Shallow Concentrated Flow Velocity

Between sheet flow and a defined channel, runoff gathers into rills and small gullies. The Federal Highway Administration's urban drainage manual (HEC-22, fourth edition, equation 4.4, from HDS 2) gives the average velocity of that stage from the slope alone: $V = K_u k \sqrt{S_p}$, where $k$ is an intercept coefficient picked from a table by land cover, $S_p$ is the slope in percent and $K_u$ is a unit constant, 3.28 for feet per second and 1.0 for metres per second. The page works in metres per second, so $K_u$ is 1 and the velocity is $k$ times the square root of the slope in percent.

The coefficient $k$ is the velocity at a slope of 1 percent. Table 4.3 runs from 0.076 for forest with heavy ground litter or hay meadow, through 0.213 for short grass pasture, 0.305 for nearly bare and untilled ground, 0.457 for a grassed waterway and 0.491 for unpaved shallow concentrated flow, up to 0.619 for paved areas or small upland gullies. The slope enters as a square root, so the velocity rises with slope but more slowly than the slope does. Watch the units: $S_p$ is in percent, so 0.6 percent is 0.006 ft/ft.

The manual's Example 4.2, step 1b, is the page's check. A grassed waterway at a slope of 0.006 ft/ft (0.6 percent) with $k = 0.457$ gives 1.16 ft/s; the page gives 0.35398 m/s, which is 1.1614 ft/s. In metric units, nearly bare ground with $k = 0.305$ at a slope of 4 percent gives $0.305 \times \sqrt{4} = 0.61$ m/s, and read backwards that velocity returns $k = 0.305$. This is the middle leg of a time of concentration, between sheet flow and open channel or pipe flow: divide its length by the velocity to get its travel time.

Say which table you used. The Natural Resources Conservation Service's National Engineering Handbook, Part 630, chapter 15 (Table 15-3), has the same relation with its own coefficient table, and its values differ from HEC-22's on the grassed waterway row. For paved ground its form $V = 20.328\, s^{0.5}$, in feet per second with $s$ in ft/ft, gives 1.2857 ft/s at $s = 0.004$, where HEC-22's $k = 0.619$ gives 1.2844 ft/s; the handbooks round their coefficients to different places. The NRCS form applies to slopes less than 0.005 ft/ft, while HEC-22 sets no limit on slope for its equation.