Acoustic absorption and generation in ducts of smoothly varying area sustaining a mean flow and a mean temperature gradient

TL;DR

This study investigates how varying cross-sectional area and temperature gradients in ducts influence acoustic energy absorption and generation, revealing that temperature gradients can control thermo-acoustic phenomena.

Contribution

It introduces an analytical model to quantify acoustic energy change in ducts with mean flow and temperature gradients, validated by numerical simulations, highlighting the impact of temperature gradients on acoustic energy.

Findings

Positive temperature gradients absorb acoustic energy.

Negative temperature gradients generate acoustic energy.

Model accurately predicts absorption for Mach numbers below 0.25.

Abstract

In ducts with varying cross-sectional area and sustaining a subsonic non-isentropic mean flow, the axially varying flow conditions affect the acoustic energy balance of the system. This is significant in understanding and controlling thermo-acoustic phenomena, particularly in combustors. This work aims at quantifying the acoustic energy change in such configurations, using the acoustic absorption coefficient, $\Delta$. The acoustic response of the duct to acoustic forcing is determined using an analytical model, neglecting the effect of entropy fluctuations on the acoustic field, and subsequently, $\Delta$ is estimated. The model predictions of $\Delta$ are validated using a linearised Euler equations (LEEs) solver. The model was found to be accurate for Mach numbers below $0.25$, provided the lower frequency limit set by the analytical solution is satisfied. For conically varying area ducts with linear mean temperature gradient, it was observed that $\Delta$ showed very little dependence on frequency, and that the absolute value of $\Delta$ tended to be maximised when the upstream boundary was anechoic rather than non-anechoic. More importantly, $\Delta$ was also observed to show stronger dependence on the mean temperature gradient than area gradient variation for such configurations. Further parametric and optimisation studies for $\Delta$ revealed a crucial finding that a positive mean temperature gradient, representing a heated duct caused acoustic energy absorption. Similarly, a negative mean temperature gradient, representing a cooled duct caused acoustic energy generation -- a key result of this analysis. This behaviour was shown to be consistent with a simplified analysis of the acoustic energy balance. Based on this finding, a linearly proportional reduction in acoustic energy generation was achieved by changing the mean temperature gradient.