Intelligent Mathematical Modeling of Discharge Coefficient of Nonlinear Weirs with Triangular Plan

Document Type : Research Article


1 Professor, Water Engineering Department, Lorestan University, Khorramabad, Iran.

2 Water Engineering Department

3 Department of Water and Soil Conservation, Ministry of Agriculture Jihad, Kerman, Iran


 In this study, the discharge coefficient (Cd) of non-linear weirs with a triangular plan was mathematically modeled using a group of method data handling (GMDH), genetic programming (GP) and multivariate adaptive regression splines (MARS) techniques. For this purpose, related datasets including parameters on Cd were collected from literature. These methods were selected since they are classified as smart function fitting (SFF) methods. The main advantages of SFF methods compared to other artificial intelligence methods are defining the most effective parameters on output and assigning more weights to them in mathematical expression process. Results of MARS indicated that this method with fifteen basic functions could achieve good accuracy for modeling and predicting Cd (R2= 0.98 and RMSE=0.024). Results of GMDH showed that this model includes two hidden layers and that there are five and four neurons at the first and second hidden layers, receptivity. Results of developed GP model declared that this model consists of three genes and has acceptable performance for modeling Cd. Evaluation of proficiency of utilized models with each other indicated that the best accuracy is related to MARS model. Calculating the discrepancy ratio index (DDR) shows that the minimum range of DDR is related to MARS model.


