1379

Prediction of pressure fluctuations on sloping stilling basins A. Güven, M. Günal, and A. Çevik

Abstract: Various types of hydraulic jump occurring on horizontal and sloping channels have been analyzed experimentally, theoretically, and numerically and the results are available in the literature. In this study, artificial neural network models were developed to simulate the mean pressure fluctuations beneath a hydraulic jump occurring on sloping stilling basins. Multilayers feed a forward neural network with a back-propagation learning algorithm to model the pressure fluctuations beneath such a type of hydraulic jump (B-jump). An explicit formula that predicts the mean pressure fluctuation in terms of the characteristics that contribute most to the hydraulic jump occurring on the sloping basins is presented. The proposed neural network models are compared with linear and nonlinear regression models that were developed using considered physical parameters. The results of the neural network modelling are found to be superior to the regression models and are in good agreement with the experimental results due to relatively small values of error (mean absolute percentage error). Key words: neural networks, pressure fluctuation, hydraulic jump, sloping stilling basin, explicit NN formulation, regression analysis. Résumé : Divers types de ressauts hydrauliques se trouvant dans les canaux horizontaux et inclinés ont été analysés expérimentalement, théoriquement et numériquement et les résultats sont disponibles dans la littérature. La présente étude développe des modèles de réseaux neuronaux artificiels pour simuler les variations moyennes de pression sous les ressauts hydrauliques le long des bassins d’amortissement inclinés. Un réseau neuronal à couches non bouclé muni d’un algorithme d’apprentissage à rétropropagation est utilisé pour modéliser les variations de pression sous de tels types de ressauts hydrauliques (B-Jump). Une formule explicite qui prédit la variation moyenne de pression en termes des caractéristiques contribuant le plus au ressaut hydraulique dans les bassins d’amortissement est présentée. Les modèles de réseaux neuronaux artificiels proposés sont comparés aux modèles de régression linéaire et non linéaire développés en utilisant les paramètres physiques arrêtés. Les résultats de la modélisation des réseaux neuronaux s’avèrent supérieurs à ceux des modèles de régression et présentent une bonne corrélation avec les résultats expérimentaux en raison des valeurs relativement petites de l’erreur (pourcentage d’erreur moyenne absolue). Mots clés : réseaux neuronaux, variation de pression, ressaut hydraulique, bassin d’amortissement incliné, formulation de réseau neuronal explicite, analyse de régression. [Traduit par la Rédaction]

Introduction Over the years, several aspects of hydraulic jumps forming in horizontal and sloping channels have been investigated. In an open channel, if the downstream depth is greater than the downstream sequent depth, the jump moves upstream to submerge the incoming flow. Such a jump is called a submerged hydraulic jump. The hydraulic jump forming in the stilling basin is a free hydraulic jump if it forms at the foot of the spillway. Usually, the jump is enclosed within a stilling basin. On the other hand, if the downstream depth of the flow in the stilling basin is greater than the sequent depth, the hydraulic jump moves towards the Received 1 May 2006. Revision accepted 19 July 2006. Published on the NRC Research Press Web site at http://cjce.nrc.ca/ on 23 January 2007. A. Güven,1 M. Günal, and A. Çevik. Department of Civil Engineering, Faculty of Engineering, University of Gaziantep, 27310 Gaziantep, Turkey. Written discussion of this article is welcomed and will be received by the Editor until 31 March 2007. 1

Corresponding author (e-mail: [email protected]).

