Thermoacoustic instability is a persistent problem in rocket engines and gas turbines because oscillations are so sensitive to many design parameters. This, however, presents an excellent opportunity for the application of adjoint methods and the combination of experiments and modelling.

Background

The first scientific reports of thermoacoustic oscillations appeared over two centuries ago. Their practical consequences have been evident since liquid rocket engine development in the 1930's: they cause thrust oscillations, structural damage, increased heat transfer, and component or payload failure. Despite decades of research by Germany from the 1930's, by the USA and USSR during the cold war, and recently by the gas turbine industry, these oscillations remain a severe problem today.

Thermoacoustic oscillations from a camping gas flame
An empty tube is placed over a camping gas flame, setting up thermo-acoustic oscillations in the tube.
Credit: Matthew Juniper
Thermoacoustic Oscillations in slow motion
A premixed bunsen flame in placed within a quartz tube. Self-excited thermo-acoustic oscillations grow to a limit cycle. This video shows the flame in slow motion during the limit cycle. Wrinkles in the flame start at the bottom and travel upwards, causing the flame to pinch off some time later. This flame pinch-off causes a sudden burst of heat release, which creates pressure waves that sustain the thermo-acoustic oscillations.
Credit: Nicholas Jamieson and Matthew Juniper

The thermoacoustic mechanism and its thermodynamic efficiency

In order to achieve high power to weight ratios and high efficiency, rocket and gas turbine engines all have low acoustic damping and high energy densities: up to 50 GW/m3 for liquid rockets, 1 GW/m3 for solid rockets, and 0.1 GW/m3 for jet engines and afterburners. Consequently, large amplitude oscillations can be sustained even if the thermoacoustic mechanism is only slightly (i.e. ~0.1%) efficient for at least one acoustic mode.

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Description of the thermoacoustic mechanism via a pressure - volume diagram
The mechanism that drives thermoacoustic oscillation is similar to that which drives a piston engine. In an idealized piston engine, work is done on a gas as it is compressed isentropically from states 1 to 2 (blue compression line). From 2 to 3, the gas combusts at fixed volume, releasing heat and raising its pressure further. From 3 to 4, the gas does work as it expands isentropically (blue expansion line). More work is done by the gas during the expansion phase than is done on it during the compression phase, leading to a net conversion of heat to work given by the area within the cycle on the pressure-volume diagram. In thermoacoustics, an acoustic wave replaces the piston and a continuous flame replaces the periodically-ignited gas. The acoustic wave independently (i) perturbs this flame and (ii) compresses and expands the gas around the flame. If the perturbed flame releases more heat than average during instants of higher local pressure, then, through the same mechanism as the piston engine, more work is done by the gas during the acoustic expansion phase than is done on it during the acoustic compression phase (red cycle). If this work is not dissipated then the oscillation amplitude grows and the system is thermoacoustically unstable (orange spiral).
Credit: Matthew Juniper

Engine development typically consists of component tests, sector tests, full combustor tests, and full engine tests. Thermoacoustic instability tends to recur during the later stages and is rarely predicted reliably by component tests and analysis. This is because the efficiency of the thermoacoustic mechanism is exceedingly sensitive to small changes to the system for reasons described in the following review paper.

Sensitivity and nonlinearity in Thermoacoustics
M. Juniper, R. I. Sujith
Annual Review of Fluid Mechanics 50, 661--689, (2018), doi:10.1146/annurev-fluid-122316-045125
pdf
Open Access
doi: https://doi.org/10.1146/annurev-fluid-122316-045125
Tutorial 1: Obtaining thermoacoustic eigenvalue sensitivities with adjoint methods
Tutorial 1 Matlab files
Tutorial 2: Tools from nonlinear dynamics
Tutorial 2 Matlab files

Nine decades of rocket engine and gas turbine development have shown that thermoacoustic oscillations are difficult to predict but can usually be eliminated with relatively small ad hoc design changes. These changes can, however, be ruinously expensive to devise. This review explains why linear and nonlinear thermoacoustic behaviour is so sensitive to parameters such as operating point, fuel composition, and injector geometry. It shows how non-periodic behaviour arises in experiments and simulations and discusses how fluctuations in thermoacoustic systems with turbulent reacting flow, which are usually filtered or averaged out as noise, can reveal useful information. Finally, it proposes tools to exploit this sensitivity in the future: adjoint-based sensitivity analysis to optimize passive control designs, and complex systems theory to warn of impending thermoacoustic oscillations and to identify the most sensitive elements of a thermoacoustic system.

