Refined Simulation Study of Hydrodynamic Properties and Flow Field Characteristics around Tandem Bridge Piers under Ice-Cover Conditions

Abstract

Ice cover is a common phenomenon in rivers in cold regions during the winter freeze-up period, leading to the formation of unsteady bypass structures around underwater piers. To reveal the variation law of the flow field around a pier under ice, a numerical calculation method is proposed to obtain the spatial and temporal characteristics of the fluid flow environment around the pier. The verification of flow conditions and convergence showed that the numerical model constructed in this study is reliable and can meet research requirements. The simulation results showed that the ice-cover condition considerably impacted the extent of a scour hole, and in the horizontal plane Z of −0.02 m, the lateral influence of the scour hole was approximately 2.6 times the diameter of the pier, which was approximately 42% wider than that of a scour hole under open-flow conditions; in the area on the side of the pier, there was a peak in longitudinal section y/D of −0.6, and the relative turbulence intensity was 0.4 and 0.51 under open-flow and ice-cover conditions, respectively, indicating that ice cover made the peak more significant in the area.

1. Introduction

The existence of underwater bridge piers causes local streamline trajectories around piers to bend, thus forming complex turbulent vortices and inducing scouring erosion on the near-bottom riverbed surface. Therefore, to study local scours around piers, analyzing the flow fields around the piers is necessary. The flow around a pier is a classic topic in hydrodynamic research and exists widely in natural environments and engineering practices [1,2].

The most direct method for obtaining the hydraulic characteristics around a pier is conducting prototype measurement experiments [3,4,5,6]. However, because of the mutual interference of various factors, such as underwater topography, the flow environment, and the series structure, the flow boundaries around piers in rivers change significantly. Moreover, prototype-based hydrodynamic–characteristic analysis results are often empirical, and the popularity of application is not high [7,8]. Physical model flume experiments directly reveal the characteristics of fluid motion under specific working conditions; however, fully reflecting the overall characteristics of the flow field remains challenging, owing to the limitations of the experimental measurement methods and boundary space [9,10]. Numerical simulation calculations can thoroughly show the characteristic changes in basin fields and can accurately capture the flow field area, which is difficult to measure using prototype observation and flume model tests. Therefore, it has been widely used to study the flow structures around piers. Early research on the flow around piers has primarily focused on the flow around a single pier. For example, Salaheldin et al. [11] simulated and analyzed the flow field around a circular pier in a clear-water scour environment and studied the influence of different turbulence models on the change in the vortex structure and velocity distribution around the pier. With the increasing research results on single-pier flows, many scholars have focused on multi-pier flows. For example, Meneghini et al. [12] considered the influence of the contraction effect of the spacing between parallel piers and used the surface vortex method to simulate and analyze the vortex structure and the change trends in the lift and drag coefficients between the parallel piers. In contrast, relatively few studies have considered bypass flow in ice-cover environments, and only individual scholars have attempted such studies. For example, Guissart et al. [13] used a two-dimensional fluid model to simulate ice-cover flow under mid-winter conditions and compared it with the low-flow case of an open channel.

Currently, related research is mostly limited to the indoor clear-water environment of the bypassing flow and involves the ice-cover environment around the multi-piers of the water flow field characteristics that have rarely been studied in depth. The novelty of the research perspective, i.e., analyzing hydrodynamic evolutionary properties from the perspective of spatial visualization, can complement the research gaps in related fields. Based on this, this study adopted the fluid dynamics software Flow3D (v11.2) to take the Yellow River Bridge in Inner Mongolia as the research object, simulated the hydrodynamic and spatial–temporal characteristics around the pier under the conditions of ice-cover and open-channel flow, and achieved groundbreaking research results for the analysis of the bypassing flow of the bridge pier under the ice-cover conditions. Meanwhile, the numerical model experiment results can reveal the flow physical mechanism and flow control characteristics of the flow around the bridge pier, and through the detailed analysis and discussion of the results of the distribution characteristics of turbulent kinetic energy and turbulent flow velocity, and the evolution trend in turbulence intensity and combined flow velocity, it can further prove the reliability and accuracy of the computational results of the present study, and it has a certain guiding significance for the study of the coupling mechanism between hydrodynamic force and sediment transport.

2. Numerical Methods and Verification

2.1. Numerical Method

The numerical discretization of the simulation software adopted the finite-difference method. By discretizing the calculation area into many small grids, the values of the physical quantities (such as velocity and pressure) and their derivatives (such as acceleration and diffusivity) at each grid point were determined. These values were used to calculate other physical quantities, and the discretization equation was used to integrate them and then interpolate the values between adjacent grids to obtain the value of the entire calculation area [14,15]. In a specific iterative process, numerous special numerical processing methods and techniques were used to improve the accuracy and efficiency of the numerical calculations, such as the Volume of Fluid (VOF) model, implicit solutions, and coupling algorithms. The solution of the pressure iterative velocity equation adopted the Generalized Minimal Residual Method (GMRES) algorithm with high computational efficiency and good convergence, and the momentum convection adopted second-order monotonicity.

