# Analyzing Wind Effects on Long-Span Bridges: A Viable Numerical Modelling Methodology Using OpenFOAM for Industrial Applications

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## Abstract

**:**

## 1. Introduction

## 2. Modelling Assumptions

_{eff}(∇U + (∇U)

^{T})] − ∇p

_{eff}is the effective kinematic viscosity, which is the summation of laminar kinematic viscosity (ν) and the turbulent kinematic viscosity (ν

_{t}). The value of the laminar kinematic viscosity, ν, can be configured in the transportProperties file in the constant/directory, which is related to the temperature of the flow. For instance, ν has a value of 1.47 × 10

^{−5}m

^{2}/s at a temperature of 15 °C. In Equation (2), the ∂U/∂t term accounts for transience in the system (URANS) but is omitted for steady-state RANS simulations.

_{t}in RANS and URANS simulations can be calculated using various turbulence models. The use of the k-ω SST turbulence model [51] has become popular in the study of bridge aerodynamics [52,53,54] due to its ability to model turbulence behavior in the boundary layer [55]. It should be noted that there are more complex models that might deliver more accurate results, such as the four-equation $\overline{{v}^{2}}$-f model [56], the four-equation γ-Re

_{θ}model [57], and the seven-equation SSG/LRR-ω full Reynolds stress model [58]. Since these models contain more transport equations than the two-equation k-ω SST model, they often require significantly more computational power and are more challenging to converge. Furthermore, the implementation of some of these models, such as the γ-Re

_{θ}, also needs empirical correlations to configure inlet parameters, which requires greater effort in calibrating the model to fit specific engineering problems. Given that the purpose of the workflow in this paper is for the engineering design of long-span bridges, computational affordability is of significant importance. In this application, the k-ω SST turbulence model is both affordable and reliable and hence has been proposed for the application of modelling wind effects on bridges in RANS and URANS simulations. To solve the aforementioned governing equations with the k-ω SST turbulence model in OpenFOAM, the simpleFoam solver should be used for RANS simulations, and the pimpleFoam or pisoFoam solver should be used for URANS simulations. In the example case of URANS simulations, the pimpleFoam solver is used because it allows for large time steps, which potentially speeds up the transient simulations. It should be noted that if accurate predictions of turbulence structures are of interest, large-eddy simulations (LESs) [59] and detached-eddy simulations (DESs) [60] are recommended. These simulations can also be performed using the pimpleFoam or pisoFoam solver. However, compared to URANS, LESs and DESs require significantly more computational power. Figure 2 visually depicts key elements of the framework, including boundary conditions, the turbulence model, and discretization methods.

## 3. Geometry Development

## 4. Creating the Mesh

^{+}value below 1 so that the use of k-ω SST turbulence is valid [51]. This is especially important for simulations with geometries containing small details. In these cases, cells within the near-wall region are refined to fully resolve the geometric details and so often correspond to a small y

^{+}value of less than 30, i.e., low-Reynolds-Number (Low-Re) wall condition [68]. While OpenFOAM provides wall functions that can be adopted in Low-Re wall condition [69], wall functions do not perform well in the buffer zone where y

^{+}ranges from 5 to 30 [70]. Therefore, the most commonly adopted approach is to refine the boundary layer so that the buffer zone is avoided and the viscous sublayer is resolved by the mesh, i.e., the y

^{+}value is less than 1. Thirdly, the maximum non-orthogonality and the maximum value of skewness of internal and boundary cells within the mesh should be below 65, 4, and 20 degrees, respectively [67]. Finally, simulations with the selected mesh scheme should also be affordable. More specifically, the mesh is expected to have a cell count of less than 64 million, estimated by multiplying the maximum number of CPU cores (128) used for the study with the maximum number of cells proposed to be distributed on each CPU core (500 thousand). It would be expected that RANS simulations of this type of study with a limit on the cell count would be expected to be completed within 20 h for approximately 8000 SIMPLE iterations. However, transient simulations require significantly more computational efforts, and so URANS simulations of 1 s in this study with the same mesh as in RANS simulations are expected to be completed within a week. Due to these requirements, the development of a suitable mesh is an iterative process. Here, Figure 6 shows a proposed final mesh developed for this example case. A mesh sensitivity study for this example is presented later in Section 9.

^{+}value at the bridge surface should be lower than 1. In this example, a refinement of six levels is applied using snappyHexMesh, where five buffer layers were added between adjacent levels to provide a smooth transition between cells of different sizes. In addition, to better capture the flow characteristics in the region that is close to the geometry, a refinement box was used where cells were firstly refined to level 4 and then refined to level 6 at the surface of the geometry. This led to a thickness of approximately 7.8125 × 10

^{−4}m for the smallest internal cells, so the smallest geometrical detail with a thickness of 1.0 × 10

^{−3}m would be captured. Afterwards, the addLayerControl in snappyHexMeshDict was enabled to insert eight layers of boundary cells. All of the above schemes would lead to a mesh with approximately 30 million cells, with most of them hexahedrons. In this case, the y

^{+}value at the bridge surface was determined to be 0.45. The aspect ratio was 3.47.

