Unraveling Debris-Enhanced Local Scour Patterns around Non-Cylindrical Bridge Piers: Experimental Insights and Innovative Modeling

Abstract

Bridge structures face a critical threat from localized scour-induced damage, prompting urgent attention to civil infrastructure resilience. Prior research has primarily focused on the influence of pier shapes on scour patterns. However, the exploration of the combined effects of various debris shapes, each possessing distinct properties, on predictive scour depth models around the non-cylindrical pier has hitherto remained less researched. This study explored the complex dynamics governing local scour around bridge piers, focusing on the influence of surface and near-surface debris. This research shed light on changes in scour depth by investigating factors like pier geometries, debris arrangements, and submersion depths. The experiments and analysis revealed the effects of various pier shapes—cylindrical, square, rectangular, oblong, oval, and lenticular—on scour patterns. Different geometries influenced primary scour zones and affected areas, with square piers causing the deepest scour and lenticular ones showing shallower instances. Scour depths typically peaked upstream across geometries, but ogival and lenticular shapes exhibited unique patterns. The research also introduced a formula that integrated debris attributes into predictive scour depth modeling, validated with favorable accuracy. Ultimately, this predictive model advances scour prediction, particularly in debris-laden flows, offering valuable insights for engineering and management practices in understanding real-world scour mechanisms and hydraulic dynamics.

1. Introduction

Bridges often fail due to loss of foundation support attributed to localized scouring at the bridge footing. Engineers attempt to predict an ultimate “long-term” scour depth to design reliable and safe foundations for bridge construction. Despite strides made over the past several decades, scientists and engineers still seek better ways to predict scouring behavior for many naturally occurring conditions. Floating debris around bridge piers exacerbates scouring mechanisms and poses the threat of catastrophic bridge collapse.

Recent investigations underscore the role of stream flow in generating a horseshoe vortex phenomenon and the flow divergence around piers, magnifying the impact of vortices recognized as primary agents of localized scouring [1,2,3,4,5,6,7]. Among these, noteworthy is the study by [1], which carefully examined average stream and turbulence parameters within a controlled flume using a cylinder-shaped pier atop a sand bed. The researcher noted the evolution of interactions as the scour hole expanded; the magnitude and kinetics of the horseshoe vortex surged while near-bed velocities reduced. The amplitude of the downflow and scour rate exhibited a strong correlation. Debris obstruction precipitates structural vulnerability [8,9,10], by enlarging the apparent pier dimensions, confining streamflow near and around the pier, and increasing streamwise and downward flow around the pier, thereby intensifying local scouring [11,12,13].

Further investigations explored the effects of pier orientation and form on local scouring. Ref. [14] revealed that the rectangular form generates the most significant scour depth while also noting that pier alignment with water flow minimizes the impact of the rectangle pier’s total length on scour depth. Ref. [15] contributed an assessment of field cases and laboratory findings, presenting the dimensions and aspects of local scour around various pier shapes. Meanwhile, ref. [16] reported on the effects of pier design and orientation on scour equilibrium surrounding individual piers during clear water flow conditions. They studied five pier shapes, including circular, square, and oval, paired with distinct size analyses. Based on both direct and indirect observations, ref. [17] suggested a workable approach for determining the likelihood of debris collection at bridge structures. The proposed methodology allows inspectors to prioritize bridges for scour assessment. Due to the significantly increased risk of scour in the presence of debris, many of the bridges that are prone to debris collections should have priority for scour assessments following floods. Ref. [18] examined the impact of debris properties and bridge pier geometry on the development and spread of wood debris jams at bridge piers. The results showed that the maximum size of jams was largely independent of the pier’s shape, with dowel jams being up to 40 times smaller than natural stick jams at higher Froude values. Wood with branching was found to be crucial for creating debris blockages.

Although extensive research exists on scour near cylindrical piers, e.g., [19,20,21], demonstrating the added impact from debris, less attention has focused on non-cylindrical piers’ physical scour patterns and their flow dynamics, where experimental studies may reveal flow–sand bed interaction. Scouring around non-cylindrical piers embedded within the sand bed presents a formidable challenge to scour depth prediction. Consequently, further laboratory investigations have focused on studying individual non-cylindrical piers’ underlying flow patterns, including interaction with various debris shapes and dimensions.

Ref. [22] conducted laboratory experiments to study scouring patterns and depths around a square pier with varying debris accumulation forms and sizes. They found that debris collection near bridge piers increases scour depth, especially when it is upstream or extends to depths near the sand substrate. The study also revealed that the collected form (clump) debris accumulation significantly influences the scour depth. Debris accumulation can significantly increase the scour depth around bridge piers, regardless of the pier shape. Similarly, ref. [23] presented an advanced study on the effects of three debris shapes on ogival piers by considering a solitary log placed at varying water depths. However, practical scenarios with river-induced debris accumulation from wave and flood actions may extend debris interaction to the riverbed [24]. The study investigated the impact of debris buildup on local scour near bridge pier groups, focusing on rectangular debris. Lab tests examined scouring patterns and depths around various debris types. A new empirical model predicted the scour depth near debris-accumulating bridge pier groups. The researchers found that upstream and full-depth debris accumulation substantially increases scour depth.

This current experimental study investigates the effects of the debris shape upstream from various pier shapes, addressing the gaps in previous studies. Notably, it delineates the scour profiles arising from debris obstruction from distinct pier shapes, determines experimentally the influence of debris depth below the surface on maximum scour depth, and introduces an innovative approach using a shape factor (S.F) to evaluate the debris block’s frontal area (%A) impact on scour, extending previous work that increased the effective pier dimensions [19,22].

The recently introduced model equation serves as a realistic tool for quantifying alterations in local scour resulting from the presence of debris. Building upon existing measurements and incorporating data from [19,25], this approach surpasses the limitations often associated with previous studies confined to examining singular log debris configurations. Additionally, this method extends its analysis to include the impact of floating debris extending downward to the sand bed. The equation allows for a more comprehensive assessment of the influence of debris in scour formation, particularly when considering the debris extension below the near-surface region. In contrast to the conventional practice of expanding apparent pier dimensions to mitigate scouring effects, this approach provides better accuracy and is more robust. This study addresses gaps in modeling scour patterns and interactions around non-cylindrical piers, thereby advancing the field’s knowledge. Incorporating a shape factor enhances our understanding of debris-related effects and significantly improves the precision of predictive models.

