## Abstract

Scouring of sediment materials under submerged vertical jets is shown at grade-control structures, downstream of weirs, and submerged vertical water jets have several uses, including seabed sediment removal and dredging in ocean engineering. Numerous studies have been conducted to demonstrate the scour process and the variation of scour profile utilizing various influencing parameters. Many researchers developed equations for predicting the scour hole characteristics. Previous studies identified two types of scour hole depth configurations: static and dynamic scour depth and the variation of scour hole variations differs for long- and short-impinging jet height conditions. In this study, extensive data on non-dimensional static and dynamic scour depth under short- and long-impinging jets acquired from prior literature, and the previously proposed equations of static and dynamic scour depth were analyzed using graphical and statistical analysis. The findings demonstrated that the relationships proposed by Aderibigbe and Rajaratnam (1996) for long-impinging jets and Amin *et al.* (2021) for short-impinging jets predict the static scour depth better than the other equations. The proposed equations for dynamic scour depth under long- and short-impinging jets are highly biased and inaccurate.

## HIGHLIGHTS

Formation of scour due to vertical jets was classified based on the impinging jet height.

Accuracy of existing equations for estimating the scour depth under various jet heights was evaluated.

Results of the analysis suggest that the equation by Amin

*et al.*(2021) is skillful in predicting the static scour depth.The performance of existing equations to predict the dynamic scour depth were comparatively less accurate.

### Graphical Abstract

## NOTATIONS

the distance from the nozzle to the upper layer of the bed sediment (L)

diameter of nozzle (L)

maximum static scour depth (L)

densimetric Froude number (–)

erosion parameter (–)

mean velocity of the jet (LT

^{−1})acceleration due to gravity (LT

^{−2})sediment particle size (L)

density of fluid (mL

^{−3})the difference between the densities of sediment and fluid (mL

^{−3})coefficient (–)

critical velocity (LT

^{−1})scour characteristics (L)

maximum dynamic scour depth (L)

coefficient (–)

*D*_{60}diameter of sediment material, 60% of particles finer by weight (L)

*D*_{10}diameter of sediment material, 10% of particles finer by weight (L)

observed non-dimensional scour depth (–)

computed non-dimensional scour depth (–)

*n*number of observations (–)

standard deviation of observed scour depth data (–)

standard deviation of computed scour depth data (–)

*CC*correlation coefficient (–)

## INTRODUCTION

*et al.*2021). However, the protection measures are sometimes too expensive to implement. As a result, estimating the scour hole characteristics produced by a submerged water jet is necessary for the safe and cost-effective construction of hydraulic structures (Ansari 1999; Kashtiban

*et al.*2021). In general, the foundation of most of the hydraulic structures is located at a deeper depth below the river bed (Jain & Kothyari 2009). However, a realistic approximation of scour hole features during the design stage will aid the designers in choosing the foundation depth. Scour by submerged water jets occurs at storm drainage pipes, downstream of weirs, and energy dissipator structures (Mazurek

*et al.*2003; Aamir

*et al.*2022). Scouring of the sediment bed occurs when the jet erosive capacity exceeds the critical bed shear stress for the instigation of sediment motion. The determination of scour features is primarily empirical due to the flow concentration conditions in a high-velocity jet and the interaction of flow with the sediment bed (Mazurek & Hossain 2007; Ahmad

*et al.*2015; Pandey

*et al.*2022). Consequently, parameters such as sediment properties, jet velocity, nozzle diameter, and jet height highly influence the estimates of scour features. Consequently, parameters such as sediment properties, jet velocity, nozzle diameter, and jet height highly influence the estimates of scour features. Consequently, parameters such as sediment properties, jet velocity, nozzle diameter, and jet height highly influence the estimates of scour features. Figure 1 depicts the scouring process under submerged circular vertical jets.

*t*is the time. The Buckingham π-theorem is applied to Equation (1) by selecting,, and as recurring variables, resulting in a functional relationship in terms of the dimensionless groups shown in Equation (2):

*t*), the scour depth varied quickly, but when

*t*was sufficiently substantial, it approximately remained constant (Shankar

*et al.*2021). Equation (2) can be expressed as follows after the aforementioned factors have been removed from it:or

Several studies were conducted to investigate the variation of the scour hole profile, such as scour depth, volume, and height of the ridge produced by submerged vertical water jets with varied initial characteristics. Rouse (1940) conducted pioneering research on scour induced by a submerged water jet in cohesionless sediment. Several studies were later conducted to investigate the response of sediment materials to various submerged water jets. Clarke (1962) identified two configurations: static and dynamic scour depths. When the jet is in running condition (dynamic scour depth), the scour depth is high because a considerable volume of sediment particles are retained and suspended in the flow. When the jet flow is shut down (static scour depth), all suspended sediment particles settle in the scour hole, resulting in a reduced scour depth. Based on the ratio of jet height and nozzle diameter Rajaratnam & Beltaos (1977) categorized the jet condition as short-impinging jet height , long-impinging height , and transition region . Rajaratnam (1982) proposed an empirical equation for estimating the maximum depth of scour hole (), which is a function of the densimetric Froude number (*F*_{0}), the diameter of the nozzle (), and jet impingement height (). Aderibigbe & Rajaratnam (1996) demonstrated that the difference between static and dynamic scour depths is greater for long-impinging jet heights. They introduced a non-dimensional parameter known as the erosion parameter (*E _{c}*), which is defined as . They classify the flow regime into two categories based on : strong deflected jet regime (SDJR) and weakly deflected jet regime (WDJR). They also proposed an equation in terms of the

*E*to predict the scour hole characteristics and similarity of the scour hole for the long-impinging jet . Previous researchers have shown that the scour hole is mostly determined by the

_{c}*E*or the

_{c}*F*. O'Donoghue

_{0}*et al.*(2001) showed that scour is also strongly dependent on the ratio of sediment size (

*D*) to jet diameter (

_{50}*d*).

