The nonlinear inhomogeneous Galbrun-Equation: Derivation and possible Ways to solve numerically Marcus GUETTLER 1 ; Steffen MARBURG 1 1 Universität der Bundeswehr München, Germany ABSTRACT In recent years, the field of aeroacoustics has gained more attention in engineering developments like low noise ventilators, HVAC systems in electric vehicles, passenger cabins of commercial aircraft as well as novel engine designs of airplanes in general, to fulfill regulations of low level noise exposure. In order to close the gap between expensive prototypes with experimental testing and short development periods under high cost pressure, numerical simulations are state of the art for virtual prototyping. Numerous theories and acoustic analogies have been developed to investigate the effect of flow combined with acoustic radi- ation and propagation. Important contributions are known as the Linearized Euler Equations (LEE) or the Acoustic Perturbation Equations (APE). A different approach is pursued following the work of Galbrun. His displacement based linear formulation of aeroacoustics is extended to account for nonlinear effects as well as acoustic sources in turbulent flow. Difficulties arise when solving the Galbrun equation numerically. Therefore some already established numerical techniques that can be used to cope with these problems such as the Finite-Element-Method and the Discontinuous-Galerkin-Method are proposed. Keywords: Aeroacoustics, Flow-Noise, Galbrun I-INCE Classification of Subjects Number(s): 04.3 1. INTRODUCTION In the field of aeroacoustics many theories and algorithms have been developed in order to give a confident estimation of sound generated and convected by moving medium such as air or water. While this engineering field originally enlivened by the famous acoustic analogy of Lighthill (1) and his followers like Curle (2) and Ffowcs Williams and Hawkings (3), many other attempts to cope with flow induced noise have been published such as the Linearized Euler Equations (LEE) (4), the Acoustic Perturbation Equations (APE) (5) or the Galbrun Equation for sound propagation in flow. While most of the theories utilize an Eulerian description of continuum mechanics, Galbrun (6) used a mixed Eulerian/Lagrange formulation in order to reduce all desired physical quantities to one - the perturbation of the particle displacement. With this approach the amount of unknowns decrease to the particle displacement perturbation vector field. All other quantities such as the fluid pressure or the particle velocity can be calculated once the particle displacement is known (7, 8). Besides this reduction of unknowns the coupling of flow acoustics with structural vibration of adjacent parts can be accomplished in a straight forward manner since in structural mechanics the unknown vector field is the particle displacement, too cf. (9). The disadvantage of this approach is that the equations become more complicated resulting in numerous difficulties when trying to solve numerically, cf. (9, 10, 11). Several publications such as (12, 13, 14) deal with these problems and give more or less attempts for a solution to the numerical difficulties. Up to this point the authors have no evidence of a successful and convincing solution when solving the Galbrun equation in its original displacement based formulation. Besides these difficulties, utilizing Galbrun’s approach seems to be suitable in order to take into account the influence of the acoustics on the flow as a back reaction, as pointed out by Bonnet-Ben Dhia (15). One can think of such a situation when dealing with Helmholtz resonators which are excited by the flow and create acoustic sound pressure levels that are in the order of the flow pressure. In this paper, following the work of Brazier and Minotti et al. (7, 8), the Galbrun equation is extended to a nonlinear formulation with source terms by means of a fluctuating volume force on the right hand side. In addition this formulation shows that neither gradients of the reference flow pressure nor the divergence of the friction related stress tensor can be a source of sound. Furthermore because of the nonlinear character of the governing equations, the influence of the sound pressure waves on the flow can be taken into account. Inter-noise 2014 Page 1 of 10