Numerical Study on Local Scour Reduction around Two Cylindrical Piers Arranged in Tandem Using Collars

Abstract

Local scour occurring near bridge piers has become a major problem all over the world, which has caused countless bridge damage events. Explorations regarding local scour reduction measurements have become a research hotspot in the field. Much effective research has been conducted on scour reduction for single piers. However, studies on local scour reduction around multiple piers that are arranged in tandem have rarely been reported. Therefore, the effect of the span and the local scour reduction measurement (collar) on the characteristics of the local scouring behavior around two piers arranged in tandem are explored in this research, with numerical simulations in clear-water conditions. The results show that the local scour depth of the downstream pier increases gradually with an increase in the pier spacing, due to the weakened sheltering effect of the upstream pier. The local scouring of both the upstream and downstream piers can be reduced if the upstream pier is protected by a collar. The local scour reduction efficiency of the upstream pier can reach 52~55%. The local scour reduction efficiency of the downstream pier decreases rapidly from 84.3% to 8.3% with an increase in the pier spacing. If the pier spacing, G, is greater than 4.0D (D is the diameter of the pier), the local scour depth around the downstream pier is larger than that around the upstream pier. Therefore, if the local scour depth of the upstream pier is considered safe and acceptable, it is used as the reduction target of the downstream local scour depth. A collar must be adapted for use around the downstream pier when G/D > 4.0. If both the piers are protected by collars of the same size (W = 3.0D), the local scour reduction efficiency of the downstream pier is about 15% more than that of the upstream pier. The local scour depth around the downstream pier is 64.5% of that around the upstream pier. Therefore, the size of the collar around the downstream pier can be decreased to save costs. The local scour reduction efficiency for the downstream pier reduces from 66.7% to 39.8% when the downstream collar size (W) decreases from 3.0D to 2.0D. To ensure that the local scour depth around the downstream pier is no greater than that of the upstream pier, the downstream collar should be larger than 2.25D. These results can serve as a reference for the local scour reduction of two piers arranged in tandem.

1. Introduction

The local scour of bridge piers entails a localized decrease in riverbed elevation around the piers under the influence of hydrodynamic forces [1]. The development of local scour around bridge piers in natural conditions is common. Piers and the foundations of bridges are located in rivers, and local scour around bridge piers occurs extremely often, representing a serious risk to the safety of the bridge. Many studies show that local scour is one of the main causes of bridge failure and collapse [2,3,4,5]. In China, more than 30% of the 106 bridge failures from 2000 to 2014 were caused by local scour [6].

A large number of studies have been carried out focusing on local scouring behavior around single bridge piers. Most research focuses on the study of local scour depth [7,8,9,10], the factors affecting local scour [11,12,13], and local scour reduction measurements [14,15,16,17,18]. The flow field and local scour morphology around multiple piers are more complex due to the vortices between piers [19]. Wang et al. [20] studied the local scour effect around tandem twin piers under clear water conditions, via indoor model tests. The results showed that the scour pit that formed around the upstream pier was almost the same as that around a single pier, while the local scour depth around the downstream pier was smaller than that around the upstream pier. Hou et al. [21] found that when the line between the two piers is parallel to the direction of the flow, the strength and velocity of the flow around the downstream pier are smaller because of the presence of the upstream pier. With an increase in bridge pier distance, the strength and velocity of the flow around the downstream pier increase. Zhou et al. [22] showed that the incidence angle of the flow (the angle between the direction of the flow and the line connecting the bridge piers) has a greater impact on the scour. Yang [23] investigated the local scour behavior around pier groups and proposed an exponential equation to quantitatively describe the temporal variation of local scour depth.

There are also many results for local scour behavior around multiple piers. Qi et al. [24] studied the effects of span on local scour depth around four columns of tandem piers in clear water. The study showed that when the span is smaller than 30.0D, local scour depths around other piers in each column decrease along the flow direction gradually and eventually become stable. If the span is greater than 30.0D, the local scour depths around the piers in each column are close in the different cases. The depths around the subsequent tandem piers can be calculated according to the reduction factor of local scour depth (R_LSD). If the span is smaller than 30.0D, the maximum R_LSD is about 0.78. If the span is greater than 30.0D, the R_LSD of the piers near the channel edge in each case is closer to each other, at about 0.85. In each case, the R_LSD of the piers near the middle line of the channel increases, along with the increment of the span, and eventually reaches 1.0. Bordbar et al. [25] investigated the evolution of local scour and flow morphology around two side-by-side piers at different distances (where G is the distance between the piers and D is the diameter of the piers) through a numerical simulation. The results showed that the depth of local scour around the piers was negatively correlated with the distance ratio of G/D. Kim [26] studied the local scour morphology around two piers in tandem. The results showed that the maximum scour depth increases and then decreases when the distance between the piers is increased. Zhang et al. [27] investigated the flow field and local scour of three adjacent piers with different placements. The results showed that pier spacing had a significant effect on the flow field and local scour around the three adjacent piers. The scour pit morphology and scour depth around piers with different relative positions were completely different. Tang et al. [28] studied the local scour morphology of three tandem piers. The results showed that the scour mechanism is more complicated than that for a single pier, due to the interaction effect between the front and middle piers. The scour depth gradually decreases from the front pier to the rear pier, due to the sheltering effect of the front pier.

