Study of Flow Characteristics and Anti-Scour Protection Around Tandem Piers Under Ice Cover

Abstract

The impact of an ice-covered environment on the local flow characteristics of a bridge pier was studied through a series of flume tests, and the dominant factors affecting the scour pattern were found to grasp the change laws of the local hydrodynamic characteristics of the bridge pier under the ice cover. At the same time, because the scour problem of the pier foundation is a technical problem throughout the life-cycle of the bridge, to determine the optimal anti-scour protection effect on the foundation of the bridge pier, active protection scour plate was used to carry out scour protection tests, and its structural shape was optimized to obtain better anti-scour performance. The test results show that the jumping movements of sediment particles in the scour hole around the pier are mainly caused by events Q2 and Q4, which are accompanied by events Q1 and Q3 and cause the particle rolling phenomenon, where Q1 and Q3 events are outward and inward interacting flow regimes, and Q2 and Q4 events are jet and sweeping flow regimes, respectively. The power spectral attenuation rate in front of the upstream pier is high without masking effects, while strong circulation at the remaining locations results in strong vorticity and high spectral density, in particular, when the sampling time series is 60 s (i.e., f = 1/60), the variance loss rates under ice-covered conditions at the front of the upstream pier, between the two piers, and at the tail end of the downstream pier are 0.5%, 4.6%, and 9.8%, respectively, suggesting a smaller contribution of ice cover to the variance loss.

1. Introduction

Bridge piers placed in a complex underwater environment for a long time will be subjected to the reciprocal elution of the current, which, in turn, constitutes a potential hazard to the safety and stability of the bridge [1]. For many years, scholars in various countries [2,3,4] have mainly used indoor experiments to study the localized bypass scour of bridge piers and to reproduce the real bypass scour environment around the piers through simulation tests. At the same time, many bridge piers around the world are destroyed every year following various damaging events [5,6,7], such as natural disasters (floods, earthquakes, typhoons, etc.), as well as manufactured factors (traffic accidents, construction misconduct, etc.), but the damage caused by hydrodynamic factors is one of the main issues. Therefore, appropriate protective measures against specific causes of damage need to be taken for bridge piers, from the microscopic fine point of view, around the underwater piers near the law of flow movement to explore and analyze [8,9]; however, the damage and protection of bridge pier structures is a complex and critical problem that still requires in-depth research.

Localized bypass scour at bridge piers is based on a coupled water–sand dynamic relationship [10,11] wherein turbulent changes in the flow regime of the surrounding waters are induced by the water-obstructing effect of the bridge piers, which, in turn, drives the transport of bed sediment particles. The fixed ice cover in the vicinity of a bridge pier after a river freezes causes the water flow structure to be redistributed under the ice, which then causes changes in the turbulent shear stress at the bottom of the river bed and produces significant local scour phenomena; at the same time, the presence of ice covers can induce flooding hazards, which can exacerbate the flow field’s complexity and instability around the pier [12]. Thus far, there are relatively few studies on the bypass scouring of tandem bridge piers under ice cover, and the existing studies mainly focus on typical pier types, such as cylindrical or rectangular piers under open-channel flow conditions [13]; therefore, we conducted this study intent on considering the influence of the ice cover factor as the evaluation criterion for the scour and pier perimeter eddy enhancement and exploring the turbulence characteristics and scouring features of the pier’s perimeter under the ice-covered flow, with the aim of providing clear knowledge toward the study of ice, water, and sand dynamics around piers.

Due to the influence of the cold northern climate, different degrees of freezing phenomena will occur near river bridge piers [14]; therefore, we carry out multifactor and multilevel orthogonal tests for the localized bypass scour of bridge pier foundations under ice cover in order to find the dominant factors affecting scour patterns. This research is of practical application value for the construction of cross-river and cross-sea bridges and is of great academic significance for the basic theoretical research of bridge piers. This paper focuses on the problem of flow characterization around bridge piers through the breakthrough of some fundamental theoretical difficulties [15,16]. Taking anti-scour characteristics as a starting point, according to the intrinsic mechanism of bridge pier protection measures [17,18], this study aims to take active protection measures, i.e., combining shaped piers for the benefit of scour plate scour protection research, to reduce the intrinsic frequency and source force of the traveling current to provide scour protection to bridge pier foundations. This study is based on the systematic exploration of the stereotypical elements of the scour plate [19] and optimizes the design of its structural shape to obtain better anti-scour performance, which can subsequently provide a reference for finding practical applications of research from on the bypass scour mechanism.

2. Experimental Setup and Program

2.1. Experimental Setup and Model

Our experiment took place in the Inner Mongolia Agricultural University Hydraulic Laboratory with a multi-functional variable-slope flume; the flume arrangement is shown in Figure 1. The sink length was 25 m, its width 1.2 m, its depth 0.65 m, the maximum variable slope 0~1%, an overflow section for the rectangular, variable-slope sink inlet rectifier was set to eliminate external interference caused by sudden changes in the current; an outlet was used to control of the loose-leaf tailgate’s opening degree in order to achieve stable flow conditions, and the front end of the sedimentation tank was convenient for recycling sand particles in the circulation of the flow. The flow velocity measurement around the bridge pier is measured using Acoustic Doppler Velocimetry (ADV). To improve the measurement accuracy, it is necessary to calibrate it before measurement, including ➀ The use of platinum resistance thermometers (uncertainty is better than 0.2 °C) to the actual temperature used for correction of the speed of sound, to control the impact of the deviation of the speed of sound to 0.1% or less; ➁ Based on the ADV to provide 0.03, 0.1, 0.3, 1.0, and 2.5 m/s and other five upper limits of the flow velocity, the upper limit of the flow velocity will directly affect the actual resolution of the measurement, to improve the accuracy of the measurement, the appropriate speed range should be selected to ensure that the most minor upper limit of the speed of the selection of the conditions do not exceed the range.

