# Parametric Study on Abutment Scour under Unsteady Flow

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{50}= 0.52 mm and 0.712 mm. The unsteadiness of the flow is considered in the form flood hydrographs of three forms, namely: advanced flood hydrograph (Type I), symmetrical flood hydrograph (Type II), and delayed flood hydrograph (Type III) with the flow maintained at clear water condition in all cases. The experimental findings are used to represent the influence of various parameters on scour depth at bridge abutments. It was observed that the scour depth at rectangular abutments is greater than that at trapezoidal and semi-circular abutments. The scour depths at abutments embedded in finer sediments are greater than those in coarser sediments. In addition, based on the study of effect of three flood hydrographs, it was noticed that the delayed flood hydrograph yields greater scour depth as compared to the other two cases. Further, based on the method of superposition and the correction of shape factor, a semi-empirical model using dimensionless parameters is proposed to compute the temporal evolution of scour depth at abutments under unsteady clear water conditions. The parameters used in this model include flow shallowness, flow intensity, sediment coarseness, and time factor. It was found that the proposed model corresponds well with the data of time-dependent scour depth in uniform sediments obtained from the present experiments (unsteady flows) and reported by different investigators (steady flows).

## 1. Introduction

## 2. Experimental Setup and Procedure

_{50}) = 0.52 mm and 0.712 mm were used for the experiments. All the experiments were run under clear water scour (V/V

_{c}< 1, where V = average approach flow velocity and V

_{c}= critical velocity) with unsteady flow conditions. Three types of hydrographs having the same base period (t

_{d}= 7 h) and the same peak flow (Q = 0.012 m

^{3}/s) but different shapes, as shown in Figure 2, were considered in this study. Type I is the advanced flood hydrograph having steep rising limb and flat recession limb, while Type III is the reverse of Type I and is called delayed flood hydrograph. However, the Type II hydrograph is a symmetrical one. The peak flow occurs at the 3rd, 4th, and 5th hour for the Type I, Type II, and Type III hydrographs, respectively. Unsteady uniform flow was developed by regulating the inlet orifice and the tailgate simultaneously, and the desired stepwise hydrographs were attained. In addition, the range of flow intensity (V/V

_{c}) for all runs under each step was maintained between 0.7 and 0.85 so that the flow was always at clear water condition (V/V

_{c}< 1). A total of eighteen experiments were conducted and the details are furnished in Table 1. As listed in Table 1, y is the approach flow depth, d

_{sf}

_{,m}and d

_{sf,c}is the measured final scour depth after the hydrograph.

## 3. Influence of Various Parameters on Scour Depth

_{50}= 0.52 mm and 0.712 mm under unsteady clear water condition (V/V

_{c}< 1) with three types of hydrographs are used to plot non-dimensional scour depth d

_{st}/L (where d

_{st}= temporal variation of scour depth and L = transverse length of abutment) versus non-dimensional time t/t

_{m}(where t

_{m}= final time of scour corresponding to peak discharge) as shown in Figure 3. Figure 3a–c illustrate the rising limbs of the three types of flood hydrographs considered (advanced, symmetrical, and delayed). The variation of the non-dimensional scour depth d

_{st}/L with non-dimensional time t/t

_{m}for rectangular, semi-circular, and trapezoidal abutments is shown in Figure 3d–f, 3g–i, and 3j–l, respectively. From Figure 3d–l it can be seen that the non-dimensional scour depth d

_{st}/L increases steeply in the initial stage of the flood hydrograph and then reaches a steady value at the end of the step. With an increase in discharge from preceding step to next step, the non-dimensional scour depth d

_{st}/L again increases steeply and follows the trend of the previous step before the next discharge being introduced. This trend repeats for all the steps of flood hydrographs. However, after the peak discharge is reached, the non-dimensional scour depth d

_{st}/L remains constant in the recession part of the hydrographs (not shown in Figure 3d–l). Further, it is also depicted in Figure 3d–l that the scour depth at abutments embedded in finer sediments (d

_{50}= 0.52 mm) is greater than that in coarser sediments (d

_{50}= 0.712 mm). This is due to the larger erosive capacity of vortexes in finer sediment than those in coarser sediments and lesser resistance of finer sediments. Further, the variation of non-dimensional scour depth d

_{st}/L with non-dimensional time t/t

_{m}at rectangular and trapezoidal abutments is quite separated for different sediment sizes (Figure 3d–f and j–l), while the curves are very close in the case of semi-circular abutments (Figure 3g–i).