However, local scour reduction around multi-pier setups is rarely reported. Most bridges commonly have more than two piers that are arranged in tandem; the upstream pier acts as a sacrificial pile, which provides shelter to reduce the local scour depth around the downstream piers. However, the upstream pier is at risk during flooding; local scour reduction for upstream piers is critical to the safety of the bridge.

Therefore, this study proposes to investigate the local scour reduction around two tandem piers with collars in clear water, using a numerical simulation, and will focus on the mechanism and characteristics of local scour reduction around two piers arranged in tandem. The results can provide reference data regarding the local scour reduction behavior of multi-pier bridges.

2. Numerical Calculation Methods

2.1. Turbulence Model

There are two turbulence models available in Flow-3D v11.2, namely, the renormalization group (RNG) k-ε model and the large eddy simulation (LES) model. Both can be applied to simulate the high strain rate and complex turbulent flow field occurring around bridge piers. The advantage of the LES model over the RNG k-ε model is that LES offers improved simulation accuracy and can simulate turbulence in more detail. The basic idea behind LES is that the turbulent motion is divided into large-scale quantity motion and small-scale quantity motion through the filter, in which the large-scale quantity motion is directly and computationally derived, and the small-scale quantity motion is established using a small-grid model. The continuity and momentum equations for the LES are as follows:

$$\frac{\partial {\overline{u}}_i}{\partial x_i}=0$$

(1)

$$\rho \frac{\partial {\overline{u}}_i}{ \partial t}+\rho \frac{\partial ({\overline{u}}_i{\overline{u}}_j)}{ \partial x_j}=-\frac{\partial \overline{p}}{\partial x_i}+\mu \frac{{\partial }^2{\overline{u}}_i}{\partial x_j^2}-\rho \frac{\partial {\tau }_{ij}}{\partial x_j}$$

(2)

$${\tau }_{ij}=\overline{u_i{u}_j}-{\overline{u}}_i{\overline{u}}_j$$

(3)

where $x_i$ (i = 1,2,3) represents the three directions of the coordinate axes; ${\overline{u}}_i$ is the velocity component along the coordinate axis; t is the time; ρ is the density of the fluid; $\overline{p}$ is the pressure; μ is the viscosity coefficient of the flow; τ_ij is the SGS stress.

2.2. Scour Model

2.2.1. Starting of Sediment

The sediment around the bridge pier is subjected to strong shear stress due to the high-velocity flow around it. When the shear stress at the bed surface exceeds the critical value, sediment particle transport, which is also known as scour, occurs. In Flow-3D, the state of the sediment particles is judged using the critical Shields number θ_cr,i. When the local Shields number θ_i is larger than the critical Shields number θ_cr,i, the sediment particles start to move. θ_i is defined using the equation below:

$${\theta }_i=\frac{\tau }{‖g‖d_i\left({\rho }_i-{\rho }_f\right)}$$

(4)

where τ is the shear stress of the riverbed; g is the acceleration of gravity; $d_i$ is the median particle size of sediment i; ${\rho }_i$ is the density of sediment i; ${\rho }_f$ is the density of the fluid.

The critical Shields number θ_cr,i can be calculated via the Soulsby–Whitehouse [29] equation:

$${\theta }_{cr,i}=\frac{0.3}{1+1.2d_{\ast }}+0.055\left[1-\exp \left(-0.02d_{\ast }\right)\right]$$

(5)

where $d_{\ast }=d{\left[{\rho }_f\left({\rho }_i-{\rho }_f\right)g{\mu }_f{}^{-2}\right]}^{1/3}$; ${\mu }_f$ is the dynamic viscosity of the fluid.

Equation (5) can be corrected according to the effect of the riverbed slope:

$${\theta }_{cr,i}^{\prime }={\theta }_{cr,i}\frac{\cos \psi \sin \beta +\sqrt{{\cos }^2\beta {\tan }^2\phi -{\sin }^2\psi {\sin }^2\beta }}{\tan \phi }$$

(6)

where ψ is the angle between the direction of flow and the direction of the riverbed slope; β is the slope of the riverbed; φ is the angle of repose of the sediment.

