Hostname: page-component-77c89778f8-m42fx Total loading time: 0 Render date: 2024-07-21T10:31:48.787Z Has data issue: false hasContentIssue false
  • English
  • Français

Roll waves on a shallow layer of mud modelled as a power-law fluid

Published online by Cambridge University Press:  26 April 2006

Chiu-On Ng  and
Chiang C. Mei
Chiu-On Ng
Affiliation:
Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Chiang C. Mei
Affiliation:
Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

We give a theory of permanent roll waves on a shallow layer of fluid mud which is modelled as a power-law fluid. Based on the long-wave approximation, Kármán's momentum integral method is applied to derive the averaged continuity and the momentum equations. Linearized instability analysis of a uniform flow shows that the growth rate of unstable disturbances increases monotonically with the wavenumber, and therefore is insufficient to suggest a preferred wavelength for the roll wave. Nonlinear roll waves are obtained next as periodic shocks connected by smooth profiles with depth increasing monotonically from the rear to the front. Among all wavelengths only those longer than a certain threshold correspond to positive energy loss across the shock, and are physically acceptable. This threshold also implies a minimum discharge, viewed in the moving system, for the roll wave to exist. These facts suggest that a roll wave developed spontaneously from infinitesimal disturbances should have the shortest wavelength corresponding to zero dissipation across the shock, though finite dissipation elsewhere. The discontinuity at the wave front is a mathematical shortcoming needing a local requirement. Predictions for the spontaneously developed roll waves in a Newtonian case are compared with available experimental data. Longer roll waves, with dissipation at the discontinuous fronts, cannot be maintained if the uniform flow is linearly stable, when the fluid is slightly non-Newtonian. However, when the fluid is highly non-Newtonian, very long roll waves may still exist even if the corresponding uniform flow is stable to infinitesimal disturbances. Numerical results are presented for the phase speed, wave height and wavenumber, and wave profiles for a representative value of the flow index of fluid mud.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 263 , 25 March 1994 , pp. 151 - 184
Copyright
© 1994 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

