Flow field characteristics analysis of vortex-induced vibration of bridge tower based on dynamic mode decomposition method

Abstract

As a high-rise structure, the tower of a suspension bridge has low stiffness due to the lack of cable system constraints when it is self-supporting. Wind-induced vibration is one of the key factors in design and construction. In this paper, the wind tunnel test of the bridge tower aeroelastic model is carried out in two different flow fields, and the wind vibration response under a self-supporting state is systematically studied. Meanwhile, numerical simulation of the unsteady flow field around the tower section is carried out, and the dynamic modal decomposition (DMD) method is used to study the modal frequency of the flow field, the surface pressure of the section, flow field reconstruction error, and the stability of the flow field around tower section is discussed. The results show that in the self-supporting state, the displacement of the tower top presents a quadratic curve relationship with the wind speed, and no galloping phenomenon occurs under the test wind speed. However, in the uniform flow, when the wind attack angle is 90°, the bridge tower has an obvious vortex-induced vibration (VIV) phenomenon. After decomposing the flow field by the DMD method, it is found that there is strong aerodynamic interference between the double rectangular column sections, the first seven orders of modal energy play a major role in the flow field, and when the main modal energy accounts for more than 90%, the reconstruction can realize the accurate restoration of the information such as the surface pressure. For the complex flow field structure, it shows that the method can more accurately identify the coherent structure and background part in the unsteady flow field. This paper provides an idea for future research on the characteristics of the flow field around complex sections and helps to further improve the understanding of the VIV mechanism.

Keywords

bridge tower vortex-induced vibration dynamic modal decomposition self-supporting state unsteady flow

Introduction

The tower is an important part of the structural system of super-large span bridges. For suspension bridges, with the increasing span of the bridge, the height of the tower is also increasing, and its damping and stiffness will continue to decrease. Meanwhile, the tower cross-section generally uses bluff section forms, it is easier to produce the airflow separation phenomenon, vortex shedding and there will be mutual interference between the upper and lower tower columns when the air passes through (Belloli et al., 2011). The wind-induced vibration problem of bridge towers may become one of the controlling factors in design and construction, especially when the bridge is in the construction stage and the towers are self-supporting with less stiffness due to the lack of cables restraint (known as Free-Standing Tower). At this time, the bridge tower as a slender high-rise structure, and the effect on the wind will be more obvious, so the wind resistance of the bridge tower will become one of the important factors for the selection of the wind resistance design of the whole bridge (Larose et al., 1998). The wind-induced vibration phenomenon of the bridge tower in the self-supporting state is mainly manifested as VIV, buffeting, and galloping. Of which VIV is a self-limiting vibration, generally will not cause catastrophic damage to the structure like flutter, but the longtime structure VIV will not only cause fatigue damage to the bridge tower structure but also affect the safety of the construction personnel and construction machinery (Pan et al., 2017; Son and Lee, 2011). Therefore, for the bridge tower in the construction stage of the occurrence of VIV phenomenon cannot be ignored. It is well known that for slender structures, aerodynamic instability is easily generated, especially in the cross-wind direction, causing the structure to vibrate substantially and causing catastrophic damage due to the aerodynamic instability of the structure (Carmo et al., 2011; Chern et al., 2014; Kim et al., 2009; Okajima, 1982). Larose et al. (1993) showed that VIV of bridge towers may sometimes dominate the aerodynamic behavior of bridge towers, stating that the vibrations of bridge towers will start at a critical wind speed and continue for a while and then disappear at a certain wind speed. Tanaka et al. (2012) conducted a series of wind tunnel tests to determine aerodynamic forces and wind pressures acting on square-plan models with various configurations which can give people a comprehensive understanding of the aerodynamic characteristics of tall buildings in various configurations. Hu et al. (2015a, 2015b) studied the air stability experiment of a slender square-section cylinder under a series of angles through aeroelastic test and pressure measurement, studied the flow field distribution characteristics around the bridge tower using large eddy simulation, and further revealed the evolution of the surrounding flow field in combination with flow visualization technology. Li et al. (2017) conducted a related study on the generation mechanism of wind-induced vibration of suspenders of suspension bridges in the wake of bridge towers and points out that the vortex shedding in the wake area of the bridge tower has a great influence on the aerodynamic force of the nearby boom. Bhatt and Alam (2018) investigated the flow field characteristics at Reynolds numbers of Re = 100 and 200 for square columns with different spacing ratios (L/D of 2 to 6), revealing that the dependence of the Strouhal number St on velocity distinguishes different branches better than the dependence of the vibration amplitude on velocity while pointing out that for wake cylinders, the gap flow has a significant effect on the vibration response, but the study did not address the essential modalities of the flow field around the whole cylinder. Fang et al. (2019) compared different copula models and selected the optimal model to simulate the correlation of wind and wave parameters, and studied the extreme response of a sea-crossing bridge tower under correlated wind and waves. Ma et al. (2019) conducted a wind tunnel test on a free-standing single-column pylon with a chamfered square cross section by using the segmental model and the full-bridge aeroelastic model. Fang et al. (2020) studied the vibration of a four-column high bridge tower by wind tunnel tests and numerical simulations and pointed out that the displacement of the tower top increased with the increase of wind speed, and different cross sections of the tower showed different vortex shedding phenomena. Hu et al. (2021) explores the dynamic characteristics and the nonlinear wind-induced buffeting response of the bridge based on a six-tower cable-stayed bridge. However, most of the above studies only focus on the vibration response of such a cross-section and the surrounding flow field distribution of the bridge tower, but seldom involve the research related to the whole surrounding flow field structure of the cross-section and its essential modalities.

