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Structure of the hydraulic jump in convergent radial flows

Published online by Cambridge University Press:  07 December 2018

K. A. Ivanova Open the ORCID record for K. A. Ivanova [Opens in a new window]  and
S. L. Gavrilyuk Open the ORCID record for S. L. Gavrilyuk [Opens in a new window]
K. A. Ivanova
Affiliation:
Aix Marseille Univ, CNRS, IUSTI, 13453 Marseille cedex 13, France
S. L. Gavrilyuk*
Affiliation:
Aix Marseille Univ, CNRS, IUSTI, 13453 Marseille cedex 13, France
*
†Email address for correspondence: [email protected]
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Abstract

We are interested in the modelling of multi-dimensional turbulent hydraulic jumps in convergent radial flow. To describe the formation of intensive eddies (rollers) at the front of the hydraulic jump, a new model of shear shallow water flows is used. The governing equations form a non-conservative hyperbolic system with dissipative source terms. The structure of equations is reminiscent of generic Reynolds-averaged Euler equations for barotropic compressible turbulent flows. Two types of dissipative term are studied. The first one corresponds to a Chézy-like dissipation rate, and the second one to a standard energy dissipation rate commonly used in compressible turbulence. Both of them guarantee the positive definiteness of the Reynolds stress tensor. The equations are rewritten in polar coordinates and numerically solved by using an original splitting procedure. Numerical results for both types of dissipation are presented and qualitatively compared with the experimental works. The results show both experimentally observed phenomena (cusp formation at the front of the hydraulic jump) as well as new flow patterns (the shape of the hydraulic jump becomes a rotating square).

JFM classification

Nonlinear Dynamical Systems: Pattern formation Geophysical and Geological Flows: Shallow water flows Waves/Free-surface Flows: Shear waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 860 , 10 February 2019 , pp. 441 - 464
Copyright
© 2018 Cambridge University Press 

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References

Andersen, A., Bohr, T. & Schnipper, T. 2010 Separation vortices and pattern formation. Theor. Comput. Fluid Dyn. 24, 329–334.Google Scholar
Bush, J. W. M. & Aristoff, J. M. 2003 The influence of surface tension on the circular hydraulic jump. J. Fluid Mech. 489, 229–238.Google Scholar
Bush, J. W. M., Aristoff, J. M. & Hosoi, A. E. 2006 An experimental investigation of the stability of the circular hydraulic jump. J. Fluid Mech. 558, 33–52.Google Scholar
Ellegaard, C., Hansen, A. E., Haaning, A., Hansen, K., Marcussen, A., Bohr, T., Hansen, J. L. & Watanabe, S. 1998 Creating corners in kitchen sinks. Nature 392, 767–768.Google Scholar
Ellegaard, C., Hansen, A. E., Haaning, A., Hansen, K., Marcussen, A., Bohr, T., Hansen, J. L. & Watanabe, S. 1999 Cover illustration: polygonal hydraulic jumps. Nonlinearity 12, 1–7.Google Scholar
Eyo, A. E., Joshua, E. E. & Udoh, P. J. 2011 Two dimensional laminar flow of a liquid with circular hydraulic jump. Mod. Appl. Sci. 5, 56–68.Google Scholar
Foglizzo, T., Kazeroni, R., Guilet, J. & Masset, F. 2015 The explosion mechanism of core-collapse supernovae: progress in supernova theory and experiments. Publ. Astron. Soc. Aust. 32, e009.Google Scholar
Foglizzo, T., Masset, F., Guilet, J. & Durand, G. 2012 Shallow water analogue of the standing accretion shock instability: experimental demonstration and a two-dimensional model. Phys. Rev. Lett. 108, 051103.Google Scholar
Gavrilyuk, S. L. & Gouin, H. 2012 Geometric evolution of the Reynolds stress tensor. Intl J. Engng Sci. 59, 65–73.Google Scholar
Gavrilyuk, S. L., Ivanova, K. A. & Favrie, N. 2018a Multidimensional shear shallow water flows: problems and solutions. J. Comput. Phys. 366, 252–280.Google Scholar
Gavrilyuk, S. L., Liapidevskii, V. Yu. & Chesnokov, A. A. 2016 Spilling breakers in shallow water: applications to Favre waves and to the shoaling and the breaking of the solitary wave. J. Fluid Mech. 808, 441–468.Google Scholar
Gavrilyuk, S. L., Liapidevskii, V. Yu. & Chesnokov, A. A. 2018b Interaction of a subsurface bubble layer with long internal waves. Eur. J. Mech. (B/Fluids), in press; doi:10.1016/j.euromechflu.2017.07.004.Google Scholar
Ivanova, K. A., Gavrilyuk, S. L., Nkonga, B. & Richard, G. L. 2017 Formation and coarsening of roll-waves in shear shallow water flows down an inclined rectangular channel. Comput. Fluids 159, 189–203.Google Scholar
Kasimov, A. R. 2008 A stationary circular hydraulic jump, the limits of its existence and its gasdynamic analogue. J. Fluid Mech. 601, 189–198.Google Scholar
Labousse, M. & Bush, J. W. M. 2013 The hydraulic bump: the surface signature of a plunging jet. Phys. Fluids 25, 229–238.Google Scholar
Liu, X. & Lienhard, J. H. 1993 The hydraulic jump in circular jet impingement and in other thin liquid films. Exp. Fluids 15, 108–116.Google Scholar
Martens, E. A., Watanabe, S. & Bohr, T. 2012 Model for polygonal hydraulic jumps. Phys. Rev. E 85, 036316.Google Scholar
Ray, A. K. & Bhattacharjee, J. K. 2007 Standing and travelling waves in the shallow-water circular hydraulic jump. Phys. Lett. A 371, 241–248.Google Scholar
Richard, G. L.2013 Elaboration d’un modèle d’écoulements turbulents en faible profondeur: application au ressaut hydraulique et aux trains de rouleaux. PhD thesis, Aix-Marseille.Google Scholar
Richard, G. L. & Gavrilyuk, S. L. 2012 A new model of roll waves: comparison with brocks experiments. J. Fluid Mech. 698, 374–405.Google Scholar
Richard, G. L. & Gavrilyuk, S. L. 2013 The classical hydraulic jump in a model of shear shallow-water flows. J. Fluid Mech. 725, 492–521.Google Scholar
Rojas, N., Argentina, M. & Tirapegui, E. 2013 A progressive correction to the circular hydraulic jump scaling. Phys. Fluids 25, 042105.Google Scholar
Shiue, M. C., Laminie, J., Temam, R. & Tribba, J. 2011 Boundary value problems for the shallow water equations with topography. J. Geophys. Res. 116, C02015.Google Scholar
Teshukov, V. M. 2007 Gas dynamic analogy for vortex free–boundary flows. J. Appl. Mech. Tech. Phys. 48, 303–309.Google Scholar
Teymourtash, A. R., Khavari, M. & Passandideh-Fard, M. 2010 Experimental and numerical investigation of circular hydraulic jump. In Proceedings of the 18th Annual International Conference on Mechanical Engineering (ed. Nouri-Borujerdi, A. & Movahhedi, M. R.), ISME2010, vol. 1, p. 35, Paper No. 3537.Google Scholar
Townsend, A. A. 1956 The Structure of Turbulent Shear Flow. Cambridge University Press.Google Scholar
Winters, K. B. 2016 The turbulent transition of a supercritical downslope flow: sensitivity to downstream conditions. J. Fluid Mech. 792, 997–1012.Google Scholar