Influence of Wind Speed, Wind Direction and Turbulence Model for Bridge Hanger: A Case Study

Abstract

Wind field (e.g., wind speed and wind direction) has the characteristics of randomness, nonlinearity, and uncertainty, which can be critical and even destructive on a long-span bridge’s hangers, such as vortex shedding, galloping, and flutter. Nowadays, the finite element method is widely used for model calculation, such as in long-span bridges and high-rise buildings. In this study, the investigated bridge hanger model was established by COMSOL Multiphysics software, which can calculate fluid dynamics (CFD), solid mechanics, and fluid–solid coupling. Regarding the wind field of bridge hangers, the influence of CFD models, wind speed, and wind direction are investigated. Specifically, the bridge hanger structure has symmetrical characteristics, which can greatly reduce the calculation efficiency. Furthermore, the von Mises stress of bridge hangers is calculated based on fluid–solid coupling.

1. Introduction

In recent years, the structure of long-span bridges has been softer, and their crossing ability has shown a continuous increase due to the advance in design and construction technologies and the emergence of novel materials [1]. However, a long-span bridge does not work properly and is even destroyed under wind effects, which include vortex shedding, galloping, flutter, and so forth [2]. For example, vortex-induced vibrations may result in long-term fatigue damage, shortening the bridge life [3]. Galloping can lead to significant amplitude oscillation of the bridge [4]. Long-span bridges may be affected by dynamic instability phenomena, that is, flutter [5]. Therefore, it is important to analyze the impact of long-span bridges under wind fields (e.g., wind speed and wind direction), which can ensure a reliable wind-resistant design of a bridge [6,7,8,9,10,11,12].

In order to study the impact of long-span bridges under wind fields (e.g., wind speed and wind direction), computational fluid dynamics [13,14,15,16,17] has been proposed based on structural health monitoring (SHM) data. Regarding the influence of wind direction, Zhu et al. (2012) [18] found variations of aerodynamic coefficients with wind direction based on the results of wind tunnel tests. However, wind tunnel tests cannot be widely used in the study of a wind-induced structural response because these tests require a substantial amount of resources, where these tests are limited to a certain number of pressure taps, and some information about separation zones, recirculation, and so forth might get lost [19,20]. On the contrary, the numerical calculation method, that is, computational fluid dynamics model, can obtain the corresponding structural stress response by setting different wind speeds and wind directions based on the finite element method [21]. For example, Huang et al. [22] ran a 3D CFD model to examine scale effects on turbulent flow and sediment scour. Shirai et al. [23] studied the applicability of the two-dimensional Reynolds-averaged numerical simulation for predicting the flow around a self-oscillating bridge deck section. Thus, the computational fluid dynamics model allows for an accurate evaluation of the impact of long-span bridges under wind fields since it directly utilizes SHM data.

The main contributions of this work are threefold: (1) The bridge hanger model was established by COMSOL Multiphysics software, which can calculate computational fluid dynamics. (2) Regarding the bridge hanger, the influence of wind speed, wind direction, and turbulence model was investigated. The rest of the paper is organized as follows: In Section 2, the standard k-ε model, realizable k-ε model, standard k–ω model, and SST k-ω model of computational fluid dynamics are described. Section 3 introduces the results of the influence of wind speed, wind direction, and turbulence model. Finally, Section 4 ends with some conclusions drawn from this study.

