Assessing Transfer Matrix Models and Measurements Using Acoustic Energy Conservation Principles

Abstract

Abstract Acoustic transfer matrices are now widely used in the analysis of combustion dynamics of gas turbines. The reliability of the analysis depends on the quality of the determination of the transfer matrices of the individual acoustic elements composing the overall acoustic network of the system. These matrices are, in some simple cases, deduced analytically, using one-dimensional acoustic modeling. However, for more complex elements, such as swirlers, perforated plates or injector units, the transfer matrix has to be obtained experimentally using an impedance tube. A few models that attempt to account for the dynamics of some key elements like the injector unit are also available in the literature. There are, however, uncertainties in the experimental determination of the transfer matrix coefficients and questions raised by the modeling so that it is worth examining experimental results and assessing transfer matrix models using acoustic energy conservation principles. The general idea is to consider the acoustic power flow in the element represented by the transfer matrix T, and compare the power input to the power output. This is best accomplished by obtaining a representation in terms of a scattering matrix S, which may be deduced from the transfer matrix T. It is first shown that standard models like those corresponding to a constant area duct or to an area change comply with acoustic energy conservation. This analysis is then employed to assess the L-ζ model that has been widely used to describe injection unit dynamics. It is shown that acoustic power is dissipated in the injection element only if a certain condition is met by the modeled transfer matrix coefficient T12. The expression that is commonly used for this coefficient fulfills this condition. The power dissipation in the injector is evaluated and shown to be linked to the head loss and typical Mach number in this unit. Finally, acoustic conservation principles are used to assess transfer matrices of a family of injectors determined experimentally and check that the data is compatible with these principles.