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

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experimental Setup

_{50}= 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 (σ

_{g}) for the three bed material samples were 1.302, 1.198, and 1.274, the uniformity coefficients (C

_{u}) were 1.611, 1.292, and 1.612, and the curvature coefficients (C

_{c}) 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 S

_{g}= 2.65. The tested bed materials are uniform because σ

_{g}< 1.4, C

_{u}< 3.0, and C

_{c}< 1.5 [15,18]. The armoring effect would not occur in this study as σ

_{g}< 1.3 [19,20].

_{a}= 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 (V

_{c}) obtained using Neill’s [21] equation, ${V}_{\mathrm{c}}=1.52\sqrt{g\left({S}_{\mathrm{g}}-1\right){d}_{50}}{\left({y}_{\mathrm{a}}/{d}_{50}\right)}^{1/6}$, is less than one (V

_{a}/V

_{c}< 1) [15,22]. In this study, the computed V

_{a}/V

_{c}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 (b

_{br}= 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.

#### 2.2. Dimensional Analysis

_{s}is the maximum scour depth, f is the functional symbol, y

_{a}is the approach flow depth, y

_{b}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, b

_{br}= (B − 2b) is the bridge width (contraction width), h

_{g}is the girder depth, h

_{s}is the submerged height of the deck, V

_{a}is the approach flow velocity, V

_{b}is the flow velocity underneath the deck, u

_{*}is the shear velocity, d

_{50}is the median bed material size, σ

_{g}is the geometric standard deviation, S

_{g}= (ρ

_{s}/ρ) is the specific gravity of the bed materials, ρ

_{s}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:

_{a}is the approach Froude number (${F}_{\mathrm{a}}={V}_{\mathrm{a}}/\sqrt{g{y}_{\mathrm{a}}}$) and R is the approach Reynolds number (R = V

_{a}y

_{a}/ν). 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 h

_{g}= 1.5 cm, respectively). The terms h

_{s}/y

_{a}and V

_{a}/V

_{b}were dependent on the approach flow depth and the depth under the bridge deck (h

_{s}= y

_{a}− y

_{b}− h

_{g}). The term u

_{*}/V

_{b}is only dependent on the height under the bridge deck (bridge opening), and the parameter y

_{b}/y

_{a}included the same effect as that of u

_{*}/V

_{b}[15]. The densimetric Froude number (${F}_{\mathrm{a}}^{*}={V}_{\mathrm{a}}/\sqrt{g\left({S}_{\mathrm{g}}-1\right){y}_{\mathrm{a}}}$) was used as three different bed materials were involved in this study, and the effect of the flow intensity V

_{a}/V

_{c}was included in ${F}_{a}^{*}$ according to Carnacina et al. [25]. The relationship in Equation (2) can thus be simplified and arranged as follows:

## 3. Results and Discussion

_{b}/y

_{a}≤ 0.85, 3.33 ≤ L/y

_{a}≤ 10, 0.733 ≤ b

_{br}/B ≤ 0.917, 0.009 ≤ d

_{50}/y

_{b}≤ 0.074, 1.198 ≤ σ

_{g}≤ 1.302, 0.128 ≤ ${F}_{a}^{*}$ ≤ 0.33). The coefficient of determination (R

^{2}) for Equation (4) was 0.92, and the adjusted R

^{2}= 0.912. This denotes that the degree of agreement between the parameters is reasonably good. B

_{br}/B, ${F}_{a}^{*}$, σ

_{g}

_{,}and y

_{b}/y

_{a}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.

_{s}/y

_{a}) 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 y

_{s}/y

_{a}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 h

_{g}/y

_{b}. The actual girder depth (h

_{g}) 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.

#### 3.1. Free Surface and Pressure-Flow Scour

_{s}/y

_{a}) at different relative bridge opening (y

_{b}/y

_{a}) for pressure-flow conditions and for the case of atmospheric flow conditions (y

_{b}/y

_{a}= 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 (h

_{s}/y

_{a}) increased (h

_{s}= y

_{a}− h

_{g}− y

_{b}). The maximum scour depth increased by up to about 77%, 73%, 69%, 58%, and 46% for range of the relative openings of y

_{b}/y

_{a}= 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 (y

_{s}+y

_{b})/(y

_{b}+

_{h}) where h= (y

_{a}-y

_{b}) 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 (${F}_{i}={V}_{a}/\sqrt{g\left({y}_{a}-{y}_{b}\right)}$), agree well with the analytical solution of Guo et al. [11].

