Marginal conditions for the onset of thermoacoustic oscillations due to instability of heat flow

Abstract

This paper derives marginal conditions for the onset of thermoacoustic oscillations of a gas due to instability of heat flow in a closed tube and a looped tube. Steady heat flow is created by installing in the tube a stack consisting of many pores axially, sandwiched by hot and cold heat exchangers to impose a temperature gradient. Assuming a time-harmonic disturbance and a temperature gradient given, an eigenvalue problem is posed on Rott's equation. Given a pore radius of the stack, a marginal condition is determined so that an unknown frequency may take a real value by a suitable choice of a temperature ratio and neutral oscillations may occur. This problem is solved numerically by a shooting method combined with Runge-Kutta method and Newton's method. The marginal conditions for the temperature ratio are displayed in curves versus the pore radius relative to a typical thickness of viscous diffusion layer. At marginal states corresponding to the conditions obtained, angular frequencies of neutral oscillations are also displayed. The same problem is solved by using the approximate equations derived from the thermoacoustic-wave equation for a case of thin or thick diffusion layer relative to a pore radius. Comparing the marginal conditions and the angular frequencies against the ones by Rott's equation, the validity of the approximate equations is checked. Influences of the porosity on the marginal conditions and the angular frequencies are examined. Also obtained are those in higher modes of oscillations in both tubes. Finally axial variations of amplitudes in pressure and mass flux density and of acoustic energy flux density (intensity) at a marginal state in the lowest mode are displayed to highlight physical differences between a standing wave in the closed tube and a travelling wave in the looped tube.