## Abstract

A two-phase 2D model that combines the volume of fluid (VOF) method with implicit staggered finite volumes discretization of the Navier-Stokes equation is presented. Staggered finite volumes are developed on the basis of nonconforming Crouzeix-Raviart finite elements, where all components of the velocity lie in the middle of the element edges and the pressure degrees of freedom are found in the centers of mass of the elements. Staggered finite Volumes extend marker and cell (MAC) regular staggered grids to unstructured mesh. A linear saddle point problem, resulting from either the discretization or the Newton method, is solved for all unknown pressures and velocities. Interface is represented with spline interpolants which follow the VOF distribution. Adaptive mesh refinement is used to obtain a high level of uniform refining at the domain of dependence of the interface. The aligned grid is obtained by irregular refining of the cells which are intersected by a curve. The boundaries of its elements coincide with the slope segments going through the intersections of the curve with the underlying regular elements boundary. The deformable computational grids are used only to discretize the Navier-Stokes equation. The advection of volume fractions is done on the advection mesh, which corresponds to highest regular refining on the computational grid. Approximation of the surface tension on spline interpolants offers a straightforward way to describe correctly the pressure jumps on interface-fitted staggered grids. This allows deletion of the anomalous currents around a statical bubble and their effective reduction in real simulations. On the aligned grid, the continuity of the viscous stress is modeled exactly due to the finite volume approach. Using the proposed numerical techniques, single bubble rise is analyzed.

Original language | English (US) |
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Pages (from-to) | 302-335 |

Number of pages | 34 |

Journal | Journal of Computational Physics |

Volume | 166 |

Issue number | 2 |

DOIs | |

State | Published - Jan 20 2001 |

Externally published | Yes |

### Bibliographical note

Funding Information:We acknowledge support from the SFB 412 and the CNRS-DFG French–German research program in computational fluid mechanics. The support of the UG numerical group has been very helpful in the implementation. I.G. is grateful to D. Gueyffier, H. Rentz-Reichert, W. J. Rider, M. Rieber, R. Scardovelli, and S. Zaleski for useful discussions. The authors thank G. Mazurkevich, W. J. Rider, and A. Schatz for their critical readings of the manuscript.

## Keywords

- Buoyant bubbles
- Cubic splines
- Finite volume method
- Laplace law
- Linear interface interpolant
- Multigrid
- Navier-Stokes equation
- Rotated elements
- Saddle point problem
- Two-phase flow
- Volume of fluid method

## ASJC Scopus subject areas

- Numerical Analysis
- Modeling and Simulation
- Physics and Astronomy (miscellaneous)
- General Physics and Astronomy
- Computer Science Applications
- Computational Mathematics
- Applied Mathematics