A comparative study of data-driven modal decomposition analysis of unforced and forced cylinder wakes

The present study on the recognition of coherent structures in flow fields was conducted using three typical data-driven modal decomposition methods: proper orthogonal decomposition (POD), dynamic mode decomposition (DMD), and Fourier mode decomposition (FMD). Two real circular cylinder wake flows (forced and unforced), obtained from two-dimensional particle image velocimetry (2D PIV) measurements, were analyzed to extract the coherent structures. It was found that the POD method could be used to extract the large-scale structures from the fluctuating velocity in a wake flow, the DMD method showed potential for dynamical mode frequency identification and linear reconstruction of the flow field, and the FMD method provided a significant computational efficiency advantage when the dominant frequency of the flow field was known. The limitations of the three methods were also identified: The POD method was incomplete in the spatial–temporal decomposition and each mode mixed multiple frequencies leading to unclear physics, the DMD method is based on the linear assumption and thus the highly nonlinear part of the flow field was unsuitable, and the FMD method is based on global power spectrum analysis while being overwhelmed by an unknown high-frequency flow field.

Incoming airflow velocity U Mean velocity of x direction V Mean velocity of y direction [u,v] Profile of flow direction and lateral velocity fluctuations Y Correlation matrix X/D, Y/D X, Y direction dimensionless length

Introduction
With the development of flow field data acquisition technology (high-frequency sampling by high-speed cameras) and the rapid development of scientific computing, flow field datasets are becoming increasingly large. Much modern analysis requires the identification of coherent structures in turbulent flows, flow field prediction and reconstruction, or the representation of an entire dataset in terms of a few characteristic features. For a two-dimensional (2D) flow field, the velocity is a spatial-temporal parameter U x; t ð Þ. A datadriven method to decompose the spatial and temporal characteristics would help to reveal flow field characteristics (Rowley and Dawson 2017). Among all of the data-driven algorithms that have been used to identify coherent structures, proper orthogonal decomposition (POD), dynamic mode decomposition (DMD), and Fourier mode decomposition (FMD) are the most widely used.
Over the past few decades, the POD method has been widely used to determine coherent structures. Lumley (1967) first introduced the POD method into flow fields to analyze large-scale structures in turbulence. Subsequently, Sirovich (1987) improved the classical POD method to produce the snapshot POD method because the data obtained from particle image velocimetry (PIV) measurements were always shorter in the temporal than in the spatial dimension. Meyer et al. (2007) used the POD method to analyze the turbulent jet in a crossflow. It was observed that the dominant dynamic structures were associated strongly with the jet core from instantaneous velocity fields, demonstrating the powerful potential of this method. Many previous studies have used the POD method to identify the coherent structures in a bluff body wake flow (Feng et al. 2011;Gao et al. 2021a, b, c), flow control (Gao et al. 2018;2021a, b, c), rain-wind induced flow (Gao et al. 2019;2021a, b, c), and turbulent jets ). There are various developments of POD methods, such as multiscale proper orthogonal decomposition (Mendez et al. 2019) and POD with discrete empirical interpolation method (Ghavamian et al. 2017).
In recent years, the DMD algorithm was developed by Schmid (2010) as an extension of the Koopman analysis of a dynamic system presented by Mezić (2013). From a mathematical viewpoint, the essence of the POD and DMD methods is associated with singular value decomposition (SVD). The DMD algorithm considers the temporary matrix V while obtaining both spatial modes and frequency information corresponding to coherent structures. The DMD method is a collective term for a series of methods, such as highorder DMD (Le Clainche and Vega 2017) and physical information DMD (Kutz et al. 2016). Because the present method is based on linear assumptions, such that continuous time predictions and flexible original data inputs are possible, DMD has been rapidly adopted (Schmid 2022).
There has also been a new mode decomposition method called Fourier mode decomposition (FMD) based on the discrete Fourier transform (DFT) (Ma et al. 2015). This method is more suitable than the POD and DMD methods when the flow field in question is in a quasi-period condition. Ma et al. (2015) presented reasons for using the FMD method that with the known external excitation frequency f e , it can be normalized by the dominant frequency f 0 and then showed that coherent structures correspond to the characteristic frequency f e =f 0 . As a circular cylinder flow field is a quasi-period flow, we use this method to extract characteristic frequency coherent structures.
Previous studies have mainly focused on one type of modal analysis method to identify the multi-scale structures in a flow field. However, it is important to choose the right analysis method based on the specific system and dataset, as well as the advantages and disadvantages of each method. Zhang et al. (2014) used both the POD method and the DMD method to conduct coherent structure identification. They showed that Kármán-like vortex structures were associated with the first two POD modes and that DMD can avoid some uncorrelated structures provided by the energy-based POD. Liu et al. (2021) analyzed POD and DMD reconstruction flow field errors in various parts of a cavitating flow. They indicated that the DMD reconstruction method may create a large error when the flow pattern is nonlinear. Ma et al. (2015) introduced the FMD method and compared FMD with POD and DMD to evaluate the advantages of FMD method in quasiperiod flow.
In this study, we used two flow demos to explain the application of mode decomposition methods. The first one is flow past a cylinder and form the Kármán vortex shedding. The second one is a flow past a cylinder embedded with a flapping jet actuator. The flapping jet actuator can be simply classified into three types: two feedback channels, one feedback channels, and free-feedback channel. The flapping jet actuator is based on a special shape and the Coanda effect, which can produce high-frequency and high-amplitude flapping jets with a small momentum input .
This study aims to give a short summary of these three methods and a thorough exploration of the POD, DMD, and FMD algorithms for a circular cylinder wake flow. The paper is arranged as follows: The experimental setup and the details of the experimental demo are introduced in Sect. 2, some properties of the time-averaged flow are provided in Sect. 3, and a comparison of the POD, DMD, and FMD analysis methods is presented in Sects. 4, 5, and 6, respectively. Finally, we discuss the advantages and disadvantages of these three methods in the hope of providing some inspiration to the reader.