_{0}Ansari *et al.* (2003) investigated the applicability of the previously described non-dimensional variables , , and in establishing the relations for the estimation of scour characteristics. The results revealed that variable represented the maximum scour depth better than or . Chakravarti *et al.* (2014) analyzed the maximum static scour depth data collected from Aderibigbe & Rajaratnam (1996), Ansari *et al.* (2003), Rajaratnam (1982), Sarma & Sivasankar (1967), and Westrich & Kobus (1973) along with their experimental data. They analyzed the transferability of the established relation between non-dimensional variables and the static scour depth for long- and short-impinging jet conditions. The findings revealed that the performance of the equations proposed by Aderibigbe & Rajaratnam (1996) and Ansari *et al.* (2003) in estimating the scour depth was satisfactory. Amin *et al.* (2021) conducted a laboratory study with short-impinging jets, and this study compares the short-impinging jet outcomes to the long-impinging jet findings based on the previous experimental data. The results demonstrated that the change of non-dimensional scour depth with *E _{c}* is logarithmic for long-impinging jets and linear for short-impinging jets. Kartal & Emiroglu (2021) evaluated downstream scour caused by a nozzle jet with or without plates. Under identical conditions, the results showed that jets with plates generated shallower scour than jets without plates. Shakya

*et al.*(2022) used artificial neural networks (ANNs) and multiple non-linear regression (MNLR) to attempt to forecast scour depth. The findings revealed that the accuracy of the ANN (

*R*

^{2}= 0.978) is more than that of multi-non-linear regression (MNLR) (

*R*

^{2}= 0.945). Using an optical method, Chen

*et al.*(2022) calculated the dynamic scour depth and its fluctuation with respect to time. The results revealed that the scour hole profile in the jet central area is mostly governed by jet momentum flux. When all other factors are held constant, the critical distance increases as the jet exit velocity increases.

Numerous studies have been conducted on the estimation of scour depths, including static and dynamic scour depths (Si *et al.* 2018, 2019). There are many formulas for predicting scour depths for short- and long-impinging jet conditions. Finding the best approach among them is extremely difficult; however, different solutions to the same issue can be found using a variety of established formulae. In general, a large number of empirical formulae were created decades ago on the basis of a small number of field and laboratory data. No study has been reported so far on the performance assessment of existing equations for the prediction of static and dynamic scour depths under short- and long-impinging jet height conditions. Hence, this study aims to evaluate the accuracy of six existing equations of static scour depth and five existing equations of dynamic scour depth for long- and short-impinging jets utilizing prior data acquired from the literature, i.e., 114 datasets of static scour depth and 115 datasets of dynamic scour depth.

## EXISTING EQUATIONS OF STATIC SCOUR DEPTH

Numerous computational and laboratory studies have been carried out to quantify the depth of scour hole produced by circular, submerged vertical jets. The majority of the investigations relied on laboratory experimentation. Numerous equations were proposed by different researchers to determine the scour depth induced by circular submerged vertical jets under various impinging jet conditions, flow parameters, and sediment bed materials (Rajaratnam 1981; Breusers & Raudkivi 1991; Aderibigbe & Rajaratnam 1996; Ansari *et al.* 2003; Chakravarti *et al.* 2014; Amin *et al.* 2021; Chen *et al.* 2022; Shakya *et al.* 2022). In the present study, six of the previously developed static scour depth equations by Aderibigbe & Rajaratnam (1996), Ansari *et al.* (2003), Chakravarti *et al.* (2014), Amin *et al.* (2021), Shakya *et al.* (2022), Chen *et al.* (2022) are chosen for verifying the reliability in the scour prediction. A brief description of these equations is given below.

*et al.*(2003). The equation for static scour depth in cohesionless sediment is provided in Equation (6).

*et al.*(2014) conducted an experiment on cohesionless soils such as sand and gravel. They distinguished the accuracy of the

*E*in estimating the scour depth. They modified Aderibigbe & Rajaratnam (1996) equation as given in Equation (7):

_{c}*et al.*(2021) carried out laboratory experiments on scour profiles generated by a submerged vertical short-impinging jet . They developed equations (Equations (8) and (9)) for determining the static scour depth for short- and long-impinging jet heights.

*et al.*(2022) suggested an equation for predicting the static scour depth under submerged vertical jets using MNLR. The suggested equation is mentioned in Equation (10).

*et al.*(2022) modified the equation (Equation (11)) proposed by Aderibigbe & Rajaratnam (1996) to calculate the non-dimensional static scour depth by using their experimental data and the experimental data obtained from previous studies.

