# Predictive Modeling the Free Hydraulic Jumps Pressure through Advanced Statistical Methods

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## Abstract

**:**

_{m}), the standard deviations of pressure fluctuations (σ*

_{X}), pressures with different non-exceedance probabilities (P*

_{k%}), and the statistical coefficient of the probability distribution (N

_{k%}) were assessed. It was found that an existing method can be used to interpret the present data, and pressure distribution in similar conditions, by using a new second-order fractional relationships for σ*

_{X}, and N

_{k%}. The values of the N

_{k%}coefficient indicated a single mean value for each probability.

## 1. Introduction

_{P}) and peak frequencies of the spatial hydraulic jumps are higher than the classical jumps. Onitsuka et al. [15] found that roller oscillations affect the instantaneous flow depth and bed pressure. In addition, the instantaneous bed pressures are associated with free surface fluctuations. Lian et al. [16] stated that the fluctuating pressure spectrum in the rolling area follows the gravity similarity law. Lopardo and Romagnoli [17] and Lopardo [18] used Cʹ

_{P}coefficient values to estimate the turbulence intensities close to the stilling basin bed for the low incident Froude numbers. Wang et al. [19] predicted the total pressure based upon the void fraction and velocity data, and the results were in good agreement with the experimental data. Firotto et al. [20] studied the stability of a plunge pool lining under the fully developed jets and proposed a design approach to determine the thickness of the linings. Barjastehmaleki et al. [21] investigated the statistical structure of fluctuating pressures within the stilling basins. Barjastehmaleki et al. [22] evaluated an approach for the structural design of stilling basins lining in the sealed and unsealed joints. Lopardo [9] recommended specific flow conditions to measure pressure fluctuations. According to this, the supercritical Reynolds number (Re

_{1}) should be more than 100,000. The minimum acquisition time must be 60 seconds. The acquisition frequency can be considered between 50 and 100 Hz. The maximum length of the plastic tube between the pressure tap and transducer is equal to 55 cm with a minimum inner diameter of 5 mm.

_{k%}) from the sample data within a stilling basin. Teixeira et al. [27] provided the cumulative curves of P*

_{k%}for characteristic points along the hydraulic jump. Souza et al. [28] investigated the behavior of the hydraulic jump concerning the longitudinal distribution of pressures near the bottom of the basin in the low Froude number zone (Fr

_{1}≤ 4.5). Prá et al. [29] investigated the influence of the vertical curve between the spillway toe and the stilling basin bed. The results showed that maximum pressure fluctuations were identified at the center of the vertical curve and assume values of 1% of the flow kinetic energy at the terminal tangency point of the curve. Novakoski et al. [30] investigated extreme pressures with different probabilities (P*

_{k%}) on a smooth basin downstream of a stepped spillway. The results showed that the values of P*

_{0.1%}and P*

_{99.9%}have lower and higher values than the values observed downstream of the smooth chute, in the region near the spillway toe, respectively.

_{2}−Y

_{1}). Analysis of S and K coefficients displays that there are several types of distributions along hydraulic jumps. Therefore, it is difficult to estimate the pressure data with a certain probability (P*

_{k%}). They proposed dimensionless relationships linking pressure data of P*

_{k%}to the mean pressure (P*

_{m}), and the standard deviation of the sample data (σ*

_{X}). Such relationships allow us to organize the results of different flow discharges or Froude numbers and characterize the interest points in hydraulic jumps.

_{1}) ranging from 7.12 to 9.46. New relationships will be proposed for the dimensionless standard deviation (σ*

_{X}), and the statistical coefficient of the probability distribution (N

_{k%}) to estimate the extreme pressures with different non-exceedance probabilities (P*

_{k%}).

