Higher order dynamic mode decomposition applied to

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Dynamic mode decomposition (DMD) is a technique introduced by Schmid11 ..... Tucker, L.R. 1996 Some mathematical notes on three-mode factor analysis.
AIAA 2017-3304 AIAA AVIATION Forum 5-9 June 2017, Denver, Colorado 47th AIAA Fluid Dynamics Conference

Higher order dynamic mode decomposition applied to post-process a limited amount of noisy PIV data Soledad Le Clainche∗, Francisco Sastre, Jos´e M. Vega and Angel Velazquez School of Aeronautics, Universidad Polit´ecnica de Madrid

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Pza. Cardenal Cisneros, E-28040, Madrid, Spain

A highly efficient method to treat experimental data is presented and applied to perform the analysis of a set of PIV measurements. The main goal of this article is to show the performance of a tool that is capable to study the flow dynamics of data coming from complex experiments, in which the number of collected data is small and highly noisy. Thus, this method has been applied to study a wall confined flow with thermal effects. This method is capable not only to clean the data, but also to find the dominant frequencies that are driving the flow.

I.

Introduction

Dynamic mode decomposition (DMD) is a technique introduced by Schmid11 that gives an approximation of the Koopman modes4, 10 and is suitable to analyze flow dynamics. This technique uses a linear operator to approximate non-linear dynamics and it can be used to study the growth rates and frequencies describing unsteady flows. Despite this method has been successfully applied to a wide range of test cases to study both linear flows2, 3 and complex non-linear dynamics,5, 6 the method could not provide the expected results if the flow studied is highly complex (i.e. transient flows1 ), or when the spatial complexity of the flow is smaller than the spectral complexity.7 Le Clainche & Vega7 proposed an extension of the DMD algorithm, called DMD-d or higher order DMD (HODMD), to expand its application to the aforementioned cases. Additionally, this algorithm has already been tested in noisy experimental data,8 and proved to be highly effective and more accurate within the frequency calculations than standard DMD. In this article HODMD is applied to analyze some Particle Image Velocimetry (PIV) experimental data. The main goal was to show the performance of a tool that is suitable not only to study the flow dynamics, but also to clean highly noisy experimental data. Thus, HODMD has been applied to find the main flow frequencies associated to a wall confined flow with thermal effects, in which the experimental time span is limited to avoid thermal contamination. Due to the high complexity of this PIV experiment, the number of snapshots collected is small and noisy. These two features make DMD-d a technique suitable to perform this analysis and to show its benefits. The article is organized as follows. Section II gives a brief summary of HODMD methodology and Section III describes the experimental facility and flow parameterization. Finally, the main results are presented in Sections IV and V and the main conclusions are presented in Section VI.

II.

Higher order dynamic mode decomposition

HODMD is an algorithm that considers to main steps. As first step, higher order singular value decomposition (HOSVD)12 is applied to reduce data redundancies and consequently to clean the noisy data, in accordance to a certain tolerance εSV D . The spatio temporal flow field is then decomposed into spatial and temporal SVD modes. As second step, a DMD-like algorithm related to the well-known sliding window process of power spectral density (PSD) is applied to the temporal SVD modes. This algorithm is called ∗ Correspondence

to: [email protected]

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DMD-d and the parameter d can be compared with the number of segments used in PSD. Thus, when d = 1 the HODMD algorithm is similar to standard DMD11 (in which SVD is used instead of HOSVD) . HODMD calculates the DMD modes um and their associated frequencies ωm , growth rates δm and amplitudes am . Thus it is possible to decompose the original data vk (snapshot) as an expansion of DMD modes in the following way vk ' vkDMD ≡

M X

am um e(δm +iωm )(k−1)∆t

for k = 1, . . . , K,

(II.1)

m=1

where the number of terms, M , can be referred to as the spectral complexity and K is the temporal dimension. The spectral complexity (number of DMD modes retained) becomes determined by two tolerance values, ε1 and ε2 . More details about this methodology can be found in the literature.7

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III.

