An interface capturing method with a continuous function: The THINC method with multi-dimensional reconstruction

An interface capturing method with a continuous function is proposed within the framework of the volume-of-fluid (VOF) method. Being different from the traditional VOF methods that require a geometrical reconstruction and identify the interface by a discontinuous Heaviside function, the present method makes use of the hyperbolic tangent function (known as one of the sigmoid type functions) in the tangent of hyperbola interface capturing (THINC) method [F. Xiao, Y. Honma, K. Kono, A simple algebraic interface capturing scheme using hyperbolic tangent function, Int. J. Numer. Methods Fluids 48 (2005) 1023-1040] to retrieve the interface in an algebraic way from the volume-fraction data of multi-component materials. Instead of the 1D reconstruction in the original THINC method, a multi-dimensional hyperbolic tangent function is employed in the present new approach. The present scheme resolves moving interface with geometric faithfulness and compact thickness, and has at least the following advantages: (1) the geometric reconstruction is not required in constructing piecewise approximate functions; (2) besides a piecewise linear interface, curved (quadratic) surface can be easily constructed as well; and (3) the continuous multi-dimensional hyperbolic tangent function allows the direct calculations of derivatives and normal vectors. Numerical benchmark tests including transport of moving interface and incompressible interfacial flows are presented to validate the numerical accuracy for interface capturing and to show the capability for practical problems such as a stationary circular droplet, a drop oscillation, a shear-induced drop deformation and a rising bubble.

[1]  L. G. Leal,et al.  Numerical solution of free-boundary problems in fluid mechanics. Part 2. Buoyancy-driven motion of a gas bubble through a quiescent liquid , 1984, Journal of Fluid Mechanics.

[2]  Yoichiro Matsumoto,et al.  A full Eulerian finite difference approach for solving fluid-structure coupling problems , 2010, J. Comput. Phys..

[3]  Markus Bussmann,et al.  Second-order accurate normals from height functions , 2008, J. Comput. Phys..

[4]  W. Rider,et al.  Reconstructing Volume Tracking , 1998 .

[5]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov's method , 1979 .

[6]  T. Yabe,et al.  The constrained interpolation profile method for multiphase analysis , 2001 .

[7]  P. Moin,et al.  Application of a Fractional-Step Method to Incompressible Navier-Stokes Equations , 1984 .

[8]  C. W. Hirt,et al.  An Arbitrary Lagrangian-Eulerian Computing Method for All Flow Speeds , 1997 .

[9]  Gretar Tryggvason,et al.  Head-on collision of drops—A numerical investigation , 1993 .

[10]  R. LeVeque,et al.  A comparison of the extended finite element method with the immersed interface method for elliptic equations with discontinuous coefficients and singular sources , 2006 .

[11]  Zhilin Li,et al.  The immersed interface method for the Navier-Stokes equations with singular forces , 2001 .

[12]  M. Lai,et al.  An Immersed Boundary Method with Formal Second-Order Accuracy and Reduced Numerical Viscosity , 2000 .

[13]  Wing Kam Liu,et al.  Lagrangian-Eulerian finite element formulation for incompressible viscous flows☆ , 1981 .

[14]  Matthew W. Williams,et al.  A balanced-force algorithm for continuous and sharp interfacial surface tension models within a volume tracking framework , 2006, J. Comput. Phys..

[15]  T. Tezduyar,et al.  A new strategy for finite element computations involving moving boundaries and interfaces—the deforming-spatial-domain/space-time procedure. I: The concept and the preliminary numerical tests , 1992 .

[16]  L. G. Leal,et al.  Numerical solution of free-boundary problems in fluid mechanics. Part 1. The finite-difference technique , 1984, Journal of Fluid Mechanics.

[17]  E. Puckett,et al.  A High-Order Projection Method for Tracking Fluid Interfaces in Variable Density Incompressible Flows , 1997 .

[18]  Julio Hernández,et al.  On reducing interface curvature computation errors in the height function technique , 2010, J. Comput. Phys..

[19]  Tomoaki Kunugi,et al.  Multi-scale modeling of the gas-liquid interface based on mathematical and thermodynamic approaches , 2010 .

[20]  S. Zaleski,et al.  Interface reconstruction with least‐square fit and split Eulerian–Lagrangian advection , 2003 .

[21]  Thomas J. R. Hughes,et al.  A space-time formulation for multiscale phenomena , 1996 .

[22]  C. W. Hirt,et al.  Volume of fluid (VOF) method for the dynamics of free boundaries , 1981 .

[23]  Kensuke Yokoi,et al.  Efficient implementation of THINC scheme: A simple and practical smoothed VOF algorithm , 2007, J. Comput. Phys..

[24]  S. Zalesak.  Fully multidimensional flux-corrected transport algorithms for fluids , 1979 .

[25]  Eugenio Aulisa,et al.  Interface reconstruction with least-squares fit and split advection in three-dimensional Cartesian geometry , 2007, J. Comput. Phys..

[26]  G. Strang.  On the Construction and Comparison of Difference Schemes , 1968 .