Experimental setup
To comprehensively compare the POD, DMD, and FMD methods, we provide an experimental demo focused on the passive control of a circular cylinder. A self-feedback flapping jet actuator is embedded in the mid-span of the test model, forming a passage in the circular cylinder, as shown in Fig. 1(a). The incoming flow through the throat gain a speed up, then the high-speed flow into the mixing chamber. Based on the Coanda effect, the fluid in the chamber flapping against the wall surface. Finally, a flapping jet is formed through the outlet with an opening angle of 60 .  and Chang et al. (2022) used a flapping jet to control the vortex shedding in a circular cylinder wake flow. The former performed POD analysis of the coherent structures in a cylindrical wake, and the latter focused on DMD analysis of the dynamic structure and spatiotemporal flow field. In subsequent research, we applied various reduced-order model (ROM) methods to analyze the advantages and disadvantages of these methods. A short introduction to the experimental setup is presented here. The experiment was completed in a closed-circle wind tunnel. The freestream turbulent intensity was less than 0.4%. The test section size was 505mm Â 505mm Â 1050mm. The experimental model was made by three-dimensional printing, with an outer diameter D ¼ 50mm and length L ¼ 505mm. In this study, the incoming flow U 0 was set to 6m=s. Based on the incoming flow speed, the diameter of the circular cylinder, and the viscosity of the local air, the Reynolds number was Re ¼ 1:99 Â 10 4 .
To better understand the characteristics of the cylindrical wake field, the flow structure of the forced and unforced cases was captured using the 2D PIV system [as shown in Fig. 1(b)]. We used a high-speed camera at a sampling frequency of 200 Hz, acquiring 3000 flow field snapshots. In this study, the size of each snapshot was 1950 Â 920 pixels. The region of interest (ROI) of the snapshots was 0 X=D 4 in the streamwise direction and 1:5 Y=D 1:5 in the crossflow direction, which was enough to show the vortex shedding structure. The images were analyzed using a cross-correlation algorithm. The interrogation window was 32 Â 32 pixels with an overlap rate of 50%. Based on Park et al. (2008), the PIV system uncertainties in this paper were approximately 1%. More detailed information on wind tunnels, the PIV equipment, and the flapping jet actuator characteristics can be found in previous works .

Statistical flow field description
The present experiment aims at comparing the wake structure of a cylindrical model with and without an embedded flapping jet actuator. We define two corresponding cases: The ''unforced condition'' has no embedded actuator, and the ''forced condition'' has an embedded actuator. For the unforced case, we focus on the identification of large-scale correlation structures, and for the forced case, we analyze the characteristics of different modal analysis methods for complex flows.
In this section, the original instantaneous velocity flow field and time-averaged results are introduced.

Original instantaneous velocity fluid field
To provide the reader with a clear picture of the velocity field of a two-dimensional cylindrical wake flow, Fig. 2 shows the velocity field of the cylindrical wake disturbance calculated directly using the crosscorrelation algorithm, which in turn gives the vorticity field x z D=U 0 , where x z ¼ r Â u ¼ ov=ox À ou=oy.
The unforced case has a high level of vorticity compared to the forced case. There is a pair vortex near the test model in the forced case. This is caused by the flapping jet actuator passage in the circular cylinder. Both cases show structures associated with alternating vortex shedding, but varying degrees of noise and small-scale turbulent structures make the distributions somewhat chaotic. There is a paradox here: On the one hand, this chaotic information is valuable if we study the structure of turbulence in the wake flow; on the other hand, too many turbulent structures can confuse the researcher, especially when they need to Fig. 2 The instantaneous vortex fields of the unforced and forced cylindrical wakes determine the main flow patterns of the overall fluid field. Therefore, data analysis tools with which to extract the main flow structures and focus on the main patterns of vortex formation and development are urgently needed.