Experimental facility and flow parameterization

Time-resolved PIV experiments have been carried out in the experimental facility of ETSIAE at Technical University of Madrid (UPM) with the aim of studying the underlying physics of a wall confined flow with thermal effects. Due to its mixing properties, this type of flow is relevant to several industrial applications, including micro-cooling or heat transfer enhancement. Figure 1 shows the experimental setup. The PIV measurements were performed using a Dantec Dynamics system. Flow illumination was achieved with a pulsed Nd:YAG 800 mJ Laser. The camera was a Dantec Dynamics Flow Sense 2ME with a resolution of 1600 × 1200 pixels. The lens was a Zeiss Makro-Planar T · 2/50 mm ZF. Flow seeding was performed with 10µm hollow glass spheres (HGS-10). Dantec Dynamics Studio software was used to synchronize image capturing and flow illumination, and to perform the analysis. Multigrid Cross-Correlation Particle Image Velocimetry (MCCDPIV) was used to resolve the in-plane velocity vector field. This calculation is an approximation that introduces a bias and random error of 5%. The size of the interrogation areas was 50 mm ×25 mm. These areas were subdivided into smaller areas by the analysis software to achieve convergence in the re-computation process of the flow field. It was estimated that the achieved spatial resolution of the flow field was of the order of 1 mm that should be enough to capture the large flow structures whose size was of the order of the prism cross-section length (10 mm). The sampling frequency was 15 Hz and the separation time between images was ∆T = 5ms.

Figure 1. Experimental setup and geometry parameters.

In this article, HODMD analysis has been performed to study three different cases in a channel with wall thermal effects (the aluminum wall temperature is 60◦ ). In the first case, the channel is empty, while in the two remaining cases, a square cylinder is located in the central part of the channel and it is either fixed or moving by the fluid-body interaction. The diameter of the cylinder is D = 1cm and the blockage ratio between the channel walls and the cylinder is L/D = 2.5. Figure 1 (right) shows an sketch of this setup. 2 of 14 American Institute of Aeronautics and Astronautics

D The experiments were carried out at different Reynolds numbers, defined as Re= Umean (where Umean , ν D and ν represent mean free stream velocity, diameter of the squared cylinder and kinematic viscosity, respectively), classified in Table 1 as function of the type of experiment. As seen, the three experiments, called for simplicity E1, E2 and E3, were performed at Re= 160 and 200. However, at Re= 133 it was only carried out the experiments that consider the squared cylinder (E2 and E3).

Table 1. Classification of the type of experiments performed to study a wall confined thermal flow as function of the Reynolds number.

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Re E1 (channel) E2 (fixed squared cylinder) E3 (moving squared cylinder)

133 X X

160 X X X

200 X X X

The error estimated within the experimental data is about 5%. Therefore, this value will lower bound the error that compares the original data with the same data reconstructed using DMD-d as an expansion of modes. In cases in which the studied flow dynamics are unknown, this error serves as a reference to detect the main frequencies and their associated values with relatively high accuracy.

IV.

Introduction to HODMD methodology

A priori, the flow dynamics are unknown. Thus it is necessary to calibrate the DMD-d relevant parameters, d and the tolerances εSV D , ε1 and ε2 , so that the reconstruction error be about 5% (estimated within the experimental data). As found in the literature,8 a good estimation for these tolerances would be to set up εSV D = ε1 = ε2 . On the one hand, it is interesting to note that the parameter εSV D is controlling the amount of data cleaning. Thus, for a similar reconstruction error, the larger this vale, the cleaner the obtained results, maximizing the benefits provided by this methodology. On the other hand, this tolerance is directly linked to the amount of collected data, being necessary to decrease the tolerance value when the number of snapshots is small. Therefore, considering a large number of snapshots, highly accurate reconstructed data could be obtained fixing εSV D = ε1 = ε2 =Experimental error (in this case 0.05). However, in the present article, the complexity of the experiment limits the number of snapshots collected and hence these tolerance should be smaller. Finally, in order to filter out noise and spurious results coming from a small tolerance, the parameter d should be increased in some cases (they are inversely proportional). The fact that the amount of data collected is small, also limits the identification of the relevant flow frequencies. However, since HODMD is a very robust methodology, it is possible to identify the main dominant frequencies by imposing the results obtained with different tolerances and values of d and similar reconstruction errors. The frequencies that are always present in the frequency spectra are the driven frequencies, associated with the main flow instabilities. A.

