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Article

Pressure-Flow Scour under a Bridge Deck in Clear Water Conditions

1
Civil Engineering Department, Benha Faculty of Engineering, Benha University, Benha 13512, Egypt
2
Civil Engineering Department, Shoubra Faculty of Engineering, Benha University, Shoubra 11629, Egypt
3
Civil Engineering Department, School of Computing and Engineering, University of West London, London W5 5RF, UK
*
Author to whom correspondence should be addressed.
Water 2023, 15(3), 404; https://doi.org/10.3390/w15030404
Submission received: 22 December 2022 / Revised: 9 January 2023 / Accepted: 13 January 2023 / Published: 18 January 2023
(This article belongs to the Section Water Erosion and Sediment Transport)

Abstract

:
The issues of scouring around a bridge have become prominent in recent research mainly due to recurrent extreme weather events. Thus, designing a bridge with the appropriate protection measures is essential to safeguard it against failure, which may take place due to scouring from high flows resulting from extreme weather events. Bridges may become partially or entirely submerged during extreme weather events such as large floods and are subject to pressure-flow scour, a condition where the flow is directed downward and under the bridge deck, creating an increase in flow velocity and a corresponding increase in bed scour. This study aims to explore the pressure-flow scour depth under a bridge deck without piers in the presence of two vertical wall abutments under clear water experiments. Sixty-six tests were conducted involving the approach flow depth, bed material size, contraction length, contraction width, and bridge opening for both pressure and free surface flow conditions. An empirical equation was deduced to determine the maximum scour depth, which could be applied as a preliminary design for bridges under pressure-flow conditions. The experimental data were used to determine the performance of the earlier models of pressure-flow scour. The results revealed that for pressure-flow conditions, the maximum scour depth increased by a factor between 2.15 and 9.81 times the maximum scour depth under free surface flow conditions. With same flow depth, when the relative bridge length was increased from 5 to 7.5 and 7.5 to 10, the maximum scour depth decreased by up to about 7.4% and 2.3%, respectively. When the relative bridge width was decreased from 5.5 to 5.2 and 5.2 to 4.4, the maximum scour depth increased by up to about 45.6% and 81.2%, respectively.

1. Introduction

Local scours at bridges are among the most common reasons for bridge failure [1]. The waterway at a bridge site may contract horizontally because of the presence of bridge abutments or bridge piers and/or vertically when a bridge deck gets submerged. Clear water scouring due to flow area contraction at the bridge site occurs because the corresponding flow velocity at the contraction increases as the cross-section area decreases. This increase in flow velocity produces additional bed shear stress, resulting in the transportation of bed materials out of the area of contraction until the maximum scour occurs when the flow velocity is equal to the critical velocity, or the bed shear stress is equal to the critical shear stress in the contraction area [2]. Pressure-flow (vertical contraction) scour occurs during floods or when a bridge deck is not high enough such that when the water surface exceeds the lower elevation of the superstructure elements of the bridge, the bridge deck and/or girder becomes a barrier to the flow. A bridge deck is considered partially submerged (pressure flow) when the flowing water reaches the lowest element of the bridge. As the water level increases, the pressurized flow under the bridge increases as the degree of submergence increases. However, when the flowing water passes over the bridge, the flow is considered fully submerged (combined weir overflow) [3,4].
Most published studies have paid attention to free surface flow scours [5,6]. Meanwhile, pressure-flow scours have received less attention [7]. Abed [8] introduced the first study on pressure-flow pier scour at a bridge with piers. She deduced experimental equations for predicting the pressure-flow scour depth at a bridge pier based on 25 experiments governed by clear water scour conditions. In such studies, differentiating between the pressure-flow scour and the pier scour was difficult as all experiments comprised both a bridge deck and a pier model. Arneson and Abt [9] developed a laboratory formula to estimate the scour hole due to the pressure flow beneath a bridge deck. This equation is also employed in HEC-18 to determine the scour depth under pressure flow conditions (see Table 1). Umbrell et al. [3] experimentally studied a scour under a bridge deck without abutments and piers under pressure flow conditions. They analyzed their experimental results by applying the continuity equation of flow passing under and over the bridge’s deck. Lyn [4] evaluated the HEC-18 equation for estimating pressure-flow scour. He reported that the experimental time to attain the equilibrium scour depth of the data sets of Arneson [10] and Umbrell et al. [3] might be insufficient. He also reanalyzed those data and developed a design equation (see Table 1). Guo et al. [11] deduced an analytical solution based on the energy and mass conservation laws for pressure-flow scour. Their theoretical solution can predict the maximum pressure-flow scour and a corresponding scour profile. Lin et al. [12] employed particle image velocimetry and flow visualization techniques to explore the flow structure under a partially submerged deck. According to the Froude number and submergence ratios, they defined four types of flow structures beneath the deck. Shan et al. [13] performed an analytical and experimental study to propose an equation for the maximum pressure-flow scour depth. They combined their experimental data with the data sets of Arneson and Abt [9] and Umbrell et al. [3] to deduce a design model. They also agreed with Lyn [4] that the pressure-flow scour at highway bridges required an improved model. Dankoo et al. [14] experimentally developed a formula to estimate the amount of scour at bridge piers in compound channels with vegetation under pressurized flow conditions. Kocyigit and Karakurt [15] experimentally investigated the pressure flow and combined weir overflow at bridge decks under clear water conditions. They considered the approach flow depths, girder depths, bed materials, and degrees of submergences and developed two equations for pressure flow and combined weir overflow to predict the maximum depth of scour (see Table 1). They reported that new experiments are required to fill a gap in the available literature.
A review of the available literature on pressure-flow scour at bridges revealed that experiments on a wider range of parameters are needed [15]. Most recent studies have not explored the direct effect of the median particle diameter of bed materials [7]. Likewise, the effects of bridge length (length of contraction) and the aspect ratio of the channel (bridge width to channel width) on the scour depth have not been reported in any study. This study aims to experimentally explore the maximum scour depth at a bridge deck with two girders and without piers under pressure flow governed by clear water conditions in the presence of two vertical wall abutments. The effects of bridge length and the aspect ratio of the channel on the pressure-flow scour were tested. A total of 66 laboratory tests were performed: 55 runs for pressure-flow conditions (partially submerged) and 11 runs for the free surface flow conditions (atmospheric flow). The approach flow depth, bed material size, bridge length, bridge width, and submergence ratios (relative bridge openings) were studied for both flow conditions.