Time-averaged analysis
One very common method of data analysis is to analyze the flow field from a time-averaged perspective. The central idea of this method is that the main structure occupies the main energy of the flow field. Timeaveraged results were calculated from 3000 pairs of images of the instantaneous flow field. Based on the sampling frequency of 200 Hz and the 3000 flow field snapshots, the sampling time for each PIV measurement was 15 s, enough to contain multiple vortex shedding complete periods. Figure 3 shows the distribution of the turbulent kinetic energy (TKE) and Reynolds shear stress (RSS) of the unforced and forced cases. In this study, the turbulent kinetic energy was calculated using the mean velocity, TKE ¼ 0:5 Â ðU 2 þ V 2 Þ=U 2 0 . Since the cylindrical wake vortex shedding is a quasi-period flow, the distributions of the TKE and streamlines show perfect symmetry. Comparing the near-wake streamlines of the forced and unforced cases as shown in Fig. 3a, c, there is a new symmetrical reflux region that pushes the original reflux region far away from the main cylinder. In addition to this, the TKE level decreases sharply as the force is applied. Figure 3b, d show the Reynolds shear stress colormap, which is also defined by the mean velocity, RSS ¼ ÀU Â V À Á =U 2 0 . Similar to the TKE, when the force is applied, the level of RSS decreases significantly. There is a mixing of the original reflux region and the new region created by the flapping jet.
Unfortunately, the information obtained from this statistical average perspective is very limited. For twodimensional flow fields, considering the spatiotemporal characteristics, the velocity of the fluid field can be described by Eq. (1). The time-averaged analysis refers to the part U. The mean velocity occupies the vast majority of the flow field but neglects the fluctuating part u 0 ðx; tÞ. This can result in the neglect of some important flow structures, for instance, the flapping jet vortex, as shown in Fig. 2.

POD analysis
Following the realization that the main structure in the flow field occupies the main energy, proper orthogonal decomposition (POD) was introduced to analyze fluid fields. The central idea of POD is to use singular value decomposition (SVD) to calculate the eigenvalue of the fluctuating velocity. The former eigenvalues, also known as modes, are used as representatives with which to reconstruct the fluctuating part.
In this section, we use the POD method to analyze the coherent structures of the fluid field and reconstruct the wake flow combined with the POD modes / n and time coefficients a n . Based on the quasiperiodicity and high correlation of the first two orders of the POD eigenvectors, we extract the phase average of the original flow field.

