Experimental Assessment of Scour Around Side-by-Side Double Piers in an S-Shaped Channel with Ice-Jammed Flow

Abstract

Through laboratory experiments in an S-shaped channel, this study analyzes how the flow Froude number, the ratio of ice-to-flow rate, pier spacing-diameter ratio, and bed material median grain size influence scour depth around side-by-side double piers under ice-jammed flow conditions. Unlike the development of a scour hole around a bridge pier in a straight channel, where the scour depth increases with the flow Froude number under ice-covered conditions, this study reveals that in an S-shaped channel, scour depth increases with the flow Froude number near the convex bank pier and decreases near the concave bank counterpart. Irrespective of ice conditions, a higher ratio of pier spacing-diameter correlates with augmented scour depth at the convex bank and diminished scour at the concave bank. As the ice-to-flow rate ratio increases, the ice jam thickness in the S-shaped channel also increases, leading to a significant decrease in the flow area and resulting in deeper scour holes around the piers. Equations have been developed to calculate the maximum scour depth around side-by-side double piers positioned in an S-shaped channel with ice-jammed flow.

1. Introduction

Ice jam is a common hydrological phenomenon in natural rivers in cold regions, occurring more frequently in areas with ice-blocking effects, such as bridges, river bends, point bars, and transitions from faster to slower flow (Beltaos et al. [1]). In the presence of either an ice cover or ice jam in natural rivers, the maximum velocity shifts closer to the riverbed, which may intensify the local scour process around bridge piers and potentially affect bridge safety.

To date, extensive research has been conducted on local scour around single bridge pier under open-flow conditions, providing key insights into scour dynamics (Chang, et al. [2]; Dey & Barbhuiya [3]; Ettma et al. [4]; Guan et al. [5]; Mia & Nago [6]; Yu & Zhu [7]; Radice & Tran [8]). In contrast to the local scour process around a single pier, around multiple piers, it is significantly more complex owing to varied pier layouts and spacing distances. With respect to local scour around multiple piers, research has mainly focused on the scour process around either tandem or side-by-side double piers. Here, tandem double piers refer to two piers arranged at two adjacent flow cross-sections, with the connecting piers side-by-side to the flow direction, while side-by-side double piers refer to two adjacent piers arranged at a single flow cross-section, perpendicular to the flow direction. In straight channels, interactions between piers significantly influence local scour depth, as demonstrated by Ashtiani et al. [9] in their study. The results showed that a pier edge distance-to-diameter ratio (G/D) of 0.25 increased scour depth around side-by-side piers by approximately 1.5 times compared to that of a single pier, while a G/D of 0.2 for tandem piers led to a 1.2-fold increase relative to single pier conditions. Kim et al. [10] employed computational modeling to investigate scour depth around cylindrical double piers in a straight channel, analyzing the effect of pier center-to-diameter ratio (L/D). For side-by-side arrangements, results indicate that maximum scour depth increases as L/D decreases within the range of 1.25 to 5. In tandem arrangements, scour depth peaks at an L/D of approximately 2 and gradually declines with increasing L/D. In another study, Kim et al. [11] investigated the effect of varying the angle α between the line connecting the double piers and the flow direction. They reported that the scour depth at the back pier increases as α increases within the range of 0° to 90°. However, when α is between approximately 45° and 60°, the scour depth at the back pier begins to decrease as α continues to increase. Regarding local scour around three cylindrical piers, Zhou et al. [12] observed that the maximum scour depth has a close relationship with the flow attack angle and pier spacing, which influence the maximum scour position, shifting it from front pier to middle pier. However, in S-shaped channels, changes in the L/D may cause local scour patterns to differ from those in straight channels.

