Numerical Simulation of Airﬂow in the Main Cable of Suspension Bridge with FPM Model

: The main cable of suspension bridges is subject to corrosion and requires advanced anti-corrosion technology. Consequently, the internal airﬂow of the main cable has become a signiﬁcant research focus. This study employs image processing and machine learning to analyze the cross-sectional images of the main cable and reveals the distribution characteristics of pores and fractures within the main cable cross-section. The numerical simulation model of the main cable is divided into inner and outer parts based on porosity, with porosity levels of 18.16% and 32.11%, respectively. Fractures randomly occurred in the inner part, with a probability of 31.37%. A simulation model based on fractured porous media (FPM) is developed, which innovatively incorporates the fracture ﬂow model into the numerical simulation of the internal airﬂow of the main cable. The numerical simulation clearly explores the intricate details of the internal ﬂow ﬁeld of the main cable, revealing that the existence of fractures has a great impact on the internal ﬂow ﬁeld of the main cable. Additionally, the relative deviation of speciﬁc frictional head loss between the ﬁeld experiment and numerical simulation is about 6.83%, indicating that the numerical simulation results are relatively reliable.


Introduction
Bridges are critical infrastructure in the process of economic development. Suspension bridges have become essential structures for spanning rivers and seas enabled by continuous advancements in bridge design and construction technology. The main cable is the principal load-bearing structure of the suspension bridge, and the quality of its anticorrosion is closely linked to the service lifetime of the bridge [1]. During transportation and construction, rainwater may wet the main cable, which cannot drain naturally after completion due to its structural limitations, leading to the retention of water between the wires of the main cable [2]. Additionally, the protective layer of the main cable may be cracked, allowing external rainwater and humid air to enter the main cable, leading to corrosion of the wires. Relevant studies [3] demonstrate that the different presence of moisture causes varying degrees of corrosion on the wires of the main cable, which subsequently affects the bearing capacity of the main cable.
In recent years, research on bridge cable anti-corrosion technology has also been continuously developing. A popular technology for corrosion resistance of the main cable is to deliver dry air into the pores of the main cable [4]. This technology operates by transporting dehumidified air into the interior of the main cable, which removes moisture and maintains low humidity inside the main cable to achieve its objectives. The anticorrosion efficiency is determined by the flow field in the main cable.
Image processing technology has experienced rapid development since its inception and is now widely used in various fields, including medicine, biology, and building materials [14][15][16]. Digital image processing transforms picture information into digital data, which can be manipulated by computers.
Therefore, preprocessing operations were first executed on the picture for subsequent image processing, as shown in Figure 1. Image segmentation algorithms were used to extract the porosity. An intricate part of the main cable cross-section served as the sample image to test image segmentation algorithms, as shown in Figure 2. Two image segmentation algorithms were assessed, and the most efficient algorithm was selected for further analysis.   Image processing technology has experienced rapid development since its inception and is now widely used in various fields, including medicine, biology, and building materials [14][15][16]. Digital image processing transforms picture information into digital data, which can be manipulated by computers.
Therefore, preprocessing operations were first executed on the picture for subsequent image processing, as shown in Figure 1. Image segmentation algorithms were used to extract the porosity. An intricate part of the main cable cross-section served as the sample image to test image segmentation algorithms, as shown in Figure 2. Two image segmentation algorithms were assessed, and the most efficient algorithm was selected for further analysis.   Researchers frequently employ the Otus algorithm and Sauvola algorithm [17] for image segmentation. Figure 3a illustrates the pores segmented by the Otus algorithm. The results show that the segmentation performance was satisfactory in areas where wire colors were well-defined. However, the Otus algorithm was susceptible to noise and non-target colors, leading to the misclassification of some noise areas as pores, resulting in errors in porosity calculation; this was due to the Otus algorithm utilizing a global threshold that was sensitive to image noise and target characteristics. When the contrast between the background and the target was indistinct, the segmentation performance might have been inadequate. As the image lacked a clear distinction between pores and wires, the segmentation effect of this algorithm was inferior.
Buildings 2023, 13, x FOR PEER REVIEW 4 of 20 Researchers frequently employ the Otus algorithm and Sauvola algorithm [17] for image segmentation. Figure 3a illustrates the pores segmented by the Otus algorithm. The results show that the segmentation performance was satisfactory in areas where wire colors were well-defined. However, the Otus algorithm was susceptible to noise and nontarget colors, leading to the misclassification of some noise areas as pores, resulting in errors in porosity calculation; this was due to the Otus algorithm utilizing a global threshold that was sensitive to image noise and target characteristics. When the contrast between the background and the target was indistinct, the segmentation performance might have been inadequate. As the image lacked a clear distinction between pores and wires, the segmentation effect of this algorithm was inferior.  In contrast, the Sauvola algorithm overcomes the limitations of the Otus algorithm by utilizing an adaptive threshold that considers the local image characteristics. Figure 3b illustrates the pores that were segmented by the Sauvola algorithm. The results show significant improvement in the segmentation accuracy of pores, particularly in areas with weak wire contrast. The Sauvola algorithm reduced the misclassification, resulting in a more accurate porosity calculation. Therefore, the Sauvola algorithm was a reliable method for segmenting pores in the main cable cross-section image.
After selecting the image segmentation algorithm, several assumptions were made to obtain the porosity:

•
This research used a 200 × 1120 pixels image to study the porosity of the main cable; • The pixels with grayscale 0 were considered as pores for subsequent calculations; • The influence of wrapping tape and other materials on the porosity was ignored; • The porosity of the main cable was about 20% [18], which was used as a standard for judging the correctness of porosity in this study.
Following the aforementioned assumptions, a representative part of the main cable cross-section was divided into five parts, labeled as a, b, c, d, and e, as shown in Figure 4. Subsequently, image segmentation and porosity calculation were conducted on each part to determine the porosity of the main cable. The formula for calculating the porosity was as follows: where ε is the porosity; poros S is the area of pores; total S is the area of the image. In contrast, the Sauvola algorithm overcomes the limitations of the Otus algorithm by utilizing an adaptive threshold that considers the local image characteristics. Figure 3b illustrates the pores that were segmented by the Sauvola algorithm. The results show significant improvement in the segmentation accuracy of pores, particularly in areas with weak wire contrast. The Sauvola algorithm reduced the misclassification, resulting in a more accurate porosity calculation. Therefore, the Sauvola algorithm was a reliable method for segmenting pores in the main cable cross-section image.
After selecting the image segmentation algorithm, several assumptions were made to obtain the porosity:

•
This research used a 200 × 1120 pixels image to study the porosity of the main cable; • The pixels with grayscale 0 were considered as pores for subsequent calculations; • The influence of wrapping tape and other materials on the porosity was ignored; • The porosity of the main cable was about 20% [18], which was used as a standard for judging the correctness of porosity in this study.
Following the aforementioned assumptions, a representative part of the main cable cross-section was divided into five parts, labeled as a, b, c, d, and e, as shown in Figure 4. Subsequently, image segmentation and porosity calculation were conducted on each part to determine the porosity of the main cable. The formula for calculating the porosity was as follows: where ε is the porosity; S poros is the area of pores; S total is the area of the image.  The porosity for each part is displayed in Figure 5. The average porosity of the selected area was 19.38%, which was close to 20%; however, the porosity abruptly increased in Part e. Using the average porosity to describe Part e would be inaccurate. Therefore, it was essential to consider the discrepancy in porosity to enhance the accuracy of the numerical simulation. As shown in Figure 6, Part e was divided into two regions based on porosity. Following partitioning, the thickness of the high porosity region was found to be 97 pixels, while The porosity for each part is displayed in Figure 5. The average porosity of the selected area was 19.38%, which was close to 20%; however, the porosity abruptly increased in Part e. Using the average porosity to describe Part e would be inaccurate. Therefore, it was essential to consider the discrepancy in porosity to enhance the accuracy of the numerical simulation.  The porosity for each part is displayed in Figure 5. The average porosity of the selected area was 19.38%, which was close to 20%; however, the porosity abruptly increased in Part e. Using the average porosity to describe Part e would be inaccurate. Therefore, it was essential to consider the discrepancy in porosity to enhance the accuracy of the numerical simulation. As shown in Figure 6, Part e was divided into two regions based on porosity. Following partitioning, the thickness of the high porosity region was found to be 97 pixels, while As shown in Figure 6, Part e was divided into two regions based on porosity. Following partitioning, the thickness of the high porosity region was found to be 97 pixels, while the radius of the cross-section was 1120 pixels. The high porosity region had a thickness proportion of 8.66% with a porosity of 32.11%. The remaining portion had a porosity of 18.16%. Through the above processing, the porosity of the FPM model was obtained. As shown in Figure 7, the model was mainly divided into two parts, the inner part with a porosity of 18.16%, and the outer part with a porosity of 32.11%.
Buildings 2023, 13, x FOR PEER REVIEW 6 of 20 the radius of the cross-section was 1120 pixels. The high porosity region had a thickness proportion of 8.66% with a porosity of 32.11%. The remaining portion had a porosity of 18.16%. Through the above processing, the porosity of the FPM model was obtained. As shown in Figure 7, the model was mainly divided into two parts, the inner part with a porosity of 18.16%, and the outer part with a porosity of 32.11%.

