The experimental data of this investigation carried out on horizontal bed were employed to test the applicability of the available equations in predicting the sequent depth ratio of a Bsminimum jump. The measured ht/h1 values and those predicted by Eqs. (2), (3) and (5) are shown in Fig. 6. The mean error, the R2 coefficient and the percentage of points in which the estimated error is less than or equal to 15% are presented in Table 1.
The mean error of Eq. (5) in its defined experimental range (0.23 < s/h1 < 1.57) is equal to 8.2%, and 100% of the predicted values are within the error band of 15%. However, the mean error of Eq. (5) increases outside the defined experimental range (s/h1 > 1.57) and reaches 13.4%, and only 66% of the predicted values fall within the \(\pm\)15% error range (Fig. 6). As shown in Fig. 6, Eq. (3) slightly underestimates the tailwater depth and 34% of the predicted values are out of \(\pm\)15% error bands. Eq. (2) shows better performance compared to Eqs. (3) and (5).
Equation

R2

Mean error (%)

% points that fall within the error band of \(\pm\)15%

Table 1
Testing the available relationships for predicting the sequent depth ratio of a Bsminimum jump
Eq. (2)

0.94

4.83

95

Eq. (3)

0.82

11.94

66

Eq. (5)

0.93

8.2

100

Eq. (13)