## 2. Materials and Methods

#### 2.1. Experimental Setup

_{b}) was considered 200 cm [35]. The basin width (B) was equal to the flume width (50 cm). The radius of the vertical curve (R) at the spillway toe was 12 cm. There was a head tank with 250 cm height to stabilize the flow upstream of the spillway. A hinged weir downstream of the flume was used to control the position of the supercritical depth (Y

_{1}) at the spillway toe. The sequent depth (Y

_{2}) was measured by an ultrasonic sensor, with an operating in the range of 10 to 100 cm, and the accuracy of the nominal value the manufacture ±0.1 mm. For the classical hydraulic jump (CHJ), the most relevant parameter is the incident Froude number (Fr

_{1}). The Froude number characterizes the balance between inertial and gravitational forces. A value of Fr

_{1}> 1 indicates the supercritical flow, and vice versa for Fr

_{1}< 1 [36,37,38].

_{1}is the mean supercritical velocity; d

_{0}is the hydraulic head upstream of the spillway crest; Z is the total water depth upstream of the spillway (Z = H + d

_{0}); and g is the gravitational acceleration. The values of Y

_{1}are calculated using the continuity low (Y

_{1}= q/V

_{1}), where q is the flow discharge per unit width. Figure 2 displays some experimental parameters. Figure 3 shows the distribution of pressure taps along the centerline of the stilling basin. The flow discharge (Q) was measured with an ultrasonic flowmeter. Experiments were carried out with different flow discharges in the range of 33 to 60.4 L/s. Table 1 presents the range of some experimental parameters along the hydraulic jumps.

#### 2.2. Statistical Data Analysis

_{k%}) at the point X can be estimated using Equation (3) [31]:

_{m}is the mean pressure at the point X (in cm of water column); N

_{k%}is the dimensionless statistical coefficient of the probability distribution at the point X; σ

_{X}is the standard deviation of pressure fluctuations at the point X (cm). The dimensionless mean pressure (P*

_{m}), and the dimensionless pressure with a certain probability (P*

_{k%}) can be expressed as a generic function of X*, and defined as follows [31]:

_{X}) is defined as follows [31]:

_{1}), and the distance of the point from the jump toe. Based on Equation (3), Teixeira [26] proposed an estimation method for the extreme pressures with different probabilities (P*

_{k%}) along free hydraulic jumps for smooth stilling basins, downstream of spillways. The method is applied to stable hydraulic jumps (4.5 < Fr

_{1}< 9), and includes the assessment of the dimensionless statistical parameters (mean pressures, standard deviation, and statistical probability distribution coefficient) as a function of X* along stilling basins with the smooth bed. These parameters are defined as follows [26]:

^{2}) [39], are provided in Table 2 [26].

_{i}is the instantaneous pressure head at each pressure tap (in cm of water column); S

_{X}is the sample standard deviation; and n is the number of data. This value represents the pressure fluctuations concerning the mean value of the sample data. A value of S < 0 refers to a longer or fatter tail on the left side of the density probability function distribution (PDF), and vice versa for S > 0.

_{X}). The value of K is a measure of the spread of data around the mean value, characterizing the flatness of the PDF curve. A value of K < 3 indicates the data distribution function is more flattened and less concentrated to the mean values compared to a normal distribution, and vice versa for K > 3. The sample kurtosis coefficient is defined as [40,41]:

_{m}, σ

_{X}, P

_{k%}, and N

_{k%}. From the analysis of the probability distribution of sample pressure data, the values of P

_{k%}were determined. Then, the dimensionless form of pressure data (P*

_{k%}) was taken to compare the results with different arrangements, obtained from a series of data with different geometries. These parameters were analyzed longitudinally, along the stilling basin, and were made dimensionless using Equations (4)–(6), respectively.

_{m}, and σ

_{X}were calculated using Equations (4) and (6), respectively. Finally, the estimated values of P

_{k%}was calculated using Equation (5). To optimize the pressure estimation method proposed by Teixeira [26], new relationships were developed for the parameters of σ*

_{X}and N

_{k%}, as a function of X* along the stilling basin. The results of P*

_{k%}, obtained from the analysis of the probability distribution of the experimental data were compared with the corresponding estimated values using the method by Teixeira [26], and the new optimized estimation method proposed in this study.

