Hostname: page-component-8448b6f56d-t5pn6 Total loading time: 0 Render date: 2024-04-16T18:59:23.442Z Has data issue: false hasContentIssue false
  • English
  • Français

Modelling turbulence generation in solitary waves on shear shallow water flows

Published online by Cambridge University Press:  14 May 2015

G. L. Richard  and
S. L. Gavrilyuk
G. L. Richard
Affiliation:
Université de Toulouse, Université Toulouse III Paul Sabatier, UMR CNRS 5219, IMT, 118 route de Narbonne, 31062 Toulouse CEDEX 9, France
S. L. Gavrilyuk*
Affiliation:
Aix-Marseille Université, UMR CNRS 7343, IUSTI, 5 rue E. Fermi, 13453 Marseille CEDEX 13, France Novosibirsk State University, 2 Pirogova 630090 Novosibirsk, Russia
*
†Email address for correspondence: sergey.gavrilyuk@univ-amu.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

We derive a dispersive model of shear shallow water flows which takes into account a non-uniform horizontal velocity. This model generalizes the Green–Naghdi model to the case of shear flows. Besides the classical dispersion term in the Green–Naghdi model related to the acceleration of the free surface, it also contains a new dispersion parameter related to the flow structure. This parameter is related to the second moment of the velocity fluctuation with respect to the vertical coordinate. The distinction between shearing and turbulence based on the scale of variation of the velocity fluctuation is proposed. In particular, an equation for the turbulence generation is derived. Solitary waves for this model are obtained in explicit form. Comparison of solitary wave profiles with experimental ones is also performed. The agreement is very good apart from the small region near the top of the wave.

