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Solitary waves in turbulent open-channel flow

Published online by Cambridge University Press:  30 May 2013

Wilhelm Schneider
Wilhelm Schneider*
Affiliation:
Institute of Fluid Mechanics and Heat Transfer, Vienna University of Technology, 1040 Vienna, Austria
*
†Email address for correspondence: [email protected]
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Abstract

Two-dimensional turbulent free-surface flow is considered. The ensemble-averaged flow quantities may depend on time. The slope of the plane bottom of the channel is assumed to be small. The roughness of the bottom is allowed to vary with the space coordinate, leading to small variations in the bottom friction coefficient. An asymptotic analysis, which is free of turbulence modelling, is performed for large Reynolds numbers and Froude numbers close to the critical value 1. As a result, an extended Korteweg–deVries (KdV) equation for the surface elevation is obtained. Other flow quantities, such as pressure, flow velocity components, and bottom shear stress, are expressed in terms of the surface elevation. The steady-state version of the extended KdV equation has eigensolutions that describe stationary solitary waves. Time-dependent solutions of the extended KdV equation provide a means for discriminating between stable and unstable stationary solitary waves. Solutions of initial value problems show that there are transient solutions that approach asymptotically the stable stationary solitary wave, whereas other transient solutions decay asymptotically with increasing time.

JFM classification

Waves/Free-surface Flows: Solitary waves Turbulent Flows: Turbulent Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 726 , 10 July 2013 , pp. 137 - 159
Copyright
©2013 Cambridge University Press 