## 3. Results and Discussion

#### 3.1. Skewness and Kurtosis Coefficient

_{σmax}), where the skewness coefficient is high, and S

_{max}is in the range of 0.5 to 1.5. The position of the flow detachment (X*

_{d}), where the skewness coefficient shifts from a positive value to a negative one (S ≈ 0). The roller endpoint (X*

_{r}) indicates the minimum skewness coefficient (S

_{min}). The hydraulic jump endpoint (X*

_{j}) is where the streamlines become parallel to the basin bed. At this position, S ≈ 0 and K ≈ 0.

_{σmax}, X*

_{d}, X*

_{r}, and X*

_{j}along the stilling basin for different Froude numbers.

#### 3.2. Cumulative Pressure Curves

_{k%}), the cumulative pressure curves were provided for each pressure tap with different Froude numbers. Figure 6 presents the cumulative pressure curves for P*

_{k%}related to the characteristic points of X*

_{σmax}, X*

_{d}, X*

_{r}, and X*

_{j}, respectively.

_{k%}values increase with increasing probability (k%). Minimum pressure data (P*

_{min}) correspond to the lowest probabilities (P*

_{1%}). On the contrary, maximum pressure data (P*

_{max}) correspond to the highest probability (P*

_{99%}). Accordingly, the maximum pressure fluctuations, and the negative pressures, are located at the positions near the spillway toe. Also, the minimum pressure fluctuations are located at the positions downstream of the hydraulic jump.

#### 3.3. Proposition of New Relationships

_{X}). Thus, a new second-order fractional relationship (rational model), as a function of the dimensionless position along the stilling basin, is introduced.

_{X}, and fitting of Equation (15), with a determination coefficient (R

^{2}) equal to 0.776.

_{k%}with different non-exceedance probabilities are determined. Figure 8 shows the longitudinal distribution of the N

_{k%}coefficient.

_{k%}are developed along the jump, especially in the case of pressures with probabilities of 5%, 10%, 90%, and 95%. Therefore, depending on the probability, the values of the coefficient of N

_{k%}indicate a single mean value for each probability. According to Wiest [42], there is no significant effect of the parameter of Fr

_{1}, and the values of N

_{k%}remain somewhat constant throughout the basin. A new second-order fractional relationship (rational model) can estimate the N

_{k%}coefficient with a determination coefficient (R

^{2}) equal to 0.98.

_{k%}for each probabilities (k%), and the proposed relationship (Equation 16).

#### 3.4. Comparison between Sample and Estimated Pressure Data

_{k%}with probabilities of 1%, 5%, 10%, 90%, 95%, and 99%. Experimental data are presented as a function of X*, together with the corresponding estimates using Teixeira [26], also modified using Equation (15).

_{1%}and P*

_{99%}, have lower and higher values, with the maximum differences than P*

_{m}. P*

_{1%}data reach negative values down to −0.2, at the position X* ≈ 2, indicating regions with low pressures. To evaluate the performance of the experimental and the estimated values of σ*

_{X}, some statistical performance criteria including determination coefficient (R

^{2}) [39], root mean squared error (RMSE) [39], mean absolute error (MAE) [39], Willmott’s index of agreement (WI) [43] are provided in Table 4. As a result, the goodness of fit statistics for the estimation of σ*

_{X}is confirmed.

^{2}and WI values should be close to the unit. According to Table 4, this relationship for σ*

_{X}provides better estimation performance as compared against Teixeira [26]. The new relationship is given in Equation (15) presents somewhat better results for P*

_{90%}, P*

_{95%}, and P*

_{99%}along the stilling basin.