2.2.2. Forms of Sediment Movement

The forms of sediment transport in the flow during scouring are divided into four types: entrainment, deposition, bed-load transport, and suspension-load transport. When the flow velocity is sufficient, the sediment on the surface of the riverbed is entrained by the flow, suspended in the flow, and moves in the direction of the flow (suspension load transport) or rolls along the riverbed surface (bed-load transport). When the flow velocity is relatively small, the sediment suspended in the flow settles onto the riverbed due to the force of gravity (deposition). The entrainment lift velocity and deposition velocity of sediment are calculated as follows:

$$u_{lift,i}={\alpha }_i{n}_s{d}_{\ast }^{0.3}{\left({\theta }_i-{\theta }_{cr,i}^{\prime }\right)}^{1.5}\sqrt{\frac{\parallel g\parallel d_i\left({\rho }_i-{\rho }_f\right)}{{\rho }_f}}$$

(7)

$$u_{settling,i}=\frac{v_f}{d_i}\left[{\left({10.36}^2+1.049d_{\ast }^3\right)}^{0.5}-10.36\right]$$

(8)

where a_i is the entrainment parameter; n_s is the unit vector of the sediment; $d_{*}$ is the dimensionless diameter of the sediment; $V_f$ is the kinematic viscosity of the flow.

The suspended sediment concentration is calculated by solving its transport equation:

$$\frac{\partial C_{s,i}}{\partial t}+\nabla \cdot \left(u_{s,i}{C}_{s,i}\right)=\nabla \cdot \nabla \left(D_f{C}_{s,i}\right)$$

(9)

where $D_f$ is the diffusion rate; $u_{s,i}$ is the velocity of the suspended sediment; $C_{s,i}$ is the sediment concentration.

For bed-load transport, the sediment volumetric transport rate per width of the riverbed can be calculated according to the three equations proposed by Meyer, Peter, and Müller [30], Nielsen [31], and Van Rijn [32], respectively. The three equations are as follows:

(1)

Meyer, Peter, and Müller:

$${\mathsf{\Phi }}_i={\beta }_{MPM,i}{\left({\theta }_i-{\theta }_{cr,i}^{\prime }\right)}^{1.5}{c}_{b,i}$$

(10)

(2)

Nielsen:

$${\mathsf{\Phi }}_i={\beta }_{Nie,i}{\theta }_i^{0.5}\left({\theta }_i-{\theta }_{cr,i}^{\prime }\right)c_{b,i}$$

(11)

(3)

Van Rijn:

$${\mathsf{\Phi }}_i={\beta }_{VR,i}{d}_{\ast ,i}^{-0.3}{\left[\frac{{\theta }_i}{{\theta }_{cr,i}^{\prime }}-1.0\right]}^{2.1}{c}_{b,i}$$

(12)

where ${\beta }_{MPM,i}$, ${\beta }_{Nie,i}$, and ${\beta }_{VR,i}$ are the model coefficients and are usually set to 8.0, 12.0, and 0.053, respectively; $C_{b,i}$ is the volume fraction of species i in the bed material.

Φ_i can be used to calculate the volumetric bed-load transport rate $q_{b,i}$, as shown in Equation (13):

$$q_{b,i}={\mathsf{\Phi }}_i{\left[\parallel g\parallel \left(\frac{{\rho }_i-{\rho }_f}{{\rho }_f}\right)d_i^3\right]}^{\frac{1}{2}}$$

(13)

where g is the acceleration of gravity; ${\rho }_i$ is the density of the sediment; ${\rho }_f$ is the density of the fluid; d_i is the median particle size of the sediment.

The selection of Φ_i affects the accuracy of the numerical simulation of local scour. Li et al. [33] studied the applicability of the three equations. The results showed that the Meyer–Peter–Müller equation was the most applicable. The scour pit morphology was the closest to that in the experiment. The relative error of the maximum scour depth was 2.3%, which is smaller than that obtained using Equation (12) (15.5%). Therefore, Equation (10) is used to calculate Φ_i in this study.

2.3. Model Validation

To verify the accuracy of the numerical simulation, the data from the physical model test conducted by Melville et al. [34] were used in this study. The validation of the numerical simulation results includes both flow morphology and local scour morphology. The layout of the physical model test is shown in Figure 1a. The test flume was 19.0 m long, 0.456 m wide, and 0.44 m high. The water depth was 0.15 m, and the average velocity was 0.25 m/s. The bottom of the test flume was covered by sediment of 0.15 m in thickness. The median grain size of the sediment was 0.385 mm, the angle of repose of the sediment was 32°, and the density was 2.65 g·cm⁻³. The model of the bridge pier was placed in the center of the flume, with a diameter of 5.08 cm. The layout of the numerical model test is shown in Figure 1b. The meshing method and the setting of boundary conditions for the numerical model were the same as those detailed in Section 3.

2.3.1. Validation of Flow Morphology

Figure 2 shows a comparison of the flow morphology in the x-y plane and y-z plane around the bridge pier in this study and in the physical model test conducted by Melville. It can be seen that the numerical simulation results are in high agreement with the physical model test, including the horseshoe vortex, caused by the downward flow in front of the bridge pier and the bypassing flow around both sides of the pier, and the tail vortex formed behind the pier.

