A three-dimensional linear accoustic analysis of gas-turbine combustion instability

by 1975- You, Danning

The present research focuses on developing a three-dimensional linear acoustic analysis to study combustion instabilities in gas turbine combustors with complex geometries and non-uniform distributions of mean flow properties. The primary objectives are: 1. establish a unified analysis to study gas-turbine combustion instability • formulate unsteady motions using a generalized wave equation; • model various source terms in the wave equation; and • solve the wave equation using spatial averaging technique 2. validate the stability analysis against benchmark data and analytical solutions 3. perform parametric studies on various instability problems The theoretical formulation is based on a generalized three-dimensional wave equation governing the acoustic pressure with various source terms. The combustor is discretized into a number of cells along the axial direction, such that the axial mean flow properties can be assumed to be uniform within each cell. Assuming that the source terms are mainly functions of the axial axis, the wave motions in the transverse plane can be approximated as the classical wave motions, which are called the normal modes. Therefore, the oscillations can be expressed as a synthesis of the normal modes in the transverse plane with axial- and time-varying amplitudes. Afterward, spatial averaging in the transverse plane is implemented to solve for the acoustic field from the derived wave equation. iv The effects of the various source terms associated with the uniform mean flow, unsteady heat release, acoustic damping device, and cooling air on the wave-equation solution are formulated or modeled. The oscillatory flow fields such as velocity and entropy fluctuations are solved from the linearized momentum equation or the equation of state. These properties are matched at the interface of each pair of adjacent cells by applying the conservation laws. The procedure will eventually yield a system equation, from which the eigen-frequencies and stability characteristics of the whole combustor can be determined while the combustor geometry and mean flow properties are given. The analysis is implemented to solve the acoustic fields in a step duct, a converging duct, a horn, and a straight duct with temperature gradient. Results show good agreement with finite element software and analytical solutions, in terms of the frequencies and mode shapes. After model validation, a parametric study is conducted to investigate the effects of mean flow, temperature, cross-sectional area, and unsteady heat release on acoustic characteristics. The wave motions including the first longitudinal, the first tangential, and the first radial modes of a model gas turbine combustor are also calculated. Cases corresponding to both stable and unstable flames are considered. The results match closely with the available experimental data and numerical solutions. The combustion response of the flame to the acoustic field is examined. The amplitude and phase angle of the response function of the unstable flame satisfy Rayleigh’s criterion, which explains the occurrence of instability. v It is demonstrated that the approximate solutions of a generalized wave equation, which characterize oscillation flow motions in the combustion chamber, can be obtained by discretization and spatial averaging techniques. The developed analysis is able to accurately predict the frequencies, damping coefficients, and mode shapes of the unsteady motions, which are the key elements of gas turbine combustion instability. The approaches are concise and efficient enough to be used at the design stage. vi