Predictive Modeling the Free Hydraulic Jumps Pressure through Advanced Statistical Methods
Abstract
:1. Introduction
2. Materials and Methods
2.1. Experimental Setup
2.2. Statistical Data Analysis
3. Results and Discussion
3.1. Skewness and Kurtosis Coefficient
3.2. Cumulative Pressure Curves
3.3. Proposition of New Relationships
3.4. Comparison between Sample and Estimated Pressure Data
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 Smax; the flow detachment point (X*d) with S ≈ 0; the roller endpoint (X*r) with Smin; 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 (Nk%). 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 R2, RMSE, MAE, and WI were achieved 0.776, 0.097, 0.075, and 0.931, respectively. The constant values of Nk% are developed along the jump. Therefore, depending on the probability, the values of the Nk% coefficient indicate a single mean value for each probability. A new second-order fractional relationship was proposed to estimate the Nk% coefficient with R2 = 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) |
Fr1 | Incident Froude number |
g | Gravitational acceleration (LT−2) |
K | Kurtosis coefficient |
L | Spillway length (L) |
Lb | Length of the USBR Type I stilling basin (L) |
MAE | Mean Absolute Error |
Nk% | Statistical coefficient of probability distribution at point X |
H | Spillway height (L) |
Pk% | Pressure with a certain non-exceedance probability (L) |
P*k% | Dimensionless pressure with a certain non-exceedance probability |
Pi | Instantaneous pressure of each pressure tap (L) |
Pm | Mean pressure of each pressure tap (L) |
P*m | Dimensionless mean pressure of each pressure tap |
Q | Flow discharge (L3T−1) |
q | Flow discharge per unit width (L2T-1) |
R1 | Hydraulic radius of the incoming flow (L) |
R2 | Determination coefficient |
Re1 | Incident Reynolds number |
RMSE | Root Mean Squared Error |
S | Skewness coefficient |
SX | Sample standard deviation |
V1 | 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/ (Y2 − Y1) |
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 |
Y1 | Supercritical depth (L) |
Y2 | Sequent depth (L) |
ΔE | Energy head loss along the hydraulic jump (L) |
σX | Standard deviation of pressure fluctuations at point x (L) |
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Q (L/s) | Y1 (cm) | Y2 (cm) | V1 (m/s) | Fr1 (−) |
---|---|---|---|---|
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 | R2 |
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 |
Fr1 | 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 | R2 | 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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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
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 StyleMousavi, 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