Research on the Flow-Induced Vibration of Cylindrical Structures Using Lagrangian-Based Dynamic Mode Decomposition

Abstract

An oscillating flow past a structure represents a complex, high-dimensional, and nonlinear flow phenomenon, which can lead to the failure of structures due to material fatigue or constraint relaxation. In order to better understand flow-induced vibration (FIV) and coupled flow fields, a numerical simulation of a two-degrees-of-freedom FIV in a cylinder was conducted. Based on the Lagrangian-based dynamic mode decomposition (L-DMD) method, the vorticity field and motion characteristics of a cylinder were decomposed, reconstructed, and predicted. A comparison was made to the traditional Eulerian-based dynamic mode decomposition (E-DMD) method. The research results show that the first-order mode in the stable phase represents the mean flow field, showcasing the slander tail vortex structure during the vortex-shedding period and the average displacement in the in-line direction. The second mode predominantly captures the crossflow displacement, with a frequency of approximately 0.43 Hz, closely matching the corresponding frequency observed in the CFD results. The higher dominant modes mainly capture outward-spreading, smaller-scale vortex structures with detail displacement characteristics. The motion of the cylinder in the in-line direction was accompanied by symmetric vortex structures, while the motion of the cylinder in the crossflow direction was associated with anti-symmetric vortex structures. Additionally, crossflow displacement will cause a symmetrical vortex structure that spreads laterally along the axis behind the cylinder. Finally, when compared with E-DMD, the L-DMD method demonstrates a notable advantage in analyzing the nonlinear characteristics of FIV.

1. Introduction

Alternating vortices may shed from either side of a structure when fluid flows past a specific geometry. These alternating vortices induce vibrations in the cylinder, which, in turn, modify the vortex-shedding pattern. This can lead to flow-induced vibration (FIV), including sustained structural resonance, and the failure of structures due to material fatigue and constraint relaxation [1]. For example, premature fatigue fractures in bridge cables and marine risers were induced due to flow-induced resonance [2,3]. Moreover, the reciprocating FIV of a foundation structure under water flow accelerated the loosening and loss of soil, resulting in local scour failure [4]. Therefore, the study of the FIV mechanism and coupled flow fields is valuable for predicting and controlling the scale of structural disasters [5,6,7,8,9]. However, FIV is a nonstationary flow process, and there are coherent flow structures between complex fluid movements involving complex nonlinear vorticity fields and difficulties in analyzing massively high-dimensional data [10,11,12]. Therefore, there is an immediate need to address the swift and efficient analysis of flow field data.

Previous studies have shown that methods based on flow field data have higher efficiency than those based on control equations in obtaining flow field mode information [13,14]. Mode decomposition technology is a method based on the above ideas, which can directly extract dynamic information from flow field data, reduce the dimensionality of dynamic models, obtain low-dimensional descriptions of flow fields [15,16], and, thus, achieve the prediction and control of a given flow field. The traditional proper orthogonal decomposition method can be used to obtain the flow field structure, but it is sensitive to the initial conditions and constrained in its ability to accurately capture time-dependent properties due to its reliance on low-rand modeling techniques [16,17]. In order to solve the problem of spatiotemporal coupling modeling, Schmid proposed the dynamic mode decomposition (DMD) method [18]. Compared to traditional intrinsic orthogonal decomposition methods, the modes obtained by the DMD method have a single growth rate and frequency [19,20].

