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# Conference_programme: 7.1 - Advances on numerical methods for computational acoustics

###### Lecture: Advanced Finite Element Formulation for the Convective Wave Equation

Author(s): Kaltenbacher Manfred, Hüppe Andreas

Summary:
The convective wave equation (CWE) describes acoustic wave propagation in flowing media. A systematic derivation of CWE has been provided in [1] starting at the full set of compressible flow equations and applying a perturbation ansatz. The final partial differential equation is similar to the standard wave equation, but instead of the second order partial time derivative it has a second order substantial time derivative. Furthermore, in computational aeroacoustics, the reformulation of the acoustic perturbation equations (APE) leads to the same convective wave operator and the substantial derivative of the incompressible flow pressure as a source term. This partial differential equation has again the scalar acoustic potential as the search for quantity and has been named perturbed acoustic wave equation (PCWE) [2]. Thereby, it can be shown that a standard Finite-Element (FE) formulation leads to eigenvalues with a strong real part. Furthermore, the magnitude of the eigenvalues scale with the Mach number of the flow. \nIn our contribution, we show the derivation of a stable FE formulation by applying an appropriate transformation of the standard FE formulation. Furthermore, we demonstrate by an eigenvalue analysis the strong reduction of the real part of the eigenvalues.\n\n[1] Allan D. Pierce: Wave equation for sound in fluids with unsteady inhomogeneous flow. J. Acoust. Soc. Am. 87, 2292, 1990\n\n[2] M. Kaltenbacher, A. Hüppe, A. Reppenhagen, F. Zenger, S. Becker: Computational Aeroacoustics for Rotating Systems with Application to an Axial Fan. AIAA 55(11), 2017\n\n