_{st}/L with non-dimensional time t/t

_{m}presented in Figure 4d–i are used to study the influence of abutment shape on scour depth. It is observed that the scour depth at rectangular abutments is greater than that at trapezoidal abutments and at semi-circular abutments in both the sediment sizes. This is because at rectangular abutments the flow separation and down flow intensity are much stronger than those at trapezoidal and semi-circular abutments. The shape of an abutment plays an important role on the equilibrium scour depth (Melville and Coleman [11]; Barbhuiya and Dey [13]). The streamline abutments, such as semi-circular and trapezoidal/45° wing-wall abutments, induce weak vortices, whereas blunt bodies, such as rectangular/vertical-wall abutments, generate strong vortices causing deeper scour depth. The three-dimensional vortex structure is stronger at rectangular abutments as compared to trapezoidal and semi-circular abutments causing larger scour depth.

_{st}/L is larger in the case of the Type III hydrograph (delayed) as compared to Type II (symmetrical) and Type I (advanced) hydrographs. This is because the time to reach peak discharge t

_{p}is 5 h in the case of the Type III hydrograph, while it is 4 h for the Type II hydrograph and 3 h for the Type I hydrograph. This indicates that the time to reach peak discharge directly influences scour depth. Further, it is also seen in general that in every step of the flood hydrograph, the Type III hydrograph (delayed) resulted in greater scour depth as compared to Type II (symmetrical) and Type I (advanced). The trend is the same for both sediment sizes of d

_{50}= 0.52 mm (Figure 4d–f) and d

_{50}= 0.712 mm (Figure 4g–i).

## 4. Computational Model for Evolution of Abutment Scour

_{50}) ranging from 26 to 844. Flow intensity (V/V

_{c}) ranges from 0.59 to 0.99, while the flow shallowness ranges (y/L) from 0.07 to 4.0, where y = approach flow depth. Based on the parameter analysis, a regression formula is established to compute the abutment scour depth under steady flow conditions:

_{d}= densimetric particle Froude number = V/(g′d

_{50})

^{0.5}; g′ = the relative gravitational acceleration = [(ρ

_{s}− ρ)/ρ]g; ρ

_{s}= density of sediment particle; ρ = density of fluid; g = gravitational acceleration; T

_{R}= t/t

_{R}= relative time; t = time; t

_{R}= reference time scale = L

_{R}/[(g′d

_{50})

^{0.5}]; L

_{R}= L

^{(2/3)}y

^{(1/3)}and a

_{0}to a

_{4}= regression constants.

