Finite Difference Time Marching in the Frequency Domain: A Parabolic Formulation for the Convective Wave Equation

Abstract

An explicit finite difference iteration scheme is developed to study harmonic sound propagation in ducts. To reduce storage requirements for large 3D problems, the time dependent potential form of the acoustic wave equation is used. To insure that the finite difference scheme is both explicit and stable, time is introduced into the Fourier transformed (steady-state) acoustic potential field as a parameter. Under a suitable transformation, the time dependent governing equation in frequency space is simplified to yield a parabolic partial differential equation, which is then marched through time to attain the steady-state solution. The input to the system is the amplitude of an incident harmonic sound source entering a quiescent duct at the input boundary, with standard impedance boundary conditions on the duct walls and duct exit. The introduction of the time parameter eliminates the large matrix storage requirements normally associated with frequency domain solutions, and time marching attains the steady-state quickly enough to make the method favorable when compared to frequency domain methods. For validation, this transient-frequency domain method is applied to sound propagation in a 2D hard wall duct with plug flow.

Capabilities of the high-order parabolic equation to predict propagation in boundary and shear layers.

“…Few applications are also devoted to duct acoustics, to predict the noise radiated from an aeroengine nacelle. In the past, Baumeister [3] proposed the numerical spatial marching techniques for in-duct propagation, where he examined the stability problem of the finite difference technique. Dougherty [9] developed a method based on the parabolic approximation to the convected Helmholtz equation in an orthogonal curvilinear coordinate system, to analyze the effects on sound propagation in non-uniform, softwall ducts.…”