## 5. Boundary Conditions

_{1}, P

_{2}, and P

_{3}are given to U, k, and ω, where P

_{2}and P

_{3}are related to P

_{1}, turbulence intensity (I), and turbulence length scale (l) and can be calculated using Equations (3) and (4):

^{2}

_{1}can be determined for the period of interest, and I and l can subsequently be determined. Another possibility is that there may be a nearby weather station from which data can be obtained. A third option is to perform complex meteorological simulations based on weather data, the results of which can also provide P

_{1}, I, and l. In some cases, for bridge construction projects, prior wind tunnel tests are sometimes performed, from which these values could be taken.

^{+}values are larger than 30, and hence kqRWallFunction, omegaWallFunction, and the nutkWallFunction are used for k, omega, and nut, respectively. At the patch of the bridge deck, the y

^{+}value is below 1, so the viscous sub-layer is resolved within the boundary cells. Theoretically, k can be configured with a value of 0 m

^{2}/s

^{2}. But it is configured with a small value (1 × 10

^{−10}m

^{2}/s

^{2}) to avoid float point issues related to division by a zero. It is noted that omegaWallFunction in OpenFOAM is a linear wall condition that fits both high-Reynolds-number and low-Reynolds-number conditions, and so it is configured for ω at the bridge patch. In contrast, the nutLowReWallFunction is configured for ν

_{t}at the bridge patch, which, according to Liu [69], is designed for ν

_{t}at the low-Reynolds-number condition.

## 6. Discretization of the Equations—fvSchemes

_{eff}(∇U)

^{T}. Laplacian terms are all discretized using the Gauss linear scheme with non-orthogonality correction. Interpolations are calculated using the linear scheme. Surface normal gradients are discretized using the non-orthogonality correction scheme, namely corrected. Wall distance is calculated using the meshWave method. It is noted that these spatial schemes are also used in the URANS simulations except that the time scheme is changed from steadyState to backward, which is a second-order implicit scheme for time discretization.

## 7. Solving Algorithms—fvSolution

^{−11}. The SIMPLE algorithm [74] is used to perform the pressure-velocity coupling. In OpenFOAM, the simpleFoam solver can adopt both the SIMPLE algorithm and the SIMPLEC algorithm [75], where the former is selected by setting the consistent variable to a false value. Before running the simpleFoam solver, the potentialFoam solver is run for 10 iterations to initialize velocity, pressure, and flux fields. The gradient of the velocity field is cached to speed up the solver.

## 8. Parallel Configurations—decomposeParDict

## 9. Mesh Sensitivity Study

^{−3}m, which corresponds to a y

^{+}value of approximately 10. The medium mesh (Figure 11b) has a background mesh that is one level finer than the coarse mesh. To better capture the flow characteristics in the region that is close to the geometry, a refinement box was used, where cells were firstly refined to level 4 and then refined to level 6 at the surface of the geometry. The smallest cell within the medium mesh has a thickness of approximately 7.8125 × 10

^{−4}m, which corresponds to a y

^{+}value of approximately 5. The finer mesh in Figure 11d has a similar scheme as the fine mesh (Figure 11c), except that it has a finer background mesh where cells within have a thickness of 0.03 m. Accordingly, the y

^{+}value of the finer mesh (approximately 0.25) is smaller than that of the fine mesh (approximately 0.45). Table 3 shows the cell counts of these meshes. It should be noted that with different configurations of the angle of attack, cell counts slightly vary for each mesh.

## 10. Post Processing

#### 10.1. Aerodynamic Force Coefficients

_{x}), lift force (F

_{z}), and pitching moment (M

_{y}), are calculated using the force function.

^{3}. The porosity variable is given a false value since porous effects are not considered in bridge deck aerodynamic studies. The CofR variable defines the center of rotation for calculating the pitching moment.

_{x}), lift forces (F

_{z}), and pitching moments (M

_{y}) are recorded at each time step and SIMPLE iteration in URANS simulations and RANS simulations, respectively. Summations of the viscous and pressure parts of each force should be taken to determine their full values. Then, aerodynamic coefficients of the bridge deck, namely the drag coefficient (C

_{Fx}), lift coefficient (C

_{Fz}), and moment coefficient (C

_{My}), are calculated in Equations (5)–(7) based on aerodynamic forces:

_{Fx}= 2 F

_{x/}ρU

^{2}HL

_{Fz}= 2 F

_{z/}ρU

^{2}BL

_{My}= 2 M

_{y/}ρU

^{2}B^2 L

_{x}, F

_{z}, M

_{y}are drag force, lift force, and pitching moment, respectively; ρ is the density of the air, which is 1.225 kg/m

^{3}and corresponds to the temperature at 15 °C; U is the reference velocity (P

_{1}); B is the effective width of the bridge deck (P

_{4}); H is the depth of the bridge deck (P

_{5}); and L is the span of the bridge deck (P

_{6}). Figure 13 shows the sample aerodynamic coefficients that would be calculated from the simulation results. These coefficients are essential to bridge design, especially in the topological optimization of bridge decks.