Numerous scholars have substantiated the efficacy of the DMD method in interpreting coherent structures and dynamics in flow fields. Wang et al. [21] employed the methodologies of proper orthogonal decomposition and DMD to investigate the wake flow generated by a propeller in a conduit. Their findings revealed that the DMD method effectively captures the complex flow characteristics, coupling multiple frequency vortex structures generated by the propeller. Zhang et al. [22] employed the DMD method to study the flow conditions during the lock-in regime of forced oscillation and fixed, square cylinders and explored the potential connection between fluid forces and coherent modes. Janocha et al. [23] investigated the coherent structures of turbulence surrounding a cylindrical object under uniform flow and identified the shedding frequencies and harmonic characteristics of vortices at different Reynolds numbers. Kou et al. [24] employed the DMD method to investigate the VIV characteristics at low Reynolds numbers and found that the shedding mode of the wake vortex exits in the subcritical flow regime and gradually diminishes as the Reynolds number decreases. From the above studies, research on FIV mainly focuses on the evolution of flow field characteristics, primarily investigating fixed or forced oscillating cylinders, without fully considering the bidirectional coupling between the flow field and structural movement. Paneer et al. [25] pointed out that the energy would transfer between a moving body and a fluid. Hence, it is essential to build the connection between the structural response and flow evolution. Mojgani et al. [26] pointed out that, by using the Lagrangian framework (computational mesh movement), as opposed to the Eulerian framework (computational mesh stationary), both flow field data and parameter/time-varying networks can be simultaneously considered. Cheung et al. [27] investigated compressible gas dynamics in complex multi-material environments, where material contact surfaces move with the mesh in the Lagrangian framework, enabling a more direct determination of material penetration distances. Lu et al. [28] compared the DMD methods in traditional Eulerian and Lagrangian frameworks, revealing that the Lagrangian-based DMD approach can incorporate more information and is more suitable for studying advection-dominated problems while considering characteristic evolution. Based on the Lagrangian-based dynamic mode decomposition (L-DMD) method, the influence of mesh deformation on flow field data can be considered, which is better suited to solve complex fluid problems [28].

In view of the above study, this study adopts the L-DMD method to study a two-degrees-of-freedom cylinder during a lock-in regime. To the best of our knowledge, this is the first piece of research that simultaneously carries out a study on the FIV characteristics of structural motion and the vortex street structure interaction based on the L-DMD method. The flow field snapshot information was obtained using the computational fluid dynamics (CFD) technique. The L-DMD method was utilized to reduce the dimensionality of the vorticity data and capture the modal structures of each mode. Additionally, the flow field and structural motion characteristics were analyzed, reconstructed, and predicted for the first 10 modes. The interplay between the displacements in the in-line and crossflow directions of a cylinder and the resulting vortex structures was investigated. Then, the accuracy of the L-DMD method was validated by comparing it with the Eulerian-based dynamic mode decomposition (E-DMD) method.

2. L-DMD Method

The goal of DMD is to decompose a dynamic system or process into several basic modes. DMD utilizes a few dominant modes to swiftly and precisely describe and predict the intricate evolution process in complex systems. The specific steps are as follows:

A sequence of flow field parameters (e.g., velocity, pressure, vorticity) in m grid nodes at moment i can be represented as a column vector ${\mathbf{x}}_i\in R^{m\times 1}$ in the spatial domain. There is a total of n moments. The matrix X of all flow field sequences can be expressed as $\mathbf{X}=({\mathbf{x}}_1,{\mathbf{x}}_2,\dots ,{\mathbf{x}}_n)\in R^{m\times n}$ at n time points, where m represents the size of the spatial domain, which is usually much larger than n.

Based on the flow field snapshot (i.e., the vorticity or pressure parameters of the flow field at a certain discrete time) data X, two submatrices are constructed:

$$X_1=[{\mathbf{x}}_1 {\mathbf{x}}_2\cdots {\mathbf{x}}_{n-1}],$$

(1)

$$X_2=[{\mathbf{x}}_2 {\mathbf{x}}_3\cdots {\mathbf{x}}_n].$$

(2)

Based on the assumptions of Koopman operator theory [29], the snapshot time interval of any two adjacent flow fields is very small, and there is a linear transformation relationship between the flow fields and at two adjacent times; thus, it can be obtained that

$$X_2=A{X}_1,$$

(3)

where A is a high-dimensional matrix that encapsulates the temporal evolution information of the system. The matrices can be approximately solved using singular value decomposition:

$$X_1=U\Sigma V^{*},$$

(4)

where U and V are unitary matrices that satisfy conditions $U^{*}U=\mathbf{I}$ and ${\mathbf{V}}^{*}\mathbf{V}=\mathbf{I}$, respectively, presenting the left singular matrix and right singular matrix; Σ is a singular value matrix.

