Keele Data Repository

Heckl, Maria (2023) Report for EU POLKA project.

This document gives a detailed derivation of the integral governing equation for the velocity potential in the following setup: A finite duct with mean flow and frequency-dependent end conditions (described by reflection coefficients) houses a compact unsteady heat source; thermoacoustic instabilities arise in the duct because of the interaction between the sound field in the duct and the unsteady heat release rate from the heat source. The derivation of the governing integral equation is based on the acoustic analogy equation, i.e. the convected wave equation with a source term that represents the heat release rate. A series of mathematical steps are performed. These include an adjoint approach with a test function. This test function is shown to be the adjoint form of the tailored Green's function of the flow duct. It is calculated with a generalised function approach and turns out to be a superposition of duct modes. Expressions for the frequencies and amplitudes of these modes are found. they depend on the reflection coefficients, duct length and speed of the flow. The final result of this document is the integral governing equation for the velocity potential; this contains the adjoint Green's function, the heat release rate and initial conditions. It can be solved with a straightforward time-stepping approach.