2. Experimental Characteristics and Setup Procedures

2.1. Flume Property

The investigation occurred at the Ministry of Water Resources-Engineering Studies and Design Center, Iraq. The experimental setup utilized a flume with dimensions measuring 12.5 m long, 0.3 m wide, and 0.55 m deep. It featured a consistent slope of 0.0004 (Figure 1a–c). Glass panels formed the flume’s lateral boundaries, while the bottom was metal. The flume comprised three distinct sections: an upstream, a downstream, and a central test section accommodating the sediment recess. The test section extended approximately 8.5 m from the inflow source. Downstream, a movable vertical gate regulated the tailwater depth. Point gauges fixed to metal bars mounted on rollers measured depths throughout the test section. These measurements provided a precision of ±0.1 mm, allowing for unobstructed surface height and depth variations. A water velocity meter with 0.1 to 4 m/s range at ±1% precision measured the flow velocity. The water circulation system traveled through interconnecting storage containers and pipes linked to a water pump. Before entering the testing section, flow stringers positioned upstream in the flume directed the water and ensured uniform and operationally favorable conditions. Moreover, a 4 m long bed reach preceded the testing section. It held a bed layered with sand to induce adequate roughness and ensure fully turbulent conditions.

Commencing from a flatbed with z = 0 denoting the bed level, a variable flow rate mechanism governing the pump allowed precise flow control. An electrical flow gauge measured water discharge with an accuracy of 0.01 m/s. The flume state positioned at the flume’s terminus controlled the flow elevation, typically at a level of 0.12 m.

2.2. Sand

Figure 1 depicts the sediment recess measured 6 m long, 0.2 m deep, and 0.3 m wide. The bed sediment’s particle size distribution curve, determined by sieve analysis, appears in Figure 2. The x-axis shows the grain size in millimeters, while the y-axis shows the percentage of sand finer than a specific grain size. D₅₀ denotes the particle diameter where 50% of the sand is coarser and 50% finer. Specific gravity tests determined the individual sand particles’ density (mass or weight). The sand homogeneity indicated how uniform the grain sizes were. The computation of uniformity required knowledge of D₈₄ and D₁₆ (corresponding to grain diameters that were 84% and 16% finer) that produces an estimated standard deviation, $\sigma =\sqrt{D_{84}/D_{16}}$, resulting in a value of 1.16. With a sand uniformity of 1.16, the sediment was generally uniform, with a limited range of particle sizes. Relatively constant streamflow conditions may transport and deposit a uniform layer of fluvial quartzite sediment. Figure 2 provides the grain size curve.

The selection of this sediment aimed to ensure that the experiments conducted for clear water scour occurred at V/Vc < 1, where V represents the flow’s average velocity and Vc denotes the critical flow velocity necessary for sand rolling initiation. Employing a D₅₀ value greater than 0.8 mm for the median sand size while avoiding ripple formation based on [26,27] mitigated the formation of bottom layers influenced by cohesive sand effects.

Concerning sand roughness (D/D₅₀), where D signifies the width of the pile, it is widely held among experts in hydraulic systems that the equilibrium scour depth generally remains unaffected, particularly when D/D₅₀ exceeds 25 [28]. Nevertheless, these studies acknowledge that experimenters cannot wholly disregard the influence of sand roughness, even when dealing with higher D/D₅₀ levels.

2.3. Pier and Debris

A comprehensive series of investigations explored scouring processes and turbulence characteristics associated with various pier configurations within the flume. The candidate pier shapes consisted of: cylindrical (C), square (S), rectangular (RE), oblong (OB), ogival (OG), and lenticular (LE), as exemplified in Figure 3. The pier widths measured 2.5 cm across all shapes. For all shapes except cylinder, and square, a uniform length of 10 cm was employed, and the angles of attack (α) were set at 0°. All tested piers were positioned at a distance of 8.5 m from the upstream strainer, a design choice intended to ensure the establishment of fully developed turbulent flow within an open channel, maintaining the required flow elevation range at 50Y–150Y, where Y is the flow depth, as stipulated by [29]. The pier’s length (L)-to-width ratio (L/D) was maintained at a value of four, a parameter adhering to the average aspect ratio recommended by [30] to nullify the impact of the (L/D) ratio on the equilibrium local scour depth when exceeding 4. Additionally, the pier width was deliberately kept below one-tenth of the flume width following [31] to mitigate undesirable sidewall effects and eliminate the occurrence of contraction scour, as emphasized by [32]. Furthermore, insights from [32] were also drawn upon to explore the relationship between flow elevation and armor-building shorelines in riverbeds. In the context of homogeneous riverbed materials, Y/D values exceeding 3 allowed the disregard of flow elevation Y. In this study, the Y/D value was set at 4.8, approximating the critical value. The critical aspect ratio value (B/Y), denoting the ratio of channel width (B) to Y, held significant importance in influencing the flow characteristics, energy dissipation mechanisms, and turbulence structure within three-dimensional channels. In the context of smooth rectangular channels, prior research has determined the critical aspect ratio to be 2 [33], indicating the pivotal point where notable changes in flow dynamics occur. Remarkably, this current investigation aligns with the established critical aspect ratio, with findings indicating a value of 2.5. This correspondence reinforces the understanding that the critical aspect ratio profoundly impacts the hydraulic behavior of smooth rectangular channels, underscoring its relevance in comprehending and predicting fluid flow phenomena. The size range and forms of debris accumulations investigated in the laboratory were informed by the testing ranges of numerous researchers. The debris configurations employed in the investigation are illustrated in Figure 2. The dimensions for all debris shapes were standardized: W (width of the debris) = 0.12 m, L_D (length of the debris) = 0.06 m, and T (thickness) ranged from 0.03 to 0.12 m. These shapes included a rectangle (R), a high wedge (HW) characterized by an upstream face with a prominent bulk directing flow downward, a triangle bow (TB) resembling a ship’s upstream-pointing bow, a low wedge (LW) featuring a substantial bulk at the low end with an upward-facing downstream face, a half cylinder (HC) showcasing a rounded upstream side, and a triangle yield sign (TY) boasting a flat upstream face broad at the top and tapering to an apex at the bottom. These configurations aimed to mirror debris geometries encountered in real-world situations involving woody debris accumulation.