2. Computational Fluid Dynamics

Generally, the computational fluid dynamics of a bridge are calculated based on the finite element method [24,25,26,27,28,29,30,31]. For an incompressible fluid, the continuity equation and momentum equation can be expressed as [32]

$$\frac{\partial U_i}{\partial x_i}=0$$

(1)

$$\begin{array}{c}{U}_j\frac{\partial U_i}{\partial x_j}=\frac{1}{\rho }\frac{\partial }{\partial x_j}(-P{\delta }_{ij}+2\upsilon S_{ij}+{\tau }_{ij})\\ S_{ij}=\frac{1}{2}\left(\frac{\partial U_i}{\partial x_j}+\frac{\partial U_j}{\partial x_i}\right)\end{array}$$

(2)

$${\tau }_{ij}={\upsilon }_t\left(\frac{\partial U_i}{\partial x_j}+\frac{\partial U_j}{\partial x_i}\right)-\frac{2}{3}k{\delta }_{ij}$$

(3)

where $\rho$ is the density of air, $\rho =1.293 {\mathrm{kg}/\mathrm{m}}^3$; $\upsilon$ is the kinematic viscosity, $\upsilon =17.9\times {10}^{-6} {\mathrm{m}}^2/\mathrm{s}$; $P$ is the pressure, Pa; $S_{ij}$ is the velocity strain rate tensor; ${\delta }_{ij}$ is the Kronecker delta; $U_i$ are the mean velocity components, m/s; and $k$ is the turbulent kinetic energy, m²/s².

Regarding the standard k-ε model, the transport equation of k can be expressed by [33]

$${\upsilon }_t=C_{\mu }\frac{k^2}{\epsilon }$$

(4)

$$\frac{\partial \rho k}{\partial t}+\frac{\partial \rho k{U}_i}{\partial x_i}=\frac{\partial }{\partial x_j}\left[\left(\upsilon +\frac{{\upsilon }_t}{{\sigma }_k}\right)\frac{\partial k}{\partial x_j}\right]+P_k-\rho \epsilon$$

(5)

$$\frac{\partial \rho \epsilon }{\partial t}+\frac{\partial \rho \epsilon U_i}{\partial x_i}=\frac{\partial }{\partial x_j}\left[\left(\upsilon +\frac{{\upsilon }_t}{{\sigma }_{\epsilon }}\right)\frac{\partial \epsilon }{\partial x_j}\right]+\frac{\epsilon }{k}\left(C_1{P}_k-C_2\rho \epsilon \right)$$

(6)

where $P_k$ is the production of turbulent kinetic energy, and $P_k={\upsilon }_t{S}^2$, $S=\sqrt{2S_{ij}{S}_{ij}}$, and other constants are $C_{\mu }=0.09$, ${\sigma }_{\epsilon }=1.3$, ${\sigma }_k=1.0$, $C_1=1.44$, and $C_2=1.92$.

Regarding the realizable k-ε model, the transport equation of k can be expressed by [34]

$$\frac{\partial \rho \epsilon }{\partial t}+\frac{\partial \rho \epsilon U_i}{\partial x_i}=\frac{\partial }{\partial x_j}\left[\left(\upsilon +\frac{{\upsilon }_t}{{\sigma }_{\epsilon }}\right)\frac{\partial \epsilon }{\partial x_j}\right]+\rho C_{\epsilon 1}S-\rho C_{\epsilon 2}\frac{{\epsilon }^2}{k+\sqrt{\epsilon \upsilon }}$$

(7)

where $\overline{{\Omega }_{ij}}$ is the mean rate-of-rotation tensor viewed in a rotating reference frame with the angular velocity ${\omega }_k$ and the constants are ${\sigma }_{\epsilon }=1.2$, $C_{\epsilon 2}=1.9$, $C_{\epsilon 1}=\max \left(0.43,\frac{k}{\epsilon }\sqrt{2S_{ij}{S}_{ij}}\right)$, $C_{\mu }=\frac{1}{4.04+\raisebox{1ex}{$A_{s}k{U}^{*}$}\!\left/ \!\raisebox{-1ex}{$\epsilon $}\right.}$, $A_s=\sqrt{6}\cos \theta$, $\theta =\frac{1}{3}\mathrm{co}{\mathrm{s}}^{-1}\sqrt{6}W$, $S^{*}=\sqrt{S_{ij}{S}_{ij}}$, $U^{*}=\sqrt{S_{ij}{S}_{ij}+{\Omega }^{*}{}_{ij}{\Omega }^{*}{}_{ij}}$, ${\Omega }^{*}{}_{ij}={\Omega }_{ij}-2{\epsilon }_{ijk}{\omega }_k$, and ${\Omega }_{ij}=\overline{{\Omega }_{ij}}-{\epsilon }_{ijk}{\omega }_k$.

