Experimental Study of Hydraulic Jump Characteristics in Trapezoidal Channels

Article History: Received: 14/05/2017 Accepted: 12/04/2018 Published:01/06/2018 In the presented study, the hydraulic jump in a trapezoidal channel has been studied experimentally. Trapezoidal channel sections of bed width 20 cm, and side slopes (1H: 1.5V) and (1H: 1V) was fixed D/S a control sluice gate. For each side slope, the discharges ranging from 5 lit/sec to 25 lit/sec were passed in the channel to form hydraulic jump. The obtained results were analyzed and compared to determine the effect of side slopes and incoming Froude number of the flow on relative energy dissipated in the jump. Empirical equations and curve relations between sequent depths, relative energy lost in the jump and incoming Froude numbers were obtained.


INTRODUCTION
Hydraulic jump is a phenomenon well known to hydraulic engineers as a useful means of dissipating excess energy and thereby prevent scour and erosion downstream of spillways, chutes, power houses and other appurtenances. It has also been used to raise the water level on the downstream to provide the requisite head for diversion into canals and rivers etc. for irrigation purpose. Hydraulic jump is a phenomenon caused by change in flow regime from supercritical to subcritical flow with energy dissipation and rise in depth of flow. Hydraulic jump serves as an energy dissipater to dissipate excess energy of flowing water downstream of hydraulic structures, such as weirs, spillway, sluice gates etc. This excess energy, if left unchecked, will have adverse effect on the banks and the bed of the channel. The below Eqns. has been widely used to study the hydraulic jumps formed on rectangular, smooth and horizontal channel beds and not much theories have been established for trapezoidal channels. 1 2 (√1 + 8 1 2 − 1) ∆ = ( 2 − 1 ) 3 4 2 1 In this study, the properties of a hydraulic jump and energy dissipation has been studied downstream sluice gate in trapezoidal channels. In the presented study, the hydraulic jump in a trapezoidal channel has been studied experimentally. Trapezoidal channel sections of bed width 20 cm, and side slopes (1H: 1.5V) and (1H: 1V) was fixed D/S a control sluice gate. For each side slope, the discharges ranging from 5 lit/sec to 25 lit/sec were passed in the channel to form hydraulic jump. The obtained results were analyzed and compared to determine the effect of side slopes and incoming Froude number of the flow on relative energy dissipated in the jump. Empirical equations and curve relations between sequent depths, relative energy lost in the jump and incoming Froude numbers were obtained.

Keywords:
Open Channel. Hydraulic Jump Sequent depth Energy Dissipation Trapezoidal section

LITERATURE REVIEW
A review of literature has shown that earlier researcher concentrated more on rectangular channel while very little information is available on trapezoidal channels. The hydraulic jump in trapezoidal open channel has not received much attention. Thus, relatively scarce literature on hydraulic jumps in trapezoidal channels is available to date. (Chow VT 1959) declared that the hydraulic jump first investigated by Bidone, an Italian, in 1818. He mentioned that Belanger obtained the explicit solutions of sequent depth ratio for rectangular and prismatic channels without bed friction. Massey and Thiruvengadam 1961 (Terry W. Sturm 2001) presented two simple charts for obtaining the change in depth of water passing through a hydraulic jump trapezoidal and circular channels. In 1987, Hager and Wanoschek (Rashwan I. Mohamed 2013) declared that, regarding the sequent depth ratio and the relative energy dissipation, trapezoidal and particularly triangular channels are much more effective than rectangular channels, provided the inflow Froude number Fr1 is fixed. (Rashwan I. Mohamed 2013) presented an Analytical solution to problems of hydraulic jump in horizontal triangular channels. (M. Debabeche, S. Cherhabil, A. Hafnaoui and B. Achour 2009) were studied Hydraulic jump in a sloped triangular channel, they obtained an empirical relation for determination of the sequent depth ratio, knowing the inflow Froude number, and the bed slope of the channel for right angle triangle channels. (Sadiq S. Muhsun 2012) were analyzed theoretically the hydraulic jump in the trapezoidal channels using Newton Raphson method.

THEORETICAL ANALYSIS
The hydraulic jump in open channels is a transitional state from an upstream supercritical to downstream subcritical flow. In this transition, water surface rises abruptly, surface rollers are formed, and intense mixing occurs, air is entrained and usually a large amount of energy is dissipated as shown in Fig. 1.
Where Q = discharge; 1 and 2 = water areas at Sections 1 and 2, respectively; ̅ 1 and ̅ 2 = the depths from the water surface to the centroids of the Sections 1 and 2, respectively; and g = the acceleration due to gravity. For trapezoidal sections: Substituting equations (2) in (1), If we define new functions as: Substituting equations (4) in (3), and simplifying: For trapezoidal sections, squared Froude number is given by: Substituting equations (4 and 5) in (6) and simplifying:

Relative Energy Dissipation:
Specific energy may be interpreted as the sum of potential and kinetic energy of fluid with respect to the bottom of the channel. Specific energy before the jump can be written as: 1 = 1 + 1 1 2 2 Where E1 = specific energy at Section 1 and α1 = energy coefficient at Section 1.
Specific energy after the jump expressed as: 2 = 2 + 2 2 2 2 For uniform velocity distribution, 1 = 2 = 1. The Relative Energy Dissipation by the jump can be written as: Substituting equations (4) and (6)  As seen in eq. (6), the squared of approached Froude number is a function of (λ and δ) and from eq. (9) the relative energy dissipation is a function of (F r1 2 , λ and δ), since F r1 2 is a function of (λ and δ), then we can say that the relative energy dissipation is a function of (λ and δ) also.