#### 3.2. Water Surface Profile and Velocity Field

_{b}/y

_{a}) 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/y

_{a}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 y

_{b}/y

_{a}= 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/y

_{a}increased as the relative bridge width (b

_{br}/y

_{a}) decreased. The relative bridge length L/y

_{a}had a relatively small effect on the water surface profiles.

_{a}) at dimensionless longitudinal distances (x/y

_{a}) 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/y

_{a}= −7.0). As the flow approached the bridge deck, the vertical velocity profile was affected near the water surface at x/y

_{a}= −5.0. A small reverse flow near the free water surface was observed at x/y

_{a}= −5.0 with y

_{b}/y

_{a}= 0.35. The velocity was negative or close to zero depending on the submergence ratios at just below the free surface at x/y

_{a}= −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/y

_{a}= 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/y

_{a}= 2.5 was similar to the velocity profile in pipe flow. The thickness of the shear layer increased as the relative opening y

_{b}/y

_{a}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/y

_{a}= 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/y

_{a}= 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/y

_{a}= 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

_{50}= 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 (h

_{s}/y

_{a}= 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.

_{50}) 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 (y

_{s}). The relative maximum scour depth y

_{s}/y

_{a}with the relative bridge opening (y

_{b}/y

_{a}) for different relative median diameters of bed materials (d

_{50}/y

_{b}) is depicted in Figure 8. It was found that for pressure-flow, as the bridge opening (y

_{b}) increased, the maximum scour depth decreased by up to about 54.8% for d

_{50}= 1.093 mm, 55.2% for d

_{50}= 1.469 mm, and 56.6% for d

_{50}= 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

_{br}) (contraction width) on the scour depth under pressure-flow conditions have not been investigated in any study.

