Fundamentals of Engineering Design
Equation (5.25) has been combined with Eqs. (5.22) through (5.24) to arrive at various logarithmic
resistance relationships. Limerinos (1970) used the contributions of Leopold and Wolman (1957) and
Chow (1959) to develop a relationship between Manning's n and relative smoothness:
n
0.0926
'
R 1/6
R
(5.26)
a % b@log
,
Limerinos found the smallest deviation between observed and computed values when D84 was used for the
equivalent sand roughness (), where D84 is the size of the minimum particle diameter that equals or exceeds
that of 84 percent of the river bed material. If D84 is used, the coefficients a and b obtain values of 0.76
and 2.00, respectively.
Often in gravel bed rivers, the banks do not have the same resistance elements as the bed. The bed
resistance is due to a rough plane boundary and the bank resistance comes from vegetation or from soil
that is different from that of the bed. Under these conditions it is ideal to calculate flow properties
separately for the bed and banks.
Einstein (1942) proposed a method of separating the hydraulic radii of the bed and the banks.
Lines perpendicular to the velocity contours are established that begin at the bed and end at the water
surface. An example of such lines can be seen in Figure 5.44. There is no velocity gradient or shear stress
across these lines. With the lines established, the cross section can be divided into three subsections. The
total area of the cross section is related to the geometry of the subsections by Einstein (1950):
AT ' PLRL % PBRB % PRRR
(5.27)
where subscripts L, B, and R indicate left bank, bed, and right bank, respectively.
VELOCITY CONTOURS (m/sec)
100
50
0
-400
-300
-200
-100
0
100
200
300
400
Distance (mm)
Assumed axis of symmetry
Water surface
Flume wall
Contour intervals of 0.05 m/sec
Figure 5.44 Velocity Contour Map With Lines Across Which There is No Shear Stress (after Gessler
et al., 1998)
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