Hostname: page-component-8448b6f56d-tj2md Total loading time: 0 Render date: 2024-04-23T23:09:45.870Z Has data issue: false hasContentIssue false
  • English
  • Français

The granular monoclinal wave

Published online by Cambridge University Press:  28 March 2018

Dimitrios Razis ,
Giorgos Kanellopoulos  and
Ko van der Weele Open the ORCID record for Ko van der Weele [Opens in a new window]
Dimitrios Razis
Affiliation:
Department of Mathematics and Center for Research and Applications of Nonlinear Systems, University of Patras, 26500 Patras, Greece
Giorgos Kanellopoulos
Affiliation:
Department of Mathematics and Center for Research and Applications of Nonlinear Systems, University of Patras, 26500 Patras, Greece Department of Mathematics, University of Ioannina, 45110 Ioannina, Greece
Ko van der Weele*
Affiliation:
Department of Mathematics and Center for Research and Applications of Nonlinear Systems, University of Patras, 26500 Patras, Greece
*
†Email address for correspondence: weele@math.upatras.gr
Get access Rights & Permissions [Opens in a new window]

Abstract

We study granular chute flow using the classic Saint-Venant approach, with the shear stresses within the granular sheet being incorporated via a friction law due to Pouliquen & Forterre (J. Fluid Mech., vol. 453, 2002, pp. 113–151) and with the in-plane stresses (which are ignored in the traditional formulation for normal fluids) being represented by a viscous-like term recently derived by Gray & Edwards (J. Fluid Mech., vol. 755, 2014, pp. 503–534). On the basis of this model, we predict that the granular sheet is able to sustain monoclinal waves, i.e. travelling shock structures that monotonically connect a thick region of uniform flow to a thinner one. We examine the balance of forces that determine the shape of this particular waveform and give the precise window of system parameters for which monoclinal waves are expected to appear in experiments.

JFM classification

Granular media: Granular media Geophysical and Geological Flows: Shallow water flows Waves/Free-surface Flows: Waves/Free-surface Flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 843 , 25 May 2018 , pp. 810 - 846
Copyright
© 2018 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

