Nowadays, the bridge structure tends to be long, thin, light, gentle and low damping, which makes the structure more sensitive to wind load. Especially when the near-ground wind blows over the bridge at low speed, the wind will flow and separate and the periodic vortex will fall off, resulting in alternating positive and negative pressures and moments on the upper and lower or left and right sides of the bridge deck, where the vibration of the bridge produces the ‘vortex vibration’ [1, 2]. Cable-stayed bridge is a kind of composite structural system bridge, which is composed of cables, towers and beams. Compared with general beam bridges, it is quite different from other bridges in seismic performance, wind resistance and vehicle vibration performance due to the differences in structural system and material mechanical performance. In addition to higher vertical, horizontal and torsional stiffness requirements, the requirements for wind resistance are also higher [3, 4].

To ensure the bridge has enough VIV stability, the critical wind speed of the bridge must be higher than the VIV test wind speed at the bridge site. At present, the critical wind speed of vortex vibration of bridge section is mainly obtained by wind tunnel test, which involves the design and manufacture of relevant models that takes a long test period and costs a lot. In addition, for the scale model test, it is difficult to obtain the flow field conditions corresponding to the real flow field [5]. With the rapid increase in storage capacity and floating-point computing speed of high-performance computers and the further development of computation fluid dynamics (CFD) theory, numerical simulation analysis has been paid more attention in the research of wind engineering in bridges [6, 7]. Compared with the wind tunnel test, although numerical calculation will introduce truncation error, it is not limited by test conditions, and can even simulate conditions that the wind tunnel test cannot achieve. Moreover, it does not need to add a two-dimensional end plate, so the angle of attack is easy to change, the wind speed is stable and it has good applicability to nonlinearity, which can simulate problems of complex geometric shape flow or turbulence [8]. As an alternative to the wind tunnel test, it can provide data that are helpful to further clarify the mechanism of related problems in bridge wind engineering. It can also assist the wind tunnel test to efficiently and accurately evaluate the vortex vibration stability of the bridge girder section, and provide a reliable basis for the design of bridge wind resistance. Because the vortex vibration response of bridge is the result of interaction between wind and structure, therefore, after the in-depth study of numerical simulation of bridge, considering the continuous deformation of numerical model, researchers began to use the fluid-solid coupling numerical simulation method to study the vortex vibration effect of bridge [9].

The solvers of CFD and computational structural dynamics (CSD) are combined with the help of System Coupling component in ANSYS software, which can realise the numerical simulation of vortex vibration for the main girder of cable-stayed bridge [10]. Therefore, in this paper, a numerical simulation method of vortex vibration in the main girder of cable-stayed bridge based on CFD/CSD bidirectional fluid-solid coupling is proposed, which is dedicated to solving the difference between the surface coupling nodes of complex bridge and the grid nodes required by CFD calculation, thus realising efficient and accurate information transmission.

The occurrence of VIV bridges is mainly related to flow separation and vortex shedding, and its fundamental reason lies in the periodic shedding vortex generated when the wind with constant velocity passes through the main girder. When there is a vortex around the structure, it will be subjected to a stable periodic force in the direction perpendicular to the fluid, while when the frequency of the force is close to the inherent frequency of a certain order of the structure, it will cause resonance [11].

When the air flows around the cylinder, it will produce a vortex that periodically falls off at the wake. In this paper, the dimensionless Storrow's number is defined to describe the periodic characteristics of the vortex, namely:

When the air flows through the cross-section of a blunt bridge structure, the wake will have periodic vortex shedding alternately, which will lead to the change in the surface pressure of the bridge and eventually lead to the vibration of the bridge, which is called VIV.

