Finite amplitude acoustic waves in variable area ducts

Abstract

This paper presents results of an investigation of the driving of finite amplitude oscillations in closed constant diameter ducts and ducts whose cross sectional area varies with axial distance. Finite amplitude oscillations are of interest in acoustic compression and thermoacoustics. The behavior of these oscillations was modeled by a non-linear wave equation in terms of velocity potential that accounted for second order nonlinearities, dissipation and periodic energy addition. Solutions were obtained using the Galerkin method, which yields considerable insight into the nature of the oscillations. The results of this study show that significantly higher amplitude pressure oscillations (-2.5 times) can be forced in horn-like shaped ducts as opposed to cylindrical ducts. Furthermore, highly dissipative shock-like waveforms are excited in the constant diameter ducts in contrast with practically continuous waveforms excited in horn-like ducts. Shock-like waveforms excited in constant diameter ducts are caused by the generation of higher harmonics through efficient non-linear coupling with the fundamental mode. In contrast, the non-linear coupling between the fundamental mode and its' harmonics is weak in ducts whose cross sectional area varies axially (e.g., the investigated horn-like ducts), which minimizes the excitation of harmonics in such ducts. This concentrates the forced power input in the fundamental mode, resulting in the forcing of continuous waveform, large amplitude oscillations. This paper has also demonstrated that the application of the Galerkin method to the solution of this non-linear problem can yield physical insight into the physics of the problem at a low computational cost.