Effects of Floating Debris on Flow Characteristics around Slotted Bridge Piers: A Numerical Simulation

: Bridge pier scouring is a significant concern in hydraulic engineering, requiring thorough investigation under various conditions to estimate maximum scour depth and mitigate the risk of bridge failure. This study aims to conduct a numerical simulation of flow around a bridge pier with slots in the presence of floating debris, with the objective of analyzing variations in parameters such as velocity, shear stress, turbulent intensity, and turbulent kinetic energy. The FLOW − 3D software package (Version 11), along with the k − ε (RNG) turbulence model, was employed for the simulation. The results indicate that the presence of a slot in the bridge pier provided a smooth pathway for the flow, resulting in a reduction in the pressure gradient and alleviating the negative impacts on the flow. This, in turn, led to a decrease in the velocity of the flow. Additionally, turbulence intensity around the pier ranges between 0 and 49, while turbulent kinetic energy varies from 0 to 0.005. The findings reveal that models without slots exhibit higher turbulence and vorticity levels, as well as greater flow separation, compared to models with slots. This disparity can be attributed to the slot’s ability to neutralize detrimental lateral and downward flows. Furthermore, the results demonstrate a gradual decrease in shear stress as the flow approaches slotted bridge piers, accompanied by a reduction in vortex intensity. These findings suggest that the accumulation of floating debris can counteract the influence of slots in reducing scour around bridge piers, necessitating thorough consideration during the design phase.


Introduction
Scouring is a crit1ical hydraulic phenomenon that plays a fundamental role in various river engineering problems, often leading to damage in river−crossing structures [1].Bridge piers are particularly vulnerable to erosion, making it crucial to estimate the maximum depth of erosion around them to prevent bridge failures [2][3][4].The flow pattern around bridge piers is highly complex, further intensified by the formation of scour holes.These scour holes develop around the piers, undermining their foundations and causing bridge failure [5].The scouring depth becomes particularly significant when it approaches the foundation of the structure, posing a severe threat to its stability [6].Additionally, in natural rivers, the flow pattern around bridge piers constantly changes due to the high velocity of the flow, exposing the riverbed around the piers to significant local scour [7].
One of the primary approaches to controlling and reducing scour in bridge piers is increasing the resistance of the bed particles by incorporating erosion-resistant materials such as riprap, concrete blocks, and geo-bags [8][9][10].Another approach is changing the flow pattern through a collar, deflectors, bed sills, sacrificial piles, or slot.The shape of the pier and the degree of scour can significantly influence the flow pattern around the pier, so square piers exhibiting downward flow and horseshoe vortex compared to circular piers [11,12].
The use of slots in bridge pier scouring has gained significant attention recent years due to its potential effectiveness in mitigating scouring around bridge foundations [13][14][15].
Water 2024, 16, 90 2 of 13 Slots are hydraulic features incorporated into the bridge piers, designed to alter the flow characteristics and reduce erosive forces.By strategically placing slots, the flow patterns can be modified.These alterations disrupt the formation and development of scour holes, effectively reducing the scour depth and protecting the stability of the bridge piers [16,17] The utilization of slots as a scour countermeasure offers a promising approach that combines hydraulic engineering principles with the aim of minimizing the detrimental effects of scour, enhancing the longevity and safety of bridges in river environments [18,19].The configuration of slots in the pier, including their location, width, and length, plays a significant role in determining the flow characteristics around bridge piers equipped with slots [20].Despite the challenges posed by construction, geotechnical considerations, and economic factors, the theoretical benefits of incorporating slots in bridge piers have been recognized, thus warranting their recommendation [21,22].
The obstruction caused by the floating debris disrupts the natural flow regime and leads to flow irregularities, including flow contraction, local turbulence, and increased flow velocities [23,24].These alterations in flow patterns can create localized areas of increased velocity and turbulence, which directly influence the scouring potential.Also, the accumulation of floating debris upstream of bridge piers intensifies the scouring process through various mechanisms [25][26][27].The turbulent flow, combined with the erosive forces exerted by the moving debris, enhances the erosion and transport of sediment particles, resulting in deeper scour depths around the bridge piers [28,29].The geometry of the floating debris influences the flow contraction, vorticity, and hydraulic gradients, which collectively contribute to enhanced scour development.Studies have shown that there is a direct correlation between the thickness of the floating debris and the depth of scour [30].
Recent advancements in numerical methods, the availability of powerful fluid dynamics processors, and the utilization of various turbulence models have facilitated the numerical analysis of hydraulic phenomena, such as scouring around and downstream of hydraulic structures [31,32].Numerical simulations have proven capable of providing logical and accurate predictions of maximum scour depth [33].
Studies focusing on the flow patterns around bridge foundations have shown that piers with variable and conical cross-sections yield similar results in flow pattern analysis and bed shear stress to those obtained from laboratory experiments, as opposed to cylindrical foundations [34][35][36].The scale effect on flow patterns, scour, and turbulence around bridge piers has been investigated using computational fluid dynamics, revealing significant changes in scour depth and turbulence over time and as the flow pattern around the pier evolves [37].Numerical studies have also highlighted the sensitivity of local scouring to the shape and orientation of the pier [38].Three-dimensional numerical simulations of local scour around circular bridge piers with ice accumulation have demonstrated good agreement with experimental observations [39].Moreover, numerical investigations on local scour around bridge piers with protective blocks have shown a substantial reduction of up to 84% and 79% in the maximum scour depth in front of and beside the foundation, respectively [40].Some studies show that mechanical factors predominantly influence the movement of ice pieces, with the presence of ice scour holes influencing the development of ice waves [41].This research's main objective is to numerically investigate the impact of floating debris accumulation upstream of bridge piers on the reduction in scour around the pier in the presence of a slot.By filling this research gap, we aim to contribute to a comprehensive understanding of the complex interactions between slot, floating debris, and scouring, providing valuable insights for the design and maintenance of bridges in riverine environments.

