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

182

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