This extreme sensitivity allows thermoacoustic systems to be stabilized with small design changes, usually at the full engine test stage. These changes can, however, be ruinously expensive to devise. The challenge is to devise and implement these design changes cheaply, quickly, and earlier in the design process.

Adjoint methods applied to thermoacoustic network models

There are many ways to model a thermoacoustic system. Acoustic network models are widely used in industry. Typically there will be many stable modes and just a handful of unstable modes:

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A typical thermoacoustic network model and corresponding natural modes
A thermoacoustic system can be modelled by combining an acoustic network with a flame model (left; adapted from Stow and Dowling 2008). The natural modes of this system (right) can be calculated. Typically a handful of these modes will be unstable and most will be stable.
Credit: Simon Stow (left) Magri et al (right)
Adjoint-based shape optimization of an annular combustion chamber in a simplified gas turbine engine
The top-right figure shows a 3D rendering of a simplified model of an annular combustion chamber in an aircraft gas turbine engine. The bottom-right shows a slice through the combustion chamber, showing the plenum on the left, the feed tubes in the middle, and the combustion chamber on the right. The figure on the left shows the acoustic natural frequencies (eigenmodes) of this system, calculated with an acoustic network model. Two modes have positive growth rate (grey region) and five have negative growth rate (white region). Adjoint methods are used to calculate the sensitivity of both unstable modes to all possible geometry changes. This gradient information is used to subtly change the shape of the plenum and combustion chamber until both modes are stable. Only small changes are required, and they are mainly in the plenum.
Credit: Jose Aguilar and Matthew Juniper
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We wish to stabilize the handful of unstable modes by altering the design parameters. There may be several thousand of these and it is impractical to work out the influence of each one by varying each in turn. With adjoint methods, however, we can evaluate the influence of every design parameter on a given eigenvalue with a single calculation. Therefore we need only as many calculations as there are unstable eigenvalues.

Our first application of this technique was to a simple model of a hot wire in a tube and the second was to a Burke-Schumann flame in a tube:

Sensitivity analysis of a time-delayed thermoacoustic system via an adjoint-based approach
L. Magri and M. P. Juniper
Journal of Fluid Mechanics 719, 183--202, (2013), doi:10.1017/jfm.2012.639
pdf
doi: https://doi.org/10.1017/jfm.2012.639

We apply adjoint-based sensitivity analysis to a time-delayed thermo-acoustic system: a Rijke tube containing a hot wire. We calculate how the growth rate and frequency of small oscillations about a base state are affected either by a generic passive control element in the system (the structural sensitivity analysis) or by a generic change to its base state (the base-state sensitivity analysis). We illustrate the structural sensitivity by calculating the effect of a second hot wire with a small heat-release parameter. In a single calculation, this shows how the second hot wire changes the growth rate and frequency of the small oscillations, as a function of its position in the tube. We then examine the components of the structural sensitivity in order to determine the passive control mechanism that has the strongest influence on the growth rate. We find that a force applied to the acoustic momentum equation in the opposite direction to the instantaneous velocity is the most stabilizing feedback mechanism. We also find that its effect is maximized when it is placed at the downstream end of the tube. This feedback mechanism could be supplied, for example, by an adiabatic mesh. We illustrate the base-state sensitivity by calculating the effects of small variations in the damping factor, the heat-release time-delay coefficient, the heat-release parameter, and the hot-wire location. The successful application of sensitivity analysis to thermo-acoustics opens up new possibilities for the passive control of thermo-acoustic oscillations by providing gradient information that can be combined with constrained optimization algorithms in order to reduce linear growth rates.
Global modes, receptivity, and sensitivity analysis of diffusion flames coupled with duct acoustics
L. Magri, M. P. Juniper
Journal of Fluid Mechanics 752, 237--265, (2014), doi:10.1017/jfm.2014.328
pdf
Open Access
doi: https://doi.org/10.1017/jfm.2014.328

In this theoretical and numerical paper, we derive the adjoint equations for a thermo-acoustic system consisting of an infinite-rate chemistry diffusion flame coupled with duct acoustics. We then calculate the thermo-acoustic system's linear global modes (i.e. the frequency/growth rate of oscillations, together with their mode shapes), and the global modes' receptivity to species injection, sensitivity to base-state perturbations and structural sensitivity to advective-velocity perturbations. Some of these could be found by finite difference calculations but the adjoint analysis is computationally much cheaper. We then compare these with the Rayleigh index. The receptivity analysis shows the regions of the flame where open-loop injection of fuel or oxidizer will have the greatest influence on the thermo-acoustic oscillation. We find that the flame is most receptive at its tip. The base-state sensitivity analysis shows the influence of each parameter on the frequency/growth rate. We find that perturbations to the stoichiometric mixture fraction, the fuel slot width and the heat-release parameter have most influence, while perturbations to the Peclet number have the least influence for most of the operating points considered. These sensitivities oscillate, e.g. positive perturbations to the fuel slot width either stabilizes or destabilizes the system, depending on the operating point. This analysis reveals that, as expected from a simple model, the phase delay between velocity and heat-release fluctuations is the key parameter in determining the sensitivities. It also reveals that this thermo-acoustic system is exceedingly sensitive to changes in the base state. The structural-sensitivity analysis shows the influence of perturbations to the advective flame velocity. The regions of highest sensitivity are around the stoichiometric line close to the inlet, showing where velocity models need to be most accurate. This analysis can be extended to more accurate models and is a promising new tool for the analysis and control of thermo-acoustic oscillations.