2. Materials and Methods

2.1. Experimental Setup

All experiments were accomplished in a 17.6 m long, 0.60 m wide, and 0.60 m deep recirculating flume with a steel horizontal bed and glass walls in the hydraulic laboratory of the Faculty of Engineering, Menoufia University, Egypt. A tailgate was located at the downstream end of the flume to adjust the selected approach flow depth. An ultrasonic flowmeter installed on the feeding pipe with a reading accuracy of ±1.0% was used to measure the flow rate. Point gauges with an accuracy of ±0.1 mm, were employed to determine the flow depth and bed levels. Moreover, the flow velocity was measured using a SonTek acoustic Doppler velocimeter (San Diego) with a side-looking 3D probe. A sampling rate of 200 Hz was used to measure the water velocity profiles. The model of the bridge deck was adjustable in the experiments to be perpendicular to the direction of flow and measured 0.50 m long, 0.60 m wide, and 0.25 m deep based on a two-lane bridge scaled at 1 to 100. Two girders beneath the bridge deck, 1.50 cm high and 0.80 cm wide, were used and kept constant in all tests. Two wood pieces, 2.5 cm wide and 0.50 m long, were glued to the flume as vertical wall abutments from both sides. Three piezometric tubes were installed on the bridge deck to observe the pressure head along the bridge deck (Figure 1). The chosen coordinate system had an origin at the surface of the bed level on the centerline of the working section where the bridge deck begins. The horizontal coordinates, x and y, are nondimensionalized by the length of the bridge in the longitudinal direction.
A working section 8.0 m long, 0.60 m wide, and 0.30 m deep was placed 6.0 m downstream of the flume inlet and filled with a 0.30 m-thick layer of sand as bed material. Three different sizes of sand with median diameters of d50 = 1.093, 1.469, and 2.575 mm were employed to examine the bed material size effect on the scour depth under pressure flow. The geometric standard deviations (σg) for the three bed material samples were 1.302, 1.198, and 1.274, the uniformity coefficients (Cu) were 1.611, 1.292, and 1.612, and the curvature coefficients (Cc) were 0.937, 0.951, and 0.983, respectively. The dry bed materials’ angle of repose was about 31°, and the specific gravity of the bed materials was Sg = 2.65. The tested bed materials are uniform because σg < 1.4, Cu < 3.0, and Cc < 1.5 [15,18]. The armoring effect would not occur in this study as σg < 1.3 [19,20].
A flume discharge of 18 l/s with five approach flow depths of ya = 8, 9, 10, 12, and 15 cm was tested. The tests were conducted in a semi-uniform flow. A clear water condition was attained in all experiments as the approach velocity (Va) to the computed critical velocity (Vc) obtained using Neill’s [21] equation, V c = 1.52 g S g 1 d 50 y a / d 50 1 / 6 , is less than one (Va/Vc < 1) [15,22]. In this study, the computed Va/Vc varied between 0.436 and 0.907. The experiments were executed at five different degrees of submergence for pressure-flow conditions and one case for the free surface flow condition. Three bridge lengths were considered (L = 50, 75, and 100 cm), and three bridge widths were tested (bbr = 55, 52, and 44 cm). Table 2 represents the values of the tested parameters in these experiments, The model of the bridge deck was installed at the middle of the working section, at 10.0 m downstream of the flume inlet where the boundary layer is fully developed (Figure 2). Guo et al. [23] and Shan et al. [13] defined the equilibrium time for the scour depth for three continuous hours, and the changes in scour at a reference point were less than 1 mm. In this study, preliminary runs showed that 10 h of test duration was adequate to attain the equilibrium condition.
For each run, bed materials in the working section were leveled. Water was released to the flume gradually from the downstream end until the flow depth was greater than the tested flow depth. Then, the pump was switched on such that the bed materials would not be disturbed or transported. Thereafter, the discharge was measured. When the flow discharge was adjusted, the considered water depth was adapted employing the tailgate. Then, the deck was slowly slid to the model. The flow discharge and depth were checked again, and the test began. For each run, the piezometric tube was used to observe the pressure head along the bridge deck. Flow depths were determined using the point gauges. Five vertical velocity profiles were measured along the centerline of the test section. At the end of the test, the pump was switched off and the water was drained very slowly; then, the deck was removed from the flume. The bed elevations were surveyed using the point gauges to define the scour profiles and determine the maximum scour depth. The experimental data for the pressure and atmospheric flows are summarized in Table 3.

