A method for analyzing dynamic mode and flow stability of double-box-girder flow field

Abstract

The application discloses a kind of double box girder flow field dynamics modal and flow stability analysis method and system, belong to bridge engineering field.This method first establishes double box girder flow field model by computational fluid dynamics, solves the momentum equation and continuity equation of two-dimensional incompressible flow;Dynamic modal decomposition is carried out again, and eigenvalue, characteristic mode and other information are obtained;Finally, based on linear stability theory, the flow field component is expressed as the sum of steady-state solution and disturbance term, and the global linear flow stability equation is obtained by substituting into the equation, and the flow stability is analyzed by discrete.This method can accurately extract the modal characteristics of each order, providing a scientific basis for wind-induced vibration analysis of double box girder, and has important application value.The method of the application occupies less computing resources, has high numerical accuracy, and can be further expanded to higher precision calculation model according to computing resources, and is suitable for application in actual bridge engineering.

Description

Technical Field

[0001] This invention belongs to the field of bridge engineering, specifically relating to a method for modal analysis of flow field dynamics and flow stability around a double box girder. Background Technology

[0002] Double-box girder bridges are a widely adopted bridge cross-section in recent years, further enhancing the spanning capacity of long-span bridges such as cable-stayed and suspension bridges. Compared to single-box girders, while achieving superior spanning capacity, double-box girders exhibit more complex flow characteristics under wind conditions. Double-box girders may experience complex wind-induced vibrations under the dynamic influence of external wind fields, classifying them as wind-sensitive structures. Dynamic modal analysis and flow stability analysis of the flow field are effective methods for analyzing the dynamic characteristics of the flow field. Therefore, conducting dynamic modal analysis and flow stability analysis of the flow field around the leading edge, trailing edge, and gaps of double-box girders has significant scientific and practical value. Summary of the Invention

[0003] To accurately analyze the dynamic modes and flow stability characteristics of the flow field around a double box girder, and to extract information such as the eigenvalues ​​and characteristic mode distributions of each mode, this invention proposes a method for analyzing the dynamic modes and flow stability of the flow field around a double box girder. The core of this method is to model the dynamic system of the flow around the double box girder and perform dynamic mode decomposition (DMD) and two-dimensional global linear flow stability analysis.

[0004] The technical solution adopted in this invention is as follows: A method for analyzing the dynamic modes and flow stability of a flow field around a double box girder, comprising the following steps:

[0005] Step 1: Modeling the flow field around the double box girder: Using computational fluid dynamics, the governing equations are the momentum equation and continuity equation of two-dimensional incompressible flow. By setting known boundary conditions, the equations are solved using the finite element method. The velocity degree of freedom is set to a second-order Lagrangian element, the pressure degree of freedom is set to a first-order Lagrangian element, and the time integration scheme of the transient term adopts the first-order backward implicit Euler method.

[0006] Step 2, Dynamic Mode Decomposition: Time t is obtained through CFD calculation. i The solution vector (u) i ,p i After that, the time-averaged velocity and pressure are calculated, the velocity fluctuations in the flow field are further calculated, and a data matrix is ​​constructed. The data matrix is ​​then subjected to r-order truncated singular value decomposition. Based on the matrix obtained from the singular value decomposition, the projection of the linear operator A of the flow dynamics around the double box girder onto the low-dimensional space U is calculated. The operator matrix is ​​then further subjected to eigenvalue decomposition to obtain the eigenvalues ​​λ of the dynamic mode decomposition (DMD). i and the corresponding characteristic mode w iThis leads to the reconstructed system DMD mode;

[0007] Step 3: Flow stability analysis: According to the linear stability theory, the velocity and pressure components of the flow field are expressed as the sum of the steady-state solution (u0, p0) and the disturbance term, where the disturbance velocity and pressure are expressed as exponential forms that increase / decrease with time. Substituting these into the momentum equation and continuity equation of the two-dimensional incompressible flow, and neglecting the higher-order terms of the disturbance, we obtain the two-dimensional global linear flow stability equation. Through finite element space discretization, we obtain the generalized eigenvalue problem and its associated problems, and then analyze the flow stability of the flow field around the double box girder.

[0008] Furthermore, in step 1, the formulas for the momentum equation and continuity equation of the two-dimensional incompressible flow are as follows:

[0009]

[0010] Where u is the flow field velocity vector (u,v), p is the flow field pressure, and ν is the fluid kinematic viscosity coefficient. The boundary conditions include the inlet boundary flow velocity, the outlet boundary pressure, the upper and lower boundary conditions, and the no-slip boundary conditions on the surface of the double box girder.