In order to clarify the local scour characteristics, we analyzed the scour holes around the pier under the ice cover based on both the morphology of the prototype piers in the field and the open-channel flow study. Given that our aim was to study the impacts of tandem piers in different scour environments, the structural dimensions of the prototype piers were simplified and scaled (see Figure 2); taking into account the limitations of the experimental flume sidewalls, three horizontal scaling ratios were used (based on experimental design considerations), i.e., 1:150, 1:75, and 1:50, wherein the heights of the modeled piers were uniform.

Considering the shading effect of the tandem piers, our proposed anti-punching plate model was divided into single-pier and combined-pier protection types, with the overall appearance of the design based on the combination of scour hole morphology characteristics around unprotected piers, of which 1 # and 2 # in Figure 3, which form the maximum outer diameter, are 2D and located in the upstream pier around the inflection point, and the simplified processing of 2 # is based on the scour hole morphology in the downstream protection; the material selected for this experiment was highly translucent acrylic board, designed and printed using a laser cutting control system.

2.2. Experimental Program Design

2.2.1. Design of Localized Scour Experiments

In this study, the model piers with a scaling ratio of 1:75 were selected for testing and analysis, and three groups of tests were carried out. The parameters of each group of test conditions can be seen in

Table 1. Experimental flow conditions.

Case	D/cm	H/cm	U/Uc	Fr	Re
C1/2/3	4.8	15	0.82	0.20	11,808
C4/5/6	4.8	15	0.94	0.23	13,536
L1/2/3	4.8	15	1.21	0.29	17,424

Note: The letter C denotes clear-water scour; the letter L denotes live-bed scour.

. Since the strength of the flow velocity magnitude directly affects the sediment exchange rate around the bridge piers, indicated with letters and serial numbers for the convenience of describing each group of tests, among which C1~3 test for the low-flow rate of clear-water scour characteristics, C4~6 test for the near-critical flow rate of clear-water scour characteristics, and L1~3 test for the high-flow rate of moving-bed scour characteristics. Meanwhile, the test conditions of each group include three types of cover conditions—open-channel current, smooth ice cover, and rough ice cover—which are used to compare the effects on the local scour depth.

2.2.2. Design of Anti-Impact Protection Test

The design of the anti-impact protection experiment, and its corresponding scour protection experiment, was mainly conducted focused on the single-pier and combined-pier protection types because we aimed to study tandem piers in different scour performance environments, simplified scaling of the prototype’s structural dimensions was undertaken, considering the limitations of the test flume side wall [20,21], whereby three horizontal scaling ratios were used to set up two types of protection tests, with a total of 54 groups of test conditions, and repetitive tests for each group of conditions, in which each group of conditions had two repeatability tests, and after the end of each group of tests, the riverbed surface sediment and the bottom layer of sediment for the flip replacement, to prevent prolonged repetition of the test, so that the sediment particle size gradation changes, which affects the depth of the scour, but also to maintain the consistency of the results of the repeat test and reliability. Finally, the scour eigenvalues of the control tests were used to identify the optimal protection design for the perimeter of the pier to resist scour.

Anti-scour plate structures play a significant role in inhibiting pier foundation scour erosion, and their installation positions significantly affect their impact protection. When the horizontal installation position h is higher than a certain height h_u from the pier perimeter bed, the influence of the submerged flow’s impact force on the anti-scour plate is small, so the plate’s ability to protect the pier perimeter bed is weak, falling into the failure zone; when the horizontal installation position h is lower than the corresponding unprotected conditions of the maximum brush depth h_n, the perimeter of the pier bed is unprotected, also falling into the failure zone; therefore, only when the anti-scour plate installation position is within the effective region, that is, within the range of h_n < h < h_u, will it play a significant protective role on the perimeter bed (as shown in Figure 4). Kumar et al. [22] concluded that the best protective effect occurs when the scour protection plate is installed on the surface of the riverbed. Based on the above conclusions, we selected the surface of the riverbed around the pier as the installation location in this study in order to further explore its anti-scour characteristics.

2.3. Experimental Phenomena

2.3.1. Localized Scour Experimental Phenomena

The underwater tandem bridge pier is a solid structure that forms an obstruction to the current and produces a bypass scour phenomenon on both sides. In addition to the transportation of sediment particles, the local scour hole around the pier is thereby gradually brushed deeper and broader and eventually reaches the limit of the scour hole’s morphology and characteristics, and the boundary contour becomes evident; therefore, through analysis of the development of the scour hole profile characteristics in different scour environments and under different cover conditions, it has been determined that observing the scour hole around the pier over time can help to deepen the understanding of the local scour mechanisms (as shown in Figure 5).