Main Subjects

[1]M. Ameri, A.A. Dehghani, A. Ahmadi, Elementary discharge coefficient of a triangular–rectangular sharp- crested side weir in subcritical flow, International Journal of River Basin Management, (2015) 1-8.
[2] J.P.  Tullis,   N.   Amanian,   D.   Waldron,   Design of Labyrinth Spillways, Journal of Hydraulic Engineering, 121(3) (1995) 247-255.
[3] S. Erpicum, F. Laugier, M. Pfister, M. Pirotton, G.M. Cicero, A.J. Schleiss, Labyrinth and Piano Key Weirs II, Taylor & Francis, 2013.
[4] B.P. Tullis, J.C. Young, M.A. Chandler, Head- Discharge Relationships for Submerged Labyrinth Weirs, Journal of Hydraulic Engineering, 133(3) (2007) 248-254.
[5] F.G. Tacail, B. Evans, A. Babb, Case study of a labyrinth weir spillway, Canadian Journal of Civil Engineering, 17(1) (1990) 1-7.
[6] M. Leite Ribeiro, M. Bieri, J.L. Boillat, A.J. Schleiss, G. Singhal, N. Sharma, Discharge Capacity of Piano Key Weirs, Journal of Hydraulic Engineering, 138(2) (2012) 199-203.
[7] R.M. Anderson, B.P. Tullis, Comparison of Piano Key and Rectangular Labyrinth Weir Hydraulics, Journal of Hydraulic Engineering, 138(4) (2012) 358-361.
[8] G. Taylor, The performance of labyrinth weirs, University of Nottingham, 1968.
[9] A.Y. Mohammed, Numerical analysis of flow over side weir, Journal of King Saud University - Engineering Sciences, 27(1) (2015) 37-42.
[10] M. Emin Emiroglu, M. Cihan Aydin, N. Kaya, Discharge Characteristics of a Trapezoidal Labyrinth Side Weir with One and Two Cycles in Subcritical Flow, Journal of Irrigation and Drainage Engineering, 140(5) (2014) 04014007.
[11] S. Bagheri, A.R. Kabiri-Samani, M. Heidarpour, Discharge coefficient of rectangular sharp-crested side weirs, Part I: Traditional weir equation, Flow Measurement and Instrumentation, 35 (2014) 109-115.
[12] H. Haddadi, M. Rahimpour, A discharge coefficient for a trapezoidal broad-crested side weir in subcritical flow, Flow Measurement and Instrumentation, 26(0) (2012) 63-67.
[13] O. Kisi, M. Emin Emiroglu, O. Bilhan, A. Guven, Prediction of lateral outflow over triangular labyrinth side weirs under subcritical conditions using soft computing approaches, Expert Systems with Applications, 39(3) (2012) 3454-3460.
[14] O. Castro-Orgaz, W. Hager, Subcritical Side-Weir Flow at High Lateral Discharge, Journal of Hydraulic Engineering, 138(9) (2012) 777-787.
[15] M.E. Emiroglu, H. Agaccioglu, N. Kaya, Discharging capacity of rectangular side weirs in straight open channels, Flow Measurement and Instrumentation, 22(4) (2011) 319-330.
[16] M. Ghodsian, Supercritical flow over a rectangular side weir, Canadian Journal of Civil Engineering, 30(3) (2003) 596-600.
[17] M. Jalili, S. Borghei, Discussion: Discharge Coefficient of Rectangular Side Weirs, Journal of Irrigation and Drainage Engineering, 122(2) (1996) 132-132.
[18] A. Corhay, S. Erpicum, F. Laugier, M.H.T. Khanh, M. Pfister, Labyrinth and Piano Key Weirs III: Proceedings of the 3rd International Workshop on Labyrinth and Piano Key Weirs (Pkw 2017), February 22-24, 2017, Qui Nhon, Vietnam, Taylor & Francis, 2017.
[19] S. Erpicum, F. Laugier, J.L. Boillat, M. Pirotton, B. Reverchon, A. Schleiss,  Labyrinth  and  piano  key weirs—PKW 2011, in: Proceedings of the International Conference on Labyrinth and Piano  Key Weirs, Balkema Liege, 2011, pp. 9-11.
[20] S. Dehdar-behbahani, A. Parsaie, Numerical modeling of flow pattern in dam spillway’s guide wall. Case study: Balaroud dam, Iran, Alexandria Engineering Journal, 55(1) (2016) 467-473.
[21] A. Parsaie, A.H. Haghiabi, Numerical Modeling of Flow Pattern in Spillway Approach Channel, Jordan Journal of Civil Engineering, 12(1) (2018) 1-9.
[22] Z. Kashkaki, H. Banejad, M. Heydari, Application of ANN in Estimating Discharge Coefficient of Circular Piano Key Spillways, Soft Computing in Civil Engineering, 2(3) (2018) 39-49.
[23] A. Parsaie, H.M. Azamathulla, A.H. Haghiabi, Prediction of discharge coefficient of cylindrical weir– gate using GMDH-PSO, ISH Journal of Hydraulic Engineering, 24(2) (2018) 116-123.
[24] E. Olyaie, H. Banejad, M. Heydari, Estimating Discharge Coefficient of PK-Weir Under Subcritical Conditions Based on High-Accuracy Machine Learning Approaches, Iranian Journal of  Science  and Technology, Transactions of Civil Engineering, (2018). 
[25] A.H. Haghiabi, A. Parsaie, S. Ememgholizadeh, Prediction of discharge coefficient of triangular labyrinth weirs using Adaptive Neuro Fuzzy Inference System, Alexandria Engineering Journal, (2017).