- (1)
- For the first flow discharge Q
_{1}of duration t_{1}, scour depth evolution follows the red line (OA curve) under the steady flow condition. The final scour depth at t_{1}is denoted as d_{s}_{1}. - (2)
- When the flow discharge increases from Q
_{1}to Q_{2}, scour depth evolution changes to follow the blue line (AB curve) under the steady flow condition, and point C is the virtual origin for the scouring process. As the scouring process can memorize the previous scour depth and because Q_{2}> Q_{1}, the time (t_{*,1}) required for the scour depth to reach d_{s}_{1}is less than t_{1}. The corresponding scour depth evolution from t_{1}to t_{2}is represented by the AB curve. To solve t_{*,1}, one can use the intersection point A of OA curve and CB curve, and if we let $\frac{{d}_{st1}}{L}={k}_{1}{\left[{\mathrm{log}}_{10}({T}_{R1}=t/{t}_{R1})\right]}^{{a}_{4}}=\frac{{d}_{st2}}{L}={k}_{2}{\left[{\mathrm{log}}_{10}({t}_{\ast ,1}/{t}_{R2})\right]}^{{a}_{4}}$, then t_{*,1}can be obtained as ${t}_{\ast ,1}={t}_{R2}\left[t/{t}_{R1}-{10}^{{({k}_{2}/{k}_{1})}^{(1/{a}_{4})}}\right]$. - (3)
- Similar to the computing procedure mentioned in step 2, when the flow rate increases from Q
_{2}to Q_{3}(>Q_{2}), scour depth evolution follows the green line under the steady flow condition, and point E is the virtual origin for the scouring process. As Q_{3}> Q_{2}, the time (t_{*,2}) required for the scour depth to reach d_{st}_{2}is less than t_{*,1}+ (t_{2}− t_{1}). The corresponding scour depth evolution from t_{2}to t_{3}is shown by the BD curve. Time t_{*,2}can be solved by using the same method as mentioned in step 2, ${t}_{\ast ,2}={t}_{R3}\left[{t}_{\ast ,1}+{t}_{2}-{t}_{1}/{t}_{R2}+{10}^{{({k}_{2}/{k}_{3})}^{(1/{a}_{4})}}\right]$. - (4)
- Repeat the procedure until all of the subdivisions are completed.
- (5)
- Obtain the temporal variation of scour depth under unsteady flow conditions.
- (6)
- Coleman et al. [18] reported that flow shallowness (y/L) has a significant effect on the evolution of abutment scour depth. Hence, for obtaining the best regression result, the data shown in Table 2 were classified into three groups as (1) y/L < 1; (2) 1 ≤ y/L < 2; and (3) 2 ≤ y/L. The coefficients of Equation (1) based on the range of flow shallowness are listed in Table 3. In general, the R
^{2}-values for all ranges of flow shallowness are very good.

## 5. Results of Computational Model

#### 5.1. Comparison Using Steady Flow Data

_{st}/L) with (t/t

_{m}) were plotted along with the corresponding measured non-dimensional abutment scour depths for comparison. In general, the agreement is not so promising for the data of Kwan (1984) as seen in Figure 6a. However, a good agreement was found between the observed and computed abutment scour depths using the proposed model for the data of other investigators (Figure 6b–d). In addition, in the initial period of scouring process (t/t

_{m}< 0.05), the proposed model gives very good agreements with the experimental data. This result is very useful for the calculation of scour depth under unsteady flow conditions. In general, for a certain unsteady flow, each flow discharge would not persist for a long period. If one would like to apply the concept of the superposition method (see earlier section), a good prediction for the initial period of scouring process is very important. Figure 7 compares the final abutment scour depth under clear-water with steady flow conditions between the measured data and the computed values using the proposed model. The dotted lines show the ±25% error boundaries, while the solid diagonal line shows the line of perfect agreement. For the data of Coleman et al. [18], the proposed model underestimates the final scour depth (d

_{sf,m}/L < 1). However, for the data of Yanmaz and Kose [20], the proposed model gives a comparatively good agreement as compared to the other data. In general, the proposed model fits well to the measured final scour depths. Hence, the accuracy of the calculated result obtained by the proposed model is considered to be reasonably good.

_{0,rec}(of rectangular abutment) is adjusted as a

_{0,trap}= 0.85a

_{0,rec}, while for the semi-circular abutment, the coefficient a

_{0,semi}is adjusted as a

_{0,semi}= 0.75a

_{0,rec}. This is similar to the result reported by Melville and Coleman [11] and Dey and Barbhuiya [19]. For the same transverse abutment length (L), approach flow (y, V), and bed sediment (d

_{50}) conditions, the abutment scour depth in sequential order is d

_{s,rec}> d

_{s,trap}> d

_{s,semi}. This result is the same as Melville and Coleman [11]. It is because the abutment scour depth mainly depended on the shape of the abutment. As shown in Figure 8a–d, in general, the comparison shows a good agreement between computed and experimental data. The accuracy of the calculated result is reasonably good. The proposed model for the temporal variation of abutment scour depth under clear water with steady flow conditions thus allows a wide range of flow intensity, flow shallowness, sediment coarseness, and abutment dimensions related to the evolution of abutment scour. In the next section, for the comparison of abutment scour depth under unsteady flows, the coefficients were kept the same as previously mentioned and details will also be discussed.

#### 5.2. Comparison Using Unsteady Flow Data

## 6. Statistical Results

_{i}and p

_{i}are the measured and predicted scour depths, respectively; N is the number of data points. In Equation (4), $\overline{\mathrm{m}}$ and $\overline{\mathrm{p}}$ are the mean values of the measured and predicted scour depths.

