109 research outputs found
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Adjoint-based sensitivity analysis of low-order thermoacoustic networks using a wave-based approach
Strict pollutant emission regulations are pushing gas turbine manufacturers to develop devices that operate in lean conditions, with the downside that combustion instabilities are more likely to occur. Methods to predict and control unstable modes inside combustion chambers have been developed in the last decades but, in some cases, they are computationally expensive. Sensitivity analysis aided by adjoint methods provides valuable sensitivity information at a low computational cost. This paper introduces adjoint methods and their application in wave-based low order network models, which are used as industrial tools, to predict and control thermoacoustic oscillations. Two thermoacoustic models of interest are analyzed. First, in the zero Mach number limit, a nonlinear eigenvalue problem is derived, and continuous and discrete adjoint methods are used to obtain the sensitivities of the system to small modifications. Sensitivities to base-state modification and feedback devices are presented. Second, a more general case with non-zero Mach number, a moving flame front and choked outlet, is presented. The influence of the entropy waves on the computed sensitivities is shown.J.G.A. is grateful to Alessandro Orchini for fruitful discussions and comments on this paper, and thankfully acknowledges CONACyT and Cambridge Trust for funding this project. L.M. gratefully acknowledges the financial support from the Royal Academy of Engineering Research Fellowships scheme
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Adjoint Methods as Design Tools in Thermoacoustics
In a thermoacoustic system, such as a flame in a combustor, heat release oscillations couple with acoustic pressure oscillations. If the heat release is sufficiently in phase with the pressure, these oscillations can grow, sometimes with catastrophic consequences. Thermoacoustic instabilities are still one of the most challenging problems faced by gas turbine and rocket motor manufacturers. Thermoacoustic systems are characterized by many parameters to which the stability may be extremely sensitive. However, often only few oscillation modes are unstable. Existing techniques examine how a change in one parameter affects all (calculated) oscillation modes, whether unstable or not. Adjoint techniques turn this around: They accurately and cheaply compute how each oscillation mode is affected by changes in all parameters. In a system with a million parameters, they calculate gradients a million times faster than finite difference methods. This review paper provides: (i) the methodology and theory of stability and adjoint analysis in thermoacoustics, which is characterized by degenerate and nondegenerate nonlinear eigenvalue problems; (ii) physical insight in the thermoacoustic spectrum, and its exceptional points; (iii) practical applications of adjoint sensitivity analysis to passive control of existing oscillations, and prevention of oscillations with ad hoc design modifications; (iv) accurate and efficient algorithms to perform uncertainty quantification of the stability calculations; (v) adjoint-based methods for optimization to suppress instabilities by placing acoustic dampers, and prevent instabilities by design modifications in the combustor's geometry; (vi) a methodology to gain physical insight in the stability mechanisms of thermoacoustic instability (intrinsic sensitivity); and (vii) in nonlinear periodic oscillations, the prediction of the amplitude of limit cycles with weakly nonlinear analysis, and the theoretical framework to calculate the sensitivity to design parameters of limit cycles with adjoint Floquet analysis. To show the robustness and versatility of adjoint methods, examples of applications are provided for different acoustic and flame models, both in longitudinal and annular combustors, with deterministic and probabilistic approaches. The successful application of adjoint sensitivity analysis to thermoacoustics opens up new possibilities for physical understanding, control and optimization to design safer, quieter, and cleaner aero-engines. The versatile methods proposed can be applied to other multiphysical and multiscale problems, such as fluid–structure interaction, with virtually no conceptual modification.</jats:p
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Thermoacoustic stabilization of a longitudinal combustor using adjoint methods
We construct a low order thermoacoustic network model that contains the most influential physical mechanisms of a thermoacoustic system. We apply it to a laboratory-scale longitudinal combustor that has been found to be thermoacoustically unstable in experiments. We model the flame, which is behind a bluff body, by a geometric level set method. We obtain the thermoacoustic eigenvalues of this configuration and examine a configuration in which six eigenmodes are unstable. We then derive the adjoint equations of this model and use the corresponding adjoint eigenmodes to obtain the sensitivities of the unstable eigenvalues to modifications of the model geometry. These sensitivities contain contributions from changes to the steady base flow and changes to the fluctuating flow. We find that these two contributions have similar magnitudes, showing that both contributions need to be considered. We then wrap these sensitivities within a gradient-based optimization algorithm and stabilize all six eigenvalues by changing the geometry. The required geometry changes are well approximated by the first step in the optimization process, showing that this sensitivity information is useful even before it is embedded within an optimization algorithm. We examine the acoustic energy balance during the optimization process and identify the physical mechanisms through which the algorithm is stabilizing the combustor. The algorithm works by, for each mode, reducing the work done by the flame, while simultaneously increasing the work done by the system on the outlet boundary. We find that only small geometry changes are required in order to stabilize every mode. The network model used in this study deliberately has the same structure as one used in the gas turbine industry in order to ease its implementation in practice.Cambridge Trust
Adjoint characteristic decomposition of one-dimensional waves
Adjoint methods enable the accurate calculation of the sensitivities of a
quantity of interest. The sensitivity is obtained by solving the adjoint
system, which can be derived by continuous or discrete adjoint strategies. In
acoustic wave propagation, continuous and discrete adjoint methods have been
developed to compute the eigenvalue sensitivity to design parameters and
passive devices (Aguilar, J. G. et al, 2017, J. Computational Physics, vol.