Calibration and dominant frequencies

As a prior step to find the main flow frequencies, it is necessary to calibrate the parameters for the analysis. The number of snapshots collected in the experiment of the fixed squared cylinder confined with thermal walls is 300, which is considered small compared to the spectral complexity that is expected to find at Re≥ 133, since the critical Reynolds number for a squared cylinder vortex shedding is Re' 45 (this value varies with the increase of the blockage or with the angle of attack13 ). Table 2 shows an example of the calibration process carried out. Since the experimental error is ∼ 0.05 (5%), the tolerances for DMD-d should be smaller than this value, so as first guess, the tolerances could be fixed to 10−2 . The Table shows that, for 10 ≤ d ≤ 50, the reconstruction error is greater than 0.08. For values of d > 50 or d < 10 this error is even higher. Hence, in order to increase the accuracy, this tolerance needs to be decreased. For a tolerance value equal to 5 · 10−3 it is possible to find an optimum value of d in which the error could be considered acceptable. Hence, for d = 35, the reconstruction error is 5.52 · 10−2 . This error increases either increasing or decreasing d. Finally, decreasing the tolerance to 4 · 10−3 it is possible to find that the reconstruction error is ∼ 0.05, smaller than in the latter case, for the range 20 ≤ d ≤ 40. However, since one of the main goals of applying DMD-d to the experimental data is to clean them up, the case with higher tolerance and acceptable reconstruction error is selected to represent the flow dynamics (d = 35 and εSV D = ε1 = ε2 = 5 · 10−3 ). It 3 of 14 American Institute of Aeronautics and Astronautics

is interesting to remark that if d = 1 is used (standard DMD) the results are completely spurious. On the contrary, if d >> 1 the number of snapshots may contain information that in time would be smaller than a period, which could also produce completely spurious results. Table 2. Reconstruction error obtained applying DMD-d to a fixed squared cylinder confined with thermal walls at Re= 133 as function of the tolerances εSV D = ε = ε1 and d.

Tolerance 10−2 5 · 10−3 4 · 10−3

d= 10 1.08 · 10−1 8.76 · 10−2 6.12 · 10−2

d= 20 1.05 · 10−1 6.50 · 10−2 5.23 · 10−2

d= 30 1.03 · 10−1 5.91 · 10−2 5.14 · 10−2

d= 35 9.88 · 10−2 5.52 · 10−2 5.12 · 10−2

d= 40 8.68 · 10−2 5.60 · 10−2 5.29 · 10−2

d= 50 8.18 · 10−2 5.98 · 10−2 6.90 · 10−2

Once the suitable parameters for the analysis are selected, the dominant frequencies are identified. In general, the driven frequencies are identified as the frequencies with higher amplitude. However, due to the noise coming from the experimental data and the reduced number of snapshots collected, in some cases the amplitude related to these frequencies might not be the highest (but the second or third highest). Nevertheless, due to the robustness of DMD-d methodology, the dominant frequencies are always found independently of the values of the tolerances or d (for a relatively small reconstruction error). Figure 2 shows the frequency spectra obtained for the cases with tolerances 5 · 10−3 and 4 · 10−3 for 30 ≤ d ≤ 40, in which the reconstruction error is considered as acceptable (< 0.06). As seen in the Figure, in all cases it can be found a frequency with highest amplitude that distinguishes the others. Therefore, at this flow conditions it is possible to conclude that there is a single driven frequency that oscillates with St' 0.42. The identification of this frequency is not as clear as in this case when the dynamics become more complex (Re> 133), but the procedure carried out for the identification of the relevant dynamics is similar to the one presented herein.

Tol.=0.004, d30 Tol.=0.004, d35 Tol.=0.004, d40 Tol.=0.005, d30 Tol.=0.005, d35 Tol.=0.005, d40

10 0

Amplitude

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Re=133

10 -1

10 -2

10 -3 -1.5

-1

-0.5

0

0.5

1

1.5

St

Figure 2. Frequencies vs. amplitudes calculated with DMD-d at Re= 133 in the flow past a squared cylinder with confined thermal walls experiments.

