Modeling nonlinear thermoacoustic instability in an electrically heated Rijke tube

Analysis of thermoacoustic instability is performed in a horizontal Rijke tube with an electrical resistance heater as the heat source. The analysis starts with the governing equations for the fluid flow. The governing equations become stiff and are difficult to solve by CFD, as the Mach number of the steady flow and the thickness of the heat source (compared to the acoustic wavelength) are small. Therefore asymptotic analysis is performed in the limit of small Mach number and compact heat source to eliminate the above stiffness problem. The unknown variables are expanded in powers of Mach number. Two systems of governing equations are obtained: one for the acoustic field and the other for the unsteady flow field in the hydrodynamic zone around the heater. The coupling between the acoustic field and the unsteady heat release rate from the heater appears naturally from asymptotic analysis. Further, a non-trivial additional term appears in the momentum equation of the hydrodynamic zone, which has serious consequences on the stability of the system. This additional term is the pseudo-acceleration term, which appears due to the change in the frame of reference from an inertial frame to an accelerating frame of reference, performed in the hydrodynamic zone during the asymptotic analysis. The presence of pseudo-acceleration term is generic, when there are two length scales involved in the problem. Subcritical transition to instability is simulated in the present paper and is related to the hysteresis in the asymptotic state of the system with the heater power, observed in Rijke tube experiments. The global asymptotic stability of the system with the system parameters is investigated using bifurcation diagram. Present numerical simulations confirm the importance of the pseudo-acceleration term. Bifurcation diagrams obtained from the simulations with and without pseudo-acceleration term are completely different. The transition to instability occurs via different types of bifurcations in the above two cases. Simulations performed with the response function for the unsteady heat release rate are compared with the simulation with pseudo-acceleration term. Numerical simulations are performed using the Galerkin technique for the acoustic zone and CFD for the hydrodynamic zone. Acoustic streaming is shown to happen during limit cycle and its effect on the unsteady heat release rate is discussed. Copyright © 2010 by Mariappan S. and Sujith R. I.