341, 163-181). In this short communication, it is shown that the continuous and
discrete adjoint characteristic decompositions, and Riemann invariants, are
connected by a similarity transformation. The results are shown in the Laplace
domain. The adjoint characteristic decomposition is applied to a
one-dimensional acoustic resonator, which contains a monopole source of sound.
The proposed framework provides the foundation to tackle larger acoustic
networks with a discrete adjoint approach, opening up new possibilities for
adjoint-based design of problems that can be solved by the method of
characteristics
Global linear stability analysis of a slit flame subject to intrinsic thermoacoustic instability
The present study makes use of the adjoint modes of the Linearized Reactive
Flow (LRF) equations to investigate the Intrinsic Thermoacoustic (ITA) feedback
loop of a laminar premixed slit flame. The analysis shows that the ITA feedback
loop is closed by vorticity generated in the boundary layer of the slit by
impinging acoustic waves penetrating the slit. In this region, adjoint
vorticity shows a high sensitivity of the flow. It is also hypothesised that
the ITA eigenmode smoothly transitions to a purely hydrodynamic mode -- vortex
shedding -- for a passive flame. The computational domain is chosen
sufficiently short so as to isolate the ITA feedback loop from cavity modes.
This analysis is made possible by the holistic character of the LRF model, i.e.
a direct linearization of the non-linear reactive flow equations, including
explicit finite rate chemistry and avoiding idealization of the flow.Comment: 11 pages, 6 figures. Presented at the International Congress on Sound
and Vibration, July 2023, Pragu
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Sensitivity analysis of thermoacoustic instability with adjoint Helmholtz solvers
Gas turbines and rocket engines sometimes suffer from violent oscillations caused by feedback between acoustic waves and flames in the combustion chamber. These are known as thermoacoustic oscillations and they often occur late in the design process. Their elimination usually requires expensive tests and re-design. Full scale tests and laboratory scale experiments show that these oscillations can usually be stabilized by making small changes to the system. The complication is that, while there is often just one unstable natural oscillation (eigenmode), there are very many possible changes to the system. The challenge is to identify the optimal change systematically, cheaply, and accurately. This paper shows how to evaluate the sensitivities of a thermoacoustic eigenmode to all possible system changes with a single calculation by applying adjoint methods to a thermoacoustic Helmholtz solver. These sensitivities are calculated here with finite difference and finite element methods, in the weak form and the strong form, with the discrete adjoint and the continuous adjoint, and with a Newton method applied to a nonlinear eigenvalue problem and an iterative method applied to a linear eigenvalue problem. This is the first detailed comparison of adjoint methods applied to thermoacoustic Helmholtz solvers. Matlab codes are provided for all methods and all figures so that the techniques can be easily applied and tested. This paper explains why the finite difference of the strong form equations with replacement boundary conditions should be avoided and why all of the other methods work well. Of the other methods, the discrete adjoint of the weak form equations is the easiest method to implement; it can use any discretization and the boundary conditions are straightforward. The continuous adjoint is relatively easy to implement but requires careful attention to boundary conditions. The Summation by Parts finite difference of the strong form equations with a Simultaneous Approximation Term for the boundary conditions (SBP--SAT) is more challenging to implement, particularly at high order or on non-uniform grids. Physical interpretation of these results shows that the well-known Rayleigh criterion should be revised for a linear analysis. This criterion states that thermoacoustic oscillations will grow if heat release rate oscillations are sufficiently in phase with pressure oscillations. In fact, the criterion should contain the adjoint pressure rather than the pressure. In self-adjoint systems the two are equivalent. In non-self-adjoint systems, such as all but a special case of thermoacoustic systems, the two are different. Finally, the sensitivities of the growth rate of oscillations to placement of a hot or cold mesh are calculated, simply by multiplying the feedback sensitivities by a number. These sensitivities are compared successfully with experimental results. With the same technique, the influence of the viscous and thermal acoustic boundary layers is found to be negligible, while the influence of a Helmholtz resonator is found, as expected, to be considerable
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Sensitivity analysis and optimization in low order thermoacoustic models
Lean combustion technologies in gas turbines reduce the generation of NOx but increase the susceptibility to thermoacoustic oscillations.
These oscillations can produce structural damage and need to be eliminated.