POD algorithm
The POD method was first introduced into the fluid dynamics field by Lumley (1967) as an attempt to decompose the fluctuating velocity into the sum of a set of deterministic functions. Formally, it is similar to a polynomial expansion or Fourier decomposition. Each part of the series represents a specific energy component. We expect to determine the flow organization from these deterministic functions. Figure 4 shows an overview of the philosophy of a POD algorithm applied to a circular cylinder wake flow. Before we do the POD, there is some preparation involving subtracting the mean velocity of the original flow to obtain the fluctuating velocity. POD is then used to obtain the coherent structures, fluid field reconstruction, and phase-averaged field.
Let us consider the fluctuating velocity u 0 ðx; tÞ, which we can obtain from Eq. (1). The core of the POD method, as mentioned before, is the representation of the fluctuating velocity as a cumulative series, so that where the / n x ð Þ are the spatial POD modes and a n t ð Þ are their time coefficients. The POD modes have suitable spatial orthogonal properties, ZZZ The orthogonality property is very useful since it indicates that each time coefficient corresponds to a specific spatial mode. This provides the theoretical foundation for fluid reconstruction.
The details of the POD algorithm calculation follow Meyer et al. (2007). Firstly, we obtain our correlation matrix Y, Fig. 4 Outline of a POD algorithm applied to a circular cylinder wake flow is a matrix of snapshots of the flow field fluctuation. Secondly, we calculate the eigenvalue of the correlation matrix. Due to Eq. (4), Y is a symmetric matrix and the eigenvalues of Y are nonnegative.
where k are the eigenvalues and A are the eigenvectors. We then sort the eigenvalues in descending order: Finally, we get the POD modes, where A i n is the nth eigenvector corresponding to the ith eigenvalue in Eq. (5), where U ¼ / 1 / 2 Á Á Á / n ½ is arranged from the first nth POD modes and a n is the time coefficient. It should be noted that we have used snapshot proper orthogonal decomposition, which was recommended by Sirovich (1987). The main difference between classical POD and snapshot POD is the correlation matrix, which is Y mÂm ¼ U mÂn U T mÂn in the former method or Y nÂn ¼ U T mÂn U mÂn in the latter method, where n is a spatial index and m is temporal. For PIV experiments, the number of spatial points is generally much larger than the number of sampling snapshots. In this study, the former is n ¼ 120 Â 56 Â 2 ¼ 13440 and the latter is m ¼ 3000. Using snapshot POD can markedly reduce the computational resources required and improve computational efficiency.
The reason why two methods approach go the same way is that they closely related to the SVD method.
Although the dimension of matrix RR T is different, we obtain the same first modes. More comments on this can be found in Weiss (2019). Figure 5 shows the cumulative energy proportion of POD modes. The contribution of each mode i is calculated as k i = P N k N . The black square points represent the unforced cases. It can be seen that the first two modes have a large energy, which means that they are responsible for explaining the main structure of the flow field. The remaining turbulent fluctuations are random distributions of higher-order modes. As the modal order increases, the growth trend of adjacent modes slows down. The forced case POD energy has a similar tendency to the unforced case. This suggests that even if the working conditions were changed, the POD method would still provide stability to the laws from the perspective of the accumulated energy. Figure 6a illustrates the first four POD modes of the unforced and forced cases. It is observed that modes 1 and 2 are symmetric about the centerline of the circular cylinder, while modes 3 and 4 are antisymmetric about the centerline. The distributions of the first two modes are quite similar, but there is a phase difference between them. We can consider mode 2 to be a continuation of the development of mode 1. As stated by Konstantinidis et al. (2007), the antisymmetric vorticity field is represented by the symmetric mode distribution. Therefore, we can assume that the first two modes refer to the Kármán vortex shedding of the main circular cylinder. Modes 3 and 4 represent the symmetrical small-scale structure (Feng et al. 2011). Mode 3 is a special mode we call the shift mode which is not associated with particular dynamics. However, mode 3 is an important sign that the mode distributions change from symmetry to antisymmetry. Mode 4 is a mode associated with high-order harmonic structures. Figure 6b, c presents the time coefficients and power spectrum of the first four modes. The POD modes may also be used to illustrate the convection of the vortex transform. For the unforced case, the time coefficients of the two modes a 1 ; a 2 are approximated in the sine and cosine forms. Combined with corresponding spatial modes to analyze, these offsets of the first-time coefficients demonstrated that there were the translating and alternating structures of the vorticities in the flow field. It is also observed that the coefficients a 3 ; a 4 are randomly fluctuating. The reduced frequency is defined as fD=U 0 , where f is the vortex shedding frequency, and the value is approximated as 0.2, the usual value for a circular cylinder.

Snapshots POD analysis
Compared to the forced case, the amplitudes of the time coefficients and power spectrum are decreased significantly. This indicates that the coherent structures of the flow field are changed by the flapping jet, but the cause of this change is not immediately clear.
In both the forced and unforced cases, the time coefficients a 1 ; a 2 have a strong correction, as shown in Fig. 6b. Van Oudheusden et al. (2005) presented the Fourier decomposition format, using the first two POD mode coefficients to define the phase w of the instantaneous evolution of the flow structures: where the phase w is the vortex-shedding angle, assumed to increase linearly as dw=dt ¼ 2pf .
Equation (11) is illustrated in Fig. 7 (left panels) as a red dashed unit circle. It is observed that the black points are surrounded by the central circle, showing a strong correction. As Pan et al. (2013) suggested, Eq. (11) can be simply rewritten as: Equation (12) is a projection, and all of the surrounding points are located on the unit circle. Under this condition, the physical meaning of the POD coefficients and phase is clearer.
Based on Eqs. (9)-(12), the phase-averaged wake flows of the unforced and forced cases are illustrated in Fig. 8. The first phase w 0 was randomly selected according to the sampling sequence. The phase-averaged method using the POD coefficients provides the flow phases from the limited snapshots. Compared to the classical cross-correlation method, this method has mathematical equivalence and reduces the dependence on a high sampling rate.
Alternative vortex shedding is shown in Fig. 8. As the phase progresses, the time within a period grows, and the main vortex structure of the fluid flowing through the circular cylinder is revealed. Compared to the original velocity flow in Fig. 2, the fluctuating turbulent structures are reduced and more coherent structures are seen. So far, we have obtained more information than from the time-averaged analysis, but averaging still greatly diminishes the role of the fluctuating components in the flow field.

