A semi-analytical solution for acoustic wave propagation in varying area ducts with mean flow

Abstract

A semi-analytical solution is developed for the propagation of plane acoustic waves in a varying area duct, sustaining a 1-D mean flow with a temperature gradient. The mean flow can be non-isentropic, such that the axial variation of the flow area and temperature can be prescribed independently. The case of an isentropic mean flow, for which the flow area and mean temperature variation are linked, is discussed. A second order differential equation (ODE) for acoustic pressure is derived from the linearised Euler equations in the frequency domain, neglecting the communication between acoustic and entropy disturbances. This ODE has axially varying coefficients and is solved using an iterative WKB approximation method. The obtained wave-like solution is expressed as the superposition of downstream and upstream propagating plane wave amplitudes. The solution thus obtained is, at any location, a function of upstream thermodynamic and mean-flow properties and wave-number, and can be applied to ducts with arbitrarily varying area and temperature profiles. For validation of the model, two shapes of area variation with linear temperature gradient are considered, and the solution is further simplified to depend only on local spatial coordinate and inlet conditions. The semi-analytical solutions are valid at “high” frequencies, thus the frequencies considered must be both low enough for a predominantly one-dimensional acoustic field, and large enough for validity of the solutions. For each geometry, the analytical solution is presented along with the frequency range of its validity. The analytical predictions are compared to numerical solutions of the linearised Euler equations (LEEs), which can either account for or neglect the acoustic - entropy wave coupling; this further allows the coupling effect to be evaluated. Within the frequency ranges of their validity, the simplified semi-analytical solutions perform well up to flow Mach-numbers around 0.3. For inlet temperature fluctuations ~ 1% of the mean, the effect of acoustic-entropy coupling on the accuracy analytical prediction was found to be insignificant for flow Mach numbers less than 0.3.