Depth-averaged model for undular hydraulic jump

ABSTRACT

An undular hydraulic jump corresponds to the weak transition from super- to subcritical-flow in the form of steady free surface undulations. Previous models on undular hydraulic jumps employed the potential flow theory, i.e., the solitary and cnoidal wave theories. Experimental observations indicate the inadequacy of this theory, which motivated the development of more advanced approximations. Basic flow features including friction effects on the velocity profile, modelling of the bed-shear stress, and Reynolds stresses are considered. However, none of the models currently available include all these aspects. In this study, a general depth-averaged model is developed based on the k-ϵ turbulence closure. The general depth-averaged equations are applied to the undular jump problem, introducing a suitable time-averaged velocity distribution based on a composite power-law model, in which both streamline curvature and vorticity are accounted for. The bed-shear stress closure is included by a boundary layer method. Predictions of the depth-averaged Reynolds averaged Navier-Stokes (RANS) model are shown to be close to the 2D RANS solution and to experimental observations.

Keywords:

• Boundary shear
• depth-averaged model
• hydraulic jump
• turbulent flow
• undular jump

Acknowledgements

The first author would like to thank Professor J.S. Montes, University of Hobart, Tasmania, for his encouragement and advice on this fascinating topic. Most of the ideas that are developed in this work were motivated by stimulating discussions with him. The first author also thanks Professor C. Apelt, University of Queensland, Australia, for having provided a copy of his work on depth-averaged turbulent flow models, and to Professor W. Schneider, T.U. Wien, for personal communications on his results regarding the asymptotic expansion of the RANS equations near the critical flow depth. This work was supported by the Spanish project CTM2013-45666-R.