Figure 6, Figure 7 and Figure 8 show the evolution of scour holes under different cover conditions in the clear-water scour environment. Based on the evolution of scour holes as determined in this study, combined with research from related scholars, and in order to investigate the influences of different current cover conditions on the local scour around the pier in the two types of scour environments, the local scour process of the pier was summarized into three phases, based on the rate of development: the initial phase, the developmental phase, and the equilibrium phase [23,24,25]. From our test results, the following processes were determined: (1) In the initial scour stage, the tandem-pier scour hole first appeared at the front of the upstream pier perimeter; then, on both sides, the scour hole developed rapidly to a maximum scour depth of about 50%; in the development stage, the upstream pier perimeter scour hole gradually developed downstream, though downstream scour development lagged behind that of the upstream pier perimeter scour hole, while the overall scour hole development rate continued to slow down, with the maximum scour depth of the upstream pier reaching 75% of the equilibrium scour depth. In the equilibrium scour stage, the upstream and downstream scour holes evolved into a whole, and the scour holes did not widen with time because the local bed shear stress around the pier was lower than the critical shear stress. (2) The local scour hole’s surface-level shape around the observed piers generally showed a tendency for longitudinal development, much larger than the transverse development rate, indicating that the influence of the vortex structure in the scour hole on the pier side of the formation of transverse-axis circulation is weaker than that of the longitudinal horseshoe vortex. (3) At the equilibrium state, the morphology of the local scour holes was similar under all three types of conditions, among which the local scour holes of the piers under the rough ice-covered condition were the most significantly amputated, followed by the smooth ice-covered condition, indicating that the influence of ice cover roughness on the scour depth and range should be addressed.

Scour holes around submerged bridge piers significantly impact the natural evolution of channel geomorphology and can provide a reliable burial depth for pier foundations during bridge construction. To more clearly portray the influences of the open-channel flow and ice-covered flow on the scour topography around the pier in different scour environments, we use this section to define the characteristic parameters of the scour hole shape, which include the upstream pier scour hole, where length L_u is the maximum distance from the upstream pier front to the scour hole in the longitudinal axis of the pier; the upstream pier surface scours hole, where width W_u is the maximum distance from the upstream pier side abutment point to the edge of the scour hole side; and the downstream pier surface, where the scour hole width W_d is the maximum distance from the downstream pier side abutment point to the edge of the scour hole side, as shown in Figure 9.

2.3.2. Reduction in Impact Protection Test Phenomena

Bridge pier foundation scour is a technical problem throughout the whole life span of the bridge and is also a significant concern for bridge construction. In order to study the protection effects and laws regarding two types of scour plate protection on the pier bed, local scour was selected under different coverage conditions, representative of the test conditions for the corresponding control analysis conducted through the flume test, to study scour plate protection of the local scour hole’s pier foundation within the hydrodynamic protection mechanism. At the same time, we explored the addition of protective measures after determining the scour characteristics from the specific protective effect on the pier perimeter, as shown in Figure 10 and Figure 11.

Figure 10 and Figure 11 show the single-pier and combined-pier scour protection tests, in which (a)~(c) indicate three current cover conditions, i.e., open-channel flow, smooth ice cover, and rough ice cover, respectively, because of the space limitation of this section, we only analyzed the test conditions for the 1:75 ratio, with a current strength of 0.94. From the test results, the following can be observed: (1) Both single-pier and combined-pier scour plates play specific protective roles on the scour around the pier foundation, as compared with the corresponding unprotected condition scour test in the previous section of the article, the scour hole surface area was significantly reduced, and the depth of the brush was also reduced year-on-year. (2) A comparison of the two types of protection scour tests can be clearly observed in Figure 10. Under open-channel current conditions, the least amount of scour was observed around the pier, indicating the best protective effect. Additionally, with the installation of ice cover, the amount of scour increased significantly, and the protective effect weakened. Scour under rough ice cover conditions was the most serious, which is consistent with our conclusion in the previous section; that is, the ice cover roughness, scour hole, and brush depth show significant positive correlations. At the same time, it can be observed that single-pier protection, placed in the front pier of the flushing plate, failed to effectively protect the rear pier, such that the rear pier scour hole was more significant and the pier tail dune accumulation was higher, compared with the effect of combination pier protection, shown in Figure 11, where the tandem pier circumference of the scour is small, the rear pier tail dune is not apparent, and the protective effect is better than that of single-pier protection. (3) Through the different types of piers protection of sediment loss and anti-scour coefficient t-test can be seen (

Table 2. Sediment loss and t-test of the coefficient of anti-scour for different pier protection types.

Protection Type	Factor	Sediment Loss at Specific Times			Anti-Scour Coefficient
		Open Channel Current	Smooth Ice Cover	Rough Ice Cover
Single pier protection	mean value	1.541	0.246	0.101	9.214
	standard deviation	0.157	0.059	0.012	0.906
Double pier protection	mean value	0.254	0.060	0.042	34.436
	standard deviation	0.096	0.006	0.015	4.759
Significance p-value		0.008	0.007	0.021	0.013

), single-pier protection than double-pier protection of anti-scour performance is significantly reduced, anti-scour coefficient of about double-pier protection of about 3.7 times, in different scour environment of a specific time under the loss of sediment is not a significant difference. (4) The significant reduction in the geometric characteristics of the scour hole indicates that the protective measures against the submerged flow around the pier were more significant, as the submerged flow failed to cause practical impacts on the sediment particles around the pier; meanwhile, due to the horseshoe vortex, sand within the range of failure was beyond the scope of protection for the impact plate, so the pier foundation was within the range of adequate protection. (5) In this study, the installation location of the anti-scour plate for the sand bed surface around the pier was based on the conclusions of Kumar et al. [22], and the results of the protection test verified the accuracy of the relevant conclusions, i.e., determining the optimal installation location of the anti-scour plate for the surface of the sand bed significantly reduces the degree of current-related scour.

Based on the results of our scour protection experiments, only the test condition with a pier scaling ratio of 1:75 was selected for analysis, and the specific scour parameters and their scour reduction rates can be seen in

Table 3. Maximum scour hole surface area rate of reduction.