Finally, the initial data have been reconstructed using the results obtained using d = 35 and tolerances 4·10−3 and 5·10−3 considering both, all the frequencies provided by DMD-d and only the dominant frequency. Figure 2 compares the reconstructed instantaneous streamwise and spanwise velocity components (Ux and Uy , respectively) using the dominant frequency or all the data calculated by the method (shown in Figure 2) with the original data and phase-averaged data (with St' 0.42). As seen, the experimental data are highly noisy, while the reconstructed instantaneous field using DMD-d is cleaner, more suitable to study the flow physics. As expected, when all the frequencies are considered, the reconstruction obtained using DMD-35 is smoother in the case with higher tolerance (5 · 10−3 ). Additionally, the reconstruction with tolerance 5 · 10−3 that considers a single frequency is cleaner and smoother and presents a reconstruction error of 1.13 · 10−1 , meaning that, despite DMD-d captures the dominant frequency with a small number of snapshots, the flow spectra is more complex. However, since the estimated error within the experimental data is ∼ 0.05, this 4 of 14 American Institute of Aeronautics and Astronautics

error can still be considered as acceptable (same order of magnitude). Finally, phase-averaged velocity shows results similar to those obtained with DMD-d when the reconstruction is performed using a single frequency, but more noisy. Besides, DMD-d detects and filters systematic inconsistent errors according to a certain tolerance value, while phase-averaged technique is only capable of filtering out noise (exhibiting zero mean) via averaging the data in accordance to a certain frequency. This frequency needs to be calculated a priori using others techniques such as fast Fourier transform (FFT) or its more accurate counterpart PSD. However, when the number of data collected are limited and highly noisy, the accurate calculation of the dominant frequency becomes more complex and it could fail in some cases, thus more sophisticated techniques should be used instead. Additionally, as seen in Figure 3, phase-averaging gives an approximation of the original data is clean, but neither reproduces the exact dynamics nor gives an estimation of the assumed error. This fact makes DMD-d a technique more suitable to study in depth the flow dynamics.

4 3.5

X

3 2.5 2 1.5 1 0.5 0.5

1

1.5

2

1.5

2

Y

3.5 3

X

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4.

2.5 2 1.5 1 0.5 0.5

1

Y

Figure 3. Instantaneous streamwise (top) and spanwise velocity (bottom) in the flow past a fixed squared cylinder at Re=133. From left to right: experimental data, phase averaged velocity (St' 0.42), DMD-d reconstruction using all the data of Figure 2, d = 35 and tolerances 4 · 10−3 (middle-left), 5 · 10−3 (middle-right), and DMD-d reconstruction using only the dominant frequency St' 0.42, d = 35 and tolerance 5 · 10−3 (right).

The same procedure has been carried out to find the dominant frequencies in the remaining experimental data. Table 3 shows the dominant frequencies, non-dimensionalized with Strouhal number defined as St= f ·D Umean , and the reconstruction error obtained with DMD-d. The table specifies the suitable parameters selected (d and tolerances εSV D = ε1 = ε2 ) and the number of snapshots used for this analysis. In the experiments carried out with both, the fixed and moving cylinder, the number of snapshots collected is 300 while in the channel this number is 100, which is a very small number of snapshots. As seen, a single dominant frequency is captured at Re= 133 in both, the case with the fixed and moving cylinder, and at Re= 160 in the case with the moving cylinder. On the contrary, two or even three different frequencies are found driving the flow in the remaining cases, thus the flow can be considered quasi-periodic. On the one hand, the number of frequencies driving the flow increases with the Reynolds number and these frequency values are different among each one of the studied cases. On the other hand, in the case of the 5 of 14 American Institute of Aeronautics and Astronautics

Table 3. Parameters used for the analysis of the experimental data. Number of snapshots, dominant frequencies, reconstruction error and selected d and tolerances εSV D = ε = ε1 . Fixed Sq. Cyl

Mov. Sq. Cyl

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Channel

Re= 133 Re = 160 Re= 200 Re= 133 Re = 160 Re= 200 Re = 160 Re= 200

Num. snap. 300 300 300 300 300 300 100 100

Frequencies St1 = 0.42 St1 = 0.43, St2 = 0.58 St1 = 0.37, St2 = 0.55, St3 = 0.27 St1 = 0.56 St1 = 0.48 St1 = 0.52, St2 = 1 St1 = 0.78, St2 = 1.7 St1 = 3.4, St2 = 0.89