## EXISTING EQUATIONS OF DYNAMIC SCOUR DEPTH

Dynamic scour depth is an important factor for the safe design and stability of hydraulic structures. There has been less research done on estimating the dynamic scour depth under submerged circular vertical jets when compared to static scour depth, as it is difficult to carry out the experiment and obtain scour depth data while the jet is flowing. Few researchers have conducted experiments and developed equations for calculating the dynamic depth of scour holes in circular, submerged vertical jets. This paper evaluates the efficacy of five of the existing dynamic scour depth equations, which are briefly described below.

*et al.*(2014) provided an equation for predicting the dynamic scour depth in sand and gravel beds by submerged circular vertical jets. The suggested equation is given in Equation (13).

*et al.*(2021) proposed an equation for predicting the dynamic scour depth for short-impinging jets based on Rajaratnam & Beltao's (1977) analysis. The proposed equation is mentioned in Equation (14).

*et al.*(2022) have performed experimental analysis to estimate the dynamic scour depth and its variation with respect to time using the optical method. They proposed an equation (Equation (16)) using theoretical and experimental analysis for estimating the dynamic scour depth.

## DATA DESCRIPTION

115 experimental data for dynamic scour depth and 114 experimental data for static scour depth were acquired from previous research (Rajaratnam 1982; Aderibigbe & Rajaratnam 1996; Ansari 1999; Ansari *et al.* 2003; Chakravarti *et al.* 2014; Amin *et al.* 2021; Kartal & Emiroglu 2021; Chen *et al.* 2022; Shakya *et al.* 2022). Previous studies have shown that for long-impinging jets, scour depth varies logarithmically with non-dimensional parameters, whereas it varies linearly for short-impinging jets (Aderibigbe & Rajaratnam 1996; Beltaos & Rajaratnam 1977; Amin *et al.* 2021). We divided the data into two parts based on the impinging height, i.e., short-impinging jet and long-impinging jet , respectively. The range of the initial parameters used in the previous studies to predict static and dynamic scour depths is shown in Table 1.

S. no. . | Authors . | Nozzle dia. (mm) . | Jet height (cm) . | Jet velocity (m/s) . | Median size D (mm)
. _{50} |
---|---|---|---|---|---|

1 | Aderibigbe & Rajaratnam (1996) | 4–12 | 4–52.3 | 2.65–4.45 | 0.88–2.42 |

2 | Ansari et al. (2003) | 8–12.5 | 15–30 | 1.3–5.75 | 0.27 |

3 | Chakravarti et al. (2014) | 8–12.5 | 15–30 | 5.12–9.84 | 2.8 |

4 | Amin et al. (2021) | 12.5 | 0–50 | 2.12–2.61 | 0.54–1.10 |

5 | Kartal & Emiroglu (2021) | 10–20 | 2–24 | 10–23 | 1.28 |

6 | Shakya et al. (2022) | 16.2 | 5–32 | 1.57–3.94 | 0.84–3.38 |

7 | Chen et al. (2022) | 8–12 | 5–45 | 1.105–7.184 | 2.54 |

S. no. . | Authors . | Nozzle dia. (mm) . | Jet height (cm) . | Jet velocity (m/s) . | Median size D (mm)
. _{50} |
---|---|---|---|---|---|

1 | Aderibigbe & Rajaratnam (1996) | 4–12 | 4–52.3 | 2.65–4.45 | 0.88–2.42 |

2 | Ansari et al. (2003) | 8–12.5 | 15–30 | 1.3–5.75 | 0.27 |

3 | Chakravarti et al. (2014) | 8–12.5 | 15–30 | 5.12–9.84 | 2.8 |

4 | Amin et al. (2021) | 12.5 | 0–50 | 2.12–2.61 | 0.54–1.10 |

5 | Kartal & Emiroglu (2021) | 10–20 | 2–24 | 10–23 | 1.28 |

6 | Shakya et al. (2022) | 16.2 | 5–32 | 1.57–3.94 | 0.84–3.38 |

7 | Chen et al. (2022) | 8–12 | 5–45 | 1.105–7.184 | 2.54 |

Aderibigbe & Rajaratnam (1996) carried out all experiments in an octagonal box with dimensions of 0.235 m length and 0.6 m height. The impinging jet nozzle with diameters of 0.4, 0.8, 1.2, and 1.9 cm was mounted to the bottom of a 150 mm diameter cylinder, and the impinging distance *h* is adjusted by holding the nozzle position constant and varying the thickness of the sediment bed. The *h* value is varied between 4 and 523 mm, while the exit velocity from the nozzle is varied from 2.6 to 4.4 m/s. The median bed particle size (*D*_{50}) of 0.88 and 2.42 mm is used in nearly uniform sand beds. The scour depths were roughly determined by placing a slender rod into the scour hole until it met the bottom.

Ansari *et al.* (2003) used a circular tank with a diameter and depth of the tank is 125 cm, that was filled with sediment having a median size (*D*_{50}) of 0.27 mm and the impinging height was varied between 0.15 and 0.30 m with varying jet velocities ranging from 1.3 to 5.75 m/s. The impinging jet is produced by a nozzle having diameters of 8 and 12.5 mm attached at the bottom of the pipe with a diameter of 0.0254 m. By using intermittent jet circumstances, the temporal fluctuation of the scour profile was observed.