0.96

3.94

100

$$\frac{{{h_0}}}{{{h_1}}}  \frac{{{z_1}}}{{{h_1}}}=0.359{F_1}$$
15
The entry of air into the rolling zone of the jump and the irregularity of the water surface affected the water depth measurement at the toe of the chute and led to the scattering of the data (Fig. 7). Figure 8 shows the distribution of the estimate error of ho/h1 by Eq. (15). The results show that the estimate error is less than or equal to ± 10% for 93.3% of the cases and the distribution of the errors is normal.
According to Ohtsu et al. (1981), the sill height s and the distance between the sill and jump toe section xs are sufficient to determine the type of jump. At the first stage of the experiments carried out in this study, only the Bsminimum jump was generated. Therefore, the distance xs was minimum for the given Froude number and sill height. In this study, the dimensionless group of s/xs(min) is defined to characterize the β coefficient. To determine the β coefficient, its value was first obtained for Bsminimum jumps (z1/h1 = 0) using Eq. (14) and experimental data. Figure 9 shows the variation of the β coefficient with s/xs(min). The results show that the relationship is power, and the following equation was obtained:
$$\beta =12.5{\left( {\frac{s}{{{x_s}}}} \right)^{1.65}}$$
16
For s = 0, the β coefficient becomes equal to zero and Eq. (16) respects the boundary conditions. The mean error and the R2 coefficient of Eq. (16) was found equal to 15.71% and 0.92, respectively.
Figure 10 shows the relationship between the β coefficient and the dimensionless groups of z1/h1 and s/xs(min) for sillcontrolled Bjumps at different chute slopes. The results show that as the dimensionless groups of z1/h1 and s/xs(min) increase, the β coefficient decreases and increases, respectively. The relationship between the β coefficient reduction and z1/h1 is linear, and at a certain point, the effect of the sill on the Bjump becomes negligible.
The results of Fig. 10 were used to obtain the maximum z1/h1, where the effect of the sill is negligible and free Bjumps are generated. The relationship between the maximum z1/h1 and s/xs (min) for different chute slopes can be represented by the following equation (Fig. 11):
$${\left( {\frac{{{z_1}}}{{{h_1}}}} \right)_{\hbox{max} }}=a\left( {\frac{s}{{{x_s}(\hbox{min} )}}} \right)$$
17
in which a is a function of chute slope. Figure 12 demonstrates that the relationship between the coefficient a and the chute slope is exponential:
$$a=8.72exp(4.3\tan \theta )$$
18
Introducing Eq. (18) into Eq. (17), the following equation was obtained to compute the maximum z1/h1:
$${\left( {\frac{{{z_1}}}{{{h_1}}}} \right)_{\hbox{max} }}=8.72exp(4.3\tan \theta )\left( {\frac{s}{{{x_s}(\hbox{max} )}}} \right)$$
19
According to Fig. 10, the general equation useful to estimate the β coefficient for a sillcontrolled Bjump can be expressed as follows:
$$\beta =12.5{\left( {\frac{s}{{{x_s}(\hbox{min} )}}} \right)^{1.65}}  \Delta \beta$$
20
in which\(\Delta \beta\)is the difference between the β coefficient of a Bsminimum jump and a sillcontrolled Bjump for the given discharge and chute slope. The analysis showed that\(\Delta \beta\) depends on tanθ, z1/h1, and s/xs (min) and can be calculated using the following equation:
$$\Delta \beta =(  1.94\tan \theta +1.38)\left( {\frac{{{z_1}}}{{{h_1}}}} \right)\left( {\frac{s}{{{x_s}(\hbox{min} )}}} \right)$$
21
Introducing Eq. (21) into Eq. (20), the general equation can be rewritten as:
$$\beta =12.5{\left( {\frac{s}{{{x_s}(\hbox{min} )}}} \right)^{1.65}}  (  1.94\tan \theta +1.38)\left( {\frac{{{z_1}}}{{{h_1}}}} \right)\left( {\frac{s}{{{x_s}(\hbox{min} )}}} \right)$$
22
Eq. (22) applies for 0\(\leqslant\) z1/h1 < (z1/h1) max. For s = 0 and z1 = 0, Eq. (22) satisfies the boundary conditions. Figure 13 shows the comparison between the experimental and estimated values of the β coefficient by Eq. (22). The mean error and the R2 coefficient of Eq. (22) were obtained equal to 32.4 % and 0.85, respectively, and 62.6% of the predicted values fall within \(\pm\)30% error bands.
Knowing the values of the ratio h0/h1 (Eq. 15) and the β coefficient (Eq. 22), the sequent depth of a sillcontrolled Bjump can be calculated by the momentum equation (Eq. 13). Table 1 shows the performance of Eq. (13) in predicting the sequent depth of a Bsminimum jump. Also, the comparison between the experimental and estimated ht/h1 values by Eq. (13) for sillcontrolled Bjumps is showed in Fig. 14. The mean error and the R2 coefficient of Eq. (13) was calculated equal to 2.25% and 0.97, respectively. The distribution of the estimate error of ht/h1 by Eq. (13) is normal, and 99.9% of the predicted points fall within the \(\pm\)1 % error bands (Fig. 15).
Figure 16 shows the comparison of the tailwater depth, and the rate of energy dissipation for the free and sillcontrolled Bjump. In this Figure, the maximum z1/h1 (calculated by Eq. 19) is also presented for different ranges of s/xs (min). The free Bjump is generated for the case of s/xs (min) = 0, in which the tailwater depth is higher (Fig. 16a), compared to the sillcontrolled Bjump (s/xs > 0). As shown, for the different ranges of s/xs (min), the maximum reduction in the tailwater depth occurs at the toe of the chute (z1/h1 = 0), and as the jump toe location moves away from the bottom kink, the difference between the tailwater depth of the sillcontrolled and free Bjump decreases and finally the effect of the sill becomes negligible for (z1/h1)max.
In addition to reducing the tailwater, increasing the jump efficiency or the energy dissipation through the jump is another significant factor in the proper design of stilling basins. Energy dissipation η for sillcontrolled and free Bjumps were calculated using the following equation proposed by Habibzadeh et al. (2011):
$$\eta =1  \frac{{{H_2}}}{{{H_1}}}$$
23
in which H1 and H2 represent the energy head at the toe and end sections of the jump, respectively.
Figure (16b) shows the variation of energy dissipation for the free and sillcontrolled Bjump at different ranges of s/xs (min). The results show that the presence of a sill increases energy dissipation, and its value increases with increasing s/xs (min). Similar to the tailwater depth, as the jump toe location moves away from the bottom kink, the difference in energy dissipation decreases.
There is no difference between the use of a free and sillcontrolled Bjump in reducing the erosion at the basin end. For ht < ht (min), the jump toe position relative to the sill becomes independent of the tailwater, and the type \({\rm I}{\rm I}\) jump is generated for both cases. Therefore, as a general result, the sillcontrolled Bjump can be used as a novel alternative to the CHJ to reduce the length of stilling basin and the tailwater, and improve the energy dissipation rate.