Analysis of Sequent Depth Ratio and Length of Sudden Expanding Hydraulic Jump under Symmetrical and Asymmetrical Developments

Introduction
In practical applications, for instance, at the downstream of hydraulic structures, in some cases the width of the basin may be larger than the upstream supercritical flow (sudden expansion in flow section). In such cases, an expanding hydraulic jump with symmetrical or asymmetrical shape will be developed at the downstream. Under the design of three parallel gates, the operation of the side or middle gate can be lead to the asymmetrical or symmetrical hydraulic jump, respectively. According to the position of jump toe, expanding hydraulic jump can be classified into four types. The present study focuses on a T-shaped hydraulic jump which the toe is established at the beginning of the divergence section.
Most of the previous studies are related to the symmetrical expanding hydraulic jump. In this case, the downstream diverging channel is symmetric on the central axis of the channel. However, a systematic study investigating the effect of symmetry and asymmetry on the expanding hydraulic jump was not found in the literature. In this study, using the momentum principle, some theoretical equations were derived to determine the sequent depths ratio of symmetrical and asymmetrical expanding hydraulic jump. Also, some regression relations were proposed to estimate the length of expanding hydraulic jump. The new proposed equations were also extended for the presence of a sill. The equations were calibrated using available experimental data from this study and the literature. This research also considered the characteristics of expanding hydraulic jump under the symmetrical and asymmetrical operation of parallel gates.

Methodology
To calibrate the new proposed relations and investigate the effects of different parameters on the expanding hydraulic jump characteristics, two experimental data sets were used. In addition to the data set from Bremen (1990), the experimental data from the present study were used. The data were collected from a hydraulic model of three parallel radial gates for operating the side or middle gate which corresponds with the asymmetrical or symmetrical expanding hydraulic jump, respectively. The experiments provided a wide range of different parameters as the approaching Froude number, hydraulic jump length, sequent depths ratio, divergence ratio, sill height and relative length of gate separator wall.



Results and discussion
Equation (4) was developed to determine the ratio of the sequent depths of expanding hydraulic jump based on the Momentum equation. For the presence of a sill, a combination of Equations (4), (7) and (8) can be used to calculate the ratio of sequent depths under the asymmetric and symmetric developments, respectively.
Determining the ratio of the sequent depths requires the calculation of the adjacent water depths of the closed gates. For this purpose, in addition to developing the regression equation (Equation 13), Equation (12) was proposed which based on the calculation of the hydraulic jump profile. It was observed that under the calibration range, the regression equation is more accurate. However, Equation (12) is recommended for the range outside of the experimental observations. The results showed that:
As the divergence ratio increases, the ratio of sequent depths approaches to the classic hydraulic jump.
By decreasing the relative length of the gate separator wall and decreasing the width of the gate on the downstream channel width, the relative depth at the side gate and consequently the ratio of the sequent depths will decrease.
For the lower length of separator wall and under operation of the middle gate, more lengths are needed to develop the jump than the side gate. However, as the length of the separator wall increases, the length and sequent depth of hydraulic jump due to the side gate increases on the middle gate.
In the presence of a sill, the relative length of hydraulic jump decreases and the ratio of secondary depths increases.
It was observed that the hydraulic jump due to the operation of the middle gate leans toward the left or right side of the channel due to oscillatory behavior.
Under operating the middle gate and in the absence of a sill, an asymmetric hydraulic jump is formed in the channel face when the length of the separator wall is less than 38% of the classical hydraulic jump length. For the presence of a sill, the minimum length of the separator wall decreases to about 26%.
By decreasing the initial depth of the hydraulic jump on the width of the gate and converting the output jet into a linear jet, the relative development length will increase.
As the sill height increases, the difference in the depths attached to the side gates will decrease and the hydraulic jump will develop more symmetrically.
As the relative height of the sill decreases, the minimum length of the separator wall to form a symmetrical hydraulic jump in the flanks, increases.

Conclusion
This research developed a set of theoretical and regression relationships for estimating the length and sequent depth ratio of expanding hydraulic jump. Moreover, the effects of sill height, divergence ratio and the length of the gate separator wall, were investigated. This study compares the effects of side and middle gate operations based on the variation of jump length and sequent depth ratio. The results can be used as a guide for the hydraulic structures operators to reduce the asymmetric severity of the expanding hydraulic jump and achieve the complete development under the minimum length.