In this paper we present a summary of numerical methods for solving free surface and two fluid flow problems. We will focus the attention on level set formulations extensively used in the context of the finite element method. In particular, numerical developments to achieve accurate solutions are described. Specific topics of the algorithms, like mass preservation and interface redefinition, are evaluated. To illustrate these aspects, numerical strategies previously developed are applied to the solution of a sloshing and a water column collapse problems. To assess the capabilities of these techniques, the numerical results are compared against each other and with experimental data. Considering these aspects, the present work is aimed to outline some well reported aspects of level set-like formulations.Fil: Cruchaga, Marcela. Universidad de Santiago de Chile; ChileFil: Battaglia, Laura. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Centro de Investigaciones En Metodos Computacionales. Universidad Nacional del Litoral. Centro de Investigaciones En Metodos Computacionales; ArgentinaFil: Storti, Mario Alberto. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Centro de Investigaciones En Metodos Computacionales. Universidad Nacional del Litoral. Centro de Investigaciones En Metodos Computacionales; ArgentinaFil: D'elia, Jorge. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Centro de Investigaciones En Metodos Computacionales. Universidad Nacional del Litoral. Centro de Investigaciones En Metodos Computacionales; Argentin

SUMMARYIn this work, a reinitialization procedure oriented to regularize the level set (LS) function field is presented. In LS approximations for two-fluid flow simulations, a scalar function indicates the presence of one or another phase and the interface between them. In general, the advection of such function produces a degradation of some properties of the LS function, such as the smoothness of the transition between phases and the correct position of the interface. The methodology introduced here, denominated bounded renormalization with continuous penalization, consists of solving by the finite element method a partial differential equation with certain distinguishing properties with the aim of keeping the desirable properties of the LS function. The performance of the strategy is evaluated for several typical cases in one, two and three-dimensional domains, for both the advection and the renormalization stages.

In this work, transient free surface flows of a viscous incompressible fluid are numerically solved through parallel computation. Transient free surface flows are boundary-value problems of the moving type that involve geometrical nonlinearities. In contrast to more conventional computational fluid dynamics problems, the computational flow domain is partially bounded by a free surface which is not known a priori, since its shape must be computed as part of the solution. In steady flow the free surface is obtained by an iterative process, but when the free surface evolves with time the problem is more difficult as it generates large distortions in the computational flow domain. The incompressible Navier-Stokes numerical solver is based on the finite element method with equal order elements for pressure and velocity (linear elements), and it uses a streamline upwind/Petrov-Galerkin (SUPG) scheme (Hughes, T. J. R., and Brooks, A. N., 1979, “A Multidimensional Upwind Scheme With no Crosswind Diffusion,” in Finite Element Methods for Convection Dominated Flows, ASME ed., 34. AMD, New York, pp. 19–35, and Brooks, A. N., and Hughes, T. J. R., 1982, “Streamline Upwind/Petrov-Galerkin Formulations for Convection Dominated Flows With Particular Emphasis on the Incompressible Navier-Stokes Equations,” Comput. Methods Appl. Mech. Eng., 32, pp. 199–259) combined with a Pressure-Stabilizing/Petrov-Galerkin (PSPG) one (Tezduyar, T. E., 1992, “Stablized Finite Element Formulations for Incompressible Flow Computations,” Adv. Appl. Mech., 28, pp. 1–44, and Tezduyar, T. E., Mittal, S., Ray, S. E., and Shih, R., 1992, “Incompressible Flow Computations With Stabilized Bilinear and Linear Equal Order Interpolation Velocity-Pressure Elements,” Comput. Methods Appl. Mech. Eng., 95, pp. 221–242). At each time step, the fluid equations are solved with constant pressure and null viscous traction conditions at the free surface and the velocities obtained in this way are used for updating the positions of the surface nodes. Then, a pseudo elastic problem is solved in the fluid domain in order to relocate the interior nodes so as to keep mesh distortion controlled. This has been implemented in the PETSc-FEM code (PETSc-FEM: a general purpose, parallel, multi-physics FEM program. GNU general public license (GPL), http://www.cimec.org.ar/petscfem) by running two parallel instances of the code and exchanging information between them. Some numerical examples are presented.

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. Highlights • Improved mass-preserving renormalization method for 3D two-fluid viscous flows • Report of new 3D experimental data for free-surface evolution in sloshing problems • 3D confirmation of 2D predictions for free and forced sloshing planar modes • Experimental validation for simulation of fully 3D sloshing cases • Effect of the initial conditions on the 3D simulation

Transient free surface flows are numerically simulated by a finite element interface capturing method based on a level set approach. The methodology consists of the solution of two-fluid viscous incompressible flows for a single domain, where the liquid phase is identified by positive values of the level set function, the gaseous phase by negative ones, and the free surface by the zero level set. The numerical solution at each time step is performed in three stages: (i) a two-fluid Navier-Stokes stage, (ii) an advection stage for the transport of the level set function, and (iii) a bounded reinitialization with continuous penalization stage for keeping smoothness of the level set function. The proposed procedure, and particularly the renormalization stage, are evaluated in three typical two and three-dimensional problems.

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SUMMARYWhen a Galerkin discretization of a boundary integral equation with a weakly singular kernel is performed over triangles, a double surface integral must be evaluated for each pair of them. If these pairs are not contiguous or not coincident, the kernel is regular and a Gauss-Legendre quadrature can be employed. When the pairs have a common edge or a common vertex, then edge and vertex weak singularities appear. If the pairs have both facets coincident, the whole integration domain is weakly singular. Taylor (IEEE Trans. Antenn. Propag. 2003; 51(7):1630-1637) proposed a systematic evaluation based on a reordering and partitioning of the integration domain, together with a use of the Duffy transformations in order to remove the singularities, in such a way that a Gauss-Legendre quadrature was performed on three coordinates with an analytic integration in the fourth coordinate. Since this scheme is a bit restrictive because it was designed for electromagnetic kernels, a full numerical quadrature is proposed in order to handle kernels with a weak singularity with a general framework. Numerical tests based on modifications of the one proposed by Wang and Atalla (Commun. Numer. Meth. Engng 1997; 13(11):885-890) are included.

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