Key geometric forms such as the optimal inverted half circle, high and low wedges, and triangle bow were found to be integral to debris development according to [34]. The debris form employed by [18,19] aligned with the characteristics of the half-cylinder shape. The percentage of blockage (%A), attributable to the obstructed frontal area of the debris, was calculated by dividing it by the unobstructed flow area, yielding values summarized in the final column of

Table 1. Testing matrix illustrating sequential variations in pier and debris geometry, symbol representation, relative thickness, and blockage ratio for experimental analysis.

Pier Shape	Debris Shape	Symbol	T/Y	A%
Cylindrical (C)	Rectangular (R)	C-R-1	0.25	7.9
		C-R-2	0.5	15.8
		C-R-3	1	31.6
	Triangle bow (TB)	C-TB-1	0.25	7.9
		C-TB-2	0.5	15.8
		C-TB-3	1	31.6
	High wedge (HW)	C-HW-1	0.25	7.9
		C-HW-2	0.5	15.8
		C-HW-3	1	31.6
	Low wedge (LW)	C-LW-1	0.25	7.9
		C-LW-2	0.5	15.8
		C-LW-3	1	31.6
	Half cylinder (HC)	C-HC-1	0.25	7.9
		C-HC-2	0.5	15.8
		C-HC-3	1	31.6
	Triangle yield sign (TY)	C-TY-1	0.25	3.1
		C-TY-2	0.5	6.2
		C-TY-3	1	12.5

, encapsulating the various pier/debris combinations. For instance, in the case of a rectangular shape in the flow direction, %A = (W − D) T/BY. Similarly, for the triangular yield sign (TY), %A = (W − D)0.5T/BY.

Other pier shape cases symbols were the same as the cylindrical pier but with different first letters, e.g., S for the square pier, RE for the rectangular pier, OB for the oblong pier, OG for the ogival pier, and LE for the lenticular pier shape.

2.4. Flow Intensity

The selection of the flow rate was meticulously undertaken to maintain a standardized bed shear stress beneath the critical threshold necessary for the commencement of sediment transport across a flat sediment bed. Specifically, the chosen flow rate was established at 0.13 l/s, as precisely measured by the flume rate meter. Concurrently, the normal flow velocity quantified as 0.29 m/s, was determined through meticulous measurements facilitated by the velocity meter and mountings system. The regulation of the flow of water depth was achieved through the meticulous control of a downstream tilting gate positioned within the flume, meticulously set to uphold a consistent level of 0.12 m throughout all experimental trials (as documented in

Table 2. Comprehensive overview of experimental flow data parameters and details.

Q (m3/s)	Y (m)	V (m/s)	Vc (m/s)	V/Vc (m/s)	Re	Fr	u* (m/s)	Re*
0.012	0.12	0.29	0.32	0.91	626400	0.27	0.022	87

).

A crucial aspect of maintaining a clear water flow was ensuring that the flow intensity remained below unity. To this end, the critical flow velocity, which signified the point of initiation for the movement of sand particles, was accurately gauged through the use of the shield diagram and supplemented by the findings derived from equations [35,36,37]. Furthermore, the fluid flow was distinctly characterized as being both rough and turbulent, a characterization substantiated by the grain Reynolds number (Re*) exceeding 70. This parameter was computed employing the formula Re* = ρuD₅₀/µ, thus affirming the presence of turbulent flow development within the stream. It is pertinent to note that the calculation of this value via the shield diagram, exceeding the threshold of 70, reinforces the establishment of a turbulent flow regime within the stream.

Additionally, the study considered a range of pier Reynolds numbers (Re_p), calculated as Re_p = (UD)/ν, where ν represents the kinematic viscosity of the water. The Reynolds number was 7250 for a pier diameter (D) of 2.5 cm. In this context, it was reasonable to assert that the influence of viscosity on the flow can be discounted. This assumption is supported by the fact that the condition proposed by [38], Rep = (UDp)/ν ≥ 7000, has been met, signifying that viscous effects are minimal in the given range of Reynolds numbers. Last, the characterization of the flow as subcritical is underscored by the Froude number (Fr) calculated as $V/\sqrt{gY}$, consistently revealing values below 1 [6].

2.5. Experimental Procedure and Methodology

The experimental process was meticulously orchestrated to ensure accurate and comprehensive insights into the dynamics of scour depth, influenced by diverse pier shapes and debris geometries. The methodological steps encompassed both preparatory measurements and the core experimental phase.

Preparatory Phase: Flow Conditions and Velocity Measurements: The initial emphasis was on establishing optimal flow conditions within the designated study section. This prerequisite was achieved through instantaneous vertical velocity measurements utilizing a velocity meter. Mounting probes were strategically positioned at three distinct horizontal locations along the flume’s centerline, extending into the flat sand bed. This preliminary step, undertaken before the main testing, was crucial to accurately capture the dynamic flow patterns that would subsequently influence scour development. For each velocity profile, the data were collected vertically, originating approximately 1 cm above the sand layer and extending to the maximum corresponding level. This comprehensive data collection encompassed a range of water depths and flow rates, thereby facilitating a thorough understanding of the velocity distribution. The decision to maintain stationary conditions over several minutes, as recommended by [39], ensured the acquisition of robust and representative velocity data.

Experimental Phase: Pier Placement and Flow Configuration: The subsequent experimental phase involved the vertical placement of the pier on the flume base, positioned at a strategic distance of 8.5 m downstream from the upstream intake. This configuration was meticulously chosen to ensure that the flowing water within the flume had attained full development, effectively eliminating any potential tailwater effects that could distort the study’s accuracy. The gradual introduction of water into the flume was orchestrated with precision to avoid any disturbance to the underlying sand material. Conspicuously, a fine screen situated 4 m from the pier’s center near the tailgate was strategically employed to prevent the entry of sediment particles into the pump. While the flow rate was sufficient to induce minor sand movement near the pier, it fell short of generating the requisite bed shear stress threshold for initiating sand transport. This approach aimed to maintain controlled conditions while exploring scour dynamics.

