Analytical solutions for the three-dimensional acoustic field in a rectangular duct with temperature gradient and mean flow

Rectangular ducts with mean temperature distribution and bulk flow are typical industrial elements sustained to high-frequency transverse thermoacoustic instabilities. However, there has been no analytical solution for the three-dimensional transverse acoustic field within these types of ducts, and the present work aims to develop it. A second-order partial differential equation in the form of only the pressure perturbation is derived from the linearised Euler equations. Then the three-dimensional analytical solutions independent on any assumptions of the temperature distribution are obtained by combining an adapted WKB approximation method and the separation of variables. Validations of these solutions are conducted by comparing them to the numerical results calculated by the commercial CFD code. Results show that the analytical solutions can accurately predict the three-dimensional transverse acoustic field for sufficiently large frequencies when both the axially varying mean temperature and the axial mean flow present.