_{b}) 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 y

_{s}/y

_{a}with the relative bridge opening y

_{b}/y

_{a}for different relative bridge widths (b

_{br}/y

_{a}). 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

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Gaudio, R.; Tafarojnoruz, A.; Calomino, F. Combined flowaltering countermeasures against bridge pier scour. J. Hydraul. Res.
**2012**, 50, 35–43. [Google Scholar] [CrossRef] - Arneson, L.A.; Zevenbergen, L.; Lagasse, P.; Clopper, P. Evaluating Scour at Bridges; Report No. FHWA-HIF-12-003; USDOT, National Highway Institute (US): Washington, DC, USA, 2012. [Google Scholar]
- Umbrell, E.R.; Young, G.; Stein, S.; Jones, J. Clearwater contraction scour under bridges in pressure flow. J. Hydraul. Eng.
**1998**, 124, 236–240. [Google Scholar] [CrossRef] - Lyn, D.A. Pressure-flow scour: A reexamination of the HEC-18 equation. J. Hydraul. Eng.
**2008**, 134, 1015–1020. [Google Scholar] [CrossRef] - Richardson, E.V.; Harrison, L.; Richardson, J.; Davis, S. Evaluating Scour at Bridges HEC 18: Hydraulic Engineering Circular, 2nd ed.; Federal Highway Administration: Washington, DC, USA, 1993; pp. 5–53. [Google Scholar]
- Carnacina, I.; Leonardi, N.; Pagliara, S. Characteristics of flow structure around cylindrical bridge piers in pressure-flow conditions. Water
**2019**, 11, 2240. [Google Scholar] [CrossRef][Green Version] - Majid, S.A.; Tripathi, S. Pressure-flow scour due to vertical contraction: A review. J. Hydraul. Eng.
**2021**, 147, 03121002. [Google Scholar] [CrossRef] - Abed, L.M. Local Scour around Bridge Piers in Pressure Flow. Ph.D. Dissertation, Department of Civil Engineering, Colorado State University, Fort Collins, CO, USA, 1991; pp. 43–47, 61–63. [Google Scholar]
- Arneson, L.A.; Abt, S.R. Vertical contraction scour at bridges with water flowing under pressure conditions. Transp. Res. Rec.
**1998**, 1647, 10–17. [Google Scholar] [CrossRef] - Arneson, L.A. The Effect of Pressure-Flow on Local Scour in Bridge Openings. Ph.D. Dissertation, Department of Civil Engineering, Colorado State University, Fort Collins, CO, USA, 1997. [Google Scholar]
- Guo, J.; Kerenyi, K.; Pagan-Ortiz, J. Bridge Pressure Flow Scour for Clear Water Conditions; Report No. FHWA-HRT-09-041; Turner-Fairbank Highway Research Center, FHWA: McLean, VA, USA, 2009; pp. 24–29. [Google Scholar]
- Lin, C.; Kao, M.J.; Hsieh, S.C.; Lo, L.F.; Raikar, R.V. On the flow structures under a partially inundated bridge deck. J. Mech.
**2012**, 28, 191–207. [Google Scholar] [CrossRef] - Shan, H.; Bojanowski, C.; Xie, Z.; Suaznabar, O.; Lottes, S.; Shen, J.; Kerenyi, K. Submerged Flow Bridge Scour under Clear Water Conditions; Publication No. FHWA-HRT-12-034; Federal Highway Administration: Washington, DC, USA, 2012. [Google Scholar]
- Dankoo, A.; Yonesi, H.; Torabipoudeh, H.; Saneie, M. The effect of pressure flow conditions on bridge pier scour in compound open channels with vegetation. J. Hydraul. Iran. Hydraul. Assoc.
**2022**, 17, 89–103. [Google Scholar] - Kocyigit, M.B.; Karakurt, O. Pressure flow and weir scour beneath a bridge deck. Can. J. Civ. Eng.
**2019**, 46, 534–543. [Google Scholar] [CrossRef] - Melville, B.W. Scour at various hydraulic structures: Sluice gates, submerged bridges, low weirs. Aust. J. Water Resour.
**2014**, 18, 101–117. [Google Scholar] [CrossRef] - Kumcu, S.Y. Steady and unsteady pressure scour under bridges at clear-water conditions. Can. J. Civ. Eng.
**2016**, 43, 334–342. [Google Scholar] [CrossRef] - Dey, S.; Raikar, R.V.; Roy, A. Scour at submerged cylindrical obstacles under steady flow. J. Hydraul. Eng.
**2008**, 134, 105–109. [Google Scholar] [CrossRef] - Raudkivi, A.J.; Ettema, R. Scour at cylindrical bridge piers in armored beds. J. Hydraul. Eng.
**1985**, 111, 713–731. [Google Scholar] [CrossRef] - Helal, E. Experimental evaluation of changes in channel bed morphology due to a defective pressure flow pipe. J. Irrig. Drain. Eng.
**2019**, 145, 04019022. [Google Scholar] [CrossRef] - Neill, C.R. Guide to Bridge Hydraulics; University of Toronto Press: Toronto, ON, Canada, 1973. [Google Scholar]
- Melville, B.W. The physics of local scour at bridge piers. In Proceedings of the 4th International Conference on Scour and Erosion (ICSE-4), Tokyo, Japan, 4–8 November 2008; Japanese Geotechnical Society: Tokyo, Japan, 2008; pp. 28–40. [Google Scholar]
- Guo, J.; Kerenyi, K.; Pagan-Ortiz, J.; Flora, K. Bridge pressure flow scour at clear water threshold condition. Trans. Tianjin Univ.
**2009**, 15, 79–94. [Google Scholar] [CrossRef] - Henderson, F.M. Open Channel Flow; The Macmillan Company: New York, NY, USA, 1966; p. 93. [Google Scholar]
- Carnacina, I.; Pagliara, S.; Leonardi, N. Bridge pier scour under pressure flow conditions. River Res. Appl.
**2019**, 35, 844–854. [Google Scholar] [CrossRef] - Picek, T.; Havlik, A.; Mattas, D.; Mares, K. Hydraulic calculation of bridges at high water stages. J. Hydraul. Res.
**2007**, 45, 400–406. [Google Scholar] [CrossRef] - Hahn, E. Clear-Water Scour at Vertically or Laterally Contracted Bridge Sections. M.Sc. Dissertation, School of Civil Engineering, Purdue University, West Lafayette, IN, USA, 2005. [Google Scholar]
- Yoon, K.S.; Lee, S.; Hong, S. Time-averaged turbulent velocity flow field through the various bridge contractions during large flooding. Water
**2019**, 11, 143. [Google Scholar] [CrossRef][Green Version] - Hahn, E.M.; Lyn, D. Anomalous contraction scour? Vertical contraction case. J. Hydraul. Eng.
**2010**, 136, 137–141. [Google Scholar] [CrossRef]