Materials and Methods
In this study, the data used for investigating scour around bridge foundations in the presence of floating debris and a slot were derived from laboratory experiments conducted by [42].The experiments took place in a flume with dimensions of 9 m in length, 0.4 m in width, and 0.6 m in height.To prevent ripple formation at threshold velocities [43], a 0.2 m-thick layer of uniform sand particles with a median diameter of 0.8 mm, having a specific gravity of 2.65 g/cm 3 and a geometric standard deviation (σg = √ (d 84 /d 16 )) of 1.3, was placed as the bed material in the flume.A cylindrical bridge pier with diameter (D) of 0.04 m was in the middle of the sediment reach.To investigate the effect of the slot on scour around the pier, a 0.1 m-wide opening (equivalent to one-fourth of the pier width) with the same height as the flow depth was utilized [44].Additionally, to study the impact of floating debris on scour, cylindrical debris with a diameter of 0.12 m and a length of 200 mm (across the flume) was attached to the pier, ensuring that the top of the debris was tangent to the water surface to make it more realistic to field conditions.The experiments were conducted at a discharge rate of 13 lit/s as it was used in the laboratory study.
FLOW−3D is one of the most important features of this model is the 3D simulation of flow in open channels, closed channels, irrigation, porous media, etc. [45].
For the numerical simulations in this study, the FLOW−3D software developed by Flow Science Inc. was employed.FLOW−3D is a computational fluid dynamics software based on the finite volume method.FLOW−3D is known for its advanced numerical algorithms and models that accurately capture complex flow phenomena.Also, it incorporates sophisticated turbulence models, allowing for a detailed examination of turbulent flow patterns.In the context of this study, where understanding turbulence is crucial for assessing scour phenomena, the use of FLOW−3D facilitates a comprehensive analysis of flow turbulence around bridge piers.Based on previous research by [46,47] regarding scour and sediment transport, it was determined that the k-ε (RNG) turbulence model provides more accurate results.Moreover, k-ε (RNG) offers a shorter simulation time compared to large-eddy simulation (LES) turbulence model [46].Therefore, the k-ε (RNG) turbulence model was selected to simulate scour in the flow modeling and erosion around the bridge foundation.
To optimize the mesh configuration, three mesh blocks with different mesh sizes were chosen: one for the upstream section of the main channel, one for the sediment section, and one for the downstream section of the main channel.By simulating the model in three different scenarios with varying mesh sizes and comparing the results to the experimental data, an optimal mesh configuration was determined.By conducting an experiment for 48 h, it was determined that 80% of the maximum scouring depth occurs in the first six hours, and therefore, the duration of the experiments was considered to be six hours [26,27].Table 1 presents the specifications of the three selected mesh sizes for sensitivity analysis.In this table, APE is the absolute percentage error that is determined by the following equation: where M E and M N are the experimentally measured and numerically simulated values, respectively, and N is the number of observations.By comparing the maximum depth of scour around a simple cylindrical bridge pier using three different mesh sizes, it was observed that reducing the mesh size resulted in a smaller disparity between the numerical and experimental values.However, the difference in APE values between the medium and fine mesh sizes was found to be less than 2%.Given that a substantial reduction in mesh size results in a notable increase in simulation time, it can be inferred that opting for a medium mesh size is optimal when choosing the most suitable mesh configuration.Each mesh block in the simulation consists of six faces, and appropriate boundary conditions need to be defined for each face.The Symmetry condition was selected for the face shared between two mesh blocks to ensure consistent hydraulic parameters at the boundaries.The inlet boundary condition was set as Inflow, with specified values for discharge and inflow depth.The outlet boundary of the computational domain adopted the Outflow condition, which maintains the flow properties (e.g., velocity, pressure) unchanged as they reach this boundary without being transmitted outside the solution domain.
The bottom boundary was assigned the Wall boundary condition to simulate a virtual wall.The Zmax boundary utilized the Specified Pressure boundary condition, with a fluid fraction of 0 to represent free surface conditions.Zmax is the maximum elevation in boundary simulating water surface.The wall boundaries were considered symmetric using the Symmetry condition (Figure 1).Each mesh block in the simulation consists of six faces, and appropriate boundary conditions need to be defined for each face.The Symmetry condition was selected for the face shared between two mesh blocks to ensure consistent hydraulic parameters at the boundaries.The inlet boundary condition was set as Inflow, with specified values for discharge and inflow depth.The outlet boundary of the computational domain adopted the Outflow condition, which maintains the flow properties (e.g., velocity, pressure) unchanged as they reach this boundary without being transmitted outside the solution domain.
The bottom boundary was assigned the Wall boundary condition to simulate a virtual wall.The Zmax boundary utilized the Specified Pressure boundary condition, with a fluid fraction of 0 to represent free surface conditions.Zmax is the maximum elevation in boundary simulating water surface.The wall boundaries were considered symmetric using the Symmetry condition (Figure 1).