Hydraulic Factors and Parameters: The experimental configuration was guided by critical hydraulic factors. The flume width-to-the-pier width ratio (B/D) was set at 12, ensuring minimal bed degradation within the constricted cross sections and effectively indicating the absence of contracting scour, in alignment with [40]. The stream water level was consistently maintained at 12 cm, yielding a shallowness factor (Y/D) of 4.8. Additionally, the sand coarseness, characterized by D/D50 = 27, was a parameter under consideration, influenced by [9,28]. Markedly, the impact of the sand coarseness on the pier form and alignment variables was negated through meticulous control of specific variable pairs, ensuring uniform sand coarseness for each set of experiments.

Experimental Duration and Scour Depth Analysis: Each scour experiment extended for approximately 360 min, during which the balanced scour depth time factor ($K_t$) was computed using the equations proposed by [41]. This equation, involving $K_t$and $t_e$ as defined in Equations (1) and (2), provided insights into the evolution of scour depth over time. Around 92% of the equilibrium scour depth was reached within the experimental time frame, substantiating the experimental conditions’ suitability.

$$K_t=EXP\left\{-0.03{\left|\frac{V}{Vc}\ln \left(\frac{t}{te}\right)\right|}^{1.6}\right\}$$

(1)

$$t_e=30.89\frac{D}{V}\left(\frac{V}{Vc}-0.4\right){\left(\frac{Y}{D}\right)}^{0.25}$$

(2)

The primary focus was to evaluate the interplay of diverse debris geometries and sizes on the scour depth around distinct pier shapes. Ensuring appropriate yet comparable durations for each test allowed the accumulation of robust scour depth data. After the 6 h mark, minimal particle rolling motions were noted, with negligible influence on scour depth alterations. The decision to conduct the experiments for a duration of 6 h was judiciously made, accounting for expansion rates below three percent of the pier width per hour [42]. Typically, the initial 10% of scour events witnessed a significant portion of the scour depth (50–80%) [41]. Furthermore, certain tests were extended for an additional 20 h period without debris, serving as a comparative reference. The semi-equilibrium conditions observed at the 6 h mark were deemed appropriate for the experimental trials, effectively capturing at least 90% of the maximum scour depth. This approach optimally expanded the scope of experimentation, leveraging the available laboratory resources.

Data Collection and Symmetry Verification: Ultimate scour patterns were meticulously recorded after each scour test, following the cessation of the pump, water drainage, and debris removal. The measurements of the scour hole dimension (X, Y, Z) were facilitated by a grid surrounding the piers, effectively illustrating scour contours. To enhance the accuracy and verification within the scour-affected area, transparent scales affixed to the piers, in conjunction with a laser device, were employed.

The symmetrical nature of the scour topography was visually confirmed for every experiment. Additionally, the study’s consistency was verified by assessing the deepest scour level and the general scour trend recorded at the 4 h mark for specific cases. This approach was bolstered by the application of the same scale to measure depths along the opposing side. The scale readings were taken at intervals, capturing variations at distinct time intervals spanning 1, 10, 20, 40, 80, 160, 240, and 360 min.

3. Results and Discussion

3.1. Scour Depth Correlation with Pier and Debris Geometries

Undoubtedly, the geometrical configurations of the bridge piers yielded a conspicuous influence on the localized scour patterns encompassing the pier structures. To illuminate this intricate phenomenon, Figure 4 meticulously illustrates the longitudinal and transverse profiles of the eroded areas encircling the bridge piers. This exposition specifically concentrated on scenarios where the perturbations due to debris were absent. The featured five distinct geometries—cylindrical, square, rectangular, oblong, and lenticular (streamlined)—each upheld uniform proportions and underwent identical flow rates and prevalent hydraulic conditions. These graphical depictions stand as tangible reflections of the magnitudes of the resultant scour depressions evident in each case.

The disposition of the principal scour regions downstream and the overarching morphology of the impacted zone distinctly succumbed to the variegated geometric constructs of the bridge piers. This observed insight harmoniously aligned with antecedent investigations, as documented in references [43,44,45]. These findings were particularly germane to cases wherein the piers remained impervious to the presence of accumulated debris.

Within the spectrum of distinctive pier geometries, the square pier, measuring 6.0 cm, exhibited the most profound scour depth (Zs), which represented the scour depth in the context of conditions without debris effects, in contrast to the lenticular shape, which spanned 2.6 cm and manifested the shallowest scour depth. The remaining geometries—rectangular, cylindrical, oblong, and ogival—elicited scour depths of 5.8 cm, 4.6 cm, 3.8 cm, and 3.2 cm, respectively. Importantly, the peak scour depths for the cylindrical, square, rectangular, and oblong piers were strategically positioned upstream at an angular orientation approximating 45°. By contrast, the ogival configuration experienced its utmost scour at the intersection of its cutwater, whereas the pinnacle of scour for the lenticular shape was symmetrically centered between its opposing sides.

This variability in the localization of maximal scour was discernibly attributed to the presence of three distinct types of vortices, most markedly the horseshoe vortices that incited upstream and lateral distortions, alongside wake vortices that induced downstream deformations [43]. It merits noting that the horseshoe vortices were discernibly less conspicuous in proximity to the leading edge of ogival and lenticular geometries. Additionally, the partitioning of flow and the bifurcation of shear layers at the lateral corners conspired to heighten the bed shear stresses, possibly contributing to the genesis of maximum scour at these designated locales.

The illustrations within Figure 5a–c provide a comprehensive understanding of the intricate dynamics governing the maximum scour depth (Zsd), representing post-debris effect conditions. These figures expound upon the interplay between six distinct pier shapes and a corresponding set of six debris configurations, sharing uniform dimensions of 12 cm in width and 6 cm in length. The spatial positioning of these configurations concerning the free surface is indicated by the dimensionless parameter (T/Y), taking values of 0.25, 0.5, and 1.0. Each set of vertical bars is associated with a specific pier geometry, while varying shades differentiate the diverse debris types under investigation. Particularly, the high wedge configuration emerged prominently, eliciting the greatest scour depths regardless of the underlying pier characteristics. In contrast, the triangular yield configuration emerged as the least detrimental, yielding scour depths comparable to those observed in debris-free conditions.

