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Shape optimization for thermoacoustic instability with an adjoint Helmholtz solver



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Falco, Stefano 


Thermoacoustic instabilities, which occur due to the feedback between confined flames and acoustics, constitute a major threat to the safe operation of gas turbines and rocket engines. These oscillations can become large enough to cause noise, vibrations or, in the worst cases, extinction of the flame or structural damage. The fact that thermoacoustic systems are sensitive to small changes to their design and operating parameters means that instabilities can appear in the late stages of the design process, requiring costly re-design. In this thesis we address the problem of finding accurately and at reduced computational cost the design changes that most stabilize a thermoacoustic system in the linear regime.

We derive a Helmholtz equation with an unsteady heat release source and acoustic impedance boundary conditions. We apply the finite element method with P2 elements to discretize the weak formulation of the Helmholtz equation. We obtain a nonlinear eigenvalue problem for the complex angular frequency, ω, that we solve with the open-source computing platform FEniCS and the library for eigenvalue problems SLEPc. We then derive the formulae for the shape derivatives of the eigenvalues of the Helmholtz equation in Hadamard form for different boundary conditions and in the cases of simple and semi-simple degenerate eigenvalues. These formulae allow us to efficiently calculate the effect of arbitrary boundary perturbations on the frequency and growth rate of the thermoacoustic oscillations, by combining the direct and adjoint eigensolutions.

We apply this model and analysis to an electrically heated Rijke tube and a turbulent swirl combustor in 2D. Both systems exhibit an unstable longitudinal mode. We parametrize the shapes using B-splines, calculate the shape derivatives and apply the most stabilizing changes until we make the systems stable.

We then study the case of a 30kW laboratory-scale symmetric annular combustor (MICCA) with an unstable azimuthal mode (two-fold degenerate). We perform a shape sensitivity analysis considering both symmetry-breaking and symmetry-preserving boundary perturbations. The second type of changes do not cause the eigenvalues to split and are the most effective at reducing the instability. We then apply shape changes to the plenum and the combustion chamber to reduce the eigenvalue growth rates.

We show how adjoint-based sensitivity analysis can be combined with a Helmholtz solver to calculate the influence of the shape of the combustor on the stability of the thermoacoustic modes. By modifying the shape accordingly, we are able to suppress the instability or at least reduce the growth rate of the oscillations. This computational method shows how to significantly alter thermoacoustic oscillations by making small geometry changes. The framework in this thesis can handle arbitrarily complex three-dimensional geometries, which could be useful for the design of industrial combustion systems.





Juniper, Matthew


Shape optimization, Thermoacoustics, Adjoint


Doctor of Philosophy (PhD)

Awarding Institution

University of Cambridge
European Commission Horizon 2020 (H2020) Marie Sk?odowska-Curie actions (765998)