Ice cover and ice jams significantly alter scour dynamics near bridge piers, with effects that differ markedly from those observed under open-flow conditions. A key factor contributing to this discrepancy is the threshold of sediment incipient velocity. Fu et al. [13] conducted experimental research using real ice to examine the impact of ice accumulation upstream of inverted siphons. Their results indicated that, under ice-covered conditions, the maximum velocity point in the vertical velocity distribution shifted closer to the riverbed and the velocity increased, making sediment more susceptible to entrainment compared to open-flow conditions. Eliisa et al. [14] investigated sediment transport under ice cover through long-term field measurements at a bend section of the Pulmanki River in northern Finland. Their findings showed that ice cover increased bed shear stress, with near-bed velocities peaking at the upstream inlet of the bend. Luo et al. [15] derived a formula for sediment incipient velocity under ice cover using mechanical analysis, demonstrating that a higher ratio of ice cover roughness to riverbed roughness facilitates sediment entrainment. Regarding scour around individual bridge piers, several studies have provided relevant insights. Ackermann et al. [16] performed flume experiments to assess the impact of ice cover on local scour around cylindrical piers under both clear-water and movable-bed conditions. They concluded that ice cover could increase local scour depth by 25% to 35% compared to open-channel flow. Hains and Zabilansky [17,18] studied scour around bridge piers under smooth and rough ice covers. Their results showed that ice cover increased flow kinetic energy near the riverbed, while higher ice roughness caused the maximum velocity point to migrate closer to the riverbed, thereby intensifying local scour around the piers. Sirianni et al. [19] further quantified how variations in the length of ice cover upstream of piers increase scour depth and volume, with a significantly greater effect than under open flow. The study reveals that scour depth and volume near bridge piers reach their peaks at 2.66 m ice cover length (up to 5.32 m), showing 46.9% and 238% increases, respectively, versus open-flow condition. Valela [20] et al. carried out experimental research on the effect of ice cover submergence depth on local scour depth around bridge piers. The results indicated a positive correlation between ice cover submergence depth and local pier scour depth: compared with open-flow conditions, when the submergence depth reached 30% of the water depth, the scour depth significantly increased by 412%. The higher Froude numbers (Fr) and sediment particle density escalate critical shear stress, affecting scour dynamics. Namaee [21] et al. conducted extensive laboratory experiments in a large-scale flume to investigate local scour processes. The research showed that the presence of ice covers significantly increases the local maximum scour depth (y_max) around piers. Compared with open-channel flow conditions, rough ice covers reduce the critical non-dimensional shear stress required for bed sediment particle initiation. Smaller pier sizes and larger pier spacing-to-diameter ratios weaken the horseshoe vortices around piers, leading to shallower scour holes. In ice-covered flow experiments within an S-shaped channel, Song et al. [22] analyzed scour around side-by-side double piers and found that the density Fr has a stronger effect on scour depth at the convex bank than the concave bank for such pier configurations. Gao et al. [23] investigated scour around tandem double piers in ice-covered flows using flume experiments and numerical simulations, identifying turbulent vortex_pier side interactions and shear stress as the primary dynamic mechanisms for pier scour. Subsequent studies by Gao et al. [24,25] revealed that ice cover expands the scour area around piers to 2.6 times the pier diameter, a 42% increase compared to open water conditions. Numerical modeling showed higher turbulent intensity around piers at equivalent positions under ice cover than under open flow. In addition, Nezaratian et al. [26] developed data-driven prediction models (M5 decision tree, PSO optimization model, GEP model) based on experimental measurements to forecast scour depth at bridge piers in ice-covered flow conditions. Results showed that the M5 decision tree and PSO optimization model achieved prediction accuracies exceeding 85%, indicating strong predictive reliability.

However, research on local scour around bridge piers under ice-jammed flow conditions is limited. Tuthill [27] et al. investigated the impact of the Milltown Dam removal in Montana on ice regime changes and analyzed the formation and breakup processes of ice jams under various scenarios using the HEC-RAS model. Results indicated that dam removal would not exacerbate downstream ice jam hazards; however, ice jams were prone to form near bridge piers, accompanied by ice jam releases, which significantly increased bed shear stress and intensified local scour around five upstream piers. Wang et al. [28] investigated scour around a single bridge pier in a straight channel under ice-jammed flow conditions. Results showed that scour intensified as the ice wave peak passed the pier downstream, while it diminished during wave trough passage. Hu et al. [29] investigated the effects of flow Froude numbers, rate of ice discharge, and pier shape coefficients on scour depth. They found that increased flow dynamics and ice jam thickness, both hydraulically and mechanically, significantly amplified the development and depth of scour holes near the bridge pier.