The interpretative framework furnished by Figure 5a–c discloses several overarching trends of note:

• Elevated risk scenarios: These critical situations posed heightened risks, often occurring during initial flood stages. Concurrently with the flood’s onset, substantial debris transport took place, preceding significant water inflow into the river channel. This escalated the water depth and intensified the debris dispersion, causing particles to travel farther from the riverbed. Consequently, the most significant debris impact on the scour depth and shape was expected at the flood’s outset.
• Depth–scour relationship: A consistent trend emerged with more pronounced debris-induced scour reduction at shallower immersion depths (T/Y = 0.25) compared with greater immersion (T/Y = 0.5, 1). Swift debris accumulation under the flow surface redirected flow profiles toward the bed, intensifying the downward flow. This fostered sediment mobilization, suspension, and elevation.
• Complete immersion effect: Outstandingly, complete immersion (T/Y = 1) exhibited heightened variability, particularly with the high wedge debris configuration. In contrast, the distinctions between T/Y = 0.5 and T/Y = 1 for different debris shapes remained relatively subtle. The high wedge’s inherent inclination, imparting a downward curvature to flow, led to increased velocities and horseshoe vortices near the sediment bed.
• Triangular yield configuration: Modest scour depths, irrespective of debris morphology, manifested when the triangular yield configuration was variably positioned at submerged depths. This effect arose from the configuration’s limited obstruction area (A%) and distinct shape. These findings aligned with observations in reference [20], highlighting the capacity of the triangular yield debris to obstruct the flow at shallower angles, thereby reducing the water velocities and bed shear stresses compared with debris shapes of similar thickness but distinct geometries.
• Shape factor dynamics: Variability in shape factor disparities among different pier geometries diminished as the debris immersion depth increased from 0.25 to 1. This trend suggests the potential for the shape factor influence to decrease at maximum submersion levels. A comprehensive exploration of this phenomenon is deferred to forthcoming discussions, focusing on an in-depth examination of shape factors.

In conclusion, Figure 5a–c provide a valuable perspective for comprehending the intricate interplay of pier shapes, debris configurations, and submersion depths in their combined influence on scour depths. These figures substantially contribute to scholarly discourse on hydraulic processes in diverse and heterogeneous scenarios.

3.2. Debris Impact on Pier Shape Factor

In light of the diverse array of scour models initially developed to address the characteristics of cylindrical pier structures, a judicious decision was made to adopt the cylindrical pier shape as the foundational archetype. This choice was further motivated by its prevalence in numerous investigative endeavors, as evidenced by prior studies such as reference [43]. Noteworthy attention is warranted to underscore the fact that all nondimensional parameters governing the analytical framework, encompassing sediment roughness (D/D₅₀) and flow depth to pier width (Y/D), remained invariant across the experimental spectrum. The influence engendered by variations in the length-to-width ratio (L/D) of the pier, spanning the range 1.33 < L/D < 4.00, was either negligible or, in some instances, absent. Similarly, the effect stemming from alterations in the angle of attack of the flow, denoted as α = 0°, was rendered irrelevant.

The shape factor, S.F, was an important measure in the study. It measured the ratio of the scour depth generated by non-cylindrical pier forms $Zs\left(particular shape\right)$ to the scour depth produced by cylindrical pier shapes $Zs\left(cylidrical pier\right)$. This shape factor was critical because it enabled the comparison of the scour depths created by different pier forms relative to the reference cylindrical pier [4]. This comparison revealed how different pier forms affected the scour depth, aiding in the understanding of hydraulic behavior around varied geometries. The research offers a clear and empirical basis for analyzing the influence of the pier form on the scour depth by utilizing Equation (3) to explain this notion quantitatively. It also examines the impact of form systematically, which aids in the creation of actual technical solutions for minimizing scour surrounding bridge piers. This is articulated mathematically as

$$\mathrm{S}.\mathrm{F}=\frac{Zs\left(particular shape\right)}{Zs\left(cylidrical pier\right)}$$

(3)

Figure 6 visually exposes the shape factor values corresponding to individual pier shapes in the absence of debris-induced influences. The experimental findings of the square, rectangular, oblong, and lenticular pier shapes aligned with the extant body of literature, corroborating prior investigations such as references [14,26,44,45,46], although demonstrating minor deviations that could be attributed to nuances in the experimental conditions. These divergences arose from distinctions in variables such as the flow intensity (V/Vc), hydraulic shallowness (Y/D), sediment roughness (D/D₅₀), and aspect ratio (B/Y).

Figure 7 presents a detailed representation of the standard deviation concerning average scour depth at a semi-equilibrium state. This depiction focuses on the impact of alterations in the obstruction area (referred to as A%) while accounting for variations in debris composition and pier geometries. The standard deviation is a crucial statistical measure that gauges the extent of data dispersion around the mean. Markedly, when A% was 0, indicating no debris effects, a substantial standard deviation of 1.02 arose due to pronounced data divergence from the mean scour depth. Conversely, at A% of 32%, signifying notable debris presence, the standard deviation dropped to 0.31, highlighting data clustering around the mean scour value. This decline was closely tied to the block ratio’s progression within the 0% to 31.1% A% range. These findings confirmed that as the block ratio parameter increased, the differences in the scour depth between various pier designs and debris configurations diminished gradually. In other words, the scour depth increased significantly with each increase in A%, regardless of the pier and debris shapes, with only minor distinctions.

Reference [20] supports these experimental deductions, bolstering the credibility of the results. In essence, the statistical tool of standard deviation plays a pivotal role in unraveling complex interactions among variables in scour-depth analysis. This understanding enhances our grasp of hydraulic processes in the context of varying debris compositions and pier geometries.

3.3. Effect of Debris Thickness on the Maximum Scour Depth

The impact of varying debris thickness on the maximum depth of scour was investigated in this study through the utilization of different debris shapes. The relationship between the relative depth of scour (Zsd/Zs) and the relative thickness of debris (T/Y) is graphically depicted in Figure 8 using experimental measurement data. The baseline measurement of the scour depth, in the absence of debris, is represented as Zs. Here T denotes the thickness of debris, measured from the debris bottom to the flow-free surface level. The specific values of T/Y were assigned as follows: T/Y = 0 signifies the absence of debris, T/Y = 0.25 corresponds to debris positioned at the initial quarter of the free surface of the approaching flow, T/Y = 0.5 indicates debris located midway within the approaching flow, and T/Y = 1 represents scenarios where the debris bottom aligns with the sand surface level. Analysis of Figure 8 revealed that the most substantial scouring occurred when debris was located at the bed bottom. Conspicuously, an increase in the debris thickness correlated with a heightened magnitude of pier scouring. Conversely, when the debris was elevated, the scour depth experienced a marked reduction. The relationship between the debris elevation and the scour depth can be clarified as follows: As debris approached the bed, it diverted downward-moving streamlines, resulting in increased contraction forces, elevated vortices, and augmented shearing forces in proximity to the bed. On the other hand, when the debris was positioned just beneath the water surface and at a distance from the bed, it partitioned the flow, causing only obstructed or lower streamlines to be directed downward. This dynamic distribution of flow proportionally depended on the extent of the debris coverage (expressed as the submerged area percentage, A%).