## 7. Discussions

## 8. Conclusions

_{50}= 0.52 mm and 0.712 mm were considered in the study. Three types of flood hydrograph were employed to study the unsteadiness of flow: they are advanced (Type I), symmetrical (Type II), and delayed (Type III) flood hydrographs with the clear water flow condition in all cases. The scour depth developed at rectangular abutment was found to be greater than that at trapezoidal and semi-circular abutments. The finer sediment resulted in larger scour depth than the coarser sediment. Further, based on the study of the effect of the three flood hydrographs, it was observed that the delayed flood hydrograph resulted in higher scour depth than the other two hydrographs. Further, a semi-empirical model was proposed for estimating the temporal variation of abutment scour uniform sediments under clear water conditions with steady and unsteady flows. The development of the proposed model uses only the data of rectangular/vertical wall abutment scour under steady flow conditions. After adopting the shape correction factor, the proposed model on time dependent scour depth can be employed to semi-circular and trapezoidal/45° wing-wall abutments. Furthermore, using the concept of superimposition, the proposed model was applied to compute the temporal variation of abutment scour depth at three types of abutments under unsteady flows. The comparisons between the experimental data and the calculated results are reasonably acceptable within ±25% error boundaries. It was demonstrated that the proposed model is accurate, practical, and suitable for computing time dependent scour depth at rectangular/vertical wall abutments and thus may be used in bridge closure for practical purposes. The proposed model does have practical applications in developing warnings at bridge locations. If the flood hydrograph at upstream locations is established from rainfall data, through flood routing the flood hydrograph at bridge sites under consideration can be derived. This will facilitate the prediction of scour depth using the proposed computational model and the design of suitable flood protection systems.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Notations

a_{0}–a_{4} | regression constants [M^{0}L^{0}T^{0}]; |

b | longitudinal length of an abutment [M^{0}L^{1}T^{0}]; |

d_{sf,c} | calculated final scour depth after a hydrograph [M^{0}L^{1}T^{0}]; |

d_{sf,m} | measured final scour depth after a hydrograph [M^{0}L^{1}T^{0}]; |

d_{st} | temporal variation of abutment scour depth [M^{0}L^{1}T^{0}]; |

d_{50} | median grain size [M^{0}L^{1}T^{0}]; |

F_{d} | densimetric particle Froude number, V/(g′d_{50})^{0.5} [M^{0}L^{0}T^{0}]; |

g | gravitational acceleration [M^{0}L^{1}T^{−2}]; |

g′ | relative gravitational acceleration, [(ρ_{s}-ρ)/ρ]g [M^{0}L^{1}T^{−2}]; |

k_{1}–k_{2} | constants [M^{0}L^{0}T^{0}]; |

L | transverse length of an abutment [M^{0}L^{1}T^{0}]; |

L_{R} | length scale, L^{(2/3)}y^{(1/3)} [M^{0}L^{1}T^{0}]; |

m | measured quantity (scour depth) |

$\overline{\mathrm{m}}$ | mean values of measured quantity (scour depth) |

N | number of data points [M^{0}L^{1}T^{0}]; |

p | predicted quantity (scour depth) |

$\overline{\mathrm{p}}$ | mean values of predicted quantity (scour depth) |

Q | flow discharge [M^{0}L^{3}T^{−1}]; |

T | time [M^{0}L^{0}T^{1}]; |

T_{R} | relative time, t/t_{R} [M^{0}L^{0}T^{0}]; |

t_{d} | base period for hydrograph [M^{0}L^{0}T^{−1}]; |

t_{m} | final time of scour experiment corresponding to peak discharge [M^{0}L^{0}T^{1}]; |

t_{R} | reference time scale, L_{R}/[( g′d_{50})^{0.5}], [M^{0}L^{0}T^{1}]; |

V | average approach flow velocity [M^{0}L^{1}T^{−1}]; |

V_{c} | critical velocity [M^{0}L^{1}T^{−1}]; |

y | approach flow depth [M^{0}L^{1}T^{0}]; |

ρ | density of fluid [M^{1}L^{−3}T^{−1}]; and |

ρ_{s} | density of sediment particle [M^{1}L^{−3}T^{−1}]. |

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**Figure 3.**Comparison of temporal variation of non-dimensional scour depth d

_{st}/L for different sediment sizes and hydrographs.