[0011] Furthermore, the formula for calculating the time-averaged velocity and pressure in step 2 is as follows:

[0012]

[0013] The constructed data matrix is ​​as follows:

[0014]

[0015] Where x i Let time t i velocity pulsation (u') i ,v' i ) T , where m is the number of modes to be analyzed;

[0016] The formula for performing r-order truncated singular value decomposition on a data matrix X is:

[0017] X≈UΣV * (4)

[0018] in, The column vectors correspond to the spatial orthogonal modes of the flow field. The diagonal elements are singular values, and are passed through a threshold r. th Cut off, The row vectors represent the time evolution information of the flow field dynamics system;

[0019] The formula for calculating the projection of the linear operator A of the flow dynamics around a double box girder onto the low-dimensional space U is as follows:

[0020]

[0021] The formula for eigenvalue decomposition of the operator matrix is:

[0022]

[0023] The formula for the reconstructed system DMD mode is:

[0024] φ i =Uw i (7).

[0025] Furthermore, the formula for expressing the velocity and pressure components of the flow field in step 3 as the sum of the steady-state solution (u0, p0) and the disturbance term is as follows:

[0026]

[0027] The disturbance velocity and pressure are expressed as exponential forms that increase / decrease with time. middle The amplitude of the disturbance velocity. Let λ be the amplitude of the disturbance pressure, and λ be the time growth rate of the disturbance wave.

[0028] The two-dimensional global linear flow stability equation is:

[0029]

[0030] The formulas for the generalized eigenvalue problem and its adjoint problem obtained by finite element spatial discretization are as follows:

[0031]

[0032] The formula for the tangent matrix of the linear operator in the corresponding linear flow stability equation is:

[0033]

[0034] Where B is the corresponding disturbance velocity amplitude term. The mass matrix, where λ is the perturbation eigenvalue and λ is the real part. r The event growth rate represents the disturbance, with the imaginary part λ. i The frequency of the disturbance wave is represented by the right eigenvector, which is the direct disturbance mode φ, and the left eigenvector is the adjoint mode ξ.

[0035] Another objective of this invention is to provide a system for analyzing the dynamic modes and flow stability of a double box girder around a flow field, comprising: a memory, a processor, and a computer program stored in the memory and executable on the processor. When the computer program is executed by the processor, it implements the method for analyzing the dynamic modes and flow stability of a double box girder around a flow field as described above.

[0036] Another object of the present invention is to provide a computer-readable storage medium storing an information transmission implementation program, characterized in that: when the program is executed by a processor, it implements the steps of the above-described method for analyzing the dynamic modes and flow stability of a double box girder flow field.

[0037] Advantages and beneficial effects of the present invention:

[0038] 1. A high-precision finite element model is adopted, which can accurately perform numerical simulation of the flow field around a double box girder;

[0039] 2. The simulation results are accurate and reliable. The time-averaged streamlines, velocity distribution, DMD modal energy and eigenvalues, and flow stability modes in the calculation results have clear physical meanings.

[0040] 3. The calculation process is simple and clear, consumes less computing resources, has high numerical accuracy, and can be further extended to a higher accuracy calculation model based on computing resources. Attached Figure Description

[0041] Figure 1 This is a cross-sectional view of a double box girder;

[0042] Figure 2 A schematic diagram showing the calculation domain and boundary conditions for numerical simulation of flow around a double box girder;

[0043] Figure 3 This is a diagram showing the time-averaged streamline distribution of the flow around a double box girder.

[0044] Figure 4 The diagram shows the time-averaged velocity distribution around the double box girder, where (a) is the downstream velocity u and (b) is the transverse velocity v.

[0045] Figure 5 The diagram shows the DMD modal energy and eigenvalue distribution of the flow field around the double box girder, where (a) is the modal energy distribution and (b) is the modal eigenvalue distribution.

[0046] Figure 6 The diagram shows the dominant DMD modes of the flow field around the double box girder, where (a) is the downstream velocity u mode and (b) is the transverse velocity v mode.

[0047] Figure 7The diagram shows the distribution of the most unstable right characteristic modes of the flow field around the double box girder, where (a) is the downstream perturbation velocity u mode and (b) is the transverse perturbation velocity v mode.

