The Chezy formula
In uniform flow in an open channel gravity is balanced by friction, the depth and velocity are constant along the flow, and the slopes of the bed, the free surface and the energy line are equal. Then by the Chezy formula v = C·√(R·i) and the flow is Q = ω·v. Here ω is the flow area, χ the wetted perimeter, R = ω/χ the hydraulic radius, i the slope and C the Chezy coefficient (m^0.5/s).
The Manning and Pavlovsky coefficients
The coefficient C is found empirically. The Manning formula is C = R^(1/6) / n, where n is the roughness coefficient (s/m^(1/3)). Russian standards often use the Pavlovsky formula C = R^y / n, y = 2.5·√n − 0.13 − 0.75·√R·(√n − 0.1). The n values in the table of the page are the normal and limit values from the reference table of V. T. Chow (1959, Open-Channel Hydraulics); they depend on the material, condition and vegetation, so a channel design adds a margin.
Normal depth
The normal depth h₀ is the depth of uniform flow for a given flow and slope. Because ω and R depend on the depth in a complicated way it is found by iteration; the flow grows with the depth monotonically. Comparing h₀ with the critical depth h_k (the Froude number Fr = v/√(g·ω/B) equals 1) tells the regime: for h₀ above h_k the