Adjoint-Based Calculation of Parametric Thermoacoustic Maps of an Industrial Combustion Chamber

2020 ◽  
Vol 143 (1) ◽  
Author(s):  
Camilo F. Silva ◽  
Laura Prieto ◽  
Maximiliano Ancharek ◽  
Pablo Marigliani ◽  
Georg A. Mensah
Keyword(s):  
Eigenvalue Problem ◽  
Helmholtz Equation ◽  
Combustion Chamber ◽  
Nonlinear Eigenvalue Problem ◽  
Industrial Test ◽  
Combustion Chambers ◽  
Large Perturbation ◽  
Flame Response ◽  
Two Parameters ◽  
Nonlinear Eigenvalue

Abstract The aim of this study is to efficiently calculate parametric thermoacoustic maps of typical combustion chambers. Two configurations are considered: an academic configuration based on a Rijke tube, and an industrial combustion chamber, which is the core of a recently developed microturbine for power generation. Such maps can be understood as the collection of loci of thermoacoustic eigenfrequencies obtained under systematic variations of some defined parameters, while considering the Helmholtz equation as the thermoacoustic model of interest. In this study we consider variations on two parameters: the gain n and time-delay τ associated with a generic flame response model. We also show the feasibility of the proposed approach when considering more realistic flame responses. A straight-forward way to calculate such a thermoacoustic map is by solving the Helmholtz equation, and, thus, the corresponding nonlinear eigenvalue problem (NLEVP), one time per parameter combination. With that approach, the nonlinear eigenvalue problem needs to be solved hundreds or thousands of times if an adequate resolution of the thermoacoustic map is sought. Such a strategy may be computationally unaffordable. In order to overcome this difficulty, this work utilizes an adjoint-based, high-order perturbation method. The actual eigenvalue problem is only solved once at a baseline point. After applying the perturbation equations at that point, a polynomial rational function—the Padé approximant—is obtained to estimate the eigenfrequency drift that results for a small or large perturbation in the flame response. It is demonstrated, for both academic and industrial test cases, that the obtained maps are accurate. Additionally, it is shown that these maps reveal a large variety of thermoacoustic features, such as stability boundaries, intrinsic thermoacoustic modes, and exceptional points. The numerical costs for such calculations are negligible even for the industrial combustion chamber investigated.

Adjoint-Based Calculation of Parametric Thermoacoustic Maps of an Industrial Combustion Chamber

2020 ◽  
Author(s):  
Camilo F. Silva ◽  
Laura Prieto ◽  
Maximiliano Ancharek ◽  
Pablo Marigliani ◽  
Georg A. Mensah
Keyword(s):  
Eigenvalue Problem ◽  
Helmholtz Equation ◽  
Combustion Chamber ◽  
Industrial Test ◽  
Linear Eigenvalue Problem ◽  
Large Perturbation ◽  
Non Linear ◽  
Flame Response ◽  
Micro Turbine ◽  
Two Parameters

Abstract The aim of the present study is to efficiently calculate parametric thermoacoustic maps of typical combustion chambers. Two configurations are considered: an academic configuration based on a Rijke tube, and an industrial combustion chamber, which is the core of a recently developed micro-turbine for power generation. Such maps can be understood as the collection of loci of thermoacoustic eigen frequencies obtained under systematic variations of some defined parameters, while considering the Helmholtz equation as the thermoacoustic model of interest. In this study we consider variations on two parameters: the gain n and time-delay τ associated with a generic flame response model. We also show the feasibility of the proposed approach when considering more realistic flame responses. A straight-forward way to calculate such a thermoacoustic map is by solving the Helmholtz equation, and, thus, the corresponding non-linear eigenvalue problem, one time per parameter combination. With that approach, the non-linear eigenvalue problem needs to be solved hundreds or thousands of times if an adequate resolution of the thermoacoustic map is sought. Such a strategy may be computationally unaffordable. In order to overcome this difficulty, the present work utilizes an adjoint-based, high-order perturbation method. The actual eigenvalue problem is only solved once at a baseline point. After applying the perturbation equations at that point, a polynomial rational function — the Padé approximant — is obtained to estimate the eigen-frequency drift that results for a small or large perturbation in the flame response. It is demonstrated, for both academic and industrial test cases, that the obtained maps are accurate. Additionally, it is shown that these maps reveal a large variety of thermoacoustic features, such as stability boundaries, intrinsic thermoacoustic modes, and exceptional points. The numerical costs for such calculations are negligible even for the industrial combustion chamber investigated.