[0048] Figure 8 The diagram shows the distribution of the most unstable left characteristic modes of the flow field around the double box girder, where (a) is the downstream perturbation velocity u mode and (b) is the transverse perturbation velocity v mode. Detailed Implementation

[0049] To make the objectives, technical solutions, and advantages of the present invention clearer, the following embodiments are given in conjunction with the accompanying drawings to further illustrate the present invention in detail. It should be understood that the specific embodiments described herein are only for explaining the present invention and are not intended to limit the present invention.

[0050] Example 1

[0051] A modal analysis method for flow field dynamics and flow stability around a double box girder, comprising the following steps:

[0052] S1. Modeling of the flow field around a double box girder

[0053] The flow field around the double box girder was modeled using computational fluid dynamics (CFD), with the governing equations being the momentum equation (Navier-Stokes equations) and continuity equation for two-dimensional incompressible flow.

[0054]

[0055] Where u is the flow field velocity vector (u,v), p is the flow field pressure, and ν is the fluid kinematic viscosity coefficient.

[0056] By setting known boundary conditions, equation (1) can be solved numerically, namely, the inlet flow velocity at the inlet boundary, the outlet pressure at the outlet boundary, the upper and lower boundary conditions, and the no-slip boundary conditions on the surface of the double box girder.

[0057] The numerical model of equation (1) is solved using the finite element method, where the velocity degree of freedom is set to a second-order Lagrangian element and the pressure degree of freedom is set to a first-order Lagrangian element, satisfying the LBB stability condition of the saddle point problem; the time integration scheme of the transient term adopts the first-order backward implicit Euler method, which satisfies the stability and accuracy requirements of the numerical solution.

[0058] S2. Dynamic Mode Decomposition

[0059] Time t was obtained through CFD calculation i The solution vector (u) i ,p i After that, calculate the time-averaged velocity and pressure:

[0060]

[0061] Further calculation of flow field velocity fluctuations And construct a data matrix:

[0062]

[0063] Where, x i Let time t i velocity pulsation (u') i ,v' i ) T , where m is the number of modes to be analyzed.

[0064] Perform r-order truncated singular value decomposition on the data matrix X:

[0065] X≈UΣV * (4)

[0066] in, The column vectors correspond to the spatial orthogonal modes of the flow field. The diagonal elements are singular values, and are passed through a threshold r. th Cut off, The row vectors represent the time evolution information of the flow field dynamics system.

[0067] Based on the matrix obtained from singular value decomposition, the projection of the linear operator A of the flow dynamics around a double box girder onto the low-dimensional space U can be calculated:

[0068]

[0069] Due to low-dimensional dynamic operators The dimensionality is significantly reduced, but the core dynamic characteristics of the flow system around a cylinder are still captured, and the operator matrix is ​​improved. Further eigenvalue decomposition is performed:

[0070]

[0071] The eigenvalues ​​λ of the dynamic mode decomposition (DMD) are obtained. i and the corresponding characteristic mode w i The reconstructed system DMD mode is then:

[0072] φ i =Uw i (7)

[0073] S3. Flow stability analysis

[0074] Based on the obtained CFD flow field data, further flow stability analysis is performed. According to linear stability theory, the velocity and pressure components of the flow field can be expressed as the steady-state solution (u0, p0) and the disturbance term. sum:

[0075]

[0076] The disturbance velocity and pressure are expressed as exponential forms that increase / decrease with time. The amplitude of the disturbance velocity. Let λ be the amplitude of the disturbance pressure, and λ be the time growth rate of the disturbance wave.

[0077] Substituting equation (8) into the momentum and continuity equations for two-dimensional incompressible flow, and neglecting higher-order terms of perturbation, we obtain the stability equation for two-dimensional global linear flow:

[0078]

[0079] Through finite element spatial discretization, the following formulas for the generalized eigenvalue problem and its adjoint problem are obtained:

[0080]

[0081] Where A is the tangent matrix of the linear operator in the corresponding linear flow stability equation:

[0082]

[0083] Where B is the corresponding disturbance velocity amplitude term. The mass matrix, where λ is the perturbation eigenvalue and λ is the real part. r The event growth rate represents the disturbance, with the imaginary part λ. i The frequency of the disturbance wave is represented by the right eigenvector, which is the direct disturbance mode φ, and the left eigenvector is the adjoint mode ξ.

