Sudden diverging channels are one of the energy dissipaters which can dissipate most of the kinetic energy of the flow through a hydraulic jump. An accurate prediction of hydraulic jump characteristics is an important step in designing hydraulic structures. This paper focuses on the capability of the support vector machine (SVM) as a meta-model approach for predicting hydraulic jump characteristics in different sudden diverging stilling basins (i.e. basins with and without appurtenances). In this regard, different models were developed and tested using 1,018 experimental data. The obtained results proved the capability of the SVM technique in predicting hydraulic jump characteristics and it was found that the developed models for a channel with a central block performed more successfully than models for channels without appurtenances or with a negative step. The superior performance for the length of hydraulic jump was obtained for the model with parameters F1 (Froude number) and (h2—h1)/h1 (h1 and h2 are sequent depth of upstream and downstream respectively). Concerning the relative energy dissipation and sequent depth ratio, the model with parameters F1 and h1/B (B is expansion ratio) led to the best results. According to the outcome of sensitivity analysis, Froude number had the most significant effect on the modeling. Also comparison between SVM and empirical equations indicated the great performance of the SVM.

INTRODUCTION

The purpose of the design of energy dissipators is to dissipate part of the kinetic energy of the inflowing flow in order to return safely the flow to the downstream channel or river and prevent scour below overflow spillways, chutes and sluices. Based on the energy dissipating action of hydraulic jumps, stilling basins are one of the possible solutions which may be adopted. The performance or efficiency of any stilling basin is usually assessed in terms of characteristics of the jump allocated (Negm 2000). The length of the hydraulic jump is mostly taken as a design parameter or as an indicator of the length of the paved downstream section, i.e. the stilling basin. The hydraulic jump is a natural phenomenon that occurs when supercritical flow is forced to change to subcritical flow by an obstruction to the flow. Depending on the geometry of the channel and tailwater conditions, the hydraulic jump can assume several distinct forms. The hydraulic jump is a useful means of dissipating the excess energy of supercritical flow so that scour in the downstream is minimized. It has also been used to raise the water level on the downstream to provide the requisite head for diversion into canals. However modeling hydraulic jump characteristics has great importance since it plays an important role in designing hydraulic structures. So far, various studies have been carried out to explain the complex phenomenon of the hydraulic jump and to estimate its characteristics. Hughes & Flack (1983) studied the effect of various roughness designs on the hydraulic jump characteristics in a stilling basin. Hager & Bremen (1989) investigated the influence of wall friction on the sequent depths ratio. Bhutto et al. (1989) developed analytical solutions for computing sequent depth and relative energy loss for free hydraulic jump in sloping and horizontal rectangular channels. Finnemore & Franzini (2002) stated that the characteristics of the hydraulic jump depend on the Froude number. Negm (2000) studied the hydraulic performance of rectangular and radial stilling basins, where the latter stand for the diverging channels. Ayanlar (2004) studied the effect of corrugated beds on hydraulic jump properties with altering flow characteristics. Bilgin (2005) studied the correlation and distribution of shear stress for turbulent flow in a smooth rectangular basin. However, due to the complexity and uncertainty of the hydraulic jump phenomenon, the results of the classical models are not general and do not show the same results under variable conditions. Therefore, it is essential to use methods which are capable of predicting hydraulic characteristics within the basins.

In recent years the artificial intelligence tools (e.g. artificial neural networks, neuro-fuzzy models, genetic programming and support vector machine (SVM)) have been used for the assessment of the accuracy of hydraulic and hydrologic complex phenomena, such as prediction of total bedload (Chang et al. 2012), prediction of suspended sediment concentration (Kisi & Shiri 2012), modeling the rainfall-runoff process (Nourani et al. 2011; Kisi et al. 2013), precipitation forecasting (Kisi & Shiri 2011), predicting of total bed material load (Roushangar et al. 2014a), developing stage-discharge curves (Azamathulla et al. 2011) and predicting of energy dissipation over spillways (Roushangar et al. 2014b).

Among others, the SVM technique has been applied in modeling various components of water resources systems. Khan & Coulibaly (2006) and Lan (2014) applied SVM for long-term prediction of lake water levels. Sivapragasam & Muttil (2005) suggested the use of SVM in the extrapolation of rating curves. Roushangar & Koosheh (2015) evaluated the genetic algorithm–support vector regression (GA-SVR) method for modeling bedload transport in gravel-bed rivers. Kisi (2012) applied least square SVM for modeling the discharge–suspended sediment relationship.

In the present study the capability of the SVM approach was assessed for modeling hydraulic jump characteristics in three different sudden diverging stilling basins. Various input combinations based on hydraulic characteristics and geometry of the appurtenances were used in order to find the most appropriate combination for modeling hydraulic jump characteristics. Finally, the accuracy of the best SVM model was compared with the existing classic approaches.

MATERIALS AND METHODS

The data sets

In this study, the experimental data presented by Bremen (1990) and Gandhi (2014) were employed for prediction goals. The experiments of Bremen (1990) were carried out at the Laboratoire de Constructions Hydrauliques of the Ecole Polytechnique Federale de Lausanne and were intended for sudden diverging basins without appurtenances, with a central sill and with a negative step (Figure 1). During experiments the 17 m3 upstream basin was supplied by two conduits of 0.30 m diameter (maximum total discharge 375 L/s). A 0.5 m wide and 10.8 m long prismatic rectangular and horizontal channel was connected to the basin. At the upstream channel extremity a standard-shaped 0.50 m wide and 0.70 m high spillway of design head H0 = 0.2 m controlled the channel inflow. According to Figure 2 two states of asyrnmetric and symmetric abrupt diverging were considered in experiments.
Figure 1

Plane view of hydraulic jump in sudden diverging basin; (a) normal, (b) with negative step and (c) with central sill (Bremen (1990), reproduced with kind permission of the author).