2.3.2. Validation of Local Scour Depth

Figure 3 shows the morphology of the local scour pit around the bridge pier in this study and the results from other scholars. All shows that the area near and upstream of the pier and the area within a range of ± 75° in front of the pier are the main areas where local scour occurs. The area downstream of the pier is dominated by sediment accumulation. The maximum local scour depth found in this study is 3.96 cm, which is very close to the results of Melville [34] (4.0 cm) and Zhang [35] (3.8 cm).

Figure 4 shows the riverbed elevations around the pier in both the longitudinal section (parallel to the flow direction) and the cross-section of the flume (perpendicular to the flow direction) at the center of the pier. As is shown, the overall trend of local scour around the pier in this study is consistent with the above research. However, the local scour depth of the upstream pier is smaller compared to the results of the model test conducted by Melville, which is 3.42 cm and is 85.5% of that in the test. Similar results were reported in the studies of Li [33], Zhang [35], and Qi [36]. Qi analyzed and explained the reasons for the above phenomena from the following four aspects: (1) the limitations of the turbulence model; (2) the interaction between sediment particles and fluid is not taken into account in the numerical simulation; (3) the transport of sediment in the actual riverbed is random, and the instantaneous shear stress is not the only standard for the initiation of sedimentation; and (4) the setup of the boundary conditions.

In summary, the simulation results have strong consistency with the test results in general, although they are not fully consistent with the test results in some respects, due to the complexity of the actual conditions and the partial simplification found in the simulation. This is accepted by most scholars in this field. The numerical simulation can be used to predict the maximum scour depth and the morphology of the local scour around the bridge pier. Above all, the numerical model in this study is reasonable and accurate.

3. Numerical Model Setup and Research Plan

3.1. Numerical Model Parameters

The numerical simulation model is shown in Figure 5. To be consistent with the referred studies [27,37], the numerical simulation model was built with a length of 24.0 m, a width of 9.0 m, and a height of 4.8 m. The velocity and depth (h_w) of flow were 2 m/s and 3 m, respectively. The riverbed was composed of three kinds of sediments, and the particle sizes were 0.005 m, 0.01 m, and 0.02 m, with a density of 2650 kg/m³. There were two bridge piers arranged in tandem. These were located in the center of the model. The diameters, D, were 1.5 m. The distance between the centers of the two piers is represented by G. The collar was placed on the surface of the sediment, and the parameters of the collar are described in Section 3.3.

3.2. Meshing and Boundary Conditions

Meshing is a key part of the numerical simulation process, which is directly related to the accuracy of the numerical simulation results. According to the results reported by Sarker et al. [38], the flow morphology around the bridge pier can be better modeled when the mesh size is less than or equal to D/20, which achieves more accurate simulation results. Also, the simulated efficiency must be considered, due to the computational resource limitation. Therefore, the mesh size of the model was changed gradually in this study. The mesh size grew gradually from around the bridge pier to the model boundary. As shown in Figure 6, the mesh size was initialized to D/20 within the center area (the yellow area) of the model and grew to D/10 near the model boundaries (inlet, outlet, and sides). The boundary conditions of the numerical model in this study were as follows: (1) the inlet was set as a grid overlay boundary, yielding a fully developed flow velocity distribution at the inlet. (2) The outlet was set for pressure, with a constant water depth. (3) Both sides were symmetrical. (4) The bottom was set as the wall. (5) The top was set as the pressure boundary, with a fluid volume fraction of 0. The roughness of the sediment surface was 2.5d₅₀ (The median particle size of sediment). The simulation time was preliminarily set to 1000 s.

3.3. Determination of Collar Parameters

The collar is a classical scour reduction measure because of its simple structure and significant effect versus its low cost. It has been used widely all over the world [36,39,40,41,42,43,44,45,46]. The sediment near the pier can be effectively protected from the downward flow and the horseshoe vortex generated around the pier with a collar. As shown in Figure 7, the main factors affecting the local scour reduction of the collar are the installation height (h/h_w) and its size (W). Secondly, the shape, the tilt angle (θ), the protection range (a/2π), and the thickness (T) of the collar also influence the local scour reduction. Therefore, the effects of the above six factors were analyzed separately, as shown below, for the selection of the collar parameters in this study.

To measure the scour reduction effect of the collar quantitatively, the scour reduction efficiency, E_P, was defined as below:

$$E_{\mathrm{p}}=\frac{d_{\max }-d_{\mathrm{col}}}{d_{\max }}\times 100\%$$

(14)

where d_max is the scour depth around the pier without a collar; d_col is the maximum local scour depth around the pier in different conditions with a collar.

The effects of the above six factors on the scour reduction effect of the collar are shown in Figure 8. Figure 8a shows that the scour reduction effect was the highest when the collar was placed near the riverbed surface. Whether the installation height of the collar was increased or decreased, the scour reduction effect was weakened. Therefore, the collar was installed on the surface of the riverbed (h/h_w = 0) in this study. Fang et al. [39] concluded that when the collar size is 3.0D, the local scour reduction effect is the best and the most economical, as shown in Figure 8b. Therefore, the diameter of the collar in this study was 3.0D. A rectangular collar is more effective in scour reduction. However, the rectangular collar does not reduce the scour depth to the same degree under different incoming flow angles, as shown in Figure 8c. A circular collar does not have these defects, while the scour reduction effect is only slightly lower than with the rectangular collar. Therefore, a circular collar was used in this study. As shown in Figure 8d–f, the laws of influence of the tilt angle, protection range, and thickness of the collar were relatively simple, all of which were monotonous changes. Therefore, this study refers to the optimal cases in their conclusions. The tilt angle of the collar was 0, the protective range of the collar was 360°, and the thickness of the collar was 0.05D.

