^{1,a)}, Oliver Lehmann

^{2}, Bernd R. Noack

^{3}and Marek Morzyński

^{4}

### 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 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.

I. INTRODUCTION

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

### Key Topics

- Mean field theory
- 81.0
- Attractors
- 49.0
- Reynolds stress modeling
- 49.0
- Flow instabilities
- 25.0
- Kinematics
- 17.0

## Figures

The actuated cylinder wake: the cylinder is represented by the black disk. The downstream circle and arrows indicate the location and orientation of a volume-force actuator. Streamlines represent a snapshot of the natural flow. Thick (thin) curves correspond to positive (negative) values of the stream-function—here and in all following flow visualizations.

The actuated cylinder wake: the cylinder is represented by the black disk. The downstream circle and arrows indicate the location and orientation of a volume-force actuator. Streamlines represent a snapshot of the natural flow. Thick (thin) curves correspond to positive (negative) values of the stream-function—here and in all following flow visualizations.

The time evolution of the fluctuation energy, , of the natural transient from the steady solution to the attractor. Circles along the curve mark the midpoints of 20 single-period (and partially overlapping) time intervals that are used to obtain local shift modes.

The time evolution of the fluctuation energy, , of the natural transient from the steady solution to the attractor. Circles along the curve mark the midpoints of 20 single-period (and partially overlapping) time intervals that are used to obtain local shift modes.

(a) The time evolution of the fluctuation energy of the forced transient from the attractor toward the steady solution and back to the attractor. This transient is longer (and slower) than the natural transient. (b) The time evolution of the slowly varying amplitude of the oscillatory actuation . A higher amplitude is needed to drive the descending trajectory toward the steady solution (—), than during the relaxation of the ascending second half (- -), as the flow gradually returns to the attractor. (c) To highlight that difference in the required actuation amplitude, is plotted as a function of the instantaneous fluctuation energy in the actuated descending trajectory (—) and the ascending trajectory (- -).

(a) The time evolution of the fluctuation energy of the forced transient from the attractor toward the steady solution and back to the attractor. This transient is longer (and slower) than the natural transient. (b) The time evolution of the slowly varying amplitude of the oscillatory actuation . A higher amplitude is needed to drive the descending trajectory toward the steady solution (—), than during the relaxation of the ascending second half (- -), as the flow gradually returns to the attractor. (c) To highlight that difference in the required actuation amplitude, is plotted as a function of the instantaneous fluctuation energy in the actuated descending trajectory (—) and the ascending trajectory (- -).

Comparison of natural transient TKE predictions by a DNS simulation (—) and a number of low order POD models: an eight-mode traditional POD model (△), a least-order mean field model with the first attractor POD mode shift mode , a model including the two attractor POD modes, two linear stability modes, and a shift mode (○), and an 11-mode model, using eight POD modes, two stability modes, and one shift mode (◇). The dashed straight line indicates the linear stability growth rate at the steady solution.

Comparison of natural transient TKE predictions by a DNS simulation (—) and a number of low order POD models: an eight-mode traditional POD model (△), a least-order mean field model with the first attractor POD mode shift mode , a model including the two attractor POD modes, two linear stability modes, and a shift mode (○), and an 11-mode model, using eight POD modes, two stability modes, and one shift mode (◇). The dashed straight line indicates the linear stability growth rate at the steady solution.

Complementing Fig. 4, this figure compares transients of a DNS simulation (●) and of the three-state Galerkin model, based on the two attractor POD modes and the global kinematic shift mode (—). The phase portrait employs the amplitudes and of the first POD and shift mode, respectively.

Complementing Fig. 4, this figure compares transients of a DNS simulation (●) and of the three-state Galerkin model, based on the two attractor POD modes and the global kinematic shift mode (—). The phase portrait employs the amplitudes and of the first POD and shift mode, respectively.

First part of an energy flow analysis of the natural transient of the cylinder wake flow. (a) The total supplied power, , is plotted as a time evolution, (b) and as a function of the TKE level . The DNS value (solid curve) is compared with cases where the exact mean field from Eq. (3) is substituted by the dynamic estimate (dotted curve), where is the shift mode from the normalized equation (14), as well as by (dash-dotted curve), and by (dashed curve). In all cases is computed with the same (exact) fluctuation field, . Notice that the TKE overshoot in (b) corresponds to the overshoot in Fig. 2.

