This paper describes a novel numerical model aiming at solving moving-boundary problems such as free-surface flows or fluid-structure interaction. This model uses a moving-grid technique to solve the Navier--Stokes equations expressed in the arbitrary Lagrangian--Eulerian kinematics. The discretization in space is based on the spectral element method. The coupling of the fluid equations and the moving-grid equations is essentially done through the conditions on the moving boundaries. Two- and three-dimensional simulations are presented: translation and rotation of a cylinder in a fluid, and large-amplitude sloshing in a rectangular tank. The accuracy and robustness of the present numerical model is studied and discussed.
Deep Dive into Solution of moving-boundary problems by the spectral element method.
This paper describes a novel numerical model aiming at solving moving-boundary problems such as free-surface flows or fluid-structure interaction. This model uses a moving-grid technique to solve the Navier–Stokes equations expressed in the arbitrary Lagrangian–Eulerian kinematics. The discretization in space is based on the spectral element method. The coupling of the fluid equations and the moving-grid equations is essentially done through the conditions on the moving boundaries. Two- and three-dimensional simulations are presented: translation and rotation of a cylinder in a fluid, and large-amplitude sloshing in a rectangular tank. The accuracy and robustness of the present numerical model is studied and discussed.
With the advent of powerful computational resources like clusters of PCs or parallel computers the numericists are able to address more challenging problems involving multi-physics and multi-scale approaches. These problems cover a large spectrum of scientific and engineering applications. However, in this paper, for the sake of conciseness, we will restrict our attention to two specific problems, namely: free-surface flows and fluid-structure interaction.
Free-surface flows occur in many industrial applications: coating flows, vertical drawing of viscous fluids, jets, die flows, etc, and in environmental flows: ocean waves, off-shore engineering, coastal habitat and management, to name a few. Two review articles have been published in recent years and report the stateof-the-art of the field [1,2]. It can be observed that free-surface flows have been tackled by direct numerical simulation at low and moderate Reynolds numbers. This reality is essentially due to the nonlinear characters of the flow. On top of the nonlinearity associated to the Navier-Stokes equations themselves, here we deal with a complicated geometry which is changing in time and which is part of the solution itself. This accumulation of difficulties calls for elaborate algorithms and numerical techniques.
Fluid-structure interaction has been recognized for a long time as a real challenge. Indeed, this interaction is present in engineering problems like turbomachinery, aerospace applications: buffeting, acoustics, and also in biomedical flows like blood flow in the coronary arteries. Fluid-structure interaction is also encountered in the field of vortex-induced vibrations having many important marine applications (e.g related to oil exploration, cable dynamics, etc.). It is only at the present time that this type of interaction for three-dimensional cases appears to be feasible as the necessary computing power becomes available. On one hand, the computational fluid dynamics (CFD) codes integrate the full steady state or transient Navier-Stokes equations which govern the dynamics of a viscous Newtonian fluid. They mostly use finite volume or finite element approximations. On the other hand, the computational solid mechanics (CSM) codes integrate the dynamics of various solid models, incorporating for example, classical infinitesimal linear elasticity, nonlinear finite elasticity with large deformations, plasticity, visco-elasticity, etc. These problems are also highly nonlinear with respect to the complicated geometries at hand. The combination of the nonlinearities of the mathematical models for the constitutive relations and for the geometrical behaviour has called for a robust approach able to deal with all the complexities and intricacies. The finite element method (FEM) with the isoparametric elements has emerged as the leading technology and methodology in CSM.
In the present paper, the methodological framework is the same for the fluid and the solid parts and rests upon the spectral element method [3][4][5][6]. With this choice the space discretization is similar for both problems. As in free-surface flows and fluid-structure interaction the geometry is deforming and moving, it is needed to use the arbitrary Lagrangian-Eulerian (ALE) formulation [7][8][9][10]. This formulation allows to treat the full geometrical problem with respect to a reference configuration that is arbitrarily chosen. A mapping is introduced to ease the description of the current configuration with respect to a reference configuration. This process leads to an ALE velocity which will be related to a grid velocity.
In Section 2, the mathematical models will be presented with the associated weak formulations in the ALE context. Section 3 will be devoted to space and time discretizations. Section 4 will describe the numerical algorithms for the moving-grid technique. Section 5 will present numerical results and the final section will draw some conclusions.
A moving boundary-fitted grid technique has been chosen to simulate the unsteady part of the boundary in our computations. In the particular cases dealt with in this paper, the unsteady part of the boundary can be either the free surface in case of free-surface flows, or for fluid-structure interaction problems, the interface between the fluid and the structure. This choice of a surface-tracking technique is primarily based on accuracy requirements. With this group of techniques, the grid is configured to conform to the shape of the interface, and thus adapts continuously-at each time step-to it and therefore provides an accurate description of the moving boundary to express the related kinematic and/or dynamic boundary conditions.
The moving-boundary incompressible Newtonian fluid flows considered in this paper, are governed by the Navier-Stokes equations comprising the momentum equation and the divergence-free condition. In the ALE formulation, a mixed kinematic description is employed: a Lagrangian descri
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