Under the average wind, the bridge structure continuously absorbs energy from the fluid flowing around it to produce vibration, and the vortex that alternately falls off at the wake reacts on the bridge structure to cause vibration, so the VIV is self-excited. While under the wind pulsation, the bridge will cause random vibration due to its pulsating components, it will also have some influence on vortex shedding, thus limiting the amplitude of bridge vortex vibration. Therefore, vortex vibration is a kind of forced vibration with limited amplitude, and it often has more than one wind speed interval. The intensification of vibration leads to strong coupling between cylinder and fluid, and controls VIV in a certain range of wind speed. At this time, the vortex shedding frequency is always consistent with the natural frequency of the cable, as shown in Figure 1.

Vortex vibration is a common phenomenon of wind-induced vibration in engineering structures, and the quality, damping ratio and natural frequency of the structure will have a greater or lesser impact on its vortex vibration performance. The natural frequency of the structure plays a decisive role in the range of wind speed locked by vortex vibration, while the magnitude of vortex amplitude is affected by the damping ratio and mass of the structure, where the larger the mass of the structure is, the more energy it needs to absorb from the airflow. Meanwhile, the damping ratio determines the energy dissipated when the structure vibrates, so that the larger the damping ratio is, the more energy dissipated [12].

Assuming all other conditions are the same, the larger the structural mass and damping ratio, the smaller is the amplitude of structural vortex vibration. The influence of the two factors on VIV can also be determined by dimensionless Scruton (

For vertical vibration:

For torsional vibration:

The amplitude of vortex vibration and its locked wind speed range can be reflected by

It can be seen from the vortex vibration principle expressed in Eq. (1) that the geometric shape of bluff body structure and the Reynolds value determine

For typical blunt body sections with sharp corners (such as rectangle, H-shape and

Flow characteristics around a flat plate with different Re values

Re = 0.3 | Viscous force takes the lead. | Laminar flow adheres to the surface of blunt body without separation. |

Re = 10 | Inertial force began to work gradually. | Separation occurs at the corner of the leading edge of the flow and two large-scale vortex clusters with the same characteristics are produced at the trailing edge, but the vortex clusters almost stagnate near the trailing edge surface without falling off. |

Re = 250 | The effect of inertia force is more obvious. | The upper and lower edges of the section periodically alternate to form antisymmetric vortices and fall off downstream. |

Re > 1,000 | Inertial force takes the lead. | There is no longer regular large-scale vortex shedding at the trailing edge, but small-scale vortex clusters are mixed in the wake. |

The change in Re has a significant influence on the flow pattern and vortex shedding mode of the fluid. For the blunt body cross section with relatively large downstream dimension, Re has a great influence on the phenomenon of re-attachment and re-separation of gas after the first separation, which leads to its great influence on the overall aerodynamic force, surface pressure distribution and

Fluid-solid coupling in wind engineering of bridge is a mechanical branch generated by the intersection of fluid mechanics and solid mechanics, which can be divided into unidirectional fluid-solid coupling analysis and bidirectional fluid-solid coupling analysis. Unidirectional fluid-solid coupling means that at the interface of two media, data transmission is unidirectional. The data results calculated by CFD analysis module of computational fluid dynamics are transmitted to the solid structure analysis module. while bidirectional fluid-solid coupling analysis means that the data transmission between the two modules is mutual.

Numerical simulation of bridge vortex vibration based on CFD/CSD can obtain the information of wind load and structural coupling motion comprehensively. Technically, it is necessary to establish CFD flow field and CSD domain model, and consider the interface coupling between flow field and domain. As CFD technology mainly adopts finite difference or finite volume discrete continuous N-S equations, while CSD technology mainly adopts finite element discrete structural dynamic equations, the bidirectional fluid–structure coupling involves the mesh matching and interface movement of the two at the fluid-solid interface.

Bidirectional fluid-solid coupling methods can be divided into strong coupling and weak coupling. In this paper, the vortex vibration of main girder of cable-stayed bridge is numerically simulated by weak coupling, that is, CFD solver and CSD solver solve and exchange data in one time step. The specific process is shown in Figure 2.

