Triggering of thermoacoustic oscillations from a linearly stable state is directly analogous to bypass transition to turbulence.

Non-modal stablity analysis

A little after the turn of the century, there was great interest in non-modal stability analysis as a way to explain bypass transition to turbulence in hydrodynamics. A conventional stability analysis considers the growth (e.g. of perturbation kinetic energy) over infinite time by considering the eigenvalues of the evolution operator. Non-modal stability analysis considers the transient growth over a defined interval of time by considering the singular values of the evolution operator (Trefethen et al, 'Hydrodynamic stability without eigenvalues' Science 261, 578--584). This could explain, for example, why round pipe flow (Hagen-Poiseuille flow) becomes turbulent even though the laminar velocity profile is stable at all Reynolds numbers.

At the time, it was conventional to linearize around a stable steady solution to the Navier--Stokes equations and to examine the non-normality of the linearized evolution operator. A strongly non-normal operator corresponds to a flow that is capable of strong transient growth, despite being linearly stable in the long time limit. Two interesting papers by Sujith's group at IIT Madras had shown that the evolution operator for thermoacoustic systems could be strongly non-normal (Balasubramanian and Sujith, 2008, Phys. Fluids 20, 044103 and Balasubramanian and Sujith, 2009, J. Fluid Mech. 594, 29–57) and therefore that one might find strong transient growth in thermoacoustics as well as in hydrodynamics.

Nonlinear adjoint looping in thermoacoustics

My first paper in this area used nonlinear adjoint looping of a time-delayed thermoacoustic system to find optimal initial states:

Triggering in the Rijke tube: non-normality, transient growth and bypass transition
M. P. Juniper
Journal of Fluid Mechanics 667 272--308 (2011) doi:10.1017/S0022112010004453

With a sufficiently large impulse, a thermoacoustic system can reach self-sustained oscillations even when it is linearly stable, a process known as triggering. In this paper, a procedure is developed to find the lowest initial energy that can trigger self-sustained oscillations, as well as the corresponding initial state. This is known as the ?most dangerous? initial state. The procedure is based on adjoint looping of the nonlinear governing equations, combined with an optimization routine. It is developed for a simple model of a thermoacoustic system, the horizontal Rijke tube, and can be extended to more sophisticated thermoacoustic models. It is observed that the most dangerous initial state grows transiently towards an unstable periodic solution before growing to a stable periodic solution. The initial energy required to trigger these self- sustained oscillations is much lower than the energy of the oscillations themselves and slightly lower than the lowest energy on the unstable periodic solution. It is shown that this transient growth arises due to non-normality of the governing equations. This is analogous to the sequence of events observed in bypass transition to turbulence in fluid mechanical systems and has the same underlying cause. The most dangerous initial state is calculated as a function of the heat-release parameter. It is found that self-sustained oscillations can be reached over approximately half the linearly stable domain. Transient growth in real thermoacoustic systems is 10^5?10^6 times greater than that in this simple model. One practical conclusion is that, even in the linearly stable regime, it may take very little initial energy for a real thermoacoustic system to trigger to high-amplitude self-sustained oscillations through the mechanism described in this paper.

The Web of Science ranked this paper 373 out of 82732 articles (top 0.45%) published between 2010 to 2014 in the field of Mechanics.

The main conclusion of this paper is that transient growth around the steady solution is almost irrelevant to bypass transition (or 'triggering', as it is known in thermoacoustics). Instead, the important characteristic is transient growth around unstable periodic solutions. Using nonlinear adjoint looping, I found the initial state that grew to self-sustained thermoacoustic oscillations from the lowest initial energy (which I called the 'most dangerous initial condition' and which hydrodynamics researchers call the 'minimal seed'). This is entirely different from the initial state that grows fastest around the steady solution. Instead, it is close to the lowest energy point on the unstable periodic solution plus a component in the direction that maximizes transient energy growth around this periodic solution:

