Abstract

Hydraulic jump has numerous applications in the field of hydraulic engineering, such as energy dissipation over spillways, chlorinating of wastewater and many others. The sequent depth ratio is one of the important characteristics of hydraulic jump useful in designing the stilling basin. Despite its importance, the exact value of sequent depth ratio is still undetermined. In the present study an attempt has been made to find out the effects of roughness heights and slopes by conducting an experimental study and artificial neural network (ANN) model. Three different roughness heights of crushed and rounded aggregates and two positive bed slopes were used. The experimental results show that the reductions in sequent depth ratios are more in the case of crushed aggregate (4%–35%) than rounded on the same slope. By increasing bed slope, the sequent depth ratios show increasing trend in the range 3%–45%. The proposed ANN model has the capability to predict the sequent depth ratio with least MAPE (mean absolute percentage error) value 3.15%. Therefore, based on the results obtained from the empirical model and ANN model, it has been concluded that the present study can be better utilized for the estimation of the sequent depth ratio of hydraulic jump.

NOTATION

     
  • b

    bed width

  •  
  • E1

    specific energy of supercritical flow

  •  
  • E2

    specific energy of subcritical flow

  •  
  • integrated bed shear stress

  •  
  • F1

    Froude number of incoming flow before jump

  •  
  • F2

    Froude number of outgoing flow after jump

  •  
  • g

    acceleration due to gravity

  •  
  • G1

    coefficient for the effect of slope

  •  
  • h1

    supercritical flow depth

  •  
  • h2

    subcritical flow depth

  •  
  • h2/h1

    sequent depth ratio

  •  
  • Ks

    bed roughness height

  •  
  • relative roughness height

  •  
  • Km

    modified parameter considering the effect of slope

  •  
  • Lj

    length of the jump

  •  
  • n

    total number of samples

  •  
  • q

    discharge intensity

  •  
  • Q

    discharge through the sharp crested weir

  •  
  • SDR

    sequent depth ratio

  •  
  • U1

    depth-average velocity of incoming flow before jump

  •  
  • U2

    depth-average velocity of outgoing flow after jump

  •  
  • mass density of water

  •  
  • slope of the bed with horizontal

  •  
  • specific weight of water

  •  
  • shear force coefficient

  •  
  • coefficient that depends upon velocity distribution

  •  
  • positive coefficient having value less than 1

  •  
  • Wsinθ

    component of weight

INTRODUCTION

A hydraulic jump is a classic example of rapidly varied flow in which a high velocity supercritical depth of flow approaches subcritical flow depth by dissipating the energy of the incoming flow in the form of turbulence and heat. Over the past many years, this phenomenon has attracted the attention of many researchers not only because of its importance in designing a stilling basin but also for its complexity. Hydraulic jump is mostly used for hydraulic structures such as drops, spillways, gates and increasing the weight of the apron and aerating water for city water supplies. The theory of jump that has been studied on horizontal or slightly inclined channels has little impact of the weight component in the analysis. But, for the sloping channel, this impact becomes so pronounced that it must be considered in the analysis and the same helps in stabilizing the location of the jump (Chow 1959; Kumar & Lodhi 2015). In the present study, the type of hydraulic jump that is of interest forms entirely on the sloping channel and is classified as a D-jump (Kindsvater 1944; Rajaratnam 1966). In addition, many of hydraulic jump events have been studied by many researchers upon rough beds. Rajaratnam & Subramanya (1968) were the first to carry out systematic investigations of hydraulic jump on rough beds. The information on hydraulic jump on a rough sloping (positive or adverse) boundary proved that the roughness has a remarkable effect on reducing the sequent depth ratio (Leutheusser & Schiller 1975; Hughes & Flack 1984; Ali 1991; Ead & Rajaratnam 2002; Carollo & Ferro 2004a, 2004b; Izadjoo & Shafai Bejestan 2007; Pagliara et al. 2008; Carollo et al. 2013; Pagliara & Palermo 2015; Mazumder 2017; Palermo & Pagliara 2017).