3.4. Numerical Simulation Plan

According to the study objects, 2 groups of tests were conducted. Tests were divided into A and B, as shown in

Table 1. Test program.

No.	G/D	Collar (W/D)
		Upstream	Downstream
A1	2.0	×	×
A2	3.0	×	×
A3	4.0	×	×
A4	5.0	×	×
A5	6.0	×	×
A6	7.0	×	×
B1	2.0	3.0	×
B2	3.0	3.0	×
B3	4.0	3.0	×
B4	5.0	3.0	×
B5	6.0	3.0	×
B6	7.0	3.0	×

. Group A was the local scour test around the bridge piers without a collar, comprising 6 tests, numbered A1–A6. Group B included tests conducted around the bridge piers when the upstream pier was protected by a collar. These also comprised 6 tests, numbered B1–B6.

4. Results and Discussion

4.1. Local Scour around Piers

4.1.1. Results of Group A

Figure 9 shows the development of local scour depth around the bridge piers over time for group A. The development of local scour depths around the upstream piers and the maximum scour depths were close in the six cases exhibiting different spacing (G/D = 2–7). However, the development rate and the maximum scour depths around the downstream piers were increased with the increase in pier spacing. The larger the spacing was, the faster the development rate and the deeper the maximum scour depth was.

According to Sheppard [47], the relationship between the local scour depth of the pier over time can be expressed as shown below:

$$S(t)=a[1-\exp (-bt)]+c[1-\exp (-dt)]$$

(15)

where S(t) represents the scour depth at time t; a, b, c, and d are the coefficients.

The development of the scour depth of the upstream pier in the cases was close; therefore, the data in case A6 (G/D = 7) were used for the fitting analysis. The results of the fitting analysis are shown in Figure 10. The parameters of the fitted curve are shown in

Table 2. The parameters in the function.

Case	a	b	c	d	Correlation Coefficient
A6	0.15731	1.43755	0.93468	0.23459	0.99899

. In Figure 10, the data matched the curve well, and the correlation coefficient was as close as 0.99899. When the fitting time was extended to 3000 s, the depth of the local scour was stable, which means that the local scour had reached a stage of scour equilibrium with the maximum scour depth of 1.09 m. Within the simulation length of t = 1000 s, the maximum scour depth was 1.06 m, which is more than 97% of that in the stage of scour equilibrium. Therefore, the length of simulation time set in this study is reasonable. Because of the sheltering effect of the upstream pier, the development of the local scour depth around the downstream pier could not fit Equation (15). It is difficult to check whether the local scour of the downstream pier was stabilized or not when using the method above. However, the method of directly observing the scour depth can be used to check the scour status. In Figure 9b, the local scour depth around the downstream pier developed quickly in the first 20 s. Then, it slowed down gradually within 800 s and became stable at 800 s. The development of the local scour depth around the downstream pier was similar to that of the upstream pier. It could, thus, be suggested that the development of local scour around the downstream pier had also reached a stage of scour equilibrium.

Table 3. Maximum scour depths around the upstream and downstream piers (Group A).

No.	G/D	Collar (W/D)		Maximum Scour Depth (m)
		Upstream	Downstream	Upstream	Downstream
A1	2.0	×	×	1.07	0.61
A2	3.0	×	×	1.09	0.69
A3	4.0	×	×	1.10	0.91
A4	5.0	×	×	1.09	1.02
A5	6.0	×	×	1.07	1.05
A6	7.0	×	×	1.07	1.06

shows the maximum scour depth around the pier when applying different spacings. Figure 11 shows the local scour pit morphology around the pier. It can be seen from Table 3 and Figure 11 that when the spacing between piers was smaller than 4.0D, the upstream pier had a significant “sheltering” effect. In the meantime, the local scour depth and range around the downstream pier were significantly smaller than that of the upstream pier. The upstream pier played a similar role as a sacrificial pile, which helped reduce the local scour around the downstream pier. With the increase in pier spacing, G/D, the local scour depth around the downstream pier increased, and the “sheltering” effect of the upstream pier was gradually weakened. When G/D = 5, the maximum local scour depth around the downstream pier was 1.02 m, which had reached 93.5% of that of the upstream pier. When G/D > 5, the local scour depth and scour pit morphology of the downstream pier and the upstream pier were the same.