In fluid mechanics research, with the improvement of computer performance and the development of fast algorithms that make the study and analysis of flow fields easier and faster, the analysis method of decomposing complex flow field structures into the modal structure is increasingly applied to the problems of flow field analysis and stability judgment. The algorithm based on flow field data is more efficient than based on control equation (Edwards et al., 1994). Currently, the construction of a Reduced-Order Model (ROM) for unsteady flow fields based on wind tunnel tests or computational samples is an important research tool (Lieu and Lesoinne, 2004). The establishment of a reduced-order model enables to obtain of the main characteristics of complex hydrodynamic phenomena, while further system mechanism analysis and development of active/passive flow control can be achieved. ROM based on flow field feature extraction methods, such as the proper orthogonal decomposition (POD) (Noack et al., 2003; Rowley, 2005) and the dynamic mode decomposition (DMD) (Rowley et al., 2009; Schmid, 2010), have been applied to the flow control of bluff bodies. Dynamic mode decomposition (DMD) is a mathematical tool for low-dimensional system decomposition developed basis on global stability Koopman analysis in recent years. Compared with POD method. Mariappan et al. (2014) found that the flow structure obtained by DMD method had certain advantages in describing the flow field in frequency domain and extracting the main unstable modes in the flow when they studied the dynamic stall phenomena of 2D and 3D wings. Wan et al. (2015) simulated the vortex flow development process of transverse jet, and the results show that the neutral stable mode and dominant frequency obtained by DMD method are consistent with the LES calculation results, while the higher-order mode obtained by POD method contains a variety of frequency components.

The essence of the flow field feature extraction method is to find a set of low-dimensional subspaces (i.e., flow modes or coherent structures) and represent the high-dimensional, complex unsteady flow field as a superposition of these subspaces on a low-dimensional coordinate system, thus describing the flow field evolution in the low-dimensional space. This method requires high-dimensional, large-scale flow field data as samples, which can visualize the evolution of unsteady flow with time and space. Therefore, it is of great importance for the analysis of the mechanism of unsteady flow fields.

Based on the above research profile, in order to ensure the wind safety of the bridge tower in the construction self-supporting stage. This paper especially carried out a more systematic model wind tunnel test on the wind resistance of the bridge tower, studied the change law of buffeting and VIV response with wind speed under the action of different wind attack angles. Numerical simulation is used to study the evolution characteristics of the flow field around the bridge tower when VIV occurs. The DMD method is used to analyze the turbulent structure of the surrounding flow field in different modes, and further reveal the vibration mechanism of the typical double rectangular section of the bridge tower, which provides a new idea for the subsequent vibration analysis of this typical bluff body section structure.

Research object

This study is based on the engineering background of a suspension bridge across the Yangtze River in China, which adopts a double-tower, double-cable suspension bridge with a span arrangement of 340 + 1038 + 305 m and a total length of 1683 m, with a tower height of 165 m. The main girder is a streamlined steel box girder, with a beam height of 3 m and a beam width of 39.6 m. The bridge tower is made of reinforced concrete structure. The general layout of the bridge is shown in Figure 1(a), and the tower structure is shown in Figure 1(b).

Figure 1.

Research object: (a) general layout of the long span suspension bridge. (b) Overall dimensions of the bridge tower.

The bridge is located in the Yangtze River valley and the surrounding ground is relatively open, thus making it likely to be subjected to wind loads from different directions during the construction process. Relevant wind tunnel test results show that the vibration response amplitude of long-span bridges under the action of oblique wind may exceed the response value under the action of normal wind with the same wind speed (Hiroshi et al., 1982). Considering that the main girders of the bridge have not yet been combined with the auxiliary piers during the construction stage, the wind resistance of the bare tower is relatively weak when it is self-supporting, as shown in Figure 1(b). Therefore, this paper focuses on the variation of the VIV and buffeting response of the bridge tower with the wind speed under different wind attack angles.

Wind tunnel test

Experimental details

The object of this study is the bridge tower in the self-supporting state, considering the dimensions of the full bridge aeroelastic model test as well as the wind tunnel test. The geometric scaling ratio of the aeroelastic model of the bridge tower was taken as 1:120, and the actual height of the bridge tower was 165 m. According to the geometric scaling ratio of 1:120, the height converted to the model was 1.375 m. The specific dimensions of the bridge tower test are shown in Table 1. The bridge tower aeroelastic model consists of a steel core beam, coat and counterweight. The core beam of the bridge tower is made of A3 steel to provide stiffness for the tower, the test model is made of steel core beam to provide stiffness, and the coat is made of plexiglass, which is divided into 15 sections, and a gap of 2 mm is left between each section to prevent the coat from providing stiffness for the system. The constraint of the aeroelastic model of the bridge tower is that the bottom surface of the tower base is completely solidified and fixed by the bottom steel plate on the wind tunnel floor to ensure that it will not produce buffeting. The wind tunnel test model is shown in Figure 2(a).

Table 1.

Bridge tower aeroelastic model control parameters.

