# Characterization of the Erosion Basin Shaped by the Jet Flow of Sky-Jump Spillways

^{*}

## Abstract

**:**

## 1. Introduction

_{2}); q is the unit flow; H is the difference in elevation between the reservoir level and the water surface downstream of the sky jump; and d the characteristic diameter of the particles. K, x, y, z are the constants to be determined. Some authors have considered additional variables, such as the flip angle (α) [5,24]. Few authors have tried to establish the size and position of the erosion basin by carrying out laboratory tests [25,26] or by applying neural networks [27]. Pagliara et al. [25,26] studied the characteristics of the erosion basin caused by a jet flowing out of pipes of different diameters and inclinations and proposed formulas for estimating the erosion depth, the position of the deepest point, and the width of the hole, which depend on the pipe diameter, the pipe inclination, the air content, the granulometry of the eroded material, and the fretwork downstream. They proposed an experimental equation for each of these variables. Azmathullah [10] applied nonlinear regression techniques to the data obtained from hydraulic models from other previous research works. He proposed formulas that allow the estimation of the total scour depth D. In addition, different techniques of computation were applied for the geometric characterization of the erosion basin, such as genetic expression programming and the adaptive neuro fuzzy inference system (ANFIS) [27]. A novel optimization algorithm HHO [28] was also proposed to improve the performance of an artificial neural network to predict scour depth; however, this study does not provide a formula for directly calculating the total scour depth.

_{2}), the total head (H), the unit flow (q), the grain diameter passing 50% or 90% of weight (d

_{50}, d

_{90}), the impingement jet angle (θ), the flip angle (α), and gravity (g). Using the total scour depth (D) widely studied by other authors, the main objective of this work is to fill a gap in the geometric characterization of the erosion basin downstream of a sky-jump spillway, providing formulas for determining the location, extent, and shape of the potential erosion basin depending of the flip bucket geometry. It should be noted that the estimation of the position, size, and shape of the erosion pit corresponds to a limit scour hole. In fact, the main goal of this study is not to predict a scour hole for a particular case but to provide a limit erosion case on the safety side regardless of soil heterogeneity and non-stationary mode of operation of the spillway. For this reason, we used an easily erodible sand, which is a material with low resistance. Furthermore, the limit scour hole is also independent of the non-stationary mode of operation of the spillway because the limit scour hole corresponds to a flow that has been maintained for a necessary time until erosion stops progressing.

## 2. Materials and Methods

#### 2.1. General Approach

#### 2.2. Relevant Parameters

_{2}), or the equivalent diameter of the eroded material (d). To study how the geometry of the flip bucket can affect the formation of the erosion basin, in addition to q, H and h

_{2}, two other parameters are defined: the radius (R) and the flip angle (α) of the flip bucket. The radius of a flip bucket is usually defined in relation to the h

_{b}, the water height at the launching point: R ≥ 4 h

_{b}or R ≥ 5 h

_{b}, respectively, according to Peterka [46] and USBR [4]. The flip bucket usually varies between 15° and 40° [46,47,48]. Although there are numerous exceptions, it is unusual for the flip angle to exceed 45°. The flip angle, together with the jet speed, influences the distance where the impact occurs in the river bed or in the erosion basin. Anyway, this distance also depends, in addition to total height (H) and the tailwater depth (h

_{2}), on the height of the lip of the flip bucket over the ground (z

_{p}). Finally, the considered parameters are flow discharge (Q), total height (H), distance between the lip of the flip bucket and the level at the reservoir (z

_{0}), flip bucket radius (R), launching angle (α), height of the flip bucket lip over the ground (z

_{p}), height of the lip of the flip bucket over the tailwater (z

_{1}), tailwater depth (h

_{2}), scour depth (t), total scour depth (D), distance of maximum scour from the bucket lip (L

_{c}) (Figure 2), semi-axis length (A

_{1}) in the direction of flow from the maximum scour point to the furthest downstream point, semi-axis length (A

_{2}) in the direction of flow from the maximum scour point towards the furthest point upstream, total length (A) in the river longitudinal direction, semi-axis length (B

_{1}) in the transverse direction to the flow from the maximum scour point towards the furthest point towards the hydraulic right (whose semi-axis length B

_{2}in the transverse direction to the flow from the maximum scour point to the furthest point to the hydraulic left), total width B, and shape parameter A/B, which we will call the circularity index (Figure 3).

#### 2.3. Physical Modeling

#### 2.3.1. Experimental Set-Up

_{50}of 0.6 mm.

#### 2.3.2. Test Procedure

#### 2.3.3. Test Program

_{2}= 0 m) and with a water cushion (h

_{2}= 0.06 m). A total of 54 tests were performed.

_{0}without water cushion and h

_{6}with a 0.06 m water cushion; and finally, “T” defines the valve opening time (17 s, 18 s, 20 s) (Table 2). For example, A1_h6_18 is the test with a sky jump with R of 0.2 m, with α of 0°, with a water cushion of 0.06 m, and valve opening time of 18 s, corresponding to an experimental flow Q of 42.00 L/s.

## 3. Results and Discussion

_{c}), maximum length (A), maximum width (B), and circularity index (A/B). For the estimation of maximum depth, many formulas are available from different authors. As the erosion process is a complex physical phenomenon and it depends on the nature of the bed, rock, or soil, this work introduces the concept of limit erosion basin, and in fact the results are here summarized and analyzed for each parameter defining the limit scour hole position, size, and shape.

_{c}), maximum length (A), maximum width (B), and circularity index (A/B). For the estimation of maximum depth, many formulas are available from different authors. The results are here summarized and analyzed for each parameter defining the scour hole position, size, and shape.

_{c}), the maximum length (A), defined as the sum of two semi-axes (A

_{1}) and (A

_{2}), and the circularity index (A/B). The explanatory variables are: the radius of the flip bucket (R), the impingement angle (θ), the scour depth (t), and the difference in height (z

_{1}) between the lip of the flip bucket and the downstream water level (h

_{2}).

^{2}), absolute mean error (MAE), and relative mean error (MRE) were determined for every formula.

#### 3.1. Erosion Depth

#### 3.1.1. Estimation of Erosion Depth

_{2}). The initial condition is the one that is established before starting the test, and the parameter h

_{2}can assume only two values, 0.00 m and 0.06 m, while the stationary condition is the one that corresponds to the instant in which the entire test channel contributes to the drainage of the flow from the spillway; in this case, the value of h

_{2}will depend on this flow. It is indicated with h

_{2}the height of the water depth cushion in the initial condition. Then, with h

_{2s}, the height of the water cushion is the stationary condition. For each experimental test, two values of experimental total scour depth (Dexp) are obtained, one relating on the initial condition and the other relating to the stationary condition. Total scour depth can also be calculated by applying the literature formulas in the two conditions indicated above. For each condition, the experimental (Dexp) and calculated (Dcal) value of total scour depth are compared using the mean absolute error (MAE) and the mean relative error (MRE) (Figure 7). In the tests without a water cushion, it was observed that the erosion developed before a height cushion h

_{2s}(stationary condition) was established, different from the initial value h

_{2}(initial condition). For the calculation of the MAE and MRE between the experimental and calculated data of total scour (D), two conditions were considered: the initial condition where h

_{2}can assume values equal to 0.00 m or 0.06 m and the stationary condition where the value of h

_{2s}also depends on the flow drained by the spillway (Table 3).

