Review Article

Adjoint Methods as Design Tools in Thermoacoustics

[+] Author and Article Information
Luca Magri

Engineering Department,
University of Cambridge,
Trumpington Street,
Cambridge CB2 1PZ, UK
e-mail: lm547@cam.ac.uk

1Corresponding author.

Manuscript received October 8, 2018; final manuscript received February 5, 2019; published online March 13, 2019. Editor: Harry Dankowicz.

Appl. Mech. Rev 71(2), 020801 (Mar 13, 2019) (42 pages) Paper No: AMR-18-1113; doi: 10.1115/1.4042821 History: Received October 08, 2018; Revised February 05, 2019

In a thermoacoustic system, such as a flame in a combustor, heat release oscillations couple with acoustic pressure oscillations. If the heat release is sufficiently in phase with the pressure, these oscillations can grow, sometimes with catastrophic consequences. Thermoacoustic instabilities are still one of the most challenging problems faced by gas turbine and rocket motor manufacturers. Thermoacoustic systems are characterized by many parameters to which the stability may be extremely sensitive. However, often only few oscillation modes are unstable. Existing techniques examine how a change in one parameter affects all (calculated) oscillation modes, whether unstable or not. Adjoint techniques turn this around: They accurately and cheaply compute how each oscillation mode is affected by changes in all parameters. In a system with a million parameters, they calculate gradients a million times faster than finite difference methods. This review paper provides: (i) the methodology and theory of stability and adjoint analysis in thermoacoustics, which is characterized by degenerate and nondegenerate nonlinear eigenvalue problems; (ii) physical insight in the thermoacoustic spectrum, and its exceptional points; (iii) practical applications of adjoint sensitivity analysis to passive control of existing oscillations, and prevention of oscillations with ad hoc design modifications; (iv) accurate and efficient algorithms to perform uncertainty quantification of the stability calculations; (v) adjoint-based methods for optimization to suppress instabilities by placing acoustic dampers, and prevent instabilities by design modifications in the combustor's geometry; (vi) a methodology to gain physical insight in the stability mechanisms of thermoacoustic instability (intrinsic sensitivity); and (vii) in nonlinear periodic oscillations, the prediction of the amplitude of limit cycles with weakly nonlinear analysis, and the theoretical framework to calculate the sensitivity to design parameters of limit cycles with adjoint Floquet analysis. To show the robustness and versatility of adjoint methods, examples of applications are provided for different acoustic and flame models, both in longitudinal and annular combustors, with deterministic and probabilistic approaches. The successful application of adjoint sensitivity analysis to thermoacoustics opens up new possibilities for physical understanding, control and optimization to design safer, quieter, and cleaner aero-engines. The versatile methods proposed can be applied to other multiphysical and multiscale problems, such as fluid–structure interaction, with virtually no conceptual modification.

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Grahic Jump Location
Fig. 1

Three main subsystems interact with each other to give rise to thermoacoustic instabilities

Grahic Jump Location
Fig. 2

(a) An example of a gas-turbine thermoacoustic network reduction taken from Ref. [51]. (b) The squares pictorially denote the eigenvalues. If the thermoacoustic network system has some eigenvalues with positive growth rate (red squares), the system is linearly unstable. Adjoint methods quantify how eigenvalues of interest change due to a small modification of any parameter (arrows). (Panel (b) is a pictorial figure and does not contain any results from calculations.)

Grahic Jump Location
Fig. 3

Traveling-wave approach as applied to a longitudinal combustor with upstream (downstream) acoustic reflection coefficient Ru (Rd) and entropic reflection coefficient, Re. The Mach number at the outlet is sonic, i.e., the combustor is choked. f is the downstream-traveling acoustic wave, g is the upstream-traveling acoustic wave, and Se is the entropic perturbation generated by the unsteady flame, whose flame front fluctuates with velocity u′s.

Grahic Jump Location
Fig. 4

Schematic of a rotationally symmetric annular combustor, which consists of a plenum and combustion chamber connected by longitudinal burners. In this case, there are 16 burners. (a) mean-flow speed of sound; ((b) and (c)) cross sections of the annular combustor. This represents the MICCA combustor [202,217219] as modeled in Ref. [171].