To sum up, the scour depth experienced a relatively smaller percentage increase when the debris was situated at the bottom, compared with the greater percentage increase seen when the debris was moved from a position of (T/Y) 0.25 to 0.5. This discrepancy was due to the substantial coverage of the bed by debris, compelling the flow to navigate around the debris on both sides. Consequently, separating the debris from the flow resulted in a more significant increase in the scour depth, in contrast to the situation where the debris was directly underneath the flow-free surface. These findings aligned with the overarching pattern documented in the study referred to as ref [20].

This investigation addressed the impact of the relative thickness of debris, denoted as T/Y, on the resulting scour depth around circular bridge piers. To achieve this, a comprehensive comparative analysis was executed using empirical data derived from diverse debris shapes, encompassing rectangle (R), triangle bow (TB), high wedge (HW), low wedge (LW), and half cylinder (HC). The objective was to elucidate the relationship between the debris thickness and the scour depth while considering the potential influence of the debris shape. The data obtained from this study were contrasted with the findings reported in [20], establishing a valuable baseline for assessment. Throughout this comprehensive analysis, each dataset was characterized by unique hydraulic parameters, such as the dimensionless sediment size to the pier diameter ratio (D/D50), relative water depth (Y/D), velocity ratio (V/Vc), and flow rate. Furthermore, specific debris attributes, including the width-to-base ratio (W/B), length-to-diameter ratio (L/D), and debris thickness-to-width ratio (T/W), varied across different scenarios. Crucially, parameters T/Y, aspect ratio (A%), and the geometrical attributes of debris shapes were maintained at uniform values throughout the analysis.

Synthesizing the diverse datasets, the combined results are visually represented in Figure 8, revealing a coherent pattern. As debris thickness increased, a corresponding augmentation in scour depth was observed. This underscored the pivotal role of the debris thickness as a determinant factor in the scour process around circular bridge piers. Notably, under shallow conditions (T/Y = 0.25), the disparity in results among different debris shapes was minimal. In this scenario, the effect of the debris shape on the final scour depth was negligible. However, as the debris depth increased (T/Y = 0.5), the influence of the debris shape became more pronounced. This was particularly evident when the debris came into contact with the bed. Under such circumstances, the specific shape of the debris significantly influenced the resulting scour depth.

In conclusion, the presence of debris, especially when situated at the bed, considerably impacted the scour depth around circular bridge piers. While the debris thickness played a primary role in determining the scour depth, the shape of the debris became more influential with increased debris thickness. These findings carried significant implications for designing effective scour mitigation strategies, necessitating the comprehensive consideration of debris attributes and their interactions with hydraulic conditions.

3.4. Effect of Debris Block Ratio (A%) on the Maximum Scour Depth

In pursuit of further interpreting the insights gleaned from this investigative study, a comprehensive analysis was undertaken with specific regard to the influence of the area coverage of debris block, denoted as %A. To this end, Figure 9 was formulated to illustrate the discernible augmentation in the pier scouring depth in direct correspondence with the escalating values of %A, encompassing data obtained from the present study, as well as those previously reported in [19,20]. Particularly, the observed data collected within the parameter range of %A < 15 demonstrated a notable alignment with the outcomes documented in [19,20].

Furthermore, the relative scour depth ratio, (Zsd/Zs)_meas, exhibited a pronounced and broad monotonic increase that was contingent upon the magnitude of %A. It is important to highlight that the investigations conducted by [19] employed circular debris configurations extending downstream from the pier structure, thereby leading to marginally diminished scouring depths in comparison with both the present study and the findings reported in [20]. The minor divergences observed in (Zsd/Zs)_meas for instances of debris accumulation where A% < 15 served to underscore the concept that given a consistent debris length (L_D) and an invariant debris shape, the parameter A% held the potential to serve as a reliable predictor for (Zsd/Zs)_meas, effectively informing the anticipated scour depth.

Building upon the insights garnered from the outcomes of the current investigation, in conjunction with the findings from previous studies [19,20], a pivotal contribution emerged in the form of a novel mathematical formulation. This formulation introduced a fresh approach that comprehensively encompassed the multifaceted influence of the debris characteristics. These encompassed the relative debris thickness (T/Y), the percentage area of debris coverage (%A), the width-to-base ratio (W/B), and the incorporation of the debris spatial arrangement via planimetric positions. Notably, the formulation was uniquely designed to accommodate a spectrum of pier configurations by incorporating a shape factor parameter, thereby accommodating the inherent diversity in pier geometries.

Specifically, the outcomes presented in [19] stemmed from experimental investigations involving cylindrical debris structures extending downstream from the cylindrical piers. The experimental scope spanned a range of %A values below 20, characterized by width-to-base ratios within the bounds of 0.08 < W/B < 0.26, and relative debris thickness ratios confined to the domain of 0.086 < T/Y < 1. The current study, for the most part, aligned closely with the data derived in [19], with deviations primarily observed in scenarios where %A surpassed 20. This discrepancy arose due to the omission of downstream debris extension in the present investigation, which may lead to conservative estimates of maximum scour depth under conditions of heightened %A.

Likewise, the research conducted in [20] contributed significant insights by investigating three distinct debris shapes: rectangular, triangular, and cylindrical configurations. The parameters under scrutiny encompassed %A values below 13, width-to-base ratios spanning the range of 0.13 < W/B < 1, and relative debris thickness ratios ranging from 0.08 < T/Y < 1. Remarkably, a substantial portion of the data generated within this study comfortably fell within the parameter ranges reported in [19,20] specifically residing within the bounds of 0 < %A < 32, 0 < W/B < 0.4, and 0 < T/Y < 1.