#### 3.1.2. Influence of the Flip Bucket Radius

_{2}) were fixed, and the absolute error (AE) and relative error (RE) were calculated taking the depth (t), marked in blue, corresponding to the greater radius (R = 0.4 m) as a reference (Table 4). In the case of a flip angle of 30°, where there is a lack of experimental dates, the relative error (RE) has been calculated taking the depth (t) corresponding to the radius equal to 0.30 m; to distinguish this situation, the results have been highlighted in yellow.

#### 3.1.3. Influence of the Flip Angle

_{2}) were fixed, and the absolute error (AE) and the relative error (RE) were calculated taking the scour depth (t) marked in blue corresponding to the greater angle (α = 45°) as a reference (Table 5).

#### 3.2. Position of the Point of Maximum Erosion Depth

_{c}defines the distance in plan from the lip of the flip bucket to the point where erosion depth is maximum. It is observed that L

_{c}increases for values of α between 0° and 30°; however, it decreases for 45° (Figure 10). We should take into consideration that there are two opposite effects: (a) when the angle increases, up to 45°, the impact on the terrain surface occurs farther away from the flip bucket; (b) also, when the angle increases, the jet impacts the ground with a more vertical direction, which implies that erosion moves less downstream and develops more in the vertical direction, causing a deeper erosion. As a consequence, an increase in the flip angle might result in a greater or lower L

_{c}, depending on the prevalence of one of the two mentioned effects

_{c}, two linear regressions were performed: one considering the parameter t/z

_{1}and the impingement angle (θ) and another considering t/z

_{1}and the radius (R). Calibration and validation sets were separated. The first formula (Equation (2)) presents an R

^{2}of 0.982, and MAEv equal to 0.19 m and an MREv of 12.76%, corresponding to the validation set. The second formula (Equation (3)) presents an R

^{2}of 0.971, an MAEv equal to 0.265 m, and an MREv of 17.44%. Although the first equation is more precise, R is directly available data, while θ must be determined [51].

_{c}/z

_{1}and θ and R, respectively. This function was represented together with the experimental data. (Figure 11 and Figure 12). We can observe a reasonable correspondence with the experimental values, as expected from the error quantification. Furthermore, the parameter L

_{c}/z

_{1}decreases when the impingement angle (θ) increases. As expected, the existence of a water cushion clearly affects the position of the point with maximum erosion depth. When there is a water cushion, the geometry of the flip bucket has a more significant influence on the parameter L

_{c}/z

_{1}(Figure 13).

#### 3.3. Length of the Erosion Basin

_{1}and a

_{2}, which are obtained by joining the point of maximum erosion with the level curve of 400 mm in the two furthest points; the same geometric criterion was applied to the transverse axis to obtain b

_{1}and b

_{2}. The results obtained with the proposed geometric criterion were compared with the results obtained from applying the morphometric theory of lakes [52] that defines the parameters Lmax, Bmax, Le, and Be. Figure 15 represents the application of the two criteria to a specific case. The results obtained with both techniques were quite similar. Comparing the values of a, obtained from the sum of a

_{1}and a

_{2}, with Lmax, and the values of b, equal to the sum of b

_{1}and b

_{2}, with Bmax, the MRE was around 2%, so it was finally decided to maintain the geometric criterion here proposed (Figure 16).

_{1}, a

_{2}, b

_{1}, and b

_{2}were measured, A

_{1}, A

_{2}, B

_{1}, and B

_{2}were determined by linear extrapolation to the 500 mm curve. To verify that the extrapolation was acceptable, we first selected the tests that presented the 500 mm contour closed and determined directly the axes A

_{1}, A

_{2}, B

_{1}, and B

_{2}(Table 6). These experimental values were then compared with those obtained by extrapolation, and the error was calculated. The MRE did not exceed 15.22%. Taking into account the uncertainties of the erosion process, the extrapolation was considered acceptable.

_{1}and A

_{2}were considered to calculate the length of the major axis A, and a linear regression type adjustment was performed. Cases where the scour hole reached the foot of the trampoline were excluded, which could distort the result. The calibration and validation sets were differentiated. The formulas obtained for A

_{1}and A

_{2}(Equations (4) and (5)), respectively, present an R

^{2}of 0.984 and 0.980. Considering the validation data, for parameter A

_{1}the MAEv is 0.13 m, and the MREv is 10.70%, and for A

_{2}, the MAEv is 0.12 m, and the MREv is 8.40%. The value of A (Equation (6)) results from the sum of A

_{1}and A

_{2}and presents a MAE equal to 0.19 m and an MRE of 7.40%.

_{1}increases when cos

^{2}θ increases, confirming that the scour hole lengthens when the impingement angle (θ) decreases. Equation (5) fits the experimental results quite well for cases that have a radius other than 0.40 m (Figure 19).

#### 3.4. Erosion Basin Shape

^{2}= 0.985. Applying the formula to validation cases results in an MAEv of 0.15 m and an MREv of 8.70%.

#### 3.5. Procedure for Estimating the Location, the Size, and the Shape of the Erosion Basin

_{0}); (e) the vertical distance from the lip of the flip bucket to the surface of the water downstream (z

_{1}); and (f) the impingement angle of incidence (θ), which can be determined with equations present in the literature [50].

_{c}, the distance on the horizontal plane from the foot of the flip bucket to point C, where the erosion is maximum. The designer can use Equations (2) or (3) for this.

_{1}and A

_{2}(Equations (4) and (5)). The total length of the longitudinal axis A is obtained from the sum of A

_{1}and A

_{2}.

_{c}> A

_{2}. If the answer is positive, it should be expected that the limit scour hole would not affect the base of the flip bucket.

_{c}parameter, the MREv is less than 17.5%. It should be noted that this procedure concerns a flip bucket with a certain width, so that width and shape cannot be extrapolated to other cases. However, the estimation of length and position of the limit erosion basin are the most relevant parameters from a design or safety point of view.

_{2}is greater than the parameter L

_{c}; in this circumstance, it should be expected that the erosion would affect the flip bucket. Equation (5) (calculation of A

_{2}) is applied using the validation tests (Table 8). It is observed that in cases where the 500 mm contour line touches the trampoline (Figure 24), the value of A

_{2}_cal must be greater than the mean value of $\overline{{L}_{c}}$ calculated with Equations (2) and (3); only in two cases (A4_h0_17; B4_h0_18) the value of A

_{2}_cal is not greater than the mean value of $\overline{{L}_{c}}$, although the 500 mm contour line touches the trampoline. In any case, this observation further validates the proposed formula because it is correct ten times out of twelve.

