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

Abstract

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.

Sensitivity analysis of thermoacoustic instability with adjoint Helmholtz solvers

“…The Program's proceedings [20] contain an outline of the mathematical principles required for adjoint Helmholtz solvers with active iteration, but no numerical implementation or results. Since then there have been some applications of adjoint Helmholtz solvers in the literature: (i) to optimize acoustic damper placement [21]; (ii) to speed up uncertainty quantification [22,23], and (iii) to investigate the effects of asymmetry in annular combustors [24]. Adjoints have been applied (in collaboration with the author) to the three dimensional finite volume-based thermoacoustic solver AVSP [25,Ch.…”

Adjoint Methods as Design Tools in Thermoacoustics

“…The adjoint sensitivity framework was applied in [162] to thermoacoustic networks, which are used in the preliminary design of aero-engines and gas turbines for power generation. Adjoint methods were developed to accelerate uncertainty quantification of thermoacoustic stability in an annular-combustor network [163] and in a turbulent swirled combustor [164], where first-and second-order corrections for nonlinear degenerate eigenproblems were used [165]. The first implementations of an adjoint Helmholtz solver can be found for a turbulent swirl combustor in [164] and a two-dimensional annular combustor in [166].…”

“…Adjoint methods were developed to accelerate uncertainty quantification of thermoacoustic stability in an annular-combustor network [163] and in a turbulent swirled combustor [164], where first-and second-order corrections for nonlinear degenerate eigenproblems were used [165]. The first implementations of an adjoint Helmholtz solver can be found for a turbulent swirl combustor in [164] and a two-dimensional annular combustor in [166]. Monte-Carlo free methods were developed by Mensah et al [167], who calculated the probability that a dump combustor becomes unstable by deriving an adjoint-based algebraic expression for the stability margin.…”

“…A detailed explanation of the Helmholtz equation as applied to thermoacoustics is given by Nicoud et al [40,197]. The advantage of the Helmholtz-equation framework is that it can tackle three-dimensional geometries, with the con that it holds for very small mean-flow Mach numbers and requires numerical discretization, such as finite elements (e.g., [40,166,172,[199][200][201][202]), finite volumes (e.g., [164]) or finite difference. This model is applied in Sec.…”

“…where N 0 is the unperturbed matrix. This is the usual approach used in hydrodynamic stability [134], which was used in thermoacoustics for the calculation of linearly optimal passive control devices [153,154], base-state sensitivities [158,160,165], uncertainty quantification both in annular [163] and longitudinal combustors [164,167] and optimal acoustic damper placements in annular combustors [172].…”