Figure 1

Plane view of hydraulic jump in sudden diverging basin; (a) normal, (b) with negative step and (c) with central sill (Bremen (1990), reproduced with kind permission of the author).

Figure 2

Schematic view of the experiments based on Bremen (1990).

Figure 2

Schematic view of the experiments based on Bremen (1990).

The experiments of Gandhi (2014) were carried out at a rectangular channel made up of Perspex sheets and the setup consisted of a constant head tank with volume of 3.6 × 3.6 × 3 m3. A symmetric-shaped channel, sized 2.1 × 0.445 × 1.2 m and flow Froude number (2 < F1 < 9) and the expansion ratio (0.4 < B < 0.8), was used for experiments. The expansion ratio (B = b1/b2 ) shows the expansion geometry in which b1 and b2 are approach channel width and expanded channel width, respectively. The ranges of experimental data used in experiments are given in Table 1.

Table 1

The range of experimental data

ResearcherSeriesF1B (b2/b1)Q (l/s)Y (h2/h1)Xs (m)
S/h1
Channel shapeNo. of data
fromtofromto
Normal 
Bremen (1990)  2.63–8.12 2–18.1 2.02–11.26 – – – – Asym 162 
2.63–8.05 5.1–26.9 2.18–10.28 – – – – Sym 178 
2.65–8.13 1.5 10–40 3.46–12.05 – – – – Asym 91 
Gandhi (2014)  2–9 0.4 6.44–15 2–8 – – – – Sym 
2–9 0.5 12.3–12.8 2–9 – – – – Sym 
2–9 0.6 6.44–10.6 2–9.5 – – – – Sym 
2–9 0.8 6.3–9.8 2–10 – – – – Sym 
Negative step 
Bremen (1990)  3.2–8.43 1.5 11–39.5 4.15–12.59 – – 0.61 1.64 Asym 105 
2.68–8.12 11–39.5 4.15–12.59 – – 0.59 1.61 Asym 112 
2.05–7.37 4.3–24 1.45–11.05 – – 0.49 1.66 Sym 129 
Central sill 
Bremen (1990)  3,5,7,9 31–225 2.89–8.81 0.2 0.8 0.6 Sym 78 
3,5,7,9 31–225 3.07–9.88 0.2 0.8 0.6 Sym 75 
3,5,7,9 1.5 31–225 3.15–10.63 0.2 0.8 0.6 Sym 60 
ResearcherSeriesF1B (b2/b1)Q (l/s)Y (h2/h1)Xs (m)
S/h1
Channel shapeNo. of data
fromtofromto
Normal 
Bremen (1990)  2.63–8.12 2–18.1 2.02–11.26 – – – – Asym 162 
2.63–8.05 5.1–26.9 2.18–10.28 – – – – Sym 178 
2.65–8.13 1.5 10–40 3.46–12.05 – – – – Asym 91 
Gandhi (2014)  2–9 0.4 6.44–15 2–8 – – – – Sym 
2–9 0.5 12.3–12.8 2–9 – – – – Sym 
2–9 0.6 6.44–10.6 2–9.5 – – – – Sym 
2–9 0.8 6.3–9.8 2–10 – – – – Sym 
Negative step 
Bremen (1990)  3.2–8.43 1.5 11–39.5 4.15–12.59 – – 0.61 1.64 Asym 105 
2.68–8.12 11–39.5 4.15–12.59 – – 0.59 1.61 Asym 112 
2.05–7.37 4.3–24 1.45–11.05 – – 0.49 1.66 Sym 129 
Central sill 
Bremen (1990)  3,5,7,9 31–225 2.89–8.81 0.2 0.8 0.6 Sym 78 
3,5,7,9 31–225 3.07–9.88 0.2 0.8 0.6 Sym 75 
3,5,7,9 1.5 31–225 3.15–10.63 0.2 0.8 0.6 Sym 60 

Support vector machine

The foundations of SVMs were developed by Vapnik (1995). The SVM is essentially used in information categorization and data set classification. This approach is known as structural risk minimization, which minimizes an upper bound on the expected risk, as opposed to the traditional empirical risk which minimizes the error on the training data. It is this difference that equips SVM with a greater ability to generalize. The main goal in statistical learning is ability to generalize (Gunn 1998). The SVM method is based on the concept of optimal hyper plane that separates samples of two classes by considering the widest gap between two classes (see Figure 3). SVR is an extension of SVM regression. The aim of SVR is to characterize a kind of function that has at most ɛ deviation from the actually obtained objectives for all training data yi and at the same time it would be as flat as possible. SVR formulation is as follows: 
formula
1
where φ(x) denotes a nonlinear function in feature of input x, b is called the bias and the vector w is known as the weight factor and expressed as Equation (2) in which αi is Lagrange multiplier, yi is forecasted value and xi is input value. 
formula
2
Figure 3

Data classification and support vectors.

Figure 3

Data classification and support vectors.

