Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-26T15:26:46.988Z Has data issue: false hasContentIssue false
  • English
  • Français

A diffuse domain method for two-phase flows with large density ratio in complex geometries

Published online by Cambridge University Press:  26 November 2020

Zhenlin Guo Open the ORCID record for Zhenlin Guo [Opens in a new window] ,
Fei Yu ,
Ping Lin ,
Steven Wise  and
John Lowengrub
Zhenlin Guo*
Affiliation:
Mechanics Division, Beijing Computational Science Research Center, Building 9, East Zone, ZPark II, No. 10 East Xibeiwang Road, Haidian District, Beijing100193, PR China Department of Mathematics, University of California, Irvine, CA92697, USA
Fei Yu
Affiliation:
Department of Mathematics, University of California, Irvine, CA92697, USA
Ping Lin
Affiliation:
Department of Mathematics, University of Dundee, DundeeDD1 4HN, UK
Steven Wise
Affiliation:
Department of Mathematics, University of Tennessee, Knoxville, TN37996, USA
John Lowengrub
Affiliation:
Department of Mathematics, University of California, Irvine, CA92697, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We present a quasi-incompressible Navier–Stokes–Cahn–Hilliard (q-NSCH) diffuse interface model for two-phase fluid flows with variable physical properties that maintains thermodynamic consistency. Then, we couple the diffuse domain method with this two-phase fluid model – yielding a new q-NSCH-DD model – to simulate the two-phase flows with moving contact lines in complex geometries. The original complex domain is extended to a larger regular domain, usually a cuboid, and the complex domain boundary is replaced by an interfacial region with finite thickness. A phase-field function is introduced to approximate the characteristic function of the original domain of interest. The original fluid model, q-NSCH, is reformulated on the larger domain with additional source terms that approximate the boundary conditions on the solid surface. We show that the q-NSCH-DD system converges to the q-NSCH system asymptotically as the thickness of the diffuse domain interface introduced by the phase-field function shrinks to zero ($\epsilon \rightarrow 0$) with $\mathcal {O}(\epsilon )$. Our analytic results are confirmed numerically by measuring the errors in both $L^{2}$ and $L^{\infty }$ norms. In addition, we show that the q-NSCH-DD system not only allows the contact line to move on curved boundaries, but also makes the fluid–fluid interface intersect the solid object at an angle that is consistent with the prescribed contact angle.

