Finite-element method to study harmonic aeroacoustics problems

Abstract

The analysis of aeroacoustics propagation is required to solve many practical problems. As an alternative to Euler’s linearized equations, an equation was established by Galbrun in 1931. It assumes the flow verifies Euler’s equations and the perturbation is small and adiabatic. It is a linear second-order vectorial equation based on the displacement. Galbrun’s equation derives from a Lagrangian density and provides conservative expressions of the aeroacoustics intensity and energy density. A (CAA) method dealing with the numerical resolution of Galbrun’s equation using the finite-element method (FEM) is presented. The exact solution for the propagation of acoustic modes inside an axisymmetric straight-lined duct in the presence of a shear flow is known and compared with the FEM solution. Comparisons are found to be in good agreement and validate a first step in the development of a CAA method based on the FEM and Galbrun’s equation. The FEM is then applied to an axisymmetric duct including a varying cross section and a nonuniform flow with respect to both the axial and the radial coordinates. The expression of the aeroacoustics intensity implemented in the FEM provides an accurate in-duct power balance.

Characterization of an acoustic liner by means of Laser Doppler Velocimetry in a subsonic flow

“…Consequently, the basic equations are Euler's equations. The perturbation expansion can be achieved in one of two ways: The first relies on the Eulerian representation, which is classically used in many books and studies [4,10,12,16,18]; the second relies on a less familiar approach, called the mixed Eulerian/Lagrangian representation [3,7,9,19,22]. These two representations are presented below.…”

“…A solution to the theoretical issue of energy conservation is the mixed Eulerian/Lagrangian representation of perturbations [7,9,19], involved in Galbrun's theory. These perturbations are considered for a given fluid particle rather than for a given position.…”

Stability and accuracy of finite element methods for flow acoustics. I: general theory and application to one‐dimensional propagation

“…In both the one-and two-dimensional cases, the numerical solution is generated by specifying the nodal variables (pressure or displacement) at x = 0 and by defining an impedance condition which is consistent with the exact solution at x = L. In the two-dimensional case a hard-walled boundary condition is weakly imposed at y = 0 and H so that the exact solution is simply a right running wave p exact = exp(−ikx) with k given by Equation (12).…”

“…With this variational formulation, originally proposed by Peyret and Élias [12], the boundary integral is the momentum flux through the boundary [13] and the volume integral yields hermitian linear systems. This formulation is solved with the standard linear triangular element, see Figure 3.…”

A strategy to implement volume sources into Galbrun equation

“…This resulted in a displacement based equation, which is known as Galbrun equation. This Galbrun equation is a second-order linear partial differential equation [4]. A number of advantages of Galbrun equation compared to LEE are presented in articles by Treyssède et al [6,7].…”