POD reconstruction analysis
We now introduce the POD reconstruction algorithm. The time coefficients and POD modes are obtained from Eqs. (7)-(8). We can reconstruct the flow field as: In this study, we use the first four POD modes to reconstruct the original fluid field. Figure 9 illustrates the reconstructed instantaneous vorticity fluid field. Firstly, compared to Fig. 2, the coherent structures are clearer. The shear layer, the vortex boundary, and the vortex shedding process can all be observed. The POD reconstruction is a filter that isolates the main coherent structures of the flow from the data that contains noise and fluctuating turbulence. For the forced case, there is a pair of small vortices near the circular cylinder. Neither time-averaged analysis nor the phase-averaged method can provide good performance.
Let us now discuss the reliability of the POD reconstruction. To determine the errors between the reconstructed flow field and the original one, Liu et al. (2021) recommended using the relative error method: where u POD and u ori are the streamwise velocity from the POD reconstruction and the original fluid field. n ¼ 120 Â 56 is introduced in Sect. 4.1 and m ¼ 200 is chosen for randomly selected one-second sequences. According to Fig. 10, the largest errors e i ð Þ occur at the shear layer and in the far wake from the main cylinder, which suggests a nonlinear part or fluctuating turbulence. We consider this acceptable since these parts are not relevant to the reconstruction. It is worth mentioning that the coherent structure and the near wake show fewer errors. This is enough to indicate that POD reconstruction can help us to understand the large-scale coherent structure of the flow field. The average velocity errors are 4.49% (for the forced case) and 7.14% (for the unforced case), respectively. As stated by Liu et al. (2021), by increasing the number of POD modes of the reconstruction reference, the errors can be reduced. This is obvious from the perspective of polynomial summation and will not be further explored in this study.

DMD analysis
We expect the POD to help us find coherent structures and interpret the physical meaning of each mode. The physical meaning of the POD modes is not immediately clear. POD modes cannot form direct mapping relationships with real physics. The idea of sorting by energy is useful but limited since each POD mode has some randomness from turbulence. Here, we introduce a new modal analysis method to try to solve this problem. The dynamic mode decomposition method is based on the assumption of a linear system and makes it theoretically possible to solve this dilemma.

DMD algorithm
For a two-dimensional flow field, Schmid (2010) recommended using DMD to obtain the physical meaning of the modes of a flow. For a temporal evolution variable x and a nonlinear system, and F represents the nonlinear governing equation, describing the mapping between x i and x iþ1 . If the Fig. 9 The distribution of the instantaneous vorticity from POD reconstruction Fig. 10 Error colormaps generated by comparing the original flow to the POD reconstruction from the first two modes nonlinear system has finite dimensionality, we can use a linear operator with infinite dimensionality to replace it.
Here, we have converted the state space x i to observable space / i using the Koopman operator K.
Assuming that the temporal evolution of the flow field can be expressed as a linear mapping of two adjacent sequences: where X ¼ u 1 u 2 Á Á Á u nÀ1 ½ is a time sequence and Y ¼ u 2 u 3 Á Á Á u n ½ is the next moment, and the interval between the two is Dt.
Interestingly, Eq. (17) is a linear equation, and the solution is known where N is the number of matrix truncations, / n are the DMD modes, and x n is the frequency of each DMD mode.
It should be noted that Eq. (18) is an approximate solution, and it is possible to use a finite number to reconstruct the whole flow field. On the other hand, X is a dynamically updating matrix. This property leads to flexibility in the use of DMD methods in areas such as active flow feedback control and flow field prediction (Kutz et al. 2016). Figure 11 presents an overview of a DMD algorithm applied to a circular cylinder wake flow. Firstly, two sequences of n À 1 snapshots sampled equidistantly in time are arranged with spacing Dt. Then, following the detailed algorithm introduced by Schmid (2011), the DMD modes and reconstruction field are plotted.