JFM classification

Multiphase and Particle-laden Flows: Multiphase flow Interfacial Flows (free surface): Contact lines Mathematical Foundations: Computational methods
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 907 , 25 January 2021 , A38
Check for updates
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Abels, H., Garcke, H. & Grun, G. 2012 Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities. Math. Models Meth. Appl. Sci. 22 (03), 1150013.CrossRefGoogle Scholar
Aki, G. L., Dreyer, W., Giesselmann, J. & Kraus, C. 2014 A quasi-incompressible diffuse interface model with phase transition. Math. Models Meth. Appl. Sci. 24 (05), 827–861.CrossRefGoogle Scholar
Aland, S., Lowengrub, J. & Voigt, A. 2010 Two-phase flow in complex geometries: a diffuse domain approach. Comput. Model. Engng Sci. 57 (1), 77–106.Google ScholarPubMed
Anjos, G., Mangiavacchi, N., Borhani, N. & Thome, J. R. 2013 3D ALE finite-element method for two-phase flows with phase change. Heat Transfer Engng 35 (5), 537–547.CrossRefGoogle Scholar
Biros, G., Ying, L. & Zorin, D. 2004 A fast solver for the Stokes equations with distributed forces in complex geometries. J. Comput. Phys. 193 (1), 317–348.CrossRefGoogle Scholar
Boyer, F. 2002 A theoretical and numerical model for the study of incompressible mixture flows. Comput. Fluids 31 (1), 41–68.CrossRefGoogle Scholar
Cao, Z., Sun, D., Wei, J., Yu, B. & Li, J. 2019 A coupled volume-of-fluid and level set method based on general curvilinear grids with accurate surface tension calculation. J. Comput. Phys. 396, 799–818.Google Scholar
Chadwick, A. F., Stewart, J. A., Enrique, R. A., Du, S. & Thornton, K. 2018 Numerical modeling of localized corrosion using phase-field and smoothed boundary methods. J. Electrochem. Soc. 165 (10), C633–C646.Google Scholar
Chen, G., Li, Z. & Lin, P. 2007 A fast finite difference method for biharmonic equations on irregular domains and its application to an incompressible Stokes flow. Adv. Comput. Maths 29 (2), 113–133.CrossRefGoogle Scholar
De Stefano, G., Nejadmalayeri, A. & Vasilyev, O. V. 2016 Wall-resolved wavelet-based adaptive large-eddy simulation of bluff-body flows with variable thresholding. J. Fluid Mech. 788, 303–336.CrossRefGoogle Scholar
Ding, H., Spelt, P. D. M. & Shu, C. 2007 Diffuse interface model for incompressible two-phase flows with large density ratios. J. Comput. Phys. 226 (2), 2078–2095.Google Scholar
Do-Quang, M. & Amberg, G. 2009 The splash of a solid sphere impacting on a liquid surface: numerical simulation of the influence of wetting. Phys. Fluids 21 (2), 022102.CrossRefGoogle Scholar
Dong, S. & Shen, J. 2012 A time-stepping scheme involving constant coefficient matrices for phase-field simulations of two-phase incompressible flows with large density ratios. J. Comput. Phys. 231 (17), 5788–5804.Google Scholar
Dziwnik, M., Münch, A. & Wagner, B. 2017 An anisotropic phase-field model for solid-state dewetting and its sharp-interface limit. Nonlinearity 30, 1465–1496.CrossRefGoogle Scholar
Franz, S., Roos, H.-G., Gärtner, R. & Voigt, A. 2012 A note on the convergence analysis of a diffuse-domain approach. Comput. Meth. Appl. Maths 12 (2), 153–167.Google Scholar
Fries, T. & Belytschko, T. 2010 The extended/generalized finite element method: an overview of the method and its applications. Intl J. Numer. Meth. Engng 84, 253–304.CrossRefGoogle Scholar
Gao, F., Ingram, D. M., Causon, D. M. & Mingham, C. G. 2007 The development of a cartesian cut cell method for incompressible viscous flows. Intl J. Numer. Meth. Fluids 54 (9), 1033–1053.CrossRefGoogle Scholar
Gilmanov, A. & Sotiropoulos, F. 2005 A hybrid Cartesian/immersed boundary method for simulating flows with 3d, geometrically complex, moving bodies. J. Comput. Phys. 207 (2), 457–492.CrossRefGoogle Scholar
Glowinski, R., Pan, T. W., Hesla, T. I., Joseph, D. D. & Périaux, J. 2001 A fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies: application to particulate flow. J. Comput. Phys. 169 (2), 363–426.Google Scholar
Gong, Y., Zhao, J. & Wang, Q. 2017 An energy stable algorithm for a quasi-incompressible hydrodynamic phase-field model of viscous fluid mixtures with variable densities and viscosities. Comput. Phys. Commun. 219, 20–34.CrossRefGoogle Scholar
Gong, Y., Zhao, J., Yang, X. & Wang, Q. 2018 Fully discrete second-order linear schemes for hydrodynamic phase field models of binary viscous fluid flows with variable densities. SIAM J. Sci. Comput. 40 (1), 138–167.Google Scholar
Gránásy, L., Pusztai, T., Saylor, D. & Warren, J. A. 2007 Phase field theory of heterogeneous crystal nucleation. Phys. Rev. Lett. 98 (3).Google ScholarPubMed
Guo, Z. & Lin, P. 2015 A thermodynamically consistent phase-field model for two-phase flows with thermocapillary effects. J. Fluid Mech. 766, 226–271.Google Scholar
Guo, Z., Lin, P. & Lowengrub, J. 2014 a 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.CrossRefGoogle Scholar
Guo, Z., Lin, P., Lowengrub, J. & Wise, S. M. 2017 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. Engng 326, 144–174.Google Scholar
Guo, Z., Lin, P. & Wang, Y. 2014 b Continuous finite element schemes for a phase field model in two-layer fluid Bénard–Marangoni convection computations. Comput. Phys. Commun. 185 (1), 63–78.CrossRefGoogle Scholar
Gutiérrez, E., Favre, F., Balcázar, N., Amani, A. & Rigola, J. 2018 Numerical approach to study bubbles and drops evolving through complex geometries by using a level set–moving mesh–immersed boundary method. Chem. Engng J. 349, 662–682.CrossRefGoogle Scholar