2.2. Dimensional Analysis

Many parameters affected the pressure-flow scouring process. A functional relationship between the maximum scour depth and independent variables associated with pressure-flow scour according to [15,19] can be expressed as
y s = f y a , y b , L , B , b , b br , h s ,   h g ,   V a , V b , u * , d 50 , σ g , S g , ρ , ν , g
where ys is the maximum scour depth, f is the functional symbol, ya is the approach flow depth, yb is the flow depth under the bridge deck, L is the bridge length (contraction length = abutment length), B is the flume width, b is the width of the wall abutment, bbr = (B − 2b) is the bridge width (contraction width), hg is the girder depth, hs is the submerged height of the deck, Va is the approach flow velocity, Vb is the flow velocity underneath the deck, u* is the shear velocity, d50 is the median bed material size, σg is the geometric standard deviation, Sg = (ρs/ρ) is the specific gravity of the bed materials, ρs is the density of the bed materials, ρ is the density of water, ν is the kinematic viscosity of water, and g is the gravitational acceleration (Figure 2). Using the Buckingham Pi theorem, the following dimensionless relationships are expressed as follows:
y s y a = f ( y b y a , L y a , b br B , h g y a , h s y a , V a V b , u * V b , R , S g , σ g , F a , d 50 y b )
where Fa is the approach Froude number ( F a = V a / g y a ) and R is the approach Reynolds number (R = Vaya). The effect of the Reynolds number can be neglected, when the flow is fully turbulence (R > 10,000), [24]. The flume width and the girder depth were kept constant in all experiments (B = 60 cm and hg = 1.5 cm, respectively). The terms hs/ya and Va/Vb were dependent on the approach flow depth and the depth under the bridge deck (hs = ya − yb − hg). The term u*/Vb is only dependent on the height under the bridge deck (bridge opening), and the parameter yb/ya included the same effect as that of u*/Vb [15]. The densimetric Froude number ( F a * = V a / g S g 1 y a ) was used as three different bed materials were involved in this study, and the effect of the flow intensity Va/Vc was included in F a * according to Carnacina et al. [25]. The relationship in Equation (2) can thus be simplified and arranged as follows:
y s y a = f ( y b y a , L y a , b br B , σ g , d 50 y b , F a * )

3. Results and Discussion

A nonlinear regression analysis was applied to fit the independent variables given on the right-hand side of Equation (3) to the laboratory data using IBM SPSS (Statistical Package for Social Sciences) advanced statistics software for defining the constant coefficients that give the best fitting. The analysis produced an empirical equation for the combined vertical contraction due to the deck of the bridge and horizontal contraction due to the vertical wall abutments governed by clear water scour conditions as follows:
y s y a = 1.55 y b y a 0.87 L y a 0.19 b b r B 4.24 d 50 y b 0.21 σ g 1.13     F a * 1.62
This relationship (Equation (4)) is valid for the considered range of parameters in this research (0.313 ≤ yb/ya ≤ 0.85, 3.33 ≤ L/ya ≤ 10, 0.733 ≤ bbr/B ≤ 0.917, 0.009 ≤ d50/yb ≤ 0.074, 1.198 ≤ σg ≤ 1.302, 0.128 ≤ F a * ≤ 0.33). The coefficient of determination (R2) for Equation (4) was 0.92, and the adjusted R2 = 0.912. This denotes that the degree of agreement between the parameters is reasonably good. Bbr/B, F a * , σg, and yb/ya were the most significant variables in Equation (4) as p < 0.001 for these variables. Figure 3 depicts the observed scour depths against the estimated values for the pressure flow. The agreement between the observed and computed values revealed that Equation (4) could estimate the pressure-flow scour depth and could be applied as a preliminary design for bridges under pressure-flow conditions.
The reliability of the present laboratory data was examined using 35 data points to compute the relative maximum scour depth (ys/ya) using the earlier pressure-flow scour equations of Arneson and Abt [9], Umbrell et al. [3], Lyn [4], Guo et al. [11], HEC-18 Equation [2], Shan et al. [13], Melville [16], Kumcu [17], and Kocyigit and Karakurt [15] (Table 1). Moreover, Figure 3 plots the measured ys/ya for pressure-flow conditions and those calculated by the abovementioned models and Equation (4). All the models except that of Kumcu [17] underpredicted the maximum scour depth for pressure-flow conditions, which is undesirable in engineering practice. Notably, the data of Arneson and Abt [9] and Umbrell et al. [3] gave many negative scour values, revealing unrealistic behaviors. The data of Kocyigit and Karakurt [15] also produced some negative values of scour depth. In this regard, Kocyigit and Karakurt [15] developed an empirical equation that involved the independent dimensionless parameter hg/yb. The actual girder depth (hg) or number of girders was not examined in the current study. All negative scour values were eliminated and were not considered in the comparison. The computed negative scour values confirmed the conclusions of Lyn [4], where, as in the experiments of Arneson and Abt [9] and Umbrell et al. [3], an equilibrium scour state was not achieved.
A statistical analysis of the predictive errors for the tested models was performed. The statistical characteristics of the errors, including the average, minimum and maximum errors, variation coefficients of the errors, and root mean square error (RMSE) are listed in Table 4. The RMSE indicated that the worst performance during the testing was obtained from the model of Arneson and Abt [9]. The other prediction models [2,3,4,11,13,15,17] performed well with the current laboratory data. The computed RMSE for the model of HEC-18 Equation [2] as an error indicator was 0.390, the lowest RMSE among those of the tested models. This implies that this model performed better using the tested data set. The equations of Lyn [4], Guo et al. [11], Shan et al. [13], and Kocyigit and Karakurt [15] yielded roughly the same RMSE values.