#### 10.2. Visualization

## 11. Validation of the Methodology

## 12. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 14.**Velocity plot on the central cross-section of the bridge deck. (

**a**) Velocity plot of a RANS simulation. (

**b**) Velocity plot of a URANS simulation.

**Figure 15.**LIC representation of the velocity plot on the central cross-section of the bridge deck. (

**a**) LIC representation of the velocity plot of a RANS simulation. (

**b**) LIC representation of the velocity plot of a URANS simulation.

**Figure 16.**Wall shear stress plot on the bridge deck surface. (

**a**) Wall shear stress on the bridge deck surface of a RANS simulation. (

**b**) Wall shear stress on the bridge deck surface of a URANS simulation.

**Figure 17.**Turbulence structures illustrated by the Q-criterion. (

**a**) Turbulence structures in a RANS simulation. (

**b**) Turbulence structures in a URANS simulation.

**Figure 18.**Effects of changes in mesh refinement level for velocities at Point CW on the QFC bridge, reproduced from [23].

**Figure 19.**Comparison of wind velocity magnitudes from the field data and CFD simulations for point A on the RFK bridge, reproduced from [46].

Patch Name | Parameter | Type of Condition in OpenFOAM | Value | Unit |
---|---|---|---|---|

Inlet | U | fixedValue | P_{1} | m/s |

p | zeroGradient | - | m^{2}/s | |

k | fixedValue | P_{2} | m^{2}/s | |

ω | fixedValue | P_{3} | s^{−1} | |

ν_{t} | calculated | - | m^{2}/s | |

Outlet | U | inletOutlet | - | m/s |

p | fixedValue | 0 | m^{2}/s^{2} | |

k | inletOutlet | - | m^{2}/s^{2} | |

ω | inletOutlet | - | s^{−1} | |

ν_{t} | calculated | - | m^{2}/s | |

Side walls | U | noSlip | 0 | m/s |

p | zeroGradient | - | m^{2}/s^{2} | |

k | kqRWallFunction | - | m^{2}/s^{2} | |

ω | omegaWallFunction | - | s^{−1} | |

ν_{t} | nutkWallFunction | - | m^{2}/s | |

Bridge | U | noSlip | 0 | m/s |

p | zeroGradient | - | m^{2}/s^{2} | |

k | fixedValue | 1 × 10^{−} | m^{2}/s^{2} | |

ω | omegaWallFunction | - | s^{−1} | |

ν_{t} | nutLowReWallFunction | - | m^{2}/s |

Number of CPU Cores | Wall-Clock Time (in h) | Speed-Up | Cells Per CPU Core |
---|---|---|---|

1 | 965.28 (estimated) | - | 33,189,094 |

16 | 60.33 | 16 | 1,937,500 |

32 | 35.88 | 26.903 | 968,750 |

64 | 16.31 | 59.172 | 484,375 |

128 | 7.84 | 123.023 | 242,188 |

256 | 3.61 | 267.175 | 121,094 |

Configurations | Coarse Mesh | Medium Mesh | Fine Mesh | Finer Mesh |
---|---|---|---|---|

−10° | 1,962,324 | 14,607,636 | 33,242,170 | 85,922,392 |

0° | 1,973,227 | 14,575,020 | 33,189,094 | 86,008,513 |

10° | 1,972,355 | 14,608,860 | 33,227,289 | 86,441,092 |

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© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhang, Y.; MacReamoinn, R.; Cardiff, P.; Keenahan, J.
Analyzing Wind Effects on Long-Span Bridges: A Viable Numerical Modelling Methodology Using OpenFOAM for Industrial Applications. *Infrastructures* **2023**, *8*, 130.
https://doi.org/10.3390/infrastructures8090130

**AMA Style**

Zhang Y, MacReamoinn R, Cardiff P, Keenahan J.
Analyzing Wind Effects on Long-Span Bridges: A Viable Numerical Modelling Methodology Using OpenFOAM for Industrial Applications. *Infrastructures*. 2023; 8(9):130.
https://doi.org/10.3390/infrastructures8090130

**Chicago/Turabian Style**

Zhang, Yuxiang, Reamonn MacReamoinn, Philip Cardiff, and Jennifer Keenahan.
2023. "Analyzing Wind Effects on Long-Span Bridges: A Viable Numerical Modelling Methodology Using OpenFOAM for Industrial Applications" *Infrastructures* 8, no. 9: 130.
https://doi.org/10.3390/infrastructures8090130