When a turbulence model is selected, its ability to calculate characteristic quantities, such as shear stress and vortex kinetic energy, is very important [11]. This study was performed under the subcritical turbulence condition (

Table 1. Geometric specifications and hydraulic parameters of the model.

Numerical Model	Scour Environment	D/cm	H/cm	V/Vc	Fr	Re
Run 1	Ice-cover flow	4.8	15	0.82	0.20	11,808
Run 2	Open-channel flow	4.8	15	0.82	0.20	11,808

Note: D—pier diameter; H—water depth; V/Vc—current intensity; Fr—Froude number; Re—Reynolds number.

), and the experimental Reynolds number ranged from 1.0 to 2.0 × 10⁴. Because the Re-Normalization Group (RNG) model is more reliable and accurate for low-viscosity fluids and less dependent on empirical coefficients, the RNG model was selected to calculate the turbulence characteristic quantities around the bridge piers. The ice cover model selected for the physical model test in this study was a polystyrene foam plate with a thickness of 5 cm; thus, the numerical simulation maintained consistency with it and was set up as a smooth ice cover of the same thickness, in which the Manning’s coefficient of the ice cover was 0.0182 and the roughness coefficient of the surface of the sand bed was 0.0163. To ensure the similarity of the computational environment, the inlet boundary was set to be the same velocity boundary as the physical model test flow velocity (the initial flow rate was 0.28 m/s), and the outlet boundary was set to be the free outflow (the pressure was 1.013 × 10⁵ Pa), so as to make the range of the boundary conditions match the actual flow field around the pier (

Table 2. Boundary conditions of turbulence models.

Software	Boundary Conditions
	Upstream	Downstream	Free Surface	Floor	Lateral
Flow3D	Grid overlay	Specified pressure	Specified pressure	Wall	Wall

). The rest of the settings, specifically the construction of the scour model, can be found in the literature.

2.2. Verification of Convergence and Flow Conditions

The grid and time steps were independently verified to ensure the credibility of the Computational Fluid Dynamics (CFD) model results [16]. The specific parameter settings and results are listed in

Table 3. Test parameter settings of grid scale and time step size.

Verification Model	Grid Size (m)	Total Cells	Time Step (s)	Pressure Iteration (max)	Run Time (h)	Elapsed Time (h)	Iteration Residual	Convergence Speed
1	0.020	12 × 104	0.010	1	0.5	3	10−4	fast
2	0.010	80 × 104	0.010	3	0.5	10	10−6	normal
3	0.005	450 × 104	0.010	10	0.5	42	10−5	slow
4	0.010	80 × 104	0.005	5	0.5	31	10−6	slow

. A convergence analysis of different grid scales was performed based on the same time step. The analytical time of Model 1 was short, and the convergence speed was fast. To ensure computational efficiency and accuracy, the mesh size of Model 2 was selected as the optimal mesh. A convergence analysis of the time step was performed according to the optimal grid scale. The convergence speed of the calculation results of Model 4 was relatively slow. Compared to Model 2, the computer time loss was approximately 67%, and there was a certain numerical dissipation.

Studying the influence of grid scale and time step on the convergence of simulation results is necessary. In this study, the pressure iteration residual was selected as the convergence criterion, and the convergence standard was set to a pressure iteration residual of less than 1 × 10⁻⁴. Figure 1 shows a comparison of the pressure–iteration residual values for the different verification models. The figure shows that, with the steady progress of the iterations, the absolute value of the residual of Model 2 tended to stabilize and reach the convergence standard, whereas Model 3 showed a large divergence of the residual value. To evaluate the accuracy, stability, and time cost of the simulation calculation, the grid scale and time step of Model 2 were selected for a case study of the flow around the scour.

The overall flow around the scour model flume was established according to the grid scale of the verification of Model 2, as shown in Figure 2. Considering the computational efficiency and accuracy of the numerical model, the structural grid was divided into global and local grids. The global grid covered the overall flow model. The main grid scale was 29 D × 13 D × 13.5 D. The local grid covered the area around the pier and was nested within the global grid. The grid subscale was 21 D × 8 D × 9 D. Grid division was completed using software-specific Fractional Area/Volume Obstacle Representation (FAVOR) technology for model viewing.

Before simulating the flow around the pier, to ensure the accuracy of the results of the flow field around the pier of the numerical simulation flume and the physical model flume, Figure 3 shows the velocity profile distribution of the empty flume under a fully developed open-flow condition. The results showed that the sidewall effect of the experiment and the simulation, as well as the distribution of the velocity profile, were consistent, and the boundary layer height was approximately 0.2 m, indicating that the calculation of the flow model around the pier in this study was reliable and can meet research requirements.