Chakravarti *et al.* (2014) conducted experimental studies to find out the scour in the sand having a median size (*D*_{50}) of 2.8 mm and a geometrical standard deviation equal to 1.21. All experiments were carried out in a circular tank having a diameter and height is 125 cm filled with desired sand up to a height of 80 cm. It was ensured that the diameter of the tank was sufficiently large so that it would not influence on scour process. Experimental runs were conducted using two types of nozzle diameters, i.e., 8 and 12.5 mm. In the case of 8 mm nozzle, the velocity of the jet varies between 6.65 and 9.84 m/s and in the case of 12.5 mm nozzle, the velocities of 7.19 and 5.12 m/s were used.

Amin *et al.* (2021) conducted laboratory experiments on the scouring of cohesionless sediment beds due to submerged vertical jets with a short-impinging height. All experimental runs were carried out in an octagonal tank measuring 578 mm in width and 610 mm in depth. A well-designed nozzle with a 1.25 mm diameter was fitted to the end of a 12 cm diameter pipe, with the velocities at the nozzle *u*_{0} being 2.12 and 2.61 m/s. The experiments were carried out for two mean sizes of *D*_{50} (0.54 and 1.10 mm), and three impinging heights (0, 2*d*_{0,} and 4*d*_{0}), to ensure short-impinging jets. The pipe was installed within the jet plenum, high above the jet nozzle, so that it did not interfere with the flow. The position of the depth rod was recorded and the dynamic depth is calculated as the difference between this measurement and the original sediment bed surface measurement.

Kartal & Emiroglu (2021) have conducted experimental studies on the scouring of sediment beds by a nozzle with and without plates. All the experiments were carried out in rectangular flumes. For all experiments, uniform quartz sand having *D*_{50} = 1.28 mm *D*_{60} = 1.47 mm and *D*_{10} = 0.84 mm with a geometric standard deviation of 1.33 is used. The nozzle diameters of 10, 15, and 20 mm were used by the impinging heights from 20 to 240 mm, and jet exit velocities from 10 to 23 m/s. These experimental runs were carried out with the constraints of 5 ≤ *h _{j}/d*

_{0}≤ 20 and 69.66 ≤

*F*

_{0}≤ 159.22.

Shakya *et al.* (2022) compared the experimental scour depth data with the predictions from the ANN model. All experiments were conducted in a rectangular steel tank with dimensions of 1.24 m × 1.24 m × 0.84 m. The nozzle had a diameter of 1.62 cm and was impinging at 90° to the sediment bed surface. The sediment size of 0.84 mm with a specific gravity of 2.65 was used. Experiments were carried out with four distinct jet velocities: 2.23, 2.87, 3.94, and 1.57 m/s as well as four different impinging heights: 50, 90, 180, and 320 mm. The maximum scour depth was measured from the start of the experiment at different time intervals.

Chen *et al.* (2022) determined to scour characteristics using nozzle diameters of 8, 10, and 12 mm, exit velocities ranging from 1.105 to 7.184 m/s, and impinging lengths ranging from 50 to 450 mm. All trials were conducted in a rectangular flume 5.8 m long, 0.8 m wide, and 0.65 m deep. The optical approach was used to capture the scour hole profiles in order to provide dynamic scour hole features and their modification over time.

Chen *et al.* (2022) used nozzle diameters of 8, 10, and 12 mm with varying exit velocities from 1.105 to 7.184 m/s and impinging distances from 50 to 450 mm for the determination of scour characteristics. All the experiments were carried out in a rectangular flume with dimensions of 580 cm long, 80 cm wide, and 65 cm deep. The optical approach was used to capture scour hole profiles in order to depict dynamic scour hole features and their fluctuation with time.

## STATISTICAL INDICES

^{2}) explains the fraction of total variance in observed data sets and it ranges from 0 to 1:

*X*is the observed non-dimensional scour depth,

_{i}*Y*is the corresponding computed non-dimensional scour depth, , and are the averages of the observed and computed non-dimensional scour depth,

_{i}*n*is the number of observations, and and are the standard deviation values of the observed and computed data. The ranges of statistical indices are listed below in Table 2.

Scores . | R^{2}
. | PBIAS (%) . | NSE . |
---|---|---|---|

Unsatisfactory | R^{2} < 0.5 | |PBIAS| > 25 | <0.5 |

Satisfactory | 0.5 < R^{2} < 0.65 | 15 < |PBIAS| < 25 | 0.5–0.65 |

Good | 0.65 < R^{2} < 0.75 | 10 < |PBIAS| < 15 | 0.65–0.75 |

Very good | 0.75 < R^{2} < 1 | |PBIAS| < 10 | >0.75 |

Scores . | R^{2}
. | PBIAS (%) . | NSE . |
---|---|---|---|

Unsatisfactory | R^{2} < 0.5 | |PBIAS| > 25 | <0.5 |

Satisfactory | 0.5 < R^{2} < 0.65 | 15 < |PBIAS| < 25 | 0.5–0.65 |

Good | 0.65 < R^{2} < 0.75 | 10 < |PBIAS| < 15 | 0.65–0.75 |

Very good | 0.75 < R^{2} < 1 | |PBIAS| < 10 | >0.75 |

Scour characteristic . | Jet condition . | X
. | . | . | AS . | RE . | RS . |
---|---|---|---|---|---|---|---|

