Extraction of head-discharge relationship for submerged standard and modified piano key weirs using intelligent algorithms

Introduction: Weirs are commonly used to measure flow, divert flow, and/or control flow water conduits. Although weirs are generally designed to run under free-flow conditions, they can become submerged under certain conditions. Weir submergence occurs when the downstream water level exceeds the crest elevation. Relative to traditional linear weirs (e.g., ogee crest), the use of nonlinear weirs to increase the flow capacity in discharge channels of limited width without much increasing the required Ho is becoming more common. Typical nonlinear weirs include labyrinth and piano key (PK) weirs which the second one has been considered in this paper. When deciding between a labyrinth and a PK weir for a channel application, the potential influence of submergence, along with the free-flow discharge capacity of both weirs, must be considered. Little works have been conducted on the submergence effects of PK weirs. This paper aims to describe an intelligent method to extract the head-discharge relationship in standard and modified Piano key weirs under submerged conditions. Support vector machine and gene expression programming intelligent algorithms were utilized to predict submerged head-discharge relationship using Dabling et al. (2014) experimental data. In the end, a comparison has been performed among SVM, GEP, and experimental based predictors using assessment criteria.
Methodology: Gathered data were from a laboratory flume measuring 0.93 m wide, 0.61 m deep, and 7.4 m long. A stilling well with a point gauge (readable to ±0.15 mm) was hydraulically connected to the flume sidewall at a distance of 4P times the weir height, (approximately 0.8 m) upstream of the weir for measuring the piezometric head level (ho and h*). A second stilling well with a point gauge connected to the flume 10P (approximately 2.0 m) downstream of the weir was used to measure the downstream piezometric head (hd). Both Ho and Hd were calculated by adding the velocity head (U2/2g) corresponding to the average cross-sectional velocity at the respective measurement locations. For the submergence investigation, variations in tailwater elevation were produced using an adjustable gate located 15P (approximately 3.0 m) downstream of the weir. A calibrated orifice meter located in the 305 mm diameter supply piping was used to accurately measure the weir discharge (± 0.2%). Repeating variables were opted as upstream flow velocity, Uu, downstream flow velocity, Ud, upstream flow depth relative to weir peak under free condition, ho, upstream flow depth relative to weir peak under submerged condition, h*, downstream flow depth relative to weir peak under submerged condition, hd, total free flow head relative to the weir crest, Ho, total submerged flow head relative to the weir crest, H*, weir height, P, gravity acceleration, g, specific mass of flow, ρ, water viscosity, µ, and water surface tension, σ. The dimensionless linear independent parameters were extracted as follow with omitting the viscosity and surface tension effects (Eq. 12)
Where Frd and Fru are downstream and upstream Froud numbers, respectively. Various combination of dimensionless parameters were examined to check the best performance of the SVM and the GEP algorithms to predict H^*/H_o using root mean square error, RMSE, determination coefficient, R2, mean normalized error, MNE, and Developed Discrepancy Ratio, DDR.
Results and Discussion: Two intelligent algorithms, i.e. SVM and GEP, were trained and tested using laboratory data from Dabling et al. (2014). The share of training and testing percent od measured data for the SVM were 60% and 40% and those of the GEP were 70% and 30%, respectively. Superior combination for the SVM to predict H^*/H_o included H_d/H_o andH_o/P for both PK-S and PK-M. The values of (RMSE, R2, MNE) for the train and the test phases for PK-S and PK-M were (0.008, 0.9996, 0.234), (0.002, 0.9989, 0.237), (0.098, 0.9833, 0.308) and (0.0918, 0.9899, 0.282), respectively. The opted dimensionless parameters for the GEP predictor included all four mentioned parameters in equation (1). The corresponding values of above mentioned criteria for train and test phases were calculated (0.0070, 0.9999, 0.3284), (0.0180, 0.9991, 0.3127), (0.0099, 0.9984, 0.2433) and (0.0097, 0.9998, 0.1825) respectively. A comparison was done between intelligent predictors and experimental equation extracted by Dabling et al., (2014). Their analogy was performed using standardized DDR values for every three predictors, i.e. ZDDR. The amount of the maximum values of ZDDR was obtained 8.517, 6.582, and 4.098 for the SVM, the GEP, and the experimental predictor, respectively. The results showed that the SVM intelligent algorithm has superiority than to the others.
Conclusion: Dispite the high values of experimental studies, the results showed that using artificial and intelligent algorithm is more practical to extract the hidden relationship among dependent and independent variables (Three-dimensional and complex flow over these weirs) for achieving high accuracy prediction. The results showed that the SVM and GEP as intelligent algorithms leads to very accurate results.