Matrix A can be represented as Equation (5) using a matrix ${\mathbf{X}}_1$ to obtain a left singular matrix:

$$\mathbf{A} \approx \mathbf{U}\tilde{\mathbf{A}}\mathbf{U}.$$

(5)

Matrix $\tilde{\mathbf{A}}$ should be determined by solving the minimization of the Frobenius norm:

$$\mathrm{Min}\left|\right|{\mathbf{X}}_2 - {\mathbf{AX}}_1\left|\right|2_F= \left|\right|{\mathbf{X}}_2 - \mathbf{U}\tilde{\mathbf{A}}\mathbf{\Sigma }\mathbf{V}*\left|\right|2_F.$$

(6)

According to Equations (5) and (6), the following can be obtained:

$$\tilde{\mathbf{A}} = {\mathbf{U}}^{\mathrm{T}}{\mathbf{X}}_2\mathbf{V}{\mathbf{\Sigma }}^{-1}.$$

(7)

Note that matrix $\tilde{\mathbf{A}}$ is a low-dimensional approximation matrix of A. Hence, the eigenvalues and eigenvectors of the matrix $\tilde{\mathbf{A}}$ can be obtained through eigenvalue decomposition of the approximation matrix.

$$\tilde{\mathbf{A}}\mathbf{W} = \mathbf{W}\mathbf{\Lambda },$$

(8)

$$\Lambda =\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left({\lambda }_1,{\lambda }_2,\cdots {\lambda }_k\right)$$

(9)

where W is the matrix composed of the eigenvectors of $\tilde{\mathbf{A}}$, $\Lambda$ is a diagonal matrix for which the diagonal elements are composed of the eigenvalues ${\lambda }_k$, and k is the chosen mode number. The mode can be obtained from

$$\Phi =\mathbf{U}W=\left[{\Phi }_1,{\Phi }_2,\cdots {\Phi }_k\right]$$

(10)

where $\mathbf{\Phi }$ characterizes the mode characteristics, and ${\mathbf{\Phi }}_k$ is the column vector, which represents the DMD decomposition of mode k.

Given the definition of mode amplitude as b, b_k denotes the amplitude of mode k, signifying the level of contribution of mode k to x₁.

$$\mathbf{b}={\Phi }^{-1}{\mathbf{x}}_1={\left[b_1,b_2,\cdots b_k\right]}^T$$

(11)

Finally, the reconstruction and prediction of flow field data are carried out to achieve the equation for reconstructing and predicting the flow field at any given moment.

$${\mathbf{x}}_i=\Phi {\Lambda }^{i-1}b,i=1,2,\cdot \cdot \cdot ,n$$

(12)

The matrix form of Equation (12) can be alternatively shown as Equation (13). This further proves that the Vandermonde matrix ${\Lambda }^{i-1}$ can be used to determine the time evolution process of dynamic modes. The matrix ${\Lambda }^{i-1}$ encompasses various modes, the initial amplitude of the object, and the number n of the time frequencies and eigenvalues of the growth/decay rate included in $\tilde{\mathbf{A}}$.

$$[x_1 x_2\cdots x_{\mathrm{n}}]\approx [{\Phi }_1 {\Phi }_2\cdots {\Phi }_k]\cdot \left[\begin{array}{cccc}{b}_1& & & \\ & b_2& & \\ & & \ddots & \\ & & & b_k\end{array}\right]\cdot \left[\begin{array}{cccc}1& {\lambda }_1& \cdots & {\lambda }_1^{n-1}\\ 1& {\lambda }_2& \cdots & {\lambda }_2^{n-1}\\ ⋮& ⋮& \ddots & ⋮\\ 1& {\lambda }_k& \cdots & {\lambda }_k^{n-1}\end{array}\right]$$

(13)

However, the structure will experience nonlinear vibrations due to fluid forces when a fluid is passing. Therefore, in addition to considering the evolution of flow field parameters, the spatial position of mesh nodes should also be considered over time [30]. The traditional DMD method based on the Euler coordinate system focuses on the analysis of flow field values while neglecting the investigation of mesh node displacements caused by unidirectional fluid coupling. The L-DMD method constructs two data matrices to describe the displacement information, P, of mesh nodes and the flow field parameter, X, respectively. The matrices P and X are combined for singular value decomposition to obtain information, such as vibration modes, frequencies, and eigenvectors, better capturing the dynamic characteristics of the column and analyzing the interaction process between a vortex street and structure. The calculation process for this method is as follows: The position matrix, P, of all mesh nodes in the time domain can be expressed as $P=\left(p_1,p_2,\dots ,p_n\right)\in R^{m\times n}$. This is combined with the corresponding flow field parameter matrix to form a snapshot matrix, X, that includes node spatial position information and flow field parameters.

$$X=\left[\begin{array}{cccc}{p}_1& p_2& \cdots & p_n\\ x_1& x_2& \cdots & x_n\end{array}\right]$$

(14)

The submatrices from Equations (15) and (16) were constructed and then brought into Equation (3) for subsequent calculations.