Static | Long | 2.12 | −0.029 | −0.0138 | −0.12 | −1.24 | |

0.055 | −0.031 | −5.69 | −0.13 | −1.33 | |||

Short | 1.924 | −0.092 | −0.0481 | −0.081 | −0.081 | ||

0.035 | −0.092 | −2.629 | −0.081 | −0.081 | |||

Dynamic | Long | 6.574 | 1.447 | 0.22 | 1.29 | 12.93 | |

0.0074 | 1.1686 | 191.19 | 1.26 | 12.68 | |||

Short | 3.748 | 2.493 | 0.66 | 0.698 | 6.985 | ||

0.038 | 2.106 | 65.10 | 0.693 | 6.93 |

Scour characteristic . | Jet condition . | X
. | . | . | AS . | RE . | RS . |
---|---|---|---|---|---|---|---|

Static | Long | 2.12 | −0.029 | −0.0138 | −0.12 | −1.24 | |

0.055 | −0.031 | −5.69 | −0.13 | −1.33 | |||

Short | 1.924 | −0.092 | −0.0481 | −0.081 | −0.081 | ||

0.035 | −0.092 | −2.629 | −0.081 | −0.081 | |||

Dynamic | Long | 6.574 | 1.447 | 0.22 | 1.29 | 12.93 | |

0.0074 | 1.1686 | 191.19 | 1.26 | 12.68 | |||

Short | 3.748 | 2.493 | 0.66 | 0.698 | 6.985 | ||

0.038 | 2.106 | 65.10 | 0.693 | 6.93 |

Scour characteristic . | Jet condition . | X
. | . | . | AS . | RE . | RS . |
---|---|---|---|---|---|---|---|

Static | Long | 2.12 | 0.0038 | 0.0018 | 0.016 | 0.163 | |

0.055 | 0.008 | 1.4551 | 0.034 | 0.341 | |||

Short | 1.924 | 0.0971 | 0.0505 | 0.085 | 0.858 | ||

0.035 | 0.097 | 2.758 | 0.085 | 0.858 | |||

Dynamic | Long | 6.574 | 1.42 | 0.177 | 1.044 | 10.447 | |

0.0074 | 1.214 | 163.6 | 1.085 | 10.857 | |||

Short | 3.748 | 2.476 | 0.561 | 0.590 | 5.90 | ||

0.0380 | 2.131 | 56.04 | 0.597 | 5.970 |

Scour characteristic . | Jet condition . | X
. | . | . | AS . | RE . | RS . |
---|---|---|---|---|---|---|---|

Static | Long | 2.12 | 0.0038 | 0.0018 | 0.016 | 0.163 | |

0.055 | 0.008 | 1.4551 | 0.034 | 0.341 | |||

Short | 1.924 | 0.0971 | 0.0505 | 0.085 | 0.858 | ||

0.035 | 0.097 | 2.758 | 0.085 | 0.858 | |||

Dynamic | Long | 6.574 | 1.42 | 0.177 | 1.044 | 10.447 | |

0.0074 | 1.214 | 163.6 | 1.085 | 10.857 | |||

Short | 3.748 | 2.476 | 0.561 | 0.590 | 5.90 | ||

0.0380 | 2.131 | 56.04 | 0.597 | 5.970 |

## RESULTS

In this study, the accuracy of six equations for predicting the static scour depth and five equations for predicting the dynamic scour depth, was assessed. The experimental data obtained from the previous studies were divided into two parts based on impinging jet height, such as long-impinging jet (84 data sets of static scour depth and 92 data sets of dynamic scour depth) and short-impinging jet (30 data sets of static scour depth and 23 data sets of dynamic scour depth). These include relationships proposed by Aderibigbe & Rajaratnam (1996), Ansari *et al.* (2003), Chakravarti *et al.* (2014), Amin *et al.* (2021), Kartal & Emiroglu (2021), Shakya *et al.* (2022) and Chen *et al.* (2022). The accuracy of the equations was analyzed graphically and statistically.

### Comparative analysis

*et al.*1969)) was used as an error metric to quantify the difference between predicted and measured scour depths. It is defined as in the following equation:when

*DR*= 0, the anticipated equals the observed value. When

*DR*is positive, the projected value of the dispersion coefficient is larger than the observed value. The frequency of occurrences in which the

*DR*is within a reasonable range for the total quantity of data is defined as accuracy. Figures 2 and 3 illustrate the discrepancy ratios for each expression in the static and dynamic laboratory datasets.

Figures 2 and 3 depict the discrepancy ratios for each empirical equation for the 115 laboratory datasets of static scour depth and 114 datasets of dynamic scour depth. The discrepancy ratios for each equation for the 115 laboratory datasets of static scour depth and the 114 datasets of dynamic scour depth are shown in figures. It can be observed from Figure 2 that the frequency of data within is 68, 57 for Aderibigbe & Rajaratnam (1996), Amin *et al.* (2021) for static scour depth under the long-impinging jet and 26, 20 for Shakya *et al.* (2022) and Amin *et al.* (2021) for static scour depth under short-impinging jet height conditions. Chen *et al.* (2022) and Chakravarti *et al.* (2014) demonstrate the same range of DR () is observed in Figure 3. This process explicitly illustrates that the relationship of Aderibigbe & Rajaratnam (1996) for static scour depth under a long-impinging jet and for a short-impinging jet described by Amin *et al.* (2021) has the highest precision. For dynamic scour depth under a long-impinging jet, Chen *et al.* (2022) and Kartal & Emiroglu (2021) equations give good accuracy.