In summary, existing research on bridge pier local scour in ice-jammed flows has been limited to straight channel investigations. However, natural river meanders may have different local scour patterns depending on the location of the bridge piers. In addition, the use of side-by-side arrangement is a common design choice in bridge engineering. Therefore, this experimental study aims to investigate local scour near side-by-side double piers under ice-jammed flow conditions in an S-shaped channel, considering the effects of key factors such as flow Fr, ratio of ice-to-flow rate, median grain size of bed material, pier spacing-to-diameter ratio.

2. Methods and Materials

2.1. Experimental Set up

The laboratory experiments were conducted in an S-shaped channel with a length of 25.17 m and a width (B) of 0.6 m, as shown in Figure 1a and Figure 2. Across the channel, 27 measurement cross-sections (CS) were uniformly spaced at 1.2 m intervals from upstream to downstream. This study utilized three types of sediment as 15 cm thick bed material, with their grain size distribution curves depicted in Figure 3.

An automatic ice feeder (Figure 1b) was set up at CS 3: a storage bin was used to store the ice particles (Figure 4a), a motor provided the power, a support frame held the remaining four components, a valve controlled the delivery of ice particles to the channel, and a vibration spring was driven by a motor to vibrate the ice particles to output them from the storage bin to the channel. Similar to other researchers (Namaee [21]; Song [22]), untreated Styrofoam panels were employed to replicate smooth and rough ice surface conditions, respectively, as illustrated in Figure 4b. For the ice-jammed flow condition, before each experimental run, a Styrofoam panel was placed on the water surface between cross-sections 25 and 26, allowing the upstream model ice particles to form an initial ice jam at this location, simulating the initial freeze-up position in natural rivers. The model ice particles used in the experiments were made of polyethylene resin and shaped as flattened ellipsoids, with a maximum diameter of 3.5 mm and a mass density of 0.918 g cm³, which is close to the density of natural ice (0.917 g cm³). Hollow cylindrical tubes made of acrylic were used to simulate the bridge piers (Figure 4c). For the ice-jammed condition, the piers located near the convex and concave banks were designated as Pier 1 and Pier 2, respectively; and in the sheet ice-covered condition, they were designated as Pier 3 and Pier 4, respectively. The layout of side-by-side double piers is depicted in Figure 5.

In total, 31 experiments were conducted (

Table 1. Experiment setting under ice-jammed and ice-covered conditions.

Number	V(m/s)	H(m)	L(m)	D(m)	Qi/Q	d50(mm)
A1	0.18	0.2	0.15	0.02	0.0010	0.713
A2	0.22	0.2	0.15	0.02	0.0016	0.713
A3	0.16	0.2	0.06	0.02	0.0016	0.713
A4	0.16	0.2	0.11	0.02	0.0016	0.713
A5	0.16	0.2	0.15	0.02	0.0016	0.713
A6	0.16	0.2	0.20	0.02	0.0016	0.713
A7	0.16	0.2	0.25	0.02	0.0016	0.713
A8	0.15	0.2	0.15	0.02	0.0016	0.713
A9	0.14	0.2	0.15	0.02	0.0016	0.713
A10	0.12	0.2	0.15	0.02	0.0016	0.713
A11	0.16	0.15	0.15	0.02	0.0016	0.713
A12	0.14	0.15	0.15	0.02	0.0016	0.713
A13	0.14	0.25	0.15	0.02	0.0016	0.713
A14	0.16	0.2	0.15	0.02	0.0015	0.713
A15	0.16	0.2	0.15	0.02	0.0018	0.713
A16	0.16	0.2	0.15	0.02	0.002	0.713
A17	0.16	0.2	0.15	0.03	0.0016	0.713
A18	0.16	0.2	0.15	0.04	0.0016	0.713
A19	0.16	0.2	0.15	0.02	0.0016	0.609
A20	0.16	0.2	0.15	0.02	0.0016	0.438
B1	0.16	0.2	0.15	0.02	—	0.713
B2	0.16	0.2	0.15	0.02	—	0.609
B3	0.16	0.2	0.15	0.02	—	0.438
B4	0.18	0.2	0.15	0.02	—	0.713
B5	0.22	0.2	0.15	0.02	—	0.713
B6	0.16	0.2	0.06	0.02	—	0.713
B7	0.16	0.2	0.08	0.02	—	0.713
B8	0.16	0.2	0.11	0.02	—	0.713
B9	0.16	0.2	0.15	0.02	—	0.713
B10	0.16	0.2	0.20	0.02	—	0.713
B11	0.16	0.5	0.25	0.02	—	0.713