To further verify the accuracy of the flow field around the pier, Figure 4 compares and verifies the free-surface morphology around the pier. The free surface calculation of the numerical model adopted the True-VOF algorithm, which accurately tracks and captures the position change in free surfaces. The physical model test indicated that the water flow was hindered by the upstream pier, and a backwater phenomenon was generated in front of the pier. Based on the Bernoulli principle [17,18,19,20], the free surface dropped at the inflection point of the pier side, resulting in a drop in the free liquid level that produces a drop phenomenon, where the free water surface was assumed to be at zero point, the depth of water in front of the upstream pier was approximately 0.75 cm, and the depth of water on the side of the upstream pier was approximately −0.21 cm; from the numerical simulation test, it can be observed that the same hydraulic phenomenon was present around the pier, in which the water depth in front of the upstream pier was approximately 0.52 cm, the water depth on the side of the upstream pier was approximately −0.43 cm, and it can be concluded that the difference in head was approximately 1 cm in both cases. The results showed that the numerical simulation test effectively simulates the interaction process between a fluid and structure, accurately analyzes the flow characteristics of free surfaces around piers, and can simulate other flow conditions based on similar parameter conditions.

2.3. Verification of Scouring Results and Error Analysis

After the model simulation of the bridge pier scour, to ensure the accuracy of the pier around the scour results, the results of the numerical calculations and experimental models were verified. Based on the comparison in Figure 5, it was intuitively found that the two types of results for scour hole geometry were basically the same; the simulation results could be an approximate representative of the measured results, while the scour terrain could be a contour comparison, as shown in Figure 6. Since the physical model experiment was susceptible to interference from external environmental factors, while the boundary conditions and initial conditions of the numerical model experiment were established by mathematical expressions or assumptions without the influence of interference factors, this leads to some differences in the scouring terrain contours in the test results of the numerical model and the physical model in Figure 6.

The height of the river bed in the study was set to 10 cm, and the modelled sand used was non-uniform sand with a density of 2650 kg/m³. The verification of the scour results showed that the simulated scour depth value was 4.9 cm and the experimental scour depth value was 5.4 cm, which can be interpreted as an error of approximately 9.3%; thus, the simulation underestimated the scour depth. To further verify the accuracy of the numerical results, various statistical indicators were evaluated using the numerical values of the scour results, where the coefficient of correlation (R²) was used to assess the degree of fit between the numerical and test results, the mean absolute error (MAE) was used to determine the average magnitude of the error in the numerical simulation, the root mean square error (RMSE) was used to measure the magnitude of the error in the simulation results, the Nash–Sutcliffe efficiency (NSE) was used to assess the model performance, and the specific equation for each statistical indicator is shown below [21,22,23]. The R², MAE, RMSE, and NSE values were 0.979, 3.58, 6.21, and 0.915, respectively, and in general, the R² and NSE values were high and the MAE and RMSE values were low, indicating that the accuracy of numerical simulation results was good.

$$R^2={\left[{\displaystyle \frac{\left(N{\displaystyle \sum M_{\exp }{M}_{sim}}\right)-\left({\displaystyle \sum M_{\exp }}\right)\left({\displaystyle \sum M_{sim}}\right)}{\sqrt{N\left({\displaystyle \sum M_{\exp }^2}\right)}-{\left({\displaystyle \sum M_{\exp }}\right)}^2\sqrt{N\left({\displaystyle \sum M_{sim}^2}\right)-{\left({\displaystyle \sum M_{sim}}\right)}^2}}}\right]}^2$$

(1)

$$MAE={\displaystyle \frac{{\displaystyle \sum \left|M_{\exp }-M_{sim}\right|}}{N}}$$

(2)

$$RMSE=\sqrt{{\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N{\left(M_{\exp }-M_{sim}\right)}^2}}$$

(3)

$$NSE=1-{\displaystyle \frac{{\displaystyle {\sum }_{i=1}^N{\left(M_{\exp }-M_{sim}\right)}^2}}{{\displaystyle {\sum }_{i=1}^N{\left(M_{\exp }-\overline{M_{\exp }}\right)}^2}}}$$

(4)

where M_exp and M_sim represent the experimental and simulated values, respectively, and N represents the total number of samples.

3. Analysis of the Influence of Hydrodynamic Characteristics around Piers

3.1. Distribution Characteristics of Turbulent Kinetic Energy and Turbulent Velocity

The change in the flow pattern around an underwater bridge pier is closely related to the development of local scour holes [24,25,26,27]. The formation of a scour hole in front of a pier causes a pressure gradient inside and outside the scour hole, resulting in the deflection of the mainstream flow pattern. With the continuous development of the scour hole, the turbulent zone outside the hole moves down to the hole, resulting in the secondary movement of the turbulence and enhanced turbulent flux. In this section, by capturing the vortex and two-dimensional streamline distribution of the local scour hole, a spatial visualization of the turbulent kinetic energy (TKE) distribution inside the scour hole was performed, and the turbulence field characteristics of the longitudinal section under different coverage conditions were compared to clarify the spatial turbulence characteristics of the flow field around the pier.

