Linear thermoacoustic instability in the time domain

An approximate time-domain description of the development of the thermoacoustic instability in gas-filled tubes is developed by exploiting the difference between the instability time scale and the period of standing waves. The perturbation results compare very favorably with the exact frequency-domain theory of Rott. The perturbation results are further simplified by introducing a short-stack approximation which is numerically much simpler and only slightly less accurate. An approximate expression for the critical temperature gradient accounting for viscous effects and other design features is also derived. In addition to the fundamental mode of a tube closed at both ends, the theory includes higher modes as well as open-end boundary conditions