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A new model of roll waves: comparison with Brock’s experiments

Published online by Cambridge University Press:  11 April 2012

G. L. Richard
Affiliation:
Aix-Marseille University, UMR CNRS 6595, IUSTI, 5 rue E. Fermi, 13453 Marseille CEDEX 13, France
S. L. Gavrilyuk*
Affiliation:
Aix-Marseille University, UMR CNRS 6595, IUSTI, 5 rue E. Fermi, 13453 Marseille CEDEX 13, France
*
Email address for correspondence: sergey.gavrilyuk@polytech.univ-mrs.fr

Abstract

We derive a mathematical model of shear flows of shallow water down an inclined plane. The non-dissipative part of the model is obtained by averaging the incompressible Euler equations over the fluid depth. The averaged equations are simplified in the case of weakly sheared flows. They are reminiscent of the compressible non-isentropic Euler equations where the flow enstrophy plays the role of entropy. Two types of enstrophies are distinguished: a small-scale enstrophy generated near the wall, and a large-scale enstrophy corresponding to the flow in the roller region near the free surface. The dissipation is then added in accordance with basic physical principles. The model is hyperbolic, the corresponding ‘sound velocity’ depends on the flow enstrophies. Periodic stationary solutions to this model describing roll waves were obtained. The solutions are in good agreement with the experimental profiles of roll waves measured in Brock’s experiments. In particular, the height of the vertical front of the waves, the shock thickness and the wave amplitude are well captured by the model.

Type
Papers
Copyright
Copyright © Cambridge University Press 2012

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References

1. Balmforth, N. J. & Mandre, S. 2004 Dynamics of roll waves. J. Fluid Mech. 514, 133.Google Scholar
2. Barker, B., Johnson, M. A., Noble, P., Rodrigues, L. M. & Zumbrun, K. 2010 Whitham averaged equations and modulational instability of periodic traveling waves of a hyperbolic–parabolic balance laws. arXiv:1008.4729v2 [math.AP] 3 Sept 2010.Google Scholar
3. Boudlal, A. & Liapidevskii, V. Yu. 2005 Stability of regular roll waves. Comput. Technol. 10 (2), 314.Google Scholar
4. Brock, R. R. 1967 Development of roll waves in open channels. PhD thesis, California Institute of Technology, Pasadena, California.Google Scholar
5. Brock, R. R. 1969 Development of roll – wave trains in open channels. J. Hydraul. Div. ASCE 95, 14011427.Google Scholar
6. Brock, R. R. 1970 Periodic permanent roll waves. J. Hydraul. Div. ASCE 96, 25652580.Google Scholar
7. Chang, H.-C., Demekhin, E. A. & Kalaidin, E. 2000 Coherent structures, self-similarity, and universal roll wave coarsening dynamics. Phys. Fluids 12, 22682278.Google Scholar
8. Charru, F. 2011 Hydrodynamic Instabilities. Cambridge University Press.CrossRefGoogle Scholar
9. Cornish, V. 1934 Ocean Waves and Kindred Geophysical Phenomena. Cambridge University Press.Google Scholar
10. Dressler, R. F. 1949 Mathematical solution of the problem of roll-waves in inclined open channels. Commun. Pure Appl. Maths 2, 149194.CrossRefGoogle Scholar
11. Gavrilyuk, S. L. & Saurel, R. 2006 Estimation of the turbulent energy production across a shock wave. J. Fluid Mech. 549, 131139.CrossRefGoogle Scholar
12. Gavrilyuk, S. L. & Saurel, R. 2007 Rankine–Hugoniot relations for shocks in heterogeneous mixtures. J. Fluid Mech. 575, 495507.Google Scholar
13. Gerbeau, J.-F. & Perthame, B. 2001 Derivation of viscous Saint-Venant system for laminar shallow water: numerical validation. Discrete Contin. Dyn. Syst. B 89102.Google Scholar
14. Hager, W. H., Bremen, K. & Kawagoshi, N. 1990 Classical hydraulic jump: length of roller. J. Hydraul. Res., IAHR 28 (5), 591608.CrossRefGoogle Scholar
15. Jeffreys, H. J. 1925 The flow of water in an inclined channel of rectangular section. Phil. Mag. 49, 793807.CrossRefGoogle Scholar
16. Kapitza, P. L. & Kapitza, S. P. 1949 Wave flow of thin layer of a viscous fluid. III. Experimental study of undulatory flow conditions. Zh. Eksp. Teor. Fiz 19, 105. Also in Collected Papers of P. L. Kapitza (ed. D. Ter Haar), vol. 2, pp. 690–709. Pergamon, 1965.Google Scholar
17. Kranenburg, C. 1992 On the evolution of roll waves. J. Fluid Mech. 245, 249261.CrossRefGoogle Scholar
18. Lawrence, D. S. L. 1997 Macroscale surface roughness and frictional resistance in overland flow. Earth Surf. Process. Landf. 22, 365382.Google Scholar
19. Liu, J., Jonathan, D. P. & Gollub, J. P. 1993 Measurements of the primary instabilities of film flows. J. Fluid Mech. 250, 69101.CrossRefGoogle Scholar
20. 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.CrossRefGoogle Scholar
21. Needham, D. J. & Merkin, J. H. 1984 On roll waves down an open inclined channel. Proc. R. Soc. Lond. A 394, 259278.Google Scholar
22. Noble, P. 2006 On spectral stability of roll waves. Indiana Univ. Math. J. 55, 795848.CrossRefGoogle Scholar
23. Noble, P. 2007 Linear stability of viscous roll waves. Commun. Part. Diff. Equ. 32 (10–12), 16911713.Google Scholar
24. Prokopiou, Th., Cheng, M. & Chang, H.-C. 1991 Long waves on inclined films at high Reynolds number. J. Fluid Mech. 222, 665691.Google Scholar
25. Ruyer-Quil, C. & Manneville, P. 1998 Modeling film flows down inclined planes. Eur. Phys. J. B 6, 277292.CrossRefGoogle Scholar
26. Teshukov, V. M. 2007 Gas-dynamics analogy for vortex free-boundary flows. J. Appl. Mech. Tech. Phys. 48 (3), 303309.Google Scholar
27. Thual, O. 2010 Hydrodynamique de l’environnement. Editions de l’Ecole Polytechnique.Google Scholar
28. Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley.Google Scholar
29. Yu, J. & Kevorkian, J. 1992 Nonlinear evolution of small disturbances into roll waves in an inclined open channel. J. Fluid Mech. 243, 575594.CrossRefGoogle Scholar
30. Zel’dovich, Ya. B. & Raizer, Yu. P. 2001 Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena. Dover.Google Scholar