The present study aims to increase the effectiveness of a sloping channel by patching gravels of the same size but different textures (rounded and crushed) on the channel bed and then investigate the influence of both channel slope and bed roughness heights on the main geometric characteristics of the hydraulic jump by giving a simple relationship that is also helpful for the practical design of hydraulic structures. In recent years artificial intelligence (AI) techniques have been in vogue in hydraulic studies (Zangooei et al. 2016; Roushangar et al. 2017). Therefore, in the present study the artificial neural network (ANN) technique has been used for the determination of the sequent depth of the hydraulic jump. The multi-layer feed-forward neural network has been trained using the Levenberg–Marquardt (LM) algorithm (Kumar & Kaur 2016).

Sequent depth ratio

For the formation of a jump on a rough sloping bed in a rectangular channel of unit width, the momentum and continuity equation is presented for the sequent depth ratio (Rajaratnam 1965) as: 
formula
(1)
where = discharge intensity, = depths of water, θ = slope of bed to the horizontal, U1 = average velocity of incoming flow at section 1, U2 = average velocity of outgoing flow at section 2, U1 = q/h1cosθ, U2=q/h2cosθ,Wsin = weight component of water in the direction of flow, = integrated bed shear stress. Rajaratnam (1965) proposed the following expression for bed shear stress: 
formula
(2)
where = specific weight of water, = shear force coefficient.
The equation used for computing the weight component within the control volume is: 
formula
(3)
where Lj = length of the hydraulic jump between section 1 and 2, and K is a function of slope which can only be found through experiment (Subramanya 2009).
Equation (1) may be simplified to (Ranga Raju 1993): 
formula
(4)
Now, using Equations (1)–(4), it is possible to derive the equation for sequent depth ratio as: 
formula
(5)
For the value of G1, Rajaratnam (1967) also proposed an empirical equation for the parameter G1 as a function of slope and Froude number as: 
formula
 
formula
(6)
where, 
formula
(7)
 
formula
(8)
Carollo & Ferro (2004b) suggested an empirical relationship between β and the relative roughness as: 
formula
(9)
where β = positive coefficient, assuming a value less than 1 to satisfy Equation (1).
Further, Govinda Rao & Ramprasad (1966) identified the solution for α= (1 − β), taking into account that both α and β are affected by the bed roughness. So, as an approach to incorporate the effect of bed roughness on a sloping channel, the modification factor α is considered as (1 − β) and Equation (5) takes the form: 
formula
(10)

Experimental setup

The experiments were conducted at the PG Irrigation & Hydraulics Engineering Laboratory of the Civil Engineering Department at PEC University of Technology, Chandigarh, India, in a tilting flume with dimensions of m respectively. Two slopes were used, viz. 19.143° and 23.901°. The bed roughnesses of the flume were calculated by maintaining a uniform flow and were found to be 0.002 m. Moreover, the different bed roughness heights were achieved by patching rounded and crushed gravels of average sizes of aggregates 0.010, 0.013, and 0.016 m on the bed of the flume. For the measurement of discharge, a rectangular sharp crested weir was installed at the downstream end of the flume. In total, 95 effective experimental runs with varying Froude numbers and the discharge were collected and the details are shown in Tables 1 and 2.

Table 1

Summary of experimental runs in the present study

SlopeExperimental setupRoughness size (mm)
No. of Runs
CrushedRounded
19.143° Smooth sloping bed – – 03 
23.901° Smooth sloping bed – – 04 
19.143° Rough sloping bed 10, 13, 16 10, 13, 16 42 
23.901° Rough sloping bed 10, 13, 16 10, 13, 16 46 
SlopeExperimental setupRoughness size (mm)
No. of Runs
CrushedRounded
19.143° Smooth sloping bed – – 03 
23.901° Smooth sloping bed – – 04 
19.143° Rough sloping bed 10, 13, 16 10, 13, 16 42 
23.901° Rough sloping bed 10, 13, 16 10, 13, 16 46 
Table 2