The coefficients of Equation (1) are predicted by minimizing the regularized risk function as expressed below: 
formula
3
where 
formula
4
The constant C is the cost factor and represents the trade-off between the weight factor and approximation error. ɛ is the radius of the tube within which the regression function must lie. The represents the loss function in which yi is forecasted value and ti is desired value in period i. The ||w|| is the norm of w vector and the term ||w||2 can be expressed in the form wT·w in which wT is the transpose form of w vector. According to Equation (4), if the forecasted value is out of ɛ -tube then the loss will be the absolute value, which is the difference between forecasted value and ɛ. Since some data may not lie inside the ɛ-tube, the slack variables (ξ, ξ*) must be introduced. These variables represent the distance from actual values to the corresponding boundary values of ɛ-tube. Therefore, it is possible to transform Equation (3) into: 
formula
5
subject to:
Using Lagrangian multipliers in Equation (5) thus yields the dual Lagrangian form: 
formula
6
subject to:
where αi and αi* are Lagrange multipliers and l(αi, αi*) represents the Lagrange function. K(xi, xj) is a kernel function to yield the inner products in the feature space φ(xi) and φ(xj) and expressed as the following: 
formula
7
The appropriate selection of kernel type is the most important step in the SVM due to its direct impact on the training and classification precision. In general, there are several types of kernel function, namely linear, polynomial, radial basis function (RBF) and sigmoid functions.

Classical models

So far, a variety of formulas have been developed to predict hydraulic jump properties, ranging from simple regressions to complex multi-parameter formulations. There are different concepts and approaches that are used in the derivation and extraction process of these formulas (i.e. different assumptions, statistical correlations, and experimental information). In the current study the applicability of several existing equations for hydraulic characteristics was assessed in order to indicate the capability of the applied approach in this study. The used equations are given in Table 2.

Table 2

Utilized equations in the study

ParameterResearcherEquationConsiderationEquation number
Sequent depth ratio Kusnetzow (1958)    (8) 
Hager (1985)    (9) 
Herbrand (1973)    (10) 
Hydraulic jump length Smetana (1934)   
For free jump in horizontal  smooth bed 
(11) 
Safranez (1929)   For free jump in horizontal smooth bed (12) 
Bakhmeteff & Matezke (1936)   2.5 < F1 < 4.5
For free jump in horizontal  smooth bed 
(13) 
Hager (1985)    2.5 < F1 < 12, 1 < B < 10, X1 > 0.1 (14) 
Silvester (1964)   For free jump in horizontal smooth bed (15) 
ParameterResearcherEquationConsiderationEquation number
Sequent depth ratio Kusnetzow (1958)    (8) 
Hager (1985)    (9) 
Herbrand (1973)    (10) 
Hydraulic jump length Smetana (1934)   
For free jump in horizontal  smooth bed 
(11) 
Safranez (1929)   For free jump in horizontal smooth bed (12) 
Bakhmeteff & Matezke (1936)   2.5 < F1 < 4.5
For free jump in horizontal  smooth bed 
(13) 
Hager (1985)    2.5 < F1 < 12, 1 < B < 10, X1 > 0.1 (14) 
Silvester (1964)   For free jump in horizontal smooth bed (15) 

Performance criteria

In the current study, the model's performance was assessed using three statistical criteria, namely, correlation coefficient (R), determination coefficient (DC), and root mean square error (RMSE), expressions for which are as follows: 
formula
16
where , , , , and N respectively represent the measured values, predicted values, mean measured values, mean predicted values and number of data samples. The RMSE describes the average difference between predicted values and measured values, R provides information for the linear dependence between observation and corresponding predicted values and DC is the coefficient used to determine the relative assessment of the model performance in dimensionless measures. The smaller the RMSE and the greater the DC and R, the higher the accuracy of the model will be. It should be noted that predicting the intended parameter using non-normalized data may lead to undesirable results; therefore, in this study all input variables were scaled to fall in the range 0.1–1 to eliminate the influence of the variables with different absolute magnitudes. This will bring about the uniformity of the data values for the network and increase the training speed and the capability of the model (Dawson & Wilby 1998). The following equation is used to normalize the data. 
formula
17

which , n, , and respectively are the normalized value, the original value, and the maximum and minimum of variable n. The normalization used only rescales the data to the range of 0.1 and 1, which after modeling can be de-normalized to real values.

Simulation and model development

Input variables

In a data-driven model, selection of appropriate parameters as input variables is a crucial step during the modeling process. In Figure 4 the quantities measured for jumps in sudden expansion basins with different appurtenances are shown. The important variables which affect the jump pattern and energy dissipation can be expressed as a function (Rajaratnam & Subramanya 1968; Hager & Bremen 1989; Gandhi 2014): 
formula
18
where h1 and h2: sequent depth of upstream and downstream, V1: upstream flow velocity, μ: dynamic viscosity of water, g: acceleration due to gravity, Lj: length of jump, ρ: density of water, b1 and b2: approach channel, and expanded channel width, EL: E1−E2 in which E1 and E2 are energy per unit weight before and after the jump and s: height of the step or sill.
Figure 4

Quantities measured and notation used for jumps in sudden expanding channel; (a) axial view, (b) plan view (Bremen (1990), reproduced with kind permission of the author).

Figure 4

Quantities measured and notation used for jumps in sudden expanding channel; (a) axial view, (b) plan view (Bremen (1990), reproduced with kind permission of the author).

From dimensional analysis and considering h1, g and μ as repeating variables, these parameters Equation (18) can be expressed as the following: 
formula
19
Equation (19) can be expressed as: 
formula
20
where F1: flow Froude number, R: flow Reynolds number, B = (b1/b2): ratio of expansion.