#### 3.6. Limitations of the Present Study

- -
- The height of the spillway is constant, so it has not allowed us to obtain a formula in order to estimate the depth of erosion depending on the flow and the geometric characteristics of the spillway; one of the formulas provided by any other author can be used for this purpose;
- -
- The spillway width is constant, so it has not allowed us to provide a direct formula for the estimation of the total width of the limit scour hole;
- -
- A wider test channel would reduce the effect of the walls, which are most likely the reason for the deviations observed in the limit scour hole.

## 4. Conclusions

- -
- Increasing the radius of the flip bucket allows a reduction in the depth of the scour hole. Although the cost of the structure is higher for a greater radius, this extra cost might in some cases be justified for the extra safety level achieved.
- -
- The greater the flip angle is, in the range of tested angles, 15° to 45°, the greater the depth of erosion is.
- -
- Scour hole is longer in the riverbed direction and less deep for low flip angles.
- -
- For the length of flip bucket tested, the scour hole is quasi-circular for a flip angle of 45° and more elongated for lower angles. However, the influence of the lip length is evident, so a different shape should be expected for different lip lengths.
- -
- The plan position of the point where the depth erosion is maximum moves away from the flip bucket with increasing flip angles between 0° and 30°. However, it is nearer the structure for a flip angle of 45°. Two opposite effects might explain this fact: increasing the angle increases the launch scope, up to 45°, but a greater angle of incidence makes the erosion more vertical, so the scour hole develops less in the direction of the riverbed and more in depth.
- -
- Empirical formulas were derived from the experimental data to estimate the position, size and shape of the scour hole. However, it should be noted that a different width, and so shape, should be expected for different lengths of the flip bucket lip, which is a parameter not considered in this experimental research. More tests are needed with different lip lengths.
- -
- A methodology is proposed, using the above mentioned formulas, to estimate position, size, and shape of the scour hole, which was fitted to a combination of two semi-ellipses.
- -
- The proposed methodology was used with success to determine whether the scour hole is likely to affect the flip bucket structure, comparing the length of the scour hole upstream of the point where the depth is maximum with the distance from that point to the flip bucket. If the scour hole overlaps with the structure, affection is likely to occur.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

A | total length of the limit scour hole in the river longitudinal direction |

A/B | circularity index |

A_{1} | semi-axis length in the direction of flow from the maximum scour point to the furthest downstream point |

A_{2} | semi-axis length in the direction of flow from the maximum scour point to the furthest upstream point |

AR | absolute error |

B | total width of the limit scour hole |

B_{1} | semi-axis length in the transverse direction to the flow from the maximum scour point towards the furthest point towards the hydraulic right |

B_{2} | semi-axis length in the transverse direction to the flow from the maximum scour point towards the furthest point towards the hydraulic left |

D | total scour depth |

D_{exp} | experimental total scour depth |

D_{cal} | calculated total scour depth |

d_{50} | grain diameter at 50% of weight |

d_{90} | grain diameter at 90% of weight |

H | total head (distance between upstream and downstream water level) |

h_{2} | tailwater depth (downstream water level) |

L_{c} | distance of maximum scour from bucket lip |

MAE | mean absolute error |

MRE | mean relative error |

Q | flow rate |

q | unit flow rate |

R | radius of flip bucket |

RE | relative error |

T | scour depth |

z_{1} | distance from the flip bucket’s lip to the downstream water level |

z_{o} | distance between flip bucket’s lip and upstream water level |

z_{p} | distance from the flip bucket’s lip to the ground |

α | flip angle |

θ | impingement jet angle |

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**Figure 5.**(

**a**) Testing area with spillway. Case: R = 0.4 m; α = 45°; (

**b**) view of the testing channel from downstream with a water cushion.

**Figure 7.**Comparing the experimental and calculated data of the total scour (D) with mean relative error in the initial condition (MRE) and in the stationary condition (MREs).

**Figure 8.**Evolution of scour depth (t) with the radius (R), separating the data, for each flip angle (α), by height of the water cushion (h

_{0}, without cushion, and h

_{6}, with cushion of 0.06 m) and by test flow rate (37.5 L/s, 42 L/s and 50 L/s).

**Figure 9.**Evolution of scour depth (t) with the flip angle (α), separating the data, for each radius, by height of the water cushion (h

_{0}, without cushion, and h

_{6}, with cushion of 0.06 m) and by test flow rate (37.5 L/s, 42 L/s and 50 L/s).

**Figure 10.**Evolution of the position of the point of maximum erosion depth L

_{c}with the flip angle (α), separating the cases without cushion h

_{0}and with cushion h

_{6}and distinguishing by size the different flows (37.5, 42.0, 50.0 L/s) and by shape and color the radii (0.2, 0.3 and 0.4 m).

**Figure 11.**Results obtained for L

_{c}/z

_{1}with the Equation (2) for several values of t/z

_{1}and impingement angle (θ). The experimental value of t/z

_{1}is indicated next to each point.

**Figure 12.**Results obtained for L

_{c}/z

_{1}with the Equation (3) for several values of t/z

_{1}and radius (R). The experimental value of t/z

_{1}is indicated next to each point.

**Figure 13.**Results obtained for L

_{c}/z

_{1}with the Equation (2) for several values of t/z

_{1}and impingement angle (θ). The experimental value of t/z

_{1}is indicated next to each point (without water cushion, h

_{0}, and with water cushion, h

_{6}).

**Figure 14.**(

**a**) For contour line at height of 400 mm is the limit scour hole semi-axis from the maximum scour point to the furthest downstream and upstream point in the direction of flow, respectively, a

_{1}and a

_{2}. In transverse direction to the flow from the maximum scour point towards the furthest point towards are the hydraulic right (b

_{1}) and hydraulic left (b

_{2}). (

**b**) For contour line at the height of 500 mm is the semi-axis in the direction of flow from the maximum scour point to the furthest downstream and upstream point, respectively, A

_{1}and A

_{2}; semi-axis length in the transverse direction to the flow from the maximum scour point is towards the furthest point towards the hydraulic right and left, respectively B

_{1}and B

_{2}.

**Figure 15.**(

**a**) Scour hole axes according to the criterion here proposed. (

**b**) According to the morphometric criterion used in lakes: Lmax is the line that connects the two points of the lakeshore that are farthest away, Bmax is the line perpendicular to Lmax that connects the two most distant points of the shore, Le is the line that connects the two points of the shore of the lake that are farthest in the direction of the current, Be is the line perpendicular to Le that connects the two most distant points. In flow direction, the joining of the point of maximum erosion with the contour level of 400 mm in the two furthest points is given by a

_{1}and a

_{2}; in cross direction, the joining of the point of maximum erosion with the contour level of 400 mm in the two furthest points is given by b

_{1}and b

_{2}.

