One often hears that 'absolutely unstable flows are insensitive to forcing'. This is only true in a linear sense. The nonlinear behaviour, which is what is observed in practice, is more elaborate.

Some flows contain large regions of absolute instability, which makes them globally unstable. They oscillate at well-defined frequencies. Two well-known examples are a helium jet discharging into air and the flow around a bluff body, such as a cylinder:

Global instability of a helium jet
This movie shows a helium jet discharging into air. The flow is globally unstable and oscillates at a well-defined frequency.
Credit: Larry Li
Global instability in the flow behind a cylinder
This movie shows a the Von-Karman vortex shedding behind a cylinder.
Credit: Physics Graphics

There is a common misconception that globally unstable flows are insensitive to forcing. Although this is true for a linear analysis, for which the long time behaviour is dominated by the intrinsic frequency (the saddle point), it is not true in a nonlinear analysis or in nature. In a series of careful experiments, Larry Li and I forced a naturally-oscillating helium jet at various frequencies and amplitudes and compared the resultant motion with that of a forced van der Pol oscillator.

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Forced response of a helium jet
Power Spectral Density (left) and Poincare map (right) of the signal from a hot wire placed inside a helium jet discharging into air. The helium jet is forced with a loudspeaker (the amplitude is shown in millivolts). At low forcing amplitudes, the transition from periodicity to quasiperiodicity occurs via a torous-birth bifurcation. At high forcing amplitudes, at which lock-in occurs, the transition from quasiperiodicity to periodicity occurs through a saddle node bifurcation with frequency-pulling.
Credit: Larry Li
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images/JFM726_Fig6e_1084_884.png
Forced response of a van der Pol oscillator
Power Spectral Density (left) and Poincare map (right) of the harmonically-forced van der Pol oscillator. At low forcing amplitudes, the transition from periodicity to quasiperiodicity occurs via a torous-birth bifurcation. At high forcing amplitudes, at which lock-in occurs, the transition from quasiperiodicity to periodicity occurs through a saddle node bifurcation with frequency-pulling. The dynamics are identical to those of the helium jet.
Credit: Larry Li
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The behaviour around the fundamental frequency (1000 Hz) is qualitatively identical (note that the van der Pol oscillator has only odd-numbered overtones). Full details are in this paper:

Lock-in and quasiperiodicity in a forced hydrodynamically self-excited jet
L. K. B. Li, M. P. Juniper
Journal of Fluid Mechanics 726, 624--655, (2013), doi:10.1017/jfm.2013.223
pdf
doi: https://doi.org/10.1017/jfm.2013.223

The ability of hydrodynamically self-excited jets to lock into strong external forcing is well known. Their dynamics before lock-in and the specific bifurcations through which they lock in, however, are less well known. In this experimental study, we acoustically force a low-density jet around its natural global frequency. We examine its response leading up to lock-in and compare this to that of a forced van der Pol oscillator. We find that, when forced at increasing amplitudes, the jet undergoes a sequence of two nonlinear transitions: (i) from periodicity to T2 quasiperiodicity via a torus-birth bifurcation; and then (ii) from T2 quasiperiodicity to 1:1 lock-in via either a saddle-node bifurcation with frequency pulling, if the forcing and natural frequencies are close together, or a torus-death bifurcation without frequency pulling, but with a gradual suppression of the natural mode, if the two frequencies are far apart. We also find that the jet locks in most readily when forced close to its natural frequency, but that the details contain two asymmetries: the jet (i) locks in more readily and (ii) oscillates more strongly when it is forced below its natural frequency than when it is forced above it. Except for the second asymmetry, all of these transitions, bifurcations and dynamics are accurately reproduced by the forced van der Pol oscillator. This shows that this complex (infinite-dimensional) forced self-excited jet can be modelled reasonably well as a simple (three-dimensional) forced self-excited oscillator. This result adds to the growing evidence that open self-excited flows behave essentially like low-dimensional nonlinear dynamical systems. It also strengthens the universality of such flows, raising the possibility that more of them, including some industrially relevant flames, can be similarly modelled.

Figures 6, 7, 10, and 11 of the above paper show the behaviour most clearly. When the jet is forced at low amplitudes, its response is quasiperiodic (in inexact terms, this corresponds to a response at both the forcing and the natural frequencies). When the jet is forced at high amplitudes, its response locks into the forcing frequency. The exact sequence of bifurcations follows that which is observed for the forced van der Pol oscillator, which can be identified with hand calculations (see Blanov et al. Synchronization; from Simple to Complex, Springer). We also observed phase trapping and slipping in our experimental measurements. These had been observed experimentally in lasers, but not before in hydrodynamics:

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Phase trapping and slipping in a forced helium jet
Experimental data from a forced helium jet. Top-right: time series data at high forcing amplitudes, where phase trapping occurs above a forcing amplitude of 1300 mV. Bottom-right: time series data at low forcing amplitudes, where phase slipping occurs below a forcing amplitude of 700 mV. The phase trapping and phase slipping regimes (centre) can be plotted on the synchronization diagram (left) and are compared with that of a forced van der Pol oscillator in our paper on the subject.
Credit: Larry Li
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Phase trapping and slipping in a forced hydrodynamically self-excited jet
L. K. B. Li, M. P. Juniper
Journal of Fluid Mechanics 735, R5, (2013), doi:10.1017/jfm.2013.533
pdf
doi: https://doi.org/10.1017/jfm.2013.533

In a recent study on a coupled laser system, Thevenin et al. (Phys. Rev. Lett., vol. 107, 2011, 104101) reported the first experimental evidence of phase trapping, a partially synchronous state characterized by frequency locking without phase locking. To determine whether this state can arise in a hydrodynamic system, we reanalyse the data from our recent experiment on a periodically forced self-excited low-density jet (J. Fluid Mech., vol. 726, 2013, pp. 624?655). We find that this jet exhibits the full range of phase dynamics predicted by model oscillators with weak nonlinearity. These dynamics include (i) phase trapping between phase drifting and phase locking when the jet is forced far from its natural frequency and (ii) phase slipping during phase drifting when it is forced close to its natural frequency. This raises the possibility that similar phase dynamics can be found in other similarly self-excited flows. It also strengthens the validity of using low-dimensional nonlinear dynamical systems based on a universal amplitude equation to model such flows, many of which are of industrial importance.

The above experiments were performed on helium jets, which have a strong but short region of absolute instability at their base. In an earlier study, we had examined jet diffusion flames, which have a strong and long region of absolute instability in their outer plume.

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Schlieren image of a forced methane jet diffusion flame
This jet diffusion flame oscillates naturally at 12 Hz. Here, it is forced at 17 Hz with forcing amplitudes of 20% of the mean jet velocity (first five images), then 50% (second five images) and 100% (last five images). At high forcing amplitudes, the flame's oscillations lock into the forcing frequency. We found, however, that the natural mode re-appears downstream.
Credit: Larry Li
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In that study, we found that the natural mode could dominate in the downstream region, even if the upstream region was oscillating at the forcing frequency. This is an interesting result because it shows that strong absolute instability could weaken the feedback loop that causes thermoacoustic oscillations.

Forcing of self-excited round jet diffusion flames
M. Juniper, L. Li, J. Nichols
Proceedings of the Combustion Institute 32 (1), 1191--1198, (2008), doi:10.1016/j.proci.2008.05.065
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doi: https://doi.org/10.1016/j.proci.2008.05.065


In this experimental and numerical study, two types of round jet are examined under acoustic forcing. The first is a non-reacting low density jet (density ratio 0.14). The second is a buoyant jet diffusion flame at a Reynolds number of 1100 (density ratio of unburnt fluids 0.5). Both jets have regions of strong absolute instability at their base and this causes them to exhibit strong self-excited bulging oscillations at well-defined natural frequencies. This study particularly focuses on the heat release of the jet diffusion flame, which oscillates at the same natural frequency as the bulging mode, due to the absolutely unstable shear layer just outside the flame.

The jets are forced at several amplitudes around their natural frequencies. In the non-reacting jet, the frequency of the bulging oscillation locks into the forcing frequency relatively easily. In the jet diffusion flame, however, very large forcing amplitudes are required to make the heat release lock into the forcing frequency. Even at these high forcing amplitudes, the natural mode takes over again from the forced mode in the downstream region of the flow, where the perturbation is beginning to saturate non-linearly and where the heat release is high. This raises the possibility that, in a flame with large regions of absolute instability, the strong natural mode could saturate before the forced mode, weakening the coupling between heat release and incident pressure perturbations, hence weakening the feedback loop that causes combustion instability.

These experiments inspired us to examine the nonlinear behaviour of thermoacoustic systems. This data was also used in one of the first papers on the Dynamic Mode Decomposition

Applications of the dynamic mode decomposition
P. J. Schmid, L. K. B. Li, M. P. Juniper, O. Pust
Theoretical and Computational Fluid Dynamics 25, 0935-4964, (2010), doi:10.1007/s00162-010-0203-9
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doi: https://doi.org/10.1007/s00162-010-0203-9

The decomposition of experimental data into dynamic modes using a data-based algorithm is applied to Schlieren snapshots of a helium jet and to time-resolved PIV-measurements of an unforced and harmonically forced jet. The algorithm relies on the reconstruction of a low-dimensional inter-snapshot map from the available flow field data. The spectral decomposition of this map results in an eigenvalue and eigenvector representation (referred to as dynamic modes) of the underlying fluid behavior contained in the processed flow fields. This dynamic mode decomposition allows the breakdown of a fluid process into dynamically revelant and coherent structures and thus aids in the characterization and quantification of physical mechanisms in fluid flow.