### Sensitivity analysis

A close examination of Table 1 shows that the *E _{C}*, which was formulated by the multiplication of and is the most likely dimensional less parameter influencing the dimensionless static and dynamic scour depth. The effect of the aforementioned non-dimensional initial parameters has been examined using sensitivity and error analysis. This is accomplished by using the average values of the dimensionless input and output parameters. The premise underlying the study is that each input variable's error is distinct. If an output error is stated as the difference between the values of the output predicted for the inputs

*X*and

*X*+ , then the error can be determined as the absolute sensitivity. Here, the input and the output are and . The error could also be expressed as a relative error. The output error is the deviation sensitivity, where represents the error that was made. The relative sensitivity is denoted by the expression (Pandey

*et al.*2017).

To perform the sensitivity and error analysis, each input value is changed by an increase of 10% in *X*. Table 3 and 4 illustrate the study's findings, which suggest that both , and are the critical factors. For long-impinging jets, is slightly more sensitive than the , whereas, for short-impinging jets, both input variables have the same effect at a 10% increase in *X*. However, the relative sensitivity is about 2.09 times that of for a 10% reduction in *X*. The prediction accuracy of the equations depends on both input parameters.

### Performances of existing equations of static scour depth

#### Long-impinging jet

*R*

^{2}), and Percentage Bias (PBIAS). The predictions obtained from the equation by Aderibigbe & Rajaratnam (1996) (Figure 4(a)) and Chen

*et al.*(2022) (Figure 4(f)) shows a good agreement (

*R*

^{2}= 0.748) between the experimental and computed data. It can be noticed that about 80% of the predicted values from these two equations were within the limits of ±20% error lines. It is also worth noting that the predictions of Chen

*et al.*(2022) slightly underestimate (PBIAS = −11.5%) the non-dimensional static scour depth when compared to predictions obtained from Aderibigbe & Rajaratnam (1996). The accuracy of predictions from the equations proposed by Ansari

*et al.*(2003) (Figure 4(b)), Chakravarti

*et al.*(2014) (Figure 4(c)), and Amin

*et al.*(2021) (Figure 4(d)) is found to be satisfactory with

*R*

^{2}value greater than 0.74. However, these equations are found to be highly overestimating the non-dimensional static scour depth values with PBIAS greater than 16%. The proposed equation by Shakya

*et al.*(2022) (Figure 4(e)) was found to be the least accurate among the six equations with a lower

*R*

^{2}value of 0.702 and an underestimation bias (PBIAS = −16.6%).

#### Short-impinging jet

*et al.*(2021) and Shakya

*et al.*(2022) are in good agreement (

*R*

^{2}= 0.959 and 0.949) with the observed data. However, the predictions using the equation by Shakya

*et al.*(2022) slightly overestimate (PBIAS = 11.1%) the observed data when compared to Amin

*et al.*(2021) predictions. The accuracy of predictions from the equations developed by Ansari

*et al.*(2003) (Figure 5(b)), and Chakravarti

*et al.*(2014) (Figure 5(c)) highly underestimates (PBIAS = −40 and −35%) the non-dimensional static scour depth for short-impinging jets. The predictions obtained from the equations by Aderibigbe & Rajaratnam (1996) (Figure 5(a)) and Chen

*et al.*(2022) (Figure 5(f)) highly underestimate the non-dimensional static scour depth for short-impinging jets due to the fact that they are only applicable for long-impinging jet conditions.

The results of the analysis suggest that the predictions of the proposed equations (Equations (5) and (11) by Aderibigbe & Rajaratnam (1996) and Chen *et al.* (2022) for long-impinging jets, as well as the Equation (10)) proposed by Amin *et al.* (2021) and Shakya *et al.* (2022) for short-impinging jets, give good agreement between observed and computed non-dimensional static scour depth data.

#### Existing equations of dynamic scour depth

*et al.*(2022) (Figure 6(e)) was found to be slightly satisfactory with

*R*

^{2}value of 0.644 and PBIAS value of 19.7%. The equations proposed by Chakravarti

*et al.*(2014) and Amin

*et al.*(2021) highly overestimate the observed data as shown in Figure 6(b) and 6(c), respectively. Aderibigbe & Rajaratnam (1996) obtained the expression for determining the dynamic scour depth similar to that of the maximum static scour depth (Equation (5)) by adding a correction term to

*E*. This is to account for the quicker decay rate of the descending jet's velocity owing to the ascending flow, which renders the velocity decay equation (Equation (12)) inapplicable, especially for large

_{c}*E*values. Hence, it can be noticed from Figure 6(a) that the computed data are highly scattered and very few estimations are within the ±20% error line showing the proposed equation does not fit to estimate the non-dimensional dynamic scour depth under long-impinging jets. The equation by Kartal & Emiroglu (2021) was also found to be underestimating the data with a PBIAS value of −45.2%.

