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A Numerical Method for Fluid Flows with Complex Free Surfaces Andrea Bonito 1∗ and Alexandre Caboussat 2 , Marco Picasso 3 , and Jacques Rappaz 3 1 Department of Mathematics, University of Maryland, College Park, MD 20742-4015, USA andrea.bonito@a3.epfl.ch 2 Department of Mathematics, University of Houston, 77204-3008, Houston, TX, USA caboussat@math.uh.edu 3 Institute of Analysis and Scientific Computing, Ecole Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland {marco.picasso;jacques.rappaz}@epfl.ch Summary. A numerical method for the simulation of fluid flows with complex free surfaces is presented. The liquid is assumed to be a Newtonian or a viscoelastic fluid. The compressible effects of the surrounding gas are taken into account, as well as surface tension forces. An Eulerian approach based on the volume-of-fluid formulation is chosen. A time splitting algorithm, together with a two-grids method, allows the various physical phenomena to be decoupled. A chronological approach is adopted to highlight the successive improvements of the model and the wide range of applications. Numerical results show the potentialities of the method. 1 Introduction Complex free surface phenomena involving Newtonian and/or non-Newtonian flows are nowadays a topic of active research in many fields of physics, engineering or bioengineering. The literature contains numerous models for complex liquid-gas free surfaces problems, see, e.g., [FCD + 06, SZ99]. For instance, when considering the injection of a liquid in a complex cavity initially filled with gas, an Eulerian approach is generally adopted in order to catch the topology changes of the liquid region. Such two-phases flows are computationally expensive in three space dimensions since (at least) both the velocity and pressure must be computed at each grid point of the whole liquid-gas domain. The purpose of this article is to review a numerical model in order to compute complex free surface flows in three space dimensions. The features ∗ Partially supported by the Swiss National Science Foundation Fellowship PBEL2– 114311
188 A. Bonito et al. of the model are the following. A volume-of-fluid method is used to track the liquid domain, which can exhibit complex topology changes. The velocity field is computed only in the liquid region. The incompressible liquid can be modeled either as a Newtonian or as a viscoelastic fluid. The ideal gas law is used to compute the external pressure in the surrounding gas and the resulting force is added on the liquid-gas free surface. Surface tension effects can also be taken into account on the liquid-gas free surface. The complete description of the model can be found in [MPR99, MPR03, CPR05, Cab06, BPL06]. The numerical model is based on a time-splitting approach [Glo03] and a two-grids method. This allows advection, diffusion and viscoelastic phenomena to be decoupled, as well as the treatment of the liquid and gas phases. Finite element techniques [FF92] are used to solve the diffusion phenomena using an unstructured mesh of the cavity containing the liquid. A forward characteristic method [Pir89] on a structured grid allows advection phenomena to be solved efficiently. The article is structured as follows. In Section 2, the simplest model is presented: the liquid is an incompressible Newtonian fluid, the effects of the surrounding gas and surface tension are neglected. The effects of the surrounding gas are described in Section 3, those of the surface tension in Section 4. Finally, the case of a viscoelastic liquid is considered in Section 5. Numerical results are presented throughout the text and illustrate the capabilities and improvements of the model. 2 Modeling of an Incompressible Newtonian Fluid with a Free Surface 2.1 Governing Equations The model presented in this section has already been published in [MPR99, MPR03]. Let Λ, with a boundary ∂Λ, be a cavity of R 3 in which a liquid must be confined, and let T>0 be the final time of simulation. For any given time t ∈ (0,T), let Ω t , with a boundary ∂Ω t , be the domain occupied by the liquid, let Γ t = ∂Ω t \ ∂Λ be the free surface between the liquid and the surrounding gas and let Q T be the space-time domain containing the liquid, i.e. Q T = {(x, t) :x ∈ Ω t , 0
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Partial Differential Equations
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Partial Differential Equations Mode
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Dedicated to Olivier Pironneau
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VIII Preface computers has been at
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Contents List of Contributors .....
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List of Contributors Yves Achdou UF
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List of Contributors XV Claude Le B
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Discontinuous Galerkin Methods Vive
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Discontinuous Galerkin Methods 5 2
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Discontinuous Galerkin Methods 7 g
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Discontinuous Galerkin Methods 9 (
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Discontinuous Galerkin Methods 15 W
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Let a h and b h denote the bilinear
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Discontinuous Galerkin Methods 19 t
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Discontinuous Galerkin Methods 21 l
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Table 1. Primal DG for transport Di
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Discontinuous Galerkin Methods 25 [
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Mixed Finite Element Methods on Pol
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Mixed FE Methods on Polyhedral Mesh
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4 Hybridization and Condensation Mi
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is symmetric and positive definite,
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with some coefficient α ∈ R wher
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Higher Order Time Stepping for Seco
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u n+1 h Optimal Higher Order Time D
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