Linearized Navier-Stokes Equations for Aeroacoustics using Stabilized Finite Elements: Boundary Conditions and Industrial Application to Aft-Fan Noise Propagation

Aloïs Bissuel (1, 2) , Grégoire Allaire (1) , Laurent Daumas (2) , Sébastien Barré (2) , Floriane Rey (2)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
2 Dassault Aviation

In this paper, a numerical method for solving the linearized Navier-Stokes equations is presented for aeroacoustic sound propagation problem. The Navier-Stokes equations are linearized in the frequency domain. The fan noise of jet engine is emitted nearly selectively at certain frequencies, which depend on the rotation velocity of the fan. A frequency domain approach is highly suitable for this kind of problem, instead of a costly time-dependent simulation which can handle a large range of frequencies depending on the time step and the mesh. The calculations presented here were all made using Aether, a Navier-Stokes code which uses finite elements stabilized with SUPG (Streamline Upwind Galerkin). Automatic code differentiation was used to linearize this code. Entropy variables bring interesting mathematical properties to the numerical scheme, but also prevent the easy implementation of boundary conditions. For instance, the pressure is a non-linear combination of the entropy variables. Imposing a pressure variation needs a linearization of this relation which is detailed herein. The performance of different types of boundary conditions used to impose the acoustic pressure variation inside the engine is studied in detail. Finally, a very surprising effect of the SUPG scheme was to transform a homogeneous Dirichlet boundary condition on all variables to a transparent one which is able to let only outgoing waves pass * Dassault Aviation 78, quai Marcel 1 through with no incoming wave. A one-dimensional toy model is given to explain how SUPG brings about this transformation. The last part of the article is dedicated to an industrial test case. The geometry of a model turbine from the Clean Sky European project was used for sound propagation of the fan exhaust noise of a jet engine. Computations on several modes with increasing complexities were done and the results compared to a boundary element method which served as a reference when no mean flow is present. Results of a computation with a mean flow are shown.

Domaines: Analyse numérique [math.NA]

Aloïs Bissuel, Grégoire Allaire, Laurent Daumas, Sébastien Barré, Floriane Rey. Linearized Navier-Stokes Equations for Aeroacoustics using Stabilized Finite Elements: Boundary Conditions and Industrial Application to Aft-Fan Noise Propagation. Computers and Fluids, inPress. ⟨hal-01695874⟩