First, the surface load result of the current time step model is obtained by CFD solver;

The result is input to the CSD solver to solve the dynamic response result of the current time step model;

The dynamic response results are transmitted back to the CFD solver, and the mesh deformation before the next time step calculation is carried out;

After the mesh deformation is completed, start the cycle of the next time step.

To facilitate CFD numerical simulation, the calculation method in this paper has the following three assumptions: conservation of mass, conservation of momentum and conservation of energy, which are the theoretical basis of the whole computational wind engineering.

The fluid mass in a unit time and in a closed area will remain constant at any moment:

Air belongs to a unit continuous medium, and its motion satisfies Newton's second law, that is, the acceleration of unit mass air is proportional to the force acting on this part of air:
_{u}_{v}_{w}

Generally, it is not considered in current wind engineering, and the temperature of the air is constant by default.

In this paper, the Gambit pre-processing software is used to calculate the bidirectional fluid-solid coupling of the steel box girder section of cable-stayed bridge, so as to explore the vortex vibration performance of the girder section. The specific process is as follows:

In this paper, the Gambit pre-processing software is used to model the steel box girder section of cable-stayed bridge and divide the grid. If the width of the calculation area is not enough, the vortex separated at the tail of the main girder structure model will move to the boundary and then return. Meanwhile, if the length of the calculation area is not enough, it is difficult to accurately define the exit boundary conditions, while too large a calculation area will increase the calculation time.

To simulate the wind environment in the natural state as much as possible, we need to reasonably define the size of the calculation domain and make the calculation basin meet the requirement that the blocking rate is <5%. In this paper, a rectangular computational domain of 28D × 17B is selected, and the computational domain is divided into rigid domain, dynamic grid domain and stable domain. The rigid domain is 1.12D × l.25B in size, while the distances from the main girder to the upper edge, the lower edge, the left edge and the right edge of the rigid body domain are 0.05B, 0.1B, 0.125D and 0.125D, respectively, and the size of the moving grid area is 3B × 10D. To fully develop the wake flow field, the size of the tail basin is selected as 9.5B × 10D.

On the premise of ensuring the accuracy of numerical simulation calculation, reducing the number of n grids s as much as possible can shorten a lot of calculation time. In this paper, quadrilateral mesh is used for division. Displacement caused by vibration in rigid area will be transferred to moving grid area, so the grid in moving grid area needs to be reconstructed and updated constantly to ensure that the grid will not be distorted. Therefore, in this paper, the triangular grid is used in the moving area, so that it can be set as the moving grid in the later coupling calculation, and the grid reconstruction method is used to update and reconstruct the grid in the area of moving grid.

Because the wake area is large, to effectively reduce the number of grids, the whole quadrilateral grid is used to divide this area. Then, the boundary layer on the main girder section and ancillary facilities are set to better simulate the fluid situation around the main girder section, so as to obtain more accurate numerical simulation results. At the same time, to verify the grid independence of fluid domain, the whole fluid domain is divided into two sets of grids, which are 182,600 and 360,000, respectively. Through trial calculation, it is found that the grids of 260,000 and 360,000 are in good match with the experimental results, which shows that the increase in the number of grids can no longer affect the calculation results. Therefore, 260,000 grids are finally selected as the final calculation grid where the first grid is 0.016 mm, with 15 floors.

In this paper, the boundary condition of the entrance is set as the velocity-inlet. In this paper, the lower boundary and the left boundary are set as the entrance boundary, the corresponding right boundary and the upper boundary are set as the exit boundary, and the exit boundary condition is set as the pressure-outlet. In addition, the main girder section and the accessory structure model surface are set as the wall boundary condition without slip. Figure 3 shows the specific boundary conditions.

Moreover, the steps of boundary setting are shown in Figure 4:

Use SST k-w turbulence model to calculate, and the turbulence intensity and turbulence viscosity ratio are set to 0.5% and 5, respectively;

Adopt SIMPLE algorithm to solve the problem in steady state;

After the steady-state calculation results converge and stabilise, the SIMPLEC algorithm is used for transient calculation;

The velocity-pressure coupling is used to solve the problem, and the second-order upwind scheme is used to control the dispersion of the calculation.

