Adjoint-free method for mean resolvent analysis of periodic flows

Abstract

The mean resolvent operator predicts, in the frequency domain, the mean linear response to forcing. As such, it provides the optimal linear time-invariant approximation of the input-output dynamics of time-varying flows in the statistically steady regime (Leclercq & Sipp 2023). In this paper, we introduce an adjoint-free projection-based method for mean resolvent analysis of periodic flows. To evaluate the convergence of the projection-based method against the subspace dimension, we also implement an adjoint-based approach based on the harmonic resolvent framework (Wereley & Hall 1990, 1991; Padovan et al. 2020). Both adjoint-free and adjoint-based approaches may also be implemented in a matrix-free paradigm, using a time-stepper. For a weakly unsteady base flow, the mean-flow resolvent qualitatively approximates the dominant receptivity peak of the mean resolvent but completely fails to capture a secondary receptivity peak. For a strongly unsteady base flow, even the dominant receptivity peak of the mean resolvent associated with vortex-pairing is incorrectly captured by the mean-flow resolvent. The projection method already converges for a subspace dimension of 10 in the weakly unsteady case, but requires at least 100 modes for quantitative predictions in the strongly unsteady case. However, even in this case, using a subspace dimension of 1 is already enough to correctly identify the dominant receptivity peak.