[26] M.C. Aydin, CFD simulation of free-surface flow over triangular labyrinth side weir, Advances in Engineering Software, 45(1) (2012) 159-166.
[27] M.C. Aydin, M.E. Emiroglu, Determination of capacity of labyrinth side weir by CFD, Flow Measurement and Instrumentation, 29(0) (2013) 1-8.
[28] G.K. Robertson, Labyrinth weir hydraulics: Validation of CFD modelling, Stellenbosch: Stellenbosch University, 2014.
[29] B.M. Crookston, B. Tullis, Labyrinth weirs, Hydraulic Structures, (2010) 59.
[30] O. Bilhan, M. Emin Emiroglu, O. Kisi, Application of two different neural network techniques to lateral outflow over rectangular side weirs located on a straight channel, Advances in Engineering Software, 41(6) (2010) 831-837.
[31] I. Ebtehaj, H. Bonakdari, A.H. Zaji, H. Azimi, A. Sharifi, Gene expression programming  to  predict  the discharge coefficient in rectangular side weirs, Applied Soft Computing, 35 (2015) 618-628.
[32] M.E. Emiroglu, O. Bilhan, O. Kisi, Neural networks for estimation of discharge capacity of triangular labyrinth side-weir located on a straight channel, Expert Systems with Applications, 38(1) (2011) 867- 874.
[33] M. Najafzadeh, A. Etemad-Shahidi, S.Y. Lim, Scour prediction in long contractions using ANFIS and SVM, Ocean Engineering, 111 (2016) 128-135.
[34] A. Parsaie, H.M. Azamathulla, A.H. Haghiabi, Physical and numerical modeling of performance of detention dams, Journal of Hydrology, (2017).
[35] A. Parsaie, A.H. Haghiabi, M. Saneie, H. Torabi, Applications of soft computing techniques for prediction of energy dissipation on stepped spillways, Neural Computing and Applications, (2016) 1-17.
[36] A.H. Haghiabi, H.M. Azamathulla, A. Parsaie, Prediction of head loss on cascade weir using ANN and SVM, ISH Journal of Hydraulic Engineering, (2016) 1-9.
[37] A.H. Zaji, H. Bonakdari, Performance evaluation of two different neural network and particle swarm optimization methods for prediction of discharge capacity of modified triangular side weirs, Flow Measurement and Instrumentation, 40 (2014) 149- 156. 
[38] K. Roushangar, M.T. Alami, J. Shiri, M.M. Asl, Determining discharge coefficient of labyrinth and arced labyrinth weirs using support vector machine, Hydrology Research, 49(3) (2018) 924-938.
[39] H.M. Azamathulla, A.H. Haghiabi, A. Parsaie, Prediction of side weir discharge coefficient by support vector machine technique, Water Science and Technology: Water Supply, 16(4) (2016) 1002-1016.
[40] J. Mohammadzadeh-Habili, M. Heidarpour, S. Samiee, Study of Energy Dissipation and Downstream Flow Regime of Labyrinth  Weirs,  Iranian  Journal of Science and Technology, Transactions of Civil Engineering, 42(2) (2018) 111-119.
[41] K. Gupta, S. Kumar, Z. Ahmad, FLOW CHARACTERISTICS OF SHARP-CRESTED W-PLANFORM WEIRS, World Applied Sciences Journal, 32(7) (2014) 1311-1317.
[42] M. Ghodsian, Stage–discharge relationship for a triangular labyrinth spillway, Proceedings of the ICE- Water Management, 162(3) (2009) 173-178.
[43] S. Kumar, Z. Ahmad, T. Mansoor, A new approach to improve the discharging capacity of sharp-crested triangular plan form weirs, Flow Measurement and Instrumentation, 22(3) (2011) 175-180.
[44] A. Haghiabi, Estimation  of  scour  downstream  of a ski-jump bucket using the multivariate adaptive regression splines, Scientia Iranica. Transaction A, Civil Engineering, 24(4) (2017) 1789-1801.
[45] A.H. Haghiabi, Prediction of longitudinal dispersion coefficient using multivariate adaptive regression splines, Journal of Earth System Science, 125(5) (2016) 985-995.
[46] A.H. Haghiabi, Modeling River Mixing Mechanism Using Data Driven Model, Water Resources Management, 31(3) (2017) 811-824.
[47] A. Parsaie, A.H. Haghiabi, M. Saneie, H. Torabi, Prediction of energy dissipation on the stepped spillway using the multivariate adaptive regression splines, ISH Journal of Hydraulic Engineering, 22(3) (2016) 281-292.
[48] A.G. Ivakhnenko, Polynomial theory of complex systems, IEEE transactions on Systems, Man, and Cybernetics, 1(4) (1971) 364-378.
[49] M.F. Brameier, W. Banzhaf, Linear Genetic Programming, Springer US, 2007.
[50] R. Noori, B. Ghiasi, H. Sheikhian, J.F. Adamowski, Estimation of the Dispersion Coefficient in Natural Rivers Using a Granular Computing Model, J. Hydraul. Eng., (2017) 04017001.