Analytical solutions for the acoustic field in a thin annular duct with temperature gradient and mean flow

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

“…The solutions provide an explicit method to get the axial distribution of acoustic perturbations in annular chambers, which can be used to predict the thermoacoustic instabilities through the low-order network models. Different from previous researches [29,35,37], frequencies and axial wave numbers are regarded as complex numbers, which allows the prediction of resonance frequencies as well as their growth rates. This is helpful in reducing the cost of calculation and analysis of physical mechanism of real annular combustion chambers.…”

“…The entropy fluctuations generated by the interaction of acoustic waves with the non-uniform cross-sectional area and mean temperature in the axial-momentum equation have been shown to have an insignificant effect on the acoustic field [29,35]. The entropy term can be expressed as,…”

“…It should be noted that the imaginary part k i of the wave number k is neglected in our previous work [29,35], as it is typically much smaller than the real part k r , especially for high frequency conditions. However, when there is sufficiently large heat source or acoustic energy sink within the thin annular combustion chamber, the absolute value of the imaginary part k i , which typically corresponds to the growth rate of the wave, can be a large value that cannot be neglected.…”

“…The validation of the analytical solution is performed by considering the perturbation terms corresponding to the acoustic field in a thin annular chamber, e.g., the pressure perturbation p, axial velocity perturbation ûx and circumferential velocity perturbation ûθ . Following our previous works [29,35], we also use the normalised perturbation terms (also called transfer functions) for the comparisons, expressed as:…”