Mean resolvent operator of statistically steady flows

TL;DR

This paper introduces the mean resolvent operator for statistically steady flows, providing a new linear approximation tool that captures the mean response to forcing, with applications in flow control and analysis of complex flow regimes.

Contribution

It defines and develops the theory of the mean resolvent operator for steady and quasiperiodic flows, extending resolvent analysis to statistically steady and turbulent regimes.

Findings

Poles of the operator correspond to Floquet exponents.

Mean transfer functions can be identified without averaging for harmonic forcing.

The operator approximates the mean flow resolvent in weakly unsteady flows.

Abstract

This paper introduces a new operator relevant to input-output analysis of flows in a statistically steady regime far from the steady base flow: the mean resolvent $\mathbf{R}_0$. It is defined as the operator predicting, in the frequency domain, the mean linear response to forcing of the time-varying base flow. As such, it provides the statistically optimal linear time-invariant approximation of the input-output dynamics, which may be useful, for instance, in flow control applications. Theory is developed for the periodic case. The poles of the operator are shown to correspond to the Floquet exponents of the system, including purely imaginary poles at multiples of the fundamental frequency. In general, evaluating mean transfer functions from data requires averaging the response to many realizations of the same input. However, in the specific case of harmonic forcings, we show that the mean transfer functions may be identified without averaging: an observation referred to as `dynamic linearity' in the literature (Dahan et al., 2012). For incompressible flows in the weakly unsteady limit, i.e. when amplification of perturbations by the unsteady part of the periodic Jacobian is small compared to amplification by the mean Jacobian, the mean resolvent $\mathbf{R}_0$ is well-approximated by the well-known resolvent operator about the mean-flow. Although the theory presented in this paper only extends to quasiperiodic flows, the definition of $\mathbf{R}_0$ remains meaningful for flows with continuous or mixed spectra, including turbulent flows. Numerical evidence supports the close connection between the two resolvent operators in quasiperiodic, chaotic and stochastic two-dimensional incompressible flows.