Adjoint-based sensitivity analysis of periodic orbits by the Fourier–Galerkin method

•Continuation of periodic orbits and assessment of their stability.•Stability of periodic orbits is assessed with the Hill's method.•Evaluation of sensitivity maps, e.g. structural sensitivity, for physical instability identification and open-loop control.•Harmonic reconstruction of sensitivity maps.•Efficient preconditioned Newton–Krylov strategy for the resolution of the nonlinear residual equation. Sensitivity of periodic solutions of time-dependent partial differential equations is commonly computed using time-consuming direct and adjoint time integrations. Particular attention must be provided to the periodicity condition in order to obtain accurate results. Furthermore, stabilization techniques are required if the orbit is unstable. The present article aims to propose an alternative methodology to evaluate the sensitivity of periodic flows via the Fourier–Galerkin method. Unstable periodic orbits are directly computed and continued without any stabilizing technique. The stability of the periodic state is determined via Hill's method: the frequency-domain counterpart of Floquet analysis. Sensitivity maps, used for open-loop control and physical instability identification, are directly evaluated using the adjoint of the projected operator. Furthermore, we propose an efficient and robust iterative algorithm for the resolution of underlying linear systems. First of all, the new approach is applied on the Feigenbaum route to chaos in the Lorenz system. Second, the transition to a three-dimensional state in the periodic vortex-shedding past a circular cylinder is investigated. Such a flow case allows the validation of the sensitivity approach by a systematic comparison with previous results presented in the literature. Finally, the transition to a quasi-periodic state past two side-by-side cylinders is considered. These last two cases also served to test the performance of the proposed iterative algorithm.