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References

Abd-el-Malek, M. B. & Helal, M. M. 2011 Group method solutions of the generalized forms of Burgers, Burgers–KdV and KdV equations with time-dependent variable coefficients. Acta Mech. 221, 281–286.Google Scholar
Binder, B. J., Vanden-Broek, J.-M. & Dias, F. 2005 Forced solitary waves and fronts past submerged obstacles. Chaos 15, 037106.Google Scholar
Bose, S. K., Castro-Orgaz, O. & Dey, S. 2012 Free surface profiles of undular hydraulic jumps. J. Hydraul. Engng 138, 362–366.Google Scholar
Brocchini, M. & Peregrine, D. H. 1998 The modelling of a spilling breaker: strong turbulence at a free surface. In Proceedings of International Conference on Coastal Engng, pp. 72–85. ASCE.Google Scholar
Caputo, J.-G. & Stepanyants, Y. A. 2003 Bore formation, evolution and disintegration into solitons in shallow inhomogeneous channels. Nonlin. Process. Geophys. 10, 407–424.Google Scholar
Castro-Orgaz, O. 2010 Weakly undular hydraulic jump: effects of friction. J. Hydraul. Res. 48, 453–465.Google Scholar
Castro-Orgaz, O. & Chanson, H. 2011 Near-critical free-surface flows: real fluid flow analysis. Environ. Fluid Mech. 11, 499–516.Google Scholar
Castro-Orgaz, O. & Hager, W. H. 2011 Observations on undular hydraulic jump in movable bed. J. Hydraul. Res. 49, 689–692.Google Scholar
Chanson, H. & Montes, J. S. 1995 Characteristics of undular hydraulic jumps: experimental apparatus and flow patterns. J. Hydraul. Engng 121, 129–144.Google Scholar
Chardard, F., Dias, F., Nguyen, H. Y. & Vanden-Broeck, J.-M. 2011 Stability of some stationary solutions to the forced KdV equation with one or two bumps. J. Engng Maths 70, 175–189.Google Scholar
Christov, C. I. & Velarde, M. G. 1995 Dissipative solitons. Physica D 86, 323–347.Google Scholar
DeKerf, F. 1988 Asymptotic analysis of a class of perturbed Korteweg-de Vries initial value problems. CWI Tract 50. Centre for Mathematics and Computer Science, Stichting Math. Centrum, Amsterdam.Google Scholar
Dias, F. & Vanden-Broeck, J.-M. 2002 Generalised critical free-surface flows. J. Engng Maths 42, 291–301.Google Scholar
Dias, F. & Vanden-Broeck, J.-M. 2004 Trapped waves between submerged obstacles. J. Fluid Mech. 509, 93–102.Google Scholar
Dutykh, D. 2009 Visco-potential free-surface flows and long wave modelling. Eur. J. Mech. (B/Fluids) 28, 430–443.Google Scholar
Dutykh, D. & Clamond, D. 2013 Efficient computation of steady solitary gravity waves (submitted).Google Scholar
Dutykh, D. & Dias, F. 2007 Viscous potential free-surface flows in a fluid layer of finite depth. C. R. Acad. Sci. Paris I 345, 113–118.Google Scholar
El, G. A., Grimshaw, R. H. J. & Kamchatkov, A. M. 2007 Evolution of solitary waves and undular bores in shallow-water flows over a gradual slope with bottom friction. J. Fluid Mech. 585, 213–244.Google Scholar
Gersten, K. & Herwig, H. 1992 Strömungsmechanik. Grundlagen der Impuls-, Wärme- und Stoffübertragung aus asymptotischer Sicht. Vieweg.Google Scholar
Gotoh, H., Yasuda, Y. & Ohtsu, I. 2005 Effect of Channel Slope on Flow Characteristics of Undular Hydraulic Jumps. In River Basin Management III (ed. Brebbia, C.A. & do Carmo, J.S.A.), WIT Transactions on Ecology and the Environment, pp. 33–42. WIT Press.Google Scholar
Grillhofer, W. 2002 Der wellige Wassersprung in einer turbulenten Kanalströmung mit freier Oberfläche. Dissertation, Technische Universität Wien, Vienna, Austria.Google Scholar
Grillhofer, W. & Schneider, W. 2003 The undular hydraulic jump in turbulent open channel flow at large Reynolds numbers. Phys. Fluids 15, 730–735.Google Scholar
Grimshaw, R. 2005 Korteweg-de Vries equation. In Nonlinear Waves in Fluids: Recent Advances and Modern Applications (ed. Grimshaw, R.). CISM Courses and Lectures, vol. 483, pp. 1–28. Springer.Google Scholar
Grimshaw, R. 2007 Internal solitary waves in a variable medium. GAMM-Mitt. 30/1, 96–109.Google Scholar
Grimshaw, R. 2010 Exponential asymptotics and generalized solitary waves. In Asymptotic Methods in Fluid Mechanics: Survey and Recent Advances (ed. Steinrück, H.). CISM Courses and Lectures, vol. 523, pp. 71–120. Springer.Google Scholar
Grimshaw, R., Pelinovsky, E. & Talipova, T. 2003 Damping of large-amplitude solitary waves. Wave Motion 37, 351–364.Google Scholar
Grimshaw, R. H. J., Zhang, D.-H. & Chow, K. W. 2007 Generation of solitary waves by transcritical flow over a step. J. Fluid Mech. 587, 235–254.Google Scholar
Hager, W. H. & Hutter, K. 1984 On pseudo-uniform flow in open channel hydraulics. Acta Mech. 53, 183–200.Google Scholar
Handler, R. A., Swean, T. F. Jr., Leighton, R. I. & Swearingen, J. D. 1993 Length scales and the energy balance for turbulence near a free surface. AIAA J. 31, 1998–2007.Google Scholar
Hassanzadeh, R., Sahin, B. & Ozgoren, M. 2012 Large eddy simulation of free-surface effects on the wake structures downstream of a spherical body. Ocean Engng 54, 213–222.Google Scholar
Johnson, R. S. 1972 Shallow water waves on a viscous fluid – the undular bore. Phys. Fluids 15, 1693–1699.Google Scholar