According to the turbulent characteristics of three-dimensional fluids, the TKE was used to reflect the intensity of turbulence to show the irregularity of turbulence. The specific expression is $0.5\left(\overline{u^{\prime }{u}^{\prime }}+\overline{v^{\prime }{v}^{\prime }}+\overline{w^{\prime }{w}^{\prime }}\right)$, where $u^{\prime }$, $v^{\prime }$, and $w^{\prime }$ represent the longitudinal, transverse, and vertical fluctuating velocities, respectively [28].

As shown in Figure 7 and Figure 8, there was a large TKE on the front wall of the pier in the scour hole under ice-cover conditions, which is consistent with the simulation results of Fuhrman et al. [29]. The core TKE values in the scour hole under the condition of open flow and ice sheet were approximately 0.0011 J/kg and 0.0013 J/kg, respectively, which were of the same order of magnitude and linearly distributed along the direction of the brush depth. The distribution trend of the TKE in the scouring hole revealed a strong correlation between the turbulence of the water flow in front of the pier and the coverage of the ice cover, reflecting that the existence of the ice cover contributed more to the TKE. From Figure 7 and Figure 8, the distributions of the TKE under the two conditions were similar overall, and the TKE increased with an increase in the scour depth. The difference was that the scour depth under the ice-cover condition intensified the momentum exchange in the hole. The ice cover weakened the constraint of the pier foundation solid wall on the water flow, leading to an enhancement of the sidewall waterflow pulsation.

Quantitative characterization provides a clearer and more accurate understanding of the development characteristics of vortices. Therefore, the long-axis size of the vortex in the scour hole [30,31] was defined as D_v. By comparing the flow field distribution results under the different environmental conditions shown in Figure 7b and Figure 8b, it can be seen that the vortex size under the condition of open flow and ice cover was 9.6 mm and 7.2 mm, respectively, and the corresponding scouring crater depths were 15.4 mm and 26.8 mm, respectively. The simulation and test results of other studies [18,32,33,34] are plotted in Figure 9. There was a power-law exponential distribution relationship between vortex size and scour depth, and the relationship between the two was obtained by fitting it as follows:

$$D_v=0.83{h_s}^{1.05}$$

(5)

where D_v is the size of the long axis of the vortex, h_s is the depth of the crater, and R² = 0.992.

Figure 7b and Figure 8b show that the flow around piers causes the formation of a periodic vortex structure in the hole, which affects water and sediment transport. There were two significant free vortices in the bottom flow of the hole under open-flow conditions. Combined with the relationship between the size of the vortex and the TKE value under different environmental conditions, it was found that the vortex in the hole was captured at the equilibrium scouring time. The vortex size under the open-flow condition was larger than that under the ice-cover condition, and the value weakened with an increase in TKE diffusion. The simulation results showed that increases in the long-axis vortex in the hole tended to reduce the depth of the scour hole, which is consistent with the conclusion of Guan et al. [35].

The velocity distribution around the pier was presented in the form of slices to fully describe the structural characteristics of the basin field around the pier. Figure 10a shows the time-averaged velocity distribution of the three longitudinal sections around the pier at the time of scouring equilibrium under open-flow conditions. From the longitudinal section y/D = 0.0, it can be seen that obstruction of the pier caused a low-speed recirculation zone to appear behind the pier, while the velocity distribution in front of the pier was consistent with the logarithmic distribution law under the condition of open-channel flow [36]. From the longitudinal section y/D = −0.6, it can be seen that the maximum contraction section was located at x/D = −0.8, which belongs to the high-speed mainstream area, and the corresponding maximum average flow velocity was approximately 0.38 m/s, which was 1.27 times the inlet flow velocity. From the longitudinal section y/D = −1.2, it can be seen that the near-bottom velocity boundary layer was redistributed (light blue area) owing to the influence of compression away from the pier, while the pier side area (dotted line part) belonged to the low-speed mainstream area, and its velocity value was significantly smaller than the corresponding high-value area at the section −0.6. Figure 10b shows the time-averaged velocity distribution nephogram of the three vertical sections around the pier at the time of scouring equilibrium under the ice-cover condition. From the vertical section y/D = 0.0, it can be observed that the law of particle motion under ice-cover flow differed from that under open-channel flow. The time-averaged velocity gradient along the water depth conformed to the double logarithmic distribution law [36], showing a ‘small up and down, large in the middle’ distribution, and the velocity profile in the scour hole in front of the pier developed faster than that under the open-flow condition, indicating that the velocity distribution was strongly correlated with the distribution law of TKE; that is, the two were similar in qualitatively describing the hydraulic characteristics in the scour hole; from the longitudinal section y/D = −0.6, it can be seen that, due to the decrease of the cross section, the maximum time-averaged velocity of the accelerated flow on the pier side was higher than that under the open-flow condition, reaching approximately 0.40 m/s, which was 1.05 times the maximum velocity under the open-flow condition. This reflected the high contribution of the ice-cover flow to the hydrodynamic characteristics around the pier from the side. From the longitudinal section y/D = −1.2, it can be seen that the overall distribution of the velocity along the flow direction on the pier side was basically the same, but the velocity value of the longitudinal section under the ice-cover condition was remarkably larger than the corresponding value under the open-flow condition, indicating that the ice cover promoted water flow. Overall, ice cover conditions significantly affect the turbulence characteristics of the flow around the bridge piers and in the scour hole, with less dimensionless shear stress required to reach the same scour depth compared to the open-flow conditions, indicating that the maximum scour depths both increase with the increase in dimensionless shear stress, and further demonstrating that the presence of the ice cover reduces the surface roughness coefficients of the riverbed and enlarges the region of maximum flow velocity.