Name	Unit	Real bridge value	Scaling ratio	Model requirement value
Height	m	165.0	1:120	1.375
Quality	kg	27,603,072	1:120³	15.974

Figure 2.

Wind tunnel test related design: (a) wind tunnel test of bridge tower. (b) Displacement measurement point arrangement.

The aeroelastic model test was conducted in the CA-01 boundary layer wind tunnel of Chang’an University Wind Tunnel Laboratory, which is a return-flow type boundary layer wind tunnel. The length, width and height of its cross-section are 15 m, 3 m, and 2.5 m, respectively. The adjustable range of wind speed is 0∼53 m/s with turbulence intensity less than 0.5%. According to the Wind-Resistance Design Specification for Highway Bridges (JTG/T 3360-01-2018), the size of the model and support meet the requirement that the blocking ratio is less than 5%. The aeroelastic model test strictly meets the requirement of a similar ratio. The test instrument adopts the HL-G125-S-J laser displacement meter sensor produced in Japan, with a measuring range of 250 ± 150 mm and a resolution of 20 um, and the accelerometer adopts a piezoelectric acceleration transducer, with a sampling frequency of 1000 Hz. Laser meters are arranged at the top of the bridge tower and 65% of the height of the main tower along the along-bridge direction and the transverse-bridge direction, respectively, as shown in Figure 2(b).

The main natural frequencies, model target frequency design values, and related measured values of the bridge tower are obtained by finite element analysis software. Among them, beam 4 element is used to simulate the tower column, beam, cap and rigid arm, and mass 21 element is used to simulate the concentrated mass of the bridge tower. The main aeroelastic model design parameters are shown in Table 2. The results show that the measured natural frequency of the bridge tower is in good agreement with the design value. All values meet the requirements of relevant Chinese specifications.

Table 2.

Natural frequencies and damping ratios of aeroelastic full model of free-standing tower.

Modes	Modal characteristics	Model target frequency (Hz)	Measured frequency (Hz)	Error (%)	Damping ratio
1	First-order bending in the direction of the bridge	2.201	2.136	−2.943	0.0157
2	First-order transverse-bridge bending	4.975	4.915	−1.206	0.0165
3	First-order bridge tower torsion	8.328	8.301	−0.322	—
4	Second-order down-bridge bending	13.820	13.794	−0.190	—
6	Second-order transverse-bridge bending	20.685	21.360	3.262	—
7	Second order bridge tower torsion	21.190	21.480	1.367	—

Wind field parameters

Determination of design basis wind speed

According to the data of meteorological stations around the bridge location and the measured research of related wind parameters, the bridge location area 100 years recurrence of the basic wind speed value V₁₀ = 31.2 m/s, the bridge deck elevation from the water surface for Z = 165 m, the bridge location in the B wind field, α = 0.16, $k_{f}$ = 1.02. The wind resistance design at the construction stage takes the coefficient value η = 0.92 according to the return period of 30 years. According to the “Wind-Resistance Design Specification for Highway Bridges (JTG/T 3360-01-2018), the design wind speed of the bridge tower in the construction stage is

V_{s d} = η * V_{d} = η * k_{f} * V_{10} {(\frac{Z}{10})}^{α} = 0.92 * * 1.02 * 31.2 * {(\frac{165}{10})}^{0.16} = 45.9 (m / s)

(1)

Wind field simulation

The model test wind field in the uniform flow field and turbulent field, according to the wind environment at the bridge site, the experiment with the spire and rough elements simulated the scaling ratio of 1:120 turbulent wind field, where the wind speed profile simulation index α is 0.16, wind tunnel simulation of the bridge site at the landscape atmospheric boundary layer wind speed profile and along the height of the turbulence profile, as shown in Figure 3. It can be seen that the experimental values at different heights are in good agreement with the target values, which can fully reflect the wind environment characteristics required for the bridge tower at different heights of the wind tunnel test.

Figure 3.

Atmospheric boundary layer of the geomorphology at the bridge site simulated by wind tunnel: (a) wind speed profile, (b) Turbulence profile.

Test wind direction and wind speed

In order to consider the uncertainty of the wind direction in the real environment, the VIV and buffeting vibration of the test bridge towers are investigated for different wind attack angles. In this study, the wind attack angle is defined as the horizontal angle between the direction of the incoming wind and the normal direction of the bridge span. Since the bridge tower is a completely symmetrical structure, the wind attack angle β is defined as 0∼90°. During the test, taking the bridge deck as a reference, the incoming flow transverse the bridge is defined as a 90° wind attack angle, that is, 90° means the incoming flow is orthogonal to the bridge span direction, and the incoming flow along the bridge is defined as 0° wind direction angle. Every 15° is a working condition. The schematic diagram of the wind attack angle is shown in Figure 4. A total of seven cases with wind attack angles β of 0°, 15°, 30°, 45°, 60°, 75°, and 90° were simulated in uniform and turbulent flows. The test wind speed ratio is $\sqrt{120}$ , which corresponds to the wind speed of 0∼74.5 m/s at the real bridge.

Figure 4.

Definition of wind attack angle of bridge tower.

Wind-induced vibration response of bridge tower in self-supporting condition

Uniform flow field test results

In the uniform flow field, the focus is on the VIV performance of the bridge tower at low wind speed. The mean and maximum values of wind-induced vibration response of the top of the tower in the self-supporting state in the along-bridge and transverse bridge direction with wind speed are shown in Figure 5.

