| Résumé | This paper presents a three-dimensional solution algorithm for solving free surface flows including the effect of near wall viscous dissipation. Slip boundary conditions that could take into account the viscous shear at the boundary are proposed in a context that is well suited to free surface flow applications. The proposed finite element method uses a Level-Set formulation on fixed meshes and, consequently, the liquid/gas interface is generally located inside mesh elements. Standard interpolation of flow variables inside partially filled elements leads to an inappropriate treatment of gravity forces, pressure gradient and inertia which can generate spurious oscillations. To alleviate these issues, we use enriched shape functions for the pressure that introduce a discontinuity in the normal pressure gradient at the interface. The additional pressure degree of freedom is eliminated at the elementary system level by static condensation thus having no impact on the structure of the linear system matrix. This approach is well-suited for gravity-driven free surface problems such as violent sloshing inside tanks. The algorithm is first verified and validated on simpler problems for which analytical or experimental results are available. Then it is applied to the study of sloshing where it is validated against a robust set of experiment from Delorme et al. [1] and Souto-Iglesias et al. [2, 3] for the water and oil flow inside an angularly oscillating rectangular tank. The proposed approach to include viscous dissipation at the wall proves to be very effective in capturing the significant differences in flow dynamics between the two liquids having different viscosities. Moreover, the enriched pressure shape functions results in stable, oscillation free solution thus representing an efficient way to accurately solve complex gravity-driven free surface problems without requiring re-meshing or mesh refinement. |
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