Andreotti, B., Forterre, Y. & Pouliquen, O. 2013 Granular Media: Between Fluid and Solid. Cambridge University Press.Google Scholar
Aranson, I. S. & Tsimring, L. S. 2006 Patterns and collective behavior in granular media: Theoretical concepts. Rev. Mod. Phys. 78, 641–692.Google Scholar
Barker, T., Schaeffer, D. G., Bohorquez, P. & Gray, J. M. N. T. 2015 Well-posed and ill-posed behaviour of the 𝜇(I)-rheology for granular flow. J. Fluid Mech. 779, 794–818.Google Scholar
Benjamin, T. B., Bona, J. L. & Mahony, J. J. 1972 Model equations for long waves in nonlinear dispersive systems. Philos. Trans. R. Soc. Lond. A 272, 47–78.Google Scholar
Bonneton, P., Filippini, A. G., Arpaia, L., Bonneton, N. & Ricchiuto, M. 2016 Conditions for tidal bore formation in convergent alluvial estuaries. Estuar. Coast. Shelf Sci. 172, 121–127.Google Scholar
Börzsönyi, T., Hasley, T. C. & Ecke, R. E. 2005 Two scenarios for avalanche dynamics in inclined granular layers. Phys. Rev. Lett. 94, 208001.Google Scholar
Boudet, J. F., Amarouchene, B., Bonnier, B. & Kellay, H. 2007 The granular jump. J. Fluid Mech. 572, 413–431.Google Scholar
Bresse, J. A. C. 1860 Cours de Mécanique Appliqué, Vol. II: Hydraulique. Mallet-Bachelier, Edit.Google Scholar
Daerr, A. & Douady, S. 1999 Two types of avalanche behaviour in granular media. Nature 399, 241–243.Google Scholar
Daerr, A. 2001 Dynamical equilibrium of avalanches on rough planes. Phys. Fluids 13, 2115–2124.Google Scholar
Drazin, P. G. & Johnson, R. S. 1988 Solitons: An Introduction. Cambridge University Press.Google Scholar
Dressler, R. F. 1949 Mathematical solutions of the problem of roll waves in inclined open channels. Commun. Pure. Appl. Maths 2, 149–194.Google Scholar
Edwards, A. N. & Gray, J. M. N. T. 2015 Erosion-deposition waves in shallow granular free-surface flows. J. Fluid Mech. 762, 35–67.Google Scholar
Edwards, A. N., Viroulet, S., Kokelaar, B. P. & Gray, J. M. N. T. 2017 Formation of levees, troughs and elevated channels by avalanches on erodible slopes. J. Fluid Mech. 823, 278–315.Google Scholar
Eggers, J. 1999 Sand as Maxwell’s demon. Phys. Rev. Lett. 83, 5322–5325.Google Scholar
Eshuis, P., van der Weele, K., van der Meer, D. & Lohse, D. 2005 Granular Leidenfrost effect: experiment and theory of floating particle clusters. Phys. Rev. Lett. 95, 258001.Google Scholar
Eshuis, P., van der Meer, D., Alam, H., van Gerner, H. J., van der Weele, K. & Lohse, D. 2010 Onset of convection on strongly shaken granular matter. Phys. Rev. Lett. 104, 038001.Google Scholar
Ferrick, M. G. 2005 Simple wave and monoclinal wave models: River flow surge applications and implications. Water Resour. Res. 41, W11402.Google Scholar
Forterre, Y. & Pouliquen, O. 2003 Long-surface-wave instability in dense granular flows. J. Fluid Mech. 486, 21–50.Google Scholar
Forterre, Y. 2006 Kapiza waves as a test for three-dimensional granular flow rheology. J. Fluid Mech. 563, 123–132.Google Scholar
Fowler, A. 2011 Mathematical Geoscience. Springer.Google Scholar
GDR-MiDi 2004 On dense granular flows. Eur. Phys. J. E 14, 341–365.Google Scholar
Gray, J. M. N. T., Tai, Y. C. & Noelle, S. 2003 Shock waves, dead-zones and particle-free regions in rapid granular free-surface flows. J. Fluid Mech. 491, 161–181.Google Scholar
Gray, J. M. N. T. & Cui, X. 2007 Weak, strong and detached oblique shocks in gravity driven granular free-surface flows. J. Fluid Mech. 579, 113–136.Google Scholar
Gray, J. M. N. T. & Ancey, C. 2009 Segregation, recirculation and deposition of coarse particles near two-dimensional avalanche fronts. J. Fluid Mech. 629, 387–423.Google Scholar
Gray, J. M. N. T. & Kokelaar, B. P. 2010 Large particle segregation, transport and accumulation in granular free-surface flows. J. Fluid Mech. 652, 105–137.Google Scholar
Gray, J. M. N. T. & Edwards, A. N. 2014 A depth-averaged 𝜇(I)-rheology for shallow granular free-surface flows. J. Fluid Mech. 755, 503–534.Google Scholar
Jan, C.-D. 2014 Gradually-varied Flow Profiles in Open Channels, Advances in Geophysical and Environmental Mechanics and Mathematics. Springer.Google Scholar
Jaeger, H., Nagel, S. & Behringer, R. 1996 Granular solids, liquids, and gases. Rev. Mod. Phys. 68, 1259–1275.Google Scholar