scholarly journals Uncertainty Quantification of Growth Rates of Thermoacoustic Instability by an Adjoint Helmholtz Solver

2016 ◽  
Vol 139 (1) ◽  
Author(s):  
Camilo F. Silva ◽  
Luca Magri ◽  
Thomas Runte ◽  
Wolfgang Polifke
Keyword(s):  
Eigenvalue Problem ◽  
Nonlinear Eigenvalue Problem ◽  
Order Term ◽  
Quantitative Agreement ◽  
Second Order ◽  
Thermoacoustic Instability ◽  
Thermoacoustic Instabilities ◽  
Acoustic Reflection ◽  
Flame Response ◽  
Nonlinear Eigenvalue

Thermoacoustic instabilities are often calculated with Helmholtz solvers combined with a low-order model for the flame dynamics. Typically, such a formulation leads to an eigenvalue problem in which the eigenvalue appears under nonlinear terms, such as exponentials related to the time delays that result from the flame model. The objective of the present paper is to quantify uncertainties in thermoacoustic stability analysis with a Helmholtz solver and its adjoint. This approach is applied to the model of a combustion test rig with a premixed swirl burner. The nonlinear eigenvalue problem and its adjoint are solved by an in-house adjoint Helmholtz solver, based on an axisymmetric finite-volume discretization. In addition to first-order correction terms of the adjoint formulation, as they are often used in the literature, second-order terms are also taken into account. It is found that one particular second-order term has significant impact on the accuracy of the predictions. Finally, the probability density function (PDF) of the growth rate in the presence of uncertainties in the input parameters is calculated with a Monte Carlo approach. The uncertainties considered concern the gain and phase of the flame response, the outlet acoustic reflection coefficient, and the plenum geometry. It is found that the second-order adjoint method gives quantitative agreement with results based on the full nonlinear eigenvalue problem, while requiring much fewer computations.

scholarly journals Lower bounds for Orlicz eigenvalues

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Ariel Salort
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Lower Bounds ◽  
Dirichlet Boundary ◽  
Nonlinear Eigenvalue Problem ◽  
Dirichlet Boundary Conditions ◽  
Young Functions ◽  
New Strategies ◽  
Nonlinear Eigenvalue

<p style='text-indent:20px;'>In this article we consider the following weighted nonlinear eigenvalue problem for the <inline-formula><tex-math id="M1">\begin{document}$ g- $\end{document}</tex-math></inline-formula>Laplacian</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ -{\text{ div}}\left( g(|\nabla u|)\frac{\nabla u}{|\nabla u|}\right) = \lambda w(x) h(|u|)\frac{u}{|u|} \quad \text{ in }\Omega\subset \mathbb R^n, n\geq 1 $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>with Dirichlet boundary conditions. Here <inline-formula><tex-math id="M2">\begin{document}$ w $\end{document}</tex-math></inline-formula> is a suitable weight and <inline-formula><tex-math id="M3">\begin{document}$ g = G' $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ h = H' $\end{document}</tex-math></inline-formula> are appropriated Young functions satisfying the so called <inline-formula><tex-math id="M5">\begin{document}$ \Delta' $\end{document}</tex-math></inline-formula> condition, which includes for instance logarithmic perturbation of powers and different power behaviors near zero and infinity. We prove several properties on its spectrum, being our main goal to obtain lower bounds of eigenvalues in terms of <inline-formula><tex-math id="M6">\begin{document}$ G $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ H $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ w $\end{document}</tex-math></inline-formula> and the normalization <inline-formula><tex-math id="M9">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> of the corresponding eigenfunctions.</p><p style='text-indent:20px;'>We introduce some new strategies to obtain results that generalize several inequalities from the literature of <inline-formula><tex-math id="M10">\begin{document}$ p- $\end{document}</tex-math></inline-formula>Laplacian type eigenvalues.</p>

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