Physical formulation and numerical algorithm for simulating N immiscible incompressible fluids involving general order parameters

We present a family of physical formulations, and a numerical algorithm, based on a class of general order parameters for simulating the motion of a mixture of N ( N ? 2 ) immiscible incompressible fluids with given densities, dynamic viscosities, and pairwise surface tensions. The N-phase formulations stem from a phase field model we developed in a recent work based on the conservations of mass/momentum, and the second law of thermodynamics. The introduction of general order parameters leads to an extremely strongly-coupled system of ( N - 1 ) phase field equations. On the other hand, the general form enables one to compute the N-phase mixing energy density coefficients in an explicit fashion in terms of the pairwise surface tensions. We show that the increased complexity in the form of the phase field equations associated with general order parameters in actuality does not cause essential computational difficulties. Our numerical algorithm reformulates the ( N - 1 ) strongly-coupled phase field equations for general order parameters into 2 ( N - 1 ) Helmholtz-type equations that are completely de-coupled from one another. This leads to a computational complexity comparable to that for the simplified phase field equations associated with certain special choice of the order parameters. We demonstrate the capabilities of the method developed herein using several test problems involving multiple fluid phases and large contrasts in densities and viscosities among the multitude of fluids. In particular, by comparing simulation results with the Langmuir-de Gennes theory of floating liquid lenses we show that the method using general order parameters produces physically accurate results for multiple fluid phases.

[1]  Junseok Kim,et al.  Buoyancy-driven mixing of multi-component fluids in two-dimensional tilted channels , 2013 .

[2]  Bruno Piar,et al.  Inertial dynamics of air bubbles crossing a horizontal fluid–fluid interface , 2012, Journal of Fluid Mechanics.

[3]  Junseok Kim,et al.  An efficient and accurate numerical algorithm for the vector-valued Allen-Cahn equations , 2012, Comput. Phys. Commun..

[4]  Jun Li,et al.  A Class of Conservative Phase Field Models for Multiphase Fluid Flows , 2014 .

[5]  Suchuan Dong,et al.  An efficient algorithm for incompressible N-phase flows , 2014, J. Comput. Phys..

[6]  Suchuan Dong,et al.  A robust and accurate outflow boundary condition for incompressible flow simulations on severely-truncated unbounded domains , 2014, J. Comput. Phys..

[7]  Jie Shen,et al.  Velocity-Correction Projection Methods for Incompressible Flows , 2003, SIAM J. Numer. Anal..

[8]  Jie Shen,et al.  A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method , 2003 .

[9]  Pedro Díez,et al.  Hierarchical X-FEM for n-phase flow (n > 2) , 2009 .

[10]  Jie Shen,et al.  An overview of projection methods for incompressible flows , 2006 .

[11]  Junseok Kim,et al.  CONSERVATIVE MULTIGRID METHODS FOR TERNARY CAHN-HILLIARD SYSTEMS ∗ , 2004 .

[12]  D. Jacqmin.  Regular Article: Calculation of Two-Phase Navier–Stokes Flows Using Phase-Field Modeling , 1999 .

[13]  Tony F. Chan,et al.  Active contours without edges , 2001, IEEE Trans. Image Process..

[14]  D. Juric,et al.  A front-tracking method for the computations of multiphase flow , 2001 .

[15]  Franck Boyer,et al.  Numerical schemes for a three component Cahn-Hilliard model , 2011 .

[16]  Junseok Kim,et al.  A second-order accurate non-linear difference scheme for the N -component Cahn–Hilliard system , 2008 .

[17]  H. Abels,et al.  Thermodynamically Consistent, Frame Indifferent Diffuse Interface Models for Incompressible Two-Phase Flows with Different Densities , 2011, 1104.1336.

[18]  L. Quartapelle,et al.  A projection FEM for variable density incompressible flows , 2000 .

[19]  S. Osher,et al.  Algorithms Based on Hamilton-Jacobi Formulations , 1988 .

[20]  D. Chopp,et al.  A projection method for motion of triple junctions by level sets , 2002 .

[21]  Suchuan Dong,et al.  An outflow boundary condition and algorithm for incompressible two-phase flows with phase field approach , 2014, J. Comput. Phys..