Figure 5.

Displacement response of bridge tower in uniform flow field: (a) average displacement along bridge, (b) average displacement transverse bridge, (c) RMS of displacement along bridge, (d) RMS of displacement transverse bridge.

The test results show that for most wind deflection angles, the displacement response of the bridge towers in the along the bridge and transverse bridge direction increases quadratically as the wind speed increase. As shown in Figure 5(a), when β = 0°, 15°, 30°, and 45°, the difference between the average displacement of the tower top along the bridge direction is small, but the average displacement at this time is the largest, and it increases with the increase of wind speed. When β = 90°, the average displacement of the along-bridge item at the top of the tower is the smallest, and it changes little with the change of wind speed. The average displacement of the tower top along the bridge direction shows a monotonically decreasing change with the increase of the wind attack angle. Figure 5(b) shows the average displacement in the transverse bridge direction of the tower under different wind attack angles. Since the tower is a frame structure in the towering plane, its transverse stiffness is significantly greater than its longitudinal stiffness, so the transverse bridge direction response of the tower is much smaller than its along-bridge direction response. As can be seen in the Figure, the average value of transverse bridge displacement at the top of the tower reaches the maximum at β = 45°. When β = 60°, the average value of the transverse bridge displacement of the bridge tower is the smallest. There is no obvious monotonic relationship between the displacement mean value of the top of the tower and the wind attack angle in the transverse bridge direction. Figure 5(c) and (d) show the results of the root mean square (RMS) test of the displacement response at the top of the bridge tower in a uniform flow field. It can be seen from Figure 5(c) that when β = 0°, 15°, 30°, and 45°, the RMS value of the displacement along the bridge at the top of the tower is the largest, probably due to the wind load is relatively perpendicular to the transverse bridge direction of the bridge tower at this time so that the effect of the bridge tower along the bridge direction is maximized.

However, when the wind attack angle β = 90° and the actual bridge wind speed is about 48 m/s, the phenomenon of VIV along the bridge direction occurs. Among them, the RMS displacement value of the bridge tower top reaches 0.3 m, which will seriously affect the construction and safety of the bridge tower structure. The results in Figure 5(d) show that when the bridge tower is at β = 45° and 75°, the RMS value of the displacement of the bridge tower in the transverse bridge direction is the largest. At the same time, since the stiffness in the transverse direction is much greater than that in the longitudinal direction, resulting in the RMS value of the displacement in the transverse direction is also smaller than that in the longitudinal direction of the bridge tower. In addition, no obvious VIV phenomenon was found in the bridge tower under the other wind direction angles. Within the wind speed range of the galloping vibration test, there is no obvious divergent galloping phenomenon under each wind direction angle.

Turbulent flow field test results

In the turbulent flow field, we focus on the buffeting performance of the bridge, and analyze whether the galloping vibration and VIV of the bridge tower may occur under the condition of self-support. In the test wind speed range, the bridge tower does not have obvious VIV and divergent galloping vibration under each wind direction angle. In the turbulent flow field, when the bridge tower is self-supporting, the average value of the wind-induced vibration response of the tower top along the bridge direction and transverse bridge direction changes with the wind speed as shown in Figure 6.

Figure 6.

Displacement response of bridge tower in turbulent flow field: (a) average displacement along bridge, (b) average displacement transverse bridge, (c) RMS of displacement along bridge, (d) RMS of displacement transverse bridge.

According to the results, it can be seen that under the action of different wind speeds, the buffeting displacement of the bridge tower in the self-supporting state along the bridge and transverse bridge direction presents a non-monotonic relationship, and the buffeting displacement increases with the increase of the wind speed, showing an approximate quadratic relationship with the wind speed. Under the turbulent wind field, the displacement along the bridge tower is larger than the displacement transverse bridge, and with the increase in wind speed, the displacement gap is increasing. At the same time, there is no VIV phenomenon in the displacement of the top of the bridge tower under the action of the orthogonal wind, no matter in the direction of the bridge or the direction of the transverse bridge. As shown in Figure 6(a) and (c), when the wind direction angle β = 0°, 15°, and 30°, the average value and RMS value of the buffeting displacement along the bridge at the top of the tower increase with the wind speed, and the trend is the same and reaches its maximum value at the same time. The results in Figure 6(b) and (d) show that when the wind direction angle is β = 45°, the average and RMS values of the transverse bridge displacement at the top of the tower reach the maximum at the same time. The reason may be that when the wind direction angle is 45°, the two bridge towers can be subjected to the wind load at the same time to the greatest extent. The degree of interference between the front and rear bridge towers reaches the maximum, and the surrounding turbulence increases accordingly so that the displacement of the bridge tower in the transverse direction reaches the maximum. The buffeting displacement of the bridge tower under the self-supporting state does not affect the construction safety performance of the bridge tower.

Numerical simulation

Settings

Computational domain mesh and flow condition settings

In this paper, based on the large-scale commercial FLUENT software, the numerical simulation of the two-dimensional cross-section of the bridge tower is carried out to study the influence of the flow field around the bridge tower cross-section on its vibration performance. In this paper, The Reynolds-averaged Navier-Stokes (RANS) simulations were performed by using the k-ω SST model, and the fluid governing equation is discretized based on the finite volume method. The pressure term and the diffusion term use the second-order central difference format, and the SIMPLE format is used to solve the pressure-velocity coupling equations. The fixed time step t = 0.005 is set in the dimensionless time unit, and the calculation convergence residual is taken as 1*10⁻⁵.