Grahic Jump Location
Fig. 5

The acoustic eigenfunctions of an open-ended duct with zero mean flow. (a) Pressure and (b) velocity with temperature jump (red-dashed lines) and without temperature jump (black-solid lines). The position of the mean-flow temperature jump is at xf = 0.25.

Grahic Jump Location
Fig. 6

Coupling between combusting hydrodynamics, governed by the low-Mach number equations and acoustics. Depending on the multiple-scale limit, the coupling terms, Fac→hyd and Fhyd→ac, have different expressions (Table 2) [161].

Grahic Jump Location
Fig. 7

Locus of eigenvalues as the flame parameters are varied as n = [0 → 1] and τ = [0 → 1] with increments of 0.05. Adapted from Ref. [175].

Grahic Jump Location
Fig. 8

(a) Spectrum and complex conjugate adjoint spectrum of a ducted flame (here a diffusion flame as an example). (b) Perturbed eigenvalues (small magenta squares) due to a small perturbation to the system. (c) The calculation of the eigenvalue shift is provided by sensitivity analysis.

Grahic Jump Location
Fig. 9

Direct eigenfunctions (left columns) and adjoint eigenfunctions (right column) for acoustic variables (top row, adapted from Ref. [162] with permission from Elsevier, xduct is the nondimensional axial coordinate of the duct), a diffusion flame (middle row, adapted from Ref. [158] with permission, light/dark colors correspond to positive/negative values), and a premixed flame (bottom row, data are courtesy of Orchini and Juniper [160]) in an open-ended duct with no mean flow. The direct eigenfunctions are the thermoacoustic modes with which the system linearly vibrates. The adjoint eigenfunctions are the receptivity of the system to open-loop sources in the equations. The nondimensional factors of the flame coordinates can be found in Refs. [158] and [160].

Grahic Jump Location
Fig. 10

Growth-rate (left column) and angular-frequency (right column) sensitivity. Top row: Sensitivities to the upstream (downstream) reflection coefficients, Ru (Rd), the flame index, n and time delay, τ, of a ducted flame for different positions of the flame's location (adapted from Ref. [162] with permission from Elsevier). Middle row: Sensitivities to the design parameters α (fuel-to-air port ratio) and Zsto of a ducted diffusion flame with the steady-flame length contours superimposed (values from 4 to 2 from bottom to top) (adapted from Ref. [158] with permission). Bottom row: Sensitivities to the flame aspect ratio and flame location of a ducted premixed flame. The maxima are marked by white lines, the minima are marked by black lines (Left: Reprinted with permission from Elsevier [160], Right: Data are courtesy of A. Orchini).

Grahic Jump Location
Fig. 11

Growth-rate drift due to the insertion of a drag device. The heat source is located at xf = 0.25. Adjoint predictions (solid line) against experimental results (circles, data are courtesy of Ref. [155]) normalized for comparison.

Grahic Jump Location
Fig. 12

Direct (left axis, thick-black lines) and adjoint (right axis, thin-red lines) eigenfunctions of a choked combustor with an nτ model (adapted from Ref. [161] with permission from Elsevier). The mean flow is not neglected; therefore, the acoustic density belongs in the state vector.

Grahic Jump Location
Fig. 13

Sensitivity to passive devices in a choked combustor that generate feedback from pressure, velocity, and density to the continuity, momentum, and energy equation. Thick lines from discrete adjoint calculations, thin lines from continuous adjoint calculations, symbols from finite difference (benchmark solution). Dark (black and red) lines and circles for growth-rate sensitivity, light (blue and cyan) lines and squares for angular-frequency sensitivity. Adapted from Ref. [162] with permission from Elsevier.

Grahic Jump Location
Fig. 14

(a) Nondimensional mean-flow temperature field obtained by large-Eddy simulation [161], (b) Growth-rate sensitivity to intrinsic hydrodynamic feedback feeding the momentum equation, the maximum straddles the recirculation-boundary (upper dashed line), and (c) Sensitivity to the hydrodynamics through acoustic mechanisms, the maximum straddles the stoichiometric line (lower dashed line). Adapted from Ref. [161] with permission from Elsevier.

