Analytical solutions for the acoustic field in thin annular combustion chambers with linear gradients of cross-sectional area and mean temperature

TL;DR

This paper develops approximate analytical solutions for the acoustic field in annular combustion chambers with linear gradients in cross-sectional area and temperature, aiding in the prediction of thermoacoustic instabilities.

Contribution

It introduces a modified WKB method to derive analytical solutions for non-uniform annular ducts, expanding the applicability of low-order acoustic models.

Findings

Derived analytical solutions for specific chamber geometries.

Applicable to low Mach number, high frequency, small non-uniformity conditions.

Enables better prediction of thermoacoustic instabilities in real chambers.

Abstract

Predictions of thermoacoustic instabilities in annular combustors are essential but difficult. Axial variations of flow and thermal parameters increase the cost of numerical simulations and restrict the application of analytical solutions. This work aims to find approximate analytical solutions for the acoustic field in annular ducts with linear gradients of axially non-uniform cross-sectional area and mean temperature. These solutions can be applied in low-order acoustic network models and enhance the ability of analytical methods to solve thermoacoustic instability problems in real annular chambers. A modified WKB method is used to solve the wave equation for the acoustic field, and an analytical solution with a wide range of applications is derived. The derivation of the equations requires assumptions such as low Mach number, high frequency and small non-uniformity. Cases with arbitrary distributions of cross-sectional surface area and mean temperature can be solved by the piecewise method as long as the assumptions are satisfied along the entire chamber.