ADJOINT BASED SHAPE OPTIMIZATION FOR THERMOACOUSTIC STABILITY OF COMBUSTORS USING FREE FORM DEFORMATION

We use the thermoacoustic Helmholtz equation to model thermoacoustic oscillations as an eigenvalue problem. We solve this with a Finite Element method. We parameterize the geometry of an annular combustor geometry using Free Form Deformation (FFD). We then use the FFD geometry, define the system parameters and impose the acoustic boundary conditions to calculate the eigenvalue and eigenvector of the problem using a Helmholtz solver. We then use adjoint methods to calculate the shape derivatives of the unstable eigenvalue with respect to the FFD control points. According to these gradients, we propose modifications to the control points that reduce the growth rate. We first demonstrate the application of this approach on the Rijke tube. Then we extend the method to a simulation of a laboratory combustor and lower the growth rate of the unstable circumferential mode. These findings show how this method could be used to reduce combustion instability in industrial annular combustors through geometric modifications.