Mean field representation of the natural and actuated cylinder wake

Abstract

The necessity to include dynamic mean field representations in low order Galerkin models, and the role and form of such representations, are explored along natural and forced transients of the cylinder wake flow. The shift mode was introduced by Noack et al. [J. Fluid Mech. 497, 335 (2003)] as a least-order Galerkin representation of mean flow variations. The need to include the shift mode was argued in that paper in terms of the dynamic properties of a low order Galerkin model. The present study revisits and elucidates this issue with a direct focus on the Navier–Stokes equations (NSEs) and on the bilateral coupling between variations in the fluctuation growth rate and mean flow variations in the NSE. A detailed transient modal energy flow analysis is introduced as a new tool to quantitatively demonstrate the indispensable role of mean field variations, as well as the capacity of the shift mode to represent that contribution. Four variants of local and global shift mode derivations are examined and compared, including the geometric approach of Noack et al. and shift modes derived by a direct appeal to the NSE. Combined with the conclusions of the energy flow analysis, the similarity of the resulting shift modes indicates that the shift mode is no accident: indeed it is an intrinsic component of transient dynamics. Mean field representations can be found as implicit components in successful low order Galerkin models. We therefore argue for the benefit of the simple and robust explicit formulation in terms of added shift modes.