Figure 7(a). shows the computational domain and grid situation used in this study. The selection of the calculation domain has a great influence on the results of numerical calculation. Under the condition of ensuring the independence of the calculation results, to improve the calculation efficiency, the distance between the entrance of the calculation domain and the center of the model is 30D, and the distance between the trailing edge of the section and the exit is 100D. The grid growth rate is set to 1.05 from the inside to the outside and the boundary layer grid is set around the model. The calculation domain is divided into blocks, with a mixed grid (structured and unstructured grid) in Zone1, and the unstructured grid in the rest of the region. A large number of vortices fall off in the cross-section wake region Zone2. In order to better capture the flow field characteristics of vortices in the structure wake region, the wake area needs to be properly encrypted (Proctor et al., 2016). The grid division and detail structure are shown in Figure 7(b)∼(d).

Figure 7.

Boundary conditions and mesh generation: (a) numerical simulation computational domain, (b) overall domain, (c) gap detail, (d) boundary layer.

Independence verification

After setting the calculation domain, grid division method and basic parameters, it is necessary to verify the parameters used in numerical simulation before CFD calculation, so as to obtain the best calculation parameters that give consideration to efficiency and accuracy. In this paper, the same section size as Liu's experiment is used for comparison (Liu et al., 2002), and the results are shown in Figure 8(a). It can be seen that the drag coefficient results of the upstream and downstream cylinders are very close to the test values of Liu, which shows that the numerical simulation parameters used in this paper have certain credibility. After the above verification, under the same CFD parameter settings, in order to make the research goal of this paper meet the accuracy requirements of numerical simulation, it is necessary to carry out the grid and time independence test on the bridge tower section. The verification results are as shown in Table 3.

Figure 8.

(a) Force coefficients of upstream and downstream square cylinders, (b) distribution of y+ values on the surface of the section.

Table 3.

Meshes of different quantities and numerical results.

Grid	Triangular cells	Quadrilateral cells	Total nodes	Total cells	Cd
Mesh1	115,944	35,632	94,275	151,576	0.403505
Mesh2	139,860	35,632	106,263	175,492	0.442381
Mesh3	169,886	35,632	121,396	205,518	0.440656
Mesh4	215,260	35,632	144,083	250,892	0.440434

After comprehensively considering the length of the section and the inlet wind speed, three different time steps are selected for time-independence test, which are 1*10⁻³, 5*10⁻⁴ and 1*10⁻⁴. The deviations of the calculation results are all within 0.3%. After considering the calculation efficiency and time efficiency, the total number of selected grids is 205,518, and the time step is 5*10⁻⁴. Meanwhile, in order to accurately simulate the separation and transition of the air flow near the section surface (defined as the wall), and to meet the requirement of dimensionless y⁺_max <1 (Zhang et al., 2020), as shown in Figure 8(b).

DMD method

Dynamic mode decomposition (DMD) method is a data-driven model, based only on flow snapshots and not constrained by the model and control equations that accurately describes the flow structure by extracting the modes in the flow, and the extracted substructures are orthogonal to each other in space-time evolution characteristics, the main characteristics of the flow field in time and space can be obtained. The data source of the DMD method is experimental or simulated data. In this paper, the flow field snapshots at N discrete moments are obtained through numerical simulation, and the time interval between any two snapshots is Δt, and the snapshot sequence is represented by a matrix.

X = {v_{1}, v_{2}, v_{3}, \dots, v_{N - 1}}

(2)

Y = {v_{2}, v_{3}, v_{4}, \dots, v_{N}}

(3)

where

v_{j}

represents the i-th flow field,

v_{j} \in C^{M}

. In the above definition, subscript 1 represents the first element in the sequence and the subscript N represents the last element entering the sequence.

The nonlinear estimation is realized based on linear assumption that is

Y = A X = {{A v}_{1}, {A v}_{2}, {A v}_{3}, \dots, v_{N - 1}}

(4)

For a matrix X of rank r, the DMD algorithm replaces the matrix A by seeking a low-dimensional approximate matrix F, This substitution can be obtained by the singular value decomposition of X, that is

X = U Σ W^{H}

(5)

A = U F U^{H}

(6)

where

Σ

is the diagonal matrix,

U^{H} U = I

, and I is the identity matrix.

Substituting equations (5) and (6) into (4), we get

A \approx F = U^{H} Y W Σ^{- 1}

(7)

The matrix F is the optimal low-dimensional estimation matrix of A. Therefore, the DMD analysis result can be obtained by solving the eigenvalues and eigenvectors of F.

Therefore, by decomposing the eigenvalues of matrix A, its eigenvalues $u_{j}$ and eigenvectors $y_{j}$ are obtained. The magnification $g_{j}$ and frequency $ω_{j}$ of the flow can be obtained by taking the logarithm of this eigenvalue, that is:

g_{j} = \frac{R e (\lg u_{j})}{V_{t} / (2 π)}

(8)

ω_{j} = \frac{I m (\lg u_{j})}{V_{t}}

(9)

where, Δt is the sampling time interval. Re and Im represent the real and imaginary parts of the eigenvalues, respectively.

