Hostname: page-component-848d4c4894-89wxm Total loading time: 0 Render date: 2024-07-07T17:09:02.009Z Has data issue: false hasContentIssue false
  • English
  • Français

Morphology of a stream flowing down an inclined plane. Part 2. Meandering

Published online by Cambridge University Press:  30 June 2008

B. BIRNIR ,
K. MERTENS ,
V. PUTKARADZE  and
P. VOROBIEFF
B. BIRNIR
Affiliation:
Department of Mathematics, University of California, Santa Barbara, CA 93106, USA
K. MERTENS
Affiliation:
Department of Mathematics, Colorado State University, Fort Collins, CO 80235, USA
V. PUTKARADZE
Affiliation:
Department of Mathematics, Colorado State University, Fort Collins, CO 80235, USA Department of Mechanical Engineering, The University of New Mexico, Albuquerque, NM 87131, USA
P. VOROBIEFF
Affiliation:
Department of Mechanical Engineering, The University of New Mexico, Albuquerque, NM 87131, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

A stream of fluid flowing down a partially wetting inclined plane usually meanders, unless the volume flow rate is maintained at a highly constant value. Here we investigate whether the meandering of this stream is an inherent instability. In our experiment, we eliminate meandering on several partially wetting substrates by reducing perturbations entering the flow. By re-introducing controlled fluctuations, we show that they are responsible for the onset of the meandering. We derive a theoretical model for the stream shape, %from first principles which includes stream dynamics and forcing by external noise. The deviation h(x) from a straight linear stream h(x)=0 shows considerable variability as a function of downstream distance x. However, for an ensemble average of stream shapes acquired at different times, the power spectrum S(k) as a function of wavenumber k has a power-law scaling S(k) ~ k5/2. Moreover, the area A(x) swept by the stream at the distance x grows as A(x) ~ x1.75.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 607 , 25 July 2008 , pp. 401 - 411
Copyright
Copyright © Cambridge University Press 2008

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

REFERENCES

Birnir, B. 2007 Turbulent rivers. Q. Appl. Maths (in press)Google Scholar
Birnir, B., Hernandez, J. & Smith, T. R. 2007 The stochastic theory of fluvial landsurfaces. J. Nonlinear Sci. 17, 13–57.CrossRefGoogle Scholar
Birnir, B., Mertens, K., Putkaradze, V. & Vorobieff, P. 2008 Meandering fluid streams in the presence of flow rate fluctuations. Physical Review Letters submitted.CrossRefGoogle Scholar
Clanet, C. 2000 Clepsydrae from Galileo to Torricelli. Phys. Fluids 12, 2743–2751.CrossRefGoogle Scholar
Culkin, J. B. & Davis, S. H. 1984 Meandering of water rivulets. AIChE J. 30, 263–267.CrossRefGoogle Scholar
Davis, S. H. 1980 Moving contact lines and rivulet instabilities. Part 1. The static rivulet. J. Fluid Mech. 98, 225–242.CrossRefGoogle Scholar
Dodds, P. S. & Rothman, D. 2000 a Geometry of river networks. I. Scaling, fluctuations, and deviations. Phys. Rev E 63, 016115.Google ScholarPubMed
Dodds, P. S. & Rothman, D. 2000 b Scaling, universality, and geomorphology. Annu. Rev. Earth Planet. Sci. 28, 571610.CrossRefGoogle Scholar
Frisch, U. 1995 Turbulence. Cambridge University Press.CrossRefGoogle Scholar
deGennes, P. G. 1985 Wetting: statics and dynamics. Rev. Mod. Phys. 57, 827–863.CrossRefGoogle Scholar
Hack, J. 1957 Studies of longitudinal stream profiles in Virginia and Maryland. US Geological Survey Professional Paper 294-B.CrossRefGoogle Scholar
Ikeda, S., Parker, G. & Sawai, K. 1981 Bend theory of river meanders. Part 1. Linear development. J. Fluid Mech. 112, 363–377.CrossRefGoogle Scholar
Kim, H.-Y., Kim, J-H. & Kang, B. H. 2004 Meandering instability of a rivulet. J. Fluid Mech. 498, 245–256.CrossRefGoogle Scholar
Le Grand-Piteira, N., Daerr, A. & Limat, L. 2006 Meandering rivulets on a plane: A simple balance between inertia and capillarity. Phys. Rev. Lett 96, 254503.CrossRefGoogle Scholar
Leopold, L. B. & Wolman, M. G. 1960 River meanders. Bull. Geol. Soc. Am. 71, 769–793.CrossRefGoogle Scholar
Maritan, A., Rinaldo, A., Rigon, R., Giacometti, A. & Rodriguez-Iturbe, I. 1996 Scaling laws for river networks. Phys. Rev. E 53, 15101515.Google ScholarPubMed
Mertens, K., Putkaradze, V. & Vorobieff, P. 2004 Braiding patterns on an inclined plane. Nature 430, 165.CrossRefGoogle Scholar
Mertens, K., Putkaradze, V. & Vorobieff, P. 2005 Morphology of a stream flowing down an inclined plane. Part 1. Braiding. J. Fluid Mech. 531, 49–58.CrossRefGoogle Scholar
Mizumura, K. 1993 Meandering water rivulet. J. Hydraul. Engng 119, 1205–1222.CrossRefGoogle Scholar
Mizumura, K. & Yamasaka, M. 1997 Analysis of meandering water rivulets of finite amplitude. J. Hydraul. Engng 123, 995–1003.CrossRefGoogle Scholar
Montgomery, D. R. & Dietrich, W. E. 1992 Channel initiation and the problem of landscape scale. Science 255, 826–830.CrossRefGoogle ScholarPubMed
Nakagawa, T. M. S. 1992 Rivulet meanders on a smooth hydrophobic surface. Intl J. Multiphase Flow 18, 455–463.CrossRefGoogle Scholar
Nakagawa, T. M. S. & Nakagawa Jr, R. 1996 A novel oscillation phenomenon of the water rivulet on a smooth hydrophobic surface. Acta Mechanica 115, 27–37.CrossRefGoogle Scholar
Nakagawa, T. & Scott, J. C. 1984 Stream meanders on a smooth hydrophobic surface. J. Fluid Mech. 149, 89–99.CrossRefGoogle Scholar
Perazzo, C. & Gratton, J. 2004 Navier-Stokes solutions for parallel flow in rivulets on an inclined plane. J. Fluid Mech. 507, 367–379.CrossRefGoogle Scholar
Rigon, R., Rodriguez-Iturbe, I., Maritan, A., Giacometti, A., Tarboton, D. G. & Rinaldo, A. 1996 On Hack's law. Water Resour. Res. 32, 3367–3374.CrossRefGoogle Scholar
Schmuki, P. & Laso, M. 1990 On the stability of rivulet flow. J. Fluid Mech. 215, 125–143.CrossRefGoogle Scholar
Seminara, G. 2006 Meanders. J. Fluid Mech. 554, 271–297.CrossRefGoogle Scholar
Simon, B. 2005 Functional Integration and Quantum Physics, 2nd edn. AMS Chelsea Publishing.CrossRefGoogle Scholar
Weiland, R. H. & Davis, S. H. 1981 Moving contact lines and rivulet instabilities. Part 2. Long waves on flat rivulets. J. Fluid Mech 107, 261–280.CrossRefGoogle Scholar
Young, G. W. & Davis, S. H. 1987 Rivulet instabilities. J. Fluid Mech 176, 1–30.CrossRefGoogle Scholar