Experimental studies by Elevatorski (2008), Wu & Rajaratnam (1996) and Ranga Raju et al. (1980) revealed that hydraulic jump characteristics only depend on Froude number, and Reynolds number has no significant impact on the prediction of hydraulic jump characteristics. Also, Hager (1992) showed that the length of hydraulic jump depends on the height of jump and Froude number. Therefore, in this study, based on upstream hydraulic data and geometric data, the models of Table 3 were considered for modeling hydraulic jump characteristics of sudden diverging basins.

Table 3

SVM developed models

Output variableModelInput variable(s)
Sequent depth ratio h2/h1 S(I) F₁ 
S(II) F₁, h₁/B 
S(III) F₁, S/h1 
S(IV) F₁, B 
Length of hydraulic jump Lj/h1 L(I) F₁ 
L(II) F₁, (h₂–h1)/h₁ 
L(III) F₁, h₂/h₁ 
L(IV) F₁, S/h1 
L(V) F₁, B 
Loss of energy EL/E1 E(I) F₁ 
E(II) F₁, h₁/B 
E(III) F₁, S/h1 
Output variableModelInput variable(s)
Sequent depth ratio h2/h1 S(I) F₁ 
S(II) F₁, h₁/B 
S(III) F₁, S/h1 
S(IV) F₁, B 
Length of hydraulic jump Lj/h1 L(I) F₁ 
L(II) F₁, (h₂–h1)/h₁ 
L(III) F₁, h₂/h₁ 
L(IV) F₁, S/h1 
L(V) F₁, B 
Loss of energy EL/E1 E(I) F₁ 
E(II) F₁, h₁/B 
E(III) F₁, S/h1 

SVM model development

For determining the best performance of SVM and selecting the best kernel function, different models were predicted via SVM using various kernels. Table 4 shows the results of statistical parameters of different kernels for model S(II) of a basin with central sill. The results of Table 4 revealed that using the model with the kernel function of RBF led to better prediction accuracy in comparison to the other kernels. According to Noori et al. (2011) the SVM model via RBF kernel is very desirable for use in prediction of hydraulic and hydrological features since: (a) unlike the linear kernel, the RBF kernel can handle the case when the relation between class labels and attributes is nonlinear, (b) the RBF kernels tend to give better performance under general smoothness assumptions, (c) it has fewer tuning parameters than the polynomial and the sigmoid kernels. Therefore, RBF kernel was selected as the core tool of SVM and was applied for the rest of the models. Implementation of SVM requires the selection of three parameters, which are constant C, ɛ and kernel parameter γ, where γ is a constant parameter of the RBF kernel. The coefficient C is a positive constant that influences a trade-off between an approximation error and the regression and must be selected by the user. ɛ has an effect on the smoothness of the SVM's response and it affects the number of support vectors, so both the complexity and the generalization capability of the network depend on its value. If epsilon is larger than the range of the target values we cannot expect a good result. If epsilon is zero, we can expect overfitting (Smola 1996). The variable parameter used with kernel function (γ) considerably affects the flexibility of function. These parameters should be selected by the user. The selection of appropriate values for the three parameters (C, ɛ, γ) has been proposed by various researchers. In the current study, according to Cherkassky & Yunqian (2002), optimization of these parameters has been done by a systematic grid search of the parameters using cross-validation on the training set. A normal range of parameter settings are investigated in this grid search. First, optimized values of C and ɛ for a specified γ were obtained and then γ was changed. Statistical parameters were used to find optimums. The statistics parameters via γ values to find SVM optimums of the testing set for the model S(II) of the basin with central sill are shown in Figure 5. In the same way, optimal parameters were obtained for all models.
Table 4

The statistical parameters of SVM method with different kernel functions; model S(II)

Kernel functionTrain
Test
RDCRMSERDCRMSE
Linear 0.885 0.818 0.138 0.882 0.722 0.149 
Polynomial 0.977 0.899 0.118 0.968 0.748 0.123 
RBF 0.994 0.987 0.026 0.993 0.985 0.029 
Sigmoid 0.574 0.108 0.203 0.343 0.095 0.273 
Kernel functionTrain
Test
RDCRMSERDCRMSE
Linear 0.885 0.818 0.138 0.882 0.722 0.149 
Polynomial 0.977 0.899 0.118 0.968 0.748 0.123 
RBF 0.994 0.987 0.026 0.993 0.985 0.029 
Sigmoid 0.574 0.108 0.203 0.343 0.095 0.273 

Bold values correspond to the superior kernel type.

Figure 5

Statistics parameters via γ values to find SVM optimums of the testing set for model S(II) of channel with central sill.

Figure 5

Statistics parameters via γ values to find SVM optimums of the testing set for model S(II) of channel with central sill.

RESULTS AND DISCUSSION

The sequent depth ratio

For evaluating hydraulic jump characteristics in sudden expanding channels with and without appurtenances, several models, according to the flow condition and geometry of the applied appurtenances, were developed. All of the SVM models were trained and tested to carry out the sequent depth ratio prediction in sudden diverging basins. The results of SVM models are shown in Table 5 and Figure 6. From the obtained results of statistical parameters (RMSE, R and DC) it can be stated that between the three types of basins, developed models for the case of the basin with a central sill in modeling of sequent depth ratio performed more successfully than the two other cases. In this state, the model S(II) with input parameters of F1 and h1/B led to more accurate outcome than the other models. This model also presented higher accuracy for basins without appurtenances and with negative step. A comparison between the results of the models S(I) and S(III) showed that for predicting the sequent depth ratio in diverging basins with appurtenances, using parameter S/h1 as input parameter improved the efficiency of the model. This parameter confirms the importance of the relative height of applied step and sill in sequent depth ratio estimating process in channels with appurtenances. Considering the results of the models S(I), S(II) and S(III), it could be inferred that the impact of parameter h1/B in increasing the accuracy of the model is more than that of parameter S/h1. Also, the model S(I) with only input parameter F1 showed desired accuracy. It could be stated that the applied method can successfully predict the sequence depth ratio using only the upstream flow characteristic as input data. Based on the results of Table 5, for basin with negative step, the models of the channel with asymmetric shape presented better results than symmetric basin, while for basin without appurtenances, the developed models for symmetric channel showed more accuracy. Figure 6 shows the verification between measured and estimated values for the best proposed model for all states.
Table 5