Can. J. Civ. Eng. 33: 1379–1388 (2006)

spillway with its toe positioned on the spillway. Such a jump is called a B-jump and a B-jump may be considered to belong to the class of submerged jumps with the entering flow at an angle to the bed of the channel (Fig. 1). Mean flow characteristics of a B-jump were investigated in detail by Hager (1989), and Ohtsu and Yasuda (1990, 1991). These studies relate to the mean velocity measurements, roller length, and length of the jump. Abdul Khader and Elongo (1974), Lopardo and Henning (1985), Toso and Bowers (1988), and Fattor et al. (2001) studied statistical properties of fluctuating pressures beneath a hydraulic jump formed downstream of the spillway. On the other hand, Vasiliev and Bukreyev (1967), Narasimhan and Bhargava (1976), Narayanan (1978), and Lopardo et al. (2004) measured the intensity of the fluctuating pressures beneath submerged and free jumps downstream of a sluice gate in the horizontal channel. Clearly, these two sets of experiments with respect to hydraulic jumps, one downstream of a spillway and the other downstream of a sluice gate, are different with respect to the upstream conditions. It is well known that different upstream conditions such as the mean velocity distribution and intensity of turbulence have a strong influence on the magnitude of the pressure fluctuations and mean flow properties. It is worth noting

doi: 10.1139/L06-101

© 2006 NRC Canada

1380 Fig. 1. Sketch of B-jump in sloping channel.

Can. J. Civ. Eng. Vol. 33, 2006




2

p
, based on the basic parameters of the geometry of the stilling basin and the incoming flow conditions. The proposed neural network models are compared with linear and nonlinear regression models that were developed using considered physical parameters. The statistical results of the proposed models are tabulated.

Numerical background that the investigations with the jump downstream of the spillway do not agree even among themselves. Abdul Khader and Elongo (1974) explained that the large intensity they obtained is the result of a large turbulence level and the upstream velocity profile. The experimental results of Narasimhan and Bhargava (1976) and Narayanan (1978) concerning the intensity of the pressure fluctuations with the hydraulic jump downstream of sluice gate collapse to a single curve. There is another important aspect relating to pressure fluctuations that is of interest to designers of a stilling basin. It is well known that the pressure fluctuations can be potentially destructive because they can cause failure due to fatigue, structural vibrations, and lifting of whole slabs. Also cavitation can be induced because of large instantaneous depressions that can lead to damage of material. Bowers et al. (1964) reported some information on the damages caused by pressure fluctuations in a hydraulic jump to the Karnafuli spillway. Fattor et al. (2001) studied the destructive action of macroturbulent flows induced by hydraulic jumps in stilling basins with a particular focus on cavitation inception and cavitation damages by severe pressure fluctuations in relatively low-velocity flows. Numerical modelling has been applied to describe the physical processes occurring in the aquatic environment for some 30 years. Physical modelling requires expensive instruments, test rigs, and skilled workers, as well as being highly timeconsuming and difficult in many cases to repeat with the same experimental conditions with high precision. Therefore, it calls for use of numerical and mathematical modelling that is well established and widely accepted technique. Despite the many successes of numerical models, their wider application is restrained by their heavy demand in computing capacity and time. It has been found that for applications like real-time and nearreal-time control, the demand on computing time and resources are of such huge magnitudes that they are sometimes far from acceptable. Recent developments in subsymbolic techniques (artificial neural networks, classifier systems, genetic programming, fuzzy logic, etc.) have produced more potential in simulating physical processes, based on measured data. Recently, artificial neural network (ANN) modelling has become the most widely used technique in predicting hydraulic data. An important advantage of ANNs compared to classical stochastic models is that they do not require variables to be stationary and normally distributed. Furthermore, ANNs are relatively stable with respect to noise in the data and have a good generalization potential to represent input–output relationships. This paper explores the use of neural networks to simulate the pressure fluctuations occurring on sloping stilling basins. Another aspect of this study is to obtain an explicit neuralnetwork formula that predicts the mean pressure fluctuations,

The instantaneous values of the pressure p are separated into mean p and fluctuating quantities p
as [1]

p = p + p


In eq. [1], the mean pressure p is expressed nondimensionally as Cp and it is defined as follows: [2]

Cp =

p ρw u21 /2

where ρw is the density of the water and u1 is the upstream velocity of the water coming from the sluice. In eq. [1], the measure of the intensity of the pressure fluctu
2

ations is their root mean square (rms) value of p
, in which p
is the fluctuating part of the pressure over
its mean value. 2