**Figure 17.**Evolution of the maximum length (A) of the scour hole with the flip angle (α), separating the data by the height water cushion (0.00 m or 0.06 m; columns), by the flow rate (37.5 L/s, 42 L/s, 50 L/s; lines), and by the radius (0.2, 0,3, 0.4 m; shape and color).

**Figure 18.**Lines obtained with Equation (4) for various values of scour depth (t), covering the range of experimental values, including in the graph (A

_{1}; cos

^{2}θ) the experimental points with the corresponding value of t.

**Figure 19.**Lines obtained with Equation (4) for various values of scour depth (t), covering the range of experimental values, including in the graph (A

_{2}; cos

^{2}θ) the experimental points with the corresponding value of t and separating the data as a function of the radius R.

**Figure 20.**Evolution of the circularity index A/B with the flip angle (α), separating in two columns the cases without water cushion h

_{0}from those with water cushion h

_{6}and separating in three rows the test flows rates (37.5 L/s, 42 L/s, 50 L/s). Radius (R) is indicated by color and shape.

**Figure 21.**Lines obtained with equation for various values of R/t, covering the range of experimental values, including in the graph (A/B; cosθ) the experimental points with the corresponding value of R/t and separating the data as a function of the flip angle (α).

**Figure 23.**Comparison between the shape, size, and position of limit scour hole obtained experimentally (continuous line) and those determined by the proposed procedure (dashed line). Some examples: (

**a**) R = 0.2 m, α = 45°, Q = 37.5 L/s and without water cushion. (

**b**) R = 0.2 m, α = 30°, Q = 50 L/s and with water cushion of 0.06 m. (

**c**) R = 0.4 m, α = 15°, Q = 42 L/s and with water cushion of 0.06 m.

**Figure 24.**Digital terrain model for some tests, with contour lines at 400 mm and 500 mm, and longitudinal (a

_{1}and a

_{2}) and transverse axes (b

_{1}and b

_{2}) closed on curve line 400 mm, for three tests: (

**a**) A4_h6_18 (R = 0.2 m, α = 45°, with a water cushion of 0.06 m and Q = 42.00 L/s), (

**b**) B2_h0_17 (R = 0.3 m, α = 15°, without a water cushion and Q = 37.50 L/s), (

**c**) C2_h0_20 (R = 0.4 m, α = 15°, without a water cushion and Q = 50.00 L/s). For every test, the calculated and measured value of A

_{2}(the semi-axis length in the direction of flow from the maximum scour point to the furthest upstream point) is provided.

Authors | Erosion Depth (m) | |
---|---|---|

Veronese B | [3] | $t+{h}_{2}=1.9{H}^{0.225}{q}^{0.54}$ |

Damle A | [12] | $t+{h}_{2}=0.652{q}^{0.5}{H}^{0.5}$ |

Damle B | [12] | $t+{h}_{2}=0.543{q}^{0.5}{H}^{0.5}$ |

Damle C | [12] | $t+{h}_{2}=0.362{q}^{0.5}{H}^{0.5}$ |

INCYTH | [30] | $t+{h}_{2}=1.413{q}^{0.5}{H}^{0.25}$ |

Schoklitsch | [2] | $t+{h}_{2}=0.521\frac{{H}^{0.2}{q}^{0.57}}{{d}_{90}^{0.32}}$ |

Veronese A | [3] | $t+{h}_{2}=0.202\frac{{q}^{0.54}{H}^{0.225}}{{d}_{50}^{0.42}}$ |

Eggenberger | [31] | $t+{h}_{2}=1.44\frac{{q}^{0.60}{H}^{0.50}}{{d}_{90}^{0.40}}$ |

Hartung | [32] | $t+{h}_{2}=1.40\frac{{q}^{0.64}{H}^{0.36}}{{d}_{90}^{0.32}}$ |

Franke | [33] | $t+{h}_{2}=1.13\frac{{q}^{0.67}{H}^{0.50}}{{d}_{90}^{0.50}}$ |

Mikhalev | [23] | $t+{h}_{2}=\frac{1.804q\mathrm{sin}\theta}{1-0.215\mathrm{cot}\theta}\left(\frac{1}{{d}_{90}^{0.33}{h}_{2}^{0.5}}-\frac{1.126}{H}\right)$ |

Mirtskhulava | [15] | $t+{h}_{2}=\left(\frac{0.97}{\sqrt{{d}_{90}}}-\frac{1.35}{\sqrt{H}}\right)\frac{q\mathrm{sin}\theta}{1-0.175\mathrm{cot}\theta}+0.25{h}_{2}$ |

Chee and Kung | [24] | $t+{h}_{2}=3.30H{\left(\frac{{q}^{2}}{g{H}^{3}}\right)}^{0.3}{\left(\frac{H}{d}\right)}^{0.1}{\alpha}^{0.1}$ |

Yildiz and Üzücek | [5] | $t+{h}_{2}=1.9{H}^{0.225}{q}^{0.54}\mathrm{sin}\theta $ |

Martins A | [34] | $t+{h}_{2}=0.14N-0.73\frac{{h}_{2}^{2}}{N}+1.7{h}_{2}$ |

Chian Min Wu | [23] | $t+{h}_{2}=1.18{H}^{0.235}{q}^{0.51}$ |

Martins B | [35] | $t+{h}_{2}=1.5\text{}{H}^{0.1}{q}^{0.6}$ |

Taraimovich | [36] | $t+{h}_{2}=0.633{H}^{0.25}{q}^{0.67}$ |

Machado B | [37] | $t+{h}_{2}=2.98{q}^{0.5}{H}^{0,25}$ |

SOFRELEC | [38] | $t+{h}_{2}=2.3{q}^{0.6}{H}^{0.1}$ |

Kotoulas | [39] | $t+{h}_{2}=0.78\frac{{q}^{0.7}{H}^{0.35}}{{d}_{90}^{0.4}}$ |

Chee and Padyar | [40] | $t+{h}_{2}=2.126\frac{{q}^{0.67}{H}^{0.18}}{{d}_{50}^{0.063}}$ |

Bisaz and Tschopp | [41] | $t+{h}_{2}=2.76{q}^{0.5}{H}^{0.25}-7.22{d}_{90}$ |

Chee and Kung | [24] | $t+{h}_{2}=1.663\frac{{q}^{0.60}{H}^{0.20}}{{d}_{50}^{0.10}}$ |

Machado A | [37] | $t+{h}_{2}=1.35\frac{{q}^{0.5}{H}^{0.3145}}{{d}_{90}^{0.0645}}$ |

Jaeger | [42] | $t+{h}_{2}=0.6{H}^{0.25}{q}^{0.5}{\left(\frac{{h}_{2}}{{d}_{50}}\right)}^{0.333}$ |