Results
Figure 2a-f present the plan view of the flow field around the bridge pier, including the vortices generated by the flow interaction with the pier.The figure illustrates four conditions: pier with and without a slot, and pier with and without floating debris.In both cases where floating debris is present (NS-D and S-D), it can be observed that the maximum velocity (0.28 m/s) occurs adjacent to and along the edge of the floating debris.This velocity increase is a result of flow contraction at the upper and lower ends of the pier.As the flow moves away from the pier, the velocity gradually decreases, indicating the presence of return flow accompanied by reduced velocity.This confirms the velocity reduction and obstruction in models without a slot.Moreover, in models with an accumulation of floating debris (Figure 2b,d), the velocity decreases further as the flow collides with the pier and starts moving downward due to the formation of horseshoe vortices.However, the wake vortices behind the pier are small.

Results
Figure 2a-f present the plan view of the flow field around the bridge pier, including the vortices generated by the flow interaction with the pier.The figure illustrates four conditions: pier with and without a slot, and pier with and without floating debris.In both cases where floating debris is present (NS-D and S-D), it can be observed that the maximum velocity (0.28 m/s) occurs adjacent to and along the edge of the floating debris.This velocity increase is a result of flow contraction at the upper and lower ends of the pier.As the flow moves away from the pier, the velocity gradually decreases, indicating the presence of return flow accompanied by reduced velocity.This confirms the velocity reduction and obstruction in models without a slot.Moreover, in models with an accumulation of floating debris (Figure 2b,d), the velocity decreases further as the flow collides with the pier and starts moving downward due to the formation of horseshoe vortices.However, the wake vortices behind the pier are small.
In Figure 3a-d, the cross-sectional view presents the distribution of flow velocity around the bridge pier.The figure shows that the range of velocity variations around the bridge pier and scour hole is lower in models with a slot compared to non-slotted piers.This indicates that the high-velocity zone around the models with a slot is less extended.The maximum velocity (0.28 m/s) is observed near the pier without a slot, specifically at the upper and lower edges of the bridge pier, leading to the formation of vortex flows upon collision with the pier.In Figure 3a-d, the cross-sectional view presents the distribution of flow velocity around the bridge pier.The figure shows that the range of velocity variations around the bridge pier and scour hole is lower in models with a slot compared to non-slotted piers.This indicates that the high-velocity zone around the models with a slot is less extended.The maximum velocity (0.28 m/s) is observed near the pier without a slot, specifically at the upper and lower edges of the bridge pier, leading to the formation of vortex flows upon collision with the pier.In contrast, when a pier includes a slot, the velocity distribution becomes slightly more uniform, resulting in reduced flow turbulence.The presence of a slot contributes to a more evenly distributed velocity field.Moreover, the maximum scour depth is observed when a floating debris is positioned near the bridge pier.Conversely, the presence of a slot in the bridge pier leads to a reduction in the scour depth, representing the minimal value.In such cases, the influence of the pier's surface resistance on the flow is diminished, resulting in a decrease in the strength of rotational vortices around the bridge pier.It is worth noting that when a floating debris is located near a bridge pier with a slot, the detrimental effect of the floating debris on increasing scour depth is neutralized, resulting in a minimal change in scour depth.The combination of a slot in the bridge pier and the presence of floating debris effectively mitigates the impact on scour depth.In contrast, when a pier includes a slot, the velocity distribution becomes slightly more uniform, resulting in reduced flow turbulence.The presence of a slot contributes to a more evenly distributed velocity field.Moreover, the maximum scour depth is observed when a floating debris is positioned near the bridge pier.Conversely, the presence of a slot in the bridge pier leads to a reduction in the scour depth, representing the minimal value.In such cases, the influence of the pier's surface resistance on the flow is diminished, resulting in a decrease in the strength of rotational vortices around the bridge pier.It is worth noting that when a floating debris is located near a bridge pier with a slot, the detrimental effect of the floating debris on increasing scour depth is neutralized, resulting in a minimal change in scour depth.The combination of a slot in the bridge pier and the presence of floating debris effectively mitigates the impact on scour depth.
