Effect of colored noise on precursors of thermoacoustic instability in model gas turbine combustors

The Van der Pol oscillator – a prototypical thermoacoustic system

In this work, we employ the Van der Pol oscillator model, which is well-established for capturing the occurrence of thermoacoustic instability within combustion chambers. Van der Pol oscillator has been previously employed to study system identification^10,11,40, stochastic bifurcations ⁴⁷ , rate-tipping delay phenomenon^20,42,48 and noise-induced CR^2,44. The governing equation corresponding to the dominant frequency of a system in a generalized form is given as ¹⁰ :

η ¨ + α η ˙ + ω 0 2 η = q ˙

(1)

For supercritical system, q ˙ can be expressed as^2,10,11,46,

q ˙ = β * η ˙ − κ η ˙ η 2

(2)

q ˙ = β * η ˙ + κ η ˙ η 2 − μ η ˙ η 4

(3)

η ¨ + ω 0 2 η = 2 v η ˙ − κ η ˙ η 2

(4)

η ¨ + ω 0 2 η = 2 v η ˙ + κ η ˙ η 2 − μ η ˙ η 4

(5)

The stochastic differential equations of the two respective systems are given as,

η ¨ + ω 0 2 η = 2 v η ˙ − κ η ˙ η 2 + ξ ( t )

(6)

η ¨ + ω 0 2 η = 2 v η ˙ + κ η ˙ η 2 − μ η ˙ η 4 + ξ ( t )

(7)

The Van der Pol systems (Equations (6) and (7)) are numerically simulated using the fourth-order Runge-Kutta method in Matlab. We use the time step of d t = 0.0001 s in the time span of 0 ≤ t ≤ 1000 s . For the analysis, we use data for last 500 s . We choose the system parameters as f 0 = ω 0 / 2 π = 120 Hz , κ = 8 s − 1 , μ = 1 s − 1 and α = 2 s − 1 following the experimental validation reported in Bonciolini et al ⁴⁶ .

In the absence of noise (i.e., ξ ( t ) = 0 ), as the control parameter, v is varied from v = − 5 to v = 4 , the two systems undergoes transition to the instability through supercritical and subcritical Hopf bifurcations respectively, with Hopf and saddle-node points observed at v H = 0 and v S N = − 2 , as shown in Figure 1. The region before the Hopf point ( v H = 0 ) in Figure 1(a) and the saddle-node point ( v S N = − 2 ) in Figure 1(b) is referred to as the subthreshold (or stable) region. In case of subcritical system (Figure 1(b)), a bistable region exists for − 2 < v < 0 where two stable states coexist: Focus and LCOs. In the present work, the region of interest in the bifurcation diagram is the subthreshold region as we intend to investigate the effects of correlated noise characteristics on the precursors of thermoacoustic instability (prior to its occurrence).

Colored noise model

To investigate the noise-induced response of the systems, we first model ξ ( t ) as the white Gaussian noise. This model simplifies analytical derivations significantly and has been used in most of the studies on stochastically forced thermoacoustic limit cycles^10,12. Secondly, to investigate the effects of correlated noise characteristics (correlation time and intensity) on the system’s response and the precursors, we model ξ ( t ) as the Ornstein Uhlenbeck (OU) process which satisfies the following Langevin equation^40,44:

ξ ˙ ( t ) = − 1 τ c ξ ( t ) + D τ c ϵ ( t )

(8)

S ξ ξ ( ω ) = Γ 2 π D 1 + ω 2 τ c 2

(9)

We present a quantitative comparison in the noise-induced response of the Van der Pol systems (Eqn. (6) and (7)) when ξ ( t ) is modelled as both white and OU noise. For this comparison, we set a criterion regarding the input power for both the type of noise: the powers provided by white and OU noise in a band (denoted by σ b ) around the system’s eigenfrequency, Δ ω = ω 2 − ω 1 , are kept equal ⁴⁰ , i.e.,

σ b = ∫ ω 1 ω 2 S ξ ξ d ω = ∫ ω 1 ω 2 Γ / 2 π d ω

(10)

D = τ c ( ω 2 − ω 1 ) atan ( ω 2 τ c ) − atan ( ω 1 τ c )

(11)

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Figure 2.

