A high-order stabilised ALE finite element formulation for the Euler equations on deformable domains

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.