A parallel cell-centered adaptive level set framework for efficient simulation of two-phase flows with subcycling and non-subcycling

Abstract We develop a unified adaptive level set (LS) framework using the multi-level collocated grid for incompressible two-phase flows. This framework allows us to advance all variables level by level using either the subcycling or the non-subcycling method such that the data advancement on each level is fully decoupled. A series of synchronization operations are designed to keep the momentum and mass conserved across all levels. A multi-level re-initialization method for the LS function is also proposed, which greatly improves the mass conservation of the two-phase flows. The collocated grid allows the use of a single set of differential schemes and interpolation operations for all variables, which greatly simplifies the numerical implementation. The capability and robustness of the computational framework are validated by a variety of canonical problems, including the inviscid shear layer, gravity wave, rising bubble, and Rayleigh-Taylor instability. It is shown that the present multi-level scheme can accurately resolve the interfaces of the two-phase flows with gravitational and surface tension effects while having good momentum and energy conservation. At last, a three-dimensional dam breaking problem is simulated to show the efficiency and significant speedup of the proposed framework.

[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]  Phillip Colella,et al.  A cell-centered adaptive projection method for the incompressible Navier-Stokes equations in three dimensions , 2007, J. Comput. Phys..

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

[4]  Ricardo Serfaty,et al.  A fully adaptive front tracking method for the simulation of two phase flows , 2014 .

[5]  Mohammad Mirzadeh,et al.  Parallel level-set methods on adaptive tree-based grids , 2016, J. Comput. Phys..

[6]  M. Al-Marouf,et al.  A versatile embedded boundary adaptive mesh method for compressible flow in complex geometry , 2017, J. Comput. Phys..

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

[8]  C. Shu,et al.  Stencil adaptive diffuse interface method for simulation of two-dimensional incompressible multiphase flows , 2010 .

[9]  M. Zingale,et al.  Meeting the Challenges of Modeling Astrophysical Thermonuclear Explosions: Castro, Maestro, and the AMReX Astrophysics Suite , 2017, 1711.06203.

[10]  Oscar Antepara,et al.  Numerical study of rising bubbles with path instability using conservative level-set and adaptive mesh refinement , 2019, Computers & Fluids.

[11]  Frédéric Gibou,et al.  A stable projection method for the incompressible Navier-Stokes equations on arbitrary geometries and adaptive Quad/Octrees , 2015, J. Comput. Phys..

[12]  Ronald Fedkiw,et al.  A review of level-set methods and some recent applications , 2018, J. Comput. Phys..

[13]  Guang Lin,et al.  A mixed upwind/central WENO scheme for incompressible two-phase flows , 2019, J. Comput. Phys..

[14]  J A Sethian,et al.  A fast marching level set method for monotonically advancing fronts. , 1996, Proceedings of the National Academy of Sciences of the United States of America.

[15]  Frédéric Gibou,et al.  The island dynamics model on parallel quadtree grids , 2018, J. Comput. Phys..

[16]  Stéphane Popinet,et al.  A quadtree-adaptive multigrid solver for the Serre-Green-Naghdi equations , 2015, J. Comput. Phys..

[17]  R. Verzicco,et al.  Effects of the wind on the breaking of modulated wave trains , 2019, European Journal of Mechanics - B/Fluids.

[18]  Carsten Burstedde,et al.  p4est: Scalable Algorithms for Parallel Adaptive Mesh Refinement on Forests of Octrees , 2011, SIAM J. Sci. Comput..

[19]  Yadong Zeng.  Modelling Wave Energy Converter (WEC) pointer absorbers using AMR techniques with both subcycling and non-subcycling , 2020 .

[20]  Carsten Burstedde,et al.  Recursive Algorithms for Distributed Forests of Octrees , 2014, SIAM J. Sci. Comput..

[21]  John B. Bell,et al.  A Numerical Method for the Incompressible Navier-Stokes Equations Based on an Approximate Projection , 1996, SIAM J. Sci. Comput..

[22]  M. Minion,et al.  Accurate projection methods for the incompressible Navier—Stokes equations , 2001 .

[23]  Lian Shen,et al.  Simulation-based study of wind loads on semi-submersed object in ocean wave field , 2016 .

[24]  T. Ménard,et al.  Coupling level set/VOF/ghost fluid methods: Validation and application to 3D simulation of the primary break-up of a liquid jet , 2007 .

[25]  Aaron L. Fogelson,et al.  Stability of approximate projection methods on cell-centered grids , 2005 .

[26]  Frédéric Gibou,et al.  A second order accurate projection method for the incompressible Navier-Stokes equations on non-graded adaptive grids , 2006, J. Comput. Phys..

[27]  Takahiko Tanahashi,et al.  Numerical analysis of moving interfaces using a level set method coupled with adaptive mesh refinement , 2004 .

[28]  M. Minion.  A Projection Method for Locally Refined Grids , 1996 .