The response stability of the system can be shown from the calculation results of the modal magnification. The modal stability is positively correlated with the magnification value, and as the magnification value increases, the modal stability also increases.

Thus, the DMD mode of each order can be defined as:

Φ_{j} = Y * W * Σ^{- 1} * y_{j}

(10)

where,

y_{j}

is the eigenvector of the low-dimensional matrix F.

Define the modal amplitude α as:

α = W^{- 1} U^{H} ν_{1}, α = [α_{1}, . . ., α_{r}]

(11)

where α is the amplitude of the i-th mode, which represents the contribution of this mode to the initial snapshot

ν_{1}

ν_{1}

represents the distribution characteristics of the flow field around the cross-section at the initial moment.

At the same time, in order to accurately reconstruct the flow field around the cross-section, it is necessary to introduce the Vandermonde matrix form, which represents the temporal distribution characteristics of the flow field, and its mathematical expression is:

V_{a n d} = [\begin{array}{l} 1 λ_{1} . λ_{1}^{N - 2} \\ 1 λ_{2} . λ_{2}^{N - 2} \\ . . . . \\ 1 λ_{r} . λ_{r}^{N - 2} \end{array}]

(12)

The snapshot sequence X can be expressed as:

X = [ν_{1}, ν_{2}, ν_{3}, . . ., ν_{N - 1}] = Φ D V_{a n d} = [Φ_{1}, Φ_{2}, . . ., Φ_{r}] [\begin{array}{l} α_{1} \\ α_{2} \\ . \\ α_{3} \end{array}] [\begin{array}{l} 1 λ_{1} . λ_{1}^{N - 2} \\ 1 λ_{2} . λ_{2}^{N - 2} \\ . . . . \\ 1 λ_{r} . λ_{r}^{N - 2} \end{array}]

(13)

Equation (13) illustrates that the flow field evolution process is mainly realized by the Vandermonde matrix $v_{a n d}$ , which contains the eigenvalues of the matrix with r number of A.

At present, different scholars have successively studied different sorting methods for the modal sorting method of DMD. For the standard DMD method, the DMD modes are sorted according to the amplitude (Jovanovic et al., 2014). Some scholars also sort according to the 2-norm of the mode (Chen et al., 2012; Zhang et al., 2014). In this paper, the modal norm is used for sorting, which is defined as follows:

‖ Φ_{j} ‖ = \frac{1}{‖ W * S^{- 1} y_{j} ‖}

(14)

Through the above DMD method, each order mode of the flow field can be extracted and the coherent structure of the flow field can be further analyzed.

Analysis of dynamic mode decomposition results

In this paper, through the numerical simulation analysis of the two-dimensional section of the bridge tower. The main modal frequencies and energy are extracted by the decomposition of DMD method, which are used to analyze the spatiotemporal distribution and evolution characteristics of the flow field in the surrounding airspace when the VIV occurs in the bridge tower section. Through the above related research, the most unfavorable wind direction angle β=90° is selected as the research object for the two-dimensional cross-section of the bridge tower. At this time, the section of the bridge tower has obvious VIV phenomenon at the test wind speed U=4.5 m/s. The height of the bridge tower at 65% of the cross-section was chosen for the numerical simulation, because this height is the most representative of the wind vibration response of the bridge, and the cross-sectional flow field at this height is the most representative. Through numerical simulation and analysis, the change of lift coefficient response of the section with time is shown in Figure 9(a).

Figure 9.

Aerodynamic lift characteristics: (a) responses of lift coefficient changing with time, (b) fourier analysis of lift coefficients of section.

From the change law of lift coefficient, it can be seen that in the first 2 s, the flow field around the section is still in the development stage and the change of lift coefficient is not very stable. After 2 s, due to the periodic movement of the surrounding flow field, the lift coefficient curve presents a sinusoidal change, and then it remains a periodic function. As shown in Figure 9(b), the magnitude of the energy at three different frequencies can be obtained after the Fourier transformation of the lift coefficients in Figure 9(a). It can be seen from the figure that strong energy at the first order frequency of 5.052 Hz, which indicates that the structure has a dominant frequency of 5.052 under the current conditions, as well as a certain energy component at both the second frequency 10.078 as well as the triple frequency 15.13. It also shows that the aerodynamic force of this section vibration has a strong nonlinear character.

Modal analysis results

According to the variation of lift force with time, the DMD modal decomposition method is used to select the pressure-time curve of the periodically stable phase in the range of 6∼10 s marked by the red dashed line in Figure 9(a), and to perform the modal decomposition of its flow field snapshot. The total pressure matrix in the flow field region of this phase was selected for the modal decomposition, and a total of 800-time steps of data were selected, which is sufficient to analyze the main features of the flow field.

When we perform the modal decomposition of the flow field data, we can choose different sorting methods to sort the modes. The analysis points out that using the amplitude or modal norm method, the results can be sorted according to the contribution and influence of each mode to the flow field, and facilitate the reconstruction analysis of the flow field by these modes.

In this paper, the modal norm definition (according to equation (13)) is used to sort the different modes of the bridge tower section. The pressure distribution of the unsteady flow field obtained above is decomposed by DMD, and then the DMD modes are arranged from high to low according to the modal norm. which is the corresponding DMD mode. The magnification and frequencies of the first five orders of modes were extracted as shown in Table 4.

Table 4.