Grahic Jump Location
Fig. 15

(a) Uniform distribution of the flame parameters centered around the mean interaction index, n¯, and mean time delay, τ¯. The standard deviations are sn and sτ, respectively. The black solid line is the stability margin, i.e., the locus of points that corresponds to a zero growth rate. The ratio between the shaded area and the total area is the probability that the mode is unstable. (b) Corresponding distribution of the eigenvalues. The integration of the PDF of the growth rate in the unstable semiplane is the probability that the mode is unstable.

Grahic Jump Location
Fig. 16

(a) Scattering of the eigenvalues calculated by standard Monte Carlo method (dark-black circles), Monte Carlo method with adjoint equations (light-blue circles), and Monte Carlo method with a surrogate algebraic model with adjoint equations (white circles). The unperturbed eigenvalue is denoted by a white square. (b) Estimates of the PDFs as histograms of the growth rates of panel (a). Units are dimensional as in the original paper [163], to which the reader is referred for more details.

Grahic Jump Location
Fig. 17

(a) Turbulent swirled combustor under investigation [302], (b) thermoacoustic eigenfunction calculated with a Helmholtz solver, (c) stability margins calculated by first-order and second-order adjoint methods with no Monte Carlo sampling. The white line is calculated by Monte Carlo sampling for reference. The rectangular represents the uniform PDF of the flame transfer functions parameters. The shaded area is the set of parameters corresponding to positive growth rates. Adapted from Refs. [164] and [167].

Grahic Jump Location
Fig. 18

Placement of the acoustic dampers in an annular combustor with 12 burners (first step of the optimization algorithm). Green and orange arrows indicate the predicted shift in growth rate for the first azimuthal mode, while red and blue arrows correspond to the second azimuthal mode. Both modes are degenerate; symmetry breaking damper placements split the degenerate eigenvalues, e.g., (1,6,7), whereas rotationally symmetric damper placements do not split the eigenvalues, e.g., (1,5,9). The asterisk denotes the growth rate that is actually computed. The triplets denote the annular sectors where acoustic dampers are applied (only a set of three dampers is considered). The most stabilizing pattern is (1,5,9). Adapted from Ref. [172].

Grahic Jump Location
Fig. 19

(a) The black (dashed) line denotes the original unstable annular combustor (a cross section of a sector is shown). The red (solid) line shows the optimized combustor. (b) Eigenvalue trajectories from the original unstable configuration (black squares) to the stabilized configuration (red circles). n indicates the wave number of the mode. Adapted from Ref. [173]. Units are dimensional as in the original paper, to which the reader may refer for details.

Grahic Jump Location
Fig. 20

(a) Pressure eigenfunction of the first plenum-dominant azimuthal mode of the MICCA combustor computed with a Helmholtz solver. (b) symmetry breaking perturbation pattern under consideration. The FTF of the burners with the same colors is perturbed by the same amount. The orange color represents positive perturbations, whereas the white color represents negative perturbations. The average perturbation to the FTF is zero. (c) Eigenvalue trajectories due to the symmetry breaking perturbation pattern (b). The star denotes the twofold semi-simple (degenerate) eigenvalue; different colors indicate the eigenvalue splitting due to the symmetry breaking perturbation; cross and plus symbols denote calculations from adjoint analysis; and black lines/symbols denote the exact solution. Third-order adjoint analysis favorably captures the actual eigenvalue splitting. For the inclination rule, it is not possible to stabilize an annular combustor (at first-order) by applying an azimuthal perturbation with zero average. The units are physical for a better comparison with the experimental data of Refs. [202] and [219]. Adapted from Ref. [171].

Grahic Jump Location
Fig. 21

Weakly nonlinear analysis at third-order (black thick line), fifth-order (red line) and numerical continuation of the fully nonlinear equations (circles). Solid and dashed lines indicate stable and unstable solutions, respectively. (a) Bifurcation diagram of the amplitude of the oscillations at the resonant frequency. (b) Frequency shift of the oscillations with respect to the marginally stable frequency. Adapted from Ref. [181] with permission.

Grahic Jump Location
Fig. 22

An example of a successful Taylor test for first- (circles) and second-order (squares) adjoint eigenvalue perturbations in a 19-burner annular combustor. Data are nondimensionalized by the modulus of the unperturbed eigenvalue. Reprinted from Ref. [165] with permission from Elsevier.



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