Study of Hydraulic Jump Length Coefficient with the Leap Generation by Canal Gate Model

The length of hydraulic jump is a parameter needed to design the stilling basin dimension in the downstream of the weir or the other water structure in the river and channel. To determine the hydraulic jump length was needed function with some flow variables associated with the hydraulic jump phenomenon. One of variable to obtain the hydraulic jump was the hydraulic jump length coefficient (Cj). A study to determine the hydraulic jump length coefficient was conducted experimentally in laboratory. The data of experiment result was analyzed by regression technic. The results are the hydraulic jump length coefficient in fixed value and function with variable is a ratio of upstream and downstream depth of the the hydraulic jump.


Introduction
In the case of the waterworks structure construction in the river or canal, so the waterworks will cause the flow pattern in the canal will change. As an example if in the canal has been built a weir or canal gate for irrigation purpose ( Figure  1), then in upstream and the downstream of the structure the flow parameters will be change. In the upstream exist the slow flow (the flow velocity will be decrease and the flow depth will be increase). In the downstream of the gate the flow velocity will be increase, the flow depth will be decrease, and the flow pattern to be complicated. Occasionally the hydraulic jump generated in downstream of the gate on flow pattern as Figure 1a or 1b.
The sketch of the flow pattern in upstream and downstream of the canal gate can be seen in Figure 1a and 1b.
The flow pattern in the downstream of the canal gate Figure 1a, imply that will be occurred flow with high velocity along hydraulic jump running with the length L 0 and the circular flow on along hydraulic jump with length L j . The condition can scour the canal base along L 0 +L j , if the place is not protected. The flow pattern in form of the hydraulic jump in the downstream of the canal gate Figure 1b shows that scouring will be occurred along the hydraulic jump only. So that the canal base that need protection is along the hydraulic jump solely. The two conditions of the downstream gate flow pattern indicate that the hydraulic jump should be brought closer to the foot gate by hydraulics analysis (Figure 1b) so Leap Generation by Canal Gate Model as the protection of the canal base can be shorter and the cost of canal protection will be cheaper. Study of hydraulic jump or hydraulic jump length (L j ) as in Figure 1 was conducted by many experts in hydraulic engineering. a. A ccording to hydraulic jump phenomenon some researchers have given the results of their study. i) Chanson Page (1935), and Posey (1941) gave the hydraulic jump length (L j ) as equation (1), (2), (3), (4), (5), and (6) respectively.
a. According to hydraulic jump length coefficient (Cj) as in equation (7) some researchers have given coefficient values as follow.
C j = 10.6 (Fr 1 2 ) -0.185 (10) In this paper the equation of hydraulic jump length coefficient was developed associated with Figure 2, equation (7), and (9). Where, Lj = length of hydraulic jump Cj = hydraulic jump length coefficient h1 = water depth in section A h2 = water depth in section B Q = discharge of the flow The equations of hydraulic jump length coefficient which desired are such as in Equation (11) And Where a and b are parameters that can be determined based on data measurement result and regression analysis and s is determined as a fixed value associated with optimum correlation coefficient from regession analysis. The variable h 1 /h 2 is the ratio of water depth in critical flow condition in upstream of hydraulic jump and water depth in subcritical flow condition in downstream of hydraulic jump, different from variable which has been used by Woycicki (1931) Equation (9) i.e. ratio of h 2 /h 1 .

Hydraulic Jump Equation
Hydraulic jump can be occurred if the flow in the canal changed from the supercritical flow to the subcritical flow ( Figure 2). Hydraulic jump equation was derived by experts with flow type was steady flow or rapid varied flow as in Figure 3. From Figure 3 can be found that the pressure force (F p ) between section I and II is, And the velocity force (F v ) between section I and II is, For the along rapid varied flow Figure 3 the two forces is balance so, Where the section I is the supercritical flow and section II is subcritical flow, and, ρ = mass density of water g = gravity acceleration V 1 = flow velocity in section I V 2 = flow velocity in section II b = width of the rectangular canal Equation (14) can be rearrange to find the hydraulic jump equation to determine downstream depth of hydraulic jump, Equation (15) and (16).

Methods
Experiments were performed in a 10-m long channel of uniform rectangular section made of glass (bottom and sidewalls), located in the Hydraulic Laboratory of the Islamic University of Indonesia (Figure 4). The channel width is 0.10 m and the sidewall height is approximately 0.40 m. The channel is supported on an elevated steel frame which spans between main supports. The channel slope can be adjusted using a geared lifting mechanism but for the study was done in horizontal channel condition. Tail water levels are controlled by a radial gate fitted at the downstream channel end.
The water discharge was measured typically using a bend, installed in below of the end of the channel flume ( Figure 5). The two quantitative measured discharges were gauged by the bend i.e. 1.072 liter/second and 1.387 liter/second. The two discharges were used in this study.  Longitudinal flow depths are measured using a rail mounted pointer gauge positioned over the channel ( Figure  4). During the experiments, the location of the hydraulic jump was controlled by the downstream gate. For one discharge value, the hydraulic jump parameters were measured for 4 times.