Rubinstein | [13] | $t+{h}_{2}={h}_{2}+2.59{h}_{2}{\left(\frac{H+{h}_{2}}{{d}_{90}}\right)}^{0.75}\frac{{q}^{1.20}}{13.70{H}^{1.80}}{\left(\frac{H}{{h}_{2}}\right)}^{1.33}$ |

Mason and Arumugam | [43] | $t+{h}_{2}=3.27\frac{{q}^{0.60}{H}^{0.05}{h}_{2}{}^{0.15}}{{g}^{0.30}{d}_{50}^{0.10}}$ |

Ghodsian et al. | [44] | $t+{h}_{2}=0.75{h}_{2}{\left(\frac{q}{{\left({h}_{2}^{3}g\right)}^{0.5}}\right)}^{0.524}{\left(\frac{{d}_{50}}{{h}_{2}}\right)}^{-0.366}{\left(\frac{H}{{h}_{2}}\right)}^{0.255}$ |

$N=0.007\sqrt[7]{\frac{{Q}^{3}{H}^{1.5}}{{d}_{90}^{2}}}$ |

Flip Bucket | A1 | A2 | A3 | A4 | B2 | B3 | B4 | C2 | C4 |
---|---|---|---|---|---|---|---|---|---|

R (m) | 0.20 | 0.20 | 0.20 | 0.20 | 0.30 | 0.30 | 0.30 | 0.40 | 0.40 |

α (°) | 0 | 15 | 30 | 45 | 15 | 30 | 45 | 15 | 45 |

h_{2} (m) | 0.00 0.06 | 0.00 0.06 | 0.00 0.06 | 0.00 0.06 | 0.00 0.06 | 0.00 0.06 | 0.00 0.06 | 0.00 0.06 | 0.00 0.06 |

Q (L/s) | 37.50 42.00 50.00 | 37.50 42.00 50.00 | 37.50 42.00 50.00 | 37.50 42.00 50.00 | 37.50 42.00 50.00 | 37.50 42.00 50.00 | 37.50 42.00 50.00 | 37.50 42.00 50.00 | 37.50 42.00 50.00 |

**Table 3.**Mean absolute error and mean relative error for the initial condition (MAE and MRE) and stationary condition (MAEs and MREs) obtained by comparing the experimental and calculated data of the total scour (D).

Author | MAE (m) | MRE | MAEs (m) | MREs | |
---|---|---|---|---|---|

Veronese B | [3] | 0.159 | 0.47 | 0.116 | 0.32 |

Damle A | [12] | 0.188 | 0.50 | 0.230 | 0.56 |

Damle B | [12] | 0.218 | 0.58 | 0.259 | 0.63 |

Damle C | [12] | 0.268 | 0.72 | 0.308 | 0.75 |

INCYTH | [30] | 0.072 | 0.22 | 0.054 | 0.14 |

Chian Min Wu | [23] | 0.050 | 0.13 | 0.069 | 0.16 |

Martins B | [35] | 0.053 | 0.15 | 0.055 | 0.13 |

Taraimovich | [36] | 0.238 | 0.64 | 0.277 | 0.68 |

Machado B | [37] | 0.528 | 1.50 | 0.479 | 1.23 |

SOFRELEC | [38] | 0.214 | 0.62 | 0.174 | 0.46 |

Schoklitsch | [2] | 0.734 | 2.07 | 0.686 | 1.75 |

Veronese A | [3] | 0.878 | 2.47 | 0.827 | 2.11 |

Eggenberger | [31] | 0.090 | 0.23 | 0.132 | 0.31 |

Hartung | [32] | 0.100 | 0.26 | 0.141 | 0.33 |

Franke | [33] | 0.184 | 0.49 | 0.225 | 0.55 |

Kotoulas | [39] | 0.237 | 0.64 | 0.275 | 0.67 |

Chee-Padiyar | [40] | 0.340 | 0.97 | 0.297 | 0.77 |

Bisaz-Tschopp | [41] | 0.443 | 1.26 | 0.396 | 1.02 |

Chee-Kung | [24] | 0.477 | 1.350 | 0.432 | 1.11 |

Machado A | [37] | 0.234 | 0.68 | 0.188 | 0.50 |

Jaeger | [42] | 0.461 | 1.299 | 0.430 | 1.099 |

Rubinstein | [13] | 0.283 | 0.765 | 0.313 | 0.756 |

Martins A | [34] | 0.293 | 0.791 | 0.282 | 0.680 |

Mason-Arumugam | [43] | 0.216 | 0.620 | 0.176 | 0.451 |

Ghodsian | [44] | 0.336 | 0.952 | 0.296 | 0.739 |

Mikhalev | [23] | 0.087 | 0.253 | 0.185 | 0.464 |

Mirtskulava | [15] | 1.687 | 4.628 | 1.493 | 3.758 |

Cheen-Kung | [24] | 0.418 | 1.151 | 0.369 | 0.937 |

Yildiz and Üzücek | [5] | 0.083 | 0.237 | 0.145 | 0.349 |

**Table 4.**Absolute errors (AR) and relative errors (RE) of the scour depth (t) with respect to the greater radius (R = 0.4 m) marked in blue for different value of flip angle (α). The relative error, highlighted in yellow, has been calculated taking the depth (t) corresponding to the radius equal to 0.30 m.