Demonstrating analytical rigor, the formulation of the new design equation emerged through the application of multiple linear regression analysis. This equation, an outcome of harmonizing diverse datasets and intricately interwoven parameters, is expressed as follows:

$$\left(\frac{\mathrm{Zsd}}{\mathrm{Zs}}\right)\mathrm{cal}.=1+\mathrm{a}\times {\left(\%\mathrm{A}\right)}^b\times \left(\mathrm{S}.\mathrm{F}\right){ }^c$$

(4)

where a, b, and c are regression coefficients based on the analysis and are equal to a = 0.136, b = 0.715, and c = −1.367.

Figure 10 presents a comprehensive analysis encompassing a comparison between the calculated (Zsd/Zs)cal and the measured (Zsd/Zs)meas values within the selected dataset. Outstandingly, the coefficient of determination (R²) attained a substantial value of 88%, thereby attesting to the robustness of the analytical framework employed. This high R² value signified a commendable degree of congruence between the calculated and observed values, which, in light of the relatively narrow diversity of scenarios under examination, was deemed satisfactory. The experimental nature of the data under scrutiny further lent credence to the acceptability of this level of deviation.

Delving into the intricacies of the findings, the observed data in conjunction with Equation (4) unveiled a distinctive pattern. Specifically, the equation tended to underestimate the magnitude of pier scouring depths in instances characterized by the presence of high wedge debris shapes possessing a rectangular profile. This tendency toward underestimation, notably prevalent when large debris blocks (A% > 14) were involved, aligned with the dominance of the (Zsd/Zs)meas values. Conversely, in cases associated with triangular yield (TY) configurations, the equation tended to overestimate the scouring depths, attributed to the smaller debris block (A% = 6.2) located on the sand bed. This positioning obviated the formation of a substantial horseshoe vortex, a key agent in the scour hole development process.

Turning attention to the overarching trends established in the outcomes of [19], it becomes evident that an overestimation bias was prevalent, with the (Zsd/Zs)cal values exerting dominance. This was primarily ascribed to the study’s focus on half-circle debris structures with downstream extensions, a design choice that consequently led to reduced final scour depths, particularly when %A exceeded 14. Concluding this comprehensive assessment, it is noteworthy to highlight a salient observation within the dataset presented by [20]. Specifically, a subset of the data points extracted from [20] manifested a notable divergence, venturing beyond the confines of the designated −15% range. It is pertinent to emphasize that this specific departure from the established range pertained exclusively to instances wherein the percentage area of debris coverage A% exceeded 12. This finding adds an interesting dimension to the overall analysis, suggesting a distinctive behavior and potentially distinctive influences under circumstances characterized by higher A% values.

In summary, this research yields a robust mathematical equation that captures the intricate interplay between debris attributes and their impacts on the scour depth around circular bridge piers. The equation, grounded in rigorous experimentation and statistical analysis, paves the way for improved design strategies that consider the nuanced effects of debris configurations on pier stability in scour-prone environments. Moreover, applying this equation offers the potential to enhance the resilience and longevity of critical infrastructure exposed to debris-laden flow conditions. The equation’s predictive capacity outperforms traditional methods, producing more accurate forecasts. Despite limited test samples and the experimental nature of the data, the observed variances are reasonable. In conclusion, this research introduces a practical tool to address debris-induced scour, facilitating safer and more effective engineering practices and opening avenues for further refinements and real-world applications.

Amidst the intricate nuances of our investigation into the prediction of pier scour in the presence of debris, Figure 11 takes on the role of unveiling underlying distributional characteristics. The unmistakable asymmetry in the bell curve’s profile provides an insightful depiction of statistical dynamics. This representation distinctly highlights a clustering of instances around the mean, punctuated by occasional deviations from this central tendency. Remarkably, when scrutinizing the residual values, a conspicuous avoidance of zero emerges, despite their clustering in proximity to the x-axis. This phenomenon reinforces the persistent influence of stochastic elements, underscoring the ongoing role of chance-driven variations in governing scour outcomes. This observation remains salient as the residuals extend toward a maximum of 0.6 on the right side of the distribution, offering further evidence of instances where overestimation of the (Zsd/Zs) ratio prevails—a manifestation that resonates with the earlier identified trend of mild overestimation. In a supplementary analysis, a set of illustrative statistical values is aimed at rigorously assessing the effectiveness of Equation (4). These metrics include the Nash Sutcliffe efficiency (NSE) of 0.884, index of agreement (d) of 0.968, R-squared correlation (R2) of 0.884, and the normalized root mean square error (NRMSE) of 0.127. Collectively, these indices serve as barometers of the equation’s prowess in faithfully reproducing real-world scour behavior.

In light of these findings, the examination of distributional traits enriches our comprehension of the intricate interplay between deterministic predictions and stochastic fluctuations inherent in modeling pier scour. The empirical validation underscored by statistical indices further solidifies the equation’s capacity to reliably project scour depths within intricate debris-laden scenarios. This nuanced comprehension propels the discourse toward more astute engineering practices, positioning the proposed equation as a potent tool for enhancing the durability and stability of crucial infrastructure within scour-prone settings.

3.5. Validation of the New Equation

Figure 12 provides a comparative assessment of the relative scour depths determined through two distinct methodologies: the established equations from [47] and the novel equation formulated in this study as Equation (4). The equations from [47] are articulated as follows:

$$Zsd=2.4\ast De when\frac{Y}{De}>2.6$$

(5)

$$Zsd=1.872\ast {\left(\frac{Y}{De}\right)}^{0.255}\ast De when\frac{Y}{De}<2.6$$

(6)

This comparative analysis was motivated by the necessity to formulate a local scour calculation formula that employed both the effective pier width (De) and the equivalent pier width (a*), as opposed to using actual dimensions. The primary purpose was to comprehensively assess the performance of Equations (5) and (6) in predicting the scour depth while accounting for the nuanced dimensions introduced by debris interactions. The effective pier width (De), which accounts for the presence of debris, was computed using two distinct methods: the approach proposed by [19], denoted as the “effectiveness pier width method”, and the technique put forth by [22], characterized as the “equivalent pier width methodology”. Specifically, the [19] method for determining De is expressed as

$$\mathrm{De}=\frac{0.52TWD+\left(Y-0.52T\right)D}{Y}$$

(7)

The [22] equivalent pier width (a*) methodology is encapsulated as

$${\mathrm{a}}^{*}=\frac{K_{d1}WT {\left(\frac{L}{Y}\right)}^{Kd2}+\left(Y-K_{d1}T\right)D}{Y}$$

(8)

The formulation of the proposed methodology incorporates several key coefficients to establish its predictive capacity. Among these, Kd1 stands as the shape factor, exhibiting distinct values of 0.39 for debris configurations characterized by a rectangular geometry and 0.14 for those presenting a triangular profile. Subsequently, Kd2, known as the flow intensity factor, assumes significance, particularly in cases where (L_D/Y) surpasses unity, attaining values of −0.79 for rectangular configurations and −0.17 for triangular counterparts.

