Some Finite Element Applications in Frequency Domain Aeroacoustics

The Linearized Euler Equations are used widely in many aerodynamic noise prediction schemes. The Linerized Euler Equations also support instability waves, which can obscure the acoustic solution. An approach to suppress the instability waves is to solve the Linearized Euler Equations in the frequency domain. Also, the frequency domain approach is far less expensive compared to the time domain approach, in terms of the computational time: though only a single frequency can be computed at one time. This paper presents the application and comparison of two unstructured grid methods for the solution of aeroacoustic problems in the frequency domain. Algorithms are developed using the Discontinuous Galerkin and the Streamline Upwind Petrov Galerkin finite element methods in two spatial dimensions. Unlike other existing frequency domain solvers, no assumption is made regarding the nature of the mean flow. The application of the two methods to a two-dimensional benchmark problem involving the suppression of instability waves is presented. The two methods are compared in terms of the computational cost, and it is concluded that the Discontinuous Galerkin method is prohibitively more expensive compared to the Streamline Upwind Petrov Galerkin method for practical aeroacoustic applications in the frequency domain. Aerodynamic noise in a turbulent flow originates from the unsteady motion of turbulent structures in the flow, usually in a shear layer or near a solid surface. The sound generated then propagates to the far field through the non-uniformly moving air. Computational fluid dynamics (CFD) codes based on some form of turbulence modeling can be used to characterize the sources of noise in the flow field. Since acoustic perturbations are small, linear methods can be used to simulate the propagation of sound to far field observer locations. One such method involves the solution of the Linearized Euler Equations (LEE). The LEE describe the propagation of sound waves in a non uniform mean flow. The linearized Euler equations also support spatially growing solutions for certain source frequencies. However, these instabilities, which are known as the KelvinHelmholtz instabilities, are limited in a real flow by the nonlinear effects present in the full governing equations, the Navier-Stokes equations. In other words, the instability wave solutions are a consequence of the simplification of the full equations of motion. Often, the acoustic solution is obscured by the non-physical instability solution. Various techniques have been attempted by several researchers to suppress the instability wave solution in the time domain solution. These techniques range from assuming zero gradient in the mean flow to numerical filtering of the instability waves. A detailed discussion of the techniques is given by Agarwal et al. 1 In the same study, an approach to suppress the instability waves was presented, by solving the LEE in the frequency domain. In the frequency domain, a harmonic response is assumed for the acoustic variables. The frequency of response is chosen as the frequency of oscillation of the source. The transportation to the frequency ⁄ Graduate Research Assistant, Member, AIAA † Boeing/A. D. Welliver Professor, Fellow, AIAA