Once the technique had been proven, we were able to move to thermoacoustic network models, which are used in industry. Using adjoint methods, we showed how to stabilize Rama Balachandran's thermoacoustic system by making small design changes:

Adjoint-based stabilization of a thermoacoustic network model
The top frame is a diagram of the combustion system analysed in Rama Balachandran thesis (2005, University of Cambridge). The flame is the red vertical line. It sits downstream of the fuel/air feed system (to the left of the flame) and upstream of the combustion chamber (to the right of the flame). The bottom frame shows the eigenvalues (black circles) of this thermoacoustic system. Those with a positive growth rate (above the dashed line) are unstable. Using adjoint methods, we calculate the sensitivity of each unstable eigenvalue to changes in the system geometry. We then shift the geometry slightly in a way that is calculated to optimally stabilize all the eigenvalues. We repeat this process until all the eigenvalues are stable. This process is very quick with adjoint methods, but prohibitively expensive with standard methods.
Credit: Jose Aguilar
Adjoint-based shape optimization of laboratory combustor, changing areas only
The top figure shows a 3D rendering of a laboratory combustor. Air enters on the left into a plenum. Air then flows through a narrow feed tube into a combustion chamber on the right. A flame sits at the left end of the combustion chamber. The bottom figure shows the acoustic natural frequencies (eigenmodes) of this system. These are calculated with an acoustic network model. Frequency is on the horizontal axis and growth rate is on the vertical axis. The eigenmodes with positive growth rate (marked in red) are unstable. In this case there are seven unstable modes. The sensitivity of the eigenmodes to changes in the area of the tube is calculated with an adjoint network model. This information is used to stabilize all seven modes by making relatively small changes to the areas.
Credit: Jose Aguilar and Matthew Juniper
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Adjoint-based sensitivity analysis of low-order thermoacoustic networks using a wave-based approach
J. G. Aguilar, L. Magri, M. P. Juniper
Journal of Computational Physics 341, 163--181, (2017), doi:10.1016/j.jcp.2017.04.013
pdf
Open Access
doi: https://doi.org/10.1016/j.jcp.2017.04.013

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 analysed. 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.

The application to annular combustors is technically more complicated because they contain degenerate eigenvalues and a second order analysis is required:

Stability analysis of thermoacoustic nonlinear eigenproblems in annular combustors. Part 1: sensitivity
L. Magri, M. Bauerheim, M. P. Juniper
Journal of Computational Physics 325, 395--410, (2016), doi:10.1016/j.jcp.2016.07.032
pdf
Open Access
doi: https://doi.org/10.1016/j.jcp.2016.07.032

We present an adjoint-based method for the calculation of eigenvalue perturbations in nonlinear, degenerate and non self-adjoint eigenproblems. This method is applied to a thermo-acoustic annular combustor network, the stability of which is governed by a nonlinear eigenproblem. We calculate the first- and second-order sensitivities of the growth rate and frequency to geometric, flow and flame parameters. Three different configurations are analysed. The benchmark sensitivities are obtained by finite difference, which involves solving the nonlinear eigenproblem at least as many times as the number of parameters. By solving only one adjoint eigenproblem, we obtain the sensitivities to any thermo-acoustic parameter, which match the finite-difference solutions at much lower computational cost.