Establishment of Fracture Network
Besides the extraction of porosity, another crucial part of the FPM model is the establishment of the fracture network. Employing machine learning techniques [19] could enhance the efficiency of this creation process. In this study, Fast Random Forest (FRF) was used to identify fractures, an improved algorithm of Random Forest [20], which has the advantages of a fast running speed and no overfitting. FRF classified each pixel and assigned it to a corresponding segment or class. After labeling the image with specified features, it became a training set for the classifier, which could then recognize and segment other image inputs. The training and recognition process is depicted in Figure 8. Buildings 2023, 13, x FOR PEER REVIEW the radius of the cross-section was 1120 pixels. The high porosity region had a proportion of 8.66% with a porosity of 32.11%. The remaining portion had a po 18.16%. Through the above processing, the porosity of the FPM model was obt shown in Figure 7, the model was mainly divided into two parts, the inner pa porosity of 18.16%, and the outer part with a porosity of 32.11%.

Establishment of Fracture Network
Besides the extraction of porosity, another crucial part of the FPM model is t lishment of the fracture network. Employing machine learning techniques [19] hance the efficiency of this creation process. In this study, Fast Random Forest ( used to identify fractures, an improved algorithm of Random Forest [20], whic advantages of a fast running speed and no overfitting. FRF classified each pixe signed it to a corresponding segment or class. After labeling the image with spec tures, it became a training set for the classifier, which could then recognize and other image inputs. The training and recognition process is depicted in Figure 8

Establishment of Fracture Network
Besides the extraction of porosity, another crucial part of the FPM model is the establishment of the fracture network. Employing machine learning techniques [19] could enhance the efficiency of this creation process. In this study, Fast Random Forest (FRF) was used to identify fractures, an improved algorithm of Random Forest [20], which has the advantages of a fast running speed and no overfitting. FRF classified each pixel and assigned it to a corresponding segment or class. After labeling the image with specified features, it became a training set for the classifier, which could then recognize and segment other image inputs. The training and recognition process is depicted in Figure 8. Buildings 2023, 13, x FOR PEER REVIEW 7 of 20 Figure 8. Training process of FRF. First, the input image undergoes feature extraction to obtain its characteristics. Next, a collection of pixel samples is established and transformed into feature vectors. A learning algorithm is then trained on these vectors to classify the rest of the image data. Finally, the accuracy of identification can be improved by correcting or adding labels.
Several assumptions were made to simplify the establishment of the fracture network: • For the convenience of the statistics, the internal fractures of the hexagonal strands were not considered; • The outer layer of the main cable cross-section had a larger porosity. The establishment of the fracture network did not involve this part; • The fractures had different lengths. Therefore, when counting the number of fractures, it was necessary to integrate the fractures. This study took a typical fracture as a standard fracture, counted the number of pixels of the standard fracture, and then calculated the number of fractures; • This study used 1/4 of the main cable cross-section for analysis, found the probability of fracture, and extended the results to the entire main cable.
As illustrated in Figure 9, based on the assumptions, three fractures were selected as fracture samples to calculate the pixel number of a standard fracture. The number of pixels for the three fractures was 387, 410, and 384, so the number of pixels for a standard fracture was 393.7. Figure 8. Training process of FRF. First, the input image undergoes feature extraction to obtain its characteristics. Next, a collection of pixel samples is established and transformed into feature vectors. A learning algorithm is then trained on these vectors to classify the rest of the image data. Finally, the accuracy of identification can be improved by correcting or adding labels.
Several assumptions were made to simplify the establishment of the fracture network: • For the convenience of the statistics, the internal fractures of the hexagonal strands were not considered; • The outer layer of the main cable cross-section had a larger porosity. The establishment of the fracture network did not involve this part; • The fractures had different lengths. Therefore, when counting the number of fractures, it was necessary to integrate the fractures. This study took a typical fracture as a standard fracture, counted the number of pixels of the standard fracture, and then calculated the number of fractures; • This study used 1/4 of the main cable cross-section for analysis, found the probability of fracture, and extended the results to the entire main cable.
As illustrated in Figure 9, based on the assumptions, three fractures were selected as fracture samples to calculate the pixel number of a standard fracture. The number of pixels for the three fractures was 387, 410, and 384, so the number of pixels for a standard fracture was 393.7. Buildings 2023, 13, x FOR PEER REVIEW 8 of 20 Figure 9. Fracture samples.
The results of identifying fractures in 1/4 of the main cable cross-section using FRF are presented in Figure 10a. The number of pixels representing fractures in this range was 13,339, and the number of fractures was calculated to be 33.9 by the following formula: Assuming that when fractures exist between all hexagonal strands in 1/4 of the main cable cross-section, as shown in Figure 10b, the total number of fractures would be 107.91. The probability of fracture occurrence was found to be 31.37%. Extrapolating to the entire fracture network, the number of fractures was 155.8. For the convenience of modeling, the number of fractures was rounded to 156. The results of identifying fractures in 1/4 of the main cable cross-section using FRF are presented in Figure 10a. The number of pixels representing fractures in this range was 13,339, and the number of fractures was calculated to be 33.9 by the following formula: where the N f racture is the number of fractures in 1/4 of the main cable cross-section; n identi f y is the number of pixels identified as fractures in 1/4 of the main cable cross-section; n standard is the number of pixels of a standard fracture. After obtaining the number of fractures, the fracture network could be established. The process is shown in Figure 11. The diameter of the main cable in this study was 855 mm, and the diameter of the wires was 5.25 mm. Each hexagonal strand comprised 127 wires, and based on the arrangement rule of the wires, the side length of the smallest circumscribed hexagon of the strand could be calculated as 34.53 mm. The fractures were  Assuming that when fractures exist between all hexagonal strands in 1/4 of the main cable cross-section, as shown in Figure 10b, the total number of fractures would be 107.91. The probability of fracture occurrence was found to be 31.37%. Extrapolating to the entire fracture network, the number of fractures was 155.8. For the convenience of modeling, the number of fractures was rounded to 156.
After obtaining the number of fractures, the fracture network could be established. The process is shown in Figure 11. The diameter of the main cable in this study was 855 mm, and the diameter of the wires was 5.25 mm. Each hexagonal strand comprised 127 wires, and based on the arrangement rule of the wires, the side length of the smallest circumscribed hexagon of the strand could be calculated as 34.53 mm. The fractures were drawn by calculating the position of the hexagon center, adhering to the rule that fractures exist between hexagonal strands. Finally, the fractures were randomly deleted, leaving only 156 as the final fracture network. After obtaining the number of fractures, the fracture network could be established. The process is shown in Figure 11. The diameter of the main cable in this study was 855 mm, and the diameter of the wires was 5.25 mm. Each hexagonal strand comprised 127 wires, and based on the arrangement rule of the wires, the side length of the smallest circumscribed hexagon of the strand could be calculated as 34.53 mm. The fractures were drawn by calculating the position of the hexagon center, adhering to the rule that fractures exist between hexagonal strands. Finally, the fractures were randomly deleted, leaving only 156 as the final fracture network. After obtaining the porosity and fracture network, the numerical simulation model can be constructed. Since the longitudinal variation in the main cable is small and can be ignored, the model can be obtained by stretching, as shown in Figure 12.  After obtaining the porosity and fracture network, the numerical simulation model can be constructed. Since the longitudinal variation in the main cable is small and can be ignored, the model can be obtained by stretching, as shown in Figure 12. After obtaining the number of fractures, the fracture network could be established. The process is shown in Figure 11. The diameter of the main cable in this study was 855 mm, and the diameter of the wires was 5.25 mm. Each hexagonal strand comprised 127 wires, and based on the arrangement rule of the wires, the side length of the smallest circumscribed hexagon of the strand could be calculated as 34.53 mm. The fractures were drawn by calculating the position of the hexagon center, adhering to the rule that fractures exist between hexagonal strands. Finally, the fractures were randomly deleted, leaving only 156 as the final fracture network. After obtaining the porosity and fracture network, the numerical simulation model can be constructed. Since the longitudinal variation in the main cable is small and can be ignored, the model can be obtained by stretching, as shown in Figure 12.