3.2. Evolution Trend in Turbulent Intensity and Combined Velocity

The turbulent characteristics of the flow around a pier are highly important for the flow field and movement of sediment particles [37]. Figure 11a shows the turbulent intensity distribution of the three vertical sections around the pier at the time of scouring equilibrium under open-flow conditions. From the vertical section y/D = 0.0, it can be seen that the turbulent intensity value between the two piers was large, and the momentum exchange in water body presented an unstable interaction process, which belonged to the turbulent core area of the basin around the pier [38]; from the longitudinal section y/D = −0.6, it can be seen that the longitudinal section was on the pier side and close to the wall. Because of the tip-type structure of the front pier, the flow pattern in the mainstream area diverged, a larger turbulent energy (red part) appeared in the free surface range of the upper layer of the water between the two piers, and the turbulent intensity fluctuated greatly. From the longitudinal section y/D = −1.2, it can be seen that the longitudinal section was far from the pier side wall surface, and the turbulence intensity was relatively stable as a whole and was less affected by the section compression effect. Figure 11b shows the distribution of the turbulence intensity of the three longitudinal sections around the pier at the time of scouring equilibrium under the condition of ice cover. From the longitudinal section y/D = 0.0, it can be observed that the turbulence intensity of the lower surface of the ice cover was intensified, the velocity gradient was significant, and the vertical distribution trend around the pier differed significantly from that under the open-flow condition [39]. From the vertical section y/D = −0.6, it can be seen that the influence of the ice cover on the hydrodynamic characteristics of the pier side (red part) was mainly concentrated in the middle and upper water bodies, the influence on the bottom water body was slightly smaller, and changes in turbulence intensity would affect the flow pattern and sediment carrying capacity of the pier side. From the longitudinal section y/D = −1.2, it can be seen that the far wall area was affected by the ice cover, and the turbulence of the upper and lower water bodies on the pier side was strengthened, which, in turn, caused horizontal and vertical diffusion to intensify. Overall, the turbulence intensity at the same position around the pier under the ice-cover condition was larger than that under the open-flow condition, which was caused by the turbulent burst of the flow around the pier. Therefore, the ice cover aggravated the turbulence of the boundary layer flow and formed a high-intensity turbulence field, which, in turn, promoted the local scour development of the pier.

To further clarify the spatial variation in the turbulent characteristic value (T_u) and the combined velocity (U_T) of the flow field around the pier with the development of the longitudinal section, the time-averaged values of each characteristic value in the upstream, downstream, and side of the pier under open-flow and ice-cover conditions were selected. The change trend with the position of the longitudinal section is shown in Figure 12. As shown in Figure 12a, in the upstream area of the pier, the variation in each characteristic value under the conditions of open flow and ice cover was consistent, and it increased with the distance from the pier axis (longitudinal section y/D = 0.0). The difference was that the turbulence intensity and combined velocity values under the ice-cover condition were higher than those in the open-flow area. From Figure 12b, in the pier side area, the longitudinal section y/D = −0.6 had a peak value, in which the relative turbulence intensity and the combined velocity values were 0.40, 0.38 m/s and 0.51,0.40 m/s under open-flow and ice-cover conditions, respectively. This indicated that the section near the pier side was strongly narrowed by the beam, thereby increasing the eigenvalue, and the existence of ice cover made the regional peak more significant; it can be seen from Figure 12c that, in the downstream area of the pier, the eigenvalues of the sections far from the pier side (y/D = −0.6, −1.2) changed slightly under the two conditions. Only at the pier shaft section (y/D = 0.0) did the relative turbulence intensity show an increasing trend compared with the upstream area, and the combined velocity value decreased significantly. This was because the water movement in this area was complex, belonging to a low-speed recirculation zone, and caused the turbulence intensity to increase.