3.1. Free Surface and Pressure-Flow Scour

To evaluate the effect of pressure-flow against atmospheric flow conditions, the laboratory data were used to present the relative maximum scour depth (ys/ya) at different relative bridge opening (yb/ya) for pressure-flow conditions and for the case of atmospheric flow conditions (yb/ya = 1.0) (Figure 4). The pressure-flow conditions produced a larger scour depth than that of the atmospheric flow conditions, which is consistent with Melville [16]. For the pressure and free surface flow conditions, the maximum scour depth increased when the densimetric Froude number increased. In addition, the relative scour depth increased as the relative bridge opening decreased and as the submergence ratios (hs/ya) increased (hs = yahg − yb). The maximum scour depth increased by up to about 77%, 73%, 69%, 58%, and 46% for range of the relative openings of yb/ya = 0.31~0.40, 0.51~0.60, 0.61~0.70, 0.71~0.80, and 0.76~0.85, respectively, compared with the maximum scour depth under atmospheric flow conditions. Decreasing the bridge openings increased the bed shear stress, which increased the scouring potential of flow. For the pressure-flow conditions, the maximum scour depth was 2.29 to 11.30 times larger than the atmospheric flow scour depending on the densimetric Froude number, the submergence ratios, and bridge openings. Abed [8] believed that the maximum scour depth increased by a factor ranging from 2.3 to 10, whereas Carnacina et al. [25] reported that the maximum scour depth increased by a factor of 2.52 times that under atmospheric flow conditions. It should be noted that these two previous investigations comprised both pressure-flow scour and pier scour. Guo et al. [11] defined the scour number (ys+yb)/(yb+h) where h= (ya-yb) as similarity numbers to describe the bridge pressure-flow scour. The present laboratory data were employed to compute the scour numbers and are listed in Table 3. The computed scour numbers at same inundation Froude number ( F i = V a / g y a y b ), agree well with the analytical solution of Guo et al. [11].

3.2. Water Surface Profile and Velocity Field

In all pressure-flow experiments, the downstream low chord of the bridge deck was found partially submerged. Thus, the bridge under pressure-flow conditions operates as an outlet orifice. The water surface level on the upstream side of the deck is higher than that on the downstream end as the deck acted as a flow barrier. During the tests, a dye injection was used and indicated that there were vortices on the upstream side of the deck extending toward the downstream part of the deck. A shear layer was formed at the lower side of the bridge deck, whereas vortices were formed between the two girders. Moreover, a reverse flow was observed just upstream and downstream of the deck. The movement of the shear layer to the free water surface on the downstream end of the bridge deck caused water surface fluctuations. The thickness of the shear layer and the strength of the vortices increased as the relative opening (yb/ya) decreased, and the approach densimetric Froude number increased. The flow observations under the pressure-flow conditions agreed with the flow descriptions of Picek et al. [26] and Lin et al. [12]. The measurements of the water surface profiles are depicted in Figure 5. According to this figure, the water surface elevation increased in the upstream face of the bridge deck and decreased just downstream of the deck (heading-up occurrence). This is the most important feature of the measured water surface profiles. The relative flow depth y/ya increased gradually as the relative opening decreased. It was observed that the water surface upstream of the deck increased under atmospheric flow, which implied the effect of the two vertical wall abutments. The relative water surface in front of the deck increased by a factor of 11, 9, 6, 5, and 4 times the relative water surface in front of the deck under atmospheric flow conditions for yb/ya = 0.35, 0.55, 0.65, 0.75, and 0.80, respectively. The densimetric Froude number had a significant influence on the water surface profile under pressure-flow conditions as the heading-up was proportional to the velocity. The relative flow depth y/ya increased as the relative bridge width (bbr/ya) decreased. The relative bridge length L/ya had a relatively small effect on the water surface profiles.
Seven vertical velocity profiles of dimensionless mean streamwise velocity (u/Va) at dimensionless longitudinal distances (x/ya) starting from the upstream to the downstream of the deck for different relative openings are plotted in Figure 6. Fifteen vertical points were measured for every vertical velocity profile along the centerline of the flume (B/2). The measured vertical velocity profile like that observed in open channels (logarithmic profile) was observed at nondimensional streamwise distances of x/ya = −7.0). As the flow approached the bridge deck, the vertical velocity profile was affected near the water surface at x/ya = −5.0. A small reverse flow near the free water surface was observed at x/ya = −5.0 with yb/ya = 0.35. The velocity was negative or close to zero depending on the submergence ratios at just below the free surface at x/ya = −2.5. This refers to the observed reverse flow near the free water surface and denotes the formation of the shear layer upstream of the bridge deck. Under the bridge deck (x/ya = 0.0 to 5.0), vortices were observed, and the thickness of the shear layer increased under the bridge deck (Figure 6). The observed velocity profile under the bridge deck at x/ya = 2.5 was similar to the velocity profile in pipe flow. The thickness of the shear layer increased as the relative opening yb/ya decreased. The vertical velocity distribution just downstream of the deck was similar to that of horizontal jet flow. After the flow passed through the bridge deck (x/ya = 6.0), negative velocities were observed toward the free surface as the shear layer moved toward the free surface, and this generated vortices at the water surface. Downstream of the bridge deck at x/ya = 6.0, the development of a boundary layer flow was dominant near the bed. The near-bed velocity gradients were clearly higher than that of x/ya= 2.5. The gradient was almost vertical. This indicated that shear stresses were generated under the bridge deck. For each vertical profile, the maximum velocity was found to be almost in the middle of the vertical profile. The observations of the velocity field in pressure-flow scour agrees well with the depictions by [12,27,28]. It is worth mentioning that the pressure flow accelerated the flow near the abutments, resulting in scour holes in front and alongside them.