Statistical parameters of the SVM models for sequent depth ratio

Performance criteria
Optimal parameters
Train
Test
ConditionSVM modelscɛγRDCRMSERDCRMSE
Basin without appurtenances 
 Sym channel S(I) 5.0 0.10 0.958 0.916 0.044 0.955 0.911 0.043 
S(II) 6.0 0.10 4 0.977 0.952 0.034 0.945 0.882 0.049 
 Asym channel S(I) 5.0 0.10 0.880 0.771 0.068 0.874 0.754 0.097 
S(II) 5.0 0.10 5 0.959 0.918 0.041 0.933 0.854 0.075 
Basin with negative step 
 Sym channel S(I) 8.0 0.02 0.900 0.805 0.063 0.893 0.744 0.075 
S(II) 10 0.10 5 0.993 0.986 0.018 0.959 0.911 0.045 
S(III) 10 0.10 0.947 0.896 0.048 0.931 0.864 0.054 
 Asym channel S(I) 10 0.01 0.963 0.922 0.052 0.962 0.908 0.056 
S(II) 8.0 0.10 4 0.993 0.985 0.029 0.989 0.977 0.028 
S(III) 8.0 0.02 0.981 0.961 0.036 0.980 0.959 0.037 
S(IV) 10 0.01 0.960 0.920 0.054 0.951 0.910 0.056 
Basin with central sill 
 Sym channel S(I) 4.0 0.01 0.984 0.967 0.039 0.975 0.944 0.057 
S(II) 5.0 0.01 2 0.994 0.987 0.026 0.993 0.985 0.029 
S(III) 5.0 0.01 0.985 0.967 0.039 0.979 0.957 0.048 
S(IV) 2.0 0.01 0.983 0.965 0.041 0.975 0.940 0.058 
Performance criteria
Optimal parameters
Train
Test
ConditionSVM modelscɛγRDCRMSERDCRMSE
Basin without appurtenances 
 Sym channel S(I) 5.0 0.10 0.958 0.916 0.044 0.955 0.911 0.043 
S(II) 6.0 0.10 4 0.977 0.952 0.034 0.945 0.882 0.049 
 Asym channel S(I) 5.0 0.10 0.880 0.771 0.068 0.874 0.754 0.097 
S(II) 5.0 0.10 5 0.959 0.918 0.041 0.933 0.854 0.075 
Basin with negative step 
 Sym channel S(I) 8.0 0.02 0.900 0.805 0.063 0.893 0.744 0.075 
S(II) 10 0.10 5 0.993 0.986 0.018 0.959 0.911 0.045 
S(III) 10 0.10 0.947 0.896 0.048 0.931 0.864 0.054 
 Asym channel S(I) 10 0.01 0.963 0.922 0.052 0.962 0.908 0.056 
S(II) 8.0 0.10 4 0.993 0.985 0.029 0.989 0.977 0.028 
S(III) 8.0 0.02 0.981 0.961 0.036 0.980 0.959 0.037 
S(IV) 10 0.01 0.960 0.920 0.054 0.951 0.910 0.056 
Basin with central sill 
 Sym channel S(I) 4.0 0.01 0.984 0.967 0.039 0.975 0.944 0.057 
S(II) 5.0 0.01 2 0.994 0.987 0.026 0.993 0.985 0.029 
S(III) 5.0 0.01 0.985 0.967 0.039 0.979 0.957 0.048 
S(IV) 2.0 0.01 0.983 0.965 0.041 0.975 0.940 0.058 

Bold values correspond to the superior model for each condition.

Figure 6

Comparison of observed and predicted sequent depth ratio for superior model; (a) symmetric basin without appurtenances, (b) symmetric basin with negative step, (c) asymmetric basin with negative step, (d) symmetric basin with central sill.

Figure 6

Comparison of observed and predicted sequent depth ratio for superior model; (a) symmetric basin without appurtenances, (b) symmetric basin with negative step, (c) asymmetric basin with negative step, (d) symmetric basin with central sill.

Length of hydraulic jump

The obtained results from SVM models for predicting the length of hydraulic jump in three different sudden diverging stilling basins are indicated in Table 6 and Figure 7. The superior performance was obtained for the model L(II) in which the basin with a central sill was used and input parameters were F1, (h₂h₁)/h₁. Based on the results of the symmetric basin with negative step demonstrated in Table 6, the model L(IV) with input parameters F1, S/h1 was more accurate than the other models, while for the case of the asymmetric basin the model L(II) was superior. According to the obtained results from the models L(I), L(III) and L(IV), it could be inferred that for modeling the length of hydraulic jump in diverging basins, adding parameters h2/h1 and S/h1 as an input parameter caused an increment in model efficiency. However, the results of the model L(II), as the superior model, indicated that parameter (h₂h₁)/h₁ was more effective than parameters h2/h1 and S/h1 in improving the model accuracy. It can be stated that the developed models for a basin with central sill, in predicting of hydraulic jump length, performed more successfully than the two other cases. The scatter plots of observed and predicted relative length of hydraulic jump for the superior model of each case are shown in Figure 7.
Table 6

