Theoretical analysis of sound propagation and entropy generation across a distributed steady heat source

TL;DR

This paper develops an asymptotic expansion method to analyze sound and entropy wave interactions in ducts with steady heat sources, providing accurate low-frequency solutions and insights into thermoacoustic mode behavior.

Contribution

It introduces a first-order correction analytical approach for thermoacoustic modes in ducts with distributed heat sources, improving upon compact models and combining with WKB for full-frequency analysis.

Findings

AE solutions match numerical results for frequencies and growth rates

AE accurately reconstructs acoustic and entropy waves

Entropy wave predictions are significantly improved over compact models

Abstract

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.