3.3. Pressure-Flow Scour Profile

The observed scour first developed in the central part of the flume upstream of the bridge deck and then progressed laterally. A scour was also observed in front of the abutments. Figure 7 shows the centerline profiles of the scour hole for the pressure and atmospheric flow conditions at F a * = 0.236 and d50 = 1.093. According to Figure 7, the observed location of the maximum scour depth was below the deck and close to its downstream side. This was because the flow was accelerated in the streamwise direction; the maximum velocity was observed under the bridge deck, which was greater than the critical velocity of the bed material particles. According to Hahn and Lyn [29], the observed location of the maximum scour depth was after the downstream end of the bridge deck. This study agrees well with the measurements of Guo et al. [23] and Shan et al. [13] but is contrary to Hahn and Lyn [29]. A scour hole was observed under the free surface flow conditions in the no-deck case (hs/ya = 0.0). This indicated that the maximum scour depth in the present experiments resulted from both pressure-flow scour and abutment scour. The maximum scour depth under the pressure-flow conditions was notably larger than those under the free surface flow conditions, which corresponded to the velocity distribution. The upstream slope of the scour hole was steeper than the downstream slope, which implied that the equilibrium scour depth was not sustained. The maximum scour depth and the upstream and downstream scour slopes increased when the relative openings decreased as scour particles were deposited at the downstream side. The increase in the relative openings decreased the bed shear, which increased the scouring potential of flow. The relative openings did not affect the location of the maximum scour depth. Although the maximum scour depth marginally decreased as the bridge length increased, the scour hole length in the streamwise direction increased as the bridge length increased. This revealed that underneath the longer deck, the velocity distribution becomes uniform along the bridge deck length and the bed elevation redistributes during the test to produce a longer and superficial scour hole. The maximum scour depth significantly increased as the bridge width decreased under pressure-flow conditions.
Three different types of sand with median diameters (d50) of 1.093, 1.469, and 2.575 mm were tested to explore the effects of the bed material size on the maximum scour depth (ys). The relative maximum scour depth ys/ya with the relative bridge opening (yb/ya) for different relative median diameters of bed materials (d50/yb) is depicted in Figure 8. It was found that for pressure-flow, as the bridge opening (yb) increased, the maximum scour depth decreased by up to about 54.8% for d50 = 1.093 mm, 55.2% for d50 = 1.469 mm, and 56.6% for d50 = 2.575 mm. It was observed that the maximum scour depth increased when finer bed materials were tested. This is because increasing the bed material size increased the critical velocity of shields and correspondingly decreased the scour depth.

3.4. Effects of the Bridge Length and Width

Majid and Tripathi [7] reported that the effects of the bridge length (L) (length of contraction) and bridge width (bbr) (contraction width) on the scour depth under pressure-flow conditions have not been investigated in any study.
The current research examined three bridge lengths: L = 50, 75, and 100 cm (Figure 9). It was observed that the scour depth decreased as the bridge opening (yb) increased. When the relative bridge length was increased from 5 to 7.5 and from 7.5 to 10, the scour depth decreased by up to about 7.4% and 2.3%, respectively. The maximum scour depth slightly decreased when a longer bridge was tested. This was because an increase in the bridge length redistributed the velocity in the streamwise direction; thus, the velocity field underneath the bridge deck became more symmetric during the test to produce a shallower scour hole. Figure 9 plots the relative scour depth ys/ya with the relative bridge opening yb/ya for different relative bridge widths (bbr/ya). Similar to the contraction length, as the bridge opening height decreased, the scour depth increased. As the relative bridge width decreased from 5.5 to 5.2 and from 5.2 to 4.4, the maximum scour depth increased by up to about 45.6% and 81.2%, respectively. The scour depth significantly increased when the bridge width decreased. This indicates that as the bridge width decreased, the velocity field underneath the deck is notably increased; consequently, the shear stress increased. The scour depth is a combined scour of both the contraction width and the pressure-flow conditions. New experimental data are needed to further evaluate the contraction length and width of the pressure-flow scour and fill the gap in the literature.

