Role of transient growth in subcritical transition to thermoacoustic instability in a horizontal Rijke tube

Abstract : A theoretical framework has been developed to understand the non-normal nature of the thermoacoustic interaction in an electrically heated horizontal Rijke tube. The ther-moacoustic system considered here includes the dynamics of the chamber acoustic field and the heat source. The eigenmodes of the system are non-orthogonal due to the nonnormal nature of the linearised evolution operator, which leads to the transient growth in the perturbations even for a linearly stable system. The transient growth is measured in a norm, which in general physically represents the energy in the disturbance. The present framework allows one to obtain a norm systematically from the definition of disturbance energy. In the present paper, adjoint optimisation technique is used to obtain the optimum initial condition (suitable for system with large degrees of freedom ~ 104) for maximum transient growth of the norm (disturbance energy). The optimum initial condition thus obtained has significant contributions from the variables governing the dynamics of the heat source. Some interesting flow structures are observed in the optimum initial condition near the heat source. The non-normal nature of the system is shown to be reflected in the reduction in the range of linearisation (threshold energy for the system to be nonlinearly unstable) of the system. © 2010 by S. Mariappan, P. J. Schmid & R. I. Sujith.