4. Analysis of Spatial and Temporal Characteristics of Flow around Pier

Figure 13 shows the result of extracting the distribution cloud diagram of the transient velocity (U⁺) of the longitudinal section around the tandem piers in the equilibrium scour hole at the horizontal plane z = 0, and it shows the three-dimensional flow streamline diagram around the pier, where the selected longitudinal sections are y/D = 0.0, −1.3, and −2.8, respectively. From the velocity cloud diagram, it was intuitively determined that the area with a large transient velocity along the flow direction was located on both sides of the pier (red area), and the velocity gradually decreased as it moved away from the pier side. This was because the pier narrowed both sides of the riverbed, thereby compressing the water flow and increasing the flow capacity. The streamline diagram clearly shows that the streamline shape in the wake region of the pier was extremely complicated and that the interference effect between the piers was large. This was because of the existence of multi-scale and multi-order turbulent vortices in the recirculation zone, and the axial direction of the vortices was complex and changeable. The above describes the spatial and temporal characteristics of the fluid flow environment around the tandem piers during the process of local scour.

Based on the local scour test of tandem piers under different scour environments, it can be seen that the front and rear piers experienced independent development of the initial scour hole until the formation of a ‘banded’ scour hole before and after the scour balance. In this section, the scour hole around the pier at the time of scour equilibrium (h_s/h_e = 1) was selected to analyze the spatial distribution characteristics of the basin around the tandem piers under different scour environments. Figure 14 shows a streamline diagram around the pier with z = 0 in the horizontal plane under the scour balance condition. The overall flow pattern distribution around the tandem piers under this condition was approximately the extended flow pattern of the upstream pier structure. The shear layer separated from the upstream pier side was reattached to the downstream pier side, which conformed to the single bluff body model and was consistent with the conclusions of Zdravkovich et al. [40].

4.1. Tangential Velocity Distribution

The flow characteristics around piers form the theoretical basis for basin simulations [41]. In this study, computational fluid dynamics (Flow3D) software was used to visualize the flow field around the tandem pier. The transient velocity field of the numerical simulation was used to extract the characteristic surface, and the longitudinal and plane velocity vector distributions of the instantaneous flow field under open-flow and ice-cover conditions were obtained.

Figure 15 shows the tangential velocity vector distribution of the longitudinal section on the central axis of the tandem piers along the water flow direction under the conditions of open flow and ice cover scour, in which longitudinal distance x and vertical flow depth z are dimensionless. The figure shows that the existence of the ice cover had a great influence on the flow field environment around the pier, which is mainly manifested in the significant reduction in the boundary velocity vector under the ice cover and the downward movement of the maximum velocity vector. By comparing the two types of scouring conditions, it was found that, for the near-wall wake region of the downstream pier, there was a negative pressure gradient along the flow depth direction. However, the velocity vector loss of the flow field in the lower layer of the pier tail under the ice-cover condition was small, and the velocity distribution behind the far-wall region tended to be consistent with that of the upper layer. In essence, the contribution of the ice cover to the flow field around the pier was mainly manifested in the redistribution of the flow structure caused by the strengthening of the flow resistance in the middle and upper layers. For the open-channel flow into the ice-cover flow, the velocity vector distribution changed from following a logarithmic distribution to following the double logarithmic distribution law [36], as shown in Figure 15a,c.

Figure 16 and Figure 17 show the velocity vector distributions of the horizontal plane under open-flow and ice-cover conditions, respectively. The length of the arrow in the figure represents the size of the instantaneous velocity vector. As shown in the figure, owing to the diversion effect of the front pier tip structure, the flow on the upstream surface was deflected, and the flow velocity in the mainstream area on the pier side increased. Meanwhile, there was a weak circulation between the front and rear piers, resulting in strong turbulence. On the horizontal plane z = 0.00 m, the overall flows of the two types of working conditions showed a similar trend. The presence of the ice cover had a significant influence on the near-bottom velocity vector of the entire sand bed. The longitudinal velocity increased from 0.28 m/s at the upstream x = −1.7 D to a maximum velocity of 0.32 m/s on the pier side, reaching 1.1 times the average velocity of the incoming flow. On the horizontal plane z = 0.02 m, the flow velocity on the side of the tandem pier in the scour hole was significant, and the ice-cover condition had a significant influence on the range of the scour hole. The lateral influence range was approximately 2.6 D of the pier diameter, which was approximately 42% wider than that of the open-flow condition.