**Figure 1.**Experimental apparatus: (

**a**) bridge deck and vertical wall abutments; (

**b**) elevation and side view.

**Figure 4.**Relative maximum scour depth (y

_{s}/y

_{a}) versus the densimetric Froude number (${F}_{a}^{*}$) for different relative bridge openings (y

_{b}/y

_{a}).

**Figure 5.**Relative flow depth y/y

_{a}in the nondimensional streamwise distance x/y

_{a}: (

**a**) for different relative bridge openings y

_{b}/y

_{a}and ${F}_{a}^{*}$ = 0.236; (

**b**) for different densimetric Froude numbers and y

_{b}/y

_{a}= 0.55; (

**c**) for different relative bridge lengths L/y

_{a}and y

_{b}/y

_{a}= 0.55; and (

**d**) for different relative bridge widths b

_{br}/y

_{a}and y

_{b}/y

_{a}= 0.55.

**Figure 6.**Vertical distributions of the nondimensional mean streamwise velocity u/V

_{a}at centerline of the channel and at different nondimensional streamwise distances x/y

_{a}, ${F}_{a}^{*}$ = 0.236 and at: (

**a**) y

_{b}/y

_{a}= 0.35; (

**b**) y

_{b}/y

_{a}= 0.55; (

**c**) y

_{b}/y

_{a}= 0.65; (

**d**) y

_{b}/y

_{a}= 0.75; (

**e**) y

_{b}/y

_{a}= 0.80; and (

**f**) y

_{b}/y

_{a}= 1.00 (no deck).

**Figure 7.**Relative scour depth y

_{s}/y

_{a}in the nondimensional streamwise distance x/y

_{a}for ${F}_{a}^{*}$ = 0.236 and for: (

**a**) different relative bridge opening y

_{b}/y

_{a}; (

**b**) different relative bridge lengths L/y

_{a}and y

_{b}/y

_{a}= 0.55; and (

**c**) different relative bridge widths (b

_{br}/y

_{a}) and y

_{b}/y

_{a}= 0.55.

**Figure 8.**Relative scour depth y

_{s}/y

_{a}with the relative bridge opening for different relative median bed materials (d

_{50}/y

_{b}).

**Figure 9.**Relative scour depth y

_{s}/y

_{a}with the relative bridge opening y

_{b}/y

_{a}for (

**a**) different relative bridge lengths (L/y

_{a}) and (

**b**) different relative bridge widths (b

_{br}/y

_{a}).

Model/ Equation | Remarks |
---|---|

Arneson and Abt [9] $\frac{{y}_{s}}{{y}_{a}}=-5.08+1.27\frac{{y}_{a}}{{y}_{b}}+4.44\frac{{y}_{b}}{{y}_{a}}+0.19\frac{{V}_{a}}{{V}_{c}}$ | V_{c} = critical velocity ${V}_{c}=C\sqrt{g\left({S}_{g}-1\right){d}_{50}}{\left({y}_{a}/{d}_{50}\right)}^{1/6}$ C = 1.52 |

Umbrell et al. [3] $\frac{{y}_{s}+{y}_{b}}{{y}_{a}}=1.102{\left[\frac{{V}_{a}}{{V}_{c}}\left(1-\frac{w}{{y}_{a}}\right)\right]}^{0.603}+0.06$ | w = flow depth overtopping bridge C = 1.58 in critical velocity, V _{c} equation |

Lyn [4] $\frac{{y}_{s}}{{y}_{a}}=min\left[0.21{\left(\frac{{V}_{b}}{{V}_{c}}\right)}^{2.95},0.6\right]$ | |

