Analytical solution to the wave equation with discrete pressure sources: A model for the Rijke tube

Despite of having been studied for several decades the phenomena of combustion instabilities are not well understood. Pressure waves due to the combustion instabilities can become violent being detrimental for both the performance and combustor life. A good prediction of the pressure distribution inside the combustor is important in order to prevent the occurrence of this phenomenon. In this work a technique for solving the wave equation with discrete sources (or sinks) using the Green's functions was developed. One and two-dimensional approaches for cylindrical and Cartesian coordinates with constant speed of sound were solved. Also the case of one-dimensional axially varying temperature is presented. This technique was validated with results found in the literature and experimental data showing excellent agreement. By combining the 2-D solution with constant speed of sound plus the 1-D with axially varying speed of sound this technique accounts for the contributions of the fuel composition since the different blends of fuel produce different temperature profiles and therefore different speeds of sound. The technique is proposed to solve the 2-D pressure distribution of the Rijke tube, which can be considered as the simplest combustor configuration. The study of the pressure distribution in the Rijke tube is fundamental for the understanding of the phenomenon of combustion instabilities.