**Figure 4.**Comparison of temporal variation of non-dimensional scour depth d

_{st}/L for different abutment shapes and hydrographs.

**Figure 5.**Method of superposition for computing the abutment scour evolution under a stepwise hydrograph.

**Figure 8.**Comparison of scour evolution between the proposed model and the experimental data of: (

**a**) Ballio and Oris [16] at a vertical wall abutment, (

**b**) Dey and Barbhuiya [19] at a semi-circular abutment, (

**c**) Dey and Barbhuiya [19] at a 45° wing-wall abutment, (

**d**) Tey [14] at a 45° wing-wall abutment.

**Figure 9.**Comparison of computed and measured abutment-scour evolution under Type-I stepwise (advanced) hydrograph. (

**a**) Rectangular abutment (

**b**) Semicircular abutment (

**c**) Trapezoidal abutment.

**Figure 10.**Comparison of computed and measured abutment-scour evolution under Type-II stepwise (symmetrical) hydrograph. (

**a**) Rectangular abutment (

**b**) Semicircular abutment (

**c**) Trapezoidal abutment.

**Figure 11.**Comparison of computed and measured abutment-scour evolution under Type-III stepwise (delayed) hydrograph. (

**a**) Rectangular abutment (

**b**) Semicircular abutment (

**c**) Trapezoidal abutment.

**Figure 12.**Comparison of final scour depth under unsteady flow conditions between the proposed model and the present experimental data.

Hydrograph Type | Abutment Type | d_{50}(mm) | V (m/s) | y (m) | Observed d _{sf,m}(m) |
---|---|---|---|---|---|

Type I | Rectangular | 0.520 | 0.20, 0.19, 0.22 | 0.09, 0.12, 0.18 | 0.0265 |

0.712 | 0.0240 | ||||

Semi-circular | 0.520 | 0.0210 | |||

0.712 | 0.0210 | ||||

Trapezoidal | 0.520 | 0.0250 | |||

0.712 | 0.0220 | ||||

Type II | Rectangular | 0.520 | 0.20, 0.19, 0.21, 0.22 | 0.09, 0.12, 0.15, 0.18 | 0.0295 |

0.712 | 0.0265 | ||||

Semi-circular | 0.520 | 0.0235 | |||

0.712 | 0.0225 | ||||

Trapezoidal | 0.520 | 0.0270 | |||

0.712 | 0.0245 | ||||

Type III | Rectangular | 0.520 | 0.20, 0.22, 0.19, 0.21, 0.22 | 0.09, 0.10, 0.12, 0.15, 0.18 | 0.0295 |

0.712 | 0.0265 | ||||

Semi-circular | 0.520 | 0.0235 | |||

0.712 | 0.0220 | ||||

Trapezoidal | 0.520 | 0.0260 | |||

0.712 | 0.0240 |

**Table 2.**Summary of rectangular/vertical-wall abutment scour under clear water steady flow conditions.

Expt. Number | L (cm) | d_{50}(mm) | V/V_{c} | y (cm) | d_{sf,m}(m) | d_{sf,c}(m) | t_{m}(min) |
---|---|---|---|---|---|---|---|

