Abstract
This paper presents a high-order accurate stabilised finite element formulation for the simulation of transient inviscid flow problems in deformable domains. This work represents an extension of the methodology described in Sevilla et al. (2013), where a high-order stabilised finite element formulation was used as an efficient alternative for the simulation of steady flow problems of aerodynamic interest. The proposed methodology combines the Streamline Upwind/Petrov-Galerkin method with the generalised-α method and employs an Arbitrary Lagrangian Eulerian (ALE) description to account for the motion of the underlying mesh. Two computational frameworks, based on the use of reference and spatial variables are presented, discussed and thoroughly compared. In the process, a tailor-made discrete geometric conservation law is derived in order to ensure that a uniform flow field is exactly reproduced. Several numerical examples are presented in order to illustrate the performance of the proposed methodology. The results demonstrate the optimal approximation properties of both spatial and temporal discretisations as well as the crucial benefits, in terms of accuracy, of the exact satisfaction of the discrete geometric conservation law. In addition, the behaviour of the proposed high-order formulation is analysed in terms of the chosen stabilisation parameter. Finally, the benefits of using high-order approximations for the simulation of inviscid flows in moving domains are discussed by comparing low and high-order approximations for the solution of the Euler equations on a deformable domain.
Highlights
The last decade has seen an increasing interest on the study of high-order finite element methods for a vast range of engineering problems
The research in high-order methods for fluid problems during the last five years has been mainly focused on the development of accurate and efficient discontinuous Galerkin methods [8], see [9,10,11,12,13,14] to name but a few, and the generation of arbitrary order meshes suitable for high Reynolds number computations [15,16,17,18,19,20,21]
The term ASUPG corresponds to the consistent Streamline Upwind/Petrov-Galerkin (SUPG) stabilisation term, which is needed to counterbalance the negative diffusion introduced by the standard Galerkin formulation [43]
Summary
The last decade has seen an increasing interest on the study of high-order finite element methods for a vast range of engineering problems These numerical techniques have shown the potential to offer the same accuracy as low-order methods yet with a reduced computational cost. In the area of computational fluid dynamics, high-order methods have significantly attracted the attention of the research community over the last years [6] This has been partially motivated by the ever increasing need of simulating high Reynolds number flows, where traditional finite volume methods, still predominant in industrial solvers, require an excessive number of degrees of freedom to achieve the desired level of accuracy [7]. Very little effort has been made on developing competitive high-order stabilised finite element formulations With this in mind, this paper advocates for the use of an ALE high-order stabilised SUPG finite element formulation for the simulation of inviscid fluid flows in deformable domains.
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