RMSE 5.52 · 10−2 5.78 · 10−2 6.30 · 10−2 5.57 · 10−2 6.45 · 10−2 6.47 · 10−2 4.02 · 10−2 5.26 · 10−2

d 35 10 10 20 35 15 10 25

Tolerance 5 · 10−3 4 · 10−3 4 · 10−3 5 · 10−3 5 · 10−3 4 · 10−3 4 · 10−3 5 · 10−3

moving cylinder, a single frequency is driving the flow at Re= 133 and Re= 160, while two frequencies are relevant at Re= 200. However, in the case of the empty channel and the fixed cylinder, two frequencies are already driving the flow at Re= 160, suggesting that the moving cylinder condition may delay the presence of the associated flow instabilities. The same statement can be concluded from the results presented in Table 4, that shows quite similar DMD-d results obtained using a smaller number of snapshots (emphasizing that DMD-d method is robust). This Table shows the RMSE error of the data reconstructed using only the dominant frequency. As seen, the reconstruction error in the moving cylinder case at Re= 133 and Re= 160 is similar to the estimated error coming from the experimental measurements. However, this error is higher (twice the experimental error) in the remaining cases. Therefore, the spectral complexity is reduced when a moving squared cylinder is used inside the channel. Table 4. Counterpart of Table 3. The reconstruction error RMSE only considers the dominant frequency.

Fixed Sq. Cyl

Mov. Sq. Cyl

V.

Re= 133 Re = 160 Re= 200 Re= 133 Re = 160 Re= 200

Num. snap. 100 100 200 100 100 200

Frequencies St1 = 0.42 St2 = 0.57 St1 = 0.38 St1 = 0.58 St1 = 0.50 St1 = 0.51

RMSE (dom freq) 1.04 · 10−1 1.36 · 10−1 1.84 · 10−1 5.39 · 10−2 8.47 · 10−2 1.48 · 10−1

d 15 5 5 5 5 10

εSV D 5 · 10−3 5 · 10−3 5 · 10−3 1 · 10−2 5 · 10−3 5 · 10−3

Analysis of flow structures: DMD modes

Figures 4, 5, 6, 7, 8 and 9 show the most relevant DMD modes presented in Table 3 in the case of the fixed and moving squared cylinders, and Figure 10 shows the first dominant mode calculated in the empty channel at Re=200 (as a representative solution). As seen, the results presented are clean, in good agreement with the tolerance selected for the DMD-d analyses. The lower this tolerance is, the higher is the level of noise found in the DMD modes presented herein. This fact is specially evident in the mode associated to the mean flow (steady mode). Therefore, the data cleaning procedure carried out by the HODMD technique is suitable to study highly noisy experimental data. DMD modes also show that the flow is parallel in the case of the empty channel, where heat transfer is driven by forced convection. The spectral complexity of the flow (number of frequencies) increases with the Reynolds number, but, the spatial structure remains almost invariant. However, the presence of either the fixed or the moving cylinder, breaks up with the spatial homogeneity. Thus, some more complex spatial structures, associated to cylinder wake instabilities, are found in the fluid flow. Such structures focus on the inner part of the channel and promote the fluid mixing and its consequent heat transfer enhancement. Nevertheless, for a fixed Reynolds number, it is possible to find some differences in the DMD modes calculated in the cases with the fixed and moving cylinder. In the former case, the spatial structures associated to each DMD mode are non-symmetric and they distribute along the entire spanwise direction (wall and inner region), while in the second case, these structures are more symmetric and focused on the inner part of the channel. This fact suggests that the mixing properties might increase with the presence of the moving cylinder, since the fluid oscillates at different frequencies in larger regions.

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Finally, as the Reynolds number increases, the flow dynamics become more complex and the wake oscillates at several different frequencies (quasi-periodic flow), leading to the laminar flow transition and consequently, enhancing heat transfer. However, it is important to note that the spectral complexity increases further with the presence of the fixed cylinder than with the moving one, which might be considered as a mechanism of flow control.9

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VI.

Concluding remarks

HODMD is a method suitable to study complex dynamics (quasi-periodic flow) using highly noisy data. This method is applied to a set of PIV experimental data with the aim at studying the main flow dynamics in a wall confined flow with thermal effects. Due to the high complexity of this experiment, the number of data collected is small, and the data are highly noisy. However, DMD-d is capable, not only to quantify the dominant frequencies that drive the flow, but also to clean the experimental data. This technique provides a good advantage with respect the common techniques usually used to analyze the main flow frequencies, such as FFT or PSD, and the techniques used to reduce the noise of experimental data, such as phaseaveraging. In contrast to FFT/PSD, HODMD captures the dominant frequencies using a small number of snapshots and, in contrast to phase-averaging, it estimates the reconstruction error of the instantaneous data if a single frequency is considered, and provides cleaner results than phase-averaging using a smaller number of snapshots. HODMD filters out inconsistent systematic errors according to a certain tolerance and reconstructs the instantaneous data cleaned, which provides a great advantage in cases in which other types of analyses will be performed (i.e.: vorticity calculations, evolution of velocity profiles, ...). Additionally, this technique also provides the modes associated with the dominant frequencies, clean of noise, which allows to study in detail what are the instability mechanisms associated with such frequencies.