), including 20 under ice-jammed conditions (as shown in table “A”) and 11 under sheet ice-covered conditions (as shown in table “B”). The Reynolds number (Re = ρVR/ν, V is the initial flow velocity; ρ is the water density, g/cm³; R is the hydraulic radius, m; ν is the coefficient of viscosity, kg/(m·s)) in this experimental study was 12,000~26,400, and the average flow Froude number (Fr = V/$\sqrt{gH}$, g is the acceleration of gravity, m/s²; H is the initial approaching flow depth, m) was 0.114. In the experiments, the ratio of bridge pier diameter (2D) to channel width (B) ranged from 6.7% to 13.3%. In accordance with Namaee et al. [21], Zafer et al. [30], Lanca et al. [31], Zhao et al. [32], Nima et al. [33], etc., it was shown that 2D/B ranged from 4% to 17%. Therefore, in the present study, the constriction effect can be disregarded.

2.2. Experiment Procedure

All experimental runs in this study were executed in a sequential manner.

(1)

Preparation: Prior to each experiment, the sand bed was leveled to ensure consistent thickness, and the approaching flow depth, velocity, and flow rate were adjusted to predefined conditions.

(2)

Setup: Once the initial flow conditions were stabilized, the model piers were carefully fixed into the sand bed at CS18. Each time an equal mass of model ice particles (m) was added to the storage bin, the ice discharge rate (Q_i = m/(ρ_iT), m is the mass of the ice particles; ρ_i is the density of the ice particles, g/cm³; T is time, minute) was controlled by adjusting the output valve of the automatic ice feeder so that the ice particles were all put into the channel within a certain time T.

(3)

Data collection: Throughout the experiment, changes in the dimensions of the scour holes were monitored and recorded through the channel’s transparent side walls. As the ice jam head (the frontal edge of the ice jam) traversed the pier section, the arrival time and upstream ice jam length were recorded for each upstream segment of the channel. Ice jam thickness was measured on both banks of the curved flume using a 0.1 cm precision steel ruler, while water surface elevations were documented via pressure gauge tubes.

(4)

Equilibrium determination: The equilibrium state was assessed through the stability of the scour hole dimensions and flow depth, as well as consistent ice input and output rates.

(5)

Final measurements: Upon the scour near the piers reaching equilibrium state, the flow velocity field (refers to the velocity value at different depths from the water surface to the riverbed) in front of the pier and the maximum scour depth were measured by ADV, and then the localized maximum depth was measured using a ram (0.01 cm precision).

3. Results

3.1. Longitudinal Velocity Distribution

To better illustrate the velocity fields at various cross-sections, the results from the experimental run with an initial approaching flow depth (H) of 0.2 m, flow velocity (V) of 0.16 m s, and an ice-to-flow rate ratio (Q_i/Q) of 0.0016 are used as an example. The longitudinal velocities (U) were measured by ADV from CS15 to CS20. Regarding the distribution of longitudinal velocity along a single bend under open-flow conditions, the longitudinal velocity is higher along the convex bank from the entrance to the apex of the bend. From the apex to the exit and further downstream, the velocity is higher near the concave bank (Duan [34]; Moradi et al. [35]; Xiao et al. [36]). In contrast, in this experiment, the pattern of change in the main flow zone was similar to the open-flow condition (shown in Figure 6). Sui et al. [37] conducted experiments to study ice accumulation along a 180° bend in a channel and found that the maximum longitudinal velocity at the cross-sections of 45°, 90°, 135°, and 180° shifted towards the concave bank. This suggests that there may be differences in flow velocity distributions on individual versus continuous bends. In addition, differences in the distribution of ice jam thickness along the course may also lead to differences in flow velocity distribution.