First part of an energy flow analysis of the natural transient of the cylinder wake flow. (a) The total supplied power, , is plotted as a time evolution, (b) and as a function of the TKE level . The DNS value (solid curve) is compared with cases where the exact mean field from Eq. (3) is substituted by the dynamic estimate (dotted curve), where is the shift mode from the normalized equation (14), as well as by (dash-dotted curve), and by (dashed curve). In all cases is computed with the same (exact) fluctuation field, . Notice that the TKE overshoot in (b) corresponds to the overshoot in Fig. 2.

Second part of an energy flow analysis of the natural transient of the cylinder wake flow, now comparing the exact value of (solid curve) to estimates including Galerkin approximations of the fluctuation field . Denoting by , , and the respective pairs of local period POD modes, stability modes, and attractor POD modes, one estimate uses the exact (period-mean) base flow, , whereas is approximated by the projection on the local (dashed curve). In the remaining three estimates, the base flow is substituted by and the fluctuation field is approximated by its projections on (dash-dotted curve), (dotted curve), and (dash-dot-plussed curve).

Second part of an energy flow analysis of the natural transient of the cylinder wake flow, now comparing the exact value of (solid curve) to estimates including Galerkin approximations of the fluctuation field . Denoting by , , and the respective pairs of local period POD modes, stability modes, and attractor POD modes, one estimate uses the exact (period-mean) base flow, , whereas is approximated by the projection on the local (dashed curve). In the remaining three estimates, the base flow is substituted by and the fluctuation field is approximated by its projections on (dash-dotted curve), (dotted curve), and (dash-dot-plussed curve).

Construction of the geometric shift mode. (a) The unstable steady solution . (b) The attractor mean flow . (c) The geometric shift mode. According to GKSM 1 this mode is defined as the normalized difference .

Construction of the geometric shift mode. (a) The unstable steady solution . (b) The attractor mean flow . (c) The geometric shift mode. According to GKSM 1 this mode is defined as the normalized difference .

POD modes of the entire natural base flow transient, as used in definition GKSM 2: the mean of the transient flow is depicted in (a) and the first five POD modes are (b)–(f), all visualized by streamlines. Note that the base flow transient mean (a) is, as expected, an intermediate form between the steady solution and the attractor’s mean, depicted in Figs. 8(a) and 8(b). Here and throughout the base flow is computed according to Eq. (5a), using the symmetrized velocity field .

POD modes of the entire natural base flow transient, as used in definition GKSM 2: the mean of the transient flow is depicted in (a) and the first five POD modes are (b)–(f), all visualized by streamlines. Note that the base flow transient mean (a) is, as expected, an intermediate form between the steady solution and the attractor’s mean, depicted in Figs. 8(a) and 8(b). Here and throughout the base flow is computed according to Eq. (5a), using the symmetrized velocity field .

(a) POD eigenvalues corresponding to Fig. 9, normalized with respect to (i.e., ). [(b) and (c)] The trajectories of the corresponding three leading Fourier coefficients.

(a) POD eigenvalues corresponding to Fig. 9, normalized with respect to (i.e., ). [(b) and (c)] The trajectories of the corresponding three leading Fourier coefficients.

Kinematic local shift modes obtained according to LKSM 1 as time derivatives of the three-modes POD approximation of the entire natural base flow transients. Henceforth, all local shift modes of the natural and forced transients will be associated both with the mean-period time and with the period-mean TKE. The figures (a)–(c) represent consecutive increments one period apart, with the respective period mean times , 41.5, and 46 and TKE levels of 1.17, 2.23, and 2.66 in Fig. 2. The TKE levels will be used to parameterize operating points and compare shift modes obtained at similar TKE levels by the various methods examined here.

Kinematic local shift modes obtained according to LKSM 1 as time derivatives of the three-modes POD approximation of the entire natural base flow transients. Henceforth, all local shift modes of the natural and forced transients will be associated both with the mean-period time and with the period-mean TKE. The figures (a)–(c) represent consecutive increments one period apart, with the respective period mean times , 41.5, and 46 and TKE levels of 1.17, 2.23, and 2.66 in Fig. 2. The TKE levels will be used to parameterize operating points and compare shift modes obtained at similar TKE levels by the various methods examined here.

POD eigenvalues of the natural transient of the base flow along 20 single periods of the natural transient marked in Fig. 2. The first nine POD eigenvalues of each of the 20 periods, normalized relative to the largest eigenvalue of each periods (i.e., for each period we present ). The eigenvalues are plotted as functions of the eigenvalue number and the mean TKE level of the respective period. These are the local kinematic shift modes described in LKSM 2. The increase in significance of , , near the attractor, is attributed to mean field variations associated with the changes in stability properties during the small, presettling overshoot in , as seen in Fig. 2.