Johnson, R. S. 1997 A Modern Introduction to the Mathematical Theory of Water Waves. Cambridge University Press.Google Scholar
Jurisits, R. & Schneider, W. 2012 Undular hydraulic jumps arising in non-developed turbulent flows. Acta Mech. 223, 1723–1738.Google Scholar
Jurisits, R., Schneider, W. & Bae, Y. S. 2007 A multiple-scales solution of the undular hydraulic jump problem. In Proc. Appl. Math. Mech. (PAMM) 7, 4120007-4120008/.Google Scholar
Kalisch, H. & Bjørkavåg, M. 2010 Energy budget in a dispersive model for undular bores. Proc. Estonian Acad. Sci. 59, 172–181.Google Scholar
Kang, T. H. 2009 The undular tidal bore in turbulent free-surface flow at large Reynolds number (in Korean). Master thesis, Changwon National University, Changwon, Korea in cooperation with Vienna University of Technology, Vienna, Austria.Google Scholar
Kluwick, A. (Ed.) 1998 Recent Advances in Boundary Layer Theory. CISM Courses and Lectures, vol. 390. Springer.Google Scholar
Kluwick, A., Cox, E. A., Exner, A. & Grinschgl, C. 2010 On the internal structure of weakly nonlinear bores in laminar high Reynolds number flow. Acta Mech. 210, 135–157.Google Scholar
Knickerbocker, C. J. & Newell, A. C. 1980 Shelves and the Korteweg-de Vries equation. J. Fluid Mech. 98, 803–818.Google Scholar
Komori, S., Nagaosa, R., Murakami, Y., Chiba, S., Ishii, K. & Kuwahara, K. 1993 Direct numerical simulation of three-dimensional open-channel flow with zero-shear gas–liquid interface. Phys. Fluids A 5, 115–125.Google Scholar
Leibovich, S. & Randall, J. D. 1971 Dissipative effects on nonlinear waves in rotating fluids. Phys. Fluids 14, 2559–2561.Google Scholar
Leibovich, S. & Randall, J. D. 1973 Amplification and decay of long nonlinear waves. J. Fluid Mech. 53, 481–493.Google Scholar
Lennon, J. M. & Hill, D. F. 2006 Particle image velocity measurements of undular and hydraulic jumps. ASCE J. Hydraul. Engng 132, 1283–1294.Google Scholar
Montes, J. S. & Chanson, H. 1998 Characteristics of undular hydraulic jumps: experiments and analysis. ASCE J. Hydraul. Engng 124, 192–205.Google Scholar
Miles, J. W. 1983a Solitary wave evolution over a gradual slope with turbulent friction. J. Phys. Oceanogr. 13, 551–553.Google Scholar
Miles, J. W. 1983b Wave evolution over a gradual slope with turbulent friction. J. Fluid Mech. 133, 207–216.Google Scholar
Mohamed, A. N. 2010 Modelling of free jumps downstream symmetric and asymmetric expansions: theoretical analysis and method of stochastic gradient boosting. J. Hydrodyn. 22, 110–120.Google Scholar
Newell, A. C. 1985 Solitons in Mathematics and Physics. Society for Industrial and Applied Mathematics.Google Scholar
Nezu, I. & Rodi, W. 1986 Open-channel flow measurements with a laser Doppler anemometer. ASCE J. Hydraul. Engng 112, 335–355.Google Scholar
Ohtsu, I., Yasuda, Y., Gotoh, H. & Iahr, M. 2001 Hydraulic condition for undular-jump formations. J. Hydraul. Res. 39, 203–209.Google Scholar
Ohtsu, I., Yasuda, Y. & Gotoh, H. 2003 Flow conditions of undular hydraulic jumps in horizontal rectangular channels. J. Hydraul. Engng 129, 948–955.Google Scholar
Ott, E. & Sudan, R. N. 1970 Damping of solitary waves. Phys. Fluids 13, 1432–1434.Google Scholar
Pelinovsky, E. N., Stepanyants, Yu. & Talipova, T. 1993 Nonlinear dispersion model of sea waves in the coastal zone. J. Korean Soc. Coast. Ocean Engrs 5, 307–317.Google Scholar
Rodi, W. 1993 Turbulence Models and their Application in Hydraulics, 3rd edn. Balkema.Google Scholar
Rostami, F., Yazdi, S. R. S., Said, M. A. M. & Sharokhi, M. 2012 Numerical simulation of undular jumps on graveled bed using volume of fluid method. Water Sci. Technol. 66.5, 909–917.Google Scholar
Schlichting, H. & Gersten, K. 2000 Boundary Layer Theory, 8th edn. Springer.Google Scholar
Schneider, W. 2005 Near-critical free-surface flows. In Proceedings of 2nd Shanghai International Symp. Nonlinear Science and Applications (Shanghai NSA’05), Section 8.6 Free and Moving Boundary Problems.Google Scholar
Schneider, W., Jurisits, R. & Bae, Y. S. 2010 An asymptotic iteration method for the numerical analysis of near-critical free-surface flows. Intl J. Heat Fluid Flow 31, 1119–1124.Google Scholar
Scott, A. 2003 Nonlinear Science, 2nd edn. Oxford University Press.Google Scholar
Steinrück, H. 2005 Multiple scales analysis of the steady-state Korteweg-de Vries equation perturbed by a damping term. Z. Angew. Math. Mech. 85, 114–121.Google Scholar
Steinrück, H. 2010 Multiple scales analysis of the turbulent undular hydraulic jump. In Asymptotic Methods in Fluid Mechanics: Survey and Recent Advances (ed. Steinrück, H.). CISM Courses and Lectures, vol. 523, pp. 197–219. Springer.Google Scholar
Steinrück, H., Schneider, W. & Grillhofer, W. 2003 A multiple scales analysis of the undular hydraulic jump in turbulent open channel flow. Fluid Dyn. Res. 33, 41–55.Google Scholar
Svendsen, I. A., Veeramony, J., Bakunin, J. & Kirby, J. T. 2000 The flow in weak turbulent hydraulic jumps. J. Fluid Mech. 418, 25–57.Google Scholar
Van Dyke, M. 1975 Perturbation Methods in Fluid Mechanics, annotated edn. Parabolic.Google Scholar
Whitham, G. B. 1999 Linear and Nonlinear Waves. Wiley.Google Scholar