Accuracy evaluation of scour depth equations under the submerged vertical jet

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 (–)

D₆₀

diameter of sediment material, 60% of particles finer by weight (L)

D₁₀

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 (–)

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 (F₀), 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_c 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 E_c or the F₀. O'Donoghue et al. (2001) showed that scour is also strongly dependent on the ratio of sediment size (D₅₀) to jet diameter (d₀).

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 (R² = 0.978) is more than that of multi-non-linear regression (MNLR) (R² = 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.

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.

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.

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.

Table 1

Range of parameters taken by researchers to estimate the scour characteristics

S. no. .	Authors .	Nozzle dia. (mm) .	Jet height (cm) .	Jet velocity (m/s) .	Median size D50 (mm) .
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 D50 (mm) .
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

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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 (D₅₀) 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 (D₅₀) 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 (D₅₀) 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 u₀ being 2.12 and 2.61 m/s. The experiments were carried out for two mean sizes of D₅₀ (0.54 and 1.10 mm), and three impinging heights (0, 2d_0, and 4d₀), 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 D₅₀ = 1.28 mm D₆₀ = 1.47 mm and D₁₀ = 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/d₀ ≤ 20 and 69.66 ≤ F₀ ≤ 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.

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.

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.

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.

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.

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.

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.

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.

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.

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

The authors declare there is no conflict.