A study of the transition to instability in a Rijke tube with axial temperature gradient

A horizontal Rijke tube with an electric heat source is a system convenient for studying the fundamental principles of thermoacoustic instabilities both experimentally and theoretically. Given the long history of the device, there is a surprising lack of accurate data defining its behavior. In this work, the main system parameters are varied in a quasi-steady fashion in order to find stability boundaries accurately. The chief purposes of this study are to obtain precise values of the system parameters at the transition to instability with specified uncertainties and to determine how well the experimental results can be explained with existing theory. Measurement errors are reported, and the influence of experimental procedures on the results is discussed. A form of hysteresis effect at stability boundaries has been observed. Mathematical modelling is based on a thermal analysis determining the temperature of the heater and the temperature field in the air inside the tube, which, consequently, affects acoustical mode shapes. Solutions of the linearized wave equation for a non-uniform medium, including losses and a heat source term, determine the stability properties of the eigen modes. Calculated results are compared with experimental data and with results of the modelling based on the common assumption of a constant temperature in the tube. The mathematical model developed here can be applied to designing thermal devices with low Mach number flows, where thermoacoustic issue is a concern.