(a) (h_{2} = 0.0 m) | ||||||

Practice | Q (l/s) | α (°) | R (m) | t (m) | AR (m) | RE |

A2_h0_17 | 37.5 | 15 | 0.2 | 0.321 | 0.062 | 0.24 |

B2_h0_17 | 37.5 | 15 | 0.3 | 0.291 | 0.032 | 0.12 |

C2_h0_17 | 37.5 | 15 | 0.4 | 0.259 | - | - |

A2_h0_18 | 42.0 | 15 | 0.2 | 0.332 | 0.061 | 0.23 |

B2_h0_18 | 42.0 | 15 | 0.3 | 0.310 | 0.039 | 0.14 |

C2_h0_18 | 42.0 | 15 | 0.4 | 0.271 | - | - |

A2_h0_20 | 50.0 | 15 | 0.2 | 0.34 | 0.055 | 0.19 |

B2_h0_20 | 50.0 | 15 | 0.3 | 0.329 | 0.044 | 0.16 |

C2_h0_20 | 50.0 | 15 | 0.4 | 0.285 | - | - |

(b) (h_{2} = 0.0 m) | ||||||

Practice | Q (l/s) | α (°) | R (m) | t (m) | AR (m) | RE |

A3_h0_17 | 37.5 | 30 | 0.2 | 0.416 | 0.049 | 0.13 |

B3_h0_17 | 37.5 | 30 | 0.3 | 0.367 | - | - |

C3_h0_17 | 37.5 | 30 | 0.4 | - | - | - |

A3_h0_18 | 42.0 | 30 | 0.2 | 0.436 | 0.048 | 0.12 |

B3_h0_18 | 42.0 | 30 | 0.3 | 0.388 | - | |

C3_h0_18 | 42.0 | 30 | 0.4 | - | - | - |

A3_h0_20 | 50.0 | 30 | 0.2 | 0.441 | 0.034 | 0.08 |

B3_h0_20 | 50.0 | 30 | 0.3 | 0.407 | - | - |

C3_h0_20 | 50.0 | 30 | 0.4 | - | - | - |

(c) (h_{2} = 0.0 m) | ||||||

Practice | Q (l/s) | α (°) | R (m) | t (m) | AR (m) | RE |

A4_h0_17 | 37.5 | 45 | 0.2 | 0.457 | 0.108 | 0.31 |

B4_h0_17 | 37.5 | 45 | 0.3 | 0.405 | 0.056 | 0.16 |

C4_h0_17 | 37.5 | 45 | 0.4 | 0.349 | - | - |

A4_h0_18 | 42.0 | 45 | 0.2 | 0.475 | 0.114 | 0.32 |

B4_h0_18 | 42.0 | 45 | 0.3 | 0.421 | 0.060 | 0.17 |

C4_h0_18 | 42.0 | 45 | 0.4 | 0.361 | - | - |

A4_h0_20 | 50.0 | 45 | 0.2 | 0.490 | 0.087 | 0.22 |

B4_h0_20 | 50.0 | 45 | 0.3 | 0.443 | 0.040 | 0.10 |

C4_h0_20 | 50.0 | 45 | 0.4 | 0.403 | - | - |

(d) (h_{2} = 0.06 m) | ||||||

Practice | Q (l/s) | α (°) | R (m) | t (m) | AR (m) | RE |

A2_h6_17 | 37.5 | 15 | 0.2 | 0.264 | 0.029 | 0.12 |

B2_h6_17 | 37.5 | 15 | 0.3 | 0.253 | 0.018 | 0.08 |

C2_h6_17 | 37.5 | 15 | 0.4 | 0.235 | - | - |

A2_h6_18 | 42.0 | 15 | 0.2 | 0.285 | 0.035 | 0.14 |

B2_h6_18 | 42.0 | 15 | 0.3 | 0.275 | 0.025 | 0.10 |

C2_h6_18 | 42.0 | 15 | 0.4 | 0.250 | - | - |

A2_h6_20 | 50.0 | 15 | 0.2 | 0.303 | 0.034 | 0.13 |

B2_h6_20 | 50.0 | 15 | 0.3 | 0.291 | 0.022 | 0.08 |

C2_h6_20 | 50.0 | 15 | 0.4 | 0.269 | - | - |

(e) (h_{2} = 0.06 m) | ||||||

Practice | Q (l/s) | α (°) | R (m) | t (m) | AR (m) | RE |

A3_h6_17 | 37.5 | 30 | 0.2 | 0.317 | 0.042 | 0.15 |

B3_h6_17 | 37.5 | 30 | 0.3 | 0.275 | - | - |

C3_h6_17 | 37.5 | 30 | 0.4 | - | - | - |

A3_h6_18 | 42.0 | 30 | 0.2 | 0.320 | 0.021 | 0.07 |

B3_h6_18 | 42.0 | 30 | 0.3 | 0.299 | - | |

C3_h6_18 | 42.0 | 30 | 0.4 | - | - | - |

A3_h6_20 | 50.0 | 30 | 0.2 | 0.328 | 0.01 | 0.03 |

B3_h6_20 | 50.0 | 30 | 0.3 | 0.318 | - | - |

C3_h6_20 | 50.0 | 30 | 0.4 | - | - | - |

(f) (h_{2} = 0.06 m) | ||||||

Practice | Q (l/s) | α (°) | R (m) | t (m) | AR (m) | RE |

A4_h6_17 | 37.5 | 45 | 0.2 | 0.385 | 0.101 | 0.36 |

B4_h6_17 | 37.5 | 45 | 0.3 | 0.367 | 0.083 | 0.29 |

C4_h6_17 | 37.5 | 45 | 0.4 | 0.284 | - | |

A4_h6_18 | 42.0 | 45 | 0.2 | 0.410 | 0.113 | 0.38 |

B4_h6_18 | 42.0 | 45 | 0.3 | 0.380 | 0.083 | 0.28 |

C4_h6_18 | 42.0 | 45 | 0.4 | 0.297 | - | - |

A4_h6_20 | 50.0 | 45 | 0.2 | 0.440 | 0.122 | 0.38 |

B4_h6_20 | 50.0 | 45 | 0.3 | 0.423 | 0.105 | 0.33 |

C4_h6_20 | 50.0 | 45 | 0.4 | 0.318 | - | - |

**Table 5.**Absolute errors (AR) and relative errors (RE) of the scour depth (t) with respect to the greater flip angle (α = 45°) marked in blue, for different values of radius (R).