$$X_1=\left[\begin{array}{cccc}{p}_1& p_2& \cdots & p_{n-1}\\ x_1& x_2& \cdots & x_{n-1}\end{array}\right]$$

(15)

$$X_2=\left[\begin{array}{cccc}{p}_{x_2}& p_{x_3}& \cdots & p_n\\ x_2& x_3& \cdots & x_n\end{array}\right]$$

(16)

3. Numerical Models

A sketch of the problem setup and boundary conditions is shown in Figure 1a. A 2D dual degree of freedom reference cylindrical structure benchmark was adopted to provide some of the components used in actual engineering, such as cables and piers. The characteristic length of the cylinder diameter is represented by D. In order to ensure an ample calculation area and minimize boundary interference, a computational domain of 20D × 45D was chosen for the model. The cylindrical surface is a nonslip boundary condition with slip boundaries at the upper and lower sides. The Dirichlet boundary velocity inlet and Neumann outlet were set as the inlet condition and outlet condition, respectively. The model parameters are as follows: D = 0.0381 m, f_n = 0.4 Hz, k = 18.72 N/m, mass ratio m* = 2.4, and mass damping ratio $m^{*}\zeta =0.0013$. All the physical parameters employed in this study remain consistent with those employed in previous studies [31,32,33]. In order to better elucidate the interaction mechanisms, the lock-in regime was selected for analysis, with a Reynolds number of 6,000, and a Reynolds-averaged Navier–Stokes (RANS) approach with the shear–stress transport (SST) of k-ω shear–stress was employed. A sensitivity validation of the mesh size for the model with a Reynolds number of 6,000 was conducted, with the results shown in

Table 1. Verification of the independence of the computational mesh.

Test	Wall Grid	Total Grid	x/D	y/D
1	60	27,600	0.4592	0.6745
2	100	39,200	0.4714	0.6913
3	120	45,400	0.4675	0.6822
4	160	62,600	0.4640	0.6803

. The displacement rates obtained by mesh 1 and mesh 2 exhibit significant discrepancies compared to mesh 4 (very fine). However, when the mesh size reaches that of mesh 3 (fine), the errors in x/D and y/D compared to mesh 4 (very fine) fall within 0.75%. Therefore, when considering both computational accuracy and efficiency, mesh 3 was selected for subsequent calculations. Figure 1b shows the mesh-partitioning scheme, employing an “O-mesh” structure. The cylinder surface was discretized into 120 nodes, with 80 layers of meshes in the vicinity of the cylinder. The surface mesh height of the first layer was set to 1 × 10⁻⁵ to ensure y⁺ ~ 1. To ensure that the Courant–Friedrichs–Lewy value remains sufficiently small to meet the computational requirements of the model, the time step was set to 0.0002 s. Reference points A–F were deliberately chosen to serve as key points for the subsequent analysis of vorticity variations, allowing for a comprehensive investigation into the complex flow patterns and a detailed understanding of the underlying fluid dynamics. In order to compare the model accuracies, the variations in crossflow amplitude rate cylinder response (y/D) over different reduced velocities were computed and validated against some of the research results of 1DOF VIV. Similar comparisons are also presented in the research [34], as shown in Figure 2. When U* = 2, the motion of the cylinder exhibited stable initial bifurcation. As U* approached 6, an upper branch emerged, reaching a peak response, followed by a rapid decline until around U* = 8, where it displayed a relatively gradual change before transitioning to a lower branch. The simulation results closely align with previous research.

In order to simulate the unsteady flow field, a moving mesh, with a total calculation time set at 60 s, was used. As shown in Figure 3a, based on the variation characteristics of displacement, the development process of the vortex can be divided into three stages. The initial stage of the process is characterized by the onset of boundary layer separation, which occurs within the first 8 s. This separation gives rise to a symmetrical vortex formation in the wake region behind the cylinder. Over time, this vortex gradually evolves downstream, eventually developing into a slender wake structure. The subsequent stage, known as the development stage, spans from approximately 8 s to 20 s. During this stage, the slender tail vortex undergoes instability and initiates oscillations. As the interaction between the wake vortices and the structure persists, the displacement of the cylinder progressively increases accordingly. Subsequently, the system enters the final stage, characterized by the stable periodic shedding of vortices, where the crossflow (x-direction) and in-line (y-direction) undergo periodic fluctuations. The in-line displacement rate exhibits fluctuating variations in the vicinity of 0.4645. As shown in Figure 3b, the corresponding stable stage spectrum is obtained through the Fourier transform, and the main frequency of the in-line displacement is about 0.8667 Hz, whereas the main frequency of the crossflow displacement is about 0.4333 Hz.