Figure 4a,b present the distribution of shear stress around both slotted and non-slotted piers.The red region indicates areas where shear stress is increased, reaching a value of 0.5 N/m², in proximity to and alongside both piers.The results demonstrate that in the upstream and downstream of the non-slotted pier, the shear stress range is closer to zero, as indicated by the presence of a dark blue region, which corresponds to the formation of rotational flows and horseshoe vortices.In contrast, for slotted piers, the shear stress gradually decreases as it approaches the pier.Figure 4 illustrates that the range of shear stress in the non-slotted configuration decreases downstream of the pier, as shown in Figure 4b with the presence of a dark blue region.Additionally, in the upstream and downstream regions of the non-slotted pier, the extent of the low shear stress zone is limited due to the formation of rotational flows and horseshoe vortices.
However, in the case of slotted piers, as the intensity of vortices decreases, the shear stress gradually decreases as it approaches the pier.This indicates a smoother and more uniform distribution of shear stress around slotted piers compared to non-slotted piers, where the presence of vortices leads to localized areas of higher shear stress.
Figure 5a-d illustrate the turbulence intensity (TI) distribution around the bridge pier under different conditions.Turbulence intensity is defined as the ratio of the root mean In contrast, for slotted piers, the shear stress gradually decreases as it approaches the pier.Figure 4 illustrates that the range of shear stress in the non-slotted configuration decreases downstream of the pier, as shown in Figure 4b with the presence of a dark blue region.Additionally, in the upstream and downstream regions of the non-slotted pier, the extent of the low shear stress zone is limited due to the formation of rotational flows and horseshoe vortices.
Water 2024, 16, 90 7 of 13 However, in the case of slotted piers, as the intensity of vortices decreases, the shear stress gradually decreases as it approaches the pier.This indicates a smoother and more uniform distribution of shear stress around slotted piers compared to non-slotted piers, where the presence of vortices leads to localized areas of higher shear stress.
Figure 5a-d illustrate the turbulence intensity (TI) distribution around the bridge pier under different conditions.Turbulence intensity is defined as the ratio of the root mean square of velocity fluctuations (u rms ) to the mean velocity (u ave ).The calculation of the root mean square velocity (u rms ) is based on the velocity values in the flow direction (u 1 , u 2 , u 3 ,. .., u n ) using the following equation: From the figures, it can be observed that the percentage change in turbulence intensity ranges from 0% to 49%.The maximum turbulence intensity occurs in the model without a slot and with floating debris, reaching 39.29% (Figure 5b).The changes in TI are most significant near the pier downstream, extending from the scour hole to the flow surface, and they decrease as the distance from the pier increases.In other words, turbulence intensity increases upstream of the pier and inside the scour hole, where vortices are formed.Furthermore, by comparing the effect of the slot in the bridge pier on the changes in TI, it can be observed that the presence of a slot reduces the values of TI downstream of the pier.This indicates that the slot mitigates turbulence and leads to a smoother flow field with lower turbulence intensity in that region.
Figure 6 illustrates the distribution of turbulent flow and the variations in turbulent kinetic energy (TKE) for different models, particularly focusing on the presence or absence of floating debris.TKE is a measure of the turbulent energy in the flow field and can be quantified by calculating the mean square of velocity fluctuations.The TKE value can be obtained using the following equation: 1/2( ) Figure 6 shows that the area of high turbulence flow is broader in models where floating debris is present, compared to models without it.Moreover, the magnitude of turbulent kinetic energy is significantly higher when floating debris is introduced, reaching a value of 0.001 J/kg (Figure 6b,d).These observations highlight the influential role of floating debris in intensifying turbulence around the bridge pier.
The interaction between the flow, bridge pier, and floating debris leads to a reduction in the mean velocity differential around the structure and a decrease in flow velocity.This phenomenon occurs as the flow repeatedly interacts with the pier and floating debris, re- Subsequently, TI can be expressed using the following equation: The models without a slot exhibit higher turbulence intensity, indicated by the yellow region in the figure, suggesting a more turbulent and dispersed flow field around the bridge piers.The presence of floating debris in the models without a slot (NS−D) intensifies this turbulence, as it generates additional destructive and turbulent flows in the downward direction.