## 4. Conclusions

- (1)
- Sample skewness (S) and kurtosis (K) coefficients indicated that the pressure distribution along the hydraulic jumps does not follow a normal distribution. Some characteristic points are the maximum pressure fluctuations point (X*
_{σmax}) with S_{max}; the flow detachment point (X*_{d}) with S ≈ 0; the roller endpoint (X*_{r}) with S_{min}; and the hydraulic jump endpoint (X*_{j}) with S ≈ 0. - (2)
- From the pressure data with different non-exceedance probabilities (P*
_{k%}), the cumulative pressure curves are presented for P*_{k%}related to the characteristic points of X*_{σmax}, X*_{d}, X*_{r}, and X*_{j}, respectively. For the positions close to the spillway toe, pressures with low and high probability (P*_{1%}and P*_{99%}), have lower and higher values, with the maximum differences than P*_{m}. P*_{1%}data, reach negative values down to −0.2, at the position X* ≈ 2, indicating regions with low pressures. - (3)
- From the analysis of the probability distribution of the sample data as collected by pressure transducers, pressures data of P*
_{k%}can be determined. - (4)
- Based on the results obtained, it was observed that the method proposed by Teixeira [26] could be optimized to be used for present data, or in similar conditions by using another relationship for the dimensionless standard deviation of pressure fluctuations (σ*
_{X}), and the statistical coefficient of the probability distribution (N_{k%}). Thus, a new second-order fractional relationship, as a function of the dimensionless position along the stilling basin (X*), is introduced for σ*_{X}. This relationship is valid for the dimensionless positions (X*) in the range of 0 to 8.4. To assess the accuracy of this relationship, some performance criteria are used. For the new proposed relationship (σ*_{X}) in this study, the values of R^{2}, RMSE, MAE, and WI were achieved 0.776, 0.097, 0.075, and 0.931, respectively. The constant values of N_{k%}are developed along the jump. Therefore, depending on the probability, the values of the N_{k%}coefficient indicate a single mean value for each probability. A new second-order fractional relationship was proposed to estimate the N_{k%}coefficient with R^{2}= 0.98. The new relationships should be validated against sample data taken in similar conditions to our case study here. - (5)
- The results contribute to enhancing the knowledge of the flow in a USBR Type I stilling basin that can be used to improve their design. This work only includes the case of free jumps. Future advancements will cover the behavior of submerged jumps with variable submergence degrees, resulting in modified pressure fields concerning those observed here for free jumps. As well, the efficiency of blocks and sills with different sizes may be investigated. A more extensive range of flow discharge will need to be explored. In the future, the more specific effort may be devoted to testing other possible distributions, fitting the observed pressure fields and their use in practice design. Also, velocity fields within the hydraulic jump may be investigated to define the turbulent components of flow fields.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Notation

B | Basin width (L) |

Fr_{1} | Incident Froude number |

g | Gravitational acceleration (LT^{−2}) |

K | Kurtosis coefficient |

L | Spillway length (L) |

L_{b} | Length of the USBR Type I stilling basin (L) |

MAE | Mean Absolute Error |

N_{k%} | Statistical coefficient of probability distribution at point X |

H | Spillway height (L) |

P_{k%} | Pressure with a certain non-exceedance probability (L) |

P*_{k%} | Dimensionless pressure with a certain non-exceedance probability |

P_{i} | Instantaneous pressure of each pressure tap (L) |

P_{m} | Mean pressure of each pressure tap (L) |

P*_{m} | Dimensionless mean pressure of each pressure tap |

Q | Flow discharge (L^{3}T^{−1}) |

q | Flow discharge per unit width (L^{2}T^{-1}) |

R_{1} | Hydraulic radius of the incoming flow (L) |

R^{2} | Determination coefficient |

Re_{1} | Incident Reynolds number |

RMSE | Root Mean Squared Error |

S | Skewness coefficient |

S_{X} | Sample standard deviation |

V_{1} | Mean supercritical velocity (LT^{−1}) |

WI | Willmott’s index of agreement |

X | Distance of each pressure tap from the spillway toe (L) |

X* | Dimensionless distance of each pressure tap from the spillway toe, i.e., X/ (Y_{2} − Y_{1}) |

X*_{d} | Point of the flow detachment |

X*_{j} | Endpoint of the hydraulic jump |

X*_{r} | Endpoint of the roller |

X*_{σmax} | Point of the maximum pressure fluctuations |

Y_{1} | Supercritical depth (L) |

Y_{2} | Sequent depth (L) |

ΔE | Energy head loss along the hydraulic jump (L) |

σ_{X} | Standard deviation of pressure fluctuations at point x (L) |

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**Figure 6.**Cumulative pressure curves for P*

_{k%}related to the characteristic points of the hydraulic jump: (

**a**) X*

_{σmax}, (

**b**) X*

_{d}, (

**c**) X*

_{r}, and (

**d**) X*

_{j}.

