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Consistent equations for open-channel flows in the smooth turbulent regime with shearing effects

Published online by Cambridge University Press:  13 October 2017

G. L. Richard Open the ORCID record for G. L. Richard [Opens in a new window] ,
A. Rambaud  and
J. P. Vila
G. L. Richard*
Affiliation:
Institut de Mathématiques de Toulouse; UMR5219, Université de Toulouse; CNRS; UPS IMT, F-31062 Toulouse CEDEX 9, France
A. Rambaud
Affiliation:
Universidad del Bío-Bío, depto de Matemática, Concepción, Chile
J. P. Vila
Affiliation:
Institut de Mathématiques de Toulouse; UMR5219, Université de Toulouse; CNRS; INSA, F-31077 Toulouse, France
*
†Email address for correspondence: gael.loic.richard@orange.fr
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Abstract

Consistent equations for turbulent open-channel flows on a smooth bottom are derived using a turbulence model of mixing length and an asymptotic expansion in two layers. A shallow-water scaling is used in an upper – or external – layer and a viscous scaling is used in a thin viscous – or internal – layer close to the bottom wall. A matching procedure is used to connect both expansions in an overlap domain. Depth-averaged equations are then obtained in the approximation of weakly sheared flows which is rigorously justified. We show that the Saint-Venant equations with a negligible deviation from a flat velocity profile and with a friction law are a consistent set of equations at a certain level of approximation. The obtained friction law is of the Kármán–Prandtl type and successfully compared to relevant experiments of the literature. At a higher precision level, a consistent three-equation model is obtained with the mathematical structure of the Euler equations of compressible fluids with relaxation source terms. This new set of equations includes shearing effects and adds corrective terms to the Saint-Venant model. At this level of approximation, energy and momentum resistances are clearly distinguished. Several applications of this new model that pertains to the hydraulics of open-channel flows are presented including the computation of backwater curves and the numerical resolution of the growing and breaking of roll waves.

JFM classification

Waves/Free-surface Flows: Hydraulics Geophysical and Geological Flows: Shallow water flows Turbulent Flows: Turbulent boundary layers
Type
Papers
Information
Journal of Fluid Mechanics , Volume 831 , 25 November 2017 , pp. 289 - 329
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© 2017 Cambridge University Press 

