Shape sensitivity of thermoacoustic oscillations in an annular combustor with a 3D adjoint Helmholtz solver

We model acoustic oscillations driven by velocity-coupled heat release rate fluctuations. We obtain an inhomogeneous wave equation and convert it to the frequency domain with a modal decomposition. We impose acoustic boundary conditions and, using a finite element discretization with Lagrange elements, express this as a nonlinear eigenvalue problem for the complex angular frequency, ω. We solve this using the open source platform FEniCS combined with the SLEPc, PETSc and OpenMPI libraries. In Hadamard form we write the derivative of the eigenvalue, ω, with respect to generic geometry changes. This requires the solution of the adjoint equation, which we obtain with the same method as the direct equation. The output is a thermoacoustic Helmholtz solver that cheaply calculates the effect of generic shape changes on the growth rate and frequency of thermoacoustic oscillations. We then consider symmetry-preserving and symmetry-breaking geometry modifications. For demonstration we model a 30kW laboratory-scale annular combustor (MICCA from EM2C). We parametrize the surfaces of the three-dimensional geometry with NURBS using control points. For the plenum and the combustion chamber, we find the eigenvalue shape derivatives with respect to the parameters of these control points. We apply two different strategies, perpendicular boundary movements and control point perturbations, to implement shape changes proportional to these shape derivatives, thereby reducing the growth rate of the unstable mode by increasing the phase shift between the pressure and the heat release rate oscillations. This computational method shows how to significantly alter thermoacoustic oscillations by making small geometry changes. The framework in this paper can handle arbitrarily complex three-dimensional geometries, which will be useful for the design of industrial combustion systems.