From the figures, it can be observed that the percentage change in turbulence intensity ranges from 0% to 49%.The maximum turbulence intensity occurs in the model without a slot and with floating debris, reaching 39.29% (Figure 5b).The changes in TI are most significant near the pier downstream, extending from the scour hole to the flow surface, and they decrease as the distance from the pier increases.In other words, turbulence intensity increases upstream of the pier and inside the scour hole, where vortices are formed.Furthermore, by comparing the effect of the slot in the bridge pier on the changes in TI, it can be observed that the presence of a slot reduces the values of TI downstream of the pier.This indicates that the slot mitigates turbulence and leads to a smoother flow field with lower turbulence intensity in that region.Figure 6 illustrates the distribution of turbulent flow and the variations in turbulent kinetic energy (TKE) for different models, particularly focusing on the presence or absence of floating debris.TKE is a measure of the turbulent energy in the flow field and can be quantified by calculating the mean square of velocity fluctuations.The TKE value can be obtained using the following equation: However, the inclusion of a slot in the pier structure mitigates the extension of TKE variations in the lower section of the pier.This implies that the presence of a slot serves to restrict the spread of turbulent kinetic energy, thereby limiting the turbulent flow in that specific region.The slot acts as a flow modifier, suppressing turbulence and minimizing the overall turbulent kinetic energy in the lower part of the pier.
The collision of flow with the bridge pier, particularly in the presence of floating debris, induces the formation of vortices that exhibit unstable oscillations.Figure 7 provides a visual representation of the vortex region's extent for different pier configurations.In models where the pier lacks a slot (Figure 7a,c), both red and blue regions are noticeable, signifying the expansion of the vortex region resulting from flow obstruction and the absence of a slot, which promotes flow turbulence.Figure 6 shows that the area of high turbulence flow is broader in models where floating debris is present, compared to models without it.Moreover, the magnitude of turbulent kinetic energy is significantly higher when floating debris is introduced, reaching a value of 0.001 J/kg (Figure 6b,d).These observations highlight the influential role of floating debris in intensifying turbulence around the bridge pier.
The interaction between the flow, bridge pier, and floating debris leads to a reduction in the mean velocity differential around the structure and a decrease in flow velocity.This phenomenon occurs as the flow repeatedly interacts with the pier and floating debris, resulting in enhanced extraction of kinetic energy from the mean flow.Turbulent kinetic energy acts as a generator of turbulence and flow fluctuations, necessitating a greater amount of energy transfer from the mean flow to sustain these fluctuations.Consequently, the presence of floating debris amplifies turbulence and requires more energy extraction from the mean flow.
However, the inclusion of a slot in the pier structure mitigates the extension of TKE variations in the lower section of the pier.This implies that the presence of a slot serves to restrict the spread of turbulent kinetic energy, thereby limiting the turbulent flow in that specific region.The slot acts as a flow modifier, suppressing turbulence and minimizing the overall turbulent kinetic energy in the lower part of the pier.
The collision of flow with the bridge pier, particularly in the presence of floating debris, induces the formation of vortices that exhibit unstable oscillations.Figure 7 provides a visual representation of the vortex region's extent for different pier configurations.In models where the pier lacks a slot (Figure 7a,c), both red and blue regions are noticeable, signifying the expansion of the vortex region resulting from flow obstruction and the absence of a slot, which promotes flow turbulence.Conversely, the inclusion of a slot in the pier diminishes the extent of vortex formation and flow turbulence, as depicted in Figure 7. Furthermore, in scenarios involving floating debris, the presence of flow obstruction leads to a reduction in flow velocity, which can even approach negative values (blue region).The model without a slot and with floating debris exhibits the greatest extent of vortex formation, reaching a value of 2.5 1/s (Figure 7b).The observed oscillations and expansion of the vortex region highlight the dynamic nature of the flow field around the pier, especially when influenced by the presence of floating debris.The absence of a slot amplifies vortex formation and turbulence, while the introduction of a slot in the pier mitigates these effects, providing a more stable flow pattern.