_{c}*et al.*(2014) (Figure 7(b)) is highly linearly correlated with the observed values (with

*R*

^{2}> 0.84), they are highly overestimating the observed values with PBIAS greater than 68.8%. Conversely, the performance of prediction by the equation proposed by Amin

*et al.*(2021) (Figure 7(c)), Kartal & Emiroglu (2021) (Figure 7(d)) and Chen

*et al.*(2022) (Figure 7(e)) is slightly better in terms of PBIAS when compared other two equations. However, the R

^{2}values of these equations are found to be unsatisfactory (

*R*

^{2}< 0.5).

It is understood that the existing equations for estimating the dynamic scour depth under long- and short-impinging jets are less accurate. In the existing equations, the performance of Chen *et al.* (2022) is satisfactory for long-impinging jets, and Amin *et al.* (2021) is the best among the selected equations for short-impinging jets in estimating the non-dimensional dynamic scour depth. There is a scope for the researchers to conduct laboratory and field experimental works on the estimation of the dynamic scour depth to generate better equations for both long- and short-impinging jets. An accurate estimation of the dynamic scour depth would help the designers to maintain the stability of hydraulic structures.

### Statistical results

Five statistical indices were taken to quantify the agreement between the observed and predicted non-dimensional static and dynamic scour depths for long- and short-impinging jets. The statistical values are listed in Tables 5–8. It can be noticed that the performance of the equation proposed by Aderibigbe & Rajaratnam (1996) for the estimation of non-dimensional static scour depth under long-impinging jets (Table 5) is found to be the best among all the statistical indicators, i.e., RMSE, MAE, MAPE, NSE, and KGE. These results are in line with the graphical analysis. The Equation (11) proposed by Chen *et al.* (2022) gives the second-highest agreements between observed and computed data. The NSE value of these two equations was greater than 0.5, indicating that the performance of the predicted values is satisfactory.

S. No. . | Author's . | RMSE . | NSE . | MAE . | MAPE . | KGE . |
---|---|---|---|---|---|---|

1. | Aderibigbe & Rajaratnam (1996) | 0.046 | 0.721 | 0.037 | 0.222 | 0.862 |

2. | Ansari et al. (2003) | 0.076 | 0.238 | 0.058 | 0.253 | 0.520 |

3. | Chakravarti et al. (2014) | 0.104 | −0.425 | 0.081 | 0.343 | 0.276 |

4. | Amin et al. (2021) | 0.072 | 0.324 | 0.055 | 0.260 | 0.600 |

5. | Shakya et al. (2022) | 0.075 | 0.267 | 0.064 | 0.311 | 0.655 |

6. | Chen et al. (2022) | 0.051 | 0.652 | 0.042 | 0.218 | 0.790 |

S. No. . | Author's . | RMSE . | NSE . | MAE . | MAPE . | KGE . |
---|---|---|---|---|---|---|

1. | Aderibigbe & Rajaratnam (1996) | 0.046 | 0.721 | 0.037 | 0.222 | 0.862 |

2. | Ansari et al. (2003) | 0.076 | 0.238 | 0.058 | 0.253 | 0.520 |

3. | Chakravarti et al. (2014) | 0.104 | −0.425 | 0.081 | 0.343 | 0.276 |

4. | Amin et al. (2021) | 0.072 | 0.324 | 0.055 | 0.260 | 0.600 |

5. | Shakya et al. (2022) | 0.075 | 0.267 | 0.064 | 0.311 | 0.655 |

6. | Chen et al. (2022) | 0.051 | 0.652 | 0.042 | 0.218 | 0.790 |

S. No. . | Authors . | RMSE . | NSE . | MAE . | MAPE . | KGE . |
---|---|---|---|---|---|---|

1. | Aderibigbe & Rajaratnam (1996) | 0.988 | −0.250 | 0.616 | 0.4133 | −0.0214 |

2. | Ansari et al. (2003) | 0.848 | 0.078 | 0.472 | 0.2783 | 0.1119 |

3. | Chakravarti et al. (2014) | 0.769 | 0.242 | 0.4109 | 0.2450 | 0.1888 |

4. | Amin et al. (2021) | 0.178 | 0.959 | 0.1343 | 0.1374 | 0.9791 |

5. | Shakya et al. (2022) | 0.302 | 0.882 | 0.1609 | 0.1291 | 0.7787 |

6. | Chen et al. (2022) | 1.042 | −0.390 | 0.6832 | 0.4896 | −0.0691 |

S. No. . | Authors . | RMSE . | NSE . | MAE . | MAPE . | KGE . |
---|---|---|---|---|---|---|

1. | Aderibigbe & Rajaratnam (1996) | 0.988 | −0.250 | 0.616 | 0.4133 | −0.0214 |