Growth rates and reduced frequencies of the first 7 DMD modes.

Modes	Growth rate/10⁻⁵	Frequency
1	−2.707	0
2-3	2.442	5.021
4-5	4.824	9.917
6-7	12.52	14.564

For the description of most unstable periodic flows, there are generally no more than 10 globally dominant modes. Therefore, we choose the first four global modes (except for the first order, which are all conjugate modes. In fact, we chose the first 7 order). From the data in Table 4, it can be seen that the magnification of the first mode is much smaller than that of other modes, and its frequency is 0. This indicates that mode 1 has the same effect on the flow field at all moments, which corresponds to the average state of the flow field, i.e., the static mode, which is the energy occupying most of the flow field. The frequency of modes 2-3 is 5.021, which is consistent with the dominant frequency in the change of lift. At the same time, the frequency doubling and third-order frequency of the flow field in the lift coefficient are also basically consistent with the modes in the flow field. It shows that through the frequency characteristics of the first 7 flow field modes, the structural characteristics of the flow field of the whole structural section can be represented. After the DMD dynamic mode decomposition method, each order mode of the flow field around the bridge tower section can be accurately separated.

As shown in Figure 10, by decomposing the flow field pressure data around the section, the structural characteristics of the first 7 flow field modes are obtained. The first order of the modes extracted by DMD is the static model, which approximates the mean flow field, and the rest of the modes appear in pairs, and only the real part of each complex mode is shown in the figure. Relevant research pointed out that the real and imaginary parts of the model do not differ much in the flow field characteristics, but there are only certain phase differences that do not affect the expression of the flow field (Chen, et al., 2012).

Figure 10.

The first 7 DMD modes. (a) DMD mode 1. (b) DMD mode 2-3. (c) DMD mode 4-5. (d) DMD mode 6-7.

Sorting the modes of the DMD, the magnitude of the percentage of energy in different modes is shown in Figure 11(a), it can be seen that when we choose the first 7 modes, its total energy is already greater than 90%, i.e.

\frac{\sum_{n = 1}^{7} e_{i}}{\sum_{n = 1}^{N} e_{i}} \times 100 % > 90 %

(15)

where

e_{i}

represents the amount of energy contained in each order of the flow field mode, N is the number of modes.

Figure 11.

Energy characteristics of different modes: (a) relationship between extracted DMD modes numbers and loss function, (b) dynamic modal decomposition energy distribution.

It shows that the main characteristics of the flow field around the bridge tower section can be captured when selecting the first 7 modes, and the flow field in the whole airspace can be reconstructed. Figure 11(b) shows the energy value of each mode according to the modal norm.

Eigenvalue analysis

Figure 12(a) shows the distribution of eigenvalues of the modes in the flow field, where the horizontal axis is the real part and the vertical axis is the imaginary part. Each discrete point in the figure represents the corresponding mode after decomposition, i.e., the number of points on each unit circle is equal to the number of flow field snapshots collected at this stage. The points outside the unit circle correspond to unstable modes, the points on the unit circle correspond to periodic modes, and the points inside the unit circle correspond to stable modes. In addition, the discrete points are approximately symmetrically distributed on the characteristic circle. It can be seen that in the period stable stage, most of the eigenvalues are regularly arranged on the unit circle, and the periodic mode occupies the main position, which corresponds to the periodic change of the lift force and the periodic shedding of the vortex in the stable stage. At the same time, about 7∼10 pairs of modal eigenvalues far away from the unit circle appear in the unit circle, corresponding to the stable modes with rapid attenuation. Figure 12(b) shows a local detail plot of the eigenvalues, and it can be seen that the first 7 orders of modes are basically on the unit circle. Figure 12(c) shows the relationship between attenuation rate and frequency. Discrete points with positive attenuation rate correspond to the divergence of modal coefficients, and conversely points with negative or zero attenuation rate correspond to the convergence of modal coefficients.

Figure 12.

Model eigenvalues distribution: (a) all Ritz values, (b) extracted Ritz eigenvalues, (c) relationship between attenuation rate and frequency.

The figure shows that most of the decay rates are in negative or zero values, and some of them are positive. The green part of the figure indicates the unstable modes in the flow field, corresponding to the discrete points outside the unit circle, although they belong to the unstable modes, their corresponding decay rates are near zero values, and their maximum amplification is 1.84. The discrete points on the corresponding characteristic circle are close to the unit circle, indicating that the divergence of the modal coefficients are very slowly and approach stability. Ye et al., (2017) showed that some of the modes have positive decay rates, but their decay rates are more close to zero values compared to other discrete points, and their corresponding modes are called weakly unstable periodic modes.

DMD main modal coefficients

After the decomposition of the flow field structure by the DMD method, the modal characteristic frequencies can be obtained, and the time evolution coefficients of different modes of the flow field can be derived at the same time. As shown in Figure 13(a).

Figure 13.

DMD mode coefficients. (a) DMD mode coefficients. (b) Fourier analysis of DMD mode coefficients.