3.2. Scour Depth and Its Affecting Factors

3.2.1. Scour Depth Development

Experimental findings revealed that the local scouring process around bridge piers in an S-shaped channel was classified into three stages by observing the ice jam head’s growth and ice particle’s downstream migration. The following description summarizes this process based on one of the experimental runs, where the V was 0.16 m/s, the L/D was 7.5, and the H was 0.20 m. In the first stage, before the ice jam head reaches the cross-section where the bridge pier is located, the local scour process around the piers is similar to that observed under open-flow conditions. After approximately 70 min (in Figure 7a, first highlighted point), the ice jam head reaches the bridge pier cross-section (see Figure 8a), marking the beginning of the second stage of scouring process around the piers. As incoming ice particles from upstream continuously accumulate under the ice jam head at the bridge pier cross-section, the maximum velocity gradually shifts toward the channel bed (see Figure 6, CS18). This results in an increase in shear stress near the channel bed. About 90 min later (in Figure 7a, second highlighted point), the scouring rate around the bridge piers accelerates significantly, as shown by the variation in scour depth over time in Figure 7a.

In the third stage of the scouring process around the piers, as the ice jam head continues to migrate upstream, the ice jam thickness around the piers remains relatively constant until the formation of the initial ice jam along the entire channel (see Figure 8b). Once the initial ice jam has developed along the entire channel, its thickness gradually increases in the upstream section as ice particles continue to accumulate at the ice jam head (see Figure 8c). The migration of ice particles beneath the ice jam leads to dynamic changes in the scouring rate around the bridge piers. When ice jam thickness increases, the overflow area compresses significantly; conversely, it expands as thickness decreases. Figure 7b shows greater changes in ice jam thickness on the convex bank than the concave bank, leading to a higher scour rate at Pier 1 than Pier 2. Specifically, the convex bank ice jam thickened during 420~570 min (marked by a red circle), and Pier 1’s scour rate in Figure 7a (same marker) also increased during this interval. This indicates a significant correlation between ice jam thickness changes and localized pier scour rates. After approximately 690 min (in Figure 7a, third highlighted point), the maximum scour depth at bridge piers is reached, though the experiment persists for 24 h to ensure full development of the maximum scour depth. Sirianni [19] further noted a critical relationship between bridge scour depth and upstream ice cover length under ice cover. Under ice jam conditions, when upstream ice particles accumulate flat at the ice jam head and develop upstream, ice jam length increases, and Figure 9 shows that bridge scour depth grows slowly; when the ice jam head stabilizes, upstream particles dive downstream of the head, thickening the ice jam and pushing it downstream to the bridge cross-section, increasing scour rate. Unlike ice cover, local pier scouring under ice jams is dynamic, closely linked to ice jam development and evolution.

3.2.2. Impact of Flow Condition on Scour Depth

Figure 10 illustrates the relationship between the scour hole relative maximum depth around the pier (the maximum depth, y_max, to the initial flow depth ratio, H) and Fr. The effect of Fr on scour depth is different due to the difference in flow velocity distribution in a straight and an S-shaped channel. According to the experimental data of Namaee et al. [21], who investigated the local scour around two side-by-side piers in a straight channel under sheet ice-covered flow conditions and, the relationship between the relative scour depth (y_max/H) and Fr in a straight channel, as depicted in Figure 10a, y_max/H at piers increases as the Fr value increases. In contrast, under ice-jammed condition, as depicted in Figure 10b, the relative maximum depth of the scour hole (y_max/H) at Pier 1 increases with an increase in Fr value, while y_max/H decreases as Fr value increases at Pier 2. As shown in Figure 11, during ice-jammed flow, the maximum longitudinal flow velocity point in front of Pier 2 migrates toward the ice jam base with an increasing Froude number (Fr). This results in a relative decline in flow shear stress near the riverbed, which causes the scour depth at Pier 2 to be smaller than that around Pier 1. As shown in the longitudinal velocity distribution in Figure 6, at CS18, under ice jam flow conditions, the main flow zone at the inlet of the S3 section is skewed toward the convex bank, while the area of maximum flow velocity shifts closer to the riverbed. The migration of the maximum velocity zone elevates shear stress near Pier 1, resulting in intensified scour. This highlights that ice jam formations restructure local flow, thereby diverging scour patterns from those under sheet ice cover.

Figure 12 illustrates the correlation between pier scour maximum depth and ice-to-flow rate ratio (Qi/Q) under equilibrium. As depicted, maximum scour depth around both piers (irrespective of position) increases with rising Qi/Q. Figure 13a shows that the increase in ice-to -flow rate ratio (Q_i/Q) makes the thickness of ice jam in the section increase, and the flow depth of the downstream channel decreases significantly (Figure 13b), and at the same time, the ice particles are easy to pile up and thicken to form an ice jam body around the bridge piers (section 18). Ice jam formation reduces the overflow area, significantly increasing local water flow velocity and enhancing its erosive power. This highlights the substantial influence of ice jams on local pier scour processes, irrespective of pier position on concave or convex banks.