Judgment of Flow Regime
Understanding the flow regime inside the main cable was crucial in selecting the appropriate physical laws for numerical simulation. The energy loss laws vary under different flow regimes. Typically, the determination of the flow regime depends on the Reynolds number (Re), which can be calculated using the following formula: where Re is the Reynolds number; v is the seepage velocity; d is the characteristic diameter of the porous medium; ν is the kinematic viscosity. Fluid flow in porous media was influenced by both the porosity and solid skeleton, resulting in complex flow conditions. In porous media, the flow regimes could be divided into four categories based on the Re. For Re values less than 1, the flow resistance was primarily dominated by viscous forces. As Re increased within the range of 1 to 10, boundary layers emerged near solid surfaces. As the Re further increased, the effect of the boundary layers on the overall flow also became more significant. When Re surpassed 150, the flow became unsteady. When the Re exceeded 300, the flow could be regarded as turbulent.
As the study of flow regime inside porous media deepens, Macdonald et al. [21] introduced a revised formula for Re: where Re * is the modified Reynolds number; ε is the porosity of the porous medium.
In this study, the flow regime inside the main cable under different flow rates was determined using the formula proposed by Macdonald, with the porosity of the main cable being 19.38%. The results are presented in Table 1, showing that the Reynolds number did not exceed 6 when the flow rate was within 0-100 m 3 /h. Although boundary layers were present, their impact on the overall flow was insignificant, and the flow regime could be considered laminar. Thus, subsequent analysis of the airflow within the main cable was conducted based on the laminar. In 1856, Darcy, a French scientist [22], conducted experiments on water passing through saturated sand and summarized Darcy's law. When water flows through soil or sand, it loses some energy because of infiltration. Darcy's law defines the corresponding relationship between energy loss and flow velocity. According to Darcy's law, the velocity field is determined by the pressure gradient, fluid viscosity, and the structure of the porous medium.
where u is the velocity vector; κ is the permeability of porous media; µ is the dynamic viscosity of the fluid; ∇p is pressure.