The rms value of the pressure fluctuations p
is expressed nondimensionally using the dynamic pressure of the incoming flow as
2 p

[3] Cp = ρw u21 /2 The numerical value of the dimensionless parameter Cp
of the fluctuating pressures is sufficiently representative for the instantaneous actions generated by the macroturbulence in a hydraulic jump (Lopardo et al. 1999). This parameter is important for assessing the effect of the uplift of bottom slabs especially for the tendency to cavitation by pressure pulses.

Dimensional analysis The magnitude and extent of the pressure fluctuations are reported to be influenced by the geometry of the stilling basin, incoming flow conditions, and Froude number (Fiorotto and Rinaldo 1992; Fattor et al. 2001).
2

Referring to Fig. 1, the mean-pressure fluctuation ( p
) beneath the B-jump, can be written as a function of gate opening (y1 ), upstream velocity of water issuing from the gate (u1 ), tail water depth (yt ), the distance from the toe of the jump to the point where the slope joins the channel bed (Ls ), the distance from the intersection of horizontal and inclined section (x) inclination angle of the sloping part of channel in degrees (φ), gravitational acceleration (g), and the density of water ρw :
2 p
= f (Ls , φ, yt , y1 , x, g, ρw ) [4] © 2006 NRC Canada

Güven et al.

Using the Buckingham π theorem, nondimensional equations in functional form can be obtained as
2   p
Ls yt x [5] , , , F , φ = f 1 y1 y1 y1 ρw u21 /2

Experimental study and data collection Gunal carried out experiments for measuring the pressure fluctuations for B-jumps. Experiments were carried out in a flume of rectangular section 0.91 m wide and 3.20 m long, having glass sides. Fluctuating pressure measurements were carried out for three different slopes of φ = 10◦ , 20◦ , and 30◦ . The jet issuing through the sluice opening converges to form a vena contracta with a constant coefficient of 0.93–0.96. Experiments were carried out for three different Froude numbers F1 at the vena contracta. To see the relation between Froude numbers and the intensity of pressure fluctuations for each slope, the hydraulic jump was formed at four different positions in the channel. The position of the jump Ls was measured from the channel junction to the toe of the jump. Further details of this experimental study can be found in Gunal (1996). The respective Froude numbers F1 and upstream and downstream depths y1 and yt , of the flow are given in Table 1.

Overview of artificial neural networks The basics and elements of artificial neural-network modelling and the back-propagation training procedure are widely presented in the literature. An interested reader who is looking for more information can consult any textbook in neural computing, e.g., (Haykin 2000) or can refer to other previously published works in related journals (Grubert 1995; Dolling and Varas 2002; Nagy et al. 2002; Azmathullah et al. 2005, 2006). Most of the ANN applications in water engineering can be found in the ASCE Task Committee (2000), Maier and Dandy (2000), Negm (2001), Negm and Shouman (2002), and Dolling and Varas (2002). Some most recent studies on ANNs were applied by Nagy et al. (2002) in a prediction of sediment-load concentration in rivers, by Sarghini et al. (2003) in the modelling of turbulent flows in co-operation with large-eddy simulation, by Azmathullah et al. (2005) in the prediction of scour downstream of a ski-jump bucket, and by Azmathullah et al. (2006) in prediction of scour below spillways.