[0084] The above steps can be used to calculate and analyze the dynamics and flow stability modes of the flow field around the double box girder, thus providing guidance for engineering applications.

[0085] Example 2

[0086] Typical cross-sectional dimensional parameters of a double box girder are as follows: Figure 1 As shown, the central section of the beam has a height D = 3.5m, the width of both sides of the bridge deck is 12m, the bridge deck slope is 2%, and the total width of the double box girder is B = 36m. The computational domain and boundary conditions for the CFD numerical simulation of this double box girder are set as follows: Figure 2 As shown, the velocity inlet is set to u=1, v=0, the pressure outlet is set to p=0, the upper and lower boundaries are set to v=0, and the boundary conditions of the double box girder surface are set to no-slip walls.

[0087] The time-averaged streamline distribution of the flow field around the double box girder is as follows: Figure 3 As shown, significant backflow was observed at both the gap between the two box girders and the downstream position of the trailing edge. The time-averaged velocity distribution around the two box girders is as follows. Figure 4 As shown, based on the downstream velocity u distribution, the distribution of the backflow zone downstream of the tail edge of the double box girder, i.e. the region where u<0, was further observed. At the same time, a clear separation phenomenon of the flow velocity v between the front and rear edges of the double box girder was observed.

[0088] The distribution of DMD modal energy and eigenvalues ​​of the flow field around the double box girder is as follows: Figure 5 As shown. Figure 5 The modal frequency in (a) is defined as St = ω i D / (2π), where ω i The imaginary part of ω = log(λ) / Δt represents the frequency of the DMD mode, and the real part ω is... r The figure represents the DMD mode growth rate. It can be observed from the figure that the peak energy of the DMD mode in the flow field around the double box girder occurs at the frequency of St = 0.20, which is equal to the frequency of vortex shedding in the wake. Furthermore, the first few modes with higher energy are all distributed around St = 0.20, indicating that the dominant DMD mode can accurately represent and predict the wake vortex shedding process of the double box girder. Figure 5 (b) Modal frequency ω i With modal growth rate ω r The relationship shows that the growth rate of most DMD modes in the flow field around the double box girder is less than 0, meaning they decay exponentially with time. The dominant mode 1, with the highest energy value, has a time growth rate ω. r =0 indicates that the DMD mode with the highest energy in the flow field around the double box girder is in a stable state and does not increase or decrease with time, i.e., neutral oscillation.

[0089] The dominant mode distribution of the flow field DMD around the double box girder is as follows: Figure 6 As shown, Figure 6 (a) represents the downstream velocity u-mode. Figure 6 (b) represents the transverse velocity mode v. The spatial distribution of mode u begins with a wave packet exhibiting a distinct modal peak at x≈20, gradually decaying downstream in an envelope distribution, showing an antisymmetric bimodal distribution along the horizontal axis. The spatial distribution of mode v exhibits a similar growth and decay trend along the downstream direction as mode u, but with a larger amplitude, indicating a higher relative intensity of the v component at the same frequency. Combined with... Figure 5 The frequency and growth rate information shown can be analyzed to obtain... Figure 6 The spatial distribution of the dominant DMD modes shown constitutes the periodic vortex shedding phenomenon in the wake of the double box girder.

[0090] The distributions of the right and left characteristic modes of the most unstable flow field around the double box girder are as follows: Figure 7and Figure 8 As shown, the right characteristic mode is called the direct mode, and the left characteristic mode is called the adjoint mode, used to analyze the sensitivity of the flow field around the double box girder. The distribution of the direct modes reveals that the modal structure located at the gap in the flow field around the double box girder is most easily amplified, reflecting momentum fluctuations and instabilities of the downstream shear layer. Meanwhile, antisymmetric distribution of instability modes is observed downstream. Based on the distribution of the adjoint modes, sensitivity analysis of the flow field around the double box girder reveals that the gap is also most sensitive to external disturbances.

[0091] The results show that the proposed analysis method can effectively model the flow field around a double box girder and obtain reliable DMD mode decomposition and flow stability results.

