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We report in this work a numerical analysis of the sloshing of a squared tank partially filled with a domestic vegetable oil. The tank is subject to controlled motions with a shake table. The free-surface evolution is captured using ultrasonic sensors and an image capturing method. Only confirmed data within the error range is reported. Filling depth, imposed amplitude and frequency effects on the sloshing wave pattern are specifically evaluated. The experiments also reveal the nonlinear wave behavior. The numerical model is based on a stabilized finite element method of the variational multi-scale type. The free-surface is captured using a level set technique developed to be used with adaptive meshes in Arbitrary Lagrangian–Eulerian framework. The numerical results are compared with the experiments for different sloshing conditions near the first sloshing mode. The simulations satisfactorily match the experiments, providing a reliable tool for the analysis of this kind of problems. [ABSTRACT FROM AUTHOR]

In this article we analyse a stabilized finite element formulation recently proposed to approximate viscoelastic fluid flows. The formulation has shown to have accuracy and robustness in the different benchmarks tested in the viscoelastic framework, and permitting the use of equal interpolation of the unknown fields. We first present results about a linearized subproblem, for which well-posedness and stability results can be proved. Then, the semidiscrete nonlinear time-dependent case is addressed using a fixed point theorem, which allows us to prove existence of a semidiscrete solution, along with error estimates. [ABSTRACT FROM AUTHOR]

In this paper we present the numerical analysis of a three-field stabilized finite element formulation recently proposed to approximate viscoelastic flows. The three-field viscoelastic fluid flow problem may suffer from two types of numerical instabilities: on the one hand we have the two infsup conditions related to the mixed nature problem and, on the other, the convective nature of the momentum and constitutive equations may produce global and local oscillations in the numerical approximation. Both can be overcome by resorting from the standard Galerkin method to a stabilized formulation. The one presented here is based on the subgrid scale concept, in which unresolvable scales of the continuous solution are approximately accounted for. In particular, the approach developed herein is based on the decomposition into their finite element component and a subscale, which is approximated properly to yield a stable formulation. The analyzed problem corresponds to a linearized version of the Navier–Stokes/Oldroyd-B case where the advection velocity of the momentum equation and the non-linear terms in the constitutive equation are treated using a fixed point strategy for the velocity and the velocity gradient. The proposed method permits the resolution of the problem using arbitrary interpolations for all the unknowns. We describe some important ingredients related to the design of the formulation and present the results of its numerical analysis. It is shown that the formulation is stable and optimally convergent for small Weissenberg numbers, independently of the interpolation used. [ABSTRACT FROM AUTHOR]

In this paper, three-field finite element stabilized formulations are proposed for the numerical solution of incompressible viscoelastic flows. These methods allow one to use equal interpolation for the problem unknowns σ – u – p (elastic deviatoric stress–velocity–pressure) and to stabilize dominant convective terms. Starting from residual-based stabilized formulations, the proposed method introduces a term-by-term stabilization which is shown to have a superior behavior when there are stress singularities. A general discontinuity-capturing technique for the elastic stress component is also proposed, which allows one to eliminate the local oscillations that can appear when the Weissenberg number We is high and the fluid flow finds an abrupt change in the geometry. The formulations are tested in the classical 4:1 planar contraction benchmark up to We = 5 in the inertial case, with Reynolds number Re = 1 , and up to We = 6.5 in the quasi non-inertial case, with Re = 0.01 . The standard Oldroyd-B constitutive model is used for the rheological behavior and linear and quadratic elements for the spatial approximation. [ABSTRACT FROM AUTHOR]

The log-conformation reformulation, originally proposed by Fattal and Kupferman (2004), allows computing incompressible viscoelastic problems with high Weissenberg numbers which are impossible to solve with the typical three-field formulation. By following this approach, in this work we develop a new stabilized finite element formulation based on the logarithmic reformulation using the Variational Multiscale (VMS) method as stabilization technique, together with a modified log-conformation formulation. Our approach follows the term-by-term stabilization proposed by Castillo and Codina (2014) for the standard formulation, which is more effective when there are stress singularities. The formulation can be used when the relaxation parameter is set to zero, and permits a direct steady numerical resolution. The formulation is validated in the classical benchmark flow past a cylinder and in the well-known planar contraction 4:1, achieving very accurate, stable and mesh independent results for highly elastic fluids. [ABSTRACT FROM AUTHOR]