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:
[1]

M. O. Abu-Al-SaudS. 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.

[2]

D. M. AndersonG. 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.

[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. 

[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.

[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.

[6]

V. BalashovE. 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.

[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.

[8]

V. BalashovA. 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.

[9] M. J. Blunt, Multiphase Flow in Permeable Media: A Pore-Scale Perspective, Cambridge University Press, 2017.  doi: 10.1017/9781316145098.
[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. 

[11]

B. N. Chetverushkin, Kinetic Schemes and Quasi-Gasdynamic System of Equations, CIMNE, Barcelona, 2008.

[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.

[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.

[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.

[15]

A. Yu. Demianov, O. Yu. Dinariev and N. V. Evseev, Introduction to the Density Functional Method in Hydrodynamics, Fizmatlit, Moscow, 2014.

[16]

T. G. Elizarova, Quasi-Gas Dynamic Equations, Computational Fluid and Solid Mechanics. Springer, Dordrecht, 2009. doi: 10.1007/978-3-642-00292-2.

[17]

F. FrankC. LiuF. 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.

[18]

Y. GongJ. 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.

[19]

L. GoudenègeD. 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.

[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.

[21]

Z. GuoP. LinJ. 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.

[22]

D. J. E. HarvieM. 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.

[23] P. C. Hiemenz and T. P. Lodge, Polymer Chemistry, 2$^{nd}$ edition, CRC Press, 2007. 
[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.

[25]

D. JametD. 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.

[26]

J. Liu, Thermodynamically Consistent Modeling and Simulation of Multiphase Flows, PhD dissertation, the University of Texas at Austin, 2014.

[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.

[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.

[29]

N. Provatas and K. Elde, Phase-Field Methods in Material Science and Engineering, Willey-VCH, Weinheim, 2010. doi: 10.1002/9783527631520.

[30]

Yu. V. Sheretov, Continuum Dynamics under Spatiotemporal Averaging, RKhD, Moscow-Izhevsk, 2009 [in Russian].

[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.

[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.

[33]

P. YueC. 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.

[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.

[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.

[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.

[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.

[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).

show all references

References:
[1]

M. O. Abu-Al-SaudS. 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.

[2]

D. M. AndersonG. 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.

[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. 

[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.

[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.

[6]

V. BalashovE. 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.

[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.

[8]

V. BalashovA. 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.

[9] M. J. Blunt, Multiphase Flow in Permeable Media: A Pore-Scale Perspective, Cambridge University Press, 2017.  doi: 10.1017/9781316145098.
[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. 

[11]

B. N. Chetverushkin, Kinetic Schemes and Quasi-Gasdynamic System of Equations, CIMNE, Barcelona, 2008.

[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.

[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.

[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.

[15]

A. Yu. Demianov, O. Yu. Dinariev and N. V. Evseev, Introduction to the Density Functional Method in Hydrodynamics, Fizmatlit, Moscow, 2014.

[16]

T. G. Elizarova, Quasi-Gas Dynamic Equations, Computational Fluid and Solid Mechanics. Springer, Dordrecht, 2009. doi: 10.1007/978-3-642-00292-2.

[17]

F. FrankC. LiuF. 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.

[18]

Y. GongJ. 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.

[19]

L. GoudenègeD. 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.

[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.

[21]

Z. GuoP. LinJ. 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.

[22]

D. J. E. HarvieM. 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.

[23] P. C. Hiemenz and T. P. Lodge, Polymer Chemistry, 2$^{nd}$ edition, CRC Press, 2007. 
[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.

[25]

D. JametD. 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.

[26]

J. Liu, Thermodynamically Consistent Modeling and Simulation of Multiphase Flows, PhD dissertation, the University of Texas at Austin, 2014.

[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.

[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.

[29]

N. Provatas and K. Elde, Phase-Field Methods in Material Science and Engineering, Willey-VCH, Weinheim, 2010. doi: 10.1002/9783527631520.

[30]

Yu. V. Sheretov, Continuum Dynamics under Spatiotemporal Averaging, RKhD, Moscow-Izhevsk, 2009 [in Russian].

[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.

[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.

[33]

P. YueC. 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.

[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.

[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.

[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.

[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.

[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).

Figure 1.  $ \tilde\Psi_0(C)\equiv\Psi_0(\tilde\rho, C) $ for $ \omega_2 > \omega_1 $ and some fixed $ \tilde\rho>0 $
Figure 4.  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 $
Figure 2.  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
Figure 3.  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
Figure 5.  Evolution of $ \sigma_L(t) $: for $ R = 0.18X $ in the cases (Ⅰ)-(Ⅲ) (left) and for several $ R $ in the case (Ⅲ) (right)
Figure 6.  Observable dependence of $ \Delta p $ on $ 1/R_a $ for different $ R $ in (46)
Figure 7.  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} $
Figure 8.  $ {\bar{E}}_{\text{kin}} (t) $ computed by schemes $ A $ (from this paper) and $ B $ (from [6])
Figure 9.  Evolution of $ C_{\min} $ and $ C_{\max} $ (for spinodal decomposition)
Figure 10.  $ \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)
Figure 12.  $ \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)
Figure 13.  Evolution of $ {\bar{E}}_{\text{kin}} $ and $ \mathcal{E}_h $ (for spinodal decomposition)
Figure 14.  Isosurfaces $ C = 0.5 $ at different time moments (for spinodal decomposition)
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