Neural network modelling application The main focus of this study is to predict the mean-pressure fluctuations beneath hydraulic jumps occurring on sloping channels by means of ANNs based on experimental results. The experimental results were derived from Gunal (1996). Two neural network models are proposed, namely, the first predicts the
2 mean pressure fluctuation, p
, as a function of basic geometrical and incoming flow variables affecting the pressure fluctuation. The second model predicts the nondimensional pressure fluctuation parameter, Cp
, as a function of nondimensional parameters derived from dimensional analysis using the same physical variables used in the first model. The variables used in the experimental study are given in Table 1. These experimental

1381 Table 1. Experimental conditions. Run No.

φ (◦ )

Ls (m)

Q (m3 /s)

y1 (m)

u1 (m/s)

yt (m)

F1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

10 10 10 10 10 10 10 10 10 10 10 10 20 20 20 20 20 20 20 20 20 20 20 20 30 30 30 30 30 30 30 30 30 30 30 30

0 10 20 30 0 10 20 30 0 10 20 30 0 10 20 30 0 10 20 30 0 10 20 30 0 15 20 27 0 15 20 27 0 15 20 27

0.04 0.04 0.0.4 0.04 0.052 0.052 0.052 0.052 0.066 0.066 0.066 0.066 0.03 0.03 0.03 0.03 0.045 0.045 0.045 0.045 0.068 0.068 0.068 0.068 0.036 0.036 0.036 0.036 0.045 0.046 0.046 0.046 0.060 0.060 0.060 0.060

0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.016 0.016 0.016 0.016 0.18 0.18 0.18 0.18 0.02 0.02 0.02 0.02 0.016 0.016 0.016 0.016 0.18 0.18 0.18 0.18 0.02 0.02 0.02 0.02

2.19 2.19 2.19 2.19 2.88 2.88 2.88 2.88 3.60 3.60 3.60 3.60 2.05 2.05 2.05 2.05 2.73 2.73 2.73 2.73 3.71 3.71 3.71 3.71 2.18 2.18 2.18 2.18 2.64 2.64 2.64 2.64 3.64 3.64 3.64 3.64

0.135 0.145 0.150 0.160 0.167 0.173 0.180 0.185 0.215 0.220 0.225 0.232 0.117 0.132 0.151 0.177 0.166 0.171 0.180 0.199 0.220 0.226 0.238 0.250 0.130 0.170 0.190 0.210 0.165 0.200 0.210 0.220 0.200 0.220 0.240 0.250

5.00 5.00 5.00 5.00 6.50 6.50 6.50 6.50 8.46 8.46 8.46 8.46 5.17 5.17 5.17 5.17 6.50 6.50 6.50 6.50 8.38 8.38 8.38 8.38 5.18 5.18 5.18 5.18 6.13 6.13 6.13 6.13 8.10 8.10 8.10 8.10

Note: φ, slope of inclined section of channel in degrees; Ls , the distance from the toe of the jump to the point where the slope joins the channel bed; Q, discharge; y1 , gate opening; u1 , upstream velocity of water issuing from the gate; yt , tail water depth; F1 , upstream Froude number.

tests were used as training and test sets for both neural network (NN) training. Among these 654 tests 131 (20% of total) tests were used as a test set and the remaining as a training set for NN training. Interested readers who are looking for whole input samples can refer to the corresponding author. The ranges of variables are presented in Table 2.

Optimal NN model selection In this study a MATLAB® NN toolbox was used for NN applications. The performance of a NN model mainly depends on the network architecture and parameter settings. One of the most difficult tasks in NN studies is to find this optimal network architecture that is based on the determination of the numbers of optimal layers and neurons in the hidden layers using a © 2006 NRC Canada

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Can. J. Civ. Eng. Vol. 33, 2006

Fig. 2. Optimum NN structure for NN model 1.

Table 2. Minimum and maximum values for canal section and incoming flow variables. Variable Inclination angle of the sloping part of channel (φ) (◦ ) Gate opening (y1 ) (mm) Upstream velocity of water issuing from the gate (u1 ) (m/s) Tail water depth (yt ) (mm) Distance from the toe of the jump to the point where the slope joins the channel bed (Ls ) (mm) Distance from the intersection of the horizontal and inclined section (x) (mm) Discharge (Q) (m3 /s) Upstream Froude number (F1 )