Guo et al. [11] ${y}_{s}=\left({y}_{b}+h\right)\sqrt{\frac{1+\frac{\u028e}{{F}_{i}^{m}}}{1+\frac{2\beta}{{F}_{i}^{2}}}}-{y}_{b}$ | h = y_{a} − y_{b} = (h_{s} + h_{g})F _{i} = inundation Froude number${F}_{i}=\frac{{V}_{a}}{\sqrt{g\left({y}_{a}-{y}_{b}\right)}}$ λ, m, β = constanta parameters |

HEC-18 Equation [2] ${y}_{s}={y}_{2}+t-{y}_{b}$ ${y}_{2}={\left[\frac{{K}_{u}{Q}^{2}}{{d}_{m}^{2/3}B}\right]}^{3/7}$ $t=0.5{\left[\frac{{y}_{b.}\left({h}_{g}+{h}_{s}\right)}{{y}_{a}^{2}}\right]}^{0.20}.{y}_{b}$ | K_{u} = 0.0077 (English units)Q = flow discharge (ft ^{3}/s)d _{m} = diameter of the smallest non-transportable particle in the bed material (= 1.25.d_{50})h _{g} = girder depth |

Shan et al. [13] ${y}_{s}={\left[\frac{{V}_{a}({y}_{a}-w)}{{K}_{u}{d}_{50}^{1/3}}\right]}^{6/7}+\left[0.5{\left(\frac{{y}_{b}{y}_{a}}{{y}_{a}^{2}}\right)}^{0.2}{\left(1-\frac{w}{{y}_{a}}\right)}^{-0.1}-1\right]{y}_{b}$ | K_{u} = constant = 6.17 m^{2}/s |

Melville [16] $\frac{{y}_{s}}{{y}_{a}}=0.75\left(\frac{{V}_{a}}{{V}_{c}}-0.4\right),0.4\frac{{V}_{a}}{{V}_{c}}\le 1$ $\frac{{y}_{s}}{{y}_{a}}=0.45,1\frac{{V}_{a}}{{V}_{c}}\le 2.5$ | |

Kumcu [17] $\frac{{y}_{s}+{y}_{b}}{{y}_{a}}=0.65+0.5\frac{{V}_{b}}{{V}_{c}},0.5\le \frac{{V}_{b}}{{V}_{c}}1$ $\frac{{y}_{s}+{y}_{b}}{{y}_{a}}=1.025+0.125\frac{{V}_{b}}{{V}_{c}},1\le \frac{{V}_{b}}{{V}_{c}}\le 1.8$ | |

Kocyigit and Karakurt [15] $\frac{{y}_{s}}{{y}_{b}}=-0.962-0.187\frac{{y}_{a}}{{y}_{b}}+0.443{F}_{b}^{*}+0.672\frac{{h}_{g}}{{y}_{b}}$ | h_{g} = girder depth${F}_{b}^{*}$ = densimetric Froude number of the flow passing under the bridge deck ${F}_{b}^{*}=\frac{{V}_{b}}{\sqrt{g\left({S}_{g}-1\right){d}_{50}}}$ |

Parameter | Values |
---|---|

Approach flow depth, y_{a} | 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, b_{br} | 55, 52, and 44 cm |

Median diameter, d_{50} | 1.093, 1.469, and 2.575 mm |

Geometric standard deviations, σ_{g} | 1.302, 1.198, and 1.274 |

Test | y_{a}(cm) | y_{b}(c) | L (cm) | b_{br}(cm) | h_{s}(cm) | V_{a}(m/s) | d_{50}(mm) | σ_{g}(-) | F_{a}(-) | ${F}_{a}^{*}$ (-) | y_{s}(cm) | Scour Number |
---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 15.0 | 6.00 | 50 | 55 | 7.50 | 0.200 | 1.093 | 1.302 | 0.165 | 0.128 | 8.22 | 0.948 |

2 | 15.0 | 9.00 | 50 | 55 | 4.50 | 0.200 | 1.093 | 1.302 | 0.165 | 0.128 | 5.86 | 0.990 |