Fracture Flow Equations
In fracture flow, the two sides of the fracture are typically assumed to be parallel. The flow regime in such fractures is laminar. The cubic law is used to describe the fracture flow model, where the flow velocity is proportional to the cube of the fracture width after calculating the average flow velocity of the fluid in the fracture and obtaining the fracture width. This law assumes that the fluid is viscous and incompressible, and its permeability is proportional to the square of the fracture width.
where κ s is the permeability; ρ is the fluid density; g is the acceleration of gravity; d F is the width of the fracture; µ is the fluid dynamic viscosity; f is the roughness coefficient; T s is the coefficient of transmissivity. After obtaining the cubic law, the governing equation for fracture flow can be obtained: where Q F is the volumetric flow rate per unit length of fracture; ∇ T is the gradient operator; D are the coordinates; S is the mass source.

Permeability Based on Kozeny-Carman Equation
The Kozeny-Carman equation [23] (KC equation) is a well-known semi-empirical formula. It is widely used to calculate the permeability of porous media in fields such as seepage, oil and gas extraction, chemical engineering, and bioelectrochemistry.
where κ is the permeability of porous media; d p is the particle size; ε is the porosity of porous media.

Permeability Based on Numerical Simulation
The fluid flow inside the main cable is shown in Figure 13. When the air flowed radially through the irregular holes, it experienced significant resistance, while the resistance was low when the air flowed longitudinally through the narrow regions; therefore, the permeability of the main cable could be different depending on the direction of the air flow.
Due to the lack of research on the multidirectional permeability of the main cable and the complexity of measuring the permeability through field experiments, numerical simulation was employed to obtain the permeability in different directions. Figure 14 shows the model used to calculate the radial and longitudinal permeability of the main cable. The radial model, as shown in Figure 14a, represents the arrangement of the wires inside the main cable in the radial. The longitudinal model, conversely, adopted a unit length of the longitudinal pore as a unit pore, as represented in Figure 14b. The calculation formula for permeability is presented below: where the κ is the permeability; Q is the flow rate of fluid through porous media per unit time; µ is the viscosity of the fluid; L is the length of fluid flowing through porous media; ∆P is the pressure difference before and after the liquid flows through the porous medium; A is the cross-sectional area of the porous medium.

Permeability Based on Numerical Simulation
The fluid flow inside the main cable is shown in Figure 13. When the air flowed radially through the irregular holes, it experienced significant resistance, while the resistance was low when the air flowed longitudinally through the narrow regions; therefore, the permeability of the main cable could be different depending on the direction of the air flow. Due to the lack of research on the multidirectional permeability of the main cable and the complexity of measuring the permeability through field experiments, numerical simulation was employed to obtain the permeability in different directions. Figure 14 shows the model used to calculate the radial and longitudinal permeability of the main cable. The radial model, as shown in Figure 14a, represents the arrangement of the wires inside the main cable in the radial. The longitudinal model, conversely, adopted a unit length of the longitudinal pore as a unit pore, as represented in Figure 14b. The calculation formula for permeability is presented below: where the κ is the permeability; Q is the flow rate of fluid through porous media per unit time; μ is the viscosity of the fluid; L is the length of fluid flowing through porous media; P Δ is the pressure difference before and after the liquid flows through the porous medium; A is the cross-sectional area of the porous medium.

Comparison and Optimization of Two Methods for Obtaining Permeability
In the radial direction, a discrepancy was observed between the permeability values obtained from the KC equation and numerical simulation, as shown in Figure 15a. In the longitudinal direction, a significant difference was observed between the permeability values calculated by the KC equation and those obtained by numerical simulation, as shown in Figure 15b. The investigation revealed that permeability was incorrectly calculated by the KC equation, so the numerical simulation should be employed to obtain permeability.