Triggering in Thermoacoustics
This is a cartoon of acoustic phase space for the thermoacoustic model used in our first investigations into triggering in the horizontal Rijke tube. The stable fixed point (the non-oscillating solution) is at the origin. The stable periodic solution (the oscillating solution) is the solid black loop. The red manifold separates the points in phase space that decay to the stable fixed point from those that are attracted to the stable periodic solution. An unstable periodic solution (dashed black line) exists exactly on the manifold. The point with lowest energy on this unstable periodic solution is marked with a white dot. Nonlinear adjoint looping is used to find points on the manifold with lower energy than this, for example the black dot. From this low energy starting point, oscillations grow to the unstable periodic solution. If they are given infinitesimally more energy initially then they grow to the stable periodic solutions. The black dot is the 'most dangerous initial state' or 'minimal seed' because it grows to stable oscillations from the lowest initial energy.
Credit: Matthew Juniper
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Pseudospectra of the monodromy matrix
The black dots are the eigenvalues of the monodromy matrix (also known as Floquet multipliers) describing the evolution of small perturbations around the unstable periodic solution for a thermoacoustic model of the horizontal Rijke tube. Eigenvalues that lie outside the unit circle have positive growth rate and are unstable. In this case there is a single unstable eigenvalue at omega ~ 1.05, showing that this periodic solution is unstable to perturbations with exactly the same period as the periodic solution - i.e. the amplitude of the periodic solution will grow. The red lines are the pseudospectra of the monodromy matrix. If this matrix were Hermitian (i.e. M M^T = M^T M) then its eigenvectors would be orthogonal and the pseudospectra would be the envelope of circles centred on the eigenvalues. The matrix is, however, slightly non-Hermitian, which means that the pseudospectra extend slightly further away from the eigenvalues. Physically, this means that some perturbations around the unstable limit cycle (those with omega ~ -0.2 +- 0.9i) will grow slightly before they decay. This means that the 'most dangerous initial condition' can exploit transient growth around the unstable limit cycle and therefore does not sit on the unstable periodic solution itself.
Credit: Matthew Juniper
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In a second paper, I repeated the analysis over a wider range of initial energies and optimization times, and came to the same conclusion:

Transient growth and triggering in the horizontal Rijke tube
M. P. Juniper
International Journal of Spray and Combustion Dynamics 3 (3) 209--224 (2011)

This theoretical paper examines a non-normal and nonlinear model of a horizontal Rijke tube. Linear and non-linear optimal initial states, which maximize acoustic energy growth over a given time from a given energy, are calculated. It is found that non-linearity and non-normality both contribute to transient growth and that, for this model, linear optimal states are only a good predictor of non-linear optimal states for low initial energies. Two types of non-linear optimal initial state are found. The first has strong energy growth during the first period of the fundamental mode but loses energy thereafter. The second has weaker energy growth during the first period but retains high energy for longer. The second type causes triggering to self-sustained oscillations from lower energy than the first and has higher energy in the fundamental mode. This suggests, for instance, that low frequency noise will be more effective at causing triggering than high frequency noise.

This is summarized in a review paper on Triggering in Thermoacoustics

Triggering in thermoacoustics
M. P. Juniper
International Journal of Spray and Combustion Dynamics 4 (3) 217--238 (2012)

Under certain conditions, the flow in a combustion chamber can sustain large amplitude oscillations even when its steady state is linearly stable. Experimental studies show that these large oscillations can sometimes be triggered by very low levels of background noise. This theoretical paper sets out the conditions that are necessary for triggering to occur. It uses a weakly nonlinear analysis to show when these conditions will be satisfied for cases where the heat release rate is a function of the acoustic velocity. The role played by non-normality is investigated. It is shown that, when a state triggers to sustained oscillations from the lowest possible energy, it exploits transient energy growth around an unstable limit cycle. The positions of these limit cycles in state space is determined by nonlinearity, but the tangled-ness of trajectories in state space is determined by non-normality. When viewed in this dynamical systems framework, triggering in thermoacoustics is seen to be directly analogous to bypass transition to turbulence in pipe flow.