Discussion
Understanding the interaction between floating debris and bridge pier scouring is crucial for effective scour countermeasures.Designing appropriate measures to mitigate the impact of floating debris requires considering both the hydraulic aspects of the flow and the characteristics of the debris.By implementing suitable strategies such as debris traps or barriers upstream of the bridge piers, it is possible to reduce the accumulation of debris and minimize its detrimental effects on scouring.Additionally, the incorporation of slots in bridge piers can further aid in the management of debris by altering the flow patterns and reducing the potential for debris accumulation.
There are definite challenges associated with the incorporation of slots in the prevention of bridge pier scouring.Construction and geotechnical considerations need to be carefully addressed to ensure the stability and durability of both the slot and the bridge pier.The design of the slots, including their width, length, and positioning, should be optimized based on site-specific conditions, such as flow characteristics, sediment properties, and hydraulic forces.Economic factors also play a role in the feasibility of slot implementation, as the costs associated with construction and maintenance need to be evaluated.Nevertheless, despite these challenges, the use of slots as a scour countermeasure holds great potential.Research studies and field applications have demonstrated the effectiveness of slots in reducing scour depths and extending the service life of bridge foundations [42].By harnessing the power of hydraulic engineering principles, slots provide a proactive and sustainable solution to mitigate scouring around bridge piers, ensuring the structural integrity and safety of bridges in riverine environments [48].
In the slotted pier model, the flow smoothly passes around the pier, mitigating the adverse effects caused by destructive flow patterns.This phenomenon is attributed to the flow passing through the slot, which induces a downward reduction in the pressure gradient ahead of the pier [3].Conversely, in cases where only floating debris exists without a slot (NS−D configuration), maximum turbulence and flow disturbance are observed downstream.The turbulence generated by the flow colliding with the pier is further intensified by the obstruction caused by the presence and accumulation of the floating debris.Additionally, the presence of the slot in the pier contributes to a decrease in shear stress in the slotted model which is in good agreement by the findings of [35].The altered flow regime caused by slots can promote the formation of vortices or eddies, which act as hydraulic barriers against scour [42].These vortices generate circulating currents that deflect sediment transport away from the bridge foundations, reducing the scour potential [48].The result of investigation of local scour is the same with the results of this article [48].
The presence of floating debris in the flow field introduces notable alterations in the behavior of the flow around the bridge pier [49].It induces a deviation of the flow towards the bed and intensifies the formation of vortices around the pier [49].However, when both a slot and a floating debris are present (S−D configuration), a more uniform flow field is observed downstream compared to the case where the floating debris accumulates upstream of the pier without a slot (NS−D configuration).The slot within the pier plays a crucial role in redirecting the downward destructive flows and acting as a barrier to direct flow collision with the pier.Consequently, the presence of the slot mitigates flow turbulence and promotes a more aligned flow pattern.Furthermore, incorporating a slot in cylindrical piers reduces the magnitude of flow velocity around the pier.
Overall, the presence of floating debris during flood events significantly influences scouring around bridge piers.The obstruction caused by the debris modifies the flow patterns, intensifies turbulence, and accelerates sediment transport, leading to increased scour depths.Recognizing the correlation between the thickness of the floating objects and scour depth helps in developing effective strategies to mitigate the impact of debris and protect the stability of bridge foundations.
The turbulence model employed in this study, which analyzes flow dynamics considering turbulence parameters, proves to be accurate in predicting the actual behavior of the flow.By employing this model, a better understanding of flow characteristics around bridge piers can be achieved, allowing for the mitigation of detrimental effects on the outlet section and downstream flow.Furthermore, the turbulence kinetic energy (TKE) parameter serves as an indicator of the rate of dissipation of flow kinetic energy, providing valuable insights into flow dynamics and turbulence levels.
One of the limitations of the present study is that it focuses on the general influence of floating debris without considering variations in debris size, shape, or density.Also, while the turbulence model employed in the study yields satisfactory predictions, it is essential to recognize that different turbulence models may provide varying results.Further studies exploring alternative turbulence models could enhance the robustness of the findings.
Overall, the combination of a slot in the pier and the presence of floating debris significantly influences the flow pattern, turbulence intensity, shear stress distribution, and velocity magnitude around the bridge pier.The findings emphasize the importance of considering these factors in the design and assessment of bridge structures to mitigate potential damage and enhance hydraulic performance.The presented analysis makes a notable contribution to the hydraulic community by offering valuable insights into the behavior of flow around bridge piers under the influence of floating debris and the incorporation of a slot.The analysis enhances our understanding of flow patterns around bridge piers by elucidating how the presence of floating debris and the inclusion of a slot can impact flow deviation, vortex intensification, and overall flow uniformity.This knowledge is crucial for optimizing hydraulic performance in bridge design.