For the rigid VIV of the steel box girder studied in this paper, the displacement, speed, acceleration, wind speed and time of the box girder will affect the instantaneous pressure acting on the surface of the girder. Without considering the influence of turbulence, VIF can be described by the function of wind speed, time, displacement, speed and acceleration of box girder. In this paper, the rigid VIV of the steel box girder mainly occurs at the main frequency, so the influence of high-order frequency on the displacement can be neglected, where the displacement and acceleration of the box girder have the same fluctuation characteristics except for the difference in phase and amplitude, so a variable can be used to replace these two physical quantities. Therefore, the rigid VIF of the steel box girder can be expressed as follows:

The vortex-excitation force consists of four kinds of components: the term containing displacement _{*} only, the cross term of displacement _{*}, and the vortex-detachment force term. Therefore, the vortex excitation force can be expressed as follows in a broad sense:
_{*} = _{*} represents the dimensionless vortex vibration velocity (_{*} = _{ij}_{s}_{s}

It can be seen from Eq. (9) that in addition to the vortex detachment force, the first-order response terms in the model only include the linear aerodynamic stiffness term _{10}_{*} and the linear aerodynamic damping term _{01}_{*}, and the other terms are all nonlinear high-order terms.

To establish the VIF mathematical model of box girder with cantilever more reasonably, it is necessary to accurately judge the influence of terms contained in the generalised expression of VIF. Therefore, in this paper, the residual error between the VIF reconstructed by the mathematical model and the actual VIF is used as the criterion, and its expression is as follows
_{VI}

In this paper, the bidirectional fluid-solid coupling calculation of the two-dimensional main girder section under +3 wind attack angle is carried out by using the same related parameters as the segmental model test, and the numerical simulation results are compared with the segmental model test results. The numerical simulation parameters are shown in Table 2.

Numerical simulation parameters

Value | 0.74 | 0.066 | 1.54 | 11.5896 | 9.745 | 0.3 |

The variation of vortex vibration amplitude of the main girder section with wind speed under the action of +3 wind attack angle is as shown in Figure 5.

Under +3 wind attack angle, the variation trend of vortex amplitude of steel box girder section at different wind speeds obtained by numerical simulation is basically the same as that obtained by the segmental model test, and the vortex vibration locking interval measured by them is basically the same, roughly between 5.5 m/s and 7.5 m/s. On the whole, however, the vortex vibration results obtained by numerical simulation are basically consistent with those measured by two-dimensional segmental model test, whose results are reliable.

The frequency spectrum analysis of numerical simulation results shows that the VIV frequency of numerical simulation is 9.498 Hz, while the natural frequency of the main beam of two-dimensional segment test is 9.745 Hz, and the difference between them is about 2.53%. Therefore, the numerical simulation method adopted in this paper can accurately simulate the VIV process of the steel box main girder, and the simulation results are accurate and reliable.

In this paper, the numerical simulation method based on bidirectional fluid-solid coupling is used to study the vortex vibration performance in steel box girder of cable-stayed bridge, where a mathematical model of VIF is put forward, and the general expression of vortex vibration amplitude in the main girder is deduced. In addition, the simulation results are compared with the two-dimensional segment model results, which verifies the reliability of the numerical simulation method used in this paper. The test results show that the difference between the vortex vibration frequency at the maximum amplitude of the steel box girder is 2.53%. Therefore, the numerical simulation method of steel box girder in cable-stayed bridge proposed can reproduce the vortex vibration process of the girder well.

