A free energy-based surface tension force model for simulation of multiphase flows by level-set method

In this paper, a free energy-based surface tension force (FESF) model is presented for accurately resolving the surface tension force in numerical simulation of multiphase flows by the level set method. By using the analytical form of order parameter along the normal direction to the interface in the phase-field method and the free energy principle, FESF model offers an explicit and analytical formulation for the surface tension force. The only variable in this formulation is the normal distance to the interface, which can be substituted by the distance function solved by the level set method. On one hand, as compared to conventional continuum surface force (CSF) model in the level set method, FESF model introduces no regularized delta function, due to which it suffers less from numerical diffusions and performs better in mass conservation. On the other hand, as compared to the phase field surface tension force (PFSF) model, the evaluation of surface tension force in FESF model is based on an analytical approach rather than numerical approximations of spatial derivatives. Therefore, better numerical stability and higher accuracy can be expected. Various numerical examples are tested to validate the robustness of the proposed FESF model. It turns out that FESF model performs better than CSF model and PFSF model in terms of accuracy, stability, convergence speed and mass conservation. It is also shown in numerical tests that FESF model can effectively simulate problems with high density/viscosity ratio, high Reynolds number and severe topological interfacial changes. A free energy-based surface tension force (FESF) model is proposed for simulation of multiphase flows by level set method.The proposed model computes the surface tension force by an explicit form.The proposed model outperforms the existing models in terms of accuracy, stability, convergence speed and mass conservation.The proposed model can effectively simulate multiphase flows.

[1]  Jian-Jun Xu,et al.  A level-set method for two-phase flows with moving contact line and insoluble surfactant , 2014, J. Comput. Phys..

[2]  Y. Wang,et al.  A mass-conserved diffuse interface method and its application for incompressible multiphase flows with large density ratio , 2015, J. Comput. Phys..

[3]  J. Waals.  The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density , 1979 .

[4]  Lin Liu,et al.  Lattice Boltzmann simulations of micron-scale drop impact on dry surfaces , 2010, J. Comput. Phys..

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

[6]  Xiaolei Yang,et al.  A smoothing technique for discrete delta functions with application to immersed boundary method in moving boundary simulations , 2009, J. Comput. Phys..

[7]  M. Jalaal,et al.  Fragmentation of falling liquid droplets in bag breakup mode , 2012 .

[8]  Peng Song,et al.  A diffuse-interface method for two-phase flows with soluble surfactants , 2011, J. Comput. Phys..

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

[10]  Robert F. Kunz,et al.  A preconditioned navier-stokes method for two-phase flows with application to Cavitation prediction , 1999 .

[11]  Jie Shen,et al.  Numerical simulations of jet pinching-off and drop formation using an energetic variational phase-field method , 2006, J. Comput. Phys..

[12]  Pier Luca Maffettone,et al.  Equation of change for ellipsoidal drops in viscous flow , 1998 .

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

[14]  Yan Wang,et al.  Multiphase lattice Boltzmann flux solver for incompressible multiphase flows with large density ratio , 2015, J. Comput. Phys..

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

[16]  Ann S. Almgren,et al.  An adaptive level set approach for incompressible two-phase flows , 1997 .

[17]  Chiang Juay Teo,et al.  Development of LBGK and incompressible LBGK‐based lattice Boltzmann flux solvers for simulation of incompressible flows , 2014 .

[18]  R. G. Cox.  The deformation of a drop in a general time-dependent fluid flow , 1969, Journal of Fluid Mechanics.

[19]  David Jacqmin,et al.  An energy approach to the continuum surface tension method , 1996 .

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

[21]  Haibo Huang,et al.  A mass-conserving axisymmetric multiphase lattice Boltzmann method and its application in simulation of bubble rising , 2014, J. Comput. Phys..

[22]  Mathieu Coquerelle,et al.  A fourth-order accurate curvature computation in a level set framework for two-phase flows subjected to surface tension forces , 2016, J. Comput. Phys..

[23]  D. Kuzmin,et al.  Quantitative benchmark computations of two‐dimensional bubble dynamics , 2009 .

[24]  N. Zhao,et al.  Dynamics of falling droplets impact on a liquid film: Hybrid lattice Boltzmann simulation , 2015 .

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

[26]  Jinsong Hua,et al.  Numerical simulation of bubble rising in viscous liquid , 2007, J. Comput. Phys..

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

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

[29]  A. H. Nikseresht,et al.  Numerical simulation of unsteady 3D cavitating flows over axisymmetric cavitators , 2012 .

[30]  Abbas Fakhari,et al.  Investigation of deformation and breakup of a falling droplet using a multiple-relaxation-time lattice Boltzmann method , 2011 .

[31]  Peter J. Mucha,et al.  A narrow-band gradient-augmented level set method for multiphase incompressible flow , 2014, J. Comput. Phys..

[32]  A. Oliva,et al.  Level-set simulations of buoyancy-driven motion of single and multiple bubbles , 2015 .

[33]  S. Osher,et al.  An improved level set method for incompressible two-phase flows , 1998 .

[34]  G BISWAS,et al.  Variant of a volume-of-fluid method for surface tension-dominant two-phase flows , 2013 .

[35]  Marcus Herrmann,et al.  Calculation of droplet deformationby surface tension effects usingthe level set method , 2002 .

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

[37]  Hongwei Zheng,et al.  A lattice Boltzmann model for multiphase flows with large density ratio , 2006, J. Comput. Phys..

[38]  Martin Grant,et al.  Phase field model of stress-induced surface instabilities: Surface diffusion , 2006 .

[39]  Axel Voigt,et al.  Benchmark computations of diffuse interface models for two‐dimensional bubble dynamics , 2012 .

[40]  S. Zaleski,et al.  Modelling Merging and Fragmentation in Multiphase Flows with SURFER , 1994 .

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

[42]  Anthony J. Robinson,et al.  Influence of surface tension implementation in Volume of Fluid and coupled Volume of Fluid with Level Set methods for bubble growth and detachment , 2013 .

[43]  Junseok Kim,et al.  A continuous surface tension force formulation for diffuse-interface models , 2005 .

[44]  Ching-Long Lin,et al.  A stable discretization of the lattice Boltzmann equation for simulation of incompressible two-phase flows at high density ratio , 2005 .

[45]  Amit Gupta,et al.  Lattice Boltzmann simulation to study multiple bubble dynamics , 2008 .

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

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