4. Conclusions

This study explored the scour depth beneath a bridge deck without piers for atmospheric and pressure flows under clear water conditions in the presence of two vertical wall abutments. Through the experiments, the effects of the flow depth, bed material size, contraction length, contraction width, and bridge opening on the maximum scour depth were examined in both flow conditions. A dimensionless relationship was deduced to predict the maximum scour depth due to the vertical deck and horizontal abutment contractions. Statistical analyses were applied to assess the agreement of the laboratory data with previously published models. The present laboratory equation can be employed in the initial design of bridges under pressure-flow conditions. The experimental data were used to analyze the predictive errors of the previous models. The results showcased that the HEC-18 Equation performed better than the other tested models and gave the lowest RMSE. Under the pressure-flow conditions, the maximum scour depth increased by a factor between 2.15 and 9.81 times that under the atmospheric flow conditions depending on the densimetric Froude number and bridge openings. The most important features of the measured water surface profiles were of the water surface elevation increase on the upstream side of the bridge deck, and the decrease just downstream of the deck. As the bridge opening increased, the maximum scour depth decreased, while it increased in the presence of finer bed materials. When the relative bridge length was increased from 5 to 7.5 and from 7.5 to 10, the maximum scour depth decreased by up to about 7.4% and 2.3%, respectively. Decreasing the relative bridge width from 5.5 to 5.2 and 5.2 to 4.4, the maximum scour depth increased by up to about 45.6% and 81.2%, respectively. Scouring at bridge abutments is the major cause of bridge collapses worldwide. Therefore, the next steps in this research would be to expand the evaluation on the contraction lengths and widths of pressure-flow scours by conducting more experimental assessments and enriching the literature on this less researched aspect. This further evaluation would contribute to a better database for computing the scour risk of bridge foundations which is key for a correct management approach and allocation of resources for maintenance and scour mitigation works.

Author Contributions

Conceptualization, F.S.A.; methodology, F.S.A.; experiments, F.S.A.; formal analysis, F.S.A., W.F. and I.M.M.; investigation, I.G.S., A.A. and A.I.; data curation, F.S.A. and W.F.; writing—original draft preparation, F.S.A., I.G.S. and A.A.; writing—review and editing, F.S.A., I.G.S. and A.A. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

The data are available from the first author upon reasonable request.

Acknowledgments

The authors express their sincere thanks to their colleagues at the Hydraulics Laboratory of the Faculty of Engineering, Menoufia University, Egypt, for facilitating the experimental work.

Conflicts of Interest

The authors have declared that no competing interests exist.