Herrmann, M. 2008 A balanced force refined level set grid method for two-phase flows on unstructured flow solver grids. J. Comput. Phys. 227 (4), 2674–2706.CrossRefGoogle Scholar
Hohenberg, P. C. 1977 Theory of dynamic critical phenomena. Rev. Mod. Phys. 49 (3), 435–479.Google Scholar
Hu, H., Patankar, N. A. & Zhu, M. Y. 2001 Direct numerical simulations of fluid–solid systems using the arbitrary Lagrangian–Eulerian technique. J. Comput. Phys. 169 (2), 427–462.Google Scholar
Jacqmin, D. 1999 Calculation of two-phase Navier-Stokes flows using phase-field modeling. J. Comput. Phys. 155 (1), 96–127.CrossRefGoogle Scholar
Jiang, Y., Lin, P., Guo, Z. & Dong, S. 2015 Numerical simulation for moving contact line with continuous finite element schemes. Commun. Comput. Phys. 18 (01), 180–202.CrossRefGoogle Scholar
Kirkpatrick, M. P., Armfield, S. W. & Kent, J. H. 2003 A representation of curved boundaries for the solution of the Navier–Stokes equations on a staggered three-dimensional Cartesian grid. J. Comput. Phys. 184 (1), 1–36.CrossRefGoogle Scholar
Lervag, K. Y. & Lowengrub, J. 2015 Analysis of the diffuse-domain method for solving PDES in complex geometries. Commun. Math. Sci. 13 (6), 1473–1500.Google Scholar
Li, X., Lowengrub, J., Rätz, A. & Voigt, A. 2009 Solving PDES in complex geometries: a diffuse domain approach. Commun. Math. Sci. 7 (1), 81–107.CrossRefGoogle ScholarPubMed
Li, Z. 2003 An overview of the immersed interface method and its applications. Taiwan. J. Math. 7 (1), 1–49.Google Scholar
Liu, C. & Wu, H. 2019 An energetic variational approach for the Cahn–Hilliard equation with dynamic boundary condition: model derivation and mathematical analysis. Arch. Rat. Mech. Anal. 233 (1), 167–247.CrossRefGoogle Scholar
Liu, H. R. & Ding, H. 2015 A diffuse-interface immersed-boundary method for two-dimensional simulation of flows with moving contact lines on curved substrates. J. Comput. Phys. 294, 484–502.CrossRefGoogle Scholar
Lowengrub, J. & Truskinovsky, L. 1998 Quasi-incompressible Cahn–Hilliard fluids and topological transitions. Proc. R. Soc. Lond. A 454 (1978), 2617–2654.CrossRefGoogle Scholar
Mittal, R., Dong, H., Bozkurttas, M., Najjar, F. M., Vargas, A. & von Loebbecke, A. 2008 A versatile sharp interface immersed boundary method for incompressible flows with complex boundaries. J. Comput. Phys. 227 (10), 4825–4852.CrossRefGoogle ScholarPubMed
O'Brien, A. & Bussmann, M. 2018 A volume-of-fluid ghost-cell immersed boundary method for multiphase flows with contact line dynamics. Comput. Fluids 165, 43–53.CrossRefGoogle Scholar
Poulsen, S. O. & Voorhees, P. W. 2018 Smoothed boundary method for diffusion-related partial differential equations in complex geometries. Intl J. Comput. Meth. 15 (3).CrossRefGoogle Scholar
Rainer, B., Steven, M. W., Marco, S. & Axel, V. 2019 Convexity splitting in a phase field model for surface diffusion. Intl J. Numer. Anal. Model. 16 (2), 192–209.Google Scholar
Schneider, K. 2015 Immersed boundary methods for numerical simulation of confined fluid and plasma turbulence in complex geometries: a review. J. Plasma Phys. 81 (6).Google Scholar
Shirokoff, D. & Nave, J.-C. 2014 A sharp-interface active penalty method for the incompressible Navier–Stokes equations. J. Sci. Comput. 62 (1), 53–77.CrossRefGoogle Scholar
Shokrpour Roudbari, M., Şimşek, G., van Brummelen, E. H. & van der Zee, K. G. 2018 Diffuse-interface two-phase flow models with different densities: a new quasi-incompressible form and a linear energy-stable method. Math. Models Meth. Appl. Sci. 28 (04), 733–770.Google Scholar
Stein, D. B., Guy, R. D. & Thomases, B. 2017 Immersed boundary smooth extension (IBSE): a high-order method for solving incompressible flows in arbitrary smooth domains. J. Comput. Phys. 335, 155–178.CrossRefGoogle Scholar
Tryggvason, G., Bunner, B., Esmaeeli, A., Juric, D., Al-Rawahi, N., Tauber, W., Han, J., Nas, S. & Jan, Y.-J. 2001 A front-tracking method for the computations of multiphase flow. J. Comput. Phys. 169 (2), 708–759.Google Scholar
Yu, F., Guo, Z. & Lowengrub, J. 2020 Higher-order accurate diffuse-domain methods for partial differential equations with Dirichlet boundary conditions in complex, evolving geometries. J. Comput. Phys. 406, 109174.CrossRefGoogle Scholar
Yu, H., Chen, H. & Thornton, K. 2012 Extended smoothed boundary method for solving partial differential equations with general boundary conditions on complex boundaries. Model. Simul. Mater. Sci. Engng 20 (7), 075008.Google Scholar
de Zélicourt, D., Ge, L., Wang, C., Sotiropoulos, F., Gilmanov, A. & Yoganathan, A. 2009 Flow simulations in arbitrarily complex cardiovascular anatomies – an unstructured Cartesian grid approach. Comput. Fluids 38 (9), 1749–1762.CrossRefGoogle Scholar
Zhang, J. & Ni, M. 2014 A consistent and conservative scheme for MHD flows with complex boundaries on an unstructured Cartesian adaptive system. J. Comput. Phys. 256, 520–542.CrossRefGoogle Scholar
Zhang, Z. L., Walayat, K., Chang, J. Z. & Liu, M. B. 2018 Meshfree modeling of a fluid-particle two-phase flow with an improved SPH method. Intl J. Numer. Meth. Engng 116 (8), 530–569.CrossRefGoogle Scholar
Zhang, Q. & Wang, X. 2016 Phase field modeling and simulation of three-phase flow on solid surfaces. J. Comput. Phys. 319, 79–107.CrossRefGoogle Scholar
Zolfaghari, H., Izbassarov, D. & Muradoglu, M. 2017 Simulations of viscoelastic two-phase flows in complex geometries. Comput. Fluids 156, 548–561.Google Scholar