Regarding the standard k-ω model, the transport equations of k and ω are defined by [35]

$${\upsilon }_{\omega t}=\frac{k}{\omega }$$

(8)

$$\frac{\partial k}{\partial t}+\frac{\partial k{U}_j}{\partial x_j}={\tau }_{ij}\frac{\partial U_j}{\partial x_j}-{\beta }^{*}k\omega \rho +\frac{\partial }{\partial x_j}\left[\left(\upsilon +{\upsilon }_{\omega t}{\sigma }_{k1}\right)\frac{\partial k}{\partial x_j}\right]$$

(9)

$$\frac{\partial \omega }{\partial t}+\frac{\partial \omega U_j}{\partial x_j}=\alpha \frac{\omega }{k}{\tau }_{ij}\frac{\partial U_j}{\partial x_j}-{\beta }_1{\omega }^2+\frac{\partial }{\partial x_j}\left[\left(\upsilon +{\upsilon }_{\omega t}{\sigma }_{\omega 1}\right)\frac{\partial \omega }{\partial x_j}\right]$$

(10)

where the constants are $\alpha =\frac{5}{9}$, ${\beta }_1=0.09$, ${\sigma }_{\omega 1}=0.5$, and ${\sigma }_{k1}=0.85$.

Regarding the SST k-ω model, the transport equations of k and ω are defined by [36,37]

$${\upsilon }_{\omega t}=\frac{a_{1}k}{\max \left(a_1\omega ,F_{2}S\right)}$$

(11)

$$\frac{\partial k}{\partial t}+\frac{\partial k{U}_j}{\partial x_j}=P_k-{\beta }^{*}k\omega +\frac{\partial }{\partial x_j}\left[\left(\upsilon +{\upsilon }_{\omega t}{\sigma }_{k2}\right)\frac{\partial k}{\partial x_j}\right]$$

(12)

$$\frac{\partial \omega }{\partial t}+\frac{\partial \omega U_j}{\partial x_j}=\alpha S^2-{\beta }_2{\omega }^2+\frac{\partial }{\partial x_j}\left[\left(\upsilon +{\upsilon }_{\omega t}{\sigma }_{\omega 2}\right)\frac{\partial \omega }{\partial x_j}\right]+2\left(1-F_1\right)\frac{{\sigma }_{\omega 2}}{\omega }\frac{\partial k}{\partial x_i}\frac{\partial \omega }{\partial x_i}$$

(13)

$$P_k=\min \left({\tau }_{ij}\frac{\partial U_i}{\partial x_j},10{\beta }^{*}k\omega \right)$$

(14)

$$F_1=\tanh \left\{{\left[\min \left(\max \left(\frac{\sqrt{k}}{{\beta }^{*}y\omega },\frac{500\upsilon }{y^2\omega }\right),\frac{4{\sigma }_{\omega 2}k}{C{D}_{k\omega }{y}^2}\right)\right]}^4\right\}$$

(15)

$$F_2=\tanh \left\{{\left[\max \left(\frac{2\sqrt{k}}{{\beta }^{*}y\omega },\frac{500\upsilon }{y^2\omega }\right)\right]}^2\right\}$$

(16)

$$C{D}_{k\omega }=\max \left(2\rho \frac{{\sigma }_{\omega 2}}{\omega }\frac{\partial k}{\partial x_i}\frac{\partial \omega }{\partial x_i},10^{-10}\right)$$

(17)

where the constants are ${\beta }_2=0.0828$, ${\sigma }_{\omega 2}=0.856$, ${\sigma }_{k2}=1$, and $a_1=0.31$.