Type	U/Uc	Surface Area of Scour Hole/cm2			Surface Area Reduction Rate/%
		Open-Channel Current	Smooth Ice Cover	Rough Ice Cover	Open-Channel Current	Smooth Ice Cover	Rough Ice Cover
Unprotected	0.82	220.5	336.5	413.4	/	/	/
	0.94	356.6	485.2	559.2	/	/	/
	1.21	488.7	525.8	756.3	/	/	/
Single-pier protection	0.82	156.4	240.2	308.5	29.1	28.6	25.4
	0.94	250.6	358.2	426.1	29.7	26.2	23.8
	1.21	281.5	426.8	531.9	42.4	18.2	29.7
Double-pier protection	0.82	108.1	183.4	254.9	50.9	45.5	38.3
	0.94	165.4	276.1	328.4	39.0	43.1	41.3
	1.21	206.7	336.4	431.9	57.7	36.0	42.9

and Figure 12. From the chart, we can conclude the following: (1) After installing the anti-punching plate, the maximum scour hole area reduction rate under the single-pier protection condition was about 42.4%, and that under the double-pier protection condition was about 57.7%; thus, single-pier protection is weaker than double-pier protection, and the protection scope is limited to the upstream pier perimeter of its installation, so it is not able to form adequate protection for downstream piers. (2) From the perspective of observing the scour hole surface area reduction rate, it can be seen that double-pier protection’s overall reduction rate is higher than that of single-pier protection, indicating that the protection effect of double-pier protection is more ideal, whereas open-channel current conditions are more significant for the reduction rate than ice-covered conditions. The reason for the existence of the ice cover was open-channel current velocity zoning, but the maximum vertical flow velocity point weakened the protective effect in a downward manner. (3) The significance of the current strength and protection effect is closely related to strong current conditions when the impact of scour reduction is more apparent, and the anti-scour plate can effectively resist the vertical current caused by the downward swirling roll to achieve scour hole area reduction.

3. Transient Analysis

For this section, based on fully developed turbulent fluid and through the study of the current cover conditions of the bypass flow pattern characteristics around the pier, the three-dimensional bypass flow field near the scour hole of an indoor underwater bridge pier was measured with the aid of Acoustic Doppler Velocimetry (ADV) to clarify the characteristics and laws of variations in the characteristic covariates of the current motion in both time and space, as well as to elucidate further the mechanism of the scour bypass phenomenon occurring around the bridge pier. The detailed distribution of flow velocity measurement lines is shown in Figure 13, where measurement lines a and f were laid in the horizontal plane around the scour hole in the direction of the current, and measurement lines b~e were laid in the direction of the vertical current; due to limitations caused by the ice cover and the measuring instruments, measurements began at the free liquid surface, and the position of about 6 cm on the lower surface of the ice cover, and the measurement depth was measured from the bottom of the scouring hole to the free liquid surface and the lower surface of the ice cover, and the spacing intervals were measured in increments of 1 cm to measure the flow field domains, and the distribution of flow velocity measurement points along the direction of the depth of the water was as dense as possible to ensure the validity of these experimental data.

3.1. Quadrant Analysis

The local bed sediment particle transport phenomena occurring near the bridge pier are closely related to the flow motion events occurring in the turbulent boundary layer around the pier, and when the water body is in a turbulent state, it will produce a sudden mutation phenomenon, specifically manifested as a migratory evolutionary process such as uplift, jetting, settling, and sweeping among the liquid mass groups; this phenomenon is the kinetic mechanism for the formation and development of the Reynolds stresses, so, regarding the current burst phenomenon, most of the quadrant analysis [26] was used to study each type of flow event quantitatively. The hole domain (Hole) is defined at the center of the plane coordinates, and the threshold parameter M defines the scale of the hole domain, the bounding region consisting of four hyperbolic curves; thus, the parameter M is utilized to clarify a quadrant that plays a dominant role.

The quadrant analysis method [26] was used to plot the measured point flow velocity in a planar coordinate system, with longitudinal pulsating flow velocity u′ and vertical pulsating flow velocity w′ as the horizontal and vertical axes, and to classify the burst events into four quadrants based on the positivity and negativity of the values of pulsating flow velocities (Figure 14). The bursts presented in each quadrant are referred to as Q-events, denoted as Qi (i = 1~4). In the first quadrant (Q1 event) and the third quadrant (Q3 event) for the outward and inward interaction flow, respectively, as well as the second quadrant (Q2 event) and the fourth quadrant (Q4 event) for the jet and sweep flow, respectively, it can be seen that the turbulence levels of Q1 and Q2 events promote the uplift of the fluid mass group. In contrast, the turbulence levels of Q3 and Q4 events promote the sedimentation of the fluid mass group. Therefore, it is essential to clarify the quadrant that plays a dominant role in predicting the turbulence intensity of the flow around the pier.

To quantitatively describe the characteristic law of burst events, we selected vertical lines a and b (Figure 15), located in front of the upstream pier at the position of 0.9D and behind the downstream pier at the position of 0.9D, respectively, as well as four points on each of the vertical lines a and b, for our measurements, and the time series of the measured longitudinal plumb flow velocity are plotted on the axes, whereby the frequency of occurrence for each of the respective Q-events was statistically determined.

Based on the identification conditions of quadrant flow states for the burst events, the characteristic time series of longitudinal and vertical pulsating flow velocity at the measuring point a2 on the plumb line, under the ice-covered flow condition were selected (Figure 16) according to the time distribution plot of the pulsating velocity, the alternation of the four flow state quadrants over time can be observed.