After conducting a sensitivity analysis of the interval comparisons, for the development stage, a total of 140 flow field snapshot data from 8 to 15 s were selected. For the stable stage, data from 25 to 40 s, totaling 300 snapshots, were chosen. Subsequently, L-DMD analysis was conducted separately for each stage to examine the respective characteristics. The calculation process is shown in Figure 4. Initially, a series of snapshots were obtained through CFD simulation. A sample matrix was then constructed based on the flow field data and nodal displacement data from the snapshots. The matrix was utilized for LDMD computation, followed by mode decomposition, reconstruction, and prediction to analyze the interaction between the flow field and the structure through the 10 prominent modes.

4. Analysis Based on L-DMD

4.1. Development Stage

The development stage mode eigenvalue circle is shown in Figure 5. During the development stage, it can be observed that the majority of the eigenvalues lie outside the unit circle and are in close proximity to it. This indicates that the vast majority of modes are in a state of slow divergence. The displacement amplification rate of Mode 1 is less than 0, and the frequency is 0, representing the average flow field. In contrast, the amplification rates of Modes 2 and 3 are greater than 0, indicating that the modal energy is continuously increasing [37]. As their frequencies are 0, their contributions to the flow field remain consistent over time. The last few modes, specifically the fourth and fifth, exhibit amplification rates greater than 0, signifying increasing modal intensity and frequencies above 0; this suggests that the influence of these modes on the flow field changes over time. The nonlinear development stage of the flow field is characterized by the divergence and convergence doping of various modes, resulting in a complex evolution process.

The L-DMD decomposition modes at 11 s and 13 s in the development stage are shown in Figure 6. The vorticity modes obtained through L-DMD exhibit a symmetrical or anti-symmetrical distribution in relation to the centerline with a level of noise. The anti-symmetric distributions observed in the L-DMD analysis are indicative of the presence of anti-symmetric vortex shedding modes, while the symmetric distributions align with the manifestation of the symmetric shedding process [38,39]. The first three modes correspond to a frequency of 0, which is approximate to the average vorticity field, and the first mode remains basically unchanged, exhibiting a static slender tail vortex structure. As vortices interact and merge, the size and quantity of vortices undergo continuous changes. The dominant modes exhibit multiscale features with the simultaneous presence of wake vortices at different scales and ongoing intensification.

In Figure 7 and Figure 8, the 11 s and 13 s flow fields reconstructed using 10 modes during the development stage are presented, respectively. It can be observed that the vortices in the development stage interfere and merge with each other, and the oscillation amplitude of the slender tail vortex increases continuously, resulting in a gradual decrease in the ability to capture the characteristics of the tail vortex.

4.2. Stable Stage

Figure 9 reveals that the majority of characteristic values during the stable stage are concentrated on or within the characteristic circle. This indicates that the developed modes predominantly exhibit periodic cyclic behavior with limited stability. A smaller number of modes are situated in the interior, distant from the unit circle, corresponding to rapid decay and convergence modes. During the development stage, the growth rate of the first five modes approximates zero, indicating that the modal energy remains essentially constant. Additionally, the frequencies of the modes exhibit a regular distribution pattern, suggesting periodic dynamic behavior at this stage.

As shown in Figure 10, mode decomposition was carried out at 30 s and 40 s. The corresponding frequency of the first dominant mode is 0, which is approximate to the average vorticity field. The vorticity field exhibits a symmetrical distribution, primarily characterized by the in-line displacement of the cylinder. Additionally, the shear layer of the flow shows significant separation. Furthermore, the second dominant mode generates a substantial alternating wake vortex structure in the crossflow direction. This structure, represented by a symmetrically distributed vorticity field, indicates the primary mode of vortex shedding, accompanied by a crossflow displacement. The third and fourth dominant modes of the cylinders capture the detailed characteristics of the wake vortex structure’s outward normal diffusion. The subsequent three modes represent the typical higher-order small-scale disturbances in the dynamic process of large-scale vortex shedding, illustrating the transverse expansion of the wake flow. The strength of the wake is basically stable at different times in the stable stage, which is mainly characterized by a change in the spatial position. The wake falls off periodically on the cylinder and moves downstream. The results of the first tenth of the dominant mode are reconstructed, and the predicted vorticity fields are shown in Figure 11. It can be found that the L-DMD method can accurately capture the low-frequency large-scale vortices downstream of the cylinder. However, due to the more complex flow changes near the cylinder and the high nonlinearity, there is a certain error in capturing the high-frequency small-scale vortices in the nearby unstable region. As a result, the reconstructed vorticity field near the cylinder has a redundant, small-scale vortex structure with outward diffusion.