The magnitude of the bed shear stress reflected the pulsation change in the turbulence field around the pier, which was a key parameter for quantitatively analyzing local scouring, and its parameter value was mainly calculated using the Reynolds stress, as shown in Equation (6); based on the velocity vector distribution of the flow rate under the ice-cover and open-flow conditions, this study calculated the maximum shear stress of the bed surface of the sand on the side of the pier under the ice-cover and open-flow scouring conditions, which was approximately 4.6 Pa and 4.3 Pa, respectively. The results indicate that the presence of an ice cover considerably affects the initiation of sediment particles and significantly increases the shear stress on the bed surface around piers.

$${\tau }_b={\left.{\left({\tau }_{\theta }^2+{\tau }_r^2\right)}^{0.5}\right|}_{at bed}$$

(6)

where ${\tau }_{\theta }=-\rho \left(\overline{w^{\prime }{u}^{\prime }}+\overline{v^{\prime }{u}^{\prime }}\right)$; ${\tau }_r=-\rho \left(\overline{u^{\prime }{v}^{\prime }}+\overline{w^{\prime }{v}^{\prime }}\right)$; and ρ is the fluid density.

4.2. Time-Averaged Eigenvalue Distribution

In this section, five characteristic surfaces (see the blue dotted line in Figure 14) at the time of scour equilibrium were selected to quantitatively analyze the spatial distribution characteristics of the transient flow field around the pier. The positions of the characteristic surfaces were x/D = −1.5 and −1.3 in the front cross-section of the upstream pier, x/D = −0.8 in the side cross-section of the upstream pier, x/D = 0.8 in the cross-section between the piers, and x/D = 3.1 in the tail cross-section of the downstream pier. Meanwhile, the positions of y/D = 0, −0.6 and −1.2 in the longitudinal section of each characteristic surface were selected for analysis (see the longitudinal section of Figure 13).

From the distribution of the longitudinal velocity in Figure 18a, the velocity component U_x along the depth direction was positive (along the water flow direction). The velocity distribution curve under the condition of open flow was approximately exponential, showing an ‘inverse L shape’, and the velocity distribution curve under the condition of ice cover was approximately double logarithmic, showing an ‘inverse C shape’. On the characteristic surface x/D = −1.3, the overall U_x value of the longitudinal velocity at the longitudinal section y/D = 0 was significantly reduced, and the U_x values at the longitudinal sections of the pier side −0.6 and −1.2 increased gradually. This was because the underwater pier caused a water-blocking effect in front of the pier, and the compression flow at the pier side caused the velocity to increase. By comparing the three-way flow velocities at U_x, U_y, and U_z, it was found that the longitudinal flow velocity played a leading role in the flow around the pier and was indispensable for the development of the scour hole.

The distribution of the transverse velocity in Figure 18b shows that the boundary layer on the pier side was separated because of the influence of the tip structure at the front end of the upstream pier, and velocity component U_y was positive (away from the pier axis). At the characteristic surface x/D = −1.5, the distribution distance of the U_y vertical line in the scour hole was shallow, owing to the close distance from the front edge of the scour hole. The characteristic surface x/D = −1.3 was located at the center of the scour hole, and the U_y vertical distribution in the scour hole was significant. The extreme value point appeared at z/D ≈ −0.2 under the open-flow condition, and the extreme value point appeared at z/D ≈ −0.5 under the ice-cover condition. This showed that the ice-cover condition had a significant influence on the position of the extreme value of the flow velocity, which was consistent with the conclusion of Robert and Tran [39]. The overall U_y value was larger at the position of x/D = −1.3 than that at the position of x/D = −1.5, indicating that the influence of boundary layer separation was more evident near the pier.

As shown in Figure 18c, the vertical distributions of the vertical velocity of the two characteristic surfaces in front of the upstream pier were basically similar, and the velocity component U_z was negative, indicating that the vertical flow pattern of the water flow in front of the pier was underflow due to the obstruction of the pier. The most significant feature outside the scour hole was that the U_z value increased monotonously with an increase in water depth, the internal scour hole gradually tended to zero with the U_z value near the bottom of the hole, and the maximum value of the overall vertical velocity was approximately near the bottom bed. Because the longitudinal section y/D was far from the pier axis, the vertical velocity was weakened by the blocking effect of the front pier; however, the overall distribution law was consistent. Compared with that under the open-flow condition, the vertical velocity change under the ice-cover condition was more intense, and its contribution to flow erosion was greater.