DMD analysis of a two-dimensional bluff body wake velocity field
Unlike the POD modes, which are ranked according to the energy defined by their eigenvalues, the DMD modes are ranked based on their contributions to the total energy in the flow field, as recommended by Rowley et al. (2009). As shown in Fig. 12, the first mode occupies an absolute percentage of the total energy, since the DMD algorithm does not subtract the mean velocity from the system, and the first mode represents the time-averaged mode. The first five modes approximately occupy the total energy. In this study, we select the truncation number to be N ¼ 21, with one mean mode and ten pairs of DMD modes. Fig. 11 An outline of a DMD algorithm applied to a circular cylinder wake flow Figure 13 illustrates the DMD characteristics of the forced and unforced cases. There is a complex plane named the Ritz circle. The dashed line is a unit circle and the axes u r ¼ Realfk j g, u i ¼ Imagfk j g represent the real and imaginary parts of the DMD modal eigenvalues, respectively. It is observed that all the modes are distributed on the circle or inside it, showing the stability of the DMD modes (Rowley et al. 2009). We colored the modes using the logarithm of the mode energy, where the energy is E j ¼ b j k/ j k 2 2 . The unforced and forced cases of the first DMD modes are presented at the bottom of Fig. 13. The meaning of the first modes is a time-averaged modal analysis. Compared to Sect. 3.2, the first DMD modes represent the mean Fig. 12 The cumulative energy proportions of different DMD modes Fig. 13 The Ritz circle and the distribution of the first six DMD eigenvalues and the first DMD modes vorticity flow field. Both the unforced and forced cases are symmetrical about the centerline of the circular cylinder. Note that there is a pair of small vortices near the main cylinder because of the flapping jet effect. Figure 14a illustrates the DMD spectrum in the unforced case, and six domains of the DMD modes are chosen and sorted by the proportion of their respective energy that makes up the total energy in descending order. The mean mode is named ''1'' and the following ''2'' to ''6.'' To focus on the stability and frequency of the DMD modes, we define G j and F j .
Corresponding to the linear solution in Eq. (18), G j is defined as the growth rate and F j is the DMD modal frequency. Based on the property of the exponential function, if the growth rate is positive, the DMD modes are unstable. Meanwhile, a negative growth rate shows the DMD modes are stable, as shown in Figs. 14(a) and (c).
As shown in Figs. 14(b) and (d), the DMD modes tend to occur in pairs. Mode 4 is a special shift mode and changes the distribution of the DMD modes from symmetric to antisymmetric. Similarly to the POD modal analysis, we assume the symmetrical mode represents the coherent structures of the flow field and the anti-symmetrical mode represents the small-scale structures.

DMD reconstruction
The flow field reconstruction using reduced-order DMD is acquired as where the U is the DMD modal matrix, and '' 0 '' is the transpose symbol. X t represents the complex plane of the DMD. Note that the t is an arbitrary variable, and thus the DMD reconstruction does not need the original flow field sequences and tedious matrix calculations. The t can be extended to exceed the sampling time range, and we can then make a DMD prediction for the future flow field for a limited time. DMD prediction is a fascinating topic that we will discuss further in a later study. Figure 15 shows the DMD reconstruction of the instantaneous vorticity flow field. The identification of the reconstruction of the coherent structures of the flow field is similar to that for the POD reconstruction in Fig. 9. This indicates that the linear assumption is reasonable in the cylindrical wake field. Unfortunately, the forced case reconstruction by DMD is unsatisfactory because the flapping jet is highly nonlinear.

DMD reconstruction error analysis
Similarly to in the POD error analysis, we also define the DMD reconstruction errors using the relative errors on the streamwise velocities.
where u DMD is the DMD reconstruction velocity and the mean spatial errors e t j À Á and mean errors e follow the definition in Eq. (14). Figure 16 depicts the reconstruction errors for 200 flow field snapshots in one second. For the unforced case, the distribution of the errors is mainly located far away from the circular cylinder. The errors are similar to the POD reconstruction errors, and the coherent structures are represented accurately. However, for the forced case, the errors are larger near the circular cylinder, especially in the flapping jet areas.
The mean errors for the unforced and the forced conditions are 8.19% and 6.06%, which are higher than the POD reconstruction errors. This may be because the POD flow field reconstruction is based on the fluctuating velocity and its modes are orthogonal, while DMD is based on a linear system of the entire flow field (Liu et al. 2021). Note that the mean error of the forces is bigger for the unforced condition. This may be because the flapping jet pushes the vortex-shedding structure far away from the cylindrical wake. Thus, the finite area of the ROI does not contain whole errors. Ma et al. (2015) were inspired by the DFT and used the FMD method to analyze a circular cylinder wake field. The FMD method was innovated from a characteristic frequency perspective. For example, the domain frequency is one of the most important properties of a flow field, and the FMD method uses it as a typical frequency. Fig. 15 The distribution of the instantaneous vorticity from DMD reconstruction 6.1 Introduction to the FMD algorithm

