Determination of Marginal Conditions for Thermoacoustic Oscillations in a Looped Tube by Evolution of an Initial Disturbance Based on the Boundary-Layer Theory

摘要

Marginal conditions are determined for the onset of thermoacoustic oscillations of a gas in a looped tube with a socalled stack sandwiched by hot and cold heat exchangers on the basis of the boundary-layer theory so far developed. Given a couple of impulses applied initially to a quiescent gas, the evolution of an infinitesimally small disturbance is studied by solving an initial-value problem to the linearized equations. Marginal states correspond to those in which the initial, disturbance neither decays nor grows with time. In the case studied by Ueda and Kato [J. Acoust. Soc. Am. 124; 851 (2008)], the marginal conditions are obtained in two cases of the temperature distribution in the thermal buffer tube and compared with their results. To identify the marginal curves, the porosity of the stack or one of the subsidiary parameters such as the wall thickness of the stack is required in addition to the squared ratio of the hydraulic radius to the typical thickness of the thermal diffusion layer in a flow passage of gas. It is found that the marginal conditions are in qualitative agreement with the experimental results and that the right branch of the marginal curve is obtainable down to such a considerably low temperature ratio that the theory may become invalid.