Figure 17 shows the quadrant analysis and frequency histograms of the burst events for different gauges around the pier under open-channel flow conditions, where Figure 17a–d are the corresponding points a1 to a4 on the a-gauge line, and Figure 17e–h are the corresponding points b1 to b4 on the b-gauge line, respectively. The following can be observed in the figure: (1) The pulsating flow velocity on upstream pier-front gauge line a is larger overall than the flow velocity at the corresponding point on downstream post-pier gauge line b, which is consistent with the conclusions of the flow velocity field distributions, presented in the previous section of this paper, suggesting that the pulsating flow velocity in front of the pier holds high momentum for the occurrence of a burst or sudden change event. (2) On plumb lines a and b, around the bridge piers, the frequency of the jet flow regime is higher for the measurement points near the free liquid surface (a1, b1), and the frequency of the sweep flow regime is higher for the measurement points near the scour pits (a4, b4), which shows that Q2 and Q4 events are the key drivers of the uplift and sinking of the fluid mass. (3) The error bars in the frequency histograms of Qi events in each quadrant are the standard deviations [27], calculated after several sets of statistical analyses, and it can be seen that the error bars of the sampled data points are small, indicating that the frequency measurement errors in this study are small and robust.

Due to the different contributions of each quadrant flow regime to the turbulence strength of the water column, the mode of movement for sediment particles changes with the change in dominance of the flow regime; therefore, to discern the contribution (S_i,M) of each quadrant flow regime to the Reynolds stress, the following equation was used [28]:

$$S_{i,M}={\displaystyle \frac{1}{T}}{\displaystyle {\int }_0^T{C}_{i,M}(t)u^{\prime }(t)w^{\prime }(t)}dt$$

(1)

where T is the sampling duration; I is the quadrant number; M is the threshold parameter; and C_i,M is the coefficient, with a value of 1 if and only if $\left|u^{\prime }(t)w^{\prime }(t)\right|>M\left|\overline{u^{\prime }{w}^{\prime }}\right|$ and $\left|u^{\prime }(t)w^{\prime }(t)\right|$ are located in quadrant Qi and a value of 0 otherwise.

Figure 18 shows the vertical line distribution of the Reynolds stress contribution values from different line burst events around the open-channel flow pier, in which the contribution values from the burst events in each quadrant can show the following characteristics: the contribution values S1 and S3, corresponding to Q1 and Q3 events, are negative; the contribution values S2 and S4, corresponding to Q2 and Q4 events, are positive. From the figure, the following can be observed: (1) The contribution of each Qi event to the Reynolds stress on the upstream pier front a-gauge line is significantly better as the relative water depth decreases, i.e., close to the scour hole, vs. close to the free liquid surface. (2) Comparing the Reynolds stress contribution values from the near-bottom pier-front survey line and the post-pier b survey line, it can be seen that the proportion of Q2 events in survey line a is about 80%, and the proportion of Q4 events is about 70%, while the proportion of Q2 events in the survey line b is about 40%, and the proportion of Q4 events is about 50%, indicating that the water mass in front of the pier is more intensely turbulent, and thus more affected by turbulent eddies. (3) Overall, the contributions of Q1 and Q3 events to the Reynolds stress are smaller than those of Q2 and Q4 events, indicating the dominant roles of jet and sweep flow regimes in the flow regime change around the pier, which is consistent with the findings of numerous studies on turbulent open-channel flow regimes [29,30,31].

Figure 19 shows the quadrant analyses and frequency histograms of burst events at different gauge lines around the pier under ice-covered current conditions, where Figure 19a–d show the corresponding points a1 to a4 on the a-gauge line, and Figure 19e–h show the corresponding points b1 to b4 on the b-gauge line, respectively. The following can be observed: (1) Combining the quadrant analysis and the corresponding frequency histograms, it is evident that most of the pulsating flow points are located in the second and fourth quadrants, suggesting that jet and sweep events play dominant roles in the flow regime around the piers. (2) Observing each measurement point on both the a and b survey lines reveals that the frequencies of the jet and sweep flow regimes were higher at the measurement points close to the scour pits (a3, a4, b3, and b4) relative to those far from the scour pits (a1, a2, b1, and b2), with the frequency of Q4 events being highest at the points nearest the scour pits (a4, b4), which accounted for about 36 percent of all Qi events. (3) Based on the ratio of the frequency of Qi events in each quadrant, it can be seen that the leading causes of sediment particles in the scour pits around the bridge piers are Q2 and Q4 events, which are accompanied by Q1 and Q3 events, and then cause the rolling phenomenon of the particles. (4) Overall, there is a higher percentage of Q4 events under ice-covered flow conditions at the measurement point near the scour pit, resulting in a deeper scour. In comparison, there is a higher percentage of Q2 events under open-channel flow conditions at the measurement point near the free-flowing liquid surface, indicating that the flow regime under ice-covered conditions significantly differs from that under open-channel conditions.

