Numerical integration of implicit functions for the initialization of the VOF function

Abstract A numerical method of initializing cell volume fraction demarcated by implicitly defined fluid interfaces is presented. Each cell of the computational domain is examined for the presence of the reference phase. When a cell is not full or empty, limits are found that allow volume fraction to be computed by numerical integration. The method enlists a number of algorithms including root finding and minimum search on an oriented segment, a preconditioned conjugate gradient minimum search on a cell face and a double Gauss–Legendre integration with a variable number of nodes, among others. Practical multi-phase fluid examples in two- and three-dimensions are presented to demonstrate the accuracy and robustness of the method.

[1]  R. Brent.  Table errata: Algorithms for minimization without derivatives (Prentice-Hall, Englewood Cliffs, N. J., 1973) , 1975 .

[2]  H. H. Rosenbrock,et al.  An Automatic Method for Finding the Greatest or Least Value of a Function , 1960, Comput. J..

[3]  S. Zaleski,et al.  Analytical relations connecting linear interfaces and volume fractions in rectangular grids , 2000 .

[4]  E. Polak,et al.  Note sur la convergence de méthodes de directions conjuguées , 1969 .

[5]  William H. Press,et al.  The Art of Scientific Computing Second Edition , 1998 .

[6]  Tim B. Swartz,et al.  Approximating Integrals Via Monte Carlo and Deterministic Methods , 2000 .

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

[8]  S. Cummins,et al.  Estimating curvature from volume fractions , 2005 .

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

[10]  William H. Press,et al.  Numerical recipes in C. The art of scientific computing , 1987 .

[11]  J. Shewchuk.  An Introduction to the Conjugate Gradient Method Without the Agonizing Pain , 1994 .

[12]  Ruben Scardovelli,et al.  Instability growth rate of two-phase mixing layers from a linear eigenvalue problem and an initial-value problem , 2010 .

[13]  Thomas Hahn,et al.  Cuba - a library for multidimensional numerical integration , 2004, Comput. Phys. Commun..

[14]  S. Zaleski,et al.  DIRECT NUMERICAL SIMULATION OF FREE-SURFACE AND INTERFACIAL FLOW , 1999 .

[15]  Sandro Manservisi,et al.  On the properties and limitations of the height function method in two-dimensional Cartesian geometry , 2011, J. Comput. Phys..