Statistical parameters of the SVM models for relative length of hydraulic jump

ConditionSVM modelsOptimal parameters
Performance criteria
Train
Test
cɛγRDCRMSERDCRMSE
Basin without appurtenances 
 Sym channel L(I) 10 0.100 0.903 0.812 0.072 0.855 0.720 0.090 
L(II) 10 0.010 5 0.908 0.819 0.069 0.856 0.722 0.088 
L(III) 10 0.010 0.907 0.818 0.071 0.855 0.721 0.089 
Basin with negative step 
 Sym channel L(I) 10 0.001 0.895 0.798 0.079 0.884 0.763 0.091 
L(II) 10 0.001 0.929 0.857 0.069 0.912 0.827 0.077 
L(III) 8.0 0.001 0.925 0.850 0.070 0.910 0.827 0.077 
L(IV) 10 0.100 6 0.935 0.877 0.065 0.930 0.848 0.065 
 Asym channel L(I) 8.0 0.100 0.930 0.861 0.057 0.920 0.845 0.066 
L(II) 10 0.100 4 0.956 0.912 0.048 0.921 0.846 0.061 
L(III) 8.0 0.001 0.922 0.849 0.065 0.892 0.820 0.078 
L(IV) 10 0.001 0.930 0.859 0.058 0.920 0.845 0.065 
L(V) 8.0 0.100 0.907 0.820 0.067 0.857 0.734 0.091 
Basin with central sill 
 Sym channel L(I) 10 0.100 0.935 0.872 0.055 0.898 0.804 0.083 
L(II) 10 0.001 2 0.958 0.915 0.035 0.930 0.860 0.048 
L(III) 10 0.100 0.936 0.879 0.052 0.921 0.833 0.073 
L(IV) 10 0.001 0.932 0.867 0.056 0.923 0.848 0.064 
L(V) 10 0.100 0.921 0.840 0.061 0.882 0.763 0.089 
ConditionSVM modelsOptimal parameters
Performance criteria
Train
Test
cɛγRDCRMSERDCRMSE
Basin without appurtenances 
 Sym channel L(I) 10 0.100 0.903 0.812 0.072 0.855 0.720 0.090 
L(II) 10 0.010 5 0.908 0.819 0.069 0.856 0.722 0.088 
L(III) 10 0.010 0.907 0.818 0.071 0.855 0.721 0.089 
Basin with negative step 
 Sym channel L(I) 10 0.001 0.895 0.798 0.079 0.884 0.763 0.091 
L(II) 10 0.001 0.929 0.857 0.069 0.912 0.827 0.077 
L(III) 8.0 0.001 0.925 0.850 0.070 0.910 0.827 0.077 
L(IV) 10 0.100 6 0.935 0.877 0.065 0.930 0.848 0.065 
 Asym channel L(I) 8.0 0.100 0.930 0.861 0.057 0.920 0.845 0.066 
L(II) 10 0.100 4 0.956 0.912 0.048 0.921 0.846 0.061 
L(III) 8.0 0.001 0.922 0.849 0.065 0.892 0.820 0.078 
L(IV) 10 0.001 0.930 0.859 0.058 0.920 0.845 0.065 
L(V) 8.0 0.100 0.907 0.820 0.067 0.857 0.734 0.091 
Basin with central sill 
 Sym channel L(I) 10 0.100 0.935 0.872 0.055 0.898 0.804 0.083 
L(II) 10 0.001 2 0.958 0.915 0.035 0.930 0.860 0.048 
L(III) 10 0.100 0.936 0.879 0.052 0.921 0.833 0.073 
L(IV) 10 0.001 0.932 0.867 0.056 0.923 0.848 0.064 
L(V) 10 0.100 0.921 0.840 0.061 0.882 0.763 0.089 

Bold values correspond to the superior model for each condition.

Figure 7

Comparison of observed and predicted relative length of hydraulic jump for superior model; (a) symmetric basin without appurtenances, (b) symmetric basin with negative step, (c) asymmetric basin with negative step, (d) symmetric basin with central sill.

Figure 7

Comparison of observed and predicted relative length of hydraulic jump for superior model; (a) symmetric basin without appurtenances, (b) symmetric basin with negative step, (c) asymmetric basin with negative step, (d) symmetric basin with central sill.

Energy dissipation

The results of SVM models for relative energy dissipation ratio are given in Table 7. According to Table 7 it can be seen that for the three cases of basins, without appurtenances, with negative step, and with central sill, the model E(II) with input parameters F1 and h1/B represented higher performance in comparison with the other models. The results of developed models for the basin with negative step and central sill revealed that using relative height of sill or step as input variables caused an increment in model efficiency. Comparison between the obtained results from the developed models for basins without appurtenances and with negative step indicated that for symmetric channels, the basin without any appurtenances presented higher accuracy. For asymmetric channels, the superior performance was obtained for the basin with negative step. However, according to Table 7, the basin with central sill, in predicting relative energy dissipation, performed more successfully than the other cases. Figure 8 shows the verification between measured and estimated values for the best SVM models for expanding channels.
Table 7