Illustration of Ornstein Uhlenbeck (OU) noise features and corresponding system response: OU noise time series (a-c) and its corresponding power spectrum (d-f) for three noise correlation times ( τ c / T 0 = 0.1 , 1 and 10 ). The dashed line in plots (d-f) shows the eigenfrequency of the Van der Pol systems. τ c is normalized by the time period of oscillations ( T 0 = 1 / f 0 ). Plots (g-l) shows the time series (system’s response) and corresponding power spectrum for supercritical Van der Pol system at v = − 0.2 ; while plots (m-r) shows the time series (system’s response) and corresponding power spectrum for subcritical Van der Pol system at v = − 2.2 .

Methodology

This section briefly presents the methodology employed to estimate the noise-induced precursors of thermoacoustic instability. We specifically focus on two types of precursors: (i) based on spectral content (coherence factor) and (ii) based on fractal signature (Hurst exponent).

Coherence factor

Coherent oscillations are induced by noise in a stable oscillator prior to the Hopf bifurcation such that the relative contribution from coherent oscillations in system response (a) increases as the system is brought closer to bifurcation and (b) exhibits a resonance-like behaviour with increasing noise intensity. This phenomenon is known as CR^50,51 and is observed in several practical systems, including thermoacoustic systems^2,21,22. The noise intensity at which peak coherence is induced is termed as the optimum noise intensity for CR. The induced coherence is quantified by coherence factor, β , ³⁵ defined as the ratio of spectral peak height to the spectral quality factor and is given as,

β = H p × ( f p Δ f )

(12)

where H p and f p / Δ f are the height and normalized width (normalized by the peak frequency) of a Lorentzian fit to the broad spectral peak; the width is measured at half the height of the peak of the fit as illustrated in Figure 3(a).

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Figure 3.

White noise driven Van der Pol systems: (a) An illustration for estimation of coherence factor, β from the power spectrum of time series in the subthreshold regime. H p represents the spectral peak, Δ f represents the full width at half maximum, and f p represents the peak frequency. The markers represent the simulation data while the solid black line is the Lorentzian fit to the data for estimation of β . Plots (b) and (c) show the variation of coherence factor ( β ) as a function of σ b i.e. white noise intensity within Δ ω / ω 0 = 0.7 and control parameter ( v ) for supercritical and subcritical systems respectively. The effect of noise color on coherence factor is shown at v = − 0.2 and v = − 2.2 (marked in red color) for both the systems respectively in the subsequent discussions. The effects of noise intensity on the two types of noise-induced precursors are discussed at σ b ( 1 ) and σ b ( 2 ) .

Effects of noise color and intensity on coherence factor

Figure 3(b) and (c) present the variation of the coherence factor ( β ) against the white noise intensity within the band Δ ω / ω 0 = 0.7 ( σ b ) in the subthreshold region for supercritical and subcritical Van der Pol systems respectively. The phenomenon of CR^2,21,35 can be observed from the two plots: (a) For each control parameter ( v ), coherence factor increases with noise intensity, peaks at a certain optimum noise intensity, and decreases thereafter; (b) The optimum noise intensity, at which the peak coherence is induced, decreases as the control parameter, v , is increased towards the Hopf point; and (c) For a given noise intensity, coherence factor increases as the two systems approach the Hopf bifurcation.

In Figure 3, although β exhibit similar variation with σ b for both the systems, the mechanisms generating the peaks in β differ between the two cases: for supercritical system, the peak in β arises from a competition between a monotonic increase in H p and Δ f with increasing σ b ; whereas the peak in β for subcritical system is a result of co-occurrence of a maximum in H p and a minimum in Δ f at optimum σ b . These results concur with the numerical investigation of Gupta et al. ² .

Figure 4(a) and (c) show the variation of coherence factor with noise color and intensity at v = − 0.2 for supercritical system and at v = − 2.2 for subcritical system, respectively (control parameter values closest to the Hopf bifurcation, marked in red color in Figure 3). The x-axis is normalized by optimum white noise intensity ( σ b , w ), and the y-axis is normalized by the peak value of coherence factor ( β p , w ) at the respective control parameters, marked in Figure 3. We observe the occurrence of CR at all values of noise correlation times: β attains a peak value at intermediate noise levels. We observe that, for a given v , the optimum noise intensity (corresponding to peak coherence) first increases with an increase in noise color up to τ c / T 0 = 1 and then decreases with a further increase in noise color. The variation of optimum noise intensity with noise color is more significant in the subcritical system than in the supercritical system because of high-order non-linearity.