Comparison and Optimization of Two Methods for Obtaining Permeability
In the radial direction, a discrepancy was observed between the permeability values obtained from the KC equation and numerical simulation, as shown in Figure 15a. In the longitudinal direction, a significant difference was observed between the permeability values calculated by the KC equation and those obtained by numerical simulation, as shown in Figure 15b. The investigation revealed that permeability was incorrectly calculated by the KC equation, so the numerical simulation should be employed to obtain permeability.
To simplify the acquisition of permeability, this study performed a fitting of permeability with porosity. The results are presented in Figure 16

Comparison and Optimization of Two Methods for Obtaining Permeability
In the radial direction, a discrepancy was observed between the permeability values obtained from the KC equation and numerical simulation, as shown in Figure 15a. In the longitudinal direction, a significant difference was observed between the permeability values calculated by the KC equation and those obtained by numerical simulation, as shown in Figure 15b. The investigation revealed that permeability was incorrectly calculated by the KC equation, so the numerical simulation should be employed to obtain permeability. To simplify the acquisition of permeability, this study performed a fitting of permeability with porosity. The results are presented in Figure 16

Simulation Models and Boundary Conditions
Previous studies have commonly made two key assumptions. Firstly, they hypothesized that the interior of the main cable was a porous medium without fractures. Secondly, they assumed that the porosity and permeability along the main cable would not change.
To compare the differences in the internal flow field between the commonly used PM model and the FPM model developed in this study, two numerical simulation models were created, as shown in Figure 17

Simulation Models and Boundary Conditions
Previous studies have commonly made two key assumptions. Firstly, they hypothesized that the interior of the main cable was a porous medium without fractures. Secondly, they assumed that the porosity and permeability along the main cable would not change.
To compare the differences in the internal flow field between the commonly used PM model and the FPM model developed in this study, two numerical simulation models were created, as shown in Figure 17.
Both models have the same length and boundary conditions, which were established based on the actual air supply scenario of a suspension bridge main cable. The inlet and outlet were annular with a width of 1 m, the two ends of the models were set as Symmetrical Boundary 1 and Symmetrical Boundary 2. The specific data of the simulation model are listed in Table 2. By comparing the discrepancies between the two models, significant conclusions were drawn.

Simulation Models and Boundary Conditions
Previous studies have commonly made two key assumptions. Firstly, they hypothesized that the interior of the main cable was a porous medium without fractures. Secondly, they assumed that the porosity and permeability along the main cable would not change. To compare the differences in the internal flow field between the commonly used PM model and the FPM model developed in this study, two numerical simulation models were created, as shown in Figure 17. Both models have the same length and boundary conditions, which were established based on the actual air supply scenario of a suspension bridge main cable. The inlet and outlet were annular with a width of 1 m, the two ends of the models were set as Symmetrical Boundary 1 and Symmetrical Boundary 2. The specific data of the simulation model are listed in Table 2. By comparing the discrepancies between the two models, significant conclusions were drawn.

Model Verification
The pressure distribution of the main cable cross-section could not be measured in field experiments. Hence, obtaining the resistance loss along the main cable and calculating the specific frictional head loss were often used to judge the reliability of numerical simulations. This study considered the pressure drop near the inlet and outlet of the main cable as the local resistance loss, while the pressure drop along the main cable was considered as the resistance loss along the main cable. As depicted in Figure 18, since the inlet and outlet of the model were circular, the air velocity flowing into the main cable could be separated into radial and longitudinal velocities. The angle θ between the velocity and longitudinal velocity remained constant despite changes in the inlet pressure. When θ was less than 0.01 • , the energy loss could be perceived as only relating to the distance along the main cable.

Model Verification
The pressure distribution of the main cable cross-section could not be measured in field experiments. Hence, obtaining the resistance loss along the main cable and calculating the specific frictional head loss were often used to judge the reliability of numerical simulations. This study considered the pressure drop near the inlet and outlet of the main cable as the local resistance loss, while the pressure drop along the main cable was considered as the resistance loss along the main cable. As depicted in Figure 18, since the inlet and outlet of the model were circular, the air velocity flowing into the main cable could be separated into radial and longitudinal velocities. The angle θ between the velocity and longitudinal velocity remained constant despite changes in the inlet pressure. When θ was less than 0.01°, the energy loss could be perceived as only relating to the distance along the main cable. In this study, pressure and velocity components were obtained at 0.1 m intervals from the inlet to the outlet, and the angle between the velocity components was calculated (see Figure 19). The orange-shaded area in the figure represented the resistance loss along the main cable, while the inlet and the outlet affected the remaining area. Upon calculation, In this study, pressure and velocity components were obtained at 0.1 m intervals from the inlet to the outlet, and the angle between the velocity components was calculated (see Figure 19). The orange-shaded area in the figure represented the resistance loss along the main cable, while the inlet and the outlet affected the remaining area. Upon calculation, the local resistance losses at the inlet and outlet were determined to be 42.90 Pa and 42.91 Pa, respectively. The specific frictional head loss of the main cable could be calculated using the following formula: where R f is the specific frictional head loss; P T is the total pressure drop; ∇P In is the local resistance loss of the inlet; ∇P Out is the local resistance loss of the outlet; L is the length between the inlet and outlet of the main cable; l In is the length affected by local resistance loss of the inlet; l Out is the length affected by local resistance loss of the outlet. After calculation, at an inlet pressure of 200 Pa (inlet flow rate of 33.13 m 3 /h), the specific frictional head loss of the main cable was found to be 19.56 Pa/m. The reliability of numerical simulations was verified by conducting field experiments on the main cable of the Xihoumen Bridge. Table 3 presents the results of the experiments, which include the resistance loss and specific frictional head loss of the main cable.  1 The length of the main cable after removing the influence of local resistance at the inlet and outlet.
The specific frictional head loss obtained from numerical simulation, converted under the same inlet conditions, was 37.75 Pa/m. The relative deviation between the field experiment and the numerical simulation was 6.83%, indicating that the numerical simulation was accurate.