References

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Figure 1. Experimental apparatus: (a) bridge deck and vertical wall abutments; (b) elevation and side view.
Figure 1. Experimental apparatus: (a) bridge deck and vertical wall abutments; (b) elevation and side view.
Water 15 00404 g001
Figure 2. Definition diagram for tested apparatus: (a) 3D view; (b) side view and sec A-A.
Figure 2. Definition diagram for tested apparatus: (a) 3D view; (b) side view and sec A-A.
Water 15 00404 g002
Figure 3. Observed relative maximum scour depth (ys/ya) versus those predicted: (a) using Equation (4); (b) using the literature equations, see Table 1. [2,3,4,9,11,13,15,16,17].
Figure 3. Observed relative maximum scour depth (ys/ya) versus those predicted: (a) using Equation (4); (b) using the literature equations, see Table 1. [2,3,4,9,11,13,15,16,17].
Water 15 00404 g003aWater 15 00404 g003b
Figure 4. Relative maximum scour depth (ys/ya) versus the densimetric Froude number ( F a * ) for different relative bridge openings (yb/ya).
Figure 4. Relative maximum scour depth (ys/ya) versus the densimetric Froude number ( F a * ) for different relative bridge openings (yb/ya).
Water 15 00404 g004
Figure 5. Relative flow depth y/ya in the nondimensional streamwise distance x/ya: (a) for different relative bridge openings yb/ya and F a * = 0.236; (b) for different densimetric Froude numbers and yb/ya = 0.55; (c) for different relative bridge lengths L/ya and yb/ya = 0.55; and (d) for different relative bridge widths bbr/ya and yb/ya = 0.55.
Figure 5. Relative flow depth y/ya in the nondimensional streamwise distance x/ya: (a) for different relative bridge openings yb/ya and F a * = 0.236; (b) for different densimetric Froude numbers and yb/ya = 0.55; (c) for different relative bridge lengths L/ya and yb/ya = 0.55; and (d) for different relative bridge widths bbr/ya and yb/ya = 0.55.
Water 15 00404 g005aWater 15 00404 g005b
Figure 6. Vertical distributions of the nondimensional mean streamwise velocity u/Va at centerline of the channel and at different nondimensional streamwise distances x/ya, F a * = 0.236 and at: (a) yb/ya = 0.35; (b) yb/ya = 0.55; (c) yb/ya = 0.65; (d) yb/ya = 0.75; (e) yb/ya = 0.80; and (f) yb/ya = 1.00 (no deck).
Figure 6. Vertical distributions of the nondimensional mean streamwise velocity u/Va at centerline of the channel and at different nondimensional streamwise distances x/ya, F a * = 0.236 and at: (a) yb/ya = 0.35; (b) yb/ya = 0.55; (c) yb/ya = 0.65; (d) yb/ya = 0.75; (e) yb/ya = 0.80; and (f) yb/ya = 1.00 (no deck).
Water 15 00404 g006aWater 15 00404 g006b
Figure 7. Relative scour depth ys/ya in the nondimensional streamwise distance x/ya for F a * = 0.236 and for: (a) different relative bridge opening yb/ya; (b) different relative bridge lengths L/ya and yb/ya = 0.55; and (c) different relative bridge widths (bbr/ya) and yb/ya = 0.55.
Figure 7. Relative scour depth ys/ya in the nondimensional streamwise distance x/ya for F a * = 0.236 and for: (a) different relative bridge opening yb/ya; (b) different relative bridge lengths L/ya and yb/ya = 0.55; and (c) different relative bridge widths (bbr/ya) and yb/ya = 0.55.
Water 15 00404 g007
Figure 8. Relative scour depth ys/ya with the relative bridge opening for different relative median bed materials (d50/yb).
Figure 8. Relative scour depth ys/ya with the relative bridge opening for different relative median bed materials (d50/yb).
Water 15 00404 g008
Figure 9. Relative scour depth ys/ya with the relative bridge opening yb/ya for (a) different relative bridge lengths (L/ya) and (b) different relative bridge widths (bbr/ya).
Figure 9. Relative scour depth ys/ya with the relative bridge opening yb/ya for (a) different relative bridge lengths (L/ya) and (b) different relative bridge widths (bbr/ya).
Water 15 00404 g009
Table 1. Prediction equations for the pressure-flow scours.
Table 1. Prediction equations for the pressure-flow scours.
Model/ EquationRemarks
Arneson and Abt [9]
y s y a = 5.08 + 1.27 y a y b + 4.44 y b y a + 0.19 V a V c
Vc = critical velocity
V c = C g S g 1 d 50 y a / d 50 1 / 6
C = 1.52
Umbrell et al. [3]
y s + y b y a = 1.102 V a V c 1 w y a 0.603 + 0.06
w = flow depth overtopping bridge
C = 1.58 in critical velocity, Vc equation
Lyn [4]
y s y a = m i n 0.21 V b V c 2.95 , 0.6
Guo et al. [11]
y s = y b + h 1 + ʎ F i m 1 + 2 β F i 2 y b
h = ya − yb = (hs + hg)
Fi = inundation Froude number
F i = V a g y a y b
λ, m, β = constanta parameters
HEC-18 Equation [2]
y s = y 2 + t y b
y 2 = K u Q 2   d m 2 / 3 B 3 / 7
t = 0.5 y b . h g + h s   y a 2 0.20 . y b
Ku = 0.0077 (English units)
Q = flow discharge (ft3/s)
dm = diameter of the smallest non-transportable particle in the bed material (= 1.25.d50)
hg = girder depth
Shan et al. [13]
y s = V a ( y a w ) K u   d 50 1 / 3 6 / 7 + 0.5 y b y a y a 2 0.2 1 w y a 0.1 1 y b
Ku = constant = 6.17 m2/s
Melville [16]
y s y a = 0.75 V a V c 0.4 ,                   0.4 < V a V c 1      
y s y a = 0.45 ,                                                                 1 < V a V c 2.5    
Kumcu [17]
y s + y b y a = 0.65 + 0.5 V b V c ,                       0.5 V b V c < 1      
y s + y b y a = 1.025 + 0.125 V b V c ,               1 V b V c 1.8      
Kocyigit and Karakurt [15]
y s y b = 0.962 0.187 y a y b + 0.443 F b * + 0.672 h g y b
hg = girder depth
F b * = densimetric Froude number of the flow passing under the bridge deck
