Soft Computing

, Volume 23, Issue 24, pp 13375–13391 | Cite as

Evaluation of conjugate depths of hydraulic jump in circular pipes using evolutionary computing

  • Mohammad NajafzadehEmail author
Methodologies and Application


Hydraulic jump phenomenon occurring in the circular pipes is a complex issue which has been paid studious attentions by hydraulic engineers. Several experimental and numerical studies were performed so as to characterize hydraulic jump in circular pipes. There are few comprehensive equations to predict the conjugate depth in circular pipes. In this investigation, three numerical models on the basis of evolutionary computing as gene-expression programming (GEP), model tree (MT), and evolutionary polynomial regression (EPR) have been utilized to evaluate the conjugate depths of the hydraulic jump in the circular pipes. Two non-dimensional parameters were yielded from conceptions of specific force to determine a functional relationship between the input and output variables. The performances of proposed approaches were compared with those obtained using conventional methods. The performance of MT indicated an accurate prediction of conjugate depths (R = 0.995 and RMSE = 0.023) in comparison with those other artificial intelligence (AI) models and empirical equations. The uncertainties prediction of the improved models were quantified and compared with those of existing models. The results of the proposed models demonstrated that linear equations provided by MT had more convenient application than empirical equations extracted from experimental investigations.


Hydraulic jump Evolutionary computing Model classification Conjugate depth Circular pipe Specific force equation 


Compliance with ethical standards

Conflict of interest

The author confirms that there is no conflict of interest in this research work.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Water Engineering, Faculty of Civil and Surveying EngineeringGraduate University of Advanced Technology, KermanKermanIran

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