Modelling roll waves with shallow water equations and turbulent closure

REFERENCES

• Balmforth, N. J., & Mander, S. (2004). Dynamics of roll waves. Journal of Fluid Mechanics, 514, 1–33. doi: 10.1017/S0022112004009930
   Web of Science ®Google Scholar
• Benjamin, T. B. (1957). Wave formation in laminar flow down an inclined plane. Journal of Fluid Mechanics, 2, 554–574. doi: 10.1017/S0022112057000373
   Web of Science ®Google Scholar
• Bouchut, F., Mangeney-Castelnau, A., Perthame, B., & Vilotte, J. P. (2003). A new model of Saint Venant and Savage–Hutter type for gravity driven shallow water flows. Comptes Rendus de l'Académie des Sciences Paris, Series I, 336(6), 531–536.
   Google Scholar
• Brock, R.R. (1967). Development of roll waves in open channels (PhD thesis), W. M. Keck Laboratory of Hydraulics and Water Resources, California Institute of Technology, USA.
   Google Scholar
• Brock, R.R. (1970). Periodic permanent roll-waves. Journal of Hydraulic Division, 96(12), 2565–2580.
   Google Scholar
• Cao, Z.X., Pender, G., Wallis, S., & Carling, P. (2004). Computational dam-break hydraulics over erodible sediment bed. Journal of Hydraulic Engineering, 130(7), 689–703. doi: 10.1061/(ASCE)0733-9429(2004)130:7(689)
   Web of Science ®Google Scholar
• Cornish, V. (1934). Ocean waves and kindred geophysical phenomena. Cambridge: Cambridge University Press.
   Google Scholar
• Dressler, R.F. (1949). Mathematical solution of the problem of roll-waves in inclined open channels. Communications on Pure and Applied Mathematics, 2, 149–194. doi: 10.1002/cpa.3160020203
   Web of Science ®Google Scholar
• Dressler, R.F., & Pohle, F.V. (1953). Resistance effects on hydraulic instability. Communications on Pure and Applied Mathematics, 6(1), 93–96. doi: 10.1002/cpa.3160060103
   Web of Science ®Google Scholar
• Duan, J.G., & Nanda, S.K. (2006). Two-dimensional depth-averaged model simulation of suspended sediment concentration distribution in a groyne field. Journal of Hydrology, 327(3), 426–437. doi: 10.1016/j.jhydrol.2005.11.055
   Web of Science ®Google Scholar
• Gharangik, A.M., & Chaudhry, M.H. (1991). Numerical simulation of hydraulic jump. Journal of Hydraulic Engineering, 117(9), 1195–1211. doi: 10.1061/(ASCE)0733-9429(1991)117:9(1195)
   Web of Science ®Google Scholar
• Iwasa, Y. (1954). The criterion for instability of steady uniform flows in open channels. Memoirs of the Faculty of Engineering, Kyoto University, Japan, 16(6), 264–275.
   Google Scholar
• Iverson, R.M., Logan, M., LaHusen, R.G., & Berti, M. (2010). The perfect debris flow? Aggregated results from 28 large-scale experiments. Journal of Geophysical Research, 115, F03005. doi: 10.1029/2009JF001514
   PubMed Web of Science ®Google Scholar
• Jeffreys, H.J. (1925). The flow of water in an inclined channel of rectangular section. Philosophical Magazine, 6(49), 793–807. doi: 10.1080/14786442508634662
   Google Scholar
• Jin, Y.C., & Steffler, P.M. (1993). Predicting flow in curved open channels by depth-averaged method. Journal of Hydraulic Engineering, 119(1), 109–124. doi: 10.1061/(ASCE)0733-9429(1993)119:1(109)
   Web of Science ®Google Scholar
• Kranenburg, C. (1992). On the evolution of roll waves. Journal of Fluid Mechanics, 245(1), 249–261. doi: 10.1017/S0022112092000442
   Web of Science ®Google Scholar
• Launder, B.E., & Spalding, D.B. (1974). The numerical computation of turbulent flows. Computer Methods in Applied Mechanics Engineering, 3, 269–289. doi: 10.1016/0045-7825(74)90029-2
   Google Scholar
• Li, J., Cao, Z.X., Pender, G., & Liu, Q.Q. (2013). A double layer-averaged model for dam-break flows over erodible bed. Journal of Hydraulic Research, 51(5), 518–534. doi: 10.1080/00221686.2013.812047
   Web of Science ®Google Scholar
• Liu, Q.Q., Chen, L., Li, J.C., & Singh, V.P. (2005). Roll waves in overland flow. Journal of Hydrologic Engineering, 10(2), 110–117. doi: 10.1061/(ASCE)1084-0699(2005)10:2(110)
   Web of Science ®Google Scholar
• Liu, K.F., & Mei, C.C. (1994). Roll waves on a layer of a muddy fluid flowing down a gentle slope-A Bingham model. Physics of Fluids, 6, 2577.