3. Calculation and Analysis

Figure 1 shows the investigated bridge. Regarding the investigated bridge hangers, their diameter is 0.077 m, their modulus of elasticity is 1.9 × 10⁸ kpa, their bulk density is 7850 kN/m³, and their Poisson’s ratio is 0.3. Regarding the boundary conditions of the bridge hangers, they have a fixed constraint on the upper and lower.

COMSOL Multiphysics is a general finite element software that can be used to calculate fluid dynamics, solid mechanics, multifield coupling, and so forth [38]. Regarding the finite element model, the left boundary is the wind speed inlet, the right boundary is the outlet, and the upper and lower boundaries are slip boundaries, which are the same boundary conditions with a wind tunnel test. Regarding the bridge section, there are 9994 domain elements and 334 boundary elements, and the mesh is encrypted at the boundary. Regarding the finite element model, it has the four slip boundaries, the wind speed inlet, and the wind speed outlet, which can be seen in Figure 2a. Finally, the finite element model is divided into 35,377,709 units based on the finite volume method to solve the fluid governing equations, which can be seen in Figure 2b.

3.1. Influence of CFD Models

In this case study, the wind speed is 12 m/s and the wind direction is 90 degree. The changes in a bridge hanger’s wind field at different CFD models are calculated, which can be seen as Figure 3. The calculation result of the SST k-ω model is different from that of the standard k-ε model, realizable k-ε model, and standard k-ω model, such as the wake flow characteristics of a hanger. Regarding the calculation result of the standard k-ε model, realizable k-ε model, and standard k-ω model, their wind field is the same. Therefore, the standard k-ε model is selected to calculate the turbulence equation.

3.2. Influence of Wind Speed

In this case study, we assume the wind direction is 0 degree and the wind speed is adjustable. The changes in the bridge hangers’ stress at different wind speeds are calculated based on the k–ε model and dynamic equations, which can be seen in Figure 4. When the wind speed increases from 3 to 12 m/s, the von Mises stress of the bridge hangers is nonlinearly increasing.

3.3. Influence of Wind Direction

In this case study, the wind speed is 12 m/s and the wind direction is adjustable. The changes in the bridge hangers’ wind field at different wind directions are calculated based on the k–ε model and dynamic equations, which can be seen in Figure 5. On the one hand, the wind field is different at different heights with the same wind speed. On the other hand, when the wind direction increases from 0 to 90 degrees, the wind speed field changes too.

The changes in the bridge hangers’ stress at different wind directions are calculated based on the k–ε model and dynamic equations, which can be seen in Figure 4d and Figure 6.

4. Discussion and Conclusions

This case analyzes the finite element model of investigated bridge hangers through COMSOL Multiphysics software. The influences of CFD models, wind speed, and wind direction on the bridge hangers are investigated through case studies, and the von Mises stress of the bridge hangers is calculated based on fluid–solid coupling. The main conclusions drawn from this study are summarized as follows:

• The calculation result of the SST k-ω model is different from that of the standard k-ε model, realizable k-ε model, and standard k-ω model, such as the wake flow characteristics of a hanger. Therefore, we will use the standard k-ε model to calculate the von Mises stress time history curve of bridge hangers and then assess the fatigue life of bridge hangers in a future work.
• The larger is the wind speed, the larger is the effect on the wind field and stress for the investigated bridge hangers. For example, when the wind speed increases from 3 to 12 m/s, the stress of the bridge hangers nonlinearly increases. Therefore, the influence of the bridge hangers should be a concern under a strong wind speed during the operation.
• Regarding the different wind directions with the same wind speed, the influence of the bridge hangers is also different. For example, when the wind direction increases from 0 to 90 degrees, the fatigue life of the bridge hangers decreases. Therefore, the influence of wind direction on the investigated bridge hangers cannot be ignored.