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References

Bailey, S. C. C., Vallikivi, M., Hultmark, M. & Smits, A. J. 2014 Estimating the value of von Kármán’s constant in turbulent pipe flow. J. Fluid Mech. 749, 79–98.CrossRefGoogle Scholar
Barré de Saint-Venant, A. J. C. 1871 Théorie et équations générales du mouvement non permanent des eaux courantes. C. R. Acad. Sci. 73, 147–154.Google Scholar
Brock, R. R. 1966 Discussion. J. Hydraul. Div. 92, 403–409.Google Scholar
Brock, R. R.1967 Development of roll waves in open channels. PhD thesis, California Institute of Technology.Google Scholar
Cao, Z., Hu, P., Hu, K., Pender, G. & Liu, Q. 2015 Modelling roll waves with shallow water equations and turbulent closure. J. Hydraul. Res. 53 (2), 161–177.CrossRefGoogle Scholar
Chow, V. T. 1959 Open Channel Hydraulics. McGraw-Hill.Google Scholar
Cousteix, J. & Mauss, J. 2004 Approximations of the Navier–Stokes equations for high Reynolds number flows past a solid wall. J. Comput. Appl. Maths 166 (1), 101–122.CrossRefGoogle Scholar
Cousteix, J. & Mauss, J. 2007 Asymptotic Analysis and Boundary Layers. Springer.CrossRefGoogle Scholar
Darcy, H. 1854 Sur des recherches expérimentales relatives au mouvement des eaux dans les tuyaux. C. R. Acad. Sci. 38, 1109–1121.Google Scholar
d’Aubuisson de Voisins, J. F. 1840 Traité d’hydraulique. Levraut.Google Scholar
Decoene, A., Bonaventura, L., Miglio, E. & Saleri, F. 2009 Asymptotic derivation of the section-averaged shallow water equations for natural river hydraulics. Math. Models Meth. Appl. Sci. 19 (3), 387–417.CrossRefGoogle Scholar
Dressler, R. F. 1949 Mathematical solution of the problem of roll-waves in inclined open channels. Commun. Pure Appl. Maths 2 (2–3), 149–194.CrossRefGoogle Scholar
Gauckler, P. 1868 Du mouvement de l’eau dans les conduites. Ann. des Ponts et Chaussées 15 (4), 229–281.Google Scholar
Henderson, F. M. 1996 Open Channel Flow. Macmillan.Google Scholar
Kranenburg, C. 1992 On the evolution of roll waves. J. Fluid Mech. 245, 249–261.CrossRefGoogle Scholar
Luchini, P. & Charru, F. 2010a Consistent section-averaged equations of quasi-one-dimensional laminar flow. J. Fluid Mech. 656, 337–341.CrossRefGoogle Scholar
Luchini, P. & Charru, F. 2010b The phase lead of shear stress in shallow-water flow over a perturbed bottom. J. Fluid Mech. 665, 516–539.CrossRefGoogle Scholar
Manning, R. 1895 On the flow of water in open channels and pipes. Trans. Inst. Civil Eng. Ireland 20, 161–207.Google Scholar
Mellor, G. L. 1972 The large Reynolds number, asymptotic theory of turbulent boundary layers. Intl J. Engng Sci. 10, 851–873.CrossRefGoogle Scholar
Noble, P. & Vila, J.-P. 2013 Thin power-law film flow down an inclined plane: consistent shallow-water models and stability under large-scale perturbations. J. Fluid Mech. 735, 29–60.CrossRefGoogle Scholar
Pouliquen, O. 1999 Scaling laws in granular flows down rough inclined planes. Phys. Fluids 11 (3), 542–548.CrossRefGoogle 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, 133–151.CrossRefGoogle Scholar
Richard, G. L.2013 Élaboration d’un modèle d’écoulements turbulents en faible profondeur. Application au ressaut hydraulique et aux trains de rouleaux. PhD thesis, Université d’Aix-Marseille.Google Scholar
Richard, G. L. & Gavrilyuk, S. L. 2012 A new model of roll waves: comparison with Brock’s experiments. J. Fluid Mech. 698, 374–405.CrossRefGoogle Scholar
Richard, G. L. & Gavrilyuk, S. L. 2013 The classical hydraulic jump in a model of shear shallow-water flows. J. Fluid Mech. 725, 492–521.CrossRefGoogle Scholar
Richard, G. L., Ruyer-Quil, C. & Vila, J. P. 2016 A three-equation model for thin films down an inclined plane. J. Fluid Mech. 804, 162–200.CrossRefGoogle Scholar
Ruyer-Quil, C. & Manneville, P. 2000 Improved modeling of flows down inclined planes. Eur. Phys. J. B 15 (2), 357–369.CrossRefGoogle 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.CrossRefGoogle Scholar
Strickler, A.1923 Beiträge zur Frage der Geschwindigkeitsformel und der Rauhigkeitszahlen für Ströme, Kanäle und geschlossene Leitungen. Mitteilungen des eidgenössischen Amtes für Wasserwirtschaft, 16, Bern.Google Scholar
Sturm, T. W. 2010 Open Channel Hydraulics. McGraw-Hill.Google Scholar
Teshukov, V. M. 2007 Gas-dynamics analogy for vortex-free boundary flows. J. Appl. Mech. Tech. Phys. 48 (3), 303–309.CrossRefGoogle Scholar
Thual, O. 2013 Modelling rollers for shallow water flows. J. Fluid Mech. 728, 1–4.CrossRefGoogle Scholar
Tracy, H. J. & Lester, C. M.1961 Resistance coefficients and velocity distribution in smooth rectangular channel. Tech. Rep. US Geological Survey.Google Scholar
Usha, R. & Uma, B. 2004 Modeling of stationary waves on a thin viscous film down an inclined plane at high Reynolds numbers and moderate Weber numbers using energy integral method. Phys. Fluids 16 (7), 2679–2696.CrossRefGoogle Scholar
Van Driest, E. R. 1951 Turbulent boundary layer in compressible fluids. J. Aeronaut. Sci. 18, 145–160, 216.CrossRefGoogle Scholar
Van Driest, E. R. 1956 On turbulent flow near a wall. J. Aeronaut. Sci. 23, 1007–1011.CrossRefGoogle Scholar
Van Driest, E. R. 1959 Convective heat transfer in gases. In Turbulent Flows and Heat Transfer (ed. Lin, C. C.), pp. 339–427. Princeton University Press.Google Scholar
Van Dyke, M. 1964 Perturbation Methods in Fluid Mechanics. Academic.Google Scholar
Weisbach, J. 1845 Lehrbuch der Ingenieur und Maschinenmechanik. Brunswick.Google Scholar
Yen, B. C. 1973 Open-channel flow equations revisited. J. Engng Mech. Div. 99 (5), 979–1009.CrossRefGoogle Scholar
Yen, B. C. 2002 Open channel flow resistance. J. Hydraul. Engng 128 (1), 20–39.CrossRefGoogle Scholar
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