Impact of Linear Coupling on Thermoacoustic Instabilities

Abstract

There have been conflicting opinions in literature, about the consequences of ignoring the linear coupling terms in finite-dimensional thermoacoustic models. We try to clarify this issue in this paper. We study consequences of ignoring the linear coupling terms on the non-normal thermoacoustic interactions in a horizontal Rijke tube. In the Galerkin formulation, we use pseudospectra and Henrici index of non-normality to study the impact of linear coupling between the Galerkin modes on the non-normality and the ensuing transient growth. This linear coupling is important for the energy transfer between the eigenmodes of the system (which are not the same as the natural acoustic modes of the duct). Although ignoring this linear coupling results in negligible shift in the eigenfrequencies, it may have serious consequences on the non-normality of the thermoacoustic system and its dynamics. Nonlinear phenomena such as triggering, boot-strapping and limit cycle oscillations may be completely missed out when this linear coupling is ignored. Ignoring the linear coupling between the Galerkin modes is justified only when the non-normality and the nonlinear dynamics of the system are not altered significantly.

Keywords:

• Combustion
• Eigenmodes
• Galerkin modes
• Linear coupling
• Non-normality
• Pseudospectra
• Rijke tube
• Thermoacoustic instability

Notes

¹The adjoint of a real matrix is the transpose of the matrix. For a differential operator T, the adjoint T† is such that 〈u, Tv〉 = 〈T ^† u, v〉 where 〈._,.〉 is the inner product. u and v are two vectors of the same length.