[29]  Elias Balaras,et al.  A direct-forcing embedded-boundary method with adaptive mesh refinement for fluid-structure interaction problems , 2010, J. Comput. Phys..

[30]  Boyce E. Griffith,et al.  A robust incompressible Navier-Stokes solver for high density ratio multiphase flows , 2018, J. Comput. Phys..

[31]  Boyce E. Griffith,et al.  An adaptive, formally second order accurate version of the immersed boundary method , 2007, J. Comput. Phys..

[32]  Frédéric Gibou,et al.  A level set based sharp interface method for the multiphase incompressible Navier-Stokes equations with phase change , 2007, J. Comput. Phys..

[33]  S. Popinet.  Gerris: a tree-based adaptive solver for the incompressible Euler equations in complex geometries , 2003 .

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

[35]  Marcus S. Day,et al.  AMReX: a framework for block-structured adaptive mesh refinement , 2019, J. Open Source Softw..

[36]  P. Colella,et al.  Local adaptive mesh refinement for shock hydrodynamics , 1989 .

[37]  S. Zaleski,et al.  Dynamics of jets produced by bursting bubbles , 2018 .

[38]  Mark Sussman,et al.  An Efficient, Interface-Preserving Level Set Redistancing Algorithm and Its Application to Interfacial Incompressible Fluid Flow , 1999, SIAM J. Sci. Comput..

[39]  Zixuan Yang,et al.  Direct numerical simulation of wind turbulence over breaking waves , 2018, Journal of Fluid Mechanics.

[40]  G. Tryggvason.  Numerical simulations of the Rayleigh-Taylor instability , 1988 .

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

[42]  Robert W. Anderson,et al.  Advances in patch-based adaptive mesh refinement scalability , 2016, J. Parallel Distributed Comput..

[43]  J. Sethian,et al.  A Fast Level Set Method for Propagating Interfaces , 1995 .

[44]  Daniel F. Martin,et al.  A Cell-Centered Adaptive Projection Method for the Incompressible Euler Equations , 2000 .

[45]  Amneet Pal Singh Bhalla,et al.  A DLM immersed boundary method based wave-structure interaction solver for high density ratio multiphase flows , 2019, J. Comput. Phys..

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

[47]  Yi-Hsiang Yu,et al.  Reynolds-Averaged Navier–Stokes simulation of the heave performance of a two-body floating-point absorber wave energy system , 2013 .

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

[49]  Jitendra Kumar,et al.  A novel consistent and well-balanced algorithm for simulations of multiphase flows on unstructured grids , 2017, J. Comput. Phys..

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

[51]  Andy R. Terrel,et al.  ForestClaw: Hybrid forest-of-octrees AMR for hyperbolic conservation laws , 2013, PARCO.

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

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

[54]  P. Colella,et al.  A Conservative Adaptive Projection Method for the Variable Density Incompressible Navier-Stokes Equations , 1998 .

[55]  P. Colella,et al.  An Adaptive Level Set Approach for Incompressible Two-Phase Flows , 1997 .

[56]  Michael Williamschen,et al.  Parallel Anisotropic Block-based Adaptive Mesh Refinement Algorithm For Three-dimensional Flows , 2013 .

[57]  P. Colella.  Multidimensional upwind methods for hyperbolic conservation laws , 1990 .

[58]  Frédéric Gibou,et al.  Level-set simulations of soluble surfactant driven flows , 2017, J. Comput. Phys..

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

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

[61]  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.

[62]  S. Sarkar,et al.  SOMAR-LES: A framework for multi-scale modeling of turbulent stratified oceanic flows , 2017 .

[63]  J. Li,et al.  Numerical simulation of moving contact line problems using a volume-of-fluid method , 2001 .

[64]  John B. Bell,et al.  An Adaptive Mesh Projection Method for Viscous Incompressible Flow , 1997, SIAM J. Sci. Comput..

[65]  Tony W. H. Sheu,et al.  Interface-preserving level set method for simulating dam-break flows , 2018, J. Comput. Phys..

[66]  Frédéric Gibou,et al.  A second order accurate level set method on non-graded adaptive cartesian grids , 2007, J. Comput. Phys..

[67]  Daniel F. Martin,et al.  Adaptive mesh, finite volume modeling of marine ice sheets , 2013, J. Comput. Phys..

[68]  Stéphane Popinet,et al.  An accurate adaptive solver for surface-tension-driven interfacial flows , 2009, J. Comput. Phys..

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

[70]  Boyce E. Griffith,et al.  A unified mathematical framework and an adaptive numerical method for fluid-structure interaction with rigid, deforming, and elastic bodies , 2013, J. Comput. Phys..

[71]  William J. Rider.  Filtering non‐solenoidal modes in numerical solutions of incompressible flows , 1998 .

[72]  Selene M.A. Guelli U. de Souza,et al.  A two-fluid model with a tensor closure model approach for free surface flow simulations , 2015 .

[73]  John B. Bell,et al.  Approximate Projection Methods: Part I. Inviscid Analysis , 2000, SIAM J. Sci. Comput..