3 | 15.0 | 10.50 | 50 | 55 | 3.0 | 0.200 | 1.093 | 1.302 | 0.165 | 0.128 | 2.49 | 0.866 |

4 | 15.0 | 12.00 | 50 | 55 | 1.5 | 0.200 | 1.093 | 1.302 | 0.165 | 0.128 | 0.42 | 0.828 |

5 | 15.0 | 12.75 | 50 | 55 | 0.8 | 0.200 | 1.093 | 1.302 | 0.165 | 0.128 | 0.45 | 0.880 |

6 | 12.0 | 4.50 | 50 | 55 | 6.0 | 0.250 | 1.093 | 1.302 | 0.230 | 0.179 | 9.73 | 1.186 |

7 | 12.0 | 6.90 | 50 | 55 | 3.6 | 0.250 | 1.093 | 1.302 | 0.230 | 0.179 | 7.51 | 1.201 |

8 | 12.0 | 8.10 | 50 | 55 | 2.4 | 0.250 | 1.093 | 1.302 | 0.230 | 0.179 | 5.58 | 1.140 |

9 | 12.0 | 9.30 | 50 | 55 | 1.2 | 0.250 | 1.093 | 1.302 | 0.230 | 0.179 | 4.01 | 1.109 |

10 | 12.0 | 9.90 | 50 | 55 | 0.6 | 0.250 | 1.093 | 1.302 | 0.230 | 0.179 | 2.14 | 1.003 |

11 | 10.0 | 3.50 | 50 | 55 | 5.0 | 0.300 | 1.093 | 1.302 | 0.303 | 0.236 | 10.39 | 1.389 |

12 | 10.0 | 5.50 | 50 | 55 | 3.0 | 0.300 | 1.093 | 1.302 | 0.303 | 0.236 | 9.27 | 1.477 |

13 | 10.0 | 6.50 | 50 | 55 | 2.0 | 0.300 | 1.093 | 1.302 | 0.303 | 0.236 | 8.17 | 1.467 |

14 | 10.0 | 7.50 | 50 | 55 | 1.0 | 0.300 | 1.093 | 1.302 | 0.303 | 0.236 | 6.120 | 1.362 |

15 | 10.0 | 8.00 | 50 | 55 | 0.5 | 0.300 | 1.093 | 1.302 | 0.303 | 0.236 | 4.69 | 1.270 |

16 | 9.00 | 3.00 | 50 | 55 | 4.5 | 0.333 | 1.093 | 1.302 | 0.355 | 0.276 | 11.62 | 1.624 |

17 | 9.00 | 4.80 | 50 | 55 | 2.7 | 0.333 | 1.093 | 1.302 | 0.355 | 0.276 | 10.44 | 1.693 |

18 | 9.00 | 5.70 | 50 | 55 | 1.8 | 0.333 | 1.093 | 1.302 | 0.355 | 0.276 | 9.26 | 1.662 |

19 | 9.00 | 6.60 | 50 | 55 | 0.9 | 0.333 | 1.093 | 1.302 | 0.355 | 0.276 | 8.03 | 1.625 |

20 | 9.00 | 7.05 | 50 | 55 | 0.5 | 0.333 | 1.093 | 1.302 | 0.355 | 0.276 | 6.51 | 1.507 |

21 | 8.00 | 2.50 | 50 | 55 | 4.0 | 0.375 | 1.093 | 1.302 | 0.423 | 0.330 | 13.13 | 1.954 |

22 | 8.00 | 4.10 | 50 | 55 | 2.4 | 0.375 | 1.093 | 1.302 | 0.423 | 0.330 | 10.73 | 1.854 |

23 | 8.00 | 4.90 | 50 | 55 | 1.6 | 0.375 | 1.093 | 1.302 | 0.423 | 0.330 | 9.76 | 1.832 |

24 | 8.00 | 5.70 | 50 | 55 | 0.8 | 0.375 | 1.093 | 1.302 | 0.423 | 0.330 | 9.23 | 1.866 |

25 | 8.00 | 6.10 | 50 | 55 | 0.4 | 0.375 | 1.093 | 1.302 | 0.423 | 0.330 | 9.08 | 1.897 |