DIMENSIONAL ANALYSIS
A dimensional analysis is applied to correlate the different factors affecting phenomena under study and the following functional relationship is obtained:

EXPERIMENTAL WORK
The experimental work was carried out in the Fluid Laboratory of the College of Engineering, Salahaddin University. The experiments were carried out in a horizontal recirculating laboratory flume 50 cm wide, 50 cm deep with a 12 m working length as shown in Fig. 2. The discharge was measured using a pre-calibrated V-notch weir (Q = 0.0195 h 2.398 , Q in lit/sec and h in cm) (Othman K. Mohammed 2010), located at the end of the overhead tank. Water drawn from the underground storage tank by an electrically driven centrifugal pump through a 6 inch diameter Steel pipe to the overhead tank providing a maximum discharge of (45 lit/sec). Trapezoidal channel sections of bed width 20 cm, and side slopes (1H: 1.5V) and (1H: 1V) was fixed D/S a control sluice gate. For each side slope, the discharges ranging from 5 lit/sec to 25 lit/sec were passed in the channel to form hydraulic jumps. The parameters y1, y2, and flow rate were measured in each experimental run.

RESULTS AND DISCUSSION
The variation of different hydraulic jump characteristics such as sequent depth ratio, relative energy loss with approach Froude number and side slope of the channel is given below. Fig. 3 shows a linear variation of sequent depth ratio (y2/y1) against the Froude number (Fr1) 2 varying from 5 to 30, for two different side slopes. The R 2 value of the linear fit shows that linear variation holds well within y2/y1 and (Fr1) 2 among the possible regression types. The plot of data shows that points of lower side slope (1H: 1V) lies above the that of higher side slope (1H: 1V), this means that lower side slope produces higher sequent depth ratio (y2/y1) for the same squared Froude number.  Fig. 4 shows a second order polynomial variation of relative energy loss (ΔE/E1) against the Froude number (F r1 2 ) varying from 5 to 30. The R 2 value of the second order polynomial variation fit shows that its variation holds well within (ΔE/E1) and (F r1 2 ) among the possible regression types. The plot of data shows that points of higher side slope (1H: 1V), lies above the that of lower side slope (1H: 1V), this means that higher side slope produces higher relative energy loss (ΔE/E1) for the same squared Froude number.  Fig. 5 shows a linear variation of relative energy loss (ΔE/E1) against the sequent depth ratio (y2/y1), for two different side slopes. The plot of data shows that points of higher side slope (1H: 1V), lies above the that of lower side slope (1H: 1V), this means that higher side slope produces higher relative energy loss (ΔE/E1) for the same sequent depth ratio (y2/y1).

Empirical Computational Models
To obtain empirical equations for (F r1 2 ) and (∆E E1 ⁄ ), (SPSS22) statistical software was used for obtaining two types of regressions between (F r1 2 ) and dimensionless parameters obtained by dimensional analysis were tested to find the best one, they are:

CONCLUSIONS
The results of the experimental and theoretical study on the hydraulic jump are presented. The discussion and analysis of the results highlighted the following conclusions: 1. Sequent depth ratio and relative energy loss increases with increase in approach Froude number. Fig. 3 and Fig. 4 2. For constant Froude number, sequent depth ratio of the jump increase with increase in side slope of the channel. Fig. 3 3. For constant Froude number, relative energy loss through the jump increase with decrease in side slope of the channel. Fig. 4 4. For constant sequent depth ratio, relative energy loss through the jump decrease with increase in side slope of the channel. Fig. 5  5. Analysis of the experimental data showed that the linear regression provides the best fit based on the measured (F r1 ) 2 versus predicted (F r1 ) 2 plots and the (R 2 ) values, which has a value of (0.814). Fig. 6 and eq. (12). 6. For the relative energy dissipation, the linear regression provides the best fit based on the measured ΔE/E1 versus predicted ΔE/E1 plots and the (R 2 ) values, which has a value of (0.652). Fig. 7 and eq. (14). 7. The applicability range of squared Froude number for proposed equations cover 2.5 -5 and side slopes (1H: 1.5V) and (1H: 1V). Further experiments are recommended to study the applicability of proposed equations for other Froude numbers and different side slopes.