Secondary Current and Classification of River Channels

Abstract

In this research, the secondary current theory is used in investigating the role of phase shift angle between the secondary current and the channel axis displacement in stability analysis of a river channel. To achieve this, a small-perturbation stability analysis is developed for investigation of the role of the secondary current accompanying channel curvature in the initiation and early development of meanders in open channels. The secondary currents are generating in planes perpendicular to the primary direction of motion. The secondary currents form a helical motion in which the water in the upper part of the river is driven outward, whereas the water near the bottom is driven inward in a bend. Force-momentum equations for longitudinal and transverse direction in open channel bends were utilized. Assuming that the transverse force contributed by the bed is negligible, the pressure force associated with the transverse surface inclination is balanced by the centripetal force. Existing equations of the transverse velocity profile were analyzed. Since the magnitude of the vertical velocity is negligible compared to the transverse velocity in secondary currents, this study concentrates on the transverse velocity which is the radial component of the secondary current. This formulation leads to a linear differential equation which is solved for its orthogonal components which give the rates of meander growth and downstream migration. It is shown that instability increases with decrease in phase shift angle. Transition from straight to meandering and then from meandering to braiding occurs when phase shift angle is reduced.

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K. Njenga, K. Kioko and G. Wanjiru, "Secondary Current and Classification of River Channels," Applied Mathematics, Vol. 4 No. 1, 2013, pp. 70-78. doi: 10.4236/am.2013.41013.

Conflicts of Interest

The authors declare no conflicts of interest.

 [1] L. B. Leopold and M. G. Wolman, “River Channel Patterns: Braided, Meandering, and Straight,” US Geological Survey Professional Paper, 1957. http://www.uvm.edu/~wbowden/Teaching/Stream_Geomorph_Assess/Resources/Private/Documents/1957_leopold_wolman_channel_patterns.pdf [2] S. A. Schumm, “The Fluvial System,” Wiley, New York, 1977. http://www.amazon.com/Fluvial-System-Stanley-Schumm/dp/1930665792 [3] G. K. Maarten, “Sorting out River Channel Patterns,” Progress in Physical Geography, Vol. 34, No. 3, 2010, pp. 287-326. doi:10.1177/0309133310365300 [4] D. L. Rosgen, “A Classification of Natural Rivers,” Catena, Vol. 22, No. 3, 1994, pp. 169-199. doi:10.1016/0341-8162(94)90001-9 [5] I. Kargapolova, “River Channel Responses to Runoff Variability,” Advances in Grosciences, Vol. 14, 2008, pp. 309-316. doi:10.5194/adgeo-14-309-2008 [6] P. F. Leggase, L. W. Zevenbergen, W. J. Spitz and L. A. Arnerson, “Stream Stability at Highway Structures,” 4th Edition, Federal Highway Administration Publication, Washington DC, 2012. http://www.trb.org/Main/Blurbs/167163.aspx [7] G. S. Parker, V. Wilkerson, E. G. Eke, J. D. Abad, J. W. Lauer, C. Paola, W. E. Dietrich and V. R. Voller, “A New Framework for Modeling the Migration of Meandering Rivers,” Earth Surface Processes and Landforms, Vol. 36, No. 1, 2011, pp. 70-86. doi:10.1002/esp.2113 [8] A. Crosato, “Simple Physics-Based Predictor for the Number of River Bars and the Transition between Meandering and Braiding,” Water Resources Research, Vol. 45, No. 3, 2009, p. 14. doi:10.1029/2008WR007242 [9] H. W. Shen, S. A. Schumm, J. D. Nelson, D. O. Doehring and M. M. Skinner, “Methods for Assessment of Stream-Related Hazards to Highways and Bridges,” Federal Highway Administration Publication, Washington DC, 1983. [10] D. B. Simons, “Theory of Design of Stable Channels in Alluvial Materials,” Ph.D. Dissertation, Colorado State University, Pueblo, 1957. http://www.worldcat.org/title/theory-and-design-of-stable-channels-in-alluvial-materials/oclc/503568203 [11] J. C. Bathurst, C. R. Thorne and R. D. Hey, “Secondary Flow and Shear Stress at River Bends,” Journal of the Hydraulics Division, Vol. 105, No. 10, 1979, pp. 1277-1295. http://cedb.asce.org/cgi/WWWdisplay.cgi?9055 [12] K. Richards, “Rivers: Form and Process in Alluvial Channels,” Methuen, London, 1982. http://books.google.co.ke/books/about/Rivers.html?id=JZJCPgAACAAJ&redir_esc=y [13] S. Won and J. J. Young, “Velocity Distribution of Secondary Currents in Curved Channels,” Science Direct, Vol. 22, No. 5, 2010, pp. 617-622. http://www.sciencedirect.com/science/journal/10016058/22/5/supp/S1 [14] K. O. Baek and S. Won, “Equation for Streamwise Variation of Secondary Flow in Sinous Channels,” Advances in Water Resources and Hydraulic Engineering, Vol. 11, 2009, pp. 580-585. [15] F. M. Henderson, “Open Channel Flow,” Mackmillan Publishing Co., London, 1966. [16] K. K. Peter and F. K. John, “Secondary Current and River-Meander Formation,” Journal of Fluid Mechanics, Vol. 144, No. 1, 1984, pp. 217-229. doi:10.1017/S0022112084001580 [17] J. M. Francisco, “Flow Resistance in Open Channels with Fixed and Movable Bed,” 2nd Joint Federal Interagency Conference, Las Vegas, 27 June-1 July 2010. http://water.usgs.gov/nrp/proj.bib/Publications/2010/simoes_2010.pdf [18] A. G. Anderson, G. Parker and A. Wood, “The Flow and Stability Characteristics of Alluvial River Channels,” Project Report No 161, University of Minnesota, USA, 1975. http://conservancy.umn.edu/handle/108222 [19] Z. B. Helmut, “Universal Equations and Constants of Turbulent Motion,” Journal of Physica Scripta, 2012. http://arxiv.org/abs/1203.5042 [20] L. Gottlieb, “Three-Dimensional Flow Pattern and Bed Topography in Meandering Channels,” Series Paper 11, Institute of Hydrodynamics, Engineering Technology University, Denmark, 1976. http://www.worldcat.org/title/three-dimensional-flow-pattern-and-bed-topography-in-meandering-channels/oclc/731721748 [21] M. A. Falcon, “Analysis of Flow in Alluvial Channel Bends,” Ph.D. Thesis, University of Lowa, Lowa City, 1979. [22] C. Zimmermann and J. F. Kennedy, “Transverse Bed Slopes in Curved Alluvial Streams,” Journal of the Hydraulics Division, Vol. 104, No. 1, 1978, pp. 33-48. http://cedb.asce.org/cgi/WWWdisplay.cgi?7831