Modelling nonlinear thermoacoustic instability in an electrically heated Rijke tube

Abstract

An analysis of thermoacoustic instability is performed for a horizontal Rijke tube with an electrical resistance heater as the heat source. The governing equations for this fluid flow become stiff and are difficult to solve by the computational fluid dynamics (CFD) technique, as the Mach number of the steady flow and the thickness of the heat source (compared to the acoustic wavelength) are small. Therefore, an 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. In this analysis, the coupling between the acoustic field and the unsteady heat release rate from the heater appears from the asymptotic analysis. Furthermore, a non-trivial additional term, referred to as the global-acceleration term, appears in the momentum equation of the hydrodynamic zone, which has serious consequences for the stability of the system. This term can be interpreted as a pressure gradient applied from the acoustic onto the hydrodynamic zone. The asymptotic stability of the system with the variation of system parameters is presented using the bifurcation diagram. Numerical simulations are performed using the Galerkin technique for the acoustic zone and CFD techniques for the hydrodynamic zone. The results confirm the importance of the global-acceleration term. Bifurcation diagrams obtained from the simulations with and without the above term are different. Acoustic streaming is shown to occur during the limit cycle and its effect on the unsteady heat release rate is discussed.

Multiple-scale thermo-acoustic stability analysis of a coaxial jet combustor

“…This enables us to reduce the complexity of the full problem governed by (1) - (5) . In thermo-acoustics, multiplescale methods were applied, among others, by [11][12][13] . The two perturbation parameters are the Mach number, M ∼ O ( ) and the flame compactness, h / L ∼ O ( n ), where 0 < 1.…”

“…In this case n = 1, i.e., there are no different time scales. The derivation for nonreacting atmospheric flows is given by [20] ; for a one-dimensional acoustic system with a flame by [11] ; and for an electrical Rijke tube by [12] . …”

“…The system is only one-way coupled. To overcome this limitation, multiple-scale methods were applied by [11][12][13] to capture the two-way interaction between the hydrodynamics and acoustics. [11] used one time and two spatial scales (1T-2S) to couple a one-dimensional acoustic network model with a one-dimensional low-Mach number flame solver.…”

“…[11] used one time and two spatial scales (1T-2S) to couple a one-dimensional acoustic network model with a one-dimensional low-Mach number flame solver. Using the same multiple scales, [12] studied the nonlinearities in an electrical Rijke-tube with onedimensional acoustics. The effect that the acoustics has on the hydrodynamics was assumed to be uniform in the hydrodynamic domain.…”

“…This quantifies the growth rate's dependence on every interaction between the two subsystems. This directly gives the drift in the dominant eigenvalue caused by the global acceleration term, which couples the acoustics to the hydrodynamics, avoiding time-consuming nonlinear simulations [12] . The ultimate aim is to set up a robust framework for stability, receptivity and sensitivity analysis of combustion-acoustic systems and thereby to control thermo-acoustic instabilities by exploiting the behavior of hydrodynamic instabilities.…”