Pressure Field
After conducting numerical simulations, the pressure distribution of different models was analyzed. Pressure is crucial for air transportation inside the main cable, and a positive pressure inside the cable was necessary to prevent the intrusion of external high- After calculation, at an inlet pressure of 200 Pa (inlet flow rate of 33.13 m 3 /h), the specific frictional head loss of the main cable was found to be 19.56 Pa/m. The reliability of numerical simulations was verified by conducting field experiments on the main cable of the Xihoumen Bridge. Table 3 presents the results of the experiments, which include the resistance loss and specific frictional head loss of the main cable.  1 The length of the main cable after removing the influence of local resistance at the inlet and outlet.
The specific frictional head loss obtained from numerical simulation, converted under the same inlet conditions, was 37.75 Pa/m. The relative deviation between the field experi-ment and the numerical simulation was 6.83%, indicating that the numerical simulation was accurate.

Pressure Field
After conducting numerical simulations, the pressure distribution of different models was analyzed. Pressure is crucial for air transportation inside the main cable, and a positive pressure inside the cable was necessary to prevent the intrusion of external high-humidity air or liquid water. Figure 20 shows the pressure distribution of the main cable section located 1.5 m from Symmetric Boundary 1.  Figure 20a shows the pressure distribution in the section of the PM model, displaying a characteristic of high outer pressure and low inner pressure. Figure 20b shows the pressure distribution in the section of the FPM model, also exhibiting high outer pressure and low inner pressure. However, the pressure has a significant difference near the fractures. When fractures appeared in an area where the outer pressure was high, the pressure around the fractures would decrease; when fractures appeared in an area where the inner pressure was low, the pressure around the fractures would increase. The existence of fractures caused the pressure on the main cable section to fluctuate. The findings from previous studies align with the pressure distribution pattern observed in the PM model, characterized by high outer pressure and low inner pressure. However, the FPM model presented the impact of fractures on the pressure distribution, an aspect that previous research failed to recognize.
Different stacking configurations of wires exist within the main cable of a suspension bridge. When air enters the main cable, it first permeates through the outer part with high porosity, resulting in a high external pressure distribution within the main cable section. In the FPM model, the addition of fractures shows more details of the internal flow field of the main cable.
The FPM model was used to analyze the longitudinal pressure distribution of the main cable. Three cross-sections positioned at 3 m, 6 m, and 9 m from Symmetric Boundary 1 were selected for the analysis. The pressure distributions are presented in Figure 21.   Figure 20a shows the pressure distribution in the section of the PM model, displaying a characteristic of high outer pressure and low inner pressure. Figure 20b shows the pressure distribution in the section of the FPM model, also exhibiting high outer pressure and low inner pressure. However, the pressure has a significant difference near the fractures. When fractures appeared in an area where the outer pressure was high, the pressure around the fractures would decrease; when fractures appeared in an area where the inner pressure was low, the pressure around the fractures would increase. The existence of fractures caused the pressure on the main cable section to fluctuate. The findings from previous studies align with the pressure distribution pattern observed in the PM model, characterized by high outer pressure and low inner pressure. However, the FPM model presented the impact of fractures on the pressure distribution, an aspect that previous research failed to recognize.
Different stacking configurations of wires exist within the main cable of a suspension bridge. When air enters the main cable, it first permeates through the outer part with high porosity, resulting in a high external pressure distribution within the main cable section. In the FPM model, the addition of fractures shows more details of the internal flow field of the main cable.
The FPM model was used to analyze the longitudinal pressure distribution of the main cable. Three cross-sections positioned at 3 m, 6 m, and 9 m from Symmetric Boundary 1 were selected for the analysis. The pressure distributions are presented in Figure 21. The figure illustrates that there are noticeable variations in the pressure distribution across different cross-sections in the FPM model. Table 4 shows the results, indicating that the initial entry of air into the main cable was met with substantial resistance. However, as the air advanced toward the outlet, there was a noticeable shift in the pressure distribution. The resistance of the main cable near the outlet limited air flow, causing a rise in internal pressure.
In the FPM model, the addition of fractures shows more details of the internal flow field of the main cable.
The FPM model was used to analyze the longitudinal pressure distribution of the main cable. Three cross-sections positioned at 3 m, 6 m, and 9 m from Symmetric Boundary 1 were selected for the analysis. The pressure distributions are presented in Figure 21. The figure illustrates that there are noticeable variations in the pressure distribution across different cross-sections in the FPM model.  Table 4 shows the results, indicating that the initial entry of air into the main cable was met with substantial resistance. However, as the air advanced toward the outlet, there was a noticeable shift in the pressure distribution. The resistance of the main cable near the outlet limited air flow, causing a rise in internal pressure.  Figure 22 shows the velocity distributions for two models at a distance of 1.5 m from Symmetric Boundary 1. In the PM model and the FPM model, the velocity was higher in the region with high porosity, which was consistent with previous studies. However, there were some differences in the FPM model, as depicted in Figure 22b. The area where the velocity inside the main cable fluctuated was near the fractures. This finding was different from results reported in the relevant literature, where the internal velocity was low and devoid of fluctuations. The presence of fractures raised the average flow velocity in the cross-section, implying that the same pressure could transport the air further and the energy for air delivery could be saved for the same flow rate. The existence of fractures increased the airflow velocity at the same pressure, improving the flow conditions.   Figure 22 shows the velocity distributions for two models at a distance of 1.5 m from Symmetric Boundary 1. In the PM model and the FPM model, the velocity was higher in the region with high porosity, which was consistent with previous studies. However, there were some differences in the FPM model, as depicted in Figure 22b. The area where the velocity inside the main cable fluctuated was near the fractures. This finding was different from results reported in the relevant literature, where the internal velocity was low and devoid of fluctuations. The presence of fractures raised the average flow velocity in the cross-section, implying that the same pressure could transport the air further and the energy for air delivery could be saved for the same flow rate. The existence of fractures increased the airflow velocity at the same pressure, improving the flow conditions.