References 1 Bagheri,

S. 2013 Koopman-mode decomposition of the cylinder wake. J. Fluid Mech. 726, 596–623. E., DeVicente, J. & Valero, E. 2014 Low cost 3D global instability analysis and flow sensitivity based on dynamic mode decomposition and high-order numerical tools Int. J. of Num. Meth. in Fluids 76 (3), 169–184. 3 Go ´ mez, F., Le Clainche, S., Paredes, P., Hermanns, M. & Theofilis, V. 2012 Four decades of studying global linear instability: progress and challenges AIAA journal 50 (12), 2731–2743. 4 Koopman, B. 1931 Hamiltonian systems and transformations in Hilbert space. Proc. Natl Acad. Sci. USA 17, 315–318. 5 Le Clainche, S., Li, I., Theofilis, V. & Soria, J. 2015 Flow around a hemisphere-cylinder at high angle of attack and low Reynolds number. Part I: Experimental and numerical investigation Aerospace Sciences and Technology 44, 77–87 6 Le Clainche, S., Rodr´ ıguez, D., Theofilis, V. & Soria, J. 2015 Flow around a hemisphere-cylinder at high angle of attack and low Reynolds number. Part II: POD and DMD applied to reduced domains Aerospace Sciences and Technology 44, 88–100 7 Le Clainche, S. & Vega, J.M. 2017 Higher order dynamic mode decomposition. SIAM J. on Appl. Dyn. Systems, (in press) DOI:10.1137/15M1054924 8 Le Clainche, S., Vega, J.M. & Soria, J. 2017 Higher order dynamic mode decomposition for noisy experimental data: the flow structure of a Zero-Net-Mass-Flux jet. Submitted to Exp. Therm. and Fluid Science 9 Prassad, K. Paramane, S. B., Agrawal, A. & Sharma, A. 2011 Effect of channel-confinement and rotation on the two-dimensional laminar flow and heat transfer across a cylinder Num. Heat Trans., Part A: applications 60(8), 699–726. 10 Rowley, C.W., Mezic ´, I., Bagheri, S., Schlatter, P. & Henningson, D. S. 2009 Spectral analysis of nonlinear flows. J. Fluid Mech. 641, 115–127. 11 Schmid, P.J. 2010 Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 5–28. 12 Tucker, L.R. 1996 Some mathematical notes on three-mode factor analysis. Psikometrica, 31, 279–311. 13 Yoon, D., Yang, K. & Choi C. 2010 Flow past a square cylinder with an angle of incidence. Physics of Fluids, 22, (043603) 1–22. 2 Ferrer,

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Mean flow (St= 0)

Fundamental frequency (St= 0.42) Figure 4. Fixed cylinder. DMD modes at Re=133. From left to right: Real(ux ), Real(uy ), Image(ux ) and Image(uy ).

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Mean flow (St= 0)

Fundamental frequency 1 (St= 0.43)

Fundamental frequency 2 (St= 0.58) Figure 5. Fixed cylinder. Counterpart of Figure 4 at Re=160.

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Mean flow (St= 0)

Fundamental frequency 1 (St= 0.37)

Fundamental frequency 2 (St= 0.55)

Fundamental frequency 3 (St= 0.27) Figure 6. Fixed cylinder. Counterpart of Figure 4 at Re=200.

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Mean flow (St= 0)

Fundamental frequency (St= 0.56) Figure 7. Moving cylinder. Counterpart of Figure 4 at Re=133.

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Mean flow (St= 0)

Fundamental frequency (St= 0.48) Figure 8. Moving cylinder. Counterpart of Figure 5 at Re=160.

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Mean flow (St= 0)

Fundamental frequency 1 (St= 0.52)

Fundamental frequency 2 (St= 1) Figure 9. Moving cylinder. Counterpart of Figure 6 at Re=200.

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Fundamental frequency 1 (St= 3.4) Figure 10. Channel flow. DMD modes at Re=200. From left to right:Real(ux ) and Real(uy ).

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