Claims

1. A method for analyzing the dynamic modes and flow stability of a flow field around a double box girder, characterized in that, Includes the following steps: Step 1: Modeling the flow field around the double box girder: Computational fluid dynamics is used, with the governing equations being the momentum and continuity equations for a two-dimensional incompressible flow. Known boundary conditions are set, and the equations are solved using the finite element method. The velocity degrees of freedom are set to second-order Lagrangian elements, and the pressure degrees of freedom are set to first-order Lagrangian elements. The time integration scheme for the transient terms uses the first-order backward implicit Euler method. The formulas for the momentum and continuity equations of the two-dimensional incompressible flow are as follows: （1） Where u is the flow field velocity vector (u, v); p is the flow field pressure; and ν is the fluid kinematic viscosity coefficient. The boundary conditions include the inlet boundary flow velocity, the outlet boundary pressure, the upper and lower boundary conditions, and the no-slip boundary conditions on the surface of the double box girder. Step 2, Dynamic Mode Decomposition: Time t is obtained through CFD calculation. i The solution vector (u) i , p i Afterwards, the time-averaged velocity and pressure are calculated, and the velocity fluctuations in the flow field are further calculated. A data matrix is ​​constructed, and an r-order truncated singular value decomposition is performed on the data matrix. Based on the matrix obtained from the singular value decomposition, the projection of the linear operator A of the flow dynamics around the double box girder onto the low-dimensional space U is calculated. The operator matrix is ​​further decomposed into eigenvalues ​​to obtain the eigenvalues ​​λ of the dynamic mode decomposition (DMD). i and the corresponding characteristic mode w i This leads to the reconstructed system DMD mode; Step 3: Flow stability analysis: Based on linear stability theory, the velocity and pressure components of the flow field are expressed as the sum of the steady-state solution (u0, p0) and the disturbance term, where the disturbance velocity and pressure are expressed as exponential forms that increase / decrease with time. Substituting these into the momentum equation and continuity equation of two-dimensional incompressible flow, and neglecting the higher-order terms of the disturbance, we obtain the two-dimensional global linear flow stability equation. Through finite element space discretization, we obtain the generalized eigenvalue problem and its associated problems, and then analyze the flow stability of the flow field around the double box girder.

2. The method for analyzing the dynamic modes and flow stability of a flow field around a double box girder according to claim 1, characterized in that, The formula for calculating the time-averaged velocity and pressure in step 2 is as follows: （2） The constructed data matrix is ​​as follows: （3） Where x i Let time t i velocity pulsation , where m is the number of modes to be analyzed; The formula for performing r-order truncated singular value decomposition on the data matrix X is: （4） in, The column vectors correspond to the spatial orthogonal modes of the flow field. The diagonal elements are singular values, and are passed through a threshold r. th Cut off, The row vectors represent the time evolution information of the flow field dynamics system; The formula for calculating the projection of the linear operator A of the flow dynamics around a double box girder onto the low-dimensional space U is as follows: （5） The formula for eigenvalue decomposition of the operator matrix is: （6） Among them, the imaginary part Indicates the frequency of the disturbance wave; The formula for the reconstructed system DMD mode is: （7）。 3. The method for analyzing the dynamic modes and flow stability of a flow field around a double box girder according to claim 2, characterized in that, In step 3, the formula for expressing the velocity and pressure components of the flow field as the sum of the steady-state solution (u0, p0) and the disturbance term is as follows: （8） The disturbance velocity and pressure are expressed as exponential forms that increase / decrease with time. , ,middle The amplitude of the disturbance velocity. Let λ be the amplitude of the disturbance pressure, and λ be the time growth rate of the disturbance wave. The two-dimensional global linear flow stability equation is: （9） The formulas for the generalized eigenvalue problem and its adjoint problem obtained by finite element spatial discretization are as follows: （10） The formula for the tangent matrix of the linear operator in the corresponding linear flow stability equation is: （11）， Where B is the corresponding disturbance velocity amplitude term. The mass matrix; λ is the time growth rate of the disturbance wave, with its real part... The imaginary part represents the event growth rate of the disturbance. The frequency of the perturbation wave is represented by the right eigenvector, which is the direct perturbation mode. The left eigenvector is the adjoint mode ξ; ν is the fluid kinematic viscosity coefficient.

4. A system for analyzing the dynamic modes and flow stability of a flow field around a double box girder, comprising: The memory, the processor, and the computer program stored in the memory and executable on the processor are characterized in that: when the computer program is executed by the processor, it implements a method for analyzing the dynamic modes and flow stability of a double box girder flow field as described in any one of claims 1-3.

5. A computer-readable storage medium storing an implementation program for information transmission, characterized in that: When the program is executed by the processor, it implements the steps of the method for analyzing the dynamic modes and flow stability of a double box girder flow field as described in any one of claims 1-3.

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