Friction factor
As with Manning’s n value and the Chézy C, the friction factor, f, is a roughness coefficient in a velocity equation, namely, the Darcy-Weisbach equation. Originally developed for pipe flow, the equation adapted for flow in open channels is:
In 1963, the ASCE Task Committee on Friction Factors in Open Channels recommended the preferential use of the Darcy-Weisbach friction factor over Manning’s n (Simons and Sentürk 1992). While Manning’s equation remains the most used equation in practice, a comparison between the two is an illustrative exercise. The equation, applicable for steady uniform flow, is a balance of downstream gravitational force and upstream boundary resistance forces. The relationship between Manning’s n and Chézy C is (Hey 1979, English units):
where:
d = hydraulic depth
To apply the velocity equation, the friction factor must be determined. As has often been discussed by researchers (Raudkivi 1990; Thorne, Hey, and Newson 2001), the vertical velocity profile can often be assumed to be logarithmic with distance from the bed. For sand and gravel channels, where the relative roughness (flow depth/bed-material size) exceeds 10, this relationship holds.
For use in gravel-bed streams, with width-to-depth ratios greater than about 15, Hey (1979) derived the following (see also Thorne, Hey, and Newson 2001):
or
where:
R = hydraulic radius
D84 = bed-material size for which 84 percent is smaller
The dimensionless a is given by (Thorne, Hey, and Newson 2001):
where:
dmax = maximum flow depth
The coefficient a varies from 11.1 to 13.46 and is a function of channel cross-sectional shape. For channels in which the width-to-depth ratio exceeds 2, the maximum flow depth is valid in the above equation. Otherwise, the value in the denominator should be the distance perpendicular from the bed surface to the point of maximum velocity. This formula for determining f may be used in gravel-bed riffle-pool streams in the riffle section, where flow is often assumed to be uniform. In general, the D84 is calculated based on a sample taken at the riffle section.
The Limerinos equation can also be used to determine the friction factor.
where:
R = hydraulic radius, in ft
D84 = particle diameter, in ft, that equals or exceeds that of 84 percent of the particles
This equation was developed from samples taken from 11 large United States rivers with bed materials ranging from small gravel to medium size boulders. This equation has been shown to work well on sand-bed streams with plane beds.