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Figure 4.

White and OU noise driven Van der Pol systems: Variation of coherence factor ( β ) as a function of σ b , i.e. noise intensity within Δ ω / ω 0 = 0.7 and noise correlation time ( τ c ) for (a) supercritical system at v = − 0.2 and (c) subcritical system at v = − 2.2 . σ b , w and β p , w correspond to the optimum white noise intensity and corresponding maximum CR at specified control parameter values, marked in red color in Figure 3. Plots (b) and (d) represent the variation of peak value of coherence factor ( β p ) in the parameter space, τ c – v , for supercritical and subcritical systems respectively. τ c is normalized by the time period of oscillations ( T 0 = 1 / f 0 ).

Figure 4(b) and (d) show the variation of the peak value of coherence factor ( β p ) with noise color and control parameter for supercritical and subcritical systems, respectively. We observe that, for a given v , the peak coherence decreases with an increase in noise correlation time. Although this trend holds for both systems, this decrease in β p is faster in the subcritical system than in the supercritical system. This decrease in β p with an increase in τ c implies that noise color constantly deteriorates the quality of induced coherence. This observation is also reported by Brugioni et al. ⁵⁴ for an excitable electronic system whose dynamics obey the FitzHugh-Nagumo model system. Furthermore, we also observe that, for a given τ c , β p increases monotonously as the two Van der Pol systems approach the Hopf bifurcation.

Figure 5 shows the variation of coherence factor with noise color and control parameter at two noise intensities ( σ b ( 1 ) and σ b ( 2 ) ) marked in Figure 3 for both supercritical (plots (a,b)) and subcritical (plots (c,d)) systems. We observe that, at low noise levels (i.e., at σ b ( 1 ) ), for a given v , β decreases with an increase in noise color for both systems. At high noise levels (i.e., at σ b ( 2 ) ), β decreases with increase in noise color at all v ≤ − 0.6 for supercritical system and v ≤ − 2.6 for subcritical system and thereafter β shows a resonance-like behaviour with τ c for all v : β first increases with increase in τ c , attains a peak near τ c / T 0 = 1 , and then decreases with a further increase in τ c . Far from the Hopf bifurcation, the variation in β is too small to distinguish a trend in the contour plots, especially at low noise levels. In such cases, the peak coherence is induced at very large noise intensities.

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Figure 5.

Variation of coherence factor ( β ) as a function of control parameter ( v ) and noise color ( τ c ) at two noise intensities ( σ b ( 1 ) and σ b ( 2 ) , marked in Figure 3) for (a, b) supercritical and (c, d) subcritical Van der Pol systems. τ c is normalized by the time period of oscillations ( T 0 = 1 / f 0 ).

Further, from Figure 5, we observe that, for a given noise color and intensity, coherence factor increases as the two systems approach the Hopf bifurcation. The coherence factor, therefore, indicates the approaching Hopf point when noise color and intensity do not change. The extent of the increase in coherence factor, however, depends on noise color and intensity, even if they do not vary. Thus, the implementation of coherence factor requires calibration for individual combustion systems based on the characteristics of the background noise.

For the subcritical system, the trends in coherence factor emerging from the Van der Pol oscillator qualitatively agree well with the experimental works of Vishnoi et al ⁴⁵ . The coherence factor decreases with increase in noise color at low noise levels, while it shows a resonance-like behaviour with noise color at high noise levels, specifically in the vicinity of Hopf bifurcation.

Effects of noise color and intensity on Hurst exponent

Figure 6 shows the variation of Hurst exponent with noise color, intensity and control parameter for both supercritical (plots (a,b)) and subcritical (plots (c,d)) systems. We can note that the value of H falls within the range of 0 − 0.5 for all τ c and σ b . This suggests that the time series from both systems exhibit an anti-correlated signature. We observe that, for a given v , Hurst exponent increases with an increase in noise color for both systems. The Hurst exponent rises as the noise intensity increases in a supercritical system but decreases in a subcritical system. Further, when noise color and intensity are kept constant, Hurst exponent decreases as both the systems approach the Hopf bifurcation.

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Figure 6.

Variation of Hurst exponent ( H ) with control parameter ( v ) and noise correlation time ( τ c ) at two noise intensities ( σ b ( 1 ) and σ b ( 2 ) , marked in Figure 3) for (a, b) supercritical and (c, d) subcritical Van der Pol systems. τ c is normalized by the time period of oscillations ( T 0 = 1 / f 0 ).