26 | 10.0 | 3.50 | 50 | 55 | 5.0 | 0.300 | 1.469 | 1.198 | 0.303 | 0.236 | 9.12 | 1.262 |

27 | 10.0 | 5.50 | 50 | 55 | 3.0 | 0.300 | 1.469 | 1.198 | 0.303 | 0.236 | 8.13 | 1.363 |

28 | 10.0 | 6.50 | 50 | 55 | 2.0 | 0.300 | 1.469 | 1.198 | 0.303 | 0.236 | 7.14 | 1.364 |

29 | 10.0 | 7.50 | 50 | 55 | 1.0 | 0.300 | 1.469 | 1.198 | 0.303 | 0.236 | 5.36 | 1.286 |

30 | 10.0 | 8.00 | 50 | 55 | 0.5 | 0.300 | 1.469 | 1.198 | 0.303 | 0.236 | 4.09 | 1.209 |

31 | 10.0 | 3.50 | 50 | 55 | 5.0 | 0.300 | 2.575 | 1.274 | 0.303 | 0.236 | 8.78 | 1.228 |

32 | 10.0 | 5.50 | 50 | 55 | 3.0 | 0.300 | 2.575 | 1.274 | 0.303 | 0.236 | 7.83 | 1.333 |

33 | 10.0 | 6.50 | 50 | 55 | 2.0 | 0.300 | 2.575 | 1.274 | 0.303 | 0.236 | 6.69 | 1.320 |

34 | 10.0 | 7.50 | 50 | 55 | 1.0 | 0.300 | 2.575 | 1.274 | 0.303 | 0.236 | 5.09 | 1.259 |

35 | 10.0 | 8.00 | 50 | 55 | 0.5 | 0.300 | 2.575 | 1.274 | 0.303 | 0.236 | 3.81 | 1.181 |

36 | 10.0 | 3.50 | 75 | 55 | 5.0 | 0.300 | 1.093 | 1.302 | 0.303 | 0.236 | 9.77 | 1.327 |

37 | 10.0 | 5.50 | 75 | 55 | 3.0 | 0.300 | 1.093 | 1.302 | 0.303 | 0.236 | 8.81 | 1.431 |

38 | 10.0 | 6.50 | 75 | 55 | 2.0 | 0.300 | 1.093 | 1.302 | 0.303 | 0.236 | 7.51 | 1.401 |

39 | 10.0 | 7.50 | 75 | 55 | 1.0 | 0.300 | 1.093 | 1.302 | 0.303 | 0.236 | 5.45 | 1.295 |

40 | 10.0 | 8.00 | 75 | 55 | 0.5 | 0.300 | 1.093 | 1.302 | 0.303 | 0.236 | 4.37 | 1.237 |

41 | 10.0 | 3.50 | 100 | 55 | 5.0 | 0.300 | 1.093 | 1.302 | 0.303 | 0.236 | 9.46 | 1.296 |

42 | 10.0 | 5.50 | 100 | 55 | 3.0 | 0.300 | 1.093 | 1.302 | 0.303 | 0.236 | 8.53 | 1.403 |

43 | 10.0 | 6.50 | 100 | 55 | 2.0 | 0.300 | 1.093 | 1.302 | 0.303 | 0.236 | 7.19 | 1.369 |

44 | 10.0 | 7.50 | 100 | 55 | 1.0 | 0.300 | 1.093 | 1.302 | 0.303 | 0.236 | 5.57 | 1.307 |

45 | 10.0 | 8.00 | 100 | 55 | 0.5 | 0.300 | 1.093 | 1.302 | 0.303 | 0.236 | 4.23 | 1.223 |

46 | 10.0 | 3.50 | 50 | 52 | 5.0 | 0.300 | 1.093 | 1.302 | 0.303 | 0.236 | 16.00 | 1.950 |

47 | 10.0 | 5.50 | 50 | 52 | 3.0 | 0.300 | 1.093 | 1.302 | 0.303 | 0.236 | 12.98 | 1.848 |

48 | 10.0 | 6.50 | 50 | 52 | 2.0 | 0.300 | 1.093 | 1.302 | 0.303 | 0.236 | 12.58 | 1.908 |