Minimum value

Maximum value

0

30

16 2.05

21.93 3.71

117 0

250 300

−35.625

36.675

0.03 5.0

0.068 8.46

trial and error approach. This process is very time consuming. Several back-propagation training algorithms are used and are given in Table 3. A program has been developed in MATLAB® that handles the trial and error process automatically. The program tries various numbers of layers and neurons in the hidden layers both for first and second hidden layers for a constant epoch for several times and selects the best NN architecture with the minimum MAPE (Mean Absolute Percentage Error). This process is repeated N times where N denotes the number of hidden nodes for the first hidden layer. This whole process is repeated for a changing number of nodes in the second hidden layer. More over, this selection process is performed for different back-propagation training algorithms such as “trainlm”, “trainscg”, and “trainbfg” given in Table 3. The optimal NN architecture in this study was found to be 6–17–1 (no. of input parameters – no. of hidden nodes – no. of output parameters) for NN Model 1, and 5–17–1 for NN Model 2, with sigmoidal transfer function. The training algorithm was Levenberg–Marquardt back propagation. The optimum NN structure for NN Model 1 is given in Fig. 2.

Results of neural network modelling Table 3. Back-propagation training algorithms used in NN training. MATLAB® function name trainbfg traincgf traincgp traingd traingda traingdx trainlm trainoss trainrp trainscg

Algorithm BFGS quasi-Newton back propagation Fletcher–Powell conjugate gradient back propagation Polak–Ribiere conjugate gradient back propagation Gradient descent back propagation Gradient descent with adaptive lr back propagation Gradient descent w/momentum and adaptive lr back propagation Levenberg–Marquardt back propagation One step secant back propagation Resilient back propagation (Rprop) Scaled conjugate gradient back propagation

Statistical parameters of normalized values of learning and training sets of NN Model 1 and Model 2 are presented in Table 4. The percentage errors and prediction of NN models and actual values of learning and testing sets are given in Figs. 3– 10. From Figs. 3–10, it can be concluded that, evidently, both NN models have a good performance in predicting the meanpressure fluctuation occurring on sloping stilling basins with relatively small error (MAPE) and high correlation coefficient (R). NN Model 1 (R = 0.980) shows moderately better performance over NN Model 1 (R = 0.944) in both learning and testing stages. The percentage error plots (Figs. 5–6 and 9–10) are also consistent with previous findings, showing relatively small errors for both models. Figures 11–13 show the predicted intensity of pressure fluctuations versus the experimental results along x/y1 for various θ , Ls /y1 , and F1 values. In Figs. 11–13, the peak value of Cp
occurs at the channel junction. The reason for this is that Cp
rises abruptly at the corner. If the experiments were carried out in a channel joined to the channel bed very smoothly, then the values © 2006 NRC Canada

Güven et al.

1383 Table 4. Statistical parameters of optimum NN models. NN model Model 1 Model 2

Training set Test set Training set Test set

MSE

RMSE

SSE

MAPE (%)

Correlation coefficient (R)

21.255 87 161.892 8 0.000 0239 0.000 0111

4.610 409 12.723 71 0.004 893 0.003 326

713 943.85 106 201.0 0.015 707 0.007 256

0.975 213 1.543 274 5.341 033 1.468 142

0.980 0.960 0.956 0.912


   x 2 y 2 ; x = X − X
; y = Y − Y
; X, observed pressure Note: R(correlation coefficient) = xy/
fluctuation parameter values; X , mean of X; Y , predicted pressure fluctuation parameter values; Y
, mean of Y ;  MSE (mean square error) = (X − Y )2 /n; n, total number of pairs X and Y values; RMSE (root mean square error)  1/2   2 ; MAPE (mean absolute percentage error) = X − Y )/Y x100 n; and SSE (sum squared error) = (X − Y ) /n  = (X − Y )2 .

Fig. 3. Prediction of NN model 1 and actual values for training set.

Fig. 5. Prediction of NN model 1 and actual values for testing set.

Fig. 4. Percentage error of training set.

