Evaluation of mean energy fluxes in thermoacoustic oscillations of a gas in a tube

Abstract

In a previous paper [N. Sugimoto and M. Yoshida, Phys. Fluids 19, 074101 (2007)], the marginal condition for the onset of thermoacoustic oscillations of a gas in a tube subjected to the parabolic temperature distribution was derived in the framework of the linear and first-order theory in the boundary-layer thickness. This paper examines the marginal oscillations from a viewpoint of mean energy fluxes averaged over one period of oscillations, aiming at understanding an action of the boundary layer under finite temperature gradient. Using the nonlinear energy equation, formulas for the acoustic energy flux and the convective heat flux (entropy flux times temperature) are derived in the main-flow region and in the boundary layer within the lowest, quadratic order in the pressure amplitude. These fluxes may be evaluated in terms of the linearized solutions and their axial distributions are displayed graphically. The boundary layer occupying nearly half of the side wall near the open end plays an active role to pump energy into the acoustic main-flow region, whereas the other half plays a passive role to dissipate energy. The acoustic energy flux in the main-flow region flows toward the closed end, while the flux in the boundary layer flows down the gradient of the pressure amplitude and toward the open end. The convective heat flux flows only in the boundary layer and directed down the temperature gradient. In the vicinity of the closed end, however, this heat flux flows against the temperature gradient. The heat flux through the side wall comes mostly into the boundary layer but goes out of it near the closed end. Such an Eulerian picture of the flow field is examined from a Lagrangian viewpoint in terms of thermodynamic relations for a gas particle fixed. While the particle in the main-flow region is subjected to the adiabatic change to yield no net power per one period of oscillations, the one in the boundary layer is subjected to a thermodynamic cycle of a prime mover or a heat pump, depending on whether the value of the product of the gradients of the logarithmic temperature and pressure amplitude is beyond a certain value. Discussions are also focused on the mechanisms to maintain the marginal state of oscillations and the efficiency as heat engines.