3.2.3. Impact of Bed Material, Pier Spacing-to-Diameter Ratio, and Diameter on Scour Depth

The influence of L/D on local pier scour in S-shaped channels manifests through two distinct mechanisms: (1) pier–pier interactions altering near-pier turbulence intensity, and (2) L/D-driven changes in pier positioning within varying flow velocity fields. While this study focuses on the second mechanism—how pier spacing modifies their exposure to differential flow velocities, the first mechanism remains underexplored under ice-jammed conditions. As shown in Figure 14, experimental data demonstrate that increasing L/D leads to deepening scour at Pier 1 (convex bank) due to heightened flow shear stress near the riverbed, as the pier becomes more aligned with the high-velocity core (CS18’s longitudinal velocity distribution in Figure 6). Conversely, scour at Pier 2 (concave bank) diminishes with larger L/D, as the pier is progressively displaced from the main flow zone, reducing sediment entrainment capacity. This behavior aligns with the velocity-driven scour mechanism observed in sheet ice-covered flows by Song et al. [22], validating the role of pier positioning in scour dynamics.

Notably, the turbulence-enhancing effect of pier interactions has not been investigated under ice jams in this study. So, further experimental work is required to quantify its contribution to scour depth in ice-jammed S-shaped channels.

As shown in Figure 15, maximum scour depth increases with pier diameter. Notably, scour depth at Pier 1 is more sensitive to pier diameter changes than that at Pier 2. This relationship arises from larger piers generating more pronounced flow obstruction, which enhances the intensity of downward flow and horseshoe vortices at the pier face, thereby deepening scour holes. Figure 16 further shows that as pier diameter increases, ice jam thickness also increases, with more pronounced changes observed on the convex bank, for example at CS18. This increase in ice jam thickness reduces the under-ice flow area, which in turn increases flow velocity. This intensified flow condition particularly exacerbates scouring at Pier 1 near the convex bank, highlighting the complex interaction between structural dimensions and hydrodynamic forces under ice-jammed conditions.

Figure 17 shows the relationship between scour depth around piers and the median grain size of bed material. Experimental results indicate that, under both ice-jammed and sheet ice-covered flow conditions, scour depth decreases as the median grain size of the bed material increases. This is because larger grain sizes require higher kinetic energy to initiate sediment movement. Therefore, under the same flow conditions, smaller particles require lower flow velocities to initiate motion, leading to deeper scour holes at the piers. The above findings are in agreement with those of Namaee [21] et al. under ice-covered flow.

3.3. Equation for Estimating the Maximum Scour Depth at Piers

Based on experimental data, factors influencing flow velocity, flow depth, balanced ice jam thickness on both banks, flow rate, incoming ice flow, bridge pier diameter, pier spacing, channel width, and sediment median grain size were considered. Channel curvature was excluded as it remained constant in the S-shaped channel. Within the experimental scope, in an S-shaped channel, a relationship equation was constructed for the maximum scour depth around side-by-side double piers under ice-jammed conditions and its influencing factors:

$$y_{max}=f\left(T_1,T_2,V,g,H,Q_i,Q,D,L,B,d_{50}\right)$$

(1)

where, y_max represents the maximum scour depth, (m). T₁ and T₂ denote the equilibrium ice jam thicknesses on the left and right banks of the pier section, respectively, (m). The variables g, V, and H denote gravitational acceleration (m·s⁻²), initial approaching flow velocity (m·s⁻¹), and flow depth (m), respectively. Q_i and Q indicate the ice discharge rate (m³ s) and flow rate (m³ s), while D and L represent the pier diameter (m) and the pier spacing distance, respectively (m). B and d₅₀ describe the wide of channel (m) and the bed sediment median grain size (m). Using dimensional analysis and assuming a dimensionless relationship can be a product of power laws, Equation (2) is expressed in the following dimensionless form:

$${\displaystyle \frac{y_{imax}}{H}}=k{\left(Fr\right)}^a{\left({\displaystyle \frac{T_i}{H}}\right)}^b{\left({\displaystyle \frac{L}{D}}\right)}^c{\left({\displaystyle \frac{Q_i}{Q}}\right)}^d{\left({\displaystyle \frac{D}{B}}\right)}^e{\left({\displaystyle \frac{d_{50}}{D}}\right)}^f$$