Figure 12 illustrates the vorticity value constructed with 10 mode variations at various reference points. Utilizing the L-DMD method and extracting information from the first 10 modes, the periodic vortex shedding characteristics are captured, and the vorticity values are reconstructed accordingly. The reconstructed vorticity values demonstrate the better reconstruction and prediction of the actual vorticity variations. It is worth noting that minor high-frequency oscillations occurred at local positions in the LDMD results, possibly due to the reconstruction using only the first 10 dominant modes while overlooking the finer detailed features.

Figure 13 shows the change curve of the mode coefficients of each mode. It can be found that there is an obvious phase difference for the mode coefficients of each mode, which indicates that the vortex structure movement corresponding to different frequencies in the wake flow is out of synchronization. Since the frequency of mode 1 is 0, regardless of consideration, the other modes show simple harmonic vibration characteristics, and the flow field is affected periodically. It is evident that the mode coefficient of Mode 2 has a frequency of 0.43 Hz, which closely aligns with the frequency of the crossflow displacement. The frequency of Mode 3 is approximately 0.87 Hz, which closely matches the frequency of variation in the in-line displacement of the cylinder. Additionally, the frequencies of the mode coefficients for Modes 4 and 5 are in close proximity to their respective multiplication frequencies.

Figure 14 presents the cylinder’s time–history displacement rate curve, revealing that the cylinder motion’s periodic and stable characteristics are captured. The reconstructed motion exhibits consistent periodicity and is more accurate in reproducing the main crossflow displacement response.

The motion trajectory curve of the cylinder is shown in Figure 15. The first dominant mode captures the main in-line displacement, displaying a motion trajectory that closely aligns with a horizontal straight. Moreover, the first-order mode motion trajectory rate extracted by L-DMD exhibits variations centered around 0.4667, which is consistent with the fluctuating pattern observed in the CFD results of the behavior of the cylinder near 0.4645. The second mode predominantly captures the main fluctuation of crossflow displacement, with a frequency of approximately 0.43 Hz, closely matching the crossflow displacement frequency observed in the CFD results of the cylinder. This emphasizes the relationship between the second mode and the primarily captured crossflow displacement. Conversely, the third mode offers a detailed characterization of the in-line displacements exhibited by the cylinder, with a frequency of around 0.87 Hz, resembling the frequency of variation in the in-line displacement. Notably, this mode contributes to the observed fluctuating in-line displacement. By integrating the L-DMD modes shown in Figure 10, the crossflow displacement of the cylinder is accompanied by axial (y = 0) vortex structures extending a certain distance downstream. As the distance increases, these vortex structures diffuse to either side, forming paired anti-symmetric vortex structures. Meanwhile, the in-line displacement is accompanied by paired symmetric vortex structures. It is evident that a close connection exists between the structural response and flow field in FIV. Moreover, the initial few modes effectively capture the predominant displacement features, and subsequent modes further extract more intricate and detailed displacement characteristics. It can be found that the motion trajectory of the cylinder restored by the first 10 dominant modes based on the L-DMD method conforms to an “8” character. The accuracy of the flow field error function was analyzed according to the loss function defined by Jovanović [40]:

$$loss=100{\displaystyle \frac{{‖X-\Phi \Lambda b‖}_F}{{‖X‖}_F}}$$

(17)

The overall loss of L-DMD is shown in Figure 16. When combined with the L-DMD loss curve, it can be seen that the loss of the reconstructed and predicted error function values in the first few modes decreases significantly with the increase in the number of modes. The vorticity field and motion are reconstructed through the first 10 modes, and L-DMD loss is reduced by about 73% compared with the use of only the first mode. However, since the higher-order mode mainly captures the small and fast-changing details of the transient characteristics in the system [41], the effect of increasing the number of modes to reduce the loss is not significant.