Velocity Field
Based on the aforementioned research, it can be inferred that by optimizing the distribution of fractures within the main cable, particularly by concentrating the fractures in the areas where residual water is present inside the main cable, the flow velocity in high-humidity areas can be improved. This optimization can accelerate the removal of water and solve the problem of high humidity in some areas of the main cable. were some differences in the FPM model, as depicted in Figure 22b. The area where the velocity inside the main cable fluctuated was near the fractures. This finding was different from results reported in the relevant literature, where the internal velocity was low and devoid of fluctuations. The presence of fractures raised the average flow velocity in the cross-section, implying that the same pressure could transport the air further and the energy for air delivery could be saved for the same flow rate. The existence of fractures increased the airflow velocity at the same pressure, improving the flow conditions.

Conclusions
This study employs image processing and FRF to obtain the porosity and fracture network of the main cable, developing the FPM model for numerical simulation. Additionally, the flow regime in the main cable is analyzed, and the permeability of the main cable is also calculated. Finally, numerical simulations are performed with the PM model and the FPM model, the results are obtained and validated through field experiments. Based on this study, the following conclusions can be drawn:

•
The FPM model for numerical simulation of the main cable mainly comprises two parts: an outer part with a porosity of 32.11% and an inner part with a porosity of 18.16%. The fractures are randomly distributed within the inner part with a probability of 31.37%; • For airflow rates between 10 m 3 /h and 100 m 3 /h, the Reynolds number ranges from 0.506 to 5.060, indicating that the flow regime inside the main cable is laminar; • The KC equation yielded different results than numerical simulations for the permeability of the porous medium. The permeability calculated through numerical simulation is more consistent with the actual situation; • The permeability demonstrates an exponential relationship with porosity, when comparing the radial permeability and the longitudinal permeability under the same porosity, a difference of approximately one order of magnitude is observed; • The pressure distribution along the main cable is not constant. Specifically, the pressure near the inlet exhibited a high external pressure and low internal pressure trend, while the opposite is observed near the outlet. This phenomenon can be attributed to the radial permeability of the main cable being lower than the longitudinal permeability, which may make it difficult for air to flow in the radial direction of the main cable; • The fractures have an impact on the distribution of airflow inside the main cable, causing fluctuations in pressure and flow velocity near the fractures, and the impact varied at different positions. The presence of fractures inside the main cable facilitated the entrance of air, increasing the airflow velocity, and promoting smoother flow conditions. Optimizing the distribution of fractures can solve the problem of high humidity in some areas.
Author Contributions: Writing-original draft preparation, W.S.; writing-review and editing, X.C. and P.P.; validation, Z.G. and P.P.; methodology, W.S. and X.C.; software, W.S. and Q.L.; data curation, Z.G. and H.G.; resources, H.G. and X.Z.; project administration, H.G. and X.Z. All authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.
Data Availability Statement: Some or all data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.