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Research Papers: Gas Turbines: Combustion, Fuels, and Emissions

Linear Growth Rate Estimation From Dynamics and Statistics of Acoustic Signal Envelope in Turbulent Combustors

[+] Author and Article Information
Nicolas Noiray

CAPS Laboratory,
Mechanical and Process
Engineering Department,
ETH Zürich,
Zürich 8092, Switzerland
e-mail: noirayn@ethz.ch

Contributed by the Combustion and Fuels Committee of ASME for publication in the JOURNAL OF ENGINEERING FOR GAS TURBINES AND POWER. Manuscript received July 19, 2016; final manuscript received July 21, 2016; published online October 18, 2016. Editor: David Wisler.

J. Eng. Gas Turbines Power 139(4), 041503 (Oct 18, 2016) (11 pages) Paper No: GTP-16-1350; doi: 10.1115/1.4034601 History: Received July 19, 2016; Revised July 21, 2016

Considerable research and development efforts are required to meet the targets of future gas turbine technologies in terms of performance, emissions, and operational flexibility. One of the recurring problems is the constructive coupling between flames and combustor's acoustics. These thermoacoustic interactions can cause high-amplitude dynamic pressure limit cycles, which reduce the lifetime of the hot gas path parts or in the worst-case scenario destroy these mechanical components as a result of a sudden catastrophic event. It is shown in this paper that the dynamics and the statistics of the acoustic signal envelope can be used to identify the linear growth rates hidden behind the observed pulsations, and the results are validated against numerical simulations. This is a major step forward and it will contribute to the development of future gas turbine combustors, because the knowledge of these linear growth rates is essential to develop robust active and passive systems to control these combustion instabilities.

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Figures

Grahic Jump Location
Fig. 1

Pulsation data from a heavy-duty gas turbine combustor at three different conditions: (a) nominal condition, (b) off-design operating points where dynamic pressure amplitude exceeds allowable limits (figure taken from Ref. [2]). Top row: joint probability density functions of modal amplitude and its time derivative. Bottom row: time traces of modal amplitude η, which is proportional to the acoustic pressure p.

Grahic Jump Location
Fig. 2

Nonlinear response of the normalized heat release rate coherent fluctuations qc as function of the amplitude η of the monochromatic acoustic forcing for a turbulent premixed flame. (∘) experimental measurements, adapted from Fig. 13(b) in Ref. [16], where qc is assumed proportional to the CH* chemiluminescence.

Grahic Jump Location
Fig. 3

Postprocessing of the signals from time domain simulations (60 s) of the stochastic Van der Pol oscillator given by Eqs. (9) and (11), for different linear growth rates ν (the signals are normalized by the maximum amplitude reached during the simulation). For these simulations, β = 60 rad/s, κ=3.4, Γ=107, and ωj=2πfj with fj = 120 Hz. Three values of the damping α are considered (72, 60, and 48 rad/s). These simulations, respectively, correspond to a linearly stable thermoacoustic system (top row), a marginally stable one when damping α balances the linear driving from the flames β (middle row), and a stochastic limit-cycle when the growth rate ν=(β−α)/2 is positive (bottom row). On each rows, from left to right: the probability density function of the simulated acoustic signal P(η), a small portion of the time series and its envelope (black curves), the probability density function of this envelope P(A), and the power spectral densities of the envelope SAA and of the acoustic pressure Sηη.

Grahic Jump Location
Fig. 4

(a) Stochastic bifurcation diagram associated with the acoustic amplitude dynamics. The bifurcation parameter is the growth rate ν that has been varied by setting different values of the damping α. A contour plot of the theoretical probability density function (p.d.f. P(A)) given by Eq. (25) is shown together with the deterministic bifurcation curve (solid line). (e)–(g) P(A) at selected linear growth rates νi with a variation from linearly stable (ν<0) to linearly unstable (ν>0) conditions. The shaded areas are the theoretical p.d.f., and the thick blue lines correspond to the p.d.f. from the normalized histograms of the simulated stochastic Van der Pol oscillators (see Figs. 3(c), 3(h), and 3(m)). (b)–(d) Corresponding potential and related contributions.

Grahic Jump Location
Fig. 5

Evolution of half the identified cutoff pulsation ωc/2 (green line), which is associated with the power spectral density SAA of the envelope of the acoustic pressure signal, as a function of the linear growth rate. The cutoff pulsation has been estimated from time-domain simulations of a stochastically forced Van der Pol oscillator. The theoretical value ωc/2=|ν| deduced from the linearized model is also plotted (red line). In the insets, one can see two example of SAA with the estimated and theoretical values of ωc.

Grahic Jump Location
Fig. 6

System identification using Eq. (35) to process the data statistics from the time domain simulations (symbols). This is done in order to estimate F(A), which can be afterward used to deduce the potential V(A). These data correspond to the ones presented in Fig. 4. The dashed lines in the upper diagrams correspond to a parametric system identification using the Van der Pol approximation, for which the identified ν, κ, and Γ give the best curve fit on the extracted amplitude-dependent transition moments (symbols colored by the probability density function of the amplitude). The theoretical curves are shown as black solid lines. The corresponding probability density functions are also given in the second row.

Grahic Jump Location
Fig. 7

Postprocessing of the signals from time domain simulations (240 s) of the self-sustained oscillator given by Eqs. (9) and (B1), for different linear growth rates ν. Same parameters as the Van der Pol case presented in Fig. 3.

Grahic Jump Location
Fig. 8

(a) Stochastic bifurcation diagram of the acoustic amplitude dynamics for an arctangent saturation. The bifurcation parameter is the growth rate ν that has been varied by setting different values of the damping α (same numerical values as for the Van der Pol case). A contour plot of the theoretical p.d.f. P(A) given by Eqs. (25) and (B4) is shown together with the deterministic bifurcation curve (solid line). (e)–(g) P(A) at selected linear growth rates νi. The shaded areas are the theoretical p.d.f., and the thick blue lines correspond to the p.d.f. from the normalized histograms of the simulated model with arctangent saturation (see Fig. 7). (b)–(d) Corresponding potential from Eq. (B4).

Grahic Jump Location
Fig. 9

Nonlinear response of the normalized heat release rate coherent fluctuations qc as function of the amplitude η of the monochromatic acoustic forcing for a turbulent premixed flame. (∘) experimental measurements of the driving, adapted from Fig. 6 in Ref. [32].

Grahic Jump Location
Fig. 10

Postprocessing of the signals from time domain simulations (240 s) of the self-sustained oscillator resulting from a coherent heat release rate feedback containing linear, cubic, and quintic terms (see Fig. 9). The arbitrarily chosen numerical values of the corresponding coefficients are ωj=2π×120 rad/s, β=60 rad/s, κ=9, γ=0.7, Γ=107, and three values for α: from top to bottom, 80 rad/s in the subthreshold region, and α: 72 and 68 rad/s in the bistable region.

Grahic Jump Location
Fig. 11

(a) Stochastic bifurcation diagram of the acoustic amplitude A for the subcritical bifurcation obtained using the nonlinear heat release approximation of Figs. 9 and 10. The bifurcation parameter is the growth rate ν that has been varied by setting different values of the damping α. A contour plot of the theoretical p.d.f. P(A) deduced from the potential (C2) is shown together with the deterministic bifurcation curve (solid line). (e)–(g) P(A) at selected linear growth rates νi corresponding toFig. 10. The shaded areas are the theoretical p.d.f., and the thick blue lines correspond to the p.d.f. from the normalized histograms of the simulations. (b)–(d) Corresponding potential from Eq. (C2).

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