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Argiris I. Delis; Hervé Guillard; Yih-Chin Tai
Numerical simulations of hydraulic jumps with the Shear Shallow Water model
SMAI-Journal of computational mathematics, 4 (2018), p. 319-344, doi: 10.5802/smai-jcm.37
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Keywords: Shear Shallow Water model, Shallow Water equations, Turbulent Hydraulic jumps, Free surface flows, Finite Volumes, Well-balancing


An extension and numerical approximation of the shear shallow water equations model, recently proposed in [25], is considered in this work. The model equations are able to describe the oscillatory nature of turbulent hydraulic jumps and as such correct the deficiency of the classical non-linear shallow water equations in describing such phenomena. The model equations, originally developed for horizontal flow or flows occurring over small constant slopes, are straightforwardly extended here for modeling flows over non-constant slopes and numerically solved by a second-order well-balanced finite volume scheme. Further, a new set of exact solutions to the extended model equations is derived and several numerical tests are performed to validate the numerical scheme and its ability to predict the oscillatory nature of hydraulic jumps under different flow conditions.


[1] A. Bermudez & M. E. Vázquez, “Upwind methods for hyperbolic conservation laws with source terms”, Computers & Fluids 23 (1994) no. 8, p. 1049-1071
[2] J. Burguete & P. García-Navarro, “Efficient construction of high-resolution TVD conservative schemes for equations with source terms: Application to shallow water flows”, International Journal for Numerical Methods in Fluids 37 (2001) no. 2, p. 209-248
[3] D. Caviedes-Voullième & G. Kesserwani, “Benchmarking a multiresolution Discontinuous Galerkin shallow water model: Implications for computational hydraulics”, Advances in Water Resources 86 (2015), p. 14-31
[4] Y. Chachereau & H. Chanson, “Free-Surface Fluctuations and Turbulence in Hydraulic Jumps”, Experimental Thermal and Fluid Science 35 (2011), p. 896-909 Article
[5] H. Chanson, “Convective transport of air bubbles in strong hydraulic jumps”, Int. J. Multiphase Flow 36 (2010), p. 798-814
[6] C.-K. Cheng, Y.-C. Tai & Y.-C. Jin, “Particle Image Velocity Measurement and Mesh-Free Method Modeling Study of Forced Hydraulic Jumps”, Journal of Hydraulic Engineering 143 (2017) no. 9
[7] O. Delestre, C. Lucas, P.-A. Ksinant, F. Darboux, C. Laguerre, T.-N.-T. Vo, F. James & S. Cordier, “SWASHES: A compilation of shallow water analytic solutions for hydraulic and environmental studies”, International Journal for Numerical Methods in Fluids 72 (2013), p. 269-300
[8] A. I. Delis, M. Kazolea & N. A. Kampanis, “A robust high-resolution finite volume scheme for the simulation of long waves over complex domains”, International Journal for Numerical Methods in Fluids 56 (2008) no. 4, p. 419-452
[9] A. I. Delis & C. P. Skeels, “TVD schemes for open channel flow”, International Journal for Numerical Methods in Fluids 26 (1998) no. 7, p. 791-809
[10] A. I. Delis, C. P. Skeels & S. C. Ryrie, “Implicit high-resolution methods for modelling one-dimensional open channel flow”, Journal of Hydraulic Research 38 (2000) no. 5, p. 369-381
[11] N. Goutal & F. Maurel, Proceedings of the 2nd Workshop on Dam-Break Wave Simulation, Technical report HE-43/97/016/B, Départment Laboratoire National d’Hydraulic, Groupe Hydraulic Fluviale Electricité de Fracne, France, 1997
[12] W. H. Graf & M. S. Altinakar, Fluvial hydraulics: flow and transport processes in channels of simple geometry, Wiley, 1998
[13] W. H. Hager & R. Bremen, “Classical hydraulic jump: sequent depths”, J. Hydraul. Res. 27 (1989), p. 565-585
[14] W. H. Hager, R. Bremen & N. Kawagoshi, “Classical hydraulic jump: lenght of roller”, J. Hydraul. Res. 28 (1990), p. 591-608
[15] F. M. Henderson, Open Channel Flow, MacMillan, 1966
[16] K. A. Ivanova, S. L. Gavrilyuk, B. Nkonga & G. L. Richard, “Formation and coarsening of roll-waves in shear shallow water flows down an inclined rectangular channel”, Computers & Fluids 159 (2017), p. 189-203
[17] G. Kesserwani, R. Ghostine, J. Vazquez, A. Ghenaim & R. Mosé, “Application of a second-order Runge-Kutta discontinuous Galerkin scheme for the shallow water equations with source terms”, International Journal for Numerical Methods in Fluids 56 (2008) no. 7, p. 805-821
[18] S.-H. Lee & N. G. Wright, “Simple and efficient solution of the shallow water equations with source terms”, International Journal for Numerical Methods in Fluids 63 (2010) no. 3, p. 313-340
[19] I. MacDonald, Analysis and computation of steady open channel flows, Ph. D. Thesis, University of Reading, 1996
[20] I. MacDonald, M. J. Baines, N. K. Nichols & P. G. Samuels, Steady open channel test problems with analytic solutions, Technical report 3/95, University of Reading, 1995
[21] I. MacDonald, M. J. Baines, N. K. Nichols & P. G. Samuels, “Analytic benchmark solutions for open-channel flows”, Journal of Hydraulic Engineering 123 (1997) no. 11, p. 1041-1044
[22] K. M. Mok, “Relation of surface roller eddy formation and surface fluctuation in hydraulic jumps”, Journal of Hydraulic Research 42 (2004) no. 2, p. 207-212 Article
[23] M. Morales-Hernandez, P. García-Navarro & J. Murillo, “A large time step 1D upwind explicit scheme (CFL$>$1): Application to shallow water equations”, Journal of Computational Physics 231 (2012) no. 19, p. 6532-6557
[24] G. L. Richard & S. L. Gavrilyuk, “A new model of roll waves: comparison with Brock’s experiments”, Journal of Fluid Mechanics 698 (2012), p. 374-405 Article
[25] G. L. Richard & S. L. Gavrilyuk, “The classical hydraulic jump in a model of shear shallow-water flows”, Journal of Fluid Mechanics 725 (2013), p. 492-521 Article
[26] E. F. Toro, Shock-Capturing Methods for Free-Surface Shallow Flows, John Wiley and Sons, Ltd, 1998
[27] M.-H. Tseng, “Improved treatment of source terms in TVD scheme for shallow water equations”, Advances in Water Resources 27 (2004) no. 6, p. 617-629
[28] M. E. Vázquez-Cendón, “Improved Treatment of Source Terms in Upwind Schemes for the Shallow Water Equations in Channels with Irregular Geometry”, Journal of Computational Physics 148 (1999) no. 2, p. 497-526