49 | 10.0 | 7.50 | 50 | 52 | 1.0 | 0.300 | 1.093 | 1.302 | 0.303 | 0.236 | 8.57 | 1.607 |

50 | 10.0 | 8.00 | 50 | 52 | 0.5 | 0.300 | 1.093 | 1.302 | 0.303 | 0.236 | 6.58 | 1.458 |

51 | 10.0 | 3.50 | 50 | 44 | 5.0 | 0.300 | 1.093 | 1.302 | 0.303 | 0.236 | 27.12 | 3.062 |

52 | 10.0 | 5.50 | 50 | 44 | 3.0 | 0.300 | 1.093 | 1.302 | 0.303 | 0.236 | 24.60 | 3.010 |

53 | 10.0 | 6.50 | 50 | 44 | 2.0 | 0.300 | 1.093 | 1.302 | 0.303 | 0.236 | 21.49 | 2.799 |

54 | 10.0 | 7.50 | 50 | 44 | 1.0 | 0.300 | 1.093 | 1.302 | 0.303 | 0.236 | 15.98 | 2.348 |

55 | 10.0 | 8.00 | 50 | 44 | 0.5 | 0.300 | 1.093 | 1.302 | 0.303 | 0.236 | 12.47 | 2.047 |

56 | 15.0 | 15.00 | 50 | 55 | 0.0 | 0.200 | 1.093 | 1.302 | 0.165 | 0.128 | 0.41 | 1.028 |

57 | 12.0 | 12.00 | 50 | 55 | 0.0 | 0.250 | 1.093 | 1.302 | 0.230 | 0.179 | 0.51 | 1.043 |

58 | 10.0 | 10.00 | 50 | 55 | 0.0 | 0.300 | 1.093 | 1.302 | 0.303 | 0.236 | 3.30 | 1.330 |

59 | 9.00 | 9.00 | 50 | 55 | 0.0 | 0.333 | 1.093 | 1.302 | 0.355 | 0.276 | 3.34 | 1.371 |

60 | 8.00 | 8.00 | 50 | 55 | 0.0 | 0.375 | 1.093 | 1.302 | 0.423 | 0.330 | 3.74 | 1.467 |

62 | 10.0 | 10.00 | 50 | 55 | 0.0 | 0.300 | 1.469 | 1.198 | 0.303 | 0.236 | 1.81 | 1.861 |

63 | 10.0 | 10.00 | 50 | 55 | 0.0 | 0.300 | 2.575 | 1.274 | 0.303 | 0.236 | 0.73 | 1.181 |

64 | 10.0 | 10.00 | 75 | 55 | 0.0 | 0.300 | 1.093 | 1.302 | 0.303 | 0.236 | 3.14 | 1.073 |

65 | 10.0 | 10.00 | 100 | 55 | 0.0 | 0.300 | 1.093 | 1.302 | 0.303 | 0.236 | 3.00 | 1.314 |

66 | 10.0 | 10.00 | 50 | 52 | 0.0 | 0.300 | 1.093 | 1.302 | 0.303 | 0.236 | 4.62 | 1.300 |

Model/ Statistical Characteristics | Average Error | Minimum Error | Maximum Error | Variance | RMSE |
---|---|---|---|---|---|

Arneson and Abt [9] | 0.522 | −0.241 | 2.477 | 0.157 | 0.650 |

Umbrell et al. [3] | 0.463 | 0.028 | 2.135 | 0.045 | 0.509 |

Lyn [4] | 0.402 | −0.017 | 2.112 | 0.047 | 0.455 |

Guo et al. [11] | 0.357 | −0.225 | 0.980 | 0.120 | 0.494 |

HEC-18 Equation [2] | 0.280 | −0.146 | 2.053 | 0.085 | 0.390 |

Shan et al. [13] | 0.385 | −0.041 | 2.190 | 0.051 | 0.445 |

Melville [16] | 0.544 | 0.001 | 2.488 | 0.088 | 0.618 |

Kumcu [17] | -0.420 | −0.919 | 1.347 | 0.082 | 0.506 |

Kocyigit and Karakurt [15] | 0.430 | 0.166 | 1.773 | 0.015 | 0.447 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

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