To quantitatively evaluate the “sheltering” effect of the upstream pier on the downstream pier, the local scour reduction efficiency of the downstream pier was calculated via Equation (14) and is shown in Figure 12. The d_max value was the maximum scour depth around a single pier without a collar. According to the conclusions of Wang et al. [20], the form and depth of the scour pits formed around the upstream pier with different pier spacing were almost the same as those formed around a single pier. Therefore, the maximum local scour depth around the upstream pier can be considered as d_max. The d_max value was 1.08 m, which was the mean value of the maximum local scour depth of the upstream piers of Group A. In Figure 12, the scour reduction efficiency of the downstream pier decreased continuously with the increase in the pier spacing G/D. When G/D < 3, the maximum scour reduction efficiency, E_p, could reach 43.5%. When G/D = 5, the scour reduction efficiency decreased rapidly to 5.6%. When G/D > 5, the scour reduction efficiency was lower than 3.0%, the “sheltering” effect of the upstream pier almost disappeared, and the local scour characteristics of the upstream and downstream piers were similar to that for a single pier.

4.1.2. Results of Group B

Figure 13 shows the variation in scour depth, within the time frame, around both upstream and downstream piers in Group B. In Figure 13, the collar of the upstream pier reduced the local scour around the upstream pier significantly. The average maximum scour depth decreased from 1.08 m to about 0.5 m without extra protection. The scour reduction effect was obvious. Meanwhile, compared with Group A, the upstream collar restricted the degree of sediment transport around the upstream pier, significantly slowing the development process of local scour around the downstream pier and the maximum scour depth.

Table 4. Maximum scour depths around the upstream and downstream piers (Group B).

No.	G/D	Collar (W/D)		Maximum Scour Depth (m)
		Upstream	Downstream	Upstream	Downstream
B1	2.0	3.0	×	0.49	0.17
B2	3.0	3.0	×	0.50	0.45
B3	4.0	3.0	×	0.50	0.53
B4	5.0	3.0	×	0.51	0.65
B5	6.0	3.0	×	0.51	0.81
B6	7.0	3.0	×	0.49	0.99

shows the maximum scour depth around the piers with different spacings. Figure 14 shows the scour pit morphology around the two piers. In Table 4 and Figure 14, the local scour morphology and maximum scour depth around the piers changed greatly compared to Group A. The scour depth around the upstream pier was reduced significantly, and the location of the maximum scour depth gradually moved away from the pier itself to the edge of the collar. For the downstream pier, the scour morphology and the scour depth around it had also been affected. The upstream collar limited the generation and development of a horseshoe vortex around the upstream pier, which then weakened the sediment-carrying capacity of the vortex system around the piers. The sediments were transported over a shorter distance and then deposited around the downstream pier, which reduced the local scour depth at the downstream pier. However, with the increase in G/D, the influence of the upstream pier and the collar on the downstream pier gradually weakened. When G/D = 7, the maximum scour depth around the downstream pier reached 0.99 m, and the scour pit morphology tended to be the same as that around the downstream pier of case A6.

Figure 15 shows the scour reduction efficiency of the upstream pier and downstream pier in Group B. As is shown, the scour reduction efficiency of the upstream pier was in the range of 52% to 55%. The trend of the scour reduction efficiency of the downstream pier was decreased with the increase in G/D. When G/D < 4, the scour reduction efficiency of the downstream pier decreased rapidly from 84.3% to 58.3%. When G/D = 4, the scour reduction efficiency of the downstream pier was close to that of the upstream pier. When G/D > 4, the scour reduction efficiency of the downstream pier decreased rapidly from 39.8% to 8.3%, which was much lower than that of the upstream pier. If the local scour depth around the upstream pier with the collar was safe and acceptable, this could be adopted as the target value for local scour depth reduction around the downstream pier. Therefore, when G/D ≥ 4, the local scour depth around the downstream pier could be reduced by using the collar.

Another group of studies, named Group C, was carried out with G/D ranges from 4.0 to 7.0, and with the collar of the downstream pier having the same size as that of the upstream pier.

4.1.3. Results of Group C

The test cases in Group C, as determined in Section 4.1.2, are shown in

Table 5. Test program of Group C.

No.	G/D	Collar (W/D)
		Upstream	Downstream
C1	4.0	3.0	3.0
C2	5.0	3.0	3.0
C3	6.0	3.0	3.0
C4	7.0	3.0	3.0

.

Figure 16 shows the maximum scour depth around the upstream and downstream piers of Group C over time. As shown in Figure 16, the scour depth of the downstream pier decreased significantly when a collar was also fitted to the downstream pier. Compared to the upstream pier, the scour depth of the downstream pier developed more slowly, and the maximum scour depth around the downstream pier was smaller.

Table 6. Maximum scour depths around the upstream and downstream piers in Group C.