   Web of Science ®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. Physics of Fluids, 20, 035106. doi: 10.1063/1.2856269
   Web of Science ®Google Scholar
• Montuori, C. (2005). Roll waves simulation using shallow water equations and weighted average flux method. Journal of Hydraulic Research, 43(1), 103–106. doi: 10.1080/00221680509500115
   Web of Science ®Google Scholar
• Needham, D.J., & Merkin, J.H. (1984). On roll waves down an open inclined channel. Proceedings of the Royal Society of London A, 394, 259–278. doi: 10.1098/rspa.1984.0079
   Google Scholar
• Ni, H.Q. (2010). Turbulence simulation and application in modern hydraulics engineering [in Chinese]. Beijing: China Water Power Press.
   Google Scholar
• Rastogi, A.K., & Rodi, W. (1978). Predictions of heat and mass transfer in open channels. Journal of Hydraulic Division, 104(3), 397–420.
   Web of Science ®Google Scholar
• Richard, G.L., & Gavrilyuk, S.L. (2012). A new model of roll waves: comparison with Brock's experiments. Journal of Fluid Mechanics, 698, 374–405. doi: 10.1017/jfm.2012.96
   Web of Science ®Google Scholar
• Rodi, W. (1993). Turbulence models and their applications in Hydraulics. IAHR Monograph, Rotterdam.
   Google Scholar
• Savage, S.B., & Hutter, K. (1991). The dynamics of avalanches of granular materials from initiation to run-out. Part I: Analysis. Acta Mechanica, 86(1–4), 201–223. doi: 10.1007/BF01175958
   Web of Science ®Google Scholar
• Shi, X.G. (1994). Turbulent flow [in Chinese]. Tianjin: Tianjin University Press.
   Google Scholar
• Stoker, J.J. (1958). Water waves, the mathematical theory with applications. New York, NY: John Wiley & Sons.
   Google Scholar
• Tamburrino, A., & Ihle, C.F. (2013). Roll wave appearance in bentonite suspensions flowing down inclined planes. Journal of Hydraulic Research, 51(3), 330–335. doi: 10.1080/00221686.2013.769468
   Web of Science ®Google Scholar
• Toro, E.F. (2001). Shock-capturing methods for free-surface shallow flows. Chichester: John Wiley & Sons.
   Google Scholar
• Wu, W. (2007). Computational river dynamics. London: Taylor & Francis.
   Google Scholar
• Yih, C. (1963). Stability of liquid flow down an inclined plane. Physics of Fluids, 6(3), 321–334. doi: 10.1063/1.1706737
   Web of Science ®Google Scholar
• Yu, J., & Kevorkian, J. (1992). Nonlinear evolution of small disturbances into roll waves in an inclined open channel. Journal of Fluid Mechanics, 243, 575–594. doi: 10.1017/S0022112092002830
   Web of Science ®Google Scholar
• Zanuttigh, B., & Lamberti, A. (2002). Roll waves simulation using shallow water equations and weighted average flux method. Journal of Hydraulic Research, 40(5), 610–622. doi: 10.1080/00221680209499905
   Web of Science ®Google Scholar
• Zanuttigh, B., & Lamberti, A. (2007). Instability and surge development in debris flows. Reviews of Geophysics, 45(3), RG3006. doi: 10.1029/2005RG000175
   Web of Science ®Google Scholar