PDF《alluvial channel》

R is hydraulic radius related to bed, g is gravitational accelera- tion, and S is energy slope. [4] The current practice is to treat the total bed shear stress as the sum of two shear stress components corresponding to the grain and bed form resistance, i.e., to ? t0 ? t00 ?2? where t0 is shear stress due to grain resistance and t00 is shear stress due to bed form resistance. These components are often assumed to be independent of one another and have been expressed in various forms by different re- searchers. For example, Einstein and Barbarossa [1952] assumed a constant rgS on both sides of equation (2) and proposed that the bed hydraulic radius is the sum of two hydraulic radii corresponding to the grain resistance R0 and bed form resistance R00 . Equation (2) becomes R ? R0 ? R00 ?3?

1 Formerly at Maritime Research Center, Nanyang Technological University, Singapore. Copyright

2005 by the American Geophysical Union. 0043-1397/05/2005WR004211 W09419 WATER RESOURCES RESEARCH, VOL. 41, W09419, doi:10.1029/2005WR004211,

2005 1 of

8 Chien and Wan [1999, p. 273] commented that equation (3) is useful only as a tool but the approach is not universally applicable because the grain and bed form resistance often affect one another. They argued that '

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. . .if bed forms form on the bed, the separation zone on the lee side of a bed form will cause a reduction of the direct contact between the flow and the bed, and correspondingly causes a reduction in the grain resistance.Instead of using the two hydraulic radii, Engelund [1966] and Smith and McLean [1977] assumed a constant rgR on both sides of equation (2) and introduced an alternative approach based on the direct summation of two energy slopes. Equation (2) becomes S ? S0 ? S00 ?4? where S0 is the energy slope due to grain resistance and S00 is the component due to the bed form resistance. The argument is that the additional energy loss associated with S00 is the result of the '

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sudden expansion'

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of flow at the lee side of the bed forms. White et al. [1981] compared the various proposed approaches and concluded that amongst the calculated overall bed resistance values which are within a factor of

2 of the measured values, Einstein and Barbarossa [1952] scored 21%, Engelund [1966] scored 83%, and White et al. [1981] scored 89%. [5] Other researchers, such as Karim [1995], Yu and Lim [2003], Wu and Wang [1999], performed the best estimate of the bed shear stress by adjusting the Manning coefficient using either the regression techniques or dimensional anal- ysis. Wu and Wang'

s [1999] method gives less than 20% errors for 91% of the

811 data used;

Karim'

s [1995] empirical equation yielded results for which 74% of the measurements has less than 20% error;

Yu and Lim'

s [2003] formula showed less than 20% error for 86% of the

4824 sets of flume and field data used. [6] The purpose of the present study is to investigate the mechanism of mobile bed shear stress or flow resistance caused by grain and bed form, or to express properly the bed shear stress with the presence of bed forms and to estimate the energy slope in alluvial channels. 2. Underlying Mechanism of Mobile Bed Resistance [7] Consider a two-dimensional bed form as shown in Figure 1, in which, L = length of bed form;

d = bed form height;

and h = flow depth. If the drop in the water level over a bed form is hf over the distance L, then we can define the energy slope as S = hf/L. Since the total head loss of a flow system is the direct summation of the component head losses, [Daugherty et al., 1985, p. 248], it follows that one could introduce two component head losses for the case shown in Figure 1, i.e., h0 f corresponding to head loss due to grain resistance, and h00 f for head loss due to the bed form resistance. Thus the total energy loss over a bed form can be expressed as follows hf ? h0 f ? h00 f ?5? We can further visualize that there are two characteristic energy slopes S0 and S00 , for which S0 = h0 f/L0 and S00 = h00 f/L00 . The physical interpretation