[22]  D. M. Anderson,et al.  DIFFUSE-INTERFACE METHODS IN FLUID MECHANICS , 1997 .

[23]  Franck Boyer,et al.  Study of a three component Cahn-Hilliard flow model , 2006 .

[24]  G. Karniadakis,et al.  Spectral/hp Element Methods for Computational Fluid Dynamics , 2005 .

[25]  J. Sethian,et al.  FRONTS PROPAGATING WITH CURVATURE DEPENDENT SPEED: ALGORITHMS BASED ON HAMILTON-JACOB1 FORMULATIONS , 2003 .

[26]  L. Formaggia,et al.  Implicit tracking for multi-fluid simulations , 2010, J. Comput. Phys..

[27]  Suchuan Dong,et al.  An eigen-based high-order expansion basis for structured spectral elements , 2011, J. Comput. Phys..

[28]  Kumbakonam R. Rajagopal,et al.  On the development and generalizations of Cahn–Hilliard equations within a thermodynamic framework , 2012 .

[29]  J. Sethian,et al.  LEVEL SET METHODS FOR FLUID INTERFACES , 2003 .

[30]  P. Tarazona.  Capillarity and Wetting , 1987 .

[31]  Jie Shen,et al.  A time-stepping scheme involving constant coefficient matrices for phase-field simulations of two-phase incompressible flows with large density ratios , 2012, J. Comput. Phys..

[32]  P. Gennes,et al.  Capillarity and Wetting Phenomena , 2004 .

[33]  Suchuan Dong,et al.  On imposing dynamic contact-angle boundary conditions for wall-bounded liquid–gas flows , 2012 .

[34]  Shouhong Wang,et al.  Unified Field Theory and Principle of Representation Invariance , 2012, 1212.4893.

[35]  Geoffrey Ingram Taylor,et al.  The formation of emulsions in definable fields of flow , 1934 .

[36]  Junseok Kim,et al.  Phase field modeling and simulation of three-phase flows , 2005 .

[37]  G. Tryggvason,et al.  A front-tracking method for viscous, incompressible, multi-fluid flows , 1992 .

[38]  George Em Karniadakis,et al.  A triangular spectral element method; applications to the incompressible Navier-Stokes equations , 1995 .

[39]  Shigeki Matsutani,et al.  Surface Tension of Multi-phase Flow with Multiple Junctions Governed by the Variational Principle , 2011, ArXiv.

[40]  J. Sethian,et al.  The Voronoi Implicit Interface Method for computing multiphase physics , 2011, Proceedings of the National Academy of Sciences.

[41]  James J. Feng,et al.  A diffuse-interface method for simulating two-phase flows of complex fluids , 2004, Journal of Fluid Mechanics.

[42]  Irving Langmuir,et al.  Oil Lenses on Water and the Nature of Monomolecular Expanded Films , 1933 .

[43]  T. Chan,et al.  A Variational Level Set Approach to Multiphase Motion , 1996 .

[44]  S. Osher,et al.  Motion of multiple junctions: a level set approach , 1994 .

[45]  Junseok Kim.  Phase-Field Models for Multi-Component Fluid Flows , 2012 .

[46]  James J. Feng,et al.  Wall energy relaxation in the Cahn–Hilliard model for moving contact lines , 2009 .

[47]  Junseok Kim,et al.  A generalized continuous surface tension force formulation for phase-field models for multi-component immiscible fluid flows , 2009 .

[48]  S. Zaleski,et al.  DIRECT NUMERICAL SIMULATION OF FREE-SURFACE AND INTERFACIAL FLOW , 1999 .

[49]  Jie Shen,et al.  An unconditionally stable rotational velocity-correction scheme for incompressible flows , 2010, J. Comput. Phys..

[50]  L. Marino,et al.  The sharp-interface limit of the Cahn–Hilliard/Navier–Stokes model for binary fluids , 2013, Journal of Fluid Mechanics.

[51]  J. Lowengrub,et al.  Quasi–incompressible Cahn–Hilliard fluids and topological transitions , 1998, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.