No.	G/D	Collar (W/D)		Maximum Scour Depth (m)
		Upstream	Downstream	Upstream	Downstream
C1	4.0	3.0	3.0	0.48	0.32
C2	5.0	3.0	3.0	0.50	0.32
C3	6.0	3.0	3.0	0.51	0.29
C4	7.0	3.0	3.0	0.51	0.36

shows the maximum scour depth around the piers at different spacings. Figure 17 shows the scour pit morphology around the piers. From Table 6 and Figure 17, it can be seen that the local scour reduction effect around the downstream pier is significantly better than that around the upstream pier. When G/D = 4, local scour is not generated in the area in front of or behind the downstream pier and is only generated on both sides of the collar. With the increase in G/D, the scour area on both sides of the downstream collar gradually expanded to the pier, and the scouring began to intensify. The reason for the above phenomenon is mainly caused by the combined effect of the upstream pier and the collar. Firstly, the upstream pier greatly weakened the strength of the flow in front of the downstream pier. Secondly, the downstream collar weakened the flow in front of the pier and inhibited the formation of the horseshoe vortex, which weakened the scouring ability and sediment-carrying ability of the water around the pier. With the increase in G/D, the shielding effect of the upstream pier gradually weakened, similar to the mechanism in Group A, resulting in the scour morphology around the downstream pier gradually converging to be the same as that of the upstream pier.

Figure 18 shows the scour reduction efficiency of the upstream pier and the downstream pier in Group C. In Figure 18, the scour reduction efficiency of the downstream pier was in the range between 66.7% and 73.1%, which was much higher than that of the upstream pier (52.8% to 55.6%). If the local scour depth around the upstream pier with the collar is safe and acceptable, it can be used as a target value for local scour depth reduction around the downstream pier. In Group C, the local scour reduction of the downstream pier was excessive compared to that of the upstream pier if using the same collar, which will result in material waste and an increase in cost. Therefore, the collar size used for the downstream pier’s local scour reduction could be reduced to save the cost. Therefore, Group D tests were carried out to investigate the change in local scour characteristics around piers with different sizes of collars.

As shown in Group A and Group B, when the spacing G/D = 7, the upstream pier and the combined form of the pier and collar had less influence on the local scour depth of the downstream pier during the scour equilibrium stage. Based on the results of Group C, the spacing G/D of the pier in Group D was set to 7, the size of the upstream collar was 3.0D, and the size of the downstream collar was 2.0D, 2.25D, 2.5D, and 2.75D, respectively.

4.1.4. Results of Group D

Based on the results discussed in Section 4.1.3, the study scenario for Group D was set as shown in

Table 7. Test program of Group D.

No.	G/D	Collar (W/D)
		Upstream	Downstream
D1	7.0	3.0	2.0
D2	7.0	3.0	2.25
D3	7.0	3.0	2.5
D4	7.0	3.0	2.75

.

Figure 19 shows the variation of scour depths around the upstream and downstream piers in Group D with time. As shown in Figure 19, the change in the size of the downstream collar did not have a significant effect on the developmental characteristics of the maximum local scour depth around the upstream pier. Figure 19b and Figure 16b show that: (1) the development of the local scour depth around the downstream pier was accelerated after the size of the downstream collar was reduced, especially when W/D = 2.0. (2) The maximum local scour depth around the downstream pier increased significantly, indicating that the collar size significantly affected the local scour depth and the development characteristics.

Table 8. Maximum scour depths around the upstream and downstream piers in Group D.

No.	G/D	Collar (W/D)		Maximum Scour Depth (m)
		Upstream	Downstream	Upstream	Downstream
D1	7.0	3.0	2.00	0.51	0.65
D2	7.0	3.0	2.25	0.51	0.55
D3	7.0	3.0	2.50	0.50	0.47
D4	7.0	3.0	2.75	0.51	0.40

shows the maximum scour depth around the piers with different sizes of collars. Figure 20 shows the morphology of the scour pit around the piers. In Table 8 and Figure 20, the local scour on both sides of the downstream pier moved from the edge of the collar to the pier when the size of the downstream collar was reduced. The smaller the collar was, the deeper the local scour was. The reason for this is that even a smaller downstream collar can effectively resist the scouring of the riverbed caused by the downstream flow in front of the pier. However, the protective effect on the two sides of the high-speed bypass flow is not so obvious with a smaller collar. Therefore, local scouring behavior can develop easily from the edge of the collar to the edge of the pier.

Figure 21 shows the local scour reduction efficiency of the upstream pier and downstream pier in Group D and for case C4. In Figure 21, the scour reduction efficiency of the downstream pier decreased from 66.7% (W/D = 3) to 39.8% (W/D = 2) with a reduction in the downstream collar size. In Table 8 and Figure 21, when the size of the downstream collar was 2.25D, the local scour depth of the downstream pier was close to that of the upstream pier. Therefore, if the local scour depth around the upstream pier with a collar is considered to be safe and acceptable, it is recommended that the downstream collar size should be larger than 2.25D (W/D ≥ 2.25).

4.2. The Flow Field around Piers

The results in Section 4.1 show that the spacing between the piers and the installation of the collar greatly affects the local scour depth and local scour topography around the piers. To reveal the mechanism of the collar to reduce the local scour depth around two piers in a tandem arrangement, the flow morphology around piers with specific cross-sections will be used, which can also be used to investigate the mechanism of scour reduction.

