Thermoacoustic oscillations induced from heated wires in a tube with air current, such as the Rijke oscillation, are investigated for the prediction of the thermoacoustic power generation and the stability of the oscillation. Based on the analysis of heat transfer response of an isothermal circular cylinder to the acoustic wave, the characteristics of the thermoacoustic power generation from a heated cylinder in cross flow are obtained and are applied to the prediction of the onset condition of the Rijke oscillation.
Firstly, the general quantitative formulation for the thermoacoustic power generation from heat input is derived by linear perturbation analyses of the continuity, momentum, energy and state equations. It is shown that the in-phase component of the fluctuating heat release from heat sources with the acoustic pressure can generate the acoustic power, and also that the mechanism of the energy conversion from thermal to acoustic energy can be explained by the engine analogy.
Secondly, the heat release response of an isothermal cylinder to the oscillating flow of an acoustic wave superimposed on the mean flow of air current is computed numerically under the assumption that the fluid is viscous and incompressible with constant properties. Variables are properly normalized by the angular velocity ω and the thermal diffusivity α. The heat transfer responses are obtained by solving numerically the unsteady full Navier-Stokes and energy equations. For a small amplitude fluctuating flow, oscillating component of heat transfer is obtained from the perturbation equations. For a large amplitude fluctuating flow, however, the instantaneous heat transfer response is simulated by integrating the unsteady governing equations. From the solution for the small amplitude flow, it is shown that the amplitude of heat transfer is decreased and the phase delay is increased with decrease in the normalized mean velocity $U_o^*$ and with increase in the normalized radius $a^*$, while, the response becomes quasi-static with increase of $U_o^*$ and decrease of $a^*$. In case of a large amplitude flow, however, the response is shown to be influenced by the instantaneous velocity rather than the mean velocity. And the time average of heat transfer, compared with that for the mean flow without fluctuation, is increased in the wake region while decreased over the forward region. Such a feature is the thermoacoustic streaming effect due to the enlarged thermal boundary layer over the forward region and the increased vorticity in the wake region driven by the streaming motion.
Thirdly, the acoustic power generation from a heated cylinder in cross flow is analyzed based on the result of the heat transfer analysis. The power generation is expressed in terms of the efficiency factor E, defined as the ratio of the complex amplitude of the fluctuating heat transfer rate to its steady component for the unit normalized amplitude of the velocity fluctuation. It is found that when the acoustic field is a traveling wave, the acoustic power generation is proportional to the real part of the efficiency factor, while, for a standing wave, it is proportional to the imaginary part. The most important feature is that the efficiency factor attains its peak value when both the normalized air current velocity and wire radius are around unity. In other words, the greatest power can be generated when the radius of the wire is the order of the (thermal) acoustic boundary layer thickness and the velocity of the air current is close to the thermal diffusion velocity.
Fourthly, the stability limit of the Rijke oscillation induced by heated wires is derived by equating the thermoacoustic power generation to the power loss. The power generation from the wire grid is estimated by correcting the free stream velocity of the analysis for a single wire, in order to take into consideration the interactions between adjacent wires. For the estimation of the power loss, the convection and the radiation losses of the acoustic energy at both ends of the tube are considered, in addition to the thermoviscous dissipation loss on the tube wall. The onset condition of the Rijke oscillation is obtained for the steady heat input to the heater, in which the effects of the heater position, the tube geometry, and the efficiency factor are shown explicitly.
Finally, the stability of the Rijke oscillation is measured by using a spiral heater of 1mm diameter in a tube of 37mm diameter. The measured heat input for the onset of the oscillation is compared with the theoretical prediction, and they agree well. This agreement is regarded as a substantial verification of theoretical analyses of this study.