Figure 20 shows the plumb line distribution of Reynolds stress contribution values from the burst events of different gauge lines around the pier under ice-covered current conditions, from which the following can be observed: (1) Under the ice-covered current condition, Q1 and Q3 events at the gauge lines in front of and behind the pier contribute less to the Reynolds stress, and bursts only occur near the bottom of the riverbed, though the overall contribution value is higher than that under open-channel current conditions, which further shows that ice cover influences the frequency of Qi events. (2) To further identify the contribution of each quadrant event in the upper and lower water column to the Reynolds stress, the contributions of Q1 and Q3 events under ice cap flow are compared with those of Q2 and Q4 events, where most of the (|S_1,0| + |S_3,0|)/(|S_2,0| + |S_4,0|) values of measurement line a are less than 1 when the relative water depth is lower than 0.5, and those of measurement line b tend to be less than 1; when the relative water depth is higher than 0.5, the (|S_1,0| + |S_3,0|)/(|S_2,0| + |S_4,0|) value of line a is usually less than 1, while that of line b is greater than 1. Overall, it is shown that the presence of ice cover weakens the frequencies of Q1 and Q3 events, while Q2 and Q4 events are strengthened. (3) The vertical distribution of bursts shows a wandering pattern for all four quadrants, and ice cover enhances the intensity of water turbulence, a crucial factor contributing to the jumping of sediment particles.

3.2. Energy Spectral Analysis

The phenomenon of turbulence prevails in rivers in nature, which is specifically expressed as the energy transition process of a series of vortex structures with very different spatial scales [32]; therefore, it is of great significance to obtain the regular characteristics of the energy distribution and vortex structure transfers around piers using energy spectral analyses, as well as to use energy cascades to study the energy transition course for vortices of different scales, for in-depth analyses of the flow field domains around bridge piers.

Based on the fully developed turbulent flow boundary layer, its one-dimensional turbulent energy spectrum [33] can be expressed as follows:

$$S(k)=\alpha {\epsilon }^{2/3}{k}^{-5/3}$$

(2)

where S(k) is the spectral density of the wave number in turbulent flow, α is the one-dimensional Kolmogrov universal constant, ԑ is the turbulent energy dissipation rate, and k is the wave number.

In turbulent fluids, where the pulsating flow rate is significantly lower than the time-averaged flow rate, the wave number spectral density is converted to a frequency spectrum according to the Taylor–Plato assumption [34], as follows:

$$k={\displaystyle \frac{2\pi f}{U}}$$

(3)

$$kS(k)=f\cdot S(f)$$

(4)

where f is the frequency, U is the time-averaged flow velocity, and S(f) is the frequency spectral density in a turbulent flow.

Equations (2)–(4) provide the following frequency spectral density relation [35]:

$$S(f)=\alpha {\epsilon }^{2/3}{f}^{-5/3}{\left(U/2\pi \right)}^{2/3}$$

(5)

The velocity threshold of the energy spectrum is chosen based on the deletion algorithm [36] in the acceleration thresholding method, which uses the frequency spectrum density relation to place the energy spectrum in the inertial subregion into good agreement with Kolmogrov’s ‘−5/3 scaling law’; meanwhile, considering the redundancy of the vertical velocity component, the turbulent kinetic energy after the frequency spectrum f^−5/3 is used as a specific feature of the energy cascade after k^−5/3.

In this segment of our study, in order to determine the energy transition and distribution laws of the multi-scale eddies within the scour holes around a bridge pier, an energy spectral analysis of the instantaneous three-dimensional flow velocity components was carried out. The Fast Fourier Transform (FFT) algorithm was used to scale the turbulent flow domain, and the Tukey 53H method was used to generate a series of smoothed time series. According to analyses from previous [37,38,39] statistical studies on the standard error of the turbulent boundary layer’s velocity, the optimal recording period for the experiment was 60 s to 120 s; therefore, we selected a sampling duration of 90 s to obtain time-independent instantaneous flow velocity turbulence analysis of three typical feature points—the upstream pier-front, inter pier, and the downstream pier-tail locations of the tandem bridge piers—and the turbulence energy spectra of the three-dimensional instantaneous flow velocity components at the corresponding feature point locations of the open-channel and ice-covered flows were plotted (Figure 21 and Figure 22).

Figure 21 shows the turbulent energy spectral analysis of different regions around the pier under open-channel flow conditions, where S_u, S_v, and S_w are the three-dimensional velocity components of the energy spectrum density, expressed as a function of dimensionless time (tU₀/D) using normalized instantaneous velocity components (u/U₀, v/U₀, and w/U₀). The following can be observed: (1) In the energy spectra at the three positions, at f ≥ 1 Hz (i.e., microscale), it can be found when sieving and filtering out the individual outliers that the turbulent kinetic energies of all velocity components exhibit very similar characteristic variations; meanwhile, in the scale range of 2~4 Hz, the turbulence shows nearly isotropic characteristics, which is in good agreement with Kolmogrov’s −5/3 power-law criterion, indicating the inertial subregion of frequency ranges; for higher frequencies (i.e., more minor scales), compared with the spectra of S_u and S_w, the spectra of the transverse velocity component S_v show a steeper slope, suggesting that the dissipation of transverse turbulence is more significant on small-scale eddies. (2) In front of the upstream pier and between the two piers, the curves of the transverse velocity component S_v have significant peaks around the frequency range of f, about 0.4 Hz, and the corresponding vortex-shedding frequency, Strauer’s number, S_t = fD/U₀, is about 0.2 and is accompanied by the distribution of the time series v/U₀, confirming obvious cyclicity (tU₀/D ≈ 10 or so); meanwhile, the curves of longitudinal velocity component S_u show weaker, but recognizable, peaks within the same frequency range, indicating that the measurement point is within the longitudinal circulation zone, while the vertical velocity component S_w shows no apparent vortex-shedding periodicity. (3) At the high-frequency section of the turbulent velocity component in front of the upstream pier in Figure 21a, the spectral contours of the three-direction velocity component have steeper slopes compared with those at the inter-pier and pier-tail locations, indicating that the energy cascade of the large-scale eddy structure here is the cause of the deeper scour crater depth, thus playing a dominant role in the turbulent kinetic energy contribution. (4) The scale–distribution relationship between energy spectral density and frequency is consistent with the decay of the energy spectral function in logarithmic coordinates, indicating that the logarithmic energy spectral characteristics of the continuous high energy can be fitted by selecting a reasonable scale region within the range of the Reynolds number for the water flow Re (2000~17,600).