#### Flow characteristics around a flat plate with different Re values

Re = 0.3 | Viscous force takes the lead. | Laminar flow adheres to the surface of blunt body without separation. |

Re = 10 | Inertial force began to work gradually. | Separation occurs at the corner of the leading edge of the flow and two large-scale vortex clusters with the same characteristics are produced at the trailing edge, but the vortex clusters almost stagnate near the trailing edge surface without falling off. |

Re = 250 | The effect of inertia force is more obvious. | The upper and lower edges of the section periodically alternate to form antisymmetric vortices and fall off downstream. |

Re > 1,000 | Inertial force takes the lead. | There is no longer regular large-scale vortex shedding at the trailing edge, but small-scale vortex clusters are mixed in the wake. |

#### Numerical simulation parameters

Value | 0.74 | 0.066 | 1.54 | 11.5896 | 9.745 | 0.3 |

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Function Model Design of Tennis Mobile Teaching Assistant System Based on Ordinary Differential Equations Financial Crisis Early Warning Model of Listed Companies Based on Fisher Linear Discriminant Analysis High Simulation Reconstruction of Crowd Animation Based on Optical Flow Constraint Equation Construction of Intelligent Search Engine for Big Data Multimedia Resource Subjects Based on Partial Least Squares Structural Equation 3D Animation Simulation of Computer Fractal and Fractal Technology Combined with Diamond-Square Algorithm Analysis of the Teaching Quality of Physical Education Class by Using the Method of Gradient Difference The Summation of Series Based on the Laplace Transformation Method in Mathematics Teaching Optimal Solution of the Fractional Differential Equation to Solve the Bending Performance Test of Corroded Reinforced Concrete Beams under Prestressed Fatigue Load Animation VR scene mosaic modeling based on generalized Laplacian equation Radial Basis Function Neural Network in Vibration Control of Civil Engineering Structure Optimal Model Combination of Cross-border E-commerce Platform Operation Based on Fractional Differential Equations The influence of accounting computer information processing technology on enterprise internal control under panel data simultaneous equation Research on Stability of Time-delay Force Feedback Teleoperation System Based on Scattering Matrix BIM Building HVAC Energy Saving Technology Based on Fractional Differential Equation Construction of comprehensive evaluation index system of water-saving irrigation project integrating penman Montei the quation Human Resource Management Model of Large Companies Based on Mathematical Statistics Equations Data Forecasting of Air-Conditioning Load in Large Shopping Malls Based on Multiple Nonlinear Regression Analysis of technical statistical indexes of college tennis players under the win-lose regression function equation Automatic extraction and discrimination of vocal main melody based on quadratic wave equation Analysis of wireless English multimedia communication based on spatial state model equation Optimization of Linear Algebra Core Function Framework on Multicore Processors Research on classification of e-commerce customers based on BP neural network The Control Relationship Between the Enterprise's Electrical Equipment and Mechanical Equipment Based on Graph Theory Mathematical Modeling and Forecasting of Economic Variables Based on Linear Regression Statistics Nonlinear Differential Equations in Cross-border E-commerce Controlling Return Rate 3D Mathematical Modeling Technology in Visualized Aerobics Dance Rehearsal System Fractional Differential Equations in Electronic Information Models BIM Engineering Management Oriented to Curve Equation Model Leakage control of urban water supply network and mathematical analysis and location of leakage points based on machine learning Analysis of higher education management strategy based on entropy and dissipative structure theory Prediction of corporate financial distress based on digital signal processing and multiple regression analysis Mathematical Method to Construct the Linear Programming of Football Training Multimedia sensor image detection based on constrained underdetermined equation The Size of Children's Strollers of Different Ages Based on Ergonomic Mathematics Design Application of Numerical Computation of Partial Differential Equations in Interactive Design of Virtual Reality Media Stiffness Calculation of Gear Hydraulic System Based on the Modeling of Nonlinear Dynamics Differential Equations in the Progressive Method Knowledge Analysis of Charged Particle Motion in Uniform Electromagnetic Field Based on Maxwell Equation Relationship Between Enterprise Talent Management and Performance Based on the Structural Equation Model Method Term structure of economic management rate based on parameter analysis of estimation model of ordinary differential equation Influence analysis of piano