FMD analysis
The details of the DFT method are provided in Ma et al. (2015). In this study, we focus on the FMD method and apply it to a global flow field to compare it with the DMD and POD methods.
For the discrete data collected from PIV experiments, the Fourier decomposition is: where Note that F n is a sequence recording the velocity field of the ROI from PIV databases, and c k is a sequence of global spectral information. To obtain the phase information, using the Euler equation, Eq. (23) can be rewritten as where A k ¼ 2 c k j j represents the global amplitude spectrum, h k ¼ Àargc k refers to the global phase spectrum, and c k indicates the Fourier dynamic modes.
An overview of the FMD algorithm as applied to the circular cylinder wake flow is shown in Fig. 17. The first step is whole domain Fourier spectrum analysis from the original PIV data via a global Fourier analysis. The second step is the selection of the domain frequency as a characteristic frequency. All frequencies extracted from the flow field are normalized by the characteristic frequency. Finally, the data post-processing mean velocity, Fourier modes, and FMD reconstruction are extracted.
In this study, the meaning of f 0 is different from that in Ma et al. (2015). Two different domain frequencies are introduced in the unforced and the forced cases because there is no excitation frequency to influence the flow field. In our opinion, the FMD method using the global Fourier analysis contains a similarity to the full domain power spectrum analysis done by Chang et al. (2022) and Xu et al. (2022). However, the FMD method is more specific and theoretically based on the flow field analysis. The global power spectrum analysis of the circular cylinder wake flow field is shown in Fig. 18. It is observed that the streamwise velocity u and crossflow velocity v are contained in the power spectrum analysis. The domain frequency acquired from the POD method was calculated as f 0 ¼ StU 0 =D, and the values were found to be f 0 ¼ 23:07Hz for the unforced case and f 0 ¼ 25:40Hz for the forced case, which are consistent with the FMD and DMD analysis. Figure 19 depicts the amplitudes of the Fourier modes for the unforced and forced cases. For the unforced condition, there is a peak-to-peak distribution of the streamwise velocity that is symmetrical around the central line of the circular cylinder. For the forced case, the peaks are pushed down to the wake. The value of the amplitude is significantly decreased and consists of the POD amplitudes as shown in Fig. 6(c).
To reveal the physical meaning of the Fourier modes, the modal phase h k and the results for the Fourier modes c k are discussed below. Note that the components of the phase, cosh k ; sinh k are also discussed because they are closely connected to the Fourier modes. Figure 20 depicts the phases of the Fourier modes in the unforced and forced cases, and the distributions of the phases are symmetrical around the central line of the cylinder. The asymmetry or symmetry characteristics of the different velocity components are related to the asymmetric shedding process of the wake vortex. The information in the amplitudes decreases and the behavior of the reflux is pushed far from the cylinder, as shown in Fig. 20. Interestingly, it is observed that the phase of the forced case is developed compared to the unforced case. For the cosh k ; Àsinh k parts, there is a symmetrical perturbation in the nearby wake of the circular cylinder. This indicates that the high-frequency flapping jet disrupts the original phase. However, the main phase does not change. Thus, the flapping jet can only change the place where the vortex shedding is created but cannot change the basic characteristics of the flow field.
The real and imaginary parts of the Fourier modes c k are illustrated in Fig. 21. The mode based on the streamwise velocity is similar to the POD modes 1 and 2. This may be because both the POD and FMD methods subtract the mean flow and focus only on the fluctuating flow. We assume that the streamwise velocity modes represent the coherent structures of the flow field. The Fourier modes based on the vertical velocity represent the shedding frequency. The wavelength can be estimated by the spacing of adjacent wave peaks. It is observed that the wavelength of the unforced case is longer than that of the forced case. This indicates that the shedding frequency and amplitudes in the unforced case are higher than those in the forced case.

FMD reconstruction
The flow field can be reconstructed using the FMD method, according to the form of the DFT: Note that c 0 represents the average value of the sequence, similar to ''Mode 1'' of the DMD method. Figure 22 represents a distribution of four typical times in the instantaneous spanwise vorticity flow field from the FMD reconstruction. Different from the POD and DMD reconstruction, the unforced case is filtered by the domain frequency. Although the quasi-periodic motion of the vortex shedding can be seen, most of the details in the flow field are missing. For the forced case reconstruction, the total flow field tends to be the time-averaged field. The high-frequency areas like the shear layer and the flapping jet recognition are inaccurate and unstable.    Fig. 22 The distribution of the instantaneous vorticity from the FMD reconstruction Modal analysis methods are effective tools for reduced-order analysis, data-driven modal decomposition, and spatial-temporary analysis. However, different methods have their own characteristics and appropriate application scenarios, and we will focus on the characteristics of the various methods in this section. We will also discuss the practical limitations of the different methods in this experiment.