Statistical parameters of the SVM models for relative energy dissipation

ConditionSVM modelsOptimal parameters
Performance criteria
Train
Test
cɛγRDCRMSERDCRMSE
Basin without appurtenances 
 Sym channel E(I) 8.0 0.120 0.957 0.909 0.046 0.950 0.898 0.047 
E(II) 10 0.100 4 0.984 0.982 0.030 0.989 0.978 0.034 
 Asym channel E(I) 8.0 0.001 0.873 0.756 0.072 0.867 0.741 0.100 
E(II) 10 0.001 0.959 0.917 0.042 0.935 0.848 0.077 
Basin with negative step 
 Sym channel E(I) 10 0.100 0.890 0.790 0.069 0.869 0.710 0.081 
E(II) 8.0 0.001 3 0.982 0.982 0.031 0.957 0.908 0.045 
E(III) 8.0 0.001 0.974 0.972 0.035 0.945 0.945 0.049 
 Asym channel E(I) 10 0.001 0.961 0.924 0.052 0.959 0.918 0.054 
E(II) 10 0.120 8 0.989 0.984 0.027 0.976 0.981 0.032 
E(III) 5.0 0.100 0.799 0.977 0.029 0.956 0.908 0.055 
Basin with central sill 
 Sym channel E(I) 8.0 0.100 0.980 0.961 0.045 0.976 0.953 0.047 
E(II) 10 0.100 4 0.993 0.985 0.027 0.992 0.984 0.030 
E(III) 10 0.001 0.981 0.965 0.042 0.980 0.964 0.044 
ConditionSVM modelsOptimal parameters
Performance criteria
Train
Test
cɛγRDCRMSERDCRMSE
Basin without appurtenances 
 Sym channel E(I) 8.0 0.120 0.957 0.909 0.046 0.950 0.898 0.047 
E(II) 10 0.100 4 0.984 0.982 0.030 0.989 0.978 0.034 
 Asym channel E(I) 8.0 0.001 0.873 0.756 0.072 0.867 0.741 0.100 
E(II) 10 0.001 0.959 0.917 0.042 0.935 0.848 0.077 
Basin with negative step 
 Sym channel E(I) 10 0.100 0.890 0.790 0.069 0.869 0.710 0.081 
E(II) 8.0 0.001 3 0.982 0.982 0.031 0.957 0.908 0.045 
E(III) 8.0 0.001 0.974 0.972 0.035 0.945 0.945 0.049 
 Asym channel E(I) 10 0.001 0.961 0.924 0.052 0.959 0.918 0.054 
E(II) 10 0.120 8 0.989 0.984 0.027 0.976 0.981 0.032 
E(III) 5.0 0.100 0.799 0.977 0.029 0.956 0.908 0.055 
Basin with central sill 
 Sym channel E(I) 8.0 0.100 0.980 0.961 0.045 0.976 0.953 0.047 
E(II) 10 0.100 4 0.993 0.985 0.027 0.992 0.984 0.030 
E(III) 10 0.001 0.981 0.965 0.042 0.980 0.964 0.044 

Bold values correspond to the superior model for each condition.

Figure 8

Comparison of observed and predicted relative energy dissipation for superior model; (a) symmetric basin without appurtenances, (b) symmetric basin with negative step, (c) asymmetric basin with negative step, (d) symmetric basin with central sill.

Figure 8

Comparison of observed and predicted relative energy dissipation for superior model; (a) symmetric basin without appurtenances, (b) symmetric basin with negative step, (c) asymmetric basin with negative step, (d) symmetric basin with central sill.

Sensitivity analysis

To investigate the impacts of different employed parameters from the best proposed models on hydraulic jump characteristics prediction via SVMs, sensitivity analysis was performed. In order to evaluate the effect of each independent parameter, the model was run with all input parameters and then one of the input parameters was eliminated and the SVM model was re-run. The RMSE criterion was used as indication of the significance of each parameter. The results are listed in Table 8. In this table ΔRMSE represents the percentages of the added values to the error criteria for each eliminated parameter. Based on the results from Table 8, it could be inferred that variable F1 has the most significant effect on the characteristics of a hydraulic jump. Also, Table 8 shows the order of the impact of each input parameter.

Table 8

Relative significance of each of input parameters of the best models (basin with central sill)

Hydraulic jump characteristicsEliminated variablePerformance criteria for test series
 
RMSEΔRMSE%The order of the impact of parameters
Sequent depth ratio 
Y F1, S/h1, B, h₁/B 0.021 – – 
F1 0.184 766 1 
B 0.023 9.5 
S/h1 0.025 19 
h1/B 0.028 34 
Length of hydraulic jump 
Lj/h1 F1, (h₂–h1)/h₁, h₂/h₁, S/h1, 0.037 – – 
F1 0.144 289 1 
(h₂–h1)/h₁ 0.046 24 
h2/h1 0.043 14 
S/h1 0.041 11 
B 0.039 
Loss of energy 
 ΔEL/E1 F1, S/h1, h₁/B 0.025 – – 
F1 0.203 712 1 
S/h1 0.030 20 
h1/B 0.044 76 
Hydraulic jump characteristicsEliminated variablePerformance criteria for test series
 
RMSEΔRMSE%The order of the impact of parameters
Sequent depth ratio 
Y F1, S/h1, B, h₁/B 0.021 – – 
F1 0.184 766 1 
B 0.023 9.5 
S/h1 0.025 19 
h1/B 0.028 34 
Length of hydraulic jump 
Lj/h1 F1, (h₂–h1)/h₁, h₂/h₁, S/h1, 0.037 – – 
F1 0.144 289 1 
(h₂–h1)/h₁ 0.046 24 
h2/h1 0.043 14 
S/h1 0.041 11 
B 0.039 
Loss of energy 
 ΔEL/E1 F1, S/h1, h₁/B 0.025 – – 
F1 0.203 712 1 
S/h1 0.030 20 
h1/B 0.044 76 

Bold values show the most significant parameter.

