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Research Papers

Thermoacoustic Modes of Quasi-One-Dimensional Combustors in the Region of Marginal Stability

[+] Author and Article Information
Camilo F. Silva

Professur für Thermofluiddynamik,
Technische Universität München,
München 85747, Germany
e-mail: silva@tfd.mw.tum.de

Kah Joon Yong

Professur für Thermofluiddynamik,
Technische Universität München,
München 85747, Germany

Luca Magri

Engineering Department,
University of Cambridge,
Cambridge CB2 1PZ, UK

1Corresponding author.

Manuscript received July 4, 2018; final manuscript received July 17, 2018; published online October 4, 2018. Editor: Jerzy T. Sawicki.

J. Eng. Gas Turbines Power 141(2), 021022 (Oct 04, 2018) (8 pages) Paper No: GTP-18-1443; doi: 10.1115/1.4041118 History: Received July 04, 2018; Revised July 17, 2018

It may be generally believed that the thermoacoustic eigenfrequencies of a combustor with fully acoustically reflecting boundary conditions depend on both flame dynamics and geometry of the system. In this work, we show that there are situations where this understanding does not strictly apply. The purpose of this study is twofold. In the first part, we show that the resonance frequencies of two premixed combustors with fully acoustically reflecting boundary conditions in the region of marginal stability depend only on the parameters of the flame dynamics but do not depend on the combustor's geometry. This is shown by means of a parametric study, where the time delay and the interaction index of the flame response are varied and the resulting complex eigenfrequency locus is shown. Assuming longitudinal acoustics and a low Mach number, a quasi-1D Helmholtz solver is utilized. The time delay and interaction index of the flame response are parametrically varied to calculate the complex eigenfrequency locus. It is found that all the eigenfrequency trajectories cross the real axis at a resonance frequency that depends only on the time delay. Such marginally stable frequencies are independent of the resonant cavity modes of the two combustors, i.e., the passive thermoacoustic modes. In the second part, we exploit the aforementioned observation to evaluate the critical flame gain required for the systems to become unstable at four eigenfrequencies located in the marginally stable region. A computationally efficient method is proposed. The key ingredient is to consider both direct and adjoint eigenvectors associated with the four eigenfrequencies. Hence, the sensitivity of the eigenfrequencies to changes in the gain at the region of marginal stability is evaluated with cheap and accurate calculations. This work contributes to the understanding of thermoacoustic stability of combustors. In the same manner, the understanding of the nature of distinct resonance frequencies in unstable combustors may be enhanced by employing the analysis of the eigenfrequency locus here reported.

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References

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Figures

Grahic Jump Location
Fig. 2

First passive thermoacoustic mode p̂1p. (Blue) Duct, (Black) BRS.

Grahic Jump Location
Fig. 1

The two configurations under investigation. Corresponding dimensions and thermodynamics parameters are given in Table 1.

Grahic Jump Location
Fig. 3

Locus of eigenfrequencies on the complex plane that corresponds to the Duct configuration. n = [0 → 1] with Δn = 0.05 and τ = [0 → 2π/ω1p]. τ = m ⋅ 2π/ω1p with m = 0 → 1 and Δm = 0.05. Numbers in the plot are values of m for the closest trajectory. The growth rate is defined as −Im(ω)/2π.

Grahic Jump Location
Fig. 4

Locus of eigenfrequencies in the complex plane of the BRS configuration. n = [0 → 4] with Δn = 0.2 and τ = [0 → 2π/ω1p]. τ = m ⋅ 2π/ω1p with m = 0 → 1 and Δm = 0.05. Numbers in the plot are values of m for the closest trajectory. The growth rate is defined as −Im(ω)/2π.

Grahic Jump Location
Fig. 5

Locus of eigenfrequencies in the complex plane of the Duct configuration. n = [0.005 → 1] with Δn = 0.005. Numbers in the plot are values of m for the closest trajectory. Note that only the trajectories defined by τ = m ⋅ 2π/ω1p with m = 0.25 → 0.45 (Δm = 0.05) have been considered for readability. Vertical lines indicate frequencies equal to j/(2τ).

Grahic Jump Location
Fig. 6

Thermoacoustic modes p̂/max(p̂) (top) and p̂†/max(p̂†) (bottom) of the Duct configuration (D1). (Gray) first iteration, (black) second iteration, (dashed blue) p̂k from direct eigenvalue problem of Eq. (8). (Dashed red) p̂k† from adjoint eigenvalue problem of Eq. (9). Note that gray and black curves overlap the blue and red curves, which verifies the proposed algorithm.

Grahic Jump Location
Fig. 7

Thermoacoustic modes p̂/max(p̂) (top) and p̂†/max(p̂†) (bottom) of the BRS configuration (B1). (Gray) first iteration, (black) second iteration, (dashed blue) p̂k from direct eigenvalue problem of Eq. (8). (Dashed red) p̂k† from adjoint eigenvalue problem of Eq. (9). Note that gray and black curves overlap the blue and red curves, which verifies the proposed algorithm.

Grahic Jump Location
Fig. 8

Locus of eigenfrequencies in the complex plane of the Duct configuration. Note that this figure is an extract of Fig. 3. The red lines indicate the slope (sensitivity ∂ω/∂n|ωg0) computed with the adjoint method. The growth rate is defined as −Im(ω)/2π.

Grahic Jump Location
Fig. 9

Locus of eigenfrequencies in the complex plane of the BRS configuration. Note that this figure is an extract of Fig. 4. The red lines indicate the slope (sensitivity ∂ω/∂n|ωg0) computed with the adjoint method. The growth rate is defined as −Im(ω)/2π.

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