Sensitivity Analysis of Thermoacoustic Instability with Adjoint Helmholtz Solvers

Thermoacoustic oscillations often occur late in the gas turbine design process. They can usually be eliminated by making a small change to the system. The challenge is to identify the optimal change systematically, cheaply, and accurately. Often there is one linearly unstable natural oscillation (eigenmode) and many possible changes to the system. This means that linear adjoint methods are ideal for identifying the optimal change. This paper applies linear adjoint methods to a thermoacoustic Helmholtz solver to evaluate, in a single calculation, the sensitivity of an eigenmode to all possible changes. These sensitivities are calculated with finite difference and finite element methods, in the weak form and the strong form, with the discrete adjoint and the continuous adjoint, and with two solution methods. This reveals that: the discrete adjoint of finite difference should be avoided; the discrete adjoint of finite element is simple and robust; if the strong form equations must be used, e.g. in order to use an existing direct solver, then the continuous adjoint is best. Finally, physical interpretation of these results shows that the well-known Rayleigh criterion should be written in terms of the adjoint pressure, not the direct pressure.