Adjoint methods applied to thermoacoustic Helmholtz solvers

Thermoacoustic Helmholtz solvers solve the wave equation in an arbitrarily complex geometry. They are more versatile and more expensive than network models. The following paper shows how to derive and implement adjoint thermoacoustic Helmholtz solvers in 1D and 2D:

Sensitivity Analysis of Thermoacoustic Instability with Adjoint Helmholtz Solvers
M. Juniper
Physical Review Fluids 3, 110509, (2018), doi:10.1103/PhysRevFluids.3.110509
pdf
Open Access
doi: https://doi.org/10.1103/PhysRevFluids.3.110509
Matlab code
Erratum

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.
Shape Optimization of Thermoacoustic Systems Using a 2D Adjoint Helmholtz Solver
S. Falco, M. P. Juniper
Journal of Engineering for Gas Turbines and Power (accepted), , (2020), doi:10.1115/1.4049305
pdf
doi: https://doi.org/10.1115/1.4049305

Thermoacoustic instabilities, which arise due to the interaction between flames and acoustics, are sensitive to small changes to the system parameters. In this paper, we apply adjoint-based shape optimization to a 2D finite element Helmholtz solver to find accurately and inexpensively the shape changes that most stabilise a 2D thermoacoustic system in the linear regime. We examine two cases: a Rijke tube and a turbulent swirl combustor. Both systems exhibit an unstable longitudinal mode and we suppress the instability by slightly modifying the geometry. In the case of the turbulent swirl combustor, the sensitivities are higher in the plenum and in the burner than in the combustion chamber, mainly due to the effect of the mean temperature. In the cooler regions, the local wavelength is shorter, which means that geometry changes of a given distance have more influence than they do where the local wavelength is longer. This is the first time adjoint-based shape optimization is applied to 2D Helmholtz solvers in thermoacoustics, after being previously applied to low-order thermoacoustic networks. But Helmholtz solvers have an intrinsic advantage: they can handle complex geometries. The easy scalability of this method to complex 3D geometries make this tool a strong candidate for the iterative design of thermoacoustically stable combustors.

Inverse Uncertainty Quantification from automated experiments

Adjoint methods can also be used in uncertainty quantification (UQ) and greatly reduce the cost of UQ, as long as the uncertainties do not vary very nonlinearly with the parameters.

Stability analysis of thermoacoustic nonlinear eigenproblems in annular combustors. Part 2: Uncertainty Quantification
L. Magri, M. Bauerheim, F. Nicoud, M. P. Juniper
Journal of Computational Physics 325, 411--421, (2016), doi:10.1016/j.jcp.2016.08.043
pdf
Open Access
doi: https://doi.org/10.1016/j.jcp.2016.08.043

Monte Carlo and Active Subspace Identification methods are combined with first- and second-order adjoint sensitivities to perform (forward) uncertainty quantification analysis of the thermo-acoustic stability of two annular combustor configurations. This method is applied to evaluate the risk factor, i.e., the probability for the system to be unstable. It is shown that the adjoint approach reduces the number of nonlinear-eigenproblem calculations by as much as the Monte Carlo samples.

Adjoint methods are ideal for identifying the small design changes that can stabilize thermoacoustic modes. They require, however, an accurate model of the system. Devising accurate thermoacoustic models is challenging because extreme sensitivity to parameters introduces considerable systematic error if parameters cannot be estimated accurately. This is arguably the most intractible problem in thermoacoustics.

One solution is to calculate parameters from automated experimental measurements and inverse uncertainty quantification. Parameters learned on laboratory-scale rigs can then be updated as tests are performed on larger scale rigs. Our approach is to measure growth and decay rates extremely accurately using automated experiments:

Experimental sensitivity analysis and control of thermoacoustic systems
G. Rigas, N. Jamieson, L. K. B. Li, M. Juniper
Journal of Fluid Mechanics 787, R1, (2016), doi:10.1017/jfm.2015.715
Experimental sensitivity analysis via a secondary heat source in an oscillating thermoacoustic system
N. Jamieson, G. Rigas, M. P. Juniper
International Journal of Spray and Combustion Dynamics 9 (4), 230--240, (2017), doi:10.1177/1756827717696325
pdf
Open Access
doi: https://doi.org/10.1177/1756827717696325

In this paper, we report the results of an experimental sensitivity analysis on a vertical electrically-heated Rijke tube. We examine the stability characteristics of the system due to the introduction of a secondary heat source. The experimental sensitivity analysis is quantified by measuring the shift in linear growth and decay rate as well as the shift in the linear frequency during periods of growth and decay of thermoacoustic oscillations. Linear growth and decay rate measurements agree qualitatively well with the theoretical predictions from adjoint-based methods of Magri & Juniper (J. Fluid Mech., vol. 719, 2013, pp. 183--202). A discrepancy in the linear frequency measurements highlight deficiencies in the model used for those predictions and shows that the experimental measurement of sensitivities is a stringent test of any thermoacoustic model. The findings suggest that adjoint-based methods are, in principle, capable of providing industry with a cheap and efficient tool for developing optimal control strategies for more complex thermoacoustic systems.

We obtain tens of thousands of datapoints and then infer the uncertainty in parameters. This work is described further in the Physics-based Statistical Learning project.