Fig. 6. Percentage error of testing set.

of Cp
should be less than the present values. Figure 11 shows that when the Froude number of the incoming jet increases, the peak value of Cp
shifts towards the inclined section and the distribution of Cp
along the channel decreases with increasing Froude number. Figure 12 shows that Ls /y1 has great affect on Cp
. The maximum Cp
occurs with larger values of Ls /y1 , the increase in Froude number with decreasing Ls /y1 causes

depletion in Cp
distribution along the channel. Figure 13 shows that the peak values and distribution of Cp
along the channel increase when the slope of the inclined channel section increases, compared to Figs. 12 and 13. It is obvious that the predictions of the NN model are strictly in good agreements with the experimental results and the NN model is capable of capturing the physical exceptions caused by experimental setup. © 2006 NRC Canada

1384 Fig. 7. Prediction of NN model 2 and actual values for training set.

Can. J. Civ. Eng. Vol. 33, 2006 Fig. 10. Percentage error of testing set.

Fig. 11. Comparison of NN model 2 and the experimental results for φ = 10◦ and Ls /y1 = 15. Fig. 8. Percentage error of training set.

Fig. 12. Comparison of NN model 2 and the experimental results for φ = 20◦ .

Fig. 9. Prediction of NN model 2 and actual values for testing set.

Fig. 13. Comparison of NN model 2 and the experimental results for φ = 30◦ .

© 2006 NRC Canada

Güven et al.

1385

Explicit formulation of NN model The main focus is to obtain the explicit formulation of mean-pressure fluctuation as a function of the geometric and inflow properties of flow through an inclined channel as follows:
2 [6]  p
= f (u1 , y1 , yt , Ls , x, φ) [7]


 2  p
= 1000 ×

1 1 + e−W



where  W = (−2.68)

        1 1 1 1 + (−2.00) + 11.20 + 5.28 + 0.093 1 + e−U2 1 + e−U3 1 + e−U4 1 + e−U5         1 1 1 1 + (−0.75) + (−5.06) + 0.05 + 0.11 1 + e−U6 1 + e−U7 1 + e−U8 1 + e−U9         1 1 1 1 + 1.66 + (−4.69) + 0.61 + 0.09 1 + e−U10 1 + e−U11 1 + e−U12 1 + e−U13         1 1 1 1 + 0.14 + (−0.16) + (−0.17) + 0.19 + 1.42 1 + e−U14 1 + e−U15 1 + e−U16 1 + e−U17 1 1 + e−U1



and the values for Ui are given as U1 = (−0.065φ) + (0.04u1 ) + (−1175.8y1 ) + (−4.03Ls ) + (44.34yt ) + (−3.28x) + 12.512 U2 = (0.15φ) + (−0.24u1 ) + (2309.85y1 ) + (1.39Ls ) + (−31.80yt ) + (3.34x) + (−29.78) U3 = (0.017φ) + (−0.003u1 ) + (5.60y1 ) + (−15.88Ls ) + (3.07yt ) + (−7.93x) + (−2.22) U4 = (−0.21φ) + (−0.12u1 ) + (432.16y1 ) + (3.92Ls ) + (5.49yt ) + (−8.28x) + (1.59) U5 = (−0.31φ) + (0.34u1 ) + (1142.7y1 ) + (30.07Ls ) + (110.30yt ) + (−5.19x) + (−48.50) U6 = (−0.26φ) + (0.01u1 ) + (−1039.95y1 ) + (6.97Ls ) + (−1.40yt ) + (−2.70x) + (22.43) U7 = (−0.21φ) + (−0.13u1 ) + (460.15y1 ) + (4.42Ls ) + (5.67yt ) + (−8.17x) + (1.39) U8 = (−0.52φ) + (−0.08u1 ) + (1242.65y1 ) + (−87.56Ls ) + (75.99yt ) + (−52.63x) + (−11.69) U9 = (0.71φ) + (0.17u1 ) + (135.81y1 ) + (36.26Ls ) + (45.85yt ) + (−10.12x) + (−26.77) U10 = (−0.18φ) + (−0.08u1 ) + (−22.19y1 ) + (−1.55Ls ) + (25.48yt ) + (−5.52x) + (1.55) U11 = (0.03φ) + (0.12u1 ) + (86.27y1 ) + (−18.74Ls ) + (−0.44yt ) + (−9.92x) + (−2.67) U12 = (0.43φ) + (0.14u1 ) + (343.88y1 ) + (1.72Ls ) + (−26.22yt ) + (6.95x) + (−13.80) U13 = (1.00φ) + (0.03u1 ) + (−2795.5y1 ) + (42.45Ls ) + (−100.68yt ) + (15.71x) + (48.76) U15 = (0.17φ) + (0.15u1 ) + (2379.75y1 ) + (−35.21Ls ) + (55.42yt ) + (−1.00x) + (−44.46) U16 = (0.11φ) + (−0.09u1 ) + (199.31y1 ) + (−25.71Ls ) + (34.62yt ) + (−26.06x) + (−3.39) U17 = (0.04φ) + (−0.32u1 ) + (−147.56y1 ) + (26.02Ls ) + (−125.34yt ) + (−9.50x) + (25.26)