Jakob, M. & Hungr, O. 2005 Debris-flow Hazards and Related Phenomena. Praxis Publ.-Springer.Google Scholar
Johnson, C. G. & Gray, J. M. N. T. 2011 Granular jets and hydraulic jumps on an inclined plane. J. Fluid Mech. 675, 87–116.Google Scholar
Hamdi, S., Schiesser, W. E. & Griffiths, G. W. 2007 Method of Lines. Scholarpedia 2 (7), 2859.Google Scholar
Kadanoff, L. P. 1999 Built upon sand: theoretical ideas inspired by granular flows. Rev. Mod. Phys. 71, 435–444.Google Scholar
Kapitza, P. L. & Kapitza, S. P. 1949 Wave flow of thin layers of a viscous fluid, III: experimental study of undulatory flow conditions. Zh. Eksperim. i Teor. Fiz. 19, 105–120; English translation in: Collected Papers of P. L. Kapitza, Volume II 1938–1964, edited by D. ter Haar (Pergamon Press, Oxford, 1965), pp. 690–709.Google Scholar
Lagrée, P.-Y., Saingier, G., Deboeuf, S., Staron, L. & Popinet, S. 2017 Granular front for flow down a rough incline: about the value of the shape factor in depths averaged models. In Proc. ‘Powders & Grains 2017’, EPJ Web of Conferences (ed. Radjai, F., Nezamabadi, S., Luding, S. & Delenne, J. Y.), vol. 140. EDP Sciences, Les Ulis, France.Google Scholar
Mejean, S., Faug, T. & Einav, I. 2017 A general relation for standing normal jumps in both hydraulic and dry granular flows. J. Fluid Mech. 816, 331–351.Google Scholar
Mejean, S., Faug, T. & Einav, I. 2017 Discrete Element Method simulations of standing jumps in granular flows down inclines. In Proc. ‘Powders & Grains 2017’, EPJ Web of Conferences (ed. Radjai, F., Nezamabadi, S., Luding, S. & Delenne, J. Y.), vol. 140. EDP Sciences, Les Ulis, France.Google Scholar
Moots, E. E. & Mavis, F. T. 1938 A Study in Flood Waves, vol. 14. University of Iowa Studies in Engineering.Google Scholar
Needham, D. J. & Merkin, J. H. 1984 On roll waves down an open inclined channel. Proc. R. Soc. Lond. 394, 259–278.Google Scholar
Peregrine, D. H. 1966 Calculations of the development of an undular bore. J. Fluid Mech. 25, 321–330.Google Scholar
Pouliquen, O. 1999 Scaling laws in granular flows down rough inclined planes. Phys. Fluids 11, 542–548.Google Scholar
Pouliquen, O. & Forterre, Y. 2002 Friction law for dense granular flows: application to the motion of a mass down a rough inclined plane. J. Fluid Mech. 453, 113–151.Google Scholar
Pudasaini, S. P. & Hutter, K. 2007 Avalanche Dynamics. Springer.Google Scholar
Pudasaini, S. P. 2011 Some exact solutions for debris and avalanche flows. Phys. Fluids 23, 043301.Google Scholar
Rayleigh, Lord 1914 On the theory on long waves and bores. Proc. R. Soc. Lond. A 90, 324–328.Google Scholar
Razis, D., Edwards, A. N., Gray, J. M. N. T. & van der Weele, K. 2014 Arrested coarsening of granular roll waves. Phys. Fluids 26, 123305.Google Scholar
Rouse, H. 1946 Elementary Mechanics of Fluids. Dover.Google Scholar
Rouse, H. & Ince, S. 1963 History of Hydraulics. Dover; unabridged and corrected republication of the work first published by the Iowa Institute of Hydraulic Research, State University of Iowa, 1957.Google Scholar
Rousseaux, G., Mougenot, J. M., Chatellier, L., David, L. & Calluaud, M. 2016 A novel method to generate tidal-like bores in the laboratory. Eur. J. Mech. (B/Fluids) 55, 31–38.Google Scholar
Saingier, G., Deboeuf, S. & Lagrée, P.-Y. 2016 On the front shape of an inertial granular flow down a rough incline. Phys. Fluids 28, 053302.Google Scholar
Savage, S. B. & Hutter, K. 1989 The motion of a finite mass of granular material down a rough incline. J. Fluid Mech. 199, 177–215.Google Scholar
Schiesser, W. E. 1991 The Numerical Method of Lines: Integration of Partial Differential Equations. Academic Press.Google Scholar
Shome, M. L. & Steffler, P. M. 2006 Flood plain filling by a monoclinal flood wave. J. Hydraul. Engng. 132, 529–532.Google Scholar
Stoker, J. J. 1958 Water Waves: The Mathematical Theory with Applications. Wiley.Google Scholar
Takahashi, T. 2014 Debris Flow: Mechanics, Prediction and Countermeasures, 2nd ed. CRC Press, Taylor and Francis.Google Scholar
Viroulet, S., Baker, J. L., Edwards, A. N., Johnson, C. G., Gjaltema, C., Clavel, P. & Gray, J. M. N. T. 2017 Multiple solutions for granular flow over a smooth two-dimensional bump. J. Fluid Mech. 815, 77–116.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley.Google Scholar