Range of data used in the present experimental study

Ks (m)Nature of aggregateSo (Degree)Q (m3/sec)h1 (m)h2 (m)(h2/h1)(Ks/h1)F1G1
0.010, 0.013 & 0.016 Rounded 19.143° & 23.901° 0.008–0.0781 0.0384–0.0856 0.1083–0.225 1.913–3.858 0.130–0.416 1.50–3.356 5.106–14.142 
0.010, 0.013 & 0.016 Crushed 19.143° & 23.901° 0.010–0.078 0.0428–0.1021 0.118–0.229 1.705–3.455 0.139–0.329 1.361–3.525 5.27–13.8 
Ks (m)Nature of aggregateSo (Degree)Q (m3/sec)h1 (m)h2 (m)(h2/h1)(Ks/h1)F1G1
0.010, 0.013 & 0.016 Rounded 19.143° & 23.901° 0.008–0.0781 0.0384–0.0856 0.1083–0.225 1.913–3.858 0.130–0.416 1.50–3.356 5.106–14.142 
0.010, 0.013 & 0.016 Crushed 19.143° & 23.901° 0.010–0.078 0.0428–0.1021 0.118–0.229 1.705–3.455 0.139–0.329 1.361–3.525 5.27–13.8 

RESULTS AND DISCUSSION

In order to investigate the effects of roughness and slopes on hydraulic jump characteristics, the sequent depth ratio (h2/h1) was considered for experimental and ANN modeling. During model development, the dimensionless jump depth (h2/h1) was selected as output and three dimensionless parameters were selected as inputs.

Experimental results of sequent depth ratio

From measured experimental data, the sequent depth (h2/h1) as a function of F1 is plotted for smooth and rough (crushed and rounded) beds on slopes of 19.143° and 23.901°, shown in Figure 1. Observations clearly show that reduction in sequent depth ratio was more in the case of crushed aggregates (4%–35%) than that of rounded aggregates on the same slope. Similarly, Figure 2 clearly shows the increasing trends of the sequent depth (h2/h1) with increase in bed slopes in the range of 4%–45% for a given value of F1.

Figure 1

Variations of (h2/h1) with F1 for different bed roughnesses: (a) slope of 19.143°; (b) slope of 23.9°.

Figure 1

Variations of (h2/h1) with F1 for different bed roughnesses: (a) slope of 19.143°; (b) slope of 23.9°.

Figure 2

Variations of (h2/h1) with F1 for different channel slopes.

Figure 2

Variations of (h2/h1) with F1 for different channel slopes.

Further, there was an attempt to check the effects of aggregate texture (crushed and rounded) separately on sequent depth ratio for the same sizes of roughness heights (10, 13 and 16 mm) on different slopes, and the observations show from Figures 3 and 4 that for both the slopes, reduction in sequent depth ratio was more in the case of crushed aggregate than that of the rounded aggregate for a given value of F1.

Figure 3

Comparison of (h2/h1) with F1 for rounded and crushed aggregates on a slope of 19.143°: (a) 10-mm-size aggregates; (b) 13-mm-size aggregates; (c) 16-mm-size aggregates.

Figure 3

Comparison of (h2/h1) with F1 for rounded and crushed aggregates on a slope of 19.143°: (a) 10-mm-size aggregates; (b) 13-mm-size aggregates; (c) 16-mm-size aggregates.

Figure 4

Comparison of (h2/h1) with F1 for rounded and crushed aggregates on a slope of 23.901°: (a) 10-mm-size aggregates; (b) 13-mm-size aggregates; (c) 16-mm-size aggregates.

Figure 4

Comparison of (h2/h1) with F1 for rounded and crushed aggregates on a slope of 23.901°: (a) 10-mm-size aggregates; (b) 13-mm-size aggregates; (c) 16-mm-size aggregates.

Hence, to consider the effects of both channel slope and bed roughness heights on sequent depth ratio contained in Km and β, there as an attempt to check the reliability of Equation (10) with the value of Km proposed by Rajaratnam (1967) (Equation (7)) and the value of β from Equation (9) proposed by Carollo & Ferro (2004b). It was observed that all the data points were under-estimated and lay below the line of agreement.

New proposed relationship

It was clearly observed from the above comparison that the existing empirical Equation (10) needs to be improved. For this new formulation, the effects of both the channel slope and bed roughness (crushed and rounded) heights on sequent depth were considered and a further two variables are present in Equation (10), so direct solution by any algebraic method is quite difficult. Hence, the solution for the same was computed by keeping one variable fixed while solving for the other and then using these new computed values as fixed and solved for another. Firstly, the effects of bed roughness heights as contained in β and were considered. The new relationships for rounded and crushed aggregates were obtained separately by substituting the observed values of (h2/h1) and G1 from Equation (6) as fixed in Equation (10). The new values of β were obtained and referred to as observed β. Now, by using this new observed value of β separately for rounded and crushed aggregates, a new relationship with a coefficient of correlation of 0.759 was obtained as indicated in Figure 5.