It should be noted that the proposed explicit formulation of the NN model presented above is valid only for the ranges of training set given in Table 2.

Prediction of pressure fluctuation using regression analysis Linear and nonlinear regression analyses were applied for characterization of the complex mapping among the parameters considered. In this context, more than 30 regression equations

were determined by linear and nonlinear analyses and the most efficient six models are presented in Table 5. In detail, y is the
2

mean pressure fluctuation ( p
), X1 is the angle of inclination of the sloping part of channel in degrees (φ), X2 is the upstream velocity of water issuing from the gate (u1 ), X3 is gate opening (y1 ), X4 is the distance from the toe of the jump to the point where the slope joins the channel bed (Ls ), X5 is the tail water depth (yt ), and X6 is the distance from the intersection of horizontal and inclined section (x). © 2006 NRC Canada

1386

Can. J. Civ. Eng. Vol. 33, 2006 Table 5. Linear and nonlinear regression models. No.

Model

1 2 3 4 5 6

p1 + p2 × X1 + p3 × X2 + p4 × X3 + p5 × X4 + p6 × X5 + p7 × X6 p1 + p2 × (X1/100) + p3 × (X2/X3) + p4 × c + p5 × (X5/X2) + p6 × (X6/X3) p1 + p2(X13 /X5) + p3 × (X22 /X3) + p4 × (X34 /X2) + p5 exp[(X42 /X22 )] + p6 × (X53 /X33 ) + p7(X62 /X3) p1 + p2 × (X12 /X5) + p3 × (X22 /X3) + p4 × (X32 /X2) + p5 × (X42 /X2) + p6 × (X53 /X3) + p7 × (X62 /X3) p1 + p2 exp[(−p3 × X1) + (−p4 × X2) + (−p5 × X3) + (−p6 × X4) + (−p7 × X5) + (−p8 × X6)] p1 + p2 × (p3X1 + p4X2 + p5X3 + p6X4 + p7X5 + p8X6 ) Table 6. Regression model parameters. Coefficients No.

p1

p2

p3

p4

p5

p6

p7

p8

1 2 3 4 5 6

−347.23 −555.20 −193.30 −130.80 2121.90 −10 508.8

5.10 406.20 4.88 1.75 −2546.60 1705.20

171.90 2.79 0.54 0.50 0.003 1.00

109.20 22 030 90.88 × 107 398 312 0.09 1.08

83.82 −13.20 157.10 191.90 5.50 432.30

−1241.50 −0.85 −0.04 −96.79 0.04 1.05

−41.34 — −6.61 −6.64 −5.60 0.46

— — — — −0.02 0.98

Table 7. Statistical parameters of linear and nonlinear regression models.