Combined data

We tried to evaluate the applicability of applied methods for a wider range of data. In other words, all data series were combined, then for predicting the hydraulic jump characteristics, as the dependent variable, the superior model of any characteristics and the model with only F1 as input variable were reanalyzed for the combined data state. The results of SVM models are given in Table 9 and Figure 9. According to the results of the combined data, it could be stated that adding h1/B and h2/h1 to input parameters caused an increment in model efficiency. However comparison between Tables 57 and 9 indicated that SVM models for the combined data set did not show the desired accuracy, and analyzing data sets separately led to more accurate results. It should be noted that for the combined date state, data series with different conditions (i.e. different hydraulic data range and different appurtenances) were used altogether. Therefore, the results are not so accurate. Figure 9 shows the verification between measured and estimated values of test series for the best proposed model for any characteristics.
Table 9

Statistical parameters of the SVM models for combined data

Output variableInput variable(s)Optimal parameters
Performance criteria
Train
Test
cɛγRDCRMSERDCRMSE
Sequent depth ratio 
h2/h1 F1 10 0.10 0.828 0.680 0.092 0.774 0.580 0.098 
F1, h1/B 10 0.01 0.873 0.760 0.079 0.816 0.660 0.088 
Length of hydraulic jump 
Lj/h1 F1 8.0 0.01 0.435 0.272 0.158 0.401 0.251 0.172 
F1,h2/h1 10 0.10 0.564 0.306 0.113 0.554 0.272 0.132 
Loss of energy 
EL/E1 F1 10 0.02 0.740 0.572 0.188 0.667 0.419 0.196 
F1, h1/B 10 0.01 0.885 0.781 0.101 0.882 0.647 0.118 
Output variableInput variable(s)Optimal parameters
Performance criteria
Train
Test
cɛγRDCRMSERDCRMSE
Sequent depth ratio 
h2/h1 F1 10 0.10 0.828 0.680 0.092 0.774 0.580 0.098 
F1, h1/B 10 0.01 0.873 0.760 0.079 0.816 0.660 0.088 
Length of hydraulic jump 
Lj/h1 F1 8.0 0.01 0.435 0.272 0.158 0.401 0.251 0.172 
F1,h2/h1 10 0.10 0.564 0.306 0.113 0.554 0.272 0.132 
Loss of energy 
EL/E1 F1 10 0.02 0.740 0.572 0.188 0.667 0.419 0.196 
F1, h1/B 10 0.01 0.885 0.781 0.101 0.882 0.647 0.118 
Figure 9

Comparison of observed and predicted hydraulic jump characteristics; combined data.

Figure 9

Comparison of observed and predicted hydraulic jump characteristics; combined data.

Comparison of SVM and classic equations

Accuracy of the best proposed models of the basin with central sill developed in this study, and some semi-theoretical formulae available in literature, were compared to evaluate the performance of the applied approach. The results of comparison are represented in Figure 10. According to three evaluated criteria (R, DC, and RMSE) which are shown in Figure 10, it can be seen that the estimated values via SVM models had more accurate results than equations. It can be observed that the formula used for estimating the sequent depth ratio provided a reasonable fit to the experimental data. In comparison with the sequent depth ratio formulas, hydraulic jump length equations did not show desired agreement between the estimated values and the observed sets, while for both hydraulic jump characteristics (Y and Lj/h1) the obtained results by the best SVM models were close to the measured data. It should be noted that the existing equations rely on a limited database, untested model assumptions, and a general lack of field data, and do not show the same results under variable flow conditions. However, despite the complexity and uncertainty of the hydraulic jump phenomenon the obtained results confirm the workability of SVM as an efficient machine learning approach in modeling of hydraulic jump characteristics in sudden diverging basins.
Figure 10

Comparison of statistical parameters between formula and best SVM model.

Figure 10

Comparison of statistical parameters between formula and best SVM model.

CONCLUSION

In the present study, the capability of the SVM approach was verified for predicting hydraulic jump characteristics (i.e. the sequent depth ratio, jump length and energy loss) in sudden diverging basins. The SVM was applied for different data sets in basins without appurtenances, with a negative step or with a central sill. The obtained results indicated that in prediction of the sequent depth ratio and relative energy dissipation the model with parameters F1 and h1/B as input variables performed more successfully than the other models. The superior performance for length of hydraulic jump was obtained for the model L(II), in which the inputs were F1 and (h2h1)/h1. Comparison between the results of three basins with sudden diverging side walls revealed that the developed models in the case of the basin with a central sill led to a more accurate outcome. It was also observed that the accuracy of the models for the basin with negative step, in a channel with asymmetric shape, was higher than for a symmetric shape channel, while for the basin without appurtenances, a symmetric channel presented better results. It showed that adding S/h1 as an input parameter improved the efficiency of the models. This issue confirmed the importance of the geometry of applied appurtenances in the hydraulic jump characteristics estimation process in basins with appurtenances. Based on the obtained results from sensitivity analysis, it was found that F1 had the most effective role in estimation of hydraulic characteristics compared with other parameters (see Table 8). From the results, it was observed that analyzing data sets separately led to a more accurate outcome. A comparison was also made between the applied technique and some classical equations. The applied technique was found to be able to predict hydraulic jump characteristics in sudden diverging basins successfully and in comparison with available classic equations is more reliable.

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