Thermoacoustic instabilities, which arise due to the interaction between flames and acoustics, are extremely sensitive to small changes to the system parameters. In an ideal situation, gas turbines and rocket engines will be linearly stable to thermoacoustic oscillations over their operating range. In this paper, we describe a tool for finding accurately and inexpensively the shape changes that most stabilise a 2D thermoacoustic system in the linear regime using adjoint-based shape optimization.

Under the hypothesis of zero Mach number mean flow, we solve a 2D Helmholtz equation for the pressure perturbation with distributed heat release and acoustic impedance boundary conditions. To do so, we use a finite element adjoint Helmholtz solver. We write in Hadamard form the eigenvalue shape derivatives, which tell us how small changes to the shape of the domain affect the stability of the thermoacoustic modes. Then we use parametric shape optimization to make the thermoacoustic system stable in the linear regime.

We consider two cases in 2D axisymmetry: the first is an electrically heated Rijke tube and the second is a turbulent swirl combustor. Both systems exhibit an unstable longitudinal mode and we suppress the instability by slightly modifying the geometry.  In the case of the turbulent swirl combustor, the sensitivities are higher in the plenum and in the burner than in the combustion chamber, mainly due to the effect of the mean temperature on the pressure eigenfunctions. In the cooler regions, the local wavelength is shorter, which means that geometry changes of a given distance have more influence than they do where the local wavelength is longer. 

This is the first time adjoint-based shape optimization is applied to 2D Helmholtz solvers in thermoacoustics. The same technique had been previously applied to low-order thermoacoustic networks and 1D Helmholtz solvers, but 2D Helmholtz solvers have an intrinsic advantage: they can handle complex geometries. The easy scalability of this method to complex 3D geometries make this tool a strong candidate for the iterative design of thermoacoustically stable combustors.


Shape Optimization of Thermoacoustic Systems Using a 2D Adjoint Helmholtz Solver

Division / Sponsor:
International Gas Turbine Institute (IGTI)

Paper Proceedings: https://asmedigitalcollection.asme.org/GT/GT2020/volume/84058

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