A weakly compressible scheme with a diffuse-interface method for low Mach number two-phase flows

Abstract A weakly compressible scheme for gas–liquid two-phase flows with low Mach number is proposed for a fully-explicit time integration. The scheme is based on the fractional-step method with the Euler equation solver and time integration for viscous, surface tension and gravity terms. For the efficient solution of the Euler equation, we introduce the directional-splitting method and the method of characteristics which is applicable to semi-Lagrangian methods. A reduction of the speed of sound is also taken into consideration to enlarge the time step intervals. The primitive flow variables are time-integrated on the node-center configuration. To capture the gas–liquid interface, the conservative phase-field equation is solved by using the multi-moment method. It is computed as a conservative form for the volume-averaged value defined at the cell center, in which the flow velocities are given on the node. We discard the exact momentum conservation, however the total mass of gas and liquid is conserved by introducing a phase-field variable. Several benchmark problems are examined and the results of the explicit scheme are in good agreement with the experimental and computational references.

[1]  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..

[2]  Ng Niels Deen,et al.  Numerical simulation of gas bubbles behaviour using a three-dimensional volume of fluid method , 2005 .

[3]  Martin Rieber,et al.  A numerical study on the mechanism of splashing , 1999 .

[4]  C. K. Thornhill,et al.  Part IV. An experimental study of the collapse of liquid columns on a rigid horizontal plane , 1952, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[5]  Massimo Bernaschi,et al.  The in-silico lab-on-a-chip: petascale and high-throughput simulations of microfluidics at cell resolution , 2015, SC15: International Conference for High Performance Computing, Networking, Storage and Analysis.

[6]  Kenji Takizawa,et al.  Conservative form of interpolated differential operator scheme for compressible and incompressible fluid dynamics , 2008, J. Comput. Phys..

[7]  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 .

[8]  Roberto Ferretti,et al.  A Weighted Essentially Nonoscillatory, Large Time-Step Scheme for Hamilton-Jacobi Equations , 2005, SIAM J. Sci. Comput..

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

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

[11]  Aoki Takayuki,et al.  Multi-GPU performance of incompressible flow computation by lattice Boltzmann method on GPU cluster , 2011, ParCo 2011.

[12]  T. Inamuro,et al.  A lattice Boltzmann method for incompressible two-phase flows with large density differences , 2004 .

[13]  S. Koshizuka,et al.  Moving-Particle Semi-Implicit Method for Fragmentation of Incompressible Fluid , 1996 .

[14]  尊之 青木,et al.  木構造に基づいた AMR 法を用いた流束項付き保存形フェーズフィールド方程式のマルチモーメント法による解法 , 2018 .

[15]  Takeshi Iwashita,et al.  Block Red-Black Ordering: A New Ordering Strategy for Parallelization of ICCG Method , 2004, International Journal of Parallel Programming.

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

[17]  B. C. Curtis,et al.  Very High Resolution Simulation of Compressible Turbulence on the IBM-SP System , 1999, ACM/IEEE SC 1999 Conference (SC'99).

[18]  Mark Sussman,et al.  A sharp interface method for incompressible two-phase flows , 2007, J. Comput. Phys..

[19]  M. Berger,et al.  Adaptive mesh refinement for hyperbolic partial differential equations , 1982 .

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

[21]  P. Woodward,et al.  The numerical simulation of two-dimensional fluid flow with strong shocks , 1984 .

[22]  Stojan Petelin,et al.  Coupling of the interface tracking and the two-fluid models for the simulation of incompressible two-phase flow , 2001 .

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

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

[25]  Jaber Almedeij,et al.  Drag coefficient of flow around a sphere: Matching asymptotically the wide trend , 2008 .

[26]  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..

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

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

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

[30]  G. Homsy,et al.  Crown-forming instability phenomena in the drop splash problem. , 2009, Journal of colloid and interface science.

[31]  S. Rebay,et al.  A High-Order Accurate Discontinuous Finite Element Method for the Numerical Solution of the Compressible Navier-Stokes Equations , 1997 .

[32]  Robin K. S. Hankin,et al.  The Euler equations for multiphase compressible flow in conservation form: simulation of shock-bubble interactions , 2001 .

[33]  James J. Quirk,et al.  On the dynamics of a shock–bubble interaction , 1994, Journal of Fluid Mechanics.

[34]  A. Colagrossi,et al.  Numerical simulation of interfacial flows by smoothed particle hydrodynamics , 2003 .

[35]  Mark H. Anderson,et al.  A computational parameter study for the three-dimensional shock–bubble interaction , 2007, Journal of Fluid Mechanics.

[36]  Hervé Guillard,et al.  Behavior of upwind scheme in the low Mach number limit: III. Preconditioned dissipation for a five equation two phase model , 2008 .

[37]  F. Xiao,et al.  Numerical simulations of free-interface fluids by a multi-integrated moment method , 2005 .

[38]  U. Ghia,et al.  High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method , 1982 .

[39]  T. Yabe,et al.  Exactly conservative semi-Lagrangian scheme for multi-dimensional hyperbolic equations with directional splitting technique , 2001 .

[40]  M. Sussman.  An Adaptive Mesh Algorithm for Free Surface Flows in General Geometries , 2001 .

[41]  Kensuke Yokoi,et al.  A density-scaled continuum surface force model within a balanced force formulation , 2014, J. Comput. Phys..

[42]  A. Colagrossi,et al.  δ-SPH model for simulating violent impact flows , 2011 .

[43]  Yoshiki Seo,et al.  14.9 TFLOPS Three-Dimensional Fluid Simulation for Fusion Science with HPF on the Earth Simulator , 2002, ACM/IEEE SC 2002 Conference (SC'02).

[44]  Siamak Kazemzadeh Hannani,et al.  A fully explicit three‐step SPH algorithm for simulation of non‐Newtonian fluid flow , 2007 .

[45]  Ronald Fedkiw,et al.  Level set methods and dynamic implicit surfaces , 2002, Applied mathematical sciences.

[46]  A. A. Amsden,et al.  A simplified MAC technique for incompressible fluid flow calculations , 1970 .

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

[48]  Rüdiger Schwarze,et al.  Numerical simulation of a single rising bubble by VOF with surface compression , 2013 .

[49]  Pao-Hsiung Chiu,et al.  A conservative phase field method for solving incompressible two-phase flows , 2011, J. Comput. Phys..

[50]  Peter Stansby,et al.  The initial stages of dam-break flow , 1998, Journal of Fluid Mechanics.