(g) (h_{2} = 0.00 m) | ||||||

Practice | Q (l/s) | α (°) | R (m) | t (m) | AR (m) | RE |

A2_h0_17 | 37.5 | 15 | 0.2 | 0.321 | 0.136 | 0.30 |

B2_h0_17 | 37.5 | 30 | 0.2 | 0.416 | 0.041 | 0.09 |

C2_h0_17 | 37.5 | 45 | 0.2 | 0.457 | - | - |

A2_h0_18 | 42.0 | 15 | 0.2 | 0.332 | 0.143 | 0.30 |

B2_h0_18 | 42.0 | 30 | 0.2 | 0.436 | 0.039 | 0.08 |

C2_h0_18 | 42.0 | 45 | 0.2 | 0.475 | - | - |

A2_h0_20 | 50.0 | 15 | 0.2 | 0.340 | 0.150 | 0.31 |

B2_h0_20 | 50.0 | 30 | 0.2 | 0.441 | 0.049 | 0.10 |

C2_h0_20 | 50.0 | 45 | 0.2 | 0.490 | - | - |

(h) (h_{2} = 0.0 m) | ||||||

Practice | Q (l/s) | α (°) | R (m) | t (m) | AR (m) | RE |

A3_h0_17 | 37.5 | 15 | 0.3 | 0.291 | 0.114 | 0.28 |

B3_h0_17 | 37.5 | 30 | 0.3 | 0.367 | 0.038 | 0.09 |

C3_h0_17 | 37.5 | 45 | 0.3 | 0.405 | - | - |

A3_h0_18 | 42.0 | 15 | 0.3 | 0.310 | 0.111 | 0.26 |

B3_h0_18 | 42.0 | 30 | 0.3 | 0.388 | 0.033 | 0.08 |

C3_h0_18 | 42.0 | 45 | 0.3 | 0.421 | - | - |

A3_h0_20 | 50.0 | 15 | 0.3 | 0.330 | 0.113 | 0.26 |

B3_h0_20 | 50.0 | 30 | 0.3 | 0.407 | 0.036 | 0.08 |

C3_h0_20 | 50.0 | 45 | 0.3 | 0.443 | - | - |

(i) (h_{2} = 0.0 m) | ||||||

Practice | Q (l/s) | α (°) | R (m) | t (m) | AR (m) | RE |

A4_h0_17 | 37.5 | 15 | 0.4 | 0.259 | 0.09 | 0.26 |

B4_h0_17 | 37.5 | 30 | 0.4 | - | - | - |

C4_h0_17 | 37.5 | 45 | 0.4 | 0.349 | - | - |

A4_h0_18 | 42.0 | 15 | 0.4 | 0.271 | 0.09 | 0.25 |

B4_h0_18 | 42.0 | 30 | 0.4 | - | - | - |

C4_h0_18 | 42.0 | 45 | 0.4 | 0.361 | - | - |

A4_h0_20 | 50.0 | 15 | 0.4 | 0.285 | 0.118 | 0.29 |

B4_h0_20 | 50.0 | 30 | 0.4 | - | - | - |

C4_h0_20 | 50.0 | 45 | 0.4 | 0.403 | - | - |

(jl) (h_{2} = 0.06 m) | ||||||

Practice | Q (l/s) | α (°) | R (m) | t (m) | AR (m) | RE |

A2_h6_17 | 37.5 | 15 | 0.2 | 0.264 | 0.121 | 0.31 |

B2_h6_17 | 37.5 | 30 | 0.2 | 0.317 | 0.068 | 0.18 |

C2_h6_17 | 37.5 | 45 | 0.2 | 0.385 | - | - |

A2_h6_18 | 42.0 | 15 | 0.2 | 0.285 | 0.125 | 0.30 |

B2_h6_18 | 42.0 | 30 | 0.2 | 0.320 | 0.09 | 0.22 |

C2_h6_18 | 42.0 | 45 | 0.2 | 0.410 | - | - |

A2_h6_20 | 50.0 | 15 | 0.2 | 0.303 | 0.137 | 0.31 |

B2_h6_20 | 50.0 | 30 | 0.2 | 0.328 | 0.112 | 0.25 |

C2_h6_20 | 50.0 | 45 | 0.2 | 0.440 | - | - |

(k) (h_{2} = 0.06 m) | ||||||

Practice | Q (l/s) | α (°) | R (m) | t (m) | AR (m) | RE |

A3_h6_17 | 37.5 | 15 | 0.3 | 0.253 | 0.114 | 0.31 |

B3_h6_17 | 37.5 | 30 | 0.3 | 0.275 | 0.092 | 0.25 |

C3_h6_17 | 37.5 | 45 | 0.3 | 0.367 | - | - |

A3_h6_18 | 42.0 | 15 | 0.3 | 0.275 | 0.105 | 0.28 |

B3_h6_18 | 42.0 | 30 | 0.3 | 0.299 | 0.081 | 0.21 |

C3_h6_18 | 42.0 | 45 | 0.3 | 0.380 | ||

A3_h6_20 | 50.0 | 15 | 0.3 | 0.291 | 0.132 | 0.31 |

B3_h6_20 | 50.0 | 30 | 0.3 | 0.318 | 0.105 | 0.25 |

C3_h6_20 | 50.0 | 45 | 0.3 | 0.423 | - | - |

(l) (h_{2} = 0.06 m) | ||||||

Practice | Q (l/s) | α (°) | R (m) | t (m) | AR (m) | RE |

A4_h6_17 | 37.5 | 15 | 0.4 | 0.235 | 0.049 | 0.17 |

B4_h6_17 | 37.5 | 30 | 0.4 | - | - | - |

C4_h6_17 | 37.5 | 45 | 0.4 | 0.284 | - | - |

A4_h6_18 | 42.0 | 15 | 0.4 | 0.250 | 0.047 | 0.16 |

B4_h6_18 | 42.0 | 30 | 0.4 | - | - | - |

C4_h6_18 | 42.0 | 45 | 0.4 | 0.297 | - | - |

A4_h6_20 | 50.0 | 15 | 0.4 | 0.269 | 0.049 | 0.15 |

B4_h6_20 | 50.0 | 30 | 0.4 | - | - | - |

C4_h6_20 | 50.0 | 45 | 0.4 | 0.318 | - | - |

**Table 6.**Absolute and relative errors between extrapolated and measured values and mean absolute and corresponding relative mean error of: (

**a**) axis A

_{1}, (

**b**) axis A

_{2}, (

**c**) axis B

_{1}, and (

**d**) axis B

_{2}.