The methodology employed in this study utilized the collected data in conjunction with data from prior investigations [19,20] to serve as the input for Equation (5) or (6). These equations facilitated the computation of two crucial parameters: the maximum scour depth (Zsd) and the equilibrium scour depth (Zs). A cornerstone of this methodology involved the utilization of an equivalent and effective pier width (De and a*, respectively), as determined by Equations (7) and (8). These values were instrumental in the calculation of Zsd, whereas D served as the pivotal variable for the Zs determinations. The foundations of this method were firmly anchored in datasets primarily rooted in the examination of rectangular and half-circle debris configurations, with limited inclusion of triangular debris shapes. The outcomes of these calculations are thoughtfully illustrated in Figure 12, offering a visual representation of the notable alignment between the projected estimates and the actual results arising from the application of this methodology.

The results obtained from this new methodology exhibited a high degree of alignment with the proposed equation (Equation (4)), evidenced by a coefficient of determination (R²) reaching 0.89, coupled with a mean absolute percentage (MAP) of 9.8%. Conversely, Equation (7) displays a tendency to underestimate the data of [20], reflected in an R² value of 0.31 and a MAP of 0.25. Conversely, overestimation characterized the outcomes for other datasets, with R² values of 0.48 and a MAP of 38% for [19] and an R² of 0.46 accompanied by a MAP of 55% for the experimental data. This disparity aligned with the prior observation that half-circle debris configurations resulted in comparatively reduced scour when compared with downstream-located rectangular debris. Remarkably, Equation (7) fails to account for the frontal debris shape and its longitudinal extension upstream of the piers.

In comparison with Equation (4), the performance of Equation (8) unveils a wider range of variance, with R² values of 0.26, 0.49, and 0.72, accompanied by MAP values of 32%, 25%, and 23% for [19,20] and the experimental dataset, respectively. While showing a tendency toward underprediction for [20] and overprediction for [19] and the experimental data, Equation (8) manages to enhance the results, albeit to a lesser extent. Significantly, Equation (4) achieves an additional 13.2% reduction in MAP, a notable improvement attributed to its incorporation of the shape factor that accounts for diverse pier geometries. Crucially, the predictive consistency demonstrated by Equation (4) extends over a broader spectrum when contrasted with the outcomes of Equation (7) and, to some extent, Equation (8), especially in scenarios characterized by elevated values of %A. The variance in data distribution, confined within a range of ±15% when computed using Equations (7) and (8), is notably constrained within this range when employing Equation (4).

It is prudent to acknowledge that the efficacy of the new equation was inherently influenced by the specific parameters under which it was developed, encompassing particular pier forms, flume dimensions, and debris morphologies. It is conceivable that in the realm of real-world situations characterized by substantial solid debris, the magnitude of scouring enhancement may exhibit disparities warranting further investigation. The emphasis on characterizing the morphology of potential debris accumulations assumes a pivotal role, with a recommendation for comprehensive evaluation encompassing diverse case study configurations to bolster the applicability and generalizability of the proposed new equation.

4. Conclusions

Impact of Pier Geometries on Scour Depth:

• The study explored various pier shapes’ influence on scour depth.
• The square pier exhibited the deepest scour depth, which was approximately 66.7% higher than that of the lenticular pier.

Interaction of Debris and Submersion Depths:

• Debris positioned closer to the bed bottom induced greater scour depths with an average 32.5% increase in scour depth.
• The block ratio parameter increased, and the differences in scour depth between various pier designs and debris configurations diminished gradually regardless of the pier and debris shapes, with only minor distinctions.

New Prediction Equation for Scour Depth:

• A novel mathematical formulation was introduced for predicting scour depth.
• The equation considered shape factor parameters for diverse pier geometries and debris shapes with superior accuracy and predictive consistency (R² = 88%).

Distributional Analysis and Stochastic Nature:

• Asymmetrical bell curve clustering around the mean highlighted the role of stochastic elements in scour outcomes. The residual values ranged from 0 to 0.6, indicating mild overestimation. Statistical metrics confirmed Equation (4) accurately reproduced real-world scour behavior.

5. Limitation and Further Research

This study presents several limitations that warrant acknowledgment when interpreting the results and considering their application in real-world contexts.

• Limited experimental data: The study’s findings were based on a relatively restricted dataset of pier shapes and debris types. While this dataset was utilized for evaluating the new equation, it is important to recognize that there may be additional variables and scenarios not accounted for in this study. Further research should consider a broader spectrum of data to enhance the comprehensiveness of predictive models.
• Inherent field study limitations: Despite employing sophisticated equipment and robust data-gathering techniques, it is essential to acknowledge that all field studies inherently possess certain limitations. Variability in environmental conditions, site-specific circumstances, and the inherent accuracy of measurement equipment can introduce uncertainties into the study. These uncertainties should be carefully considered when applying the research findings to practical scenarios.
• Measurement uncertainty: The measurements obtained during the experimental phase represent a significant source of uncertainty. Despite meticulous efforts to ensure precision, field instruments may inherently possess limitations in sensitivity and accuracy. These limitations may result in variations in the collected data, contributing to uncertainty in the analysis. It is crucial to recognize that measurements of the scour depth, flow velocity, sediment properties, and debris accumulation are all subject to potential errors.
• Stationary debris focus: The study primarily focused on stationary debris and did not account for dynamic debris responses to flow, which may limit the generalizability of the findings to real-world situations. However, it is important to emphasize that the research deliberately concentrated on worst-case scenarios, specifically scenarios involving pre-existing upstream debris. The study aimed to investigate the severe consequences of incrementally increasing debris submergence from 3 cm to 6 cm to 12 cm, with the intention of exploring extreme conditions rather than encompassing the full spectrum of dynamic debris scenarios.