Figure 13(a) depicts the time-dependent characteristics of the different modal coefficients. Among them, DMD mode one is a static mode, whose amplitude does not change with the eigenvalue and therefore is not given. It can be seen that the modal coefficients decomposed by DMD basically conform to the simple harmonic characteristics. However, for the results of POD decomposition, some studies have pointed out that, the modal coefficients may contain multiple frequency components, resulting in time-frequency mixing, which may not accurately express the time-evolution characteristics of the flow field at each order (Kou and Zhang, 2016). Compared with the POD method, DMD method can more accurately express the time evolution characteristics of each modal coefficient, which plays an important role in the subsequent analysis of flow field characteristics and flow field reconstruction. By performing Fourier analysis on the modal coefficients, the main frequency characteristics of different modes can be seen. As shown in Figure 13(b), the Fourier transform results of modal coefficients are shown. Each flow field mode corresponds to a frequency component. The flow field modal coefficients are arranged from small to large according to the frequency. It can be seen that the dominant frequency of the flow field is 4.89 Hz, which is closer to the results of the Fourier transform analysis for the lift force in Figure 9(b), and also the high-order frequency components are also consistent.

Flow field restoration

To further investigate the accuracy of the DMD modal decomposition method for the extraction of flow field features, the flow field is reconstructed by the DMD decomposition method. The flow field around the double rectangular bridge tower is restored by a few flow field modes. As shown in Figure 14, the degree of restoration of the flow field by different flow field modes is compared.

Figure 14.

(a) Initial flow field; (b)∼(d) the reduction degree of the flow field at the initial time by using different modal orders.

Figure 14(a) shows the flow field at the initial moment of the double rectangular section of the bridge tower, and Figure 14(b)–(d) are the degrees of flow field restoration at the initial moment using flow field modes of different orders, respectively. It can be seen that the flow field at the initial moment can be basically restored when the first 7 orders of flow field modes are selected.

In order to further study the reconstruction accuracy of the DMD modal decomposition method for the flow field, this paper chooses to compare the surface pressure of the double rectangular section of the bridge tower at a certain time to study the accuracy of the DMD decomposition method, as shown in Figure 15.

Figure 15.

Sectional pressure reconstruction diagram. (a) Front section. (b) Behind section.

It can be seen that the flow field structure of the entire section can be reconstructed by selecting several modes with a relatively large contribution to the flow field, and the pressure values on the surface of the section also basically match with the real values, which proves that it can express the pressure distribution characteristics on the surface of the section. However, only from the surface pressure distribution of the section, it cannot express the spatial and temporal evolution characteristics of the flow field in the entire section airspace, so further analysis is needed for the flow field in the airspace around the section.

As shown in Figure 16, the pressure changes of several representative characteristic points in the flow field are selected for comparison in this paper, in which two characteristic points A and B are selected in the region of periodic vortex shedding after the first section and points C and D in the wake region, as shown in Figure 14(a) above. The evolution of the flow field at these four characteristic points has relatively rich change characteristics.

Figure 16.

Time evolution of pressure at four observation points: (a) time evolution pressure of at A, (b) time evolution pressure of at B, (c) time evolution pressure of at C, (d) time evolution pressure of at D.

From Figure 16, we can see that the pressure variation at each point shows a nonharmonic variation characteristic, the curve has obvious high-order harmonic components, and the flow field evolution has strong nonlinear characteristics. By selecting the main flow field modes after superposition, it can be seen that the pressure curves at points A, B, C, and D can accurately grasp the high-frequency components at the peak, and at the same time, their pressure changes can all match well with the original flow field. It is shown that the DMD modal decomposition can reflect the Spatio-temporal evolution characteristics of each variable in the flow field, and then can accurately express the coherent structure and background structure of different frequency components in the flow field. At the same time, it also shows that the method has certain reversibility and can accurately restore the flow field through the main mode of the flow field.

Conclusion

In this paper, the vibration response of the bridge tower under different wind speeds and wind attack angles in the self-supporting state are studied by a combination of wind tunnel tests and numerical simulation. The DMD method is used to decompose the flow field around the double rectangular bluff body section. The coherent structure of different flow field modes around the section are accurately analyzed. The main conclusions are as follows:

1. Wind tunnel tests have shown that the bridge tower undergoes significant VIV in its self-supporting state, with its maximum amplitude significantly higher than the allowable value in the specifications. In the construction of ultra-high bridge towers, especially with the use of double column sections, in addition to focusing on the buffeting response generated by the structure, we should also focus on the vortex vibration response under uniform flow.

2. The decomposition results of the 2D cross-sectional flow field of the bridge tower using the DMD method show that the first 7 main modes of the flow field contribute more than 90% to the entire cross-sectional flow field. At the same time, when periodic flow occurs, the reconstruction of the original cross-sectional vortex distribution can be achieved. In addition, the time evolution coefficient of the flow field mode is consistent with the spectral analysis of the aerodynamic lift, indicating that this method can accurately analyze the spatiotemporal evolution characteristics of the surrounding flow field when VIV occurs on the bridge tower section.

3. For analyzing the flow field of complex blunt body cross-sections, compared to traditional flow field display methods, using the DMD method can obtain the evolution rules of each order of flow field modal characteristics, and can conduct relevant analysis based on the contribution value to the cross-sectional flow field, greatly reducing the dimensionality reduction of complex flow fields and more helpful in revealing the triggering mechanisms of various flow induced vibrations.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The work described in this paper was fully supported by a grant from the National Natural Science Foundation of China (No 51978077) and the Natural Science Research Project of Colleges and Universities in Anhui Province (No KJ2021A0949).

ORCID iDs

Zhenxing Ma

Zhengfeng Shen

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