Reconstruction of acoustic transfer matrices by instationary computational fluid dynamics

Thermoacoustic combustion instabilities are a frequently encountered problem in the operation of combustion equipment. The "brute-force" application of computational fluid dynamics to the analysis of thermoacoustic instabilities is estimated to be forbiddingly expensive for many systems of technical interest due to the high computational demands of a time- and space-accurate simulation of a (low Mach number) compressible reacting flow in a complex geometry. Thermoacoustic systems can be modelled efficiently as networks of acoustic multi-ports, where each multi-port corresponds to a certain component of the system, e.g., air or fuel supply, burner, flame, combustor and suitable terminations, and is represented mathematically by its transfer matrix. For some multi-ports, the transfer matrix can be derived analytically from first principles: i.e., the equations of fluid motions and suitable approximations. However, the acoustic behavior of more complicated components, e.g., a burner of a flame, has to be determined by empirical methods, by using a "black box" approach common in communications engineering. In this work, a method is introduced which allows one to reconstruct the transfer matrix of an acoustic two-port from an instationary computation of the response of the two-port to an imposed perturbation of the steady state. Firstly, from the time series data of fluctuating velocity and pressure on both sides of the two-port, the auto- and cross-correlations of the fluctuations are estimated. Then, the unit impulse responses of the multi-port are computed by inverting the Wiener-Hopf equation. Finally, the unit impulse responses are z-transformed to yield the coefficients of the transfer matrix. The method is applied to the one-dimensional model of a heat source with time delay placed in a low-Mach-number compressible flow, for which an analytical description can be derived from first principles. Computational predictions of the transfer matrix have been validated successfully against these analytical results. Furthermore, a comparison of the novel approach presented in this paper with a method for computing the frequency response of a flame by Laplace-transforming its step response is carried out.