Alekseenko, S. V., Nakoryakov, V. Ye. & Pokusaev, B. G. 1985 Wave formation on a vertical falling liquid film. AIChE J. 31, 1446–1460.Google Scholar
Benjamin, T. B. 1957 Wave formation in laminar flow domain an inclined plane. J. Fluid Mech. 2, 554–574.Google Scholar
Bryant, A. E. J. & Williams, D. J. A. 1980 Rheology of cohesive suspensions. In Industrialized Embayments and Their Environmental Problems (ed. M. B. Collins) Pergamon.
Chang, H.-C., Demekhin, E. A. & Kopelevich, D. I. 1993 Nonlinear evolution of waves on a vertically falling film. J. Fluid Mech. 250, 433–480.Google Scholar
Cornish, V. 1934 Ocean Waves and Kindred Geophysical Phenomena. Cambridge University Press.
Dai, J., Wan, Z., Wang, W., Cheng, W. & Li, X. 1980 An experimental study of slurry transport in pipes. In Proc. Intl Symp. River Sedimentation, Beijing, China, Vol. 1, pp. 195–204 (in chinese).
Darby, R. 1986 Laminar and turbulent pipe flows of non-Newtonian fluids. In Encyclopedia of Fluid Mechanics, Vol. 7 (ed. N. P. Cheremisinoff), pp. 19–53. Gulf.
Davies, T. R. H. 1986 Large debris flows: a macro-viscous phenomenon. Acta Mechanica 63, 161–178.Google Scholar
Dressler, R. F. 1949 Mathematical solution of the problem of roll-waves in inclined open channels. Commun. Pure Appl. Maths 2, 149–194.Google Scholar
Dressler, R. F. & Pohle, F. V. 1953 Resistance effects on hydraulic instability. Commun. Pure Appl. Maths 6, 93–96.Google Scholar
Hwang, S.-H. & Chang, H.-C. 1987 Turbulent and inertial roll waves in inclined film flow. Phys. Fluids 30, 1259–1268.Google Scholar
Ishihara, T., Iwagaki, Y. & Iwasa, Y. 1954 Theory of the roll wave train in laminar water flow on a steep slope surface. Trans. JSCE 19, 46–57 (in Japanese).Google Scholar
Jeffreys, H. 1925 The flow of water in an inclined channel of rectangular section. Phil. Mag. 49 (6), 793–807.Google Scholar
Johnson, A. M. 1970 Physical Processes in Geology. Freeman, Cooper and Co.
Julien, P. Y. & Hartley, D. M. 1985 Formation of roll waves in laminar sheet flow. Rep. CER84-85PYJ-DMH18, Department of Civil Engineering, Colorado State University.Google Scholar
Julien, P. Y. & Hartley, D. M. 1986 Formation of roll waves in laminar sheet flow. J. Hydraul. Res. 24, 5–17.Google Scholar
Kajiuchi, T. & Saito, A. 1984 Flow enhancement of laminar pulsating flow of Bingham plastic fluids. J. Chem. Engng Japan 17, 34–38.Google Scholar
Kapitza, P. L. 1948 Wave flow of thin layers of a viscous fluid. Zh. Eksp. Teor. Fiz. 18, 3–28. (See also Collected Papers of P. L. Kapitza, Vol. II, 1938–1964, Pergamon Press (1965). pp. 662–709.)Google Scholar
Li, J., Yuan, J., Bi, C. & Luo, D. 1983 The main features of the mudflow in Jiang-jia Ravine. Zeit. Geomorph. 27, 325–341.Google Scholar
Liu, J., Paul, J. D. & Gollub, J. P. 1993 Measurements of the primary instabilities of film flows. J. Fluid Mech. 250, 69–101.Google Scholar
Liu, K. F. 1990 Dynamics of a shallow layer of fluid mud. PhD thesis, Civil Engineering Department, MIT.
Mayer, P. G. 1959 Roll waves and slug flows in inclined open channels. J. Hydraul. Div., ASCE 85 (HY7), 99–141.Google Scholar
Merkin, J. H. & Needham, D. J. 1986 An infinite period bifurcation arising in roll waves down an open inclined channel. Proc. R. Soc. Lond. A 405, 103–116.Google Scholar
Needham, D. J. 1988 The development of a bedform disturbance in an alluvial river of channel. Z. Angew. Math. Phys. 39, 28–49.Google Scholar
Needham, D. J. 1990 Wave hierarchies in alluvial river flows. Geophys. Astrophys. Fluid Dyn. 51, 167–194.Google Scholar
Needham, D. J. & Merkin, J. H. 1984 On roll waves down an open inclined channel. Proc. R. Soc. Lond. A 394, 259–278.Google Scholar
O’Brien, J. S. & Julien, P. Y. 1988 Laboratory analysis of mudflow properties. J. Hydraul. Engng ASCE 114(8), 877–887.Google Scholar
Okuda, S., Suwa, H., Okunishi, K., Yokoyama, K. & Nakano, M. 1980 Observations on the motion of a debris flow and its geomorphological effects. Zeit. Geomorph. Suppl.-Bd. 35, 142–163.Google Scholar
Pierson, T. C. 1980 Erosion and deposition by debris flows at Mt Thomas, North Canterbury, New Zealand. Earth Surf. Proc. 5, 227–247.Google Scholar
Prokopiou, Th., Cheng, M. & Chang, H.-C. 1991 Long waves on inclined films at high Reynolds number. J. Fluid Mech. 222, 665–691.Google Scholar
Qian, N. & Wan, Z. 1986 A Critical Review of the Research on the Hyperconcentrated Flow in China. International Research and Training Centre on Erosion and Sedimentation, Beijing, China.
Schlichting, H. 1979 Boundary-Layer Theory. McGraw Hill.
Sharp, R. P. & Nobles, L. H. 1953 Mudflow of 1941 at Wrightwood, southern California. Geol. Soc. Am. Bull. 64, 547–560.Google Scholar
Tamada, K. & Tougou, H. 1979 Stability of roll waves on thin laminar flow down an inclined plane wall. J. Phys. Soc. Japan 47, 1992–1998.Google Scholar
Tanner, R. I. 1985 Engineering Rheology. Clarendon.
Wan, Z. 1982 Bed material movement in hyperconcentrated flow. Series Paper 31. Inst. Hydrodynamics and Hydraulic Engineering, Technical University of Denmark.
Whitham, G. B. 1974 Linear and Nonlinear Waves. John Wiley & Sons.
Yano, K. & Daido, A. 1965 Fundamental study on mudflows. Ann. Disaster Prevention Res. Inst., Kyoto Univ. 7, 340–347.Google Scholar
Yih, C.-S. 1963 Stability of liquid flow down an inclined plane. Phys. Fluids 6, 321–334.Google Scholar