2. | Ansari et al. (2003) | 0.848 | 0.078 | 0.472 | 0.2783 | 0.1119 |

3. | Chakravarti et al. (2014) | 0.769 | 0.242 | 0.4109 | 0.2450 | 0.1888 |

4. | Amin et al. (2021) | 0.178 | 0.959 | 0.1343 | 0.1374 | 0.9791 |

5. | Shakya et al. (2022) | 0.302 | 0.882 | 0.1609 | 0.1291 | 0.7787 |

6. | Chen et al. (2022) | 1.042 | −0.390 | 0.6832 | 0.4896 | −0.0691 |

S. No. . | Authors . | RMSE . | NSE . | MAE . | MAPE . | KGE . |
---|---|---|---|---|---|---|

1. | Aderibigbe & Rajaratnam (1996) | 0.84 | −0.426 | 0.673 | 0.700 | 0.362 |

2. | Chakravarti et al. (2014) | 1.762 | −5.273 | 1.192 | 1.113 | −0.896 |

3. | Amin et al. (2021) | 2.428 | −10.908 | 1.493 | 1.057 | −1.867 |

4. | Kartal & Emiroglu (2021) | 0.692 | 0.032 | 0.536 | 0.584 | 0.497 |

5. | Chen et al. (2022) | 0.609 | 0.248 | 0.425 | 0.389 | 0.553 |

S. No. . | Authors . | RMSE . | NSE . | MAE . | MAPE . | KGE . |
---|---|---|---|---|---|---|

1. | Aderibigbe & Rajaratnam (1996) | 0.84 | −0.426 | 0.673 | 0.700 | 0.362 |

2. | Chakravarti et al. (2014) | 1.762 | −5.273 | 1.192 | 1.113 | −0.896 |

3. | Amin et al. (2021) | 2.428 | −10.908 | 1.493 | 1.057 | −1.867 |

4. | Kartal & Emiroglu (2021) | 0.692 | 0.032 | 0.536 | 0.584 | 0.497 |

5. | Chen et al. (2022) | 0.609 | 0.248 | 0.425 | 0.389 | 0.553 |

S. No. . | Authors . | RMSE . | NSE . | MAE . | MAPE . | KGE . |
---|---|---|---|---|---|---|

1. | Aderibigbe & Rajaratnam (1996) | 62.79 | −466.9 | 28.976 | 4.337 | −19.79 |

2. | Chakravarti et al. (2014) | 2.89 | 0.0069 | 2.454 | 0.944 | 0.256 |

3. | Amin et al. (2021) | 3.309 | −0.299 | 2.099 | 0.749 | 0.320 |

4. | Kartal & Emiroglu (2021) | 3.106 | −0.145 | 2.136 | 0.576 | 0.105 |

5. | Chen et al. (2022) | 2.660 | 0.160 | 1.560 | 0.301 | 0.182 |

S. No. . | Authors . | RMSE . | NSE . | MAE . | MAPE . | KGE . |
---|---|---|---|---|---|---|

1. | Aderibigbe & Rajaratnam (1996) | 62.79 | −466.9 | 28.976 | 4.337 | −19.79 |

2. | Chakravarti et al. (2014) | 2.89 | 0.0069 | 2.454 | 0.944 | 0.256 |

3. | Amin et al. (2021) | 3.309 | −0.299 | 2.099 | 0.749 | 0.320 |

4. | Kartal & Emiroglu (2021) | 3.106 | −0.145 | 2.136 | 0.576 | 0.105 |

5. | Chen et al. (2022) | 2.660 | 0.160 | 1.560 | 0.301 | 0.182 |

The statistical performance evaluation metrics of the equations for predicting the non-dimensional static scour depth under short-impinging jets are tabulated in Table 6. The results show that the performance of the equation (Equation (9)) proposed by Amin *et al.* (2021) in terms of the chosen statistical indicators has optimal values, whereas the equation by Shakya *et al.* (2022) has the second-best performance.

*et al.*(2022) shows comparatively better results (NSE > 0.16) than other equations in estimating the dynamic scour depth under both long- and short-impinging jets. Figure 8 shows the percentage of error given by the proposed equations of static and dynamic scour depths. In Figure 8, it can be seen that the proposed equation of Shakya

*et al.*(2022) and Chen

*et al.*(2022) produce less percent of error as compared to the previously proposed equations.

## CONCLUSIONS

Six relationships for static scour depth and five equations for dynamic scour depth were used for verifying the accuracy of scour depth equations under long- and short-impinging jet conditions. The equations proposed by Aderibigbe & Rajaratnam (1996) and Chen *et al.* (2022) under long-impinging jet give better agreements graphically and statistically, as shown in Figure 4(a) and 4(f) and Table 5, with Aderibigbe & Rajaratnam (1996) performing slightly better than the latter. For short-impinging jets, the equations proposed by Amin *et al.* (2021) and Shakya *et al.* (2022) show better conformity between observed and computed data, as shown in Figure 5(d) and 5(e). The computed statistical indicators for Amin *et al.* (2021) indicate a good performance than the other equations. The existing equations for predicting the dynamic scour depth under long- and short-impinging jets are less accurate and highly biased shown in Figures 6 and 7, and Tables 7 and 8. However, the equation proposed by Chen *et al.* (2022) is comparatively better among the chosen equations. Finally, it was concluded by the authors that after graphical and statistical analysis, equations proposed by Aderibigbe & Rajaratnam (1996) and Amin *et al.* (2021) predicted the static scour depth under long- and short-impinging jets with the least errors among all the relationships. The effect of each initial parameter on scour process is determined by each initial parameter is changed by of average value of the initial parameter. As per previous studies, analysis suggests that the and is the most likely dimensional less parameter influencing the dimensionless static and dynamic scour depth. The outcomes showed that both and are the critical factors when the 10% increment of initial parameters. However, the relative sensitivity is about 2.09 times that of for a 10% reduction in the initial parameters.

## DATA AVAILABILITY STATEMENT

All data, models, and code generated or used during the study appear in the published articles.

## CONFLICT OF INTEREST

The authors declare there is no conflict.