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Bayesian inference of flame impulse responses

Matthew Yoko, Wolfgang Polifke

Abstract

The impulse response of a flame to acoustic velocity perturbations is a key quantity for predicting thermoacoustic stability, but its identification from sparse, noisy observations requires solving an ill-posed inverse convolution problem. This is typically achieved with system identification methods, which require hand-tuning of regularization, model order, and sampling parameters, and provide no principled mechanism for incorporating prior physical knowledge. In this paper, we reformulate the identification problem within a Bayesian framework. The impulse response is represented as a physically motivated distributed time delay model, whose parameters correspond to convective delays and dispersive broadening. For a given number of pulses, the model parameters are inferred from the data using Bayesian parameter inference. The number of pulses is then selected using Bayesian model comparison, which balances data fit against model complexity to identify the simplest model capable of explaining the data. The framework is demonstrated on broadband-forced large eddy simulation data from a turbulent swirl-stabilized burner. Bayesian model comparison selects a three-Gaussian impulse response for this flame, consistent with physical interpretations in previous work. Compared with system identification, the Bayesian approach produces impulse responses with fewer spurious features and enables straightforward enforcement of a known low-frequency gain. Finally, we show that the Bayesian approach is robust to significant reductions in recording length, making it appealing for impulse response identification from costly simulations, where there is an incentive to minimize computational cost.

Bayesian inference of flame impulse responses

Abstract

The impulse response of a flame to acoustic velocity perturbations is a key quantity for predicting thermoacoustic stability, but its identification from sparse, noisy observations requires solving an ill-posed inverse convolution problem. This is typically achieved with system identification methods, which require hand-tuning of regularization, model order, and sampling parameters, and provide no principled mechanism for incorporating prior physical knowledge. In this paper, we reformulate the identification problem within a Bayesian framework. The impulse response is represented as a physically motivated distributed time delay model, whose parameters correspond to convective delays and dispersive broadening. For a given number of pulses, the model parameters are inferred from the data using Bayesian parameter inference. The number of pulses is then selected using Bayesian model comparison, which balances data fit against model complexity to identify the simplest model capable of explaining the data. The framework is demonstrated on broadband-forced large eddy simulation data from a turbulent swirl-stabilized burner. Bayesian model comparison selects a three-Gaussian impulse response for this flame, consistent with physical interpretations in previous work. Compared with system identification, the Bayesian approach produces impulse responses with fewer spurious features and enables straightforward enforcement of a known low-frequency gain. Finally, we show that the Bayesian approach is robust to significant reductions in recording length, making it appealing for impulse response identification from costly simulations, where there is an incentive to minimize computational cost.
Paper Structure (21 sections, 38 equations, 10 figures, 1 table, 1 algorithm)

This paper contains 21 sections, 38 equations, 10 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Physically motivated model of the impulse response
  3. A probabilistic formulation of the inverse problem
  4. Bayesian parameter inference
  5. The prior
  6. The likelihood
  7. The posterior
  8. Bayesian model selection
  9. Uncertainty estimation
  10. Summary
  11. Robust solution of the Bayesian inverse problem
  12. Parameterization and the prior
  13. Solution of the optimization problem
  14. Edge effects and the valid convolution region
  15. Choosing the impulse response support
  16. ...and 6 more sections

Figures (10)

  • Figure 1: Prior probability density functions for the impulse response parameters: (a) amplitude $n_i$, (b) non-dimensionalized time delay gap $\alpha_i$, (c) non-dimensionalized width $\tilde{\sigma}_i$.
  • Figure 2: Schematic of the BRS burner, showing the swirler in the forward position (black) and aft postion (grey).
  • Figure 3: Example input-output data from LES of a turbulent swirl flame: (blue) velocity perturbation, (orange) heat release rate fluctuation.
  • Figure 4: Model ranking metrics for the baseline case, showing the three model ranking metrics described in Section \ref{['sec:modelRanking']}. The best-fit likelihood (BFL) rewards models that fit the data well, while the Occam factor (OF) penalizes models that are too flexible. The marginal likelihood (ML) balances these two effects, and is used to select the most likely model. The best-fit likelihood and marginal likelihood are shown relative to the marginal likelihood of the three-Gaussian model, which is the most likely model.
  • Figure 5: Impulse responses inferred from LES data using system identification (SI) and Bayesian inference (BI) with and without enforcing the low-frequency limit (LFL).
  • ...and 5 more figures