**Figure 10.**Distributions of P*

_{k%}with different probabilities: (

**a**) P*

_{1%}, (

**b**) P*

_{5%}, (

**c**) P*

_{10%}, (

**d**) P*

_{90%}, (

**e**) P*

_{95%}, and (

**f**) P*

_{99%}.

Q (L/s) | Y_{1} (cm) | Y_{2} (cm) | V_{1} (m/s) | Fr_{1} (−) |
---|---|---|---|---|

60.4 | 3.04 | 27.55 | 3.89 | 7.12 |

55.0 | 2.78 | 26.49 | 3.88 | 7.44 |

52.7 | 2.66 | 26.05 | 3.88 | 7.59 |

47.5 | 2.41 | 24.87 | 3.87 | 7.96 |

43.0 | 2.18 | 23.70 | 3.86 | 8.34 |

33.0 | 1.68 | 20.65 | 3.84 | 9.46 |

k% | α | b | c | R^{2} |

1% | +0.0512 | −0.4480 | −1.6601 | 0.92 |

5% | +0.0130 | −0.1323 | −1.3061 | 0.73 |

10% | +0.0032 | −0.0450 | −1.0869 | 0.59 |

90% | +0.0048 | −0.0325 | +1.2695 | 0.26 |

95% | +0.0171 | −0.1393 | +1.8624 | 0.81 |

99% | +0.0317 | −0.3598 | +3.3008 | 0.86 |

Fr_{1} | X*_{σmax} | X*_{d} | X*_{r} | X*_{j} |
---|---|---|---|---|

7.12 | 1.734 | 3.98 | 5.81 | 7.71 |

7.44 | 1.79 | 4.11 | 6.01 | 7.97 |

7.59 | 1.60 | 3.95 | 6.09 | 8.08 |

7.96 | 1.67 | 3.89 | 5.45 | 8.41 |

8.34 | 1.74 | 3.83 | 5.69 | 7.55 |

9.46 | 2.00 | 4.09 | 5.40 | 7.51 |

[31] | 1.75 | 4.00 | 6.00 | 8.50 |

Method | R^{2} | RMSE | MAE | WI |
---|---|---|---|---|

Equation (15) | 0.776 | 0.097 | 0.075 | 0.931 |

[26] | 0.674 | 0.140 | 0.115 | 0.875 |

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**MDPI and ACS Style**

Mousavi, S.N.; Júnior, R.S.; Teixeira, E.D.; Bocchiola, D.; Nabipour, N.; Mosavi, A.; Shamshirband, S.
Predictive Modeling the Free Hydraulic Jumps Pressure through Advanced Statistical Methods. *Mathematics* **2020**, *8*, 323.
https://doi.org/10.3390/math8030323

**AMA Style**

Mousavi SN, Júnior RS, Teixeira ED, Bocchiola D, Nabipour N, Mosavi A, Shamshirband S.
Predictive Modeling the Free Hydraulic Jumps Pressure through Advanced Statistical Methods. *Mathematics*. 2020; 8(3):323.
https://doi.org/10.3390/math8030323

**Chicago/Turabian Style**

Mousavi, Seyed Nasrollah, Renato Steinke Júnior, Eder Daniel Teixeira, Daniele Bocchiola, Narjes Nabipour, Amir Mosavi, and Shahabodin Shamshirband.
2020. "Predictive Modeling the Free Hydraulic Jumps Pressure through Advanced Statistical Methods" *Mathematics* 8, no. 3: 323.
https://doi.org/10.3390/math8030323