Figure 22 shows the turbulent energy spectral analysis of different regions around the pier under ice-covered flow conditions, where S_u, S_v, and S_w are the three-direction velocity components of the energy spectrum density, expressed using the normalized instantaneous velocity components (u/U₀, v/U₀, and w/U₀) as a function of dimensionless time (uU₀/D). The following can be observed: (1) The energy spectral distributions in the range of low-frequency vortex scales (f < 1 Hz) under ice-covered current conditions share a similar quasi-periodic oscillation law, and there are apparent peak fluctuations in the frequency range of 0.2~0.8 Hz, as well as a multi-peak phenomenon, which is likely to be caused by the detachment of vortices at the dominant frequency. (2) The curve of the transverse velocity component S_v in the upstream pier-front turbulence in Figure 22a has a steeper slope in the frequency range of f ≥ 1 Hz compared with the inter-pier and pier-tail locations, indicating that the dissipation of transverse turbulence in the pier-front segment is more significant, thus presenting a more pronounced effect on the width of the pier-front scour hole explicitly; for the exact inter-pier measurement point in Figure 22b, the gap between the frequency spectrum energy density of each velocity component is smaller compared with the upstream pier front and the downstream pier tail, indicating that the turbulent energy inside the pier is subjected to a weaker perturbation, which has less influence on the changes within the scour hole. As can be seen in Figure 22c, the reduction factor of the current flow disturbance on the riverbed surface after the pier tail demonstrates an apparent negative correlation between the vertical velocity component w and the lateral velocity component v. (3) The turbulent instantaneous velocity energy spectrum distribution describes the turbulent pulsation characteristics of the multi-scale vortex, and the energy spectral analysis shows that the large-scale vortex causes low-frequency pulsation, while the small-scale vortex determines high-frequency pulsation; thus, the velocity pulsation energy is in transition, from the large-scale vortex structure to the small-scale vortex structure, and the final coherent structure vortex energy dissipates due to viscous forces. (4) With regard to riverbed sediment particles, the existence of an ice cap produces periodic disturbances (i.e., orderly turbulence) to the flow around the pier, which enhances the sand hostage and mobility of the bypassing flow, resulting in the intensification of the depth and breadth of the scour hole. Previous scholars [40,41,42] researching fluid bypass have reached similar conclusions, a positive outcome for studying viscous incompressible turbulence field.

4. Impact Analysis of Scour Reduction and Protective Properties

4.1. Standard Errors

In order to characterize the turbulence structure, improve the accuracy and robustness of the measured velocity vectors, and ensure better smoothness and consistency in the turbulence parameters captured from the flow velocity time series, turbulence statistics, and error analyses were performed for the sampling period time series in different coverage environments. With the help of ADV to measure the peri-pier feature points of the combined bridge pier, the velocity vectors in the measured time series were filtered based on the threshold criteria [43] of frequency signal correlation and signal-to-noise ratio. Figure 23 depicts a comparison of the normalized time series variation for the 3D velocity components under the two coverage conditions, where the light yellow area indicates the time series segment of the error analysis, i.e., t* is 1 min (this time interval was chosen to ensure the smoothness of the turbulence parameters captured from the velocity time series to obtain a higher spatial resolution), and the gray line indicates the correlation variation in these measured data; it can be observed that the longitudinal velocity component fluctuates at a larger magnitude, while the transverse and vertical velocity components show steady-state values.

Standard error analysis of the relevant characteristic parameters obtained from flow velocity measurements helps to measure the reliability of current turbulence intensity on short time scales; the accuracy of the relative velocity measurements on short time scales is the short-term error, as expressed with the following metric [44]:

$$SE={\displaystyle \frac{\sigma }{\sqrt{N}}}={\displaystyle \frac{\sigma }{\sqrt{R\Delta t}}}$$

(6)

where SE is the standard error, σ is the pulse standard deviation, N is the number of samples, and ∆t is the sampling time step.

Figure 24 shows the standard error of the near-bed relative height, time-averaged velocity, and turbulence intensity for the 60 s record length, corresponding to the pier circumference in the selected time series segment; red represents the downstream velocity, blue represents the transverse velocity, the solid circle represents the open-channel current, and the hollow circle represents the ice-covered current. From the figure, the following can be determined: (1) The selected record length, i.e., 1 min, is most significant for the standard error value at the near-bed region, reflecting its better prediction error value, as the uncertainty of its index parameters increases further away from the sand bed surface, which applies to downstream velocity U and lateral velocity V in both coverage environments. This outcome is consistent with the conclusion of the correlation between the standard errors at the near-bed region and the outflow region, as drawn by Nikora et al. [45]. (2) The downstream time-averaged velocity U is positively correlated with the standard error SE₆₀, and the open-channel and ice-covered currents have comparable levels of influence. The correlation between the transverse time-averaged velocity V and the standard error SE₆₀ is governed by the flow environment, with a positive correlation under open-channel current conditions and a negative correlation under ice-covered current conditions; this discrepancy is likely to be related to the ice-cover altering the distribution of the flow velocity’s boundary layer. (3) The relationship between standard error and turbulence intensity is linearly correlated, i.e., a more significant standard error, independent of the relevance of other variables, accompanies more vigorous turbulence intensity.

