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Interpolation between Darcy–Weisbach and Darcy for laminar and turbulent flows

William L. Hogarth 1 Jean-Yves Parlange 2 Calvin W. Rose 3 Carlos Fuentes 4 Randel Haverkamp 5 M. Todd Walter 2
1 Faculty of Science and Information Technology
2 Department of Biological and Environmental Engineering
3 Faculty of Environmental Sciences
4 IMTA - Instituto Mexicano de Tecnología del Agua
5 LTHE - Laboratoire d'étude des transferts en hydrologie et environnement
Abstract : An equation describing flow in an open channel with obstacles is derived, following the conservation of momentum approach used by Bélanger and St. Venant. When the obstacles are all submerged the result yields the Darcy–Weisbach equation for turbulent flow in pipes and open channels. When the obstacles are only partially submerged the result leads to the governing equation in a porous medium. If the flow is turbulent the square of the velocity is proportional to the hydraulic gradient and if the flow is laminar, which is the usual case, the velocity is proportional to the hydraulic gradient. This last result is in agreement with Darcy's law in porous media. Thus our equation interpolates between and reduces to, the two fundamental results of Darcy. In general our equation should prove useful in practice for open flow in a channel with both submerged and emerging obstacles.
Keywords : Darcy Darcy–Weisbach Porous media Open channels Turbulent flow Laminar flow
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https://hal-insu.archives-ouvertes.fr/insu-00383169
Contributor : Thierry Pellarin <>
Submitted on : Tuesday, May 12, 2009 - 11:25:25 AM
Last modification on : Friday, September 25, 2020 - 11:02:44 AM

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William L. Hogarth, Jean-Yves Parlange, Calvin W. Rose, Carlos Fuentes, Randel Haverkamp, et al.. Interpolation between Darcy–Weisbach and Darcy for laminar and turbulent flows. Advances in Water Resources, Elsevier, 2005, 28 (10), pp.1028 à 1031. ⟨10.1016/j.advwatres.2004.10.012⟩. ⟨insu-00383169⟩

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