  Acoustic and entropy waves interacting in a duct with a steady heat source
and mean flow are analysed using an asymptotic expansion (AE) for low
frequencies. The analytical AE solutions are obtained by taking advantage of
flow invariants and applying a multi-step strategy. The proposed solutions
provide first-order corrections to the compact model in the form of integrals
of mean flow variables. An eigenvalue system is then built to predict the
thermoacoustic modes of a duct containing a distributed heat source or sink.
Predictions from the AE solutions agree well with the numerical results of the
linearised Euler equations for both frequencies and growth rates, as long as
the low-frequency condition is satisfied. The AE solutions are able to
accurately reconstruct the acoustic and entropy waves and correct the
significant errors in the predicted entropy wave associated with the compact
model. The analysis illustrates that the thermoacoustic system needs to account
for the entropy wave generated by the interaction of acoustic wave and the
distributed steady heat source, especially when density- or entropy-dependent
boundary conditions are prescribed at the duct ends. Furthermore, a combination
of the AE method and the modified WKB approximation method is discussed for a
cooling case. The AE solutions remedy the disadvantage of the WKB solution in
the low and very low-frequency domain and facilitate full-frequency theoretical
analyses of sound propagation and entropy generation in inhomogeneous duct flow
fields.