(2)

where Fr, k, a, b, c, d, e, f are coefficients

The relative maximum scour depth around piers along the convex bank (i.e., pier 1) is as follows:

$${\displaystyle \frac{y_{1max}}{H}}=2.8\times {10}^2{\left(Fr\right)}^{0.471}{\left({\displaystyle \frac{T_1}{H}}\right)}^{0.03}{\left({\displaystyle \frac{L}{D}}\right)}^{0.174}{\left({\displaystyle \frac{Q_i}{Q}}\right)}^{1.088}{\left({\displaystyle \frac{D}{B}}\right)}^{0.321}{\left({\displaystyle \frac{d_{50}}{D}}\right)}^{-0.305}$$

(3)

(With R² = 0.80).

The relative maximum scour depth around piers along the concave bank (i.e., pier 2) is as follows:

$${\displaystyle \frac{y_{2max}}{H}}=7.7\times {10}^{-4}{\left(Fr\right)}^{-1.125}{\left({\displaystyle \frac{T_2}{H}}\right)}^{0.015}{\left({\displaystyle \frac{L}{D}}\right)}^{-0.106}{\left({\displaystyle \frac{Q_i}{Q}}\right)}^{0.606}{\left({\displaystyle \frac{D}{B}}\right)}^{-0.832}{\left({\displaystyle \frac{d_{50}}{D}}\right)}^{-0.985}$$

(4)

(With R² = 0.83).

From Equations (3) and (4), under ice-jammed flow, the relative maximum scour depth (y_max/H) at side-by-side piers in S-shaped channels increases with relative ice jam thickness (T/H) and ice-to-flow rate ratio (Qᵢ/Q), irrespective of pier location (convex or concave bank). Additionally, for both pier positions, y_max/H rises as bed sediment median grain size (d₅₀/D) decreases. At the convex bank pier, y_max/H is positively correlated with flow Froude number, pier spacing-to-diameter ratio (L/D), and pier diameter (D/B). Conversely, at the concave bank pier, y_max/H increases when L/D and D/B decrease.

Equations (3) and (4) enable the calculation of maximum scour depth at side-by-side double piers in S-shaped channels under ice-jammed flow. Defining Y₁ = y_1max/H and Y₂ = y_2max/H, partial derivatives of the equations with respect to each input parameter Xi (i = 1, 2, 3, 4, 5, 6) are derived to assess model output sensitivity to parameters, with results tabulated in

Table 2. Parameter local sensitivity analysis.

Input	${\displaystyle \frac{\mathcal{\partial }{Y}_1}{\mathcal{\partial }(Fr)}}$	${\displaystyle \frac{\mathcal{\partial }{Y}_1}{\mathcal{\partial }(\raisebox{1ex}{$T_1$}\!\left/ \!\raisebox{-1ex}{$H$}\right.)}}$	${\displaystyle \frac{\mathcal{\partial }{Y}_1}{\mathcal{\partial }(\raisebox{1ex}{$L$}\!\left/ \!\raisebox{-1ex}{$D$}\right.)}}$	${\displaystyle \frac{\mathcal{\partial }{Y}_1}{\mathcal{\partial }(\raisebox{1ex}{$Q_i$}\!\left/ \!\raisebox{-1ex}{$Q$}\right.)}}$	${\displaystyle \frac{\mathcal{\partial }{Y}_1}{\mathcal{\partial }(\raisebox{1ex}{$D$}\!\left/ \!\raisebox{-1ex}{$B$}\right.)}}$	${\displaystyle \frac{\mathcal{\partial }{Y}_1}{\mathcal{\partial }(\raisebox{1ex}{$d_{50}$}\!\left/ \!\raisebox{-1ex}{$D$}\right.)}}$
Result	0.41	0.01	0.01	67.36	0.95	−0.85
Input	${\displaystyle \frac{\mathcal{\partial }{Y}_2}{\mathcal{\partial }(Fr)}}$	${\displaystyle \frac{\mathcal{\partial }{Y}_2}{\mathcal{\partial }(\raisebox{1ex}{$T_2$}\!\left/ \!\raisebox{-1ex}{$H$}\right.)}}$	${\displaystyle \frac{\mathcal{\partial }{Y}_2}{\mathcal{\partial }(\raisebox{1ex}{$L$}\!\left/ \!\raisebox{-1ex}{$D$}\right.)}}$	${\displaystyle \frac{\mathcal{\partial }{Y}_2}{\mathcal{\partial }(\raisebox{1ex}{$Q_i$}\!\left/ \!\raisebox{-1ex}{$Q$}\right.)}}$	${\displaystyle \frac{\mathcal{\partial }{Y}_2}{\mathcal{\partial }(\raisebox{1ex}{$D$}\!\left/ \!\raisebox{-1ex}{$B$}\right.)}}$	${\displaystyle \frac{\mathcal{\partial }{Y}_2}{\mathcal{\partial }(\raisebox{1ex}{$d_{50}$}\!\left/ \!\raisebox{-1ex}{$D$}\right.)}}$
Result	−0.68	0.01	−0.003	26.33	−1.74	−1.92