In real systems, a minimum threshold value of the Hurst exponent at a suitable distance from the Hopf point needs to be defined to track the system’s proximity to the Hopf bifurcation ²⁸ . There are no standard criteria to set the minimum threshold value; hence, it varies among systems, whether experimental or simulation. From Figure 6, we observe that for all τ c / T 0 ≤ 0.1 , the decrease in Hurst exponent is too small to distinguish a clear trend. Only at high noise correlation times could the threshold values be effectively defined. Further, we observe that a slight change in noise color (for example, τ c / T 0 = 0.1 to 0.4 ) as v is increased towards the Hopf point, causes significant variation in H , which can result in non-monotonous trends. This has also been reported in the experimental works of Vishnoi et al. ⁴⁵ for the subcritical system: in the vicinity of the saddle-node point, the Hurst exponent show non-monotonous trends with changes in the control parameter up to τ c / T 0 = 1.2 .

Robustness of the noise-induced precursors

In practical combustion systems, inherent noise may deviate from the white noise assumption. Moreover, inherent noise, characterized by its color and intensity, may vary as the operating conditions are varied. Both these effects must be considered to detect the system’s proximity to thermoacoustic instability. Figure 7 shows the comparison between the two types of precursors investigated in this work – coherence factor and Hurst exponent – as a 2D map in the τ c – v plane for both supercritical (Figure 7(a), (c) and subcritical (Figure 7(b), (d) systems at σ b ( 2 ) in the subthreshold region. The dashed arrows roughly indicate the prominent direction of variation in β and H as the control parameter and noise color are varied. We observe from Figure 7(a) and (b) that changes in coherence factor are predominantly due to control parameter variation for both, supercritical and subcritical systems. Hence, the coherence factor will serve as a robust precursor even when noise characteristics vary with simultaneous changes in the control parameter.

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Figure 7.

Robustness of the noise-induced precursors: Variation of coherence factor (a, b) and Hurst exponent (c, d) as a function of τ c and v at σ b ( 2 ) for supercritical (a, c) and subcritical (b, d) Van der Pol systems in the subthreshold region. τ c is normalized by the time period of oscillations ( T 0 = 1 / f 0 ). The arrows roughly indicate whether the variation in β and H is mainly in the direction of noise color or control parameter.

We observe from Figure 7(c) and (d) that changes in Hurst exponent are predominantly due to changes in noise color rather than the control parameter for τ c / T 0 ≤ 1 . The observation implies that the Hurst exponent can only be employed as a precursor at large noise correlation times, i.e. when τ c / T 0 > 1 . Examples for previous successful implementation of the Hurst exponent as a precursor include Nair et al.,^28,55 Unni et al., ⁵⁶ and Fu et al., ³⁰ where the transition to periodic oscillations is reported to occur via intermittency. In these aforementioned studies, noise color may have satisfied the criterion for correlation time. It is also possible that the change in Hurst exponent is due to a combined effect of noise color variation and parameter variation, but was ascribed to the parameter. For instance, with reference to Figure 7(d), simultaneous increase in control parameter and decrease in noise color will result in a prominent monotonous decrease in the Hurst exponent. Whereas, the use of the Hurst exponent in an experiment where noise color increases with increase in parameter will lead to a much smaller decrease – or even an increase – in the Hurst exponent with increase in control parameter. Such an experiment on the Hurst exponent as an instability precursor would likely be considered as inconclusive, but as we see here, such trends are a manifestation of the effects of noise characteristics.

In practical combustion systems, the flame response to the acoustic forcing can occur after a time delay ( τ ), as shown in Bonciolini et al.^49,57,58. In Appendices A and B, we show the effect of the time delay on the trends in the noise-induced precursors. In case the time delay of flame response is not negligible, we find that the coherence factor will work at most time delays and noise color, while the trends in the Hurst exponent are mostly dependent on the noise color of additive noise (i.e., τ c / T 0 ≥ 1 ) than the time delay. We emphasize, however, that the inclusion of time delay as a parameter of the study and the additional (in addition to additive noise) source of multiplicative noise through noise in the time delay significantly increases the complexity of the stochastic dynamics of the system. More comprehensive dedicated studies are necessary; the results presented in the appendix are thus of a preliminary nature.