The levels of transient energy growth found in the above analysis, which were of order 1 to 10, were significantly lower than those quoted by Balasubramanian and Sujith. This discrepancy was resolved when Luca Magri found an error in their original paper, showing that transient energy growth in their systems is indeed of order 1 to 10:

Non-normality in combustion-acoustic interaction in diffusion flames: a critical revision
L. Magri, K. Balasubramanian, R. I. Sujith, M. P. Juniper
Journal of Fluid Mechanics 733 681--683 (2013) doi:10.1017/jfm.2013.468

Perturbations in a non-normal system can grow transiently even if the system is linearly stable. If this transient growth is sufficiently large, it can trigger self-sustained oscillations from small initial disturbances. This has important practical consequences for combustion-acoustic oscillations, which are a persistent problem in rocket and aircraft engines. Balasubramanian and Sujith (Journal of Fluid Mechanics 2008, 594, 29--57) modelled an infinite-rate chemistry diffusion flame in an acoustic duct and found that the transient growth in this system can amplify the initial energy by a factor, G_{max}, of order 10^5 to 10^7. However, recent investigations by L. Magri M. P. Juniper have brought to light certain errors in that paper. When the errors are corrected, G_{max} is found to be of order 1 to 10, revealing that non-normality is not as influential as it was thought to be.

Triggering by background noise

With the above analysis, we found the initial states that cause triggering from the smallest initial energy. We found that these states needed most of their energy to be at lower frequencies (around the frequency of the least stable mode) but with some perturbations at higher frequencies. A more interesting experimental question is to find the type of noise that causes triggering from the lowest sound levels. We postulated that 'pink noise', which has more energy at lower frequencies, would be more effective than 'white noise', which has even energy across all frequencies. This did indeed prove to be the case:

Triggering, bypass transition and the effect of noise on a linearly stable thermoacoustic system
I. C. Waugh, M. Geuss, M. P. Juniper
Proceedings of the Combustion Institute 33 2945--2952 (2011) doi:10.1016/j.proci.2010.06.018

This paper explores the analogy between triggering in thermoacoustics and bypass transition to turbulence in hydrodynamics. These are both mechanisms through which a small perturbation causes a system to develop large self-sustained oscillations, despite the system being linearly stable. For example, it explains why round pipe flow (Hagen-Poiseuille flow) can become turbulent, even though all its eigenvalues are stable at all Reynolds numbers.

In hydrodynamics, bypass transition involves transient growth of the initial perturbation, which arises due to linear non-normality of the stability operator, followed by attraction towards a series of unstable periodic solutions of the Navier-Stokes equations, followed by repulsion either to full turbulence or re-laminarization. This paper shows that the triggering process in thermoacoustics is directly analogous to this. In thermoacoustics, the linearized stability operator is also non-normal and also gives rise to transient growth. The system then evolves towards an unstable periodic solution of the governing equations, followed by repulsion either to a stable periodic solution or to the zero solution. The paper demonstrates that initial perturbations that have higher amplitudes at low frequencies are more effective at triggering self-sustained oscillations than perturbations that have similar amplitudes at all frequencies.

This paper then explores the effect that different types of noise have on triggering. Three types of noise are considered: pink noise (higher amplitudes at low frequencies), white noise (similar amplitudes at all frequencies) and blue noise (higher amplitudes at high frequencies). Different amplitudes of noise are applied, both as short bursts and continuously. Pink noise is found to be more effective at causing triggering than white noise and blue noise, in line with the results found in the first part of the paper.

In summary, this paper investigates the triggering mechanism in thermoacoustics and demonstrates that some types of noise cause triggering more effectively than others.

We then examined the influence of noise on the bifurcation diagram and showed, through brute force simulations, that a subcritical bifurcation becomes like a supercritical bifurcation at high noise levels. This was shown later by Tony et al. (Physical Review E 2005 92:062902) by analysing the stochastic governing equations.

Triggering in a thermoacoustic system with stochastic noise
I. C. Waugh, M. P. Juniper
International Journal of Spray and Combustion Dynamics 3 (3) 225--242 (2011)

This paper explores the mechanism of triggering in a simple thermoacoustic system, the Rijke tube. It is demonstrated that additive stochastic perturbations can cause triggering before the linear stability limit of a thermoacoustic system. When triggering from low noise amplitudes, the system is seen to evolve to self-sustained oscillations via an unstable periodic solution of the governing equations. Practical stability is introduced as a measure of the stability of a linearly stable state when finite perturbations are present. The concept of a stochastic stability map is used to demonstrate the change in practical stability limits for a system with a subcritical bifurcation, once stochastic terms are included. The practical stability limits are found to be strongly dependent on the strength of noise.