Conclusions
This study focused on investigating the influence of floating debris and the presence of a slot within bridge piers on flow characteristics.The findings have provided valuable insights into the behavior of flow around piers and emphasized the significance of incor-porating a slot in the pier design for improved hydraulic performance.The presence of floating debris was observed to have a considerable impact, causing flow deviation towards the sediment bed and intensifying vortices around the pier.When a slot was present in addition to the floating debris, the flow downstream of the pier exhibited a more uniform pattern compared to the configuration without a slot.This effect was attributed to the slot's ability to reduce flow turbulence, redirect destructive flows, and align flow vectors.Consequently, the presence of the slot facilitated a smoother passage of flow around the pier, minimizing flow disturbances and associated risks.Furthermore, incorporating a slot in cylindrical piers reduced the velocity magnitude around the pier.The smooth flow passage through the slot decreased the pressure gradient and mitigated the adverse effects of the flow, leading to a decrease in flow velocity.This reduction in velocity has important implications for the stability and longevity of bridge structures, as it helps minimize scouring effects.Moreover, the slot's presence contributed to a decrease in shear stress, particularly in regions with higher shear stress.By reducing shear stress, sedimentation and scouring potential in these areas can be effectively controlled, enhancing the durability of the bridge pier.
The turbulence model employed in this study, which considered turbulence parameters and the dissipation of kinetic energy, demonstrated satisfactory predictions of flow behavior.This model provided a better understanding of scour phenomena around bridge piers and offered a valuable tool for assessing and managing potential detrimental effects on the downstream flow.In conclusion, this study highlights the importance of considering the presence of floating debris and incorporating a slot within bridge piers to optimize flow characteristics, reduce turbulence, and mitigate scouring effects.These findings contribute to the design and construction of more efficient and resilient bridge structures, ensuring their long-term stability and safety.By accounting for these factors, engineers and designers can improve the performance of bridge piers, enhance their resistance to flow-induced forces, and minimize the risk of damage and failure.Further research in this area can continue to refine and expand our understanding of flow behavior around bridge piers, leading to even more effective design strategies and improved bridge performance in diverse hydraulic conditions.