Figure 5

Relationship between β and for (a) rounded and (b) crushed aggregates.

Figure 5

Relationship between β and for (a) rounded and (b) crushed aggregates.

Again, by using the measured values of (h2/h1) from Equation (10) and keeping the new values of β from Figure 5 fixed, the new values of G1 were computed. Now, using these new computed values of G1 and F1 from Equation (8), the new values of the modified parameter Km were computed. By taking the average of all values of Km for both the slopes, a new relationship between Km and channel slope θ was proposed as shown in Figure 6. The new formulated relationships for sequent depth are, for rounded aggregates: 
formula
(11)
and for crushed aggregates: 
formula
(12)
The comparison of computed values of (h2/h1) from Equations (11) and (12) with observed values of sequent depth (h2/h1) lay close to the line of agreement with a maximum error of 25%.
Figure 6

Relationship between modified parameter Km and channel slope θ.

Figure 6

Relationship between modified parameter Km and channel slope θ.

Sequent depth estimation using the ANN model

An ANN model has been used for the prediction of sequent depth from other parameters such as and θ. In the present study a multi-layer feed-forward neural network was trained using the LM algorithm. Different neural network architectures have been trained to estimate sequent depth by varying hidden layer neurons. The parameters and θ were used as inputs to estimate the output h2 through the neural network. Of the data, 60% were used for training, 20% for testing and the remaining 20% were used for validations. The neurons in the hidden layer were varied in the network from eight to 13, to check network accuracy and performance in terms of mean square error (MSE). For accuracy, the networks were run a number of times and the mean absolute percentage error (MAPE) was calculated using Equation (13). 
formula
(13)
The accuracies of each network for training and testing are shown in Table 3. The network with four inputs, ten hidden neurons and one output has the least MAPE value of 3.15. The best model obtained during the study was also used to estimate the crushed sequent depth. The MAPE value for the crushed sequent depth was 4.09. The value of R (correlation coefficient) represents the link between targets and output values of the neural network model. The value of R lies between 0 and 1, with values nearer to 1 representing a strong relationship. The obtained values for training and testing accuracies are shown in Figure 7. The overall correlation coefficient value is 0.96 and the value of the slope is 0.95, attained throughout the entire dataset showing that the ANN model 4-10-1 can predict the sequent depth ratio of hydraulic jump to the measured value.
Table 3

Training and testing occurrences of each network with their MAPE value

ANN networkCorrelation coefficient
MAPE
TrainingTesting
4-8-1 0.97 0.87 4.29 
4-9-1 0.96 0.95 3.475 
4-10-1 0.98 0.93 3.151 
4-11-1 0.92 0.95 3.652 
4-12-1 0.94 0.90 3.723 
4-13-1 0.34 0.62 5.023 
ANN networkCorrelation coefficient
MAPE
TrainingTesting
4-8-1 0.97 0.87 4.29 
4-9-1 0.96 0.95 3.475 
4-10-1 0.98 0.93 3.151 
4-11-1 0.92 0.95 3.652 
4-12-1 0.94 0.90 3.723 
4-13-1 0.34 0.62 5.023 
Figure 7

Regression plot for best network model 4-10-1.

Figure 7

Regression plot for best network model 4-10-1.

CONCLUSIONS

This study aimed to analyze the effects of crushed and rounded aggregate along with channel bed slope on sequent depth ratio as playing a significant role in stabilizing the hydraulic jump. Based on the experimental observation, it has been observed that maximum reduction in sequent depth ratio in the case of crushed aggregate was reported as 35% while in the case of increasing slope condition it was observed as 45% (maximum value). Further, the existing relationships for slopes and roughness were examined and found to be under-estimated. Hence, an attempt was made to develop new relationships for the sequent depth ratio based on the experimental observations and showing a good correlation with calculated values from proposed relationships. Thereafter, an artificial neural network was applied to the experimental data to estimate sequent depth ratio of the hydraulic jump. The comparison of sequent depth ratio between the observed and the calculated values obtained by both the empirical as well as ANN model shows a good agreement with satisfactory evaluation criteria. Therefore, these models can be considered for estimation of sequent depth of hydraulic jump.

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