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 flowanalysis 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 flowanalysis, 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.
The authors gratefully acknowledge funding from the U.S. National Science Foundation (NSF) under Grant Nos. 0524070 and 0410246, from the U.S. Air Force Office of Scientific Research (AFOSR) under Grant Nos. FA9550-0610373 and FA9550-0510399, and from the Deutsche Forschungsgemeinschaft (DFG) under Grant Nos. 258/1-1 and 258/2-3 and via the Collaborative Research Center (Grant No. Sfb 557) “Control of Complex Turbulent Shear Flows” at the Berlin Institute of Technology, and from CNRS, e.g., via Invited Researcher grants. The authors acknowledge stimulating discussions with Katarina Aleksic, Laurent Cordier, Mark Luchtenburg, Rudibert King, Mark Pastoor, Michael Schlegel, Jon Scouten, Stefan Siegel, Tino Weinkauf, and Jose-Eduardo Wesfreid. We are grateful for outstanding hardware and software support by Lars Oergel and Martin Franke at the Berlin Institute of Technology. Finally, we wish to thank two anonymous referees for very insightful comments.
II. THE CYLINDER WAKE BENCHMARK
A. The laminar 2D cylinder wake flow
B. Simulation data
C. Empirical base flow and fluctuation trajectories
III. MEAN FIELD THEORY
A. A simple motivating example
B. The NSE perspective
1. An axiomatic framework and filtered partition of the NSE
2. The need for a mean field representation
3. An NSE-based shift mode definition
4. The generality of our conclusions
C. Modal energy flowanalysis
IV. SHIFT MODE DEFINITIONS
A. Global kinematic shift modes (GKSMs)
1. GKSM 1: Geometric global correction
2. GKSM 2: POD-based global correction
B. Local kinematic shift modes (LKSMs)
1. LKSM 1: POD base flow gradient approximation
2. LKSM 2: Local POD analysis of base flow increments
C. Global dynamic shift mode (GDSM) definition
1. GDSM 1: A Reynolds equation based global shift mode
D. Local dynamic shift mode (LDSM) definitions
1. LDSM 1: Linearized corrections from period means
2. LDSM 2: Shift mode based on local increments in solutions of the RANSE
V. QUANTITATIVE COMPARISON
VI. CONCLUDING REMARKS
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