POD eigenvalues of the natural transient of the base flow along 20 single periods of the natural transient marked in Fig. 2. The first nine POD eigenvalues of each of the 20 periods, normalized relative to the largest eigenvalue of each periods (i.e., for each period we present ). The eigenvalues are plotted as functions of the eigenvalue number and the mean TKE level of the respective period. These are the local kinematic shift modes described in LKSM 2. The increase in significance of , , near the attractor, is attributed to mean field variations associated with the changes in stability properties during the small, presettling overshoot in , as seen in Fig. 2.

Counterpart of Fig. 12 for the forced transient: POD eigenvalues of the base flow along each of 155 single periods of the forced transient, normalized with respect of the dominant eigenvalue (i.e., for each period we present ). (a) POD eigenvalues of periods from descending first half of the forced trajectory, from the attractor toward the steady solution. (b) Same, for the ascending second half of the forced trajectory, as it is relaxed from the steady solution to the attractor. Eigenvalues are shown as functions of the eigenvalue number and that period’s TKE level .

Counterpart of Fig. 12 for the forced transient: POD eigenvalues of the base flow along each of 155 single periods of the forced transient, normalized with respect of the dominant eigenvalue (i.e., for each period we present ). (a) POD eigenvalues of periods from descending first half of the forced trajectory, from the attractor toward the steady solution. (b) Same, for the ascending second half of the forced trajectory, as it is relaxed from the steady solution to the attractor. Eigenvalues are shown as functions of the eigenvalue number and that period’s TKE level .

First POD modes of five selected analyzed periods indicated in Fig. 2 along the natural base flow transient. The respective mean-period times for plots (a)–(e) are , 26.4, 35, 40.2, and 48.9, and the corresponding period-mean TKE levels are 0.002, 0.1, 1, 1.97, and 2.81.

First POD modes of five selected analyzed periods indicated in Fig. 2 along the natural base flow transient. The respective mean-period times for plots (a)–(e) are , 26.4, 35, 40.2, and 48.9, and the corresponding period-mean TKE levels are 0.002, 0.1, 1, 1.97, and 2.81.

POD analysis of 155 periods of the actuated transient, as described in Sec. II B. [(a)–(g)] The first (dominant) POD modes of the base flow from seven periods of the actuated transient, computed as in LKSM 2. The respective mean period times are , 147.1, 255.1, 399.7, 556.6, 714.2, and 856.4. The corresponding TKE levels read 2.65, 1.9, 1.49, 0.64, 0.18, 1, and 1.8. POD mode streamlines have opposite signs for periods of the descending and ascending trajectories. Quantitative comparisons with other computations in Sec. V take into account the relative proximity to the steady solution or the natural attractor.

POD analysis of 155 periods of the actuated transient, as described in Sec. II B. [(a)–(g)] The first (dominant) POD modes of the base flow from seven periods of the actuated transient, computed as in LKSM 2. The respective mean period times are , 147.1, 255.1, 399.7, 556.6, 714.2, and 856.4. The corresponding TKE levels read 2.65, 1.9, 1.49, 0.64, 0.18, 1, and 1.8. POD mode streamlines have opposite signs for periods of the descending and ascending trajectories. Quantitative comparisons with other computations in Sec. V take into account the relative proximity to the steady solution or the natural attractor.

Six local shift modes computed according to LDSM 1 from short simulations initiated at points along the natural base flow transient. Relating to Fig. 2, plots (a)–(f) correspond to trajectories initiated at with , 19, 26.5, 34, 39, and 61.4, and approximate TKE levels of 0, 0.06, 0.1, 0.8, 1.78, and 2.69. The clear difference from the LDSM 2 modes in Fig. 17 is explained in the text.

Six local shift modes computed according to LDSM 1 from short simulations initiated at points along the natural base flow transient. Relating to Fig. 2, plots (a)–(f) correspond to trajectories initiated at with , 19, 26.5, 34, 39, and 61.4, and approximate TKE levels of 0, 0.06, 0.1, 0.8, 1.78, and 2.69. The clear difference from the LDSM 2 modes in Fig. 17 is explained in the text.

Six local shift modes computed according to LDSM 2 from evaluation of local increments in RANSE solutions. The respective midperiod times in plots (a)–(f) are the same as the initial times used in the examples in Fig. 16 above; i.e., they are , 19, 26.5, 34, 39, and 61.4, associated with corresponding approximate TKE levels of 0, 0.06, 0.1, 0.8, 1.78, and 2.69.