4.2. Loss of Variance

The formation of the scour hole around the pier is caused by the current’s turbulent shear stress, while turbulent changes in the current are reflected in the superposition of high- and low-frequency eddies with different scales in the fluid mixing layer. Turbulence spectral analysis [44] was used to demonstrate the strength of energy transport and dissipation in eddies of different frequencies, where high frequencies indicate small-scale eddies and low frequencies indicate large-scale eddies. Meanwhile, Escauriaza [32] found that the experimental measurements of turbulence intensity around piers have variance loss associated with low-frequency fluctuations, whose values are too large to lead to an underestimation of the Reynolds shear stress, based on which we employed the cumulative power spectral density (CSD) to focus on the peri-pier variance loss for different coverage environments in order to clarify the influence of the shading effect on the eddy frequency.

Loss of variance means that the dispersion of flow measurement data increases, leading to large fluctuations in the measurement results at different points in time under the same measurement conditions, thus reducing the accuracy of the flow measurement. Figure 25 and Figure 26 show the power spectral density and cumulative power spectral density distributions around the piers under open-flow and ice-covered conditions, respectively, where S_vv represents the transverse pulsating flow power; S_fv, S_mv and S_bv denote those at the upstream pier-front, inter-piers, and downstream pier-tail locations, respectively; V_f, V_m and V_b denote the variance loss in the same locations; RL₆₀ denotes the selected time series segment in 60 s record length; and f is the frequency. The figures show the following: (1) Comparing the attenuation degree slope and the two characteristic slopes of the power spectral density distribution in Figure 25 and Figure 26, the power spectral slopes in the upstream pier-front location are closer to a straight line, with a characteristic slope of −5/3, and the power spectral slopes at the downstream pier-tail and inter-pier locations are closer to the characteristic slopes of −1, under both open-flow and ice-covered conditions; meanwhile, S_mv and S_bv are significantly higher than S_fv, indicating that the power spectral attenuation rate is larger at the upstream pier-front location unaffected by the mask effect, while the rest of the location is strongly circulated via the mask effect, resulting in strong vorticity and high spectral density. (2) According to the cumulative spectral density obtained from the selected time series, the differences in variance loss at different regional locations can be derived from Figure 25b and Figure 26b. When the sampling time series is 60 s (i.e., f = 1/60), the variance losses at the upstream pier-front, inter-pier, and downstream pier-tail locations under the open-flow condition are 5.2%, 12.4%, and 20.3%, respectively; on the other hand, the variance losses at the upstream pier front, inter-pier, and downstream pier-tail locations under the ice-covered condition are 0.5%, 4.6%, and 9.8%, respectively, indicating that the presence of ice cover contributes somewhat to the variance loss. (3) From the spectral density distribution in the figure, it can be seen that the spatial power spectrum is largely confined to the low-frequency region, which cannot show the correspondence between frequency and time more clearly.

5. Conclusions

Using a prototype bridge pier as the research object, a series of physical modeling tests were carried out to analyze the role of an ice-covered environment in the local scour of the bridge pier, and the following conclusions were obtained:

(1) Through the comparison of several combinations of bridge piers and local scour characteristic parameters, we found that the use of tandem protection for the rectangular pier perimeter caused deeper and broader local scour depth than combined protection; the test results show that the combination of bridge pier structures for the flow of anti-scour performance is more optimal and therefore recommended for river bridge piers.

(2) From the perspective of observing the scour hole surface area reduction rate, we found that the overall reduction rate of double-pier protection is higher than that of single-pier protection, indicating that the protection effect of a double-pier scour plate is more ideal, as the reduction rate is higher under open-channel current conditions. Under ice-covered conditions, the existence of the ice cover shifts the maximum vertical flow velocity point downward, weakening the protective effect.

(3) Based on the ratio of the frequency of events in each quadrant, it can be seen that most of the sediment particle jumping in the scour hole around the bridge pier is caused by Q2 and Q4 events, which are accompanied by Q1 and Q3 events and then cause the particle rolling phenomenon. Because quadrant analysis is limited by dimensionality, studies are usually confined to the same region to characterize bursts.

(4) The scale–distribution relationship between the energy spectral density and frequency is consistent with the decay of the energy spectral function in logarithmic coordinates, indicating that the logarithmic energy spectral characteristics of continuously high energy can be fitted by selecting a reasonable scale region within the range of the Reynolds number for the water flow, Re (2000~17,600).

(5) The relationship between standard error and turbulence intensity is linearly correlated, i.e., a more significant standard error, independent of the relevance of other variables, accompanies more vigorous turbulence intensity; at the same time, based on the coefficient of loss of variance, it is possible to identify the strength of the frequency vortex in the fluid mixing layer around the bridge pier. The structural form of the bridge pier protection mechanism significantly affects the quantification of the vortex frequency, and the decrease in spectral density values under open-flow conditions further validates the low-frequency variance loss.

(6) According to the cumulative spectral density obtained from the selected time series, the differences in variance loss at different regional locations can be discerned; when the sampling time series is 60 s (i.e., f = 1/60), the variance losses at the upstream pier-front, inter-pier, and downstream pier-tail locations under the open-flow condition were 5.2%, 12.4%, and 20.3%, respectively, while those at the same locations under the ice-covered condition were 0.5%, 4.6%, and 9.8%, respectively, indicating that the presence of ice cover contributes somewhat to the variance loss.