music immersion virtual reality cooperation based on mapping equation Chinese painting and calligraphy image recognition technology based on pseudo linear directional diffusion equation Label big data compression in Internet of things based on piecewise linear regression Animation character recognition and character intelligence analysis based on semantic ontology and Poisson equation Design of language assisted learning model and online learning system under the background of artificial intelligence Study on the influence of adolescent smoking on physical training vital capacity in eastern coastal areas Application of machine learning in stock selection Comparative analysis of CR of ideological and political education in different regions based on improved fuzzy clustering Action of Aut( G ) on the set of maximal subgroups ofp -groupsThe internal mechanism of corporate social responsibility fulfillment affecting debt risk in China: analysis of intermediary transmission effect based on degree of debt concentration and product market competitive advantage Study on transmission characteristics in three kinds of deformed finlines based on edge-based finite element method Asymptotic stability problem of predator–prey system with linear diffusion Research on loyalty prediction of e-commerce customer based on data mining Algebraic Equations in Educational Model of College Physical Education Course Education Professional English Translation Corpus Under the Binomial Theorem Coefficient Geometric Tolerance Control Method for Precision Machinery Based on Image Modeling and Novel Saturation Function Retrieval and Characteristic Analysis of Multimedia Tester Based on Bragg Equation Semiparametric Spatial Econometric Analysis of Household Consumption Based on Ordinary Linear Regression Model Video adaptive watermark embedding and detection algorithm based on phase function equation English Learning Motivation of College Students Based on probability Distribution Scientific Model of Vocational Education Teaching Method in Differential Nonlinearity Research on mobile Awareness service and data privacy Protection based on Linear Equations computing protocol Vocal Music Teaching Model Based on Finite Element Differential Mathematical Equations Studying a matching method combining distance proximity and buffer constraints The trend and influence of media information Propagation based on nonlinear Differential equation Research on the construction of early warning model of customer churn on e-commerce platform Evaluation and prediction of regional human capital based on optimised BP neural network Study on inefficient land use determination method for cities and towns from a city examination perspective A sentiment analysis method based on bidirectional long short-term memory networks A study of local smoothness-informed convolutional neural network models for image inpainting Mathematical Calculus Modeling in Improving the Teaching Performance of Shot Put Application of Nonlinear Differential Equation in Electric Automation Control System Higher Mathematics Teaching Curriculum Model Based on Lagrangian Mathematical Model Computational Algorithm to Solve Two–Body Problem Using Power Series in Geocentric System Decisions of competing supply chain with altruistic retailer under risk aversion Optimization of Color Matching Technology in Cultural Industry by Fractional Differential Equations The Marketing of Cross-border E-commerce Enterprises in Foreign Trade Based on the Statistics of Mathematical Probability Theory Application of Linear Partial Differential Equation Theory in Guiding Football Scientific Training Nonlinear Channel Estimation for Internet of Vehicles Some Necessary Conditions for Feedback Functions of de Bruijn Sequences The Evolution Model of Regional Tourism Economic Development Difference Based on Spatial Variation Function System Model of Shipping Enterprise Safety Culture Based on Dynamic Calculation Matrix Model An empirical research on economic growth from industrial structure optimisation in the Three Gorges Reservoir area The Inner Relationship between Students' Psychological Factors and Physical Exercise Based on Structural Equation Model (SEM) Analysis and Research on Influencing Factors of Ideological and Political Education Teaching Effectiveness Based on Linear Equation Study of agricultural finance policy information extraction based on ELECTRA-BiLSTM-CRF Fractional Differential Equations in Sports Training in Universities Examination and Countermeasures of Network Education in Colleges and Universities Based on Ordinary Differential Equation Model Innovative research of vertical video creation under the background of mobile communication Higher Education Agglomeration Promoting Innovation and Entrepreneurship Based on Spatial Dubin Model Chinese-English Contrastive Translation System Based on Lagrangian Search Mathematical Algorithm Model Genetic algorithm-based congestion control optimisation for mobile data network