Double-Averaging Concept for Rough-Bed Open-Channel and Overland Flows: Theoretical Background
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VIEW THE ORIGINAL ARTICLEPublication: Journal of Hydraulic Engineering
Volume 133, Issue 8
Abstract
The goal of this paper is to discuss the spatial averaging concept in environmental hydraulics and develop it further by considering transport equations for fluid momentum, passive substances, and suspended sediments. The averaging theorems, the double-averaged (in time and in space) fluid momentum equation, and advection-diffusion equations for a passive substance and suspended sediments are introduced and their limitations and applications for modeling rough-bed flows, experimental design, and data interpretation are discussed. The suggested equations differ from those considered in terrestrial canopy aerodynamics and porous media hydrodynamics by accounting for roughness mobility, change in roughness density in space and time, and particle settling effects for the case of suspended sediments. We show that the form of the double-averaged equations may depend on the type of decomposition of flow variables and that this difference may have important implications for modeling. We also show that the suggested methodology offers better definitions for hydraulic characteristics, variables, and parameters such as flow uniformity, flow two dimensionality, and bed shear stress.
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Acknowledgments
The research was partly funded by the Foundation for Research Science and Technology (C01X0307 and C01X0308) and the Marsden Fund (UOA220, LCR203) administered by the New Zealand Royal Society (New Zealand), University of Aberdeen (Scotland), Grant No. GR/R51865 from the Engineering and Physical Sciences Research Council of the UK, and Grant Nos. CTS-8911359, CTS-9217804, and CTS-9634261 from the National Science Foundation (USA). The writers are grateful for useful discussions and suggestions to J. Finnigan, D. Goring, G. Katul, V.C. Patel, M. Raupach, R. Spigel, and to all participants of the double-averaging workshops in Belgium/Wavre (2002), Greece/Thessalonica (2003), Spain/Madrid (2004), Scotland/Aberdeen (2005), and Portugal/Lisbon (2006). Three reviewers and the Associate Editor provided useful comments that helped to improve this manuscript.
References
Antohe, B. V., and Lage, J. L. (1997). “A general two equation macroscopic turbulence model for incompressible flow in porous media.” Int. J. Heat Mass Transfer, 40(13), 3013–3024.
Ayotte, K. W., Finnigan, J., and Raupach, M. R. (1999). “A second-order closure for neutrally stratified vegetative canopy flows.” Boundary-Layer Meteorol., 90, 189–216.
Bohm, M., Finnigan, J. J., and Raupach, M. R. (2000). “Dispersive fluxes and canopy flows: Just how important are they?” Proc., of 24th Conf. on Agricultural and Forest Meteorology (AFME), Davis, Calif., 106–107.
Campbell, L., et al. (2005). “Bed-load effects on hydrodynamics of rough-bed open-channel flows.” J. Hydraul. Eng., 131(7), 576–585.
Cheng, H., and Castro, I. P. (2002). “Near wall flow near urban-like roughness.” Boundary-Layer Meteorol., 104, 229–259.
Coceal, O., Thomas, T. G., Castro, I. P., and Belcher, S. E. (2006). “Mean flow and turbulence statistics over groups of urban-like cubical obstacles.” Boundary-Layer Meteorol., 121(3), 491–519.
de Lemos, M. J. S. (2006). “Turbulence in porous media.” Modeling and applications, Elsevier, Amsterdam, The Netherlands.
de Lemos, M. J. S., and Pedras, M. H. J. (2001). “Recent mathematical models for turbulent flow in saturated rigid porous media.” J. Fluids Eng., 123, 935–940.
Finnigan, J. J. (1985). “Turbulent transport in flexible plant canopies.” The forest-atmosphere interactions, B. A. Hutchinson and B. B. Hicks, eds., D. Reidel, 443–480.
Finnigan, J. J. (2000). “Turbulence in plant canopies.” Annu. Rev. Fluid Mech., 32, 519–571.
Galmarini, S., and Thunis, P. (1999). “On the validity of Reynolds assumptions for running-mean filters in the absence of a spectral gap.” J. Atmos. Sci., 56(12), 1785–1796.
Gimenez-Curto, L. A., and Corniero Lera, M. A. (1996). “Oscillating turbulent flow over very rough surfaces.” J. Geophys. Res., 101(C9), 20745–20758.