Fig. 14. Prediction of regression model 3 and actual values.

Model No.

MSE

RMSE

SSE

MAPE

R

1 2 3 4 5 6

4409.55 4006.81 3538.11 3556.69 4019.99 4015.82

66.40 63.29 59.52 59.63 63.40 63.37

2 892 662 2 628 467 2 324 098 2 323 189 2 637 112 2 634 379

23.39 23.03 22.35 22.27 23.26 23.23

0.78 0.79 0.82 0.82 0.80 0.79

Results of regression analysis Resulting regression coefficients (pi) are given in Table 6. Referring to Table 5, nonlinear regression models produced slightly better outcomes over linear models, namely, identification of target mapping with linear and nonlinear behaviours produced almost similar outcomes in terms of correlation coefficient (R) and mean absolute percentage errors (MAPE). In this sense, Model 2 (R = 0.79) and Model 3 (R = 0.82) were selected to be representative linear and nonlinear models, respectively. Statistical parameters of regression models are given in Table 7. To evaluate the relative performances of linear and nonlinear regression models, scatter plots (Figs. 14 and 15) and residual plots (Figs. 16 and 17) are illustrated. As can be seen from Figs. 14 and 15, discrepancies from best fit line in the nonlinear model (Model 3) are slight smaller than in the linear model (Model 2). Residual plots are also consistent with previous findings, therefore, it can be concluded that the nonlinear analysis exhibited better performance over the linear approach.

Fig. 15. Prediction of regression model 2 and actual values.

Neural network modelling versus regression analysis The overall performances of neural network modelling and regression analysis are compared through residual plots of NN Model 1 and the regression Models 2 and 3 (Fig. 18). In Fig. 18, discrepancies of the NN Model 1 are considerably smaller than

those of the regression models. Referring to Fig. 18, it can be concluded that neural network modelling exhibited considerable dominance over regression analysis in predicting the meanpressure fluctuation beneath hydraulic jump. © 2006 NRC Canada

Güven et al. Fig. 16. Residual plots of regression model 3.

1387

sure fluctuations beneath this type of hydraulic jump (Bjump). (3) The experimental data were derived from Gunal (1996). Among these experimental data, a number were used for an NN training test and the rest as a test set. (4) An explicit formula that predicts the mean-pressure fluctuation in terms of the most contributing variables of hydraulic jumps occurring in stilling basins is presented. (5) Artificial neural networks are found to be successively capable of modelling the pressure fluctuations beneath hydraulic jumps occurring in sloping stilling basins.

Fig. 17. Residual plots of regression model 2.

(6) Linear and nonlinear regression models are developed using considered physical parameters and the proposed neural network and regression models are compared. Results show that neural network models show much better performance than regression models. (7) Artificial neural networks can be used as an effective tool to establish the correlation among physical variables and they have a generalized potential to represent input and output relationships. (8) It was observed that there is a good agreement between the neural network predictions and the measured values and they are close to the line of the perfect agreement due to relatively small values of the error (MAPE).

References

Fig. 18. Comparison of NN model 1 and regression models 2, 3 results.

Conclusions (1) Artificial neural network models are developed to predict the pressure fluctuations beneath a hydraulic jump occurring in sloping stilling basins. (2) A multilayer feed-forward neural network with a backpropagation learning algorithm is used to model the pres-

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List of symbols Cp Cp
g Ls p p p

pi 2

p
Ui u1 x Xi yt y1 ρw φ

nondimensional mean pressure parameter nondimensional pressure fluctuation parameter gravitational acceleration distance from the toe of the jump to the point where the slope joins the channel bed instantaneous pressure mean part of pressure fluctuating part of pressure regression model parameters root mean square value of pressure fluctuation sum up of weighted inputs upstream velocity of water issuing from the gate distance from the intersection of the horizontal and inclined section physical variables used in regression analysis tail water depth gate opening density of water inclination angle of the sloping part of channel

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