A theory is developed to describe high frequency three-dimensional thermoacoustic waves in a simplified geometry representing a typical premix combustor. The theory considers linear modes of frequency ω and circumferential mode number m i.e. proportional to eiωt+imθ. The radial and axial dependence is determined for a cylindrical combustor.
Simple geometries are investigated systematically to analyze the effect of different inlet boundary conditions to the combustion chamber on the frequency of oscillation and on the susceptibility to instability, both near and away from the cut-off frequencies.
The model includes a one-dimensional mean flow, radial mode coupling and idealized combustion processes, which are added in stages to build up an understanding of the complicated acoustics of the premix combustor geometry.
It is demonstrated that the flow through the premix ducts provides a frequency-dependent boundary condition at combustor inlet and causes modal coupling.
Generalized linear equations of conservation of mass, momentum and energy, together with boundary conditions, are solved to identify the eigenfrequencies, ω, of the total system. Then Real ω determines the frequency of the oscillation, while Imaginary ω indicates the growth rate of the disturbance.
It is found that strong resonant peaks in the pressure waves exist close to the cut-off condition for acoustic waves and that the relationship between the unsteady rate of heat release and the flow significantly influences the instability of oscillation.