Six local shift modes computed according to LDSM 2 from evaluation of local increments in RANSE solutions. The respective midperiod times in plots (a)–(f) are the same as the initial times used in the examples in Fig. 16 above; i.e., they are , 19, 26.5, 34, 39, and 61.4, associated with corresponding approximate TKE levels of 0, 0.06, 0.1, 0.8, 1.78, and 2.69.

The correlation matrices between different local shift modes computed according to LKSM 2. Color represents the amplitude of the correlation between two shift modes. Axes parameterize local shift modes by the associated periods TKE levels. (a) Correlation between shift modes of the natural transient (horizontal axis) and those computed along the descending forced transient (vertical axis). (b) Correlation between shift modes along the natural transient (horizontal axis) and along the ascending forced transient (vertical axis). (c) Correlation between shift modes along the descending forced transient (horizontal axis) and along the ascending forced transient (vertical axis). The solid and dashed white lines indicate the quadratic polynomial approximations of the lines connecting peak values in each row and column of the matrix and the equal TKE levels line, respectively.

The correlation matrices between different local shift modes computed according to LKSM 2. Color represents the amplitude of the correlation between two shift modes. Axes parameterize local shift modes by the associated periods TKE levels. (a) Correlation between shift modes of the natural transient (horizontal axis) and those computed along the descending forced transient (vertical axis). (b) Correlation between shift modes along the natural transient (horizontal axis) and along the ascending forced transient (vertical axis). (c) Correlation between shift modes along the descending forced transient (horizontal axis) and along the ascending forced transient (vertical axis). The solid and dashed white lines indicate the quadratic polynomial approximations of the lines connecting peak values in each row and column of the matrix and the equal TKE levels line, respectively.

Same as Fig. 18, but for the natural transient, computed by different methods. Color codes, axes parameterizations, and white/dashed white lines are as in Fig. 18. (a) Correlation of shift modes computed according to LKSM 2 (horizontal) and those computed according to LDSM 1 (vertical). (b) Correlation of shift modes computed according to LKSM 2 (horizontal) and those computed according to LDSM 2 (vertical).

Same as Fig. 18, but for the natural transient, computed by different methods. Color codes, axes parameterizations, and white/dashed white lines are as in Fig. 18. (a) Correlation of shift modes computed according to LKSM 2 (horizontal) and those computed according to LDSM 1 (vertical). (b) Correlation of shift modes computed according to LKSM 2 (horizontal) and those computed according to LDSM 2 (vertical).

Plots of the LKSM 2 mode of the natural transient at (a), followed by the best matches of that modes by LKSM 2 modes from the descending [(b), ] and ascending [(c), ] actuated transients. Next are the best matches of (a) by local shift modes along the natural transient, computed according to LDSM 1 [(d), ] and LDSM 2 [(e), ]. The last plot was computed by a simplified version of LDSM 2 [(f), ], where the fluctuation is modeled by the dominant POD mode pair of the attractor, scaled by the local TKE, instead of by the POD mode pair of the transient period under consideration.

Plots of the LKSM 2 mode of the natural transient at (a), followed by the best matches of that modes by LKSM 2 modes from the descending [(b), ] and ascending [(c), ] actuated transients. Next are the best matches of (a) by local shift modes along the natural transient, computed according to LDSM 1 [(d), ] and LDSM 2 [(e), ]. The last plot was computed by a simplified version of LDSM 2 [(f), ], where the fluctuation is modeled by the dominant POD mode pair of the attractor, scaled by the local TKE, instead of by the POD mode pair of the transient period under consideration.

Correlations of the best matching shift modes in Fig. 20. Labels indicate the LKSM 2 mode of the natural transient (LKSM 2 natural), and the best matches with LKSM 2 modes from the descending and ascending actuated trajectories (LKSM 2 descending and LKSM 2 ascending, respectively), the shift modes obtained from the natural transient according to LDSM 1, LDSM 2, and the simplified version of the latter (LDSM 2 simple), where the scaled attractor POD modes where used instead of the local POD modes, used in LDSM 2.

Correlations of the best matching shift modes in Fig. 20. Labels indicate the LKSM 2 mode of the natural transient (LKSM 2 natural), and the best matches with LKSM 2 modes from the descending and ascending actuated trajectories (LKSM 2 descending and LKSM 2 ascending, respectively), the shift modes obtained from the natural transient according to LDSM 1, LDSM 2, and the simplified version of the latter (LDSM 2 simple), where the scaled attractor POD modes where used instead of the local POD modes, used in LDSM 2.

## Tables

An overview of Sec. IV and the shift mode definitions described in Secs. IV A–IV D.

An overview of Sec. IV and the shift mode definitions described in Secs. IV A–IV D.

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