F b * = V b g S g 1 d 50
Table 2. Values of experimental parameters.
Table 2. Values of experimental parameters.
ParameterValues
Approach flow depth, ya 8, 9, 10, 12, and 15 cm
Girder dimension 1.5 cm height and 0.8 cm width
Bridge length, L 50, 75, and 100 cm
Bridge width, bbr 55, 52, and 44 cm
Median diameter, d501.093, 1.469, and 2.575 mm
Geometric standard deviations, σg1.302, 1.198, and 1.274
Table 3. Laboratory data and computed scour number.
Table 3. Laboratory data and computed scour number.
Testya
(cm)
yb
(c)
L
(cm)
bbr
(cm)
hs
(cm)
Va
(m/s)
d50
(mm)
σg
(-)
Fa
(-)
F a *
(-)
ys
(cm)
Scour Number
115.06.0050557.500.2001.0931.3020.1650.1288.220.948
215.09.0050554.500.2001.0931.3020.1650.1285.860.990
315.010.5050553.00.2001.0931.3020.1650.1282.490.866
415.012.0050551.50.2001.0931.3020.1650.1280.420.828
515.012.7550550.80.2001.0931.3020.1650.1280.450.880
612.04.5050556.00.2501.0931.3020.2300.1799.731.186
712.06.9050553.60.2501.0931.3020.2300.1797.511.201
812.08.1050552.40.2501.0931.3020.2300.1795.581.140
912.09.3050551.20.2501.0931.3020.2300.1794.011.109
1012.09.9050550.60.2501.0931.3020.2300.1792.141.003
1110.03.5050555.00.3001.0931.3020.3030.23610.391.389
1210.05.5050553.00.3001.0931.3020.3030.2369.271.477
1310.06.5050552.00.3001.0931.3020.3030.2368.171.467
1410.07.5050551.00.3001.0931.3020.3030.2366.1201.362
1510.08.0050550.50.3001.0931.3020.3030.2364.691.270
169.003.0050554.50.3331.0931.3020.3550.27611.621.624
179.004.8050552.70.3331.0931.3020.3550.27610.441.693
189.005.7050551.80.3331.0931.3020.3550.2769.261.662
199.006.6050550.90.3331.0931.3020.3550.2768.031.625
209.007.0550550.50.3331.0931.3020.3550.2766.511.507
218.002.5050554.00.3751.0931.3020.4230.33013.131.954
228.004.1050552.40.3751.0931.3020.4230.33010.731.854
238.004.9050551.60.3751.0931.3020.4230.3309.761.832
248.005.7050550.80.3751.0931.3020.4230.3309.231.866
258.006.1050550.40.3751.0931.3020.4230.3309.081.897
2610.03.5050555.00.3001.4691.1980.3030.2369.121.262
2710.05.5050553.00.3001.4691.1980.3030.2368.131.363
2810.06.5050552.00.3001.4691.1980.3030.2367.141.364
2910.07.5050551.00.3001.4691.1980.3030.2365.361.286
3010.08.0050550.50.3001.4691.1980.3030.2364.091.209
3110.03.5050555.00.3002.5751.2740.3030.2368.781.228
3210.05.5050553.00.3002.5751.2740.3030.2367.831.333
3310.06.5050552.00.3002.5751.2740.3030.2366.691.320
3410.07.5050551.00.3002.5751.2740.3030.2365.091.259
3510.08.0050550.50.3002.5751.2740.3030.2363.811.181
3610.03.5075555.00.3001.0931.3020.3030.2369.771.327
3710.05.5075553.00.3001.0931.3020.3030.2368.811.431
3810.06.5075552.00.3001.0931.3020.3030.2367.511.401
3910.07.5075551.00.3001.0931.3020.3030.2365.451.295
4010.08.0075550.50.3001.0931.3020.3030.2364.371.237
4110.03.50100555.00.3001.0931.3020.3030.2369.461.296
4210.05.50100553.00.3001.0931.3020.3030.2368.531.403
4310.06.50100552.00.3001.0931.3020.3030.2367.191.369
4410.07.50100551.00.3001.0931.3020.3030.2365.571.307
4510.08.00100550.50.3001.0931.3020.3030.2364.231.223
4610.03.5050525.00.3001.0931.3020.3030.23616.001.950
4710.05.5050523.00.3001.0931.3020.3030.23612.981.848
4810.06.5050522.00.3001.0931.3020.3030.23612.581.908
4910.07.5050521.00.3001.0931.3020.3030.2368.571.607
5010.08.0050520.50.3001.0931.3020.3030.2366.581.458
5110.03.5050445.00.3001.0931.3020.3030.23627.123.062
5210.05.5050443.00.3001.0931.3020.3030.23624.603.010
5310.06.5050442.00.3001.0931.3020.3030.23621.492.799
5410.07.5050441.00.3001.0931.3020.3030.23615.982.348
5510.08.0050440.50.3001.0931.3020.3030.23612.472.047
5615.015.0050550.00.2001.0931.3020.1650.1280.411.028
5712.012.0050550.00.2501.0931.3020.2300.1790.511.043
5810.010.0050550.00.3001.0931.3020.3030.2363.301.330
599.009.0050550.00.3331.0931.3020.3550.2763.341.371
608.008.0050550.00.3751.0931.3020.4230.3303.741.467
6210.010.0050550.00.3001.4691.1980.3030.2361.811.861
6310.010.0050550.00.3002.5751.2740.3030.2360.731.181
6410.010.0075550.00.3001.0931.3020.3030.2363.141.073
6510.010.00100550.00.3001.0931.3020.3030.2363.001.314
6610.010.0050520.00.3001.0931.3020.3030.2364.621.300
Table 4. Error statistics for different models using current laboratory data sets.
Table 4. Error statistics for different models using current laboratory data sets.
Model/ Statistical Characteristics Average
Error
Minimum
Error
Maximum
Error
VarianceRMSE
Arneson and Abt [9]0.522−0.2412.4770.1570.650
Umbrell et al. [3]0.4630.0282.1350.0450.509
Lyn [4]0.402−0.0172.1120.0470.455
Guo et al. [11]0.357−0.2250.9800.1200.494
HEC-18 Equation [2]0.280−0.1462.0530.0850.390
Shan et al. [13]0.385−0.0412.1900.0510.445
Melville [16]0.5440.0012.4880.0880.618
Kumcu [17]-0.420−0.9191.3470.0820.506
Kocyigit and Karakurt [15]0.4300.1661.7730.0150.447
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Abdelhaleem, F.S.; Mohamed, I.M.; Shaaban, I.G.; Ardakanian, A.; Fahmy, W.; Ibrahim, A. Pressure-Flow Scour under a Bridge Deck in Clear Water Conditions. Water 2023, 15, 404. https://doi.org/10.3390/w15030404

AMA Style

Abdelhaleem FS, Mohamed IM, Shaaban IG, Ardakanian A, Fahmy W, Ibrahim A. Pressure-Flow Scour under a Bridge Deck in Clear Water Conditions. Water. 2023; 15(3):404. https://doi.org/10.3390/w15030404

Chicago/Turabian Style

Abdelhaleem, Fahmy Salah, Ibrahim M. Mohamed, Ibrahim G. Shaaban, Atiyeh Ardakanian, Wael Fahmy, and Amir Ibrahim. 2023. "Pressure-Flow Scour under a Bridge Deck in Clear Water Conditions" Water 15, no. 3: 404. https://doi.org/10.3390/w15030404

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