Estimation of waterfall height in the downstream of ogee spillways to hydraulic jump control

The purpose of this study is to generate the relationships to directly calculate waterfall height at the downstream of the spillway to form a jump at the toe of the spillway and to prevent erosion and destruction of the downstream river bed or body of the spillway in a submerged and free hydraulic jump.

The purpose of this study is to provide a relationship to directly calculate the waterfall height (h) without the need for a trial and error procedure. For this purpose, the application of the momentum relationship between two sections (1and 2) yields the waterfall height (h) according to Eq. (1).   h=y1 .....   (1) Since in Eq. (1), Fr1 and y1 are themselves a function of the waterfall height (h), so this equation must be solved by trial and error or by using design charts provided by other researchers. To calculate the waterfall height in oscillating jump conditions (2.5<Fr1<4.5) and also for steady and strong jump conditions (4.5<Fr1<15.5), by assuming different values ​​for p, y0, yt and C (where C is the discharge coefficient), 300 and 1146 series of numbers have been generated by trial and error, respectively. Then, the parameters in Eq. 1 (y1, Fr1) were also calculated. The final waterfall height (h) was obtained for each series of numbers. Using the existing variables, dimensionless parameters ( ) and Fr1 were extracted. Then, multiple linear and nonlinear multiple regression relationships were tested for direct calculation of . Finally, the best relationships with the least error were selected. The results of multiple regression (MR) relationships are also compared with the results of ANN and SVM methods.

The proposed non-linear multiple regression relationship (MR-3) shows higher accuracy in estimating the waterfall height, compared to MR-1 and MR-2 regression models based on three statistical indices (RE%, RMSE, R2) for 2.5<Fr1<4.5 and 4.5<Fr1<15.5. Therefore with having the Froude number related to the initial hydraulic jump depth (Fr1), tail water depth (yt), weir height (P) and initial jump depth (y1), the calculation of waterfall height (h) will be possible using MR-3 model without any trial and error procedures. Moreover, the results show that among the intelligent models, the ANN model has very close results to the proposed nonlinear regression relation (MR-3) based on the statistical indices.

A new method for calculating the height of a waterfall at the toe of the ogee spillway was presented to control hydraulic jump. To calculate the height of the waterfall directly, multiple nonlinear regression (MR) relationships were presented for two ranges of different Froude numbers. The MR, ANN, and SVM models showed good performance in predicting the height of the waterfall downstream of spillways, but the ability of the two MR and ANN models were better than the SVM.