Non-dimensional groups for similarity analysis of thermoacoustic instabilities

An approach for similarity analysis of thermoacoustic combustion instabilities is proposed. A set of non-dimensional Π-groups is deduced from a solution based on modal expansion of the quasi-1D Helmholtz equation with a time-lagged heat source. Although thermoacoustic stability depends on a considerable number of parameters, which describe the geometry and thermodynamic characteristics of a combustor, the acoustic boundary conditions, the strength of flow-flame-acoustic coupling, etc., it is found that the respective non-dimensional frequencies and growth rates of thermoacoustic modes are dominated by only two Π-groups: Firstly, flame potency, which is governed by the intensity of mean and fluctuating rate of heat release and a parameter that quantifies the strength of flame-acoustic interaction for a given mode. The second Π-group is the non-dimensional time lag of the flame. Eigenfrequency locus plots for three combustor configurations are generated by varying interaction index and time lag of the flame model. It is found that similarity analysis explicates universal features of thermoacoustic instability and helps to classify thermoacoustic modes. Moreover, similarity analysis establishes not only the stability of eigenmodes, but also the sensitivity of eigenfrequency or growth rate to changes of any parameter contained in any of the two Π groups. In closing, the relevance of flame potency for the design of stable combustors is discussed.