Figure 1 .
Figure 1.(a) Boundary conditions in the numerical model; and (b) floating debris in front of the slotted pier.

Figure 1 .
Figure 1.(a) Boundary conditions in the numerical model; and (b) floating debris in front of the slotted pier.

Figure 2 .
Figure 2. Flow velocity vectors around the bridge pier in different models: (a) NS−ND, (b) NS−D, (c) S−ND, (d) S−D, (e) velocity contours for the NS−ND experiment, and (f) velocity contours for the S−ND experiment.

Figure 2 .
Figure 2. Flow velocity vectors around the bridge pier in different models: (a) NS−ND, (b) NS−D, (c) S−ND, (d) S−D, (e) velocity contours for the NS−ND experiment, and (f) velocity contours for the S−ND experiment.

Figure 3 .
Figure 3. Velocity distribution around the bridge pier from a cross-sectional view: (a) NS−ND, (b) NS−D, (c) S−ND, and (d) S−D.

Figure 3 .
Figure 3. Velocity distribution around the bridge pier from a cross-sectional view: (a) NS−ND, (b) NS−D, (c) S−ND, and (d) S−D.

Figure
Figure 4a,b present the distribution of shear stress around both slotted and non-slotted piers.The red region indicates areas where shear stress is increased, reaching a value of 0.5 N/m 2 , in proximity to and alongside both piers.The results demonstrate that in the upstream and downstream of the non-slotted pier, the shear stress range is closer to zero, as indicated by the presence of a dark blue region, which corresponds to the formation of rotational flows and horseshoe vortices.Water 2024, 16, x FOR PEER REVIEW 7 of 14

Figure 7 .
Figure 7. Variations in flow vortices around the bridge pier in different configurations: (a) NS−ND, (b) NS−D, (c) S−ND, and (d) S−D.

Figure 7 .
Figure 7. Variations in flow vortices around the bridge pier in different configurations: (a) NS−ND, (b) NS−D, (c) S−ND, and (d) S−D.

Table 1 .
Specifications of the three selected mesh sizes for mesh sensitivity analysis.