KW-1 | 71.7 | 0.85 | 0.936 | 5 | 0.231 | 0.183 | 6944 |

KW-2 | 31.4 | 0.85 | 0.936 | 5 | 0.157 | 0.138 | 3095 |

KW-3 | 16.4 | 0.85 | 0.936 | 5 | 0.084 | 0.124 | 4904 |

KW-4 | 16.4 | 0.85 | 0.864 | 10 | 0.171 | 0.135 | 1430 |

CO-1 | 30 | 0.82 | 0.742 | 20 | 0.272 | 0.186 | 4740 |

CO-3 | 30 | 0.82 | 0.742 | 20 | 0.112 | 0.172 | 2277 |

CO-8 | 60 | 0.82 | 0.599 | 12 | 0.140 | 0.112 | 3711 |

CO-14 | 60 | 0.82 | 0.557 | 20 | 0.274 | 0.128 | 4760 |

CO-17 | 60 | 0.82 | 0.742 | 20 | 0.367 | 0.223 | 5571 |

CO-23 | 30 | 0.82 | 0.848 | 20 | 0.266 | 0.239 | 4868 |

CO-25 | 5 | 1.02 | 0.588 | 20 | 0.064 | 0.059 | 3698 |

CO-30 | 5 | 1.02 | 0.731 | 10 | 0.079 | 0.076 | 4096 |

CO-34 | 30 | 0.8 | 0.949 | 10 | 0.270 | 0.209 | 5285 |

CO-37 | 5 | 0.85 | 0.989 | 20 | 0.182 | 0.180 | 5429 |

CO-40 | 5 | 0.85 | 0.980 | 10 | 0.148 | 0.125 | 3713 |

DB-1 | 8 | 0.26 | 0.950 | 20 | 0.127 | 0.136 | 6795 |

DB-2 | 10 | 0.26 | 0.950 | 20 | 0.141 | 0.159 | 4164 |

DB-3 | 10 | 0.52 | 0.950 | 20 | 0.176 | 0.147 | 3542 |

DB-4 | 8 | 0.91 | 0.950 | 20 | 0.170 | 0.136 | 4205 |

DB-5 | 6 | 1.86 | 0.950 | 20 | 0.188 | 0.206 | 3083 |

DB-6 | 8 | 3.1 | 0.950 | 20 | 0.250 | 0.256 | 3976 |

YK-1 | 12.5 | 1.8 | 0.777 | 8.9 | 0.126 | 0.133 | 360 |

YK-2 | 12.5 | 1.8 | 0.777 | 8.3 | 0.123 | 0.128 | 360 |

YK-3 | 12.5 | 1.8 | 0.741 | 7.5 | 0.118 | 0.106 | 360 |

YK-4 | 12.5 | 1.8 | 0.713 | 6.8 | 0.116 | 0.097 | 360 |

YK-5 | 12.5 | 1.8 | 0.682 | 6.1 | 0.105 | 0.084 | 360 |

YK-6 | 12.5 | 1.8 | 0.640 | 5.3 | 0.074 | 0.069 | 360 |

YK-7 | 10 | 1.8 | 0.777 | 8.9 | 0.120 | 0.125 | 360 |

YK-8 | 10 | 1.8 | 0.751 | 8.3 | 0.115 | 0.114 | 360 |

YK-9 | 10 | 1.8 | 0.741 | 7.5 | 0.110 | 0.105 | 360 |

YK-10 | 10 | 1.8 | 0.713 | 6.8 | 0.097 | 0.092 | 360 |

YK-11 | 10 | 1.8 | 0.682 | 6.1 | 0.078 | 0.080 | 360 |

YK-12 | 10 | 1.8 | 0.640 | 5.3 | 0.050 | 0.065 | 360 |

YK-13 | 5 | 1.8 | 0.777 | 8.9 | 0.083 | 0.089 | 360 |

YK-14 | 5 | 1.8 | 0.751 | 8.3 | 0.073 | 0.079 | 360 |

YK-15 | 5 | 1.8 | 0.741 | 7.5 | 0.062 | 0.070 | 360 |

YK-16 | 5 | 1.8 | 0.713 | 6.8 | 0.053 | 0.060 | 360 |

YK-17 | 12.5 | 0.9 | 0.985 | 5.2 | 0.095 | 0.100 | 360 |

YK-18 | 12.5 | 0.9 | 0.899 | 4.4 | 0.066 | 0.077 | 360 |

YK-19 | 10 | 0.9 | 0.985 | 5.2 | 0.089 | 0.094 | 360 |

YK-20 | 10 | 0.9 | 0.899 | 4.4 | 0.063 | 0.073 | 360 |

YK-21 | 5 | 0.9 | 0.985 | 5.2 | 0.062 | 0.059 | 360 |

Coefficients | $\mathit{y}/\mathit{L}\le 1$ | $1<\mathit{y}/\mathit{L}\le 2$ | $2<\mathit{y}/\mathit{L}$ |
---|---|---|---|