Figure 19 shows the vertical distribution of the flow velocity on the characteristic surface of the upstream pier. As shown in Figure 19a, the vertical gradient of the longitudinal velocity inside the scour hole on the pier side was large, indicating that the scouring ability of the characteristic surface was strong and the longitudinal turbulent diffusion speed was fast. Meanwhile, the outermost longitudinal section y/D = −1.2 around the pier had a large loss of fluid energy, resulting in a decrease in the value of velocity component U_x and a decrease in the downstream scouring force. As shown in Figure 19b, the characteristic surface was most affected by the boundary layer separation, and the horizontal axis circulation was enhanced, resulting in a lateral momentum exchange. This led to the lateral widening of the scour hole. In the longitudinal section y/D = −1.2, compared with −0.6, the value of velocity component U_y inside the scour hole decreased remarkably, which was due to the gradual attenuation of the lateral flow to the side wall of the scour hole, and continued to shift to the downstream diffusion with the superposition of longitudinal motion. According to Figure 19c, because the characteristic surface was located in the turbulent vortex influence area on the upstream pier side, the vertical development scale of the scour hole was the largest, and velocity component U_z was the largest compared with the overall values of the other characteristic surfaces. The vertical distribution of U_z in the scour hole first increased and then decreased with an increase in y/D. The vertical velocity enhances the turbulent vortex on the pier side and promotes the vertical development of the scour hole. In extreme environments, this leads to direct destruction of the pier foundation.

Figure 20 shows the vertical distribution of the flow velocity at the characteristic surfaces of the front and rear piers. By comparing the flow velocity distribution of each characteristic surface, it was found that the characteristic surface behind the pier was clearly different from the other positions. The most significant feature was that the flow velocity vertical line of the longitudinal section y/D = 0 at the characteristic surface x/D = 3.1 was distributed above the sand bed surface, indicating that the sediment particles accumulated to form sand dunes. As shown in Figure 20a, the two characteristic surfaces exhibited a reverse flow in the upper water body behind the pier at y/D = 0 in the longitudinal section. This was because the turbulent vortex on the pier side disturbed the wake field of the pier, forming a negative-pressure backflow area behind the pier. For the characteristic surface x/D = 0.8, the backflow scale between the two piers was significantly larger. As shown in Figure 20b, velocity component U_y at the characteristic surface x/D = 3.1 was negative, and the transverse velocity was deflected to the pier axis, indicating that the upstream region was hindered by the pier foundation, and the separated boundary layer was reattached to the pier tail. The value of velocity component U_y at the characteristic surface x/D = 0.8 was small, and the distribution is not remarkable, indicating that the flow structure between the two piers was complex and the flow pattern was disordered. It can be concluded from Figure 20c that, because the characteristic surface x/D = 3.1 was located in the wake vortex motion area behind the pier and the values of velocity component U_z on all longitudinal sections were positive, the lateral influence area of the characteristic surface was large, which will have a suction effect on the sediment particles and form a siltation zone. In general, the distribution trend of the scour around the pier under ice-cover conditions was consistent with that under open-flow conditions, with the only difference being the size of the velocity vector and the scale of the scour hole.

5. Conclusions

In this study, based on a low Reynolds number condition, simulation analyses of the flow around tandem piers in open-flow and ice-cover environments were conducted. The results of the simulated flow field were in good agreement with the results of existing experimental data, which verified the accuracy and efficiency of the flow model constructed in this study. This indicates that the hydrodynamic and flow field characteristics around the piers could be accurately simulated. Through a comparative analysis of the results, the following conclusions were drawn:

(1) The core TKE values in the scouring hole under open-flow and ice-cover conditions were approximately 0.0011 J/kg and 0.0013 J/kg, respectively. The TKE distributions were similar, and the TKE was enhanced with the gradient as the scouring depth of the scouring hole increased. However, the scouring depth of the scouring hole under the ice-cover condition intensified the momentum exchange in the hole. The difference was that the presence of ice cover weakens the constraint of water flow on the solid wall of the pier foundation, which in turn leads to the pulsation of water flow on the sidewall, reflecting the greater contribution of the presence of ice cover to the turbulent energy.

(2) The contribution of the ice cover to the flow field around the pier was mainly manifested in the strengthening of the upper and middle water flow resistance, resulting in the redistribution of the water flow structure. For the flow of the open-channel into the ice-cover flow, the flow velocity vector distribution went from following a logarithmic distribution to following the double logarithmic distribution law.

(3) The intensity of turbulence at the same location around the pier under ice-cover conditions was greater than that under open-flow conditions because of the turbulent bursts of the bypassing flow, and the inhomogeneity of the flow distribution in the bypassing field led to the enrichment of the vortex regime in the bypassing region. As a result, the presence of the ice cover increases the turbulence of the boundary layer currents, creating a high-intensity turbulence field, which in turn promotes the development of localized scour at the base of the piers.

(4) The maximum shear stress on the sand bed surface on the pier side was calculated for the ice-cover and open-flow scour conditions, which were approximately 4.6 Pa and 4.3 Pa, respectively, and the results showed that the presence of the ice cover had a greater impact on sediment particle initiation, significantly increasing the shear stress on the bed surface around the pier.

(5) The turbulence model developed in this study can be derived and applied to hydrological modeling to assess the ecological impacts of different river structures by numerically simulating the changes in hydrological processes in watersheds.