Hydraulic Jump Parameters Measurement
The three parameters of the hydraulic jump which were measured are upstream depth (h 1 ) and downstream depth (h 2 ) and length (L j ) of hydraulic jump. The measurement result can be seen in Table 1, 2, 3, 4, and 5. Table 5 is correlation of hydraulic jump depth and length. Average of hydraulic jump length coefficient (Cj) is 6.454.

Data Analysis and Results
As mentioned in above paragraphs that the hydraulic jump length coefficient functions which desired are such as in equation (11) and (12). In those equations C j is a function of h 1 /h 2 ratio, therefore to find the coefficient and power of the both equations are required regression analysis. The data for the analysis is in Table 7 which derived from Table 5. Results of regression process in linear form are Equation (17) and in Figure 6.

Discussion
Based on the literature review which mention above that researchers such as Bakhmeteff and Matzke, Smetana, Kinney, Riegel and Beeba, Aravin, Triatmojo, Wu, Woycicki, Ivanchenko have used Equation (7) to compute the length of hydraulic jump. Each of the researches gave the C j quantity that different beetwen a researcher to another according to Equation (7).

C j with Specific Quantity
Bakhmeteff and Matzke, Smetana, Kinney, Riegel and Beeba, Aravin gave a fixed value of C j i.e. 5.0, 6.0, 6.2, 5.0, 5.4 respectively and Triatmojo gave the C j value in range from 5 to 7. Based on experiments in laboratory in this study was found range of the C j from 5.569 to 8.157. The range is greater than C j value which given by researchers before i.e. 5 to 7. From the experiment the average value of C j = 6.454. The value bigger than C j which were given by all researchers above and the closer value of C j = 6.454 is from Kinney with C j = 6.2.

C j in the Linear Function Form
In this study the independent variable of C j function was ratio of h 1 /h 2 , different from was given by other researchers which use ratio of h 2 /h 1 or Fr 1 . Fr 1 is a Froude number in upstream of hydraulic jump with depth h 1 . The result of the study in linear function (see Equation 17) is, From Woicicki, the equation is.  Comparison of Equation (19) to (20) can be seen in Figure  8 and 9. Figure 8 show that correlation between C j and h 1 give straight line trench for Woicicki formula and arch line for the result of the study and Figure 9 show that correlation between C j and h 2 give straight line trench for the result of the study and arch line for Woicicki formula.
Based on Figure 8 and 9 in general calculation result of C j with Woicicki formula is greater than the study formula.

C j in Power Form
The result of the study in power function (see Equation 18) is,  Based on Figure 10, in general computation result of C j with Wu or Ivanchenko formula is smaller than the study formula.

Conclusion
The coefficient of hydraulic jump length (C j ) associated with Equation (7) has been gained in form of fixed value, linear function, and power function with independent variable was ratio of h 1 /h 2 .
The average value (fixed value) of hydraulic jump length coefficient from the experiment was C j = 6.454. The value bigger than which were provided by researchers before. The closer value to the C j = 6.454 was given by Kinney (1935) with value C j = 6.2. The fixed value of C j = 6.454 can be used in practical purpose because the value close to Kenney C j = 6.2 and in the range of 5 to 7 propose by Triatmojo (1993).
In the study independent variable that was utilized associated with Equation (7) is a ratio of h 1 /h 2 . A function which can be obtained from regression analysis in linear form was C j = 6.3273 + 0.5974(h 1 /h 2 ). The function is different from Woicicki formula which used variable h 2 /h 1 , Equation (20). In power form the function is C j = 3.8274 + 3.0883(h 1 /h 2 ) 0.1 . Wu and Ivanchenko gave formula in variable of Froude number (Fr 1 ) i.e. C j = 10Fr 1 -0. 16 and C j = 10.6Fr 1 -0.185 respectively. Wu and Ivanchenko formula gave smaller value than the value of the study equation.
The two functions can be utilized in practical purpose to calculate C j coefficient to determine the length of hydraulic jump because the C j value from the function within the range of 5 to 7 (Triatmojo 1993).
In general hydraulic jump length coefficient was found from the study appropriate to be applied to calculate the hydraulic jump length to design a stilling basin in downstream of the weir.