.

As presented in Table 2, the ice-to-flow rate ratio (Qi/Q) exhibits the largest partial derivative, signifying that maximum scour depth around side-by-side double piers is highly sensitive to Qi/Q fluctuations, consistent with experimental observations. In contrast, the partial derivative of L/D is relatively low, indicating its variation has a minor effect on maximum scour depth around these piers.

Using Equations (3) and (4), the computed maximum scour depths around side-by-side double piers in the channel are validated against laboratory experimental data. Figure 18 demonstrates close alignment between calculated results and measured values, indicating that Equations (3) and (4) provide a basis for estimating maximum scour depth around such piers under ice-jammed flow in natural rivers. However, equation coefficients and parameters require calibration using field-measured data from natural river environments.

4. Questions and Discussion

This study presents a preliminary investigation into the local scour mechanisms of side-by-side double piers during ice-jammed flow in S-shaped channels. However, the local scour law of piers in an S-shaped channel is different from that in a straight channel, and further experimental research needs to be carried out.

(1)

In the experiment, only three types of bed material median grain size were conducted, which may result in insufficient representativeness of the parameter term related to the median grain size of the bed material in the fitted formula. Further experimental research should be carried out in the future. Additionally, prototype observations are needed to validate and correct the coefficients in the equation.

(2)

The experiment only conducted research on cylindrical side-by-side double piers. However, in practical engineering, pier shapes and arrangement patterns are diverse, and the scour laws of different pier types under ice jam conditions may vary. Therefore, further experimental research should be carried out in the future.

(3)

In the S-shaped channel under ice jam conditions, the current study has only explored the influence of pier spacing L/D changes (causing piers to be in different velocity fields) on scour depth, while the impact of interactions between piers on scour depth still needs further experimental investigation.

5. Conclusions

Through experimental investigations in S-shaped channels, this study analyzed scour law of side-by-side double piers during ice-jammed flow conditions, with key findings summarized below:

(1)

Within the experimental scope, under ice jam conditions, the longitudinal velocity distribution in an S-shaped channel resembles that of open flow: the main flow zone shifts toward the convex bank upon entering the bend and toward the concave bank when exiting. Based on ice jam development, local pier scour can be divided into three stages: (1) open-flow scour, (2) scour induced by the ice jam head reaching the pier section, and (3) dynamic scour changes due to ice jam evolution. Unlike ice-covered conditions, scour depth under ice jams positively correlates with upstream ice jam length; as the ice jam accumulates and thickens, a secondary increase in scour rate occurs.

(2)

Under ice-jammed flow conditions, as pier spacing-to-diameter ratio (L/D) and flow Froude number (Fr) increase, the maximum scour depth around the piers follows the trend of ‘increasing along the convex bank and decreasing along the concave bank’. Increases in ice-to-flow ratio (Q_i/Q) and bridge pier diameter (D) both result in deeper scour depths; the smaller the median grain size of the bed material (d₅₀), the more likely to form deeper scour holes.

(3)

Equations (3) and (4), derived through multiple regression analysis, characterize the peak local scour depth for side-by-side double piers within ice-jammed S-shaped channels. These equations demonstrate strong consistency with experimental measurements.