${a}_{0}$ | 0.209 | 0.100 | 0.022 |

${a}_{1}$ | 0.263 | 0.424 | 0.700 |

${a}_{2}$ | −0.427 | −0.523 | −0.469 |

${a}_{3}$ | 1.857 | 2.168 | 2.274 |

${a}_{4}$ | 1.269 | 1.594 | 1.938 |

R^{2}-value | 0.892 | 0.916 | 0.935 |

**Table 4.**Comparison of observed and computed (Equation (1)) scour depth at abutment under unsteady flow.

Hydrograph Type | Abutment Type | d_{50}(mm) | Observed d _{sf,m}(m) | Computed d _{sf,c}(m) |
---|---|---|---|---|

Type I | Rectangular | 0.520 | 0.0265 | 0.0312 |

0.712 | 0.0240 | 0.0264 | ||

Semi-circular | 0.520 | 0.0210 | 0.0234 | |

0.712 | 0.0210 | 0.0196 | ||

Trapezoidal | 0.520 | 0.0250 | 0.0265 | |

0.712 | 0.0220 | 0.0223 | ||

Type II | Rectangular | 0.520 | 0.0295 | 0.0332 |

0.712 | 0.0265 | 0.0278 | ||

Semi-circular | 0.520 | 0.0235 | 0.0249 | |

0.712 | 0.0225 | 0.0209 | ||

Trapezoidal | 0.520 | 0.0270 | 0.0282 | |

0.712 | 0.0245 | 0.0237 | ||

Type III | Rectangular | 0.520 | 0.0295 | 0.0339 |

0.712 | 0.0265 | 0.0281 | ||

Semi-circular | 0.520 | 0.0235 | 0.0255 | |

0.712 | 0.0220 | 0.0213 | ||

Trapezoidal | 0.520 | 0.0260 | 0.0289 | |

0.712 | 0.0240 | 0.0241 |

Hydrograph Type | Abutment Shape | d_{50}(mm) | RMSE (m) | MAPE (%) | Correlation Coefficient R |
---|---|---|---|---|---|

Type-I | Rectangular | 0.520 | 0.00289 | 24.79 | 0.95 |

0.712 | 0.00293 | 31.63 | 0.94 | ||

Semi-circular | 0.520 | 0.00160 | 14.53 | 0.98 | |

0.712 | 0.00145 | 13.09 | 0.98 | ||

Trapezoidal | 0.520 | 0.00181 | 14.94 | 0.97 | |

0.712 | 0.00171 | 16.99 | 0.97 | ||

Type-II | Rectangular | 0.520 | 0.00272 | 22.41 | 0.98 |

0.712 | 0.00197 | 17.95 | 0.97 | ||

Semi-circular | 0.520 | 0.00126 | 10.62 | 0.99 | |

0.712 | 0.00189 | 15.36 | 0.97 | ||

Trapezoidal | 0.520 | 0.00158 | 13.98 | 0.99 | |

0.712 | 0.00220 | 20.63 | 0.97 | ||

Type-III | Rectangular | 0.520 | 0.00265 | 19.43 | 0.98 |

0.712 | 0.00200 | 15.33 | 0.98 | ||

Semi-circular | 0.520 | 0.00143 | 11.04 | 0.98 | |

0.712 | 0.00120 | 7.46 | 0.98 | ||

Trapezoidal | 0.520 | 0.00168 | 12.14 | 0.97 | |

0.712 | 0.00180 | 16.13 | 0.98 |

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**MDPI and ACS Style**

Raikar, R.V.; Hong, J.-H.; Deshmukh, A.R.; Guo, W.-D.
Parametric Study on Abutment Scour under Unsteady Flow. *Water* **2022**, *14*, 1820.
https://doi.org/10.3390/w14111820

**AMA Style**

Raikar RV, Hong J-H, Deshmukh AR, Guo W-D.
Parametric Study on Abutment Scour under Unsteady Flow. *Water*. 2022; 14(11):1820.
https://doi.org/10.3390/w14111820

**Chicago/Turabian Style**

Raikar, Rajkumar V., Jian-Hao Hong, Anandrao R. Deshmukh, and Wen-Dar Guo.
2022. "Parametric Study on Abutment Scour under Unsteady Flow" *Water* 14, no. 11: 1820.
https://doi.org/10.3390/w14111820