Figure 22 shows the flow morphology around the piers in Group A. When the flow passed the upstream pier, a rising tail flow was formed behind the upstream pier. When the pier spacing G/D was smaller than 4.0, the downstream pier was located in the rising tail flow of the upstream pier, which caused a much lower velocity of the downward flow in front of the downstream pier and lesser strength of the horseshoe vortex than that around the upstream pier. Therefore, the scour depth of the downstream pier was smaller than that of the upstream pier. With the increase in G/D, the downstream pier gradually detached from the wake region behind the upstream pier, the “shielding” effect of the upstream pier was weakened, and the velocity of the downward flow in front of the downstream pier and the strength of the horseshoe vortex increased. The local scour around the piers increased. When the spacing between the piers was greater to a certain extent, the local scour depth and the shape of the scour pit of the downstream pier changed more gradually in response to the upstream pier, and the behavior of the two tended to be independent.

Figure 23 shows the streamline of the longitudinal section at the center of the pier in cases A6, B6, C4, and D1. Figure 23a shows that when a collar was not placed around the upstream or downstream pier, both sides of the upstream and downstream piers produced high flow-velocity areas due to the disturbed flow, which is one of the main factors that produce local scour. Meanwhile, the vortex systems behind the upstream and downstream piers were similar. In Figure 23b, when only the upstream pier was protected, the collar isolated the downstream flow effectively, which reduced the direct scouring of the riverbed caused by the downward flow in front of the pier and the high-speed bypass flow on both sides. In the meantime, the existence of the collar made the vortex system behind the upstream pier move toward the water’s surface and dissipate rapidly. Hence, the local scour around the upstream pier was significantly reduced. As for the downstream pier, the lack of protection by the collar resulted in an obvious local scouring effect. However, the upstream pier and the collar changed the flow field around the pier, especially that of the vortex system behind the upstream pier, which weakened the vortex system behind the downstream pier and reduced the local scour around the downstream pier. In Figure 23c,d, the local scour around the downstream pier showed a significant reduction when the collar was used to protect both the upstream and downstream piers. The flow forms and vortex systems around both piers were similar. However, with a shrinkage of the size of the downstream collar, the area protected by the collar was smaller than the influence area of the horseshoe vortex, which resulted in serious scouring at the edge of the collar. The flow and velocity below the collar were increased, which exacerbated the local scour of the downstream pier. Therefore, the location of the maximum local scour depth, as well as the maximum local scour depth, needed to be considered when deciding local scour reduction measures and the dimensions of the downstream pier.

5. Conclusions

This study investigated the characteristics of the influence of pier spacing and the use of a collar on the local scour around two piers built in tandem via numerical simulation. The conclusions are as follows:

(1)

The local scour depth of the downstream pier was not greater than that of the upstream pier because of the sheltering effect of the upstream pier without the use of a collar. With an increase in the pier spacing (from 2.0D to 7.0D), the sheltering effect of the upstream pier gradually weakened. The local scour depth of the downstream pier gradually became closer to that of the upstream pier. When the pier spacing was 5.0D, the maximum scour depth of the downstream pier reached 93.5% of that around the upstream pier.

(2)

The local scour of the upstream pier could be reduced effectively if the upstream pier was protected by a collar. The local scour reduction efficiency could reach 52~55%. Meanwhile, the local scour depth around the downstream pier could be reduced significantly as well. The local scour reduction efficiency of the downstream pier was decreased from 84.3% to 8.3% rapidly with an increase in the pier spacing. If G/D < 4.0, the local scour depth around the downstream pier would be smaller than that around the upstream pier. If G/D > 4.0, the local scour depth around the downstream pier would be greater. Therefore, if the local scour depth around the upstream pier with the collar is considered to be safe and acceptable, local scour reduction around the downstream pier by the collar must be considered if G/D > 4.0.

(3)

If both the upstream and downstream piers were protected by collars of the same size (W/D = 3.0), the scour reduction effect would be significant around the downstream pier. The local scour reduction efficiency of the downstream pier could reach 66.7~73.1% with a decrease in G/D from 7.0 to 4.0, which would be significantly higher than that of the upstream pier (52~55%). The local scour depth around the downstream pier was 64.5% of that around the upstream pier. Therefore, the size of the collar of the downstream pier can be reduced if the local scour depth around the upstream pier with the collar is considered to be safe and acceptable, which can also reduce the cost.

(4)

The local scour reduction efficiency of the downstream pier decreased from 66.7% (W/D = 3.0) to 39.8% (W/D = 2.0) with a reduction in the downstream collar size. When the size of the downstream collar was 2.25D, the local scour depth of the downstream pier was close to that of the upstream pier. Therefore, it is recommended that the downstream collar size should be bigger than 2.25D (W/D ≥ 2.25) if the local scour depth around the upstream pier with the collar is considered to be safe and acceptable.