth. Appl. Mech. Eng., 326 (2017), 144-174.  doi: 10.1016/j.cma.2017.08.011.  Google Scholar [22] D. J. E. Harvie, M. R. Davidson and M. Rudman, An analysis of parasitic current generation in volume of fluid simulations, Appl. Math. Model., 30 (2006), 1056-1066.  doi: 10.1016/j.apm.2005.08.015.  Google Scholar [23] P. C. Hiemenz and T. P. Lodge, Polymer Chemistry, 2$^{nd}$ edition, CRC Press, 2007.   Google Scholar [24] D. Jacqmin, Calculation of two-phase Navier–Stokes flows using phase-field modeling, J. Comput. Phys., 155 (1999), 96-127.  doi: 10.1006/jcph.1999.6332.  Google Scholar [25] D. Jamet, D. Torres and J. U. Brackbill, 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 (2002), 262-276.  doi: 10.1006/jcph.2002.7165.  Google Scholar [26] J. Liu, Thermodynamically Consistent Modeling and Simulation of Multiphase Flows, PhD dissertation, the University of Texas at Austin, 2014. Google Scholar [27] J. Lowengrub and L. Truskinovsky, Quasi-incompressible Cahn-Hilliard fluids and topological transitions, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 454 (1978), 2617-2654.  doi: 10.1098/rspa.1998.0273.  Google Scholar [28] S. Minjeaud, An adaptive pressure correction method without spurious velocities for diffuse-interface models of incompressible flows, J. Comput. Phys., 236 (2013), 143-156.  doi: 10.1016/j.jcp.2012.11.022.  Google Scholar [29] N. Provatas and K. Elde, Phase-Field Methods in Material Science and Engineering, Willey-VCH, Weinheim, 2010. doi: 10.1002/9783527631520.  Google Scholar [30] Yu. V. Sheretov, Continuum Dynamics under Spatiotemporal Averaging, RKhD, Moscow-Izhevsk, 2009 [in Russian]. Google Scholar [31] M. Svärd, A new Eulerian model for viscous and heat conducting compressible flows, Phys. A, 506 (2018), 350-375.  doi: 10.1016/j.physa.2018.03.097.  Google Scholar [32] G. Tierra and F. Guillén-González, Numerical methods for solving the Cahn-Hilliard equation and its applicability to related energy-based models, Arch. Comput. Meth. Eng., 22 (2015), 269-289.  doi: 10.1007/s11831-014-9112-1.  Google Scholar [33] P. Yue, C. Zhou and J. J. Feng, Spontaneous shrinkage of drops and mass conservation in phase-field simulations, J. Comput. Phys., 223 (2007), 1-9.  doi: 10.1016/j.jcp.2006.11.020.  Google Scholar [34] I. Zacharov et al., "Zhores" – Petaflops supercomputer for data-driven modeling, machine learning and artificial intelligence installed in Skolkovo Institute of Science and Technology, preprint, arXiv: 1902.07490. Google Scholar [35] A. Zlotnik, On the energy dissipative spatial discretization of the barotropic quasi-gasdynamic and compressible Navier–Stokes systems of equations in polar coordinates, Russ. J. Numer. Anal. Math. Model., 33 (2018), 199-210.  doi: 10.1515/rnam-2018-0017.  Google Scholar [36] A. A. Zlotnik, On conservative spatial discretizations of the barotropic quasi-gasdynamic system of equations with a potential body force, Comput. Math. Math. Phys., 56 (2016), 303-319.  doi: 10.1134/S0965542516020160.  Google Scholar [37] A. A. Zlotnik and T. A. Lomonosov, Conditions for $L^2$-dissipativity of linearized explicit difference schemes with regularization for 1D barotropic gas dynamics equations, Comput. Math. Math. Phys., 59 (2019), 452-464.  doi: 10.1134/S0965542519030151.  Google Scholar [38] A. A. Zlotnik and T. A. Lomonosov, On $L^2$-dissipativity of a linearized explicit finite-difference scheme with quasi-gasdynamic regularization for the barotropic gas dynamics system of equations, Dokl. Math., 101 (2020) (in press). Google Scholar
$\tilde\Psi_0(C)\equiv\Psi_0(\tilde\rho, C)$ for $\omega_2 > \omega_1$ and some fixed $\tilde\rho>0$
Distributions of $C$ and $\rho$ along the segment $x_1\in[0.5X, X]$ and $x_2 = x_3 = 0.5X$, at the vicinity of the interface at $t = 20\cdot10^3\Delta t$
Location of nodes of $\omega_{h\bar{k},\bar{l}^*,\bar{m}^*}$ (thick dots) and $\omega_{h\bar{k}^*,\bar{l},\bar{m}}$ (red crosses), where $u_k$ and $\Pi_{lk}$, $l\neq k$, are respectively defined
Droplet interface evolution in the case (Ⅰ) for $R = 0.25X$. Distribution of $C(x)$ in the section $x_1,x_2\in[0.5X,0.87X]$, $x_3 = 0.5X$, is represented
Evolution of $\sigma_L(t)$: for $R = 0.18X$ in the cases (Ⅰ)-(Ⅲ) (left) and for several $R$ in the case (Ⅲ) (right)
Observable dependence of $\Delta p$ on $1/R_a$ for different $R$ in (46)
Evolution of ${\bar{E}}_{\text{kin}}$ and $\mathcal{E}_h-\tilde{E}$ for the droplet with $R = 0.25X$ and $\tilde{E} = 9.16034\cdot 10^{-5}\, \text{J}$
${\bar{E}}_{\text{kin}} (t)$ computed by schemes $A$ (from this paper) and $B$ (from [6])
Evolution of $C_{\min}$ and $C_{\max}$ (for spinodal decomposition)
$\bar{E}_\text{kin}(t)$ for $\alpha^\ast = 0$ and $0.5$ and some $\Delta t$ (the break in the graph line means that computations collapse due to instability)
$\bar{E}_{\text{kin}}(t)$ for some $\alpha^\ast\geq0.5$ and $\Delta t$ (the break in the graph line means that computations collapse due to instability)
Evolution of ${\bar{E}}_{\text{kin}}$ and $\mathcal{E}_h$ (for spinodal decomposition)
Isosurfaces $C = 0.5$ at different time moments (for spinodal decomposition)
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