[27]  Gretar Tryggvason,et al.  Numerical simulations of three-dimensional drop collisions , 1996 .

[28]  Chang Shu,et al.  Diffuse interface model for incompressible two-phase flows with large density ratios , 2007, J. Comput. Phys..

[29]  P. Woodward,et al.  SLIC (Simple Line Interface Calculation) , 1976 .

[30]  S. Osher,et al.  A level set approach for computing solutions to incompressible two-phase flow , 1994 .

[31]  P. Colella,et al.  A second-order projection method for the incompressible navier-stokes equations , 1989 .

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

[33]  C. Peskin.  The immersed boundary method , 2002, Acta Numerica.

[34]  Ted Belytschko,et al.  Fluid-structure interaction , 1980 .

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

[36]  James J. Feng,et al.  An arbitrary Lagrangian-Eulerian method for simulating bubble growth in polymer foaming , 2007, J. Comput. Phys..

[37]  I. Hutchings,et al.  Numerical studies of the influence of the dynamic contact angle on a droplet impacting on a dry surface , 2009 .

[38]  M. Sussman,et al.  A Coupled Level Set and Volume-of-Fluid Method for Computing 3D and Axisymmetric Incompressible Two-Phase Flows , 2000 .

[39]  S. Cummins,et al.  Estimating curvature from volume fractions , 2005 .

[40]  J. G. Hnat,et al.  Spherical cap bubbles and skirt formation , 1976 .

[41]  F. Harlow,et al.  Numerical Calculation of Time‐Dependent Viscous Incompressible Flow of Fluid with Free Surface , 1965 .

[42]  C. Peskin.  Flow patterns around heart valves: A numerical method , 1972 .

[43]  D. Fletcher,et al.  A New Volume of Fluid Advection Algorithm , 2000 .

[44]  S. Zaleski,et al.  Volume-of-Fluid Interface Tracking with Smoothed Surface Stress Methods for Three-Dimensional Flows , 1999 .

[45]  T. Sundararajan,et al.  A Quadratic Spline based Interface (QUASI) reconstruction algorithm for accurate tracking of two-phase flows , 2009, J. Comput. Phys..

[46]  Feng Xiao,et al.  Description of Complex and Sharp Interface during Shock Wave Interaction with Liquid Drop , 1993 .

[47]  肖锋,et al.  Revisit To the Thinc Scheme: A Simple Algebraic VOF Algorithm , 2011 .

[48]  Feng Xiao,et al.  An efficient method for capturing free boundaries in multi‐fluid simulations , 2003 .

[49]  S. Zaleski,et al.  A geometrical area-preserving volume-of-fluid advection method , 2003 .

[50]  David J. Torres,et al.  The point-set method: front-tracking without connectivity , 2000 .

[51]  Feng Xiao,et al.  A simple algebraic interface capturing scheme using hyperbolic tangent function , 2005 .

[52]  M. Renardy,et al.  PROST: a parabolic reconstruction of surface tension for the volume-of-fluid method , 2002 .

[53]  J. M. Rallison.  The Deformation of Small Viscous Drops and Bubbles in Shear Flows , 1984 .

[54]  R. LeVeque.  High-resolution conservative algorithms for advection in incompressible flow , 1996 .

[55]  S. Zaleski,et al.  Analytical relations connecting linear interfaces and volume fractions in rectangular grids , 2000 .

[56]  Qiang Zhang,et al.  Three-Dimensional Front Tracking , 1998, SIAM J. Sci. Comput..

[57]  J. Brackbill,et al.  A continuum method for modeling surface tension , 1992 .

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

[59]  Kensuke Yokoi,et al.  A Numerical Method for Free-Surface Flows and Its Application to Droplet Impact on a Thin Liquid Layer , 2008, J. Sci. Comput..

[60]  Stéphane Popinet,et al.  A front-tracking algorithm for accurate representation of surface tension , 1999 .

[61]  J. Sethian,et al.  Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations , 1988 .

[62]  Héctor D. Ceniceros,et al.  Computation of multiphase systems with phase field models , 2002 .

[63]  L YoungsD,et al.  Time-dependent multi-material flow with large fluid distortion. , 1982 .

[64]  Feng Xiao,et al.  Revisit to the THINC scheme: A simple algebraic VOF algorithm , 2011, J. Comput. Phys..

[65]  E. Puckett,et al.  Second-Order Accurate Volume-of-Fluid Algorithms for Tracking Material Interfaces , 2013 .

[66]  Mark Sussman,et al.  A Stable and Efficient Method for Treating Surface Tension in Incompressible Two-Phase Flow , 2009, SIAM J. Sci. Comput..

[67]  M. Rudman.  INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, VOL. 24, 671–691 (1997) VOLUME-TRACKING METHODS FOR INTERFACIAL FLOW CALCULATIONS , 2022 .

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

[69]  Yoichiro Matsumoto,et al.  Numerical Analysis of a Single Rising Bubble Using Boundary-Fitted Coordinate System , 1995 .