An Adaptive Level Set Approach for Incompressible Two-Phase Flows

We present a numerical method using the level set approach for solving incompressible two-phase flow with surface tension. In the level set approach, the free surface is represented as the zero level set of a smooth function; this has the effect of replacing the advection of density, which has steep gradients at the free surface, with the advection of the level set function, which is smooth. In addition, the free surface can merge or break up with no special treatment. We maintain the level set function as the signed distance from the free surface in order to accurately compute flows with high density ratios and stiff surface tension effects. In this work, we couple the level set scheme to an adaptive projection method for the incompressible Navier?Stokes equations, in order to achieve higher resolution of the free surface with a minimum of additional expense. We present two-dimensional axisymmetric and fully three-dimensional results of air bubble and water drop computations.

[1]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[2]  H. J.,et al.  Hydrodynamics , 1924, Nature.

[3]  Michael Selwyn Longuet-Higgins,et al.  The deformation of steep surface waves on water - I. A numerical method of computation , 1976, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

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

[5]  A. Brandt,et al.  The Multi-Grid Method for the Diffusion Equation with Strongly Discontinuous Coefficients , 1981 .

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

[7]  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.

[8]  P. Colella.  A Direct Eulerian MUSCL Scheme for Gas Dynamics , 1985 .

[9]  Thomas S. Lundgren,et al.  Oscillations of drops in zero gravity with weak viscous effects , 1988, Journal of Fluid Mechanics.

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

[11]  ShuChi-Wang,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes, II , 1989 .

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

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

[14]  A. Harten.  ENO schemes with subcell resolution , 1989 .

[15]  Narayan V. Deshpande.  Fluid Mechanics of Bubble Growth and Collapse in a Thermal Ink Jet Printhead , 1989, Photonics West - Lasers and Applications in Science and Engineering.

[16]  S. Osher,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes,II , 1989 .

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

[18]  A. Prosperetti,et al.  Bubble entrainment by the impact of drops on liquid surfaces , 1990, Journal of Fluid Mechanics.

[19]  Phillip Colella,et al.  An efficient second-order projection method for viscous incompressible flow , 1991 .

[20]  Isidore Rigoutsos,et al.  An algorithm for point clustering and grid generation , 1991, IEEE Trans. Syst. Man Cybern..

[21]  Thomas S. Lundgren,et al.  Vortex ring bubbles , 1991, Journal of Fluid Mechanics.

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

[23]  Numerical Solution of Free Boundary Problems in Solids Mechanics , 1992 .

[24]  E. Rouy,et al.  A viscosity solutions approach to shape-from-shading , 1992 .

[25]  J. Bell,et al.  A Second-Order Projection Method for Variable- Density Flows* , 1992 .

[26]  U. Ozen,et al.  A front-tracking method for viscous, incompressible, multi-fluid flows , 1992 .

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

[28]  J. Best.  The formation of toroidal bubbles upon the collapse of transient cavities , 1993, Journal of Fluid Mechanics.

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

[30]  Osamu Tatebe,et al.  The multigrid preconditioned conjugate gradient method , 1993 .

[31]  Louis H. Howell,et al.  A multilevel adaptive projection method for unsteady incompressible flow , 1993 .

[32]  Joel C. W. Rogers,et al.  A numerical algorithm for hydrodynamic free boundary problems , 1993 .

[33]  John R. Blake,et al.  Gas bubbles bursting at a free surface , 1993, Journal of Fluid Mechanics.

[34]  Marsha Berger,et al.  Three-Dimensional Adaptive Mesh Refinement for Hyperbolic Conservation Laws , 1994, SIAM J. Sci. Comput..

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

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

[37]  W. Shyy,et al.  ELAFINT - A mixed Eulerian-Lagrangian method for fluid flows with complex and moving boundaries , 1994 .

[38]  D. Chambers,et al.  Relaxation spectra of surface waves , 1995 .

[39]  John B. Bell,et al.  Cartesian grid method for unsteady compressible flow in irregular regions , 1995 .

[40]  R. Milne.  An Adaptive Level Set Method , 1995 .

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

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

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

[44]  Vladimir Hlavacek,et al.  Numerical Solution of Free-Boundary Problems , 1996 .

[45]  S. Osher,et al.  A Level Set Formulation of Eulerian Interface Capturing Methods for Incompressible Fluid Flows , 1996 .

[46]  Michael L. Minion.  On the Stability of Godunov-Projection Methods for Incompressible Flow , 1996 .

[47]  Asghar Esmaeeli,et al.  An inverse energy cascade in two-dimensional low Reynolds number bubbly flows , 1996, Journal of Fluid Mechanics.

[48]  John B. Bell,et al.  A Cartesian Grid Projection Method for the Incompressible Euler Equations in Complex Geometries , 1994, SIAM J. Sci. Comput..

[49]  Peter Smereka,et al.  Axisymmetric free boundary problems , 1997, Journal of Fluid Mechanics.

[50]  Ali Borhan,et al.  Thermocapillary motion of deformable drops at finite Reynolds and Marangoni numbers , 1997 .

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

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

[53]  D. Juric,et al.  Computations of Boiling Flows , 1998 .

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

[55]  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..