An energy dissipative spatial discretization for the regularized compressible Navier-Stokes-Cahn-Hilliard system of equations
Abstract
We consider the regularized 3D Navier-Stokes-Cahn-Hilliard equations describing isothermal flows of viscous compressible two-component fluids with interphase effects. We construct for them a new energy dissipative finite-difference discretization in space, i.e., with the non-increasing total energy in time. This property is preserved in the absence of a regularization. In addition, the discretization is well-balanced for equilibrium flows and the potential body force. The sought total density, mixture velocity and concentration of one of the components are defined at nodes of one and the same grid. The results of computer simulation of several 2D test problems are presented. They demonstrate advantages of the constructed discretization including the absence of the so-called parasitic currents.
Keyword : two-component two-phase isothermal flows, interphase effects, regularized viscous compressible Navier-Stokes-Cahn-Hilliard equations, finite-difference discretization in space, energy dissipativeness
How to Cite
Balashov, V., & Zlotnik, A. (2020). An energy dissipative spatial discretization for the regularized compressible Navier-Stokes-Cahn-Hilliard system of equations. Mathematical Modelling and Analysis, 25(1), 110-129. https://doi.org/10.3846/mma.2020.10577
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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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(3):199–210, 2018. https://doi.org/10.1515/rnam-2018-0017
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(2):303–319, 2016. https://doi.org/10.1134/S0965542516020160
A.A. Zlotnik. Entropy-conservative spatial discretization of the multidimensional quasi-gasdynamic system of equations. Comput. Math. Math. Phys., 57(4):706– 725, 2017. https://doi.org/10.1134/S0965542517020166
A.A. Zlotnik and T.A. Lomonosov. Conditions for L2-dissipativity of linearized explicit difference schemes with regularization for 1D barotropic gas dynamics equations. Comput. Math. Math. Phys., 59(3):452–464, 2019. https://doi.org/10.1134/S0965542519030151
A.A. Zlotnik and T.A. Lomonosov. On L2-dissipativity of a linearized explicit finite-difference scheme with QGD-regularization for the barotropic gas dynamics system of equations. Dokl. Math., (in press).
D.M. Anderson, G.B. McFadden and A.A. Wheeler. Diffuse-interface methods in fluid mechanics. Ann. Rev. Fluid Mech., 30(1):139–165, 1998. https://doi.org/10.1146/annurev.fluid.30.1.139
V. Balashov and E. Savenkov. Quasi-hydrodynamic model of multiphase fluid flows taking into account phase interaction. J. Appl. Mech. Tech. Phys., 59(3):434–444, 2018. https://doi.org/10.1134/S0021894418030069
V. Balashov, E. Savenkov and A. Zlotnik. Numerical method for 3D two-component isothermal compressible flows with application to digital rock physics. Russ. J. Numer. Anal. Math. Model., 34(1):1–13, 2019. https://doi.org/10.1515/rnam-2019-0001
V. Balashov, A. Zlotnik and E. Savenkov. Analysis of a regularized model for the isothermal two-component mixture with the diffuse interface. Russ. J. Numer. Anal. Math. Model., 32(6):347–358, 2017. https://doi.org/10.1515/rnam-20170033
V.A. Balashov and E.B. Savenkov. Thermodynamically consistent spatial discretization of the one-dimensional regularized system of the Navier-Stokes-CahnHilliard equations. J. Comput. Appl. Math., (in press).
B.N. Chetverushkin. Kinetic Schemes and Quasi-Gasdynamic System of Equations. CIMNE, Barcelona, 2008.
K. Connington and T. Lee. A review of spurious currents in the lattice Boltzmann method for multiphase flows. J. Mech. Sci. Technology, 26(12):3857–3863, 2012. https://doi.org/10.1007/s12206-012-1011-5
M.I.M. Copetti and C.M. Elliott. Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy. Numer. Math., 63(1):39–65, 1992. https://doi.org/10.1007/BF01385847
T.G. Elizarova. Quasi-Gas Dynamic Equations. Springer, Berlin-Heidelberg, 2009.
T.G. Elizarova, A.A. Zlotnik and B.N. Chetverushkin. On quasi-gasdynamic and quasi-hydrodynamic equations for binary gas mixtures. Dokl. Math., 90(3):719– 723, 2014. https://doi.org/10.1134/S1064562414070217
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(2):543–563, 2018. https://doi.org/10.1007/s10596-017-9709-1
J.-L. Guermond and B. Popov. Viscous regularization of the Euler equations and entropy principles. SIAM J. Appl. Math., 74(2):284–305, 2014. https://doi.org/10.1137/120903312
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(10):1056–1066, 2006. https://doi.org/10.1016/j.apm.2005.08.015
D. Jacqmin. Calculation of two-phase Navier-Stokes flows using phase-field modeling. J. Comput. Phys., 155(1):96–127, 1999. https://doi.org/10.1006/jcph.1999.6332
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(1):262–276, 2002. https://doi.org/10.1006/jcph.2002.7165
J. Kim. A continuous surface tension force formulation for diffuse-interface models. J. Comput. Phys., 204(2):784–804, 2005. https://doi.org/10.1016/j.jcp.2004.10.032
J. Liu. Thermodynamically consistent modeling and simulation of multiphase flows. PhD thesis, The University of Texas at Austin, 2014.
J. Lowengrub and L. Truskinovsky. Quasi-incompressible Cahn-Hilliard fluids and topological transitions. Proc. Royal Soc. London. A., 454:2617–2654, 1998. https://doi.org/10.1098/rspa.1998.0273
N. Provatas and K. Elde. Phase-field methods in material science and engineering. Weinheim: Willey-VCH, 2010.
Yu. V. Sheretov. Continuum dynamics under spatiotemporal averaging. R&C Dynamics, Moscow-Izhevsk, 2009. (in Russian)
M. Svärd. A new Eulerian model for viscous and heat conducting compressible flows. Phys. A: Stat. Mech. Appl., 506:350–375, 2018. https://doi.org/10.1016/j.physa.2018.03.097
G. Tierra and F. Guillén-González. Numerical methods for solving the CahnHilliard equation and its applicability to related energy-based models. Arch. Comput. Meth. Eng., 22(2):269–289, 2015. https://doi.org/10.1007/s11831-0149112-1
P. Yue, C. Zhou and J.J. Feng. Spontaneous shrinkage of drops and mass conservation in phase-field simulations. J. Comput. Phys., 223(1):1–9, 2007. https://doi.org/10.1016/j.jcp.2006.11.020
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(3):199–210, 2018. https://doi.org/10.1515/rnam-2018-0017
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(2):303–319, 2016. https://doi.org/10.1134/S0965542516020160
A.A. Zlotnik. Entropy-conservative spatial discretization of the multidimensional quasi-gasdynamic system of equations. Comput. Math. Math. Phys., 57(4):706– 725, 2017. https://doi.org/10.1134/S0965542517020166
A.A. Zlotnik and T.A. Lomonosov. Conditions for L2-dissipativity of linearized explicit difference schemes with regularization for 1D barotropic gas dynamics equations. Comput. Math. Math. Phys., 59(3):452–464, 2019. https://doi.org/10.1134/S0965542519030151
A.A. Zlotnik and T.A. Lomonosov. On L2-dissipativity of a linearized explicit finite-difference scheme with QGD-regularization for the barotropic gas dynamics system of equations. Dokl. Math., (in press).