(a) | |||||

Practice | A_{1 extr} | A_{1 exp} | AE (m) | RE | |

A4_h0_17 | 0.891 | 0.895 | 0.004 | 0.005 | |

A4_h0_18 | 1.035 | 1.027 | 0.008 | 0.007 | |

A4_h0_18 | 0.983 | 0.974 | 0.008 | 0.009 | |

A4_h6_17 | 0.873 | 1.085 | 0.211 | 0.195 | |

A4_h6_18 | 0.991 | 0.916 | 0.075 | 0.082 | |

A4_h6_20 | 1.027 | 1.035 | 0.007 | 0.007 | |

B3_h0_20 | 1.158 | 1.210 | 0.051 | 0.042 | |

B3_h6_17 | 1.026 | 0.982 | 0.044 | 0.045 | |

B4_h6_18 | 0.989 | 1.052 | 0.063 | 0.060 | |

B4_h6_20 | 1.062 | 1.095 | 0.032 | 0.029 | |

C4_h0_17 | 1.032 | 1.001 | 0.031 | 0.031 | |

C4_h0_20 | 1.058 | 1.120 | 0.062 | 0.056 | |

C4_h6_17 | 0.920 | 0.878 | 0.042 | 0.048 | |

C4_h6_20 | 0.901 | 0.931 | 0.030 | 0.033 | |

MAEv | 0.048 | ||||

MREv | 0.046 | ||||

(b) | |||||

Practice | A_{2 extr} | A_{2 exp} | AE (m) | RE | |

A4_h0_17 | 1.163 | 1.244 | 0.081 | 0.065 | |

A4_h0_18 | 1.075 | 1.114 | 0.038 | 0.034 | |

A4_h0_18 | 1.285 | 1.516 | 0.231 | 0.152 | |

A4_h6_17 | 1.082 | 1.137 | 0.056 | 0.049 | |

A4_h6_18 | 1.042 | 1.280 | 0.238 | 0.186 | |

A4_h6_20 | 1.067 | 1.189 | 0.122 | 0.102 | |

B3_h0_20 | 1.468 | 1.519 | 0.051 | 0.034 | |

B3_h6_17 | 2.008 | 1.525 | 0.483 | 0.316 | |

B4_h6_18 | 1.027 | 0.997 | 0.030 | 0.030 | |

B4_h6_20 | 1.137 | 1.764 | 0.627 | 0.356 | |

C4_h0_17 | 1.426 | 1.245 | 0.181 | 0.145 | |

C4_h0_20 | 1.518 | 1.559 | 0.042 | 0.027 | |

C4_h6_17 | 1.650 | 1.517 | 0.133 | 0.088 | |

C4_h6_20 | 1.468 | 1.200 | 0.269 | 0.224 | |

MAEv | 0.184 | ||||

MREv | 0.129 | ||||

(c) | |||||

Practice | B_{1 extr} | B_{1 exp} | AE (m) | RE | |

A4_h0_17 | 0.787 | 0.795 | 0.009 | 0.011 | |

A4_h0_18 | 0.810 | 0.830 | 0.020 | 0.024 | |

A4_h0_18 | 0.915 | 0.813 | 0.102 | 0.125 | |

A4_h6_17 | 0.725 | 0.753 | 0.027 | 0.036 | |

A4_h6_18 | 0.914 | 0.844 | 0.070 | 0.083 | |

A4_h6_20 | 0.930 | 0.884 | 0.046 | 0.052 | |

B3_h0_20 | 0.930 | 1.256 | 0.326 | 0.259 | |

B3_h6_17 | 0.752 | 0.846 | 0.094 | 0.111 | |

B4_h6_18 | 0.736 | 0.734 | 0.003 | 0.004 | |

B4_h6_20 | 0.861 | 0.837 | 0.023 | 0.028 | |

C4_h0_17 | 1.033 | 1.087 | 0.054 | 0.050 | |

C4_h0_20 | 0.768 | 0.809 | 0.041 | 0.051 | |

C4_h6_17 | 0.579 | 0.694 | 0.115 | 0.166 | |

C4_h6_20 | 0.752 | 1.110 | 0.358 | 0.322 | |

MAEv | 0.092 | ||||

MREv | 0.094 | ||||

(d) | |||||

Practice | B_{2 extr} | B_{2 exp} | AE (m) | RE | |

A4_h0_17 | 0.900 | 0.979 | 0.079 | 0.080 | |

A4_h0_18 | 0.989 | 1.147 | 0.159 | 0.139 | |

A4_h0_18 | 0.999 | 1.113 | 0.114 | 0.103 | |

A4_h6_17 | 0.877 | 1.085 | 0.208 | 0.191 | |

A4_h6_18 | 0.892 | 1.136 | 0.244 | 0.215 | |

A4_h6_20 | 0.910 | 0.993 | 0.083 | 0.083 | |

B3_h0_20 | 0.794 | 0.921 | 0.127 | 0.138 | |

B3_h6_17 | 0.765 | 0.991 | 0.226 | 0.228 | |

B4_h6_18 | 0.772 | 1.043 | 0.271 | 0.260 | |

B4_h6_20 | 0.967 | 1.206 | 0.239 | 0.198 | |

C4_h0_17 | 0.708 | 0.813 | 0.105 | 0.129 | |

C4_h0_20 | 1.221 | 1.343 | 0.123 | 0.091 | |

C4_h6_17 | 0.744 | 0.920 | 0.175 | 0.191 | |

C4_h6_20 | 0.652 | 0.712 | 0.060 | 0.084 | |

MAEv | 0.158 | ||||

MREv | 0.152 |

**Table 7.**MAEv and MREv of the calculated parameters of the scour hole taking the measured data as a reference, considering all the validation tests.

Parameter | L_{c}(θ) (m) | L_{c}(R) (m) | A_{1} (m) | A_{2}(m) (m) | A(m) (m) | A/B | B(m) (m) |
---|---|---|---|---|---|---|---|

MAEv (m) | 0.19 | 0.26 | 0.126 | 0.12 | 0.19 | 0.15 | 0.15 |

MREv (%) | 12.76 | 17.44 | 10.70 | 8.40 | 7.40 | 8.70 | 9.90 |

**Table 8.**Comparison of the value of A

_{2}, calculated with Equation (5), with the experimental and calculated values of L

_{c}for validation tests.

Validation Test | t (m) | cos^{2}θ | A_{2_exp} (m) | A_{2_cal} (m) | L_{c exp} (m) | L_{c} (θ) (m) | L_{c} (R) (m) | Reached Trampoline | A_{2_cal} > $\overline{{L}_{c}}$ |
---|---|---|---|---|---|---|---|---|---|

A3_h0_20 | 0.441 | 0.62 | 1.568 | 1.785 | 1.594 | 2.061 | 2.228 | no | no |

A3_h6_18 | 0.320 | 0.68 | 1.694 | 1.800 | 1.607 | 1.612 | 1.637 | yes | yes |

A4_h0_17 | 0.457 | 0.39 | 1.163 | 1.271 | 1.596 | 1.819 | 2.287 | yes | no |

A4_h6_18 | 0.410 | 0.43 | 1.042 | 1.314 | 1.572 | 1.877 | 2.087 | no | no |

B2_h0_17 | 0.291 | 0.79 | 1.970 | 2.037 | 1.429 | 1.377 | 1.405 | yes | yes |

B2_h6_20 | 0.291 | 0.87 | 2.262 | 2.202 | 1.704 | 1.568 | 1.477 | yes | yes |

B3_h0_17 | 0.367 | 0.61 | 1.666 | 1.681 | 1.641 | 1.569 | 1.771 | yes | = |

B3_h6_20 | 0.318 | 0.66 | 1.627 | 1.769 | 1.756 | 1.548 | 1.584 | yes | yes |

B4_h0_18 | 0.421 | 0.37 | 1.098 | 1.202 | 1.514 | 1.443 | 1.999 | yes | no |

B4_h6_17 | 0.367 | 0.41 | 1.216 | 1.228 | 1.594 | 1.469 | 1.786 | no | no |

C2_h0_20 | 0.285 | 0.79 | 1.885 | 2.033 | 1.507 | 1.334 | 1.324 | yes | yes |

C2_h6_18 | 0.250 | 0.86 | 2.246 | 2.154 | 1.497 | 1.322 | 1.235 | yes | yes |

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**MDPI and ACS Style**

Pellegrino, R.; Toledo, M.Á.
Characterization of the Erosion Basin Shaped by the Jet Flow of Sky-Jump Spillways. *Water* **2023**, *15*, 2930.
https://doi.org/10.3390/w15162930

**AMA Style**

Pellegrino R, Toledo MÁ.
Characterization of the Erosion Basin Shaped by the Jet Flow of Sky-Jump Spillways. *Water*. 2023; 15(16):2930.
https://doi.org/10.3390/w15162930

**Chicago/Turabian Style**

Pellegrino, Raffaella, and Miguel Á. Toledo.
2023. "Characterization of the Erosion Basin Shaped by the Jet Flow of Sky-Jump Spillways" *Water* 15, no. 16: 2930.
https://doi.org/10.3390/w15162930