JFM classification

Geophysical and Geological Flows: Shallow water flows Waves/Free-surface Flows: Shear waves Waves/Free-surface Flows: Surface gravity waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 773 , 25 June 2015 , pp. 49 - 74
Check for updates
Copyright
© 2015 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Antuono, M. & Brocchini, M. 2013 Beyond Boussinesq-type equations: semi-integrated models for coastal dynamics. Phys. Fluids 25, 016603.Google Scholar
Antuono, M., Liapidevskii, V. & Brocchini, M. 2009 Dispersive nonlinear shallow-water equations. Stud. Appl. Maths 122, 1–28.Google Scholar
Barré de Saint-Venant, A. J. C. 1871 Théorie du mouvement non-permanent des eaux, avec application aux crues des rivières et à l’introduction des marées dans leur lit. C. R. Acad. Sci. Paris 73, 147–154.Google Scholar
Barros, R., Gavrilyuk, S. & Teshukov, V. 2007 Dispersive nonlinear waves in two-layer flows with free surface. Part I. Model derivation and general properties. Stud. Appl. Maths 119, 191–211.Google Scholar
Benney, D. J. 1973 Some properties of long nonlinear waves. Stud. Appl. Maths 52 (1), 45–50.Google Scholar
Bonneton, P., Chazel, F., Lannes, D., Marche, F. & Tissier, M. 2011 A splitting approach for the fully nonlinear and weakly dispersive Green–Naghdi model. J. Comput. Phys. 230, 1479–1498.Google Scholar
Boussinesq, J. 1872 Théorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, en communiquant au liquide contenu dans le canal des vitesses sensiblement pareilles de la surface au fond. J. Math. Pures Appl. 17, 55–108.Google Scholar
Bridges, T. J. & Needham, D. J. 2011 Breakdown of the shallow water equations due to growth of the horizontal velocity. J. Fluid Mech. 679, 655–666.Google Scholar
Burns, J. C. 1953 Long waves in running water. Proc. Camb. Phil. Soc. 49, 695–706.Google Scholar
Carter, J. D. & Cienfuegos, R. 2011 The kinematics and stability of solitary and cnoidal wave solutions of the Serre equations. Eur. J. Mech. B 30, 259–268.Google Scholar
Castro, A. & Lannes, D. 2014 Fully nonlinear long-wave models in the presence of vorticity. J. Fluid Mech. 759, 642–675.CrossRefGoogle Scholar
Chachereau, Y. & Chanson, H. 2011 Free-surface fluctuations and turbulence in hydraulic jumps. Exp. Therm. Fluid Sci. 35, 896–909.CrossRefGoogle Scholar
Cienfuegos, R., Barthelemy, E. & Bonneton, P. 2010 A wave-breaking model for Boussinesq-type equations including mass-induced effects. J. Waterway Port Coastal Ocean Engng 136, 10–26.Google Scholar
Dabiri, D. & Gharib, M. 1997 Experimental investigation of the vorticity generation within a spilling water wave. J. Fluid Mech. 330, 113–139.Google Scholar
Daily, J. W. & Stephan, S. C.1952 The solitary wave: its celerity, profile, internal velocities and amplitude attenuation in a horizontal smooth channel. Proceedings of the 3rd Conference on Coastal Engineering, pp. 13–30.Google Scholar
El, G. A., Grimshaw, R. H. J. & Smyth, N. F. 2006 Unsteady undular bores in fully nonlinear shallow-water theory. Phys. Fluids 18, 027104.Google Scholar
Fenton, J. 1972 A ninth order solution for the solitary wave. J. Fluid Mech. 53, 257–271.Google Scholar
Friedrichs, K. O. & Hyers, D. H. 1954 The existence of solitary waves. Commun. Pure Appl. Maths 7, 517–550.Google Scholar
Gavrilyuk, S. & Teshukov, V. 2001 Generalized vorticity for bubbly liquid and dispersive shallow water equations. Contin. Mech. Thermodyn. 13, 365–382.Google Scholar
Green, A. E., Laws, N. & Naghdi, P. M. 1974 On the theory of water waves. Proc. R. Soc. Lond. A 338, 43–55.Google Scholar
Green, A. E. & Naghdi, P. M. 1976 A derivation of equations for wave propagation in water of variable depth. J. Fluid Mech. 78, 237–246.Google Scholar
Grimshaw, R. 1971 The solitary wave in water of variable depth. Part 2. J. Fluid Mech. 46, 611–622.Google Scholar
Hornung, H. G., Willert, C. & Turner, S. 1995 The flow field downstream of a hydraulic jump. J. Fluid Mech. 287, 299–316.Google Scholar
Kim, J. W., Bai, K. J., Ertekin, R. C. & Webster, W. C. 2003 A strongly-nonlinear model for water waves in water of variable depth: the irrotational Green–Naghdi model. J. Offshore Mech. Arctic 125, 25–32.CrossRefGoogle Scholar
Korteweg, D. J. & de Vries, G. 1895 On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Phil. Mag. 39 (5), 422–443.CrossRefGoogle Scholar
Lannes, D. 2013 The Water Waves Problem, Mathematical Surveys and Monographs, vol. 188. American Mathematical Society.Google Scholar
Lavrent’ev, M. A. 1947 On the theory of long waves (in Russian). Akad. Nauk Ukrain. RSR, Zb. Prac’ Inst. Mat. 8, 13–69.Google Scholar
Le Metayer, O., Gavrilyuk, S. & Hank, S. 2010 A numerical scheme for the Green–Naghdi model. J. Comput. Phys. 229, 2034–2045.Google Scholar
Li, M., Guyenne, P., Li, F. & Xu, L. 2014 High order well-balanced CDG-FE methods for shallow water waves by a Green–Naghdi model. J. Comput. Phys. 257, 169–192.Google Scholar
Makarenko, N. 1986 A second long-wave approximation in the Cauchy–Poisson problem (in Russian). Dyn. Contin. Media 77, 56–72.Google Scholar
McCowan, J. 1894 On the highest wave of permanent type. Phil. Mag. 38, 351–358.Google Scholar
Mignot, E. & Cienfuegos, R. 2010 Energy dissipation and turbulent production in weak hydraulic jumps. J. Hydraul. Engng 136 (2), 116–121.Google Scholar
Mignot, E. & Cienfuegos, R. 2011 Spatial evolution of turbulence characteristics in weak hydraulic jumps. J. Hydraul. Res. 49 (2), 222–230.Google Scholar
Misra, S. K., Kirby, J. T., Brocchini, M., Veron, F., Thomas, M. & Kambhamettu, C. 2008 The mean and turbulent flow structure of a weak hydraulic jump. Phys. Fluids 20, 035106.Google Scholar
Rayleigh 1876 On waves. Phil. Mag. 5, 257–279.Google Scholar
Richard, G. L. & Gavrilyuk, S. L. 2012 A new model of roll waves: comparison with Brock’s experiments. J. Fluid Mech. 698, 374–405.CrossRefGoogle 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
Russell, J. S. 1844 Report on waves. Br. Assoc. Adv. Sci. 14, 311–390.Google Scholar
Serre, F. 1953 Contribution à l’étude des écoulements permanents et variables dans les canaux. La Houille Blanche 8, 374–388.CrossRefGoogle Scholar
Shields, J. J. & Webster, W. C. 1988 On direct methods in water-wave theory. J. Fluid Mech. 197, 171–199.Google Scholar
Su, C. H. & Gardner, C. S. 1969 Korteweg–de Vries equation and generalizations. Part III. Derivation of the Korteweg–de Vries equation and Burgers equation. J. Math. Phys. 10, 536–539.Google Scholar
Svendsen, I. A., Veeramony, J., Bakunin, J. & Kirby, J. T. 2000 The flow in weak turbulent hydraulic jumps. J. Fluid Mech. 418, 25–57.Google Scholar
Teshukov, V. M. 2007 Gas-dynamics analogy for vortex free-boundary flows. J. Appl. Mech. Tech. Phys. 48 (3), 303–309.Google Scholar
Tissier, M., Bonneton, P., Marche, F., Chazel, F. & Lannes, D. 2012 A new approach to handle wave breaking in fully non-linear Boussinesq models. Coast. Engng 67, 54–66.Google Scholar
Watanabe, S.2007 History of soliton experiments. In International Meeting on Perspectives of Soliton Physics, 16–17 February 2007.Google Scholar