Comparison between POD and DMD
These two methods are mathematically identical because both are based on the ''SVD'' method, but the same mathematical essence does not necessarily imply the same physical laws. The classical POD method extracts only the spatial characteristics of the original fluctuating velocity and neglects the matrix V in SVD. The matrix V contains the temporal information on a POD orthogonal basic subspace. On the contrary, the standard DMD method considers the matrix V, and so is a spatial-temporal decomposition.
When using the POD method, we expect to obtain the corresponding flow field dynamics structure from modal analysis. Unfortunately, even a real flow as ''simple'' as a fluid passing a circular cylinder does not guarantee function mapping relationships with real physics at a high Reynolds number. The accumulation of the product by time coefficients and POD modes assumes that the maximum energy implies the strongest correlation. This is often unsatisfactory in the identification of fluctuating turbulent structures since these high-energy modes are likely due to the high randomness of the turbulence. In comparison, each DMD modal contribution is determined by both energy percentage and temporal bases, or in other words, each order of DMD mode has a unique frequency corresponding to it. While this is a convenience arising from the linear assumption, it also imposes the limitation that DMD cannot account for the high nonlinearity generated by the flapping jet.

Comparison between DMD and FMD
In general, the FMD method is more suitable than the DMD method for handling a quasi-period flow field. FMD is more efficiently computed than DMD based on the FFT algorithm. The FMD is also highly adaptable to the form of the original data since it has fewer limitations in terms of dimension. In other words, the discrete Fourier transform is a powerful tool. However, one issue it faces is that FMD cannot provide a linear continuity system like DMD, while its analysis is only based on the available data.

Comparison between POD and FMD
For the phase analysis of a flow field, the FMD and POD methods are similar. As the Kármán vortex shedding structures represent a quasi-period flow, FMD can be used to extract the characteristic frequency. Coherent structures based on the characteristic frequency f =f 0 can then be extracted. If the external excitation frequency is known, the FMD method shows unprecedented potential. However, for external excitation from a flapping jet with high frequency, its value is unknown. FMD, therefore, provides a less accurate reconstruction flow field than POD. If we only want to extract the coherent structures of the fluid field, the energy-based POD method is essentially the best choice.

Conclusions
In the present study, we have performed a comparison of the ability of the POD, DMD, and FMD algorithms to identify the coherent structures and small-scale structures in a flow field. The influences of data noise, nonlinear structures, and complex frequencies were taken into consideration. All three modal analysis methods were applied to an experiment involving an unforced circular cylinder flow field and a forced flow field. A circular cylinder wake with an embedded flapping jet actuator was investigated in a close-circle wind tunnel at Re ¼ 1:99 Â 10 4 . The 2D PIV system was used to acquire the original velocity field.
• Time-averaged analysis showed that the mean turbulent kinetic energy and Reynolds shear stress were significantly decreased when the flapping jet was applied. A pair of refluxes were located near the circular cylinder, and the forced case pushed the refluxes far away from the wake. • POD mode analysis showed the coherent structures of the first two modes. The Kármán vortex shedding structures were relative to the first two POD modes. For higher POD modes, it was observed that the symmetry was changed significantly. The instantaneous vorticity flow field was reconstructed using the first four POD modes, and the reconstruction errors were 4.49% for the forced case and 7.74% for the unforced case. The coherent structures and reconstruction accuracy were clearly shown. Although the physical meaning of the POD modes was unclear, this was still helpful in the identification of large-scale structures in the flow field. • DMD mode analysis was used to examine the flow stability using the growth rate and Ritz circle. In this study, the growth rate was negative and the DMD modes were distributed inside or on the unit circle, indicating the stability of the flow field development. The first DMD mode occupied most of the energy of the flow field and was referred to as the mean velocity U. The Kármán vortex shedding was associated with the first two modes, and the higher DMD modes represented the higher-order harmonics. Based on the assumption of linear estimation, DMD reconstruction was applied to the wake field. The errors associated with the reconstruction were 8.19% for the unforced case and 6.06% for the forced case, which were higher than the POD reconstruction errors. This is a natural limitation of the linear assumption. • FMD mode analysis was applied to provide global Fourier decomposition and the domain frequency of the flow field. The domain characteristics of the FMD modes were well defined with physical meanings. The DMD modes had a strong relationship with the FMD modes, which could be used for characteristic frequency recognition. The phases of the FMD modes showed that the coherent structures moved far away from the wake.
The modal analysis methods considered here will be a powerful tool for modern data-driven flow field analysis. Meanwhile, reduced-order modal analysis can solve the problems of memory crises and computing efficiency. We should fully understand the essence of each method and select different methods appropriately for different systems. In the future, the improvement of traditional methods and their application in multiple disciplines are likely to be highly valuable.