Global thermoacoustic oscillations in a thermally driven pulse tube

We obtain linearized, BiGlobal thermoacoustic solutions in a pulse tube driven via an imposed mean temperature gradient. Here, the pulse tube is treated as a key unit of a thermoacoustic heat engine, in which the conversion of thermal energy to useful acoustic fluctuations occurs. A primary goal of this work is to understand the hydrodynamic efficiency of the energy conversion process and how it depends upon some of the important operating parameters, including the geometry of the device which in the limit of long length-to-diameter ratio approaches the so-called narrow tube approximation. As this limit is frequently imposed in the wave propagation analyses of thermoacoustic devices, it is critical to investigate the physical connections of such a model to more realistic finite-length pulse tube configurations, which we do here. The mean flow is quiescent with an analytic mean temperature profile that still models the necessary physical details of the hot heat exchanger and regenerator. The computed thermoacoustic oscillations are found to be globally stable, approaching neutral stability conditions at the narrow tube limit. In finite-length tubes, three distinct types of modes are identified and analyzed. Here, within a linear framework, radial modes do appear to act as key enablers for longitudinal modes to be the primary carriers of acoustic energy from the pulse tube section, while the identified boundary modes, essentially numerical constructs, are ignored in the analysis. Further, a disturbance energy-based efficiency metric is constructed that provides mechanistic understanding of some of the key parameters in pulse tube operation. For finite-length tubes, it shows oscillations of the first asymmetric mode to be the most efficient, while the axisymmetric perturbations dominate for longer tubes that eventually lead to the idealized plane wave propagation.