Uncertainty Quantification of Thermoacoustic Instabilities in a Swirled Stabilized Combustor

Abstract

Combustion instabilities can develop in modern gas-turbines as large amplitude pressure oscillations coupled with heat release fluctuations. In extreme cases, they lead to irreversible damage which can destroy the combustor. Prediction and control of all acoustic modes of the configuration at the design stage are therefore required to avoid these instabilities. This is a challenging task because of the large number of parameters involved. This situation becomes even more complex when considering uncertainties of the underlying models and input parameters. The forward uncertainty quantification problem is addressed in the case of a single swirled burner combustor. First, a Helmholtz solver is used to analyze the thermoacoustic modes of the combustion chamber. The Flame Transfer Function measured experimentally is used as a flame model for the Helmholtz solver. Then, the frequency of oscillation and the growth rate of the first thermoacoustic mode are computed in 24 different operating points. Comparisons between experimental and numerical results show good agreements except for modes which are marginally stable/unstable. The main reason is that the uncertainties can arbitrary change the nature of these modes (stable vs unstable); in other words, the usual mode classification stable/unstable must be replaced by a more continuous description such as the risk factor, i.e. the probability for a mode to be unstable given the uncertainties on the input parameters. To do so, a Monte Carlo analysis is performed using 4000 Helmholtz simulations of a single experimental operating point but with random perturbations on the FTF parameters. This allows the computation of the risk factor associated to this acoustic mode. Finally, the analysis of the Monte Carlo database suggests that a reduced two-step UQ strategy may be efficient to deal with thermoacoustics in such a system. First, two bilinear surrogate models are tuned from a moderate number of Helmholtz solutions (a few tens). Then, these algebraic models are used to perform a Monte Carlo analysis at reduced cost and approximate the risk factor of the mode. The accuracy and efficiency of this reduced UQ strategy are assessed by comparing the reference risk factor given by the full Monte Carlo database and the approximate risk factor obtained by the surrogate models. It shows a good agreement which proves that reduced efficient methods can be used to predict unstable modes.

Uncertainty Quantification of Growth Rates of Thermoacoustic Instability by an Adjoint Helmholtz Solver

“…is modeled by a gain G 0 and time delay τ 0 , which are characteristic of flame B for the frequencies of interest. The same values were considered in Ndiaye et al [6]. In addition, a plenum length l 1 = 0.096 m is considered (see Fig.…”

“…However, there is only a small number of methodologies proposed in the literature to perform Uncertainty Quantification (UQ) of thermoacoustic instabilities when considering Helmholtz solvers [6]. UQ may be crucial in cases where input design parameters contain a certain degree of uncertainty around a given mean value.…”

“…In order to decrease the computational costs required to construct an accurate PDF, surrogate models for the eigenfrequency, which are algebraic expressions obtained by regression, may be used instead of the full Helmholtz equation [6]. Additionally, surrogate models can be combined with Active Subspace techniques so that the number of uncertain input parameters is reduced [7].…”

Stability analysis of thermo-acoustic nonlinear eigenproblems in annular combustors. Part II. Uncertainty quantification

“…uncertain region is measured by the risk factor [3], which corresponds to the probability that the mode is unstable. Although the probabilistic estimation (uncertainty quantification) of the thermo-acoustic stability is paramount for practitioners, there are only a few studies in the literature [4,5,6,3].…”

“…Uncertainty Quantification (UQ) of thermo-acoustic stability was performed in longitudinal academic configurations containing one turbulent flame in Ndiaye et al [6]. Assuming that only the flame model was uncertain, i.e., the gain and the time delay, the risk factor of the system was obtained by combining a Helmholtz solver [7] with a Monte-Carlo analysis.…”