Adjoint Accelerated Bayesian Inference of Acoustically-Forced Premixed Flames

¹ Department of Mechanics, Mathematics and Management, Politecnico di Bari, Via Re David 200, 70125 Bari, Italy

² Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK

A. Giannotta¹, M. Yoko², S. Cherubini¹, P. De Palma¹, M. Juniper²

What are Thermoacoustic Instabilities?�Acoustic oscillations induce flow and mixture perturbations, which turn into heat release rate oscillations

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Why do we study it? �Thermoacoustic oscillations depend on the phase difference between the heat release rate and pressure oscillations.

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With well-chosen experiments, we can

• tune the parameters of candidate models
• compare candidate models against each other and select the one with most evidence, given the experimental data.

Juniper MP and Sujith RI. Sensitivity and Nonlinearity of Thermoacoustic Oscillations. Annual Review of Fluid Mechanics 2018; 50: 661–689

• Difficult to predict
• Extremely sensitive to parameters

This extreme sensitivity means that thermoacoustic systems can often be stabilized by making small changes.

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Prior

Posterior Likelihood

Evidence

Likelihood

The best estimate of the parameters is that which maximises the left-hand side of the equation

What is Bayesian Inference?�We assume that all of the distributions are Gaussian and define the Cost Function J as the negative log of the numerator

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Prior

Posterior Likelihood

Evidence

Likelihood

Contribution from

model and data discrepancy

Contribution from the Prior

How can adjoint be helpful for Bayesian Inference?�Using adjoint methods we can calculate the derivative of the cost function J with respect to the model parameters

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Model Sensitivity

Prior

Posterior Likelihood

Likelihood

How can adjoint be helpful for Bayesian Inference?�The errors in the parameters (the posterior parameter covariance matrix) can be estimated very cheaply with Laplace’s approximation, using adjoint methods

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Gaussian

Gaussian

Gaussian

Hessian of the cost function

Second order Adjoint

or approximated using the BFGS method

What do we do? We assimilate experimental data into flame simulations

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Qualitatively-accurate Physics-based Flame Model

Experimental Data

Quantitatively-accurate Physics-based Flame Model

Data Assimilation

Best set of parameters estimation

What have we done so far?�We perform experiments on acoustically-forced ducted Bunsen flames and record the flame natural emission

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Yoko, Matthew, and Matthew P. Juniper. "Adjoint-accelerated Bayesian inference applied to the thermoacoustic behaviour of a ducted conical flame." Journal of Fluid Mechanics 985 (2024): A38.

What have we done so far?�The ROM is derived from the G-Equation

• Steady velocity field:

linear combination of a uniform and a Poiseuille flow¹

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• Each point on the flame-front moves according to

Velocity Field

Flame Speed

*Matalon, M. (1983). On flame stretch. Combustion science and technology, 31(3-4):169–181.

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The model has a handful of parameters that we don’t know a-priori:�it is not quantitatively accurate ‘out-of-the-box’. We infer the most-likely model parameters and their uncertainties using a Bayesian inverse-problem methodology

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Before

Data Assimilation

After

Data Assimilation

This effectively gives us a digital twin of the flame that we can explore in ways that aren’t possible experimentally.

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Why do we need

?

We have demonstrated that Bayesian Inference, accelerated by adjoint methods, is a powerful tool for developing quantitatively accurate physics-based models and for rigorously quantifying uncertainties in model parameters.

However, industrial-scale combustion systems present significantly greater complexity!

INDIAN INSTITUTE OF SPACE SCIENCE AND TECHNOLOGY, THIRUVANANTHAPURAM

Self-excited thermo-acoustic instability in a non-premixed swirl stabilized burner

Reynolds number 30,000;

Global equivalence ratio ~ 1.0

Fuel: Methane

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Our project roadmap

We need a CFD code that meets the following needs

• Harmonic inlet velocity or harmonic balance
• Axisymmetric flow
• Flamelet progress variable (FPV) approach
• URANS turbulence model
• Premixed flame and partially premixed with a secondary inlet stream of pure air.

We need the adjoint sensitivity with respect to:

• Boundary conditions
• Look up Table
• Turbulence parameters
• Burner geometry (secondary)

First step: simulate a harmonically forced premixed flame

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• Stoichiometric Methane/Air premixed flame
• Flamelet Model (FGM)
• LUT generated by a MATLAB script and CANTERA
• python wrapper to impose a time varying inlet TestCases/py_wrapper/custom_inlet/run.py

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Next Step: find the sensitivity of the h.h.r. and the flame front position with respect to the BCs and burner geometry

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Conclusions

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• We take a qualitatively-accurate physics-based model and render it quantitatively-accurate by assimilating data. This requires less data, is interpretable, and extrapolates to situations that share the same physics.
• We have demonstrated an application to thermoacoustic instabilities. However, this approach is versatile and can be extended to address a wide range of physical problems.
• SU2 adjoint capabilities can be extremely beneficial for solving these inverse problems.
• Further work is needed to simulate turbulent flames and to extend adjoint sensitivities to boundary conditions, turbulence model parameters and Look-Up-Table.

Thank you!

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Further reading about inverse problems in thermoacoustics and fluid dynamics:

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1. Giannotta, Alessandro, et al. "Bayesian inference of physics-based models of acoustically-forced laminar premixed conical flames." arXiv preprint arXiv:2407.03701 (2024).
2. Juniper, Matthew P., and Matthew Yoko. "Generating a physics-based quantitatively-accurate model of an electrically-heated Rijke tube with Bayesian inference." Journal of Sound and Vibration 535 (2022): 117096.
3. Kontogiannis, Alexandros, et al. "Joint reconstruction and segmentation of noisy velocity images as an inverse Navier–Stokes problem." Journal of Fluid Mechanics 944 (2022): A40.