A higher-order finite element method for the linearised Euler equations

The propagation of sound in complex flows is a critical issue for many industries. Most Computational Aero Acoustics (CAA) propagation solvers currently in use in industry are based on the full potential theory which cannot take into account the propagation through complex sheared flows. To better represent the physics at hand, particularly when dealing with turbomachinery noise radiating through engine exhausts including sound refraction through the jet shear layer, it is necessary to solve the Linearised Euler Equations (LEEs). The most popular method for solving the LEEs in the time domain is the high-order Discontinuous Galerkin Method (DGM). Most of the issues associated with this approach (stability, time step design, impedance boundary condition) can be avoided by resolving the LEEs in the frequency domain. However, the method of choice when dealing with this type of equations on unstructured grids is the Finite Element Method (FEM). At high frequency, the standard FEM is known to suffer from large dispersion errors and its straightforward application to the LEEs, which involve up to five unknowns in 3D, is computationally inefficient. An alternative approach based on a higher-order FEM is proposed in this paper. The use of higher-order shape functions allows to reduce significantly the number of degrees of freedom, while maintaining a good level of accuracy. An axisymmetric formulation of the LEEs is developed in conjunction with perfectly matched layers at the inlet and at the far-field boundaries. The solver is validated with the problem of sound radiation from a semi-infinite duct in the presence of flow.