The stability of a given configuration can be examined with a thermoacoustic model.
In this thesis a wave-based network model is used.
Using adjoint methods the gradients of the eigenvalue with respect to system parameters can be obtained at a low computational cost.
This information is used as an input to an optimization routine to find stable thermoacoustic configurations.
In this thesis thermocaoustic oscillations are analysed using a linear low order network model.
This modelling approach is used to predict the unstable modes of five different configurations: a Rijke tube, a choked combustor, a longitudinal combustor, a generic lean premix prevaporized annular combustor and the laboratory scale annular combustor built in Cambridge University Engineering Department.
The continuous and discrete adjoint equations for the low order network model are derived.
Using the adjoint equations the sensitivities of the eigenvalues to changes in base state parameters such as time delays, areas, lengths and mean radii are computed.
Similarly, the sensitivity of the eigenvalues to the introduction of a feedback device such as a drag mesh or a secondary heat source is investigated.
By fitting experimental data to a low Mach number model of the Rijke tube, the predictions of the growth rate and frequency shifts due to the presence of these mechanisms are improved.
Finally, using the sensitivity information, two different optimization algorithms are developed to stabilize the thermoacoustic systems.
Different stabilization scenarios are presented, showing the changes required in each section of the configurations to eliminate thermoacoustic oscillations.
The techniques presented as part of this thesis are readily scalable to more complex models and geometries and the inclusion of further constraints.
This demonstrates that adjoint-based sensitivity analysis and optimization could become an indispensable tool for the design of thermoacoustically-stable combustors.Cambridge Trust and CONACy
Hard-constrained neural networks for modelling nonlinear acoustics
We model acoustic dynamics in space and time from synthetic sensor data. The
tasks are (i) to predict and extrapolate the spatiotemporal dynamics, and (ii)
reconstruct the acoustic state from partial observations. To achieve this, we
develop acoustic neural networks that learn from sensor data, whilst being
constrained by prior knowledge on acoustic and wave physics by both informing
the training and constraining parts of the network's architecture as an
inductive bias. First, we show that standard feedforward neural networks are
unable to extrapolate in time, even in the simplest case of periodic
oscillations. Second, we constrain the prior knowledge on acoustics in
increasingly effective ways by (i) employing periodic activations (periodically
activated neural networks); (ii) informing the training of the networks with a
penalty term that favours solutions that fulfil the governing equations
(soft-constrained); (iii) constraining the architecture in a
physically-motivated solution space (hard-constrained); and (iv) combination of
these. Third, we apply the networks on two testcases for two tasks in nonlinear
regimes, from periodic to chaotic oscillations. The first testcase is a twin
experiment, in which the data is produced by a prototypical time-delayed model.
In the second testcase, the data is generated by a higher-fidelity model with
mean-flow effects and a kinematic model for the flame source. We find that (i)
constraining the physics in the architecture improves interpolation whilst
requiring smaller network sizes, (ii) extrapolation in time is achieved by
periodic activations, and (iii) velocity can be reconstructed accurately from
only pressure measurements with a combination of physics-based hard and soft
constraints. In and beyond acoustics, this work opens strategies for
constraining the physics in the architecture, rather than the training
Degenerate perturbation theory in thermoacoustics: High-order sensitivities and exceptional points
In this study, we connect concepts that have been recently developed in
thermoacoustics, specifically, (i) high-order spectral perturbation theory,
(ii) symmetry induced degenerate thermoacoustic modes, (iii) intrinsic
thermoacoustic modes, and (iv) exceptional points. Their connection helps gain
physical insight into the behaviour of the thermoacoustic spectrum when
parameters of the system are varied. First, we extend high-order adjoint-based
perturbation theory of thermoacoustic modes to the degenerate case. We provide
explicit formulae for the calculation of the eigenvalue corrections to any
order. These formulae are valid for self-adjoint, non-self-adjoint or even
non-normal systems; therefore, they can be applied to a large range of
problems, including fluid dynamics. Second, by analysing the expansion
coefficients of the eigenvalue corrections as a function of a parameter of
interest, we accurately estimate the radius of convergence of the power series.
Third, we connect the existence of a finite radius of convergence to the
existence of singularities in parameter space. We identify these singularities
as exceptional points, which correspond to defective thermoacoustic
eigenvalues, with infinite sensitivity to infinitesimal changes in the
parameters. At an exceptional point, two eigenvalues and their associated
eigenvectors coalesce. Close to an exceptional point, strong veering of the
eigenvalue trajectories is observed. As demonstrated in recent work,
exceptional points naturally arise in thermoacoustic systems due to the
interaction between modes of acoustic and intrinsic origin. The role of
exceptional points in thermoacoustic systems sheds new light on the physics and
sensitivity of thermoacoustic stability, which can be leveraged for passive
control by small design modifications
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