For the moving mesh, the flow field and structural motion at each moment can also be interpolated into the stationary Eulerian mesh [28], and DMD analysis can be carried out under the improved Eulerian framework, which is also used by Lian [42] to investigate the VIV of wind turbine airfoils. Specifically, the location information of grid nodes at each time step is interpolated to generate the corresponding sequence matrix of the flow field. Subsequently, the constructed flow field snapshots are also interpolated accordingly. The results of the E-DMD analysis are shown in Figure 17. The E-DMD analysis method generates redundant flow field information when reducing the vorticity field near the cylinder. The loss obtained is averaged from each snapshot to obtain the mean loss, and the results are presented in

Table 2. L-DMD/E-DMD average loss.

Number of Modes	5 Modes	10 Modes	15 Modes	30 Modes	50 Modes
Error of L-DMD	0.1998	0.1651	0.1648	0.1603	0.1606
Error of E-DMD	1.09	1.0663	1.0750	1.0357	0.9821

. The overall error for the E-DMD method is significantly higher compared to the L-DMD method. Additionally, increasing the number of reconstructed modes can also reduce the E-DMD error. When the fluid bypasses the cylinder, the shear layer separates, and vortices are alternately formed on the two sides of the cylinder. This causes the cylinder to undergo periodic motion, and there is an interaction between the cylinder and the vortices, resulting in a strong nonlinear effect on the flow field both temporally and spatially near the cylinder. On the other hand, interpolating the flow field information from the moving grid to the fixed grid leads to the loss of motion information of the cylinder. As a result, it is difficult to capture the dynamic characteristics of the flow field near the cylinder with the E-DMD analysis method, and the capture of the displacement characteristics of the node is also lost.

5. Conclusions

In this study, the L-DMD method was used to analyze the flow field and structural motion of FIV under a lock-in regime, considering the interaction between cylinder motion and vortex street structure and analyzing the capture and reconstruction effects of L-DMD characteristics in the development and stable stages. The following conclusions were obtained:

(1) The L-DMD method considers cylinder motion and vortex street–structure interaction, which can better characterize the linear characteristics of the flow field evolution process. Through a comprehensive analysis of the vorticity field and motion of structure using L-DMD, the dominant flow modes were identified, along with their respective frequencies. The reconstructed vorticity field near the cylinder also reveals an excessive prevalence of outward diffusion vortex structures. There was a marginal discrepancy in accurately capturing high-frequency, small-scale vortices. In future work, this method can be used to analyze FIV over a longer sampled time duration to achieve more accurate results. Additionally, the method can be further extended to 3D models and combined with particle image velocimetry technology to conduct predictive maintenance for practical engineering applications.

(2) The first dominant mode of the FIV of the cylinder extracted using L-DMD is mainly represented by the dominant stationary slender wake structure in the evolution process of the vortex; the main in-line displacement rate of the cylinder is captured, and the motion path is nearly horizontal. Moreover, the mode exhibits variations centered around 0.4667, which is close to the fluctuating pattern observed in the CFD results of the cylinder near 0.4645. The second mode predominantly captures crossflow displacement, with a frequency of approximately 0.43 Hz, closely matching the corresponding frequency observed in the CFD results. Furthermore, the third mode provides a periodic change feature from the in-line displacements, with a frequency of approximately 0.87 Hz, which is near the frequency of variation in the in-line displacement. The corresponding frequency of the higher dominant modes is the high dominant frequency multiplier, mainly capturing the vortex structure at a smaller scale and obtaining the displacement characteristics of the cylinder in more detail. Importantly, this method can accurately capture the prevailing crossflow displacements.

(3) For the development phase, more intensive data snapshots are required to capture more detailed flow field characteristics. During FIV stabilization, the flow field distribution is relatively regular, the number of modes required for the reconstruction of the flow field is fewer, and the reconstruction effect is better. The L-DMD method can capture the relationship between structural response and flow field changes during the flow-induced vibration process. The results of each mode indicate that the in-line displacement is accompanied by symmetrical vortex structures in the wake, while crossflow displacement is associated with anti-symmetric vortex structures, forming vortices that gradually diffuse to both sides along the axis of symmetry behind the cylinder. When compared with the E-DMD method, the L-DMD method has a better ability to capture the nonlinear characteristics of the relatively strong spatio-temporal flow field near the cylinder. Additionally, during the stable stage, L-DMD exhibits superior accuracy in reproducing the vorticity field near the cylinder, with minimal deviations.