Thermoacoustic velocity and pressure oscillations in one-dimensional combustors with spatially varying heat sources and thermal distributions

Abstract

In this work, an asymptotically based framework using a naturally occurring perturbation parameter is further developed in the context of a one-dimensional tube with an open–open end point configuration without and with a spatially variable heat source. This approach is shown to produce accurate predictions of not only pressure waveforms, as previously demonstrated, but also of velocity mode shapes and frequencies for arbitrary temperature distributions that mimic a wide variety of flow heating arrangements. These include those associated with a Rijke tube as well as other thermoacoustic sound-generation devices. The underlying formulation consists of two linearly coupled partial differential equations that can be solved simultaneously while using a Green's function to capture the thermoacoustically induced velocity. The strategy leading to accurate predictions of the thermoacoustic velocity is described and then applied to several representative cases. Results pertaining to the acoustic velocity and pressure are systematically discussed and compared to other models in the literature. Overall, we find the axially dependent thermal gain to have the most significant impact on the mode shape structure, frequencies, and nodal locations. For example, higher thermal gains lead to elevated wave propagation speeds in the downstream segment of the tube. As such, they cause the velocity nodes to gradually shift upstream with successive increases in the thermal gain. Moreover, due in part to a non-homogeneous pressure–temperature pumping term that appears exclusively in the acoustic velocity formulation, the main difference between the acoustic pressure and velocity oscillation frequencies can be directly attributed to specific derivatives of the spatially varying thermal profiles.