Gimenez-Curto, L. A., and Corniero Lera, M. A. (2003). “Highest natural bed forms.” J. Geophys. Res., 108(C2), 3046.
Graf, W. H., and Altinakar, M. S. (1998). Fluvial hydraulics, Wiley, New York.
Gray, W. G. (1975). “A derivation of the equation for multiphase transport.” Chem. Eng. Sci., 30, 229–233.
Gray, W. G., and Lee, P. C. Y. (1977). “On the theorems for local volume averaging of multiphase systems.” Int. J. Multiphase Flow, 3, 333–340.
Hoffmann, M. R. (2004). “Application of a simple space-time averaged porous media model to flow in densely vegetated channels.” J. Porous Media, 7(3), 183–191.
Howes, F. A., and Whitaker, S. (1985). “The spatial averaging theorem revisited.” Chem. Eng. Sci., 40(8), 1,387–1,392.
Hsu, T.-J., Sakakiyama, T., and Liu, P. L.-F. (2002). “A numerical model for wave motions and turbulent flows in front of a composite breakwater.” Coastal Eng., 46, 25–50.
Kaimal, J. C., and Finnigan, J. J. (1994). Atmospheric boundary layer flows, Oxford University Press, New York.
Katul, G. G., Mahrt, L., Poggi, D., and Sanz, C. (2004). “One- and two-equation models for canopy turbulence.” Boundary-Layer Meteorol., 113, 81–109.
Lage, J. L., de Lemos, M. J. S., and Nield, D. A. (2002). “Modelling turbulence in porous media.” Transport phenomena in porous media, D. B. Ingham and I. Pop, eds., Pergamon, Amsterdam, The Netherlands, 198–230.
Lien, F.-S., Yee, E., and Wilson, J. D. (2005). “Numerical modeling of the turbulent flow developing within and over a 3-D building array. Part II: A mathematical foundation for a distributed drag force approach.” Boundary-Layer Meteorol., 114(2), 245–285.
Lopez, F., and Garcia, M. H. (1998). “Open-channel flow through simulated vegetation: Suspended sediment transport modelling.” Water Resour. Res., 34(9), 2,341–2,352.
Lopez, F., and Garcia, M. H. (2001). “Mean flow and turbulence structure of open-channel flow through nonemergent vegetation.” J. Hydraul. Eng., 127(5), 392–402.
Maddux, T. B., McLean, S. R., and Nelson, J. M. (2003). “Turbulent flow over three-dimensional dunes: 2, Fluid and bed stresses.” J. Geophys. Res., 108(F1), 6010.
McLean, S. R., Wolfe, S. R., and Nelson, J. M. (1999). “Spatially averaged flow over a wavy boundary revisited.” J. Geophys. Res., 104(C7), 15743–15753.
Monin, A. S., and Yaglom, A. M. (1971). Statistical fluid mechanics: Mechanics of turbulence. Vol. 1, MIT Press, Boston.
Murota, A., Fukuhara, T., and Sato, M. (1984). “Turbulence structure in vegetated open channel flows.” J. Hydrosci. Hydr. Eng., 2(1), 47–61.
Naot, D., Nezu, I., and Nakagawa, H. (1996). “Hydrodynamic behavior of partly vegetated open channels.” J. Hydraul. Eng., 122(11), 625–633.
Neary, V. S. (2003). “Numerical solution of fully developed flow with vegetative resistance.” J. Eng. Mech., 129(5), 558–563.
Nezu, I., and Nakagawa, H. (1993). Turbulence in open-channel flows, Balkema, Rotterdam, Brookfield, The Netherlands.
Nield, D. A. (2001). “Alternative models of turbulence in porous medium and related matters.” J. Fluids Eng., 123, 928–931.
Nikora, V. (2004). “Spatial averaging concept for rough-bed, open-channel, and overland flows.” Proc., of the 6th Int. Conf. on HydroScience and Engineering (CD-ROM), Brisbane.
Nikora, V., et al. (2007). “Double-averaging concept for rough-bed open-channel and overland flows: Applications.” J. Hydraul. Eng., 133(8), 884–895.
Nikora, V., Goring, D., McEwan, I., and Griffiths, G. (2001). “Spatially averaged open-channel flow over rough bed.” J. Hydraul. Eng., 127(2), 123–133.
Nikora, V., Koll, K., McEwan, I., McLean, S., and Dittrich, A. (2004a). “Velocity distribution in the roughness layer of rough-bed flows.” J. Hydraul. Eng., 130(10), 1,036–1,042.
Nikora, V., Larned, S., and Biggs, B. (2004b). “Periphyton-flow interactions and mass-transfer-uptake processes.” Proc., of the 5th Int. Symp. on Ecohydraulics, Aquatic Habitats: Analysis and Restoration (CD-ROM), Vol. 2, Madrid, 1141–1142.
Pedras, M. H. J., and de Lemos, M. J. S. (2000). “On the definition of turbulent kinetic energy for flow in porous media.” Int. Commun. Heat Mass Transfer, 27(2), 211–220.
Pedras, M. H. J., and de Lemos, M. J. S. (2001). “Macroscopic turbulence modeling for incompressible flow through undeformable porous media.” Int. J. Heat Mass Transfer, 44, 1,081–1,093.
Poggi, D., Katul, G. G., and Albertson, J. D. (2004a). “Momentum transfer and turbulent kinetic energy budgets within a dense canopy.” Boundary-Layer Meteorol., 111, 589–614.
Poggi, D., Katul, G. G., and Albertson, J. D. (2004b). “A note on the contribution of dispersive fluxes to momentum transfer within canopies.” Boundary-Layer Meteorol., 111, 615–621.
Prat, M., Plouraboue, F., and Letalleur, N. (2002). “Averaged Reynolds equation for flows between rough surfaces in sliding motion.” Transp. Porous Media, 48, 291–313.
Raupach, M. R., Antonia, R. A., and Rajagopalan, S. (1991). “Rough-wall turbulent boundary layers.” Appl. Mech. Rev., 44(1), 1–25.
Raupach, M. R., and Shaw, R. H. (1982). “Averaging procedures for flow within vegetation canopies.” Boundary-Layer Meteorol., 22, 79–90.
Rowinski, P. M., and Kubrak, J. (2002). “A mixing-length model for predicting vertical velocity distribution in flows through emergent vegetation.” Hydrol. Sci. J., 47(6), 893–904.
Shimizu, Y., and Tsujimoto, S. (1994). “Numerical analysis of turbulent open-channel flow over vegetation layer using a turbulence model.” J. Hydrosci. Hydr. Eng., 11(2), 57–67.
Siqueira, M., and Katul, G. (2002). “Estimating heat sources and fluxes in thermally stratified canopy flows using higher-order closure models.” Boundary-Layer Meteorol., 103, 125–142.
Slattery, J. C. (1999). Advanced transport phenomena, Cambridge University Press, Cambridge, U.K.
Smith, J. D., and McLean, S. R. (1977). “Spatially averaged flow over a wavy surface.” J. Geophys. Res., 83(12), 1,735–1,746.
Vu, T. C., Ashie, Y., and Asaeda, T. (2002). “A turbulence closure model for the atmospheric boundary layer including urban canopy boundary-layer meteorology.” Boundary-Layer Meteorol., 102(3), 459–490.
Walters, R. A. (2002). “From river to ocean: A unified modeling approach.” Estuarine and coastal modeling, M. L. Spaulding, ed., ASCE, New York, 683–694.
Whitaker, S. (1985). “A simple geometrical derivation of the spatial averaging theorem.” Chem. Eng. Educ., 19, 18–21 and 50–52.
Whitaker, S. (1992). Introduction to fluid mechanics, Krieger, Malabar, Fla.
Whitaker, S. (1999). The method of volume averaging, Kluwer Academic, Dordrecht.
Wilson, N. R., and Shaw, R. H. (1977). “A higher order closure model for canopy flow.” J. Appl. Meteorol., 16, 1197–1205.
Winterwerp, J. C., and van Kesteren, W. G. M. (2004). Introduction to the physics of cohesive sediment in the marine environment, Elsevier, Amsterdam, The Netherlands.
Yalin, M. S. (1977). Mechanics of sediment transport, Pergamon, Oxford, U.K.
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Published In
Journal of Hydraulic Engineering
Volume 133 • Issue 8 • August 2007
Pages: 873 - 883
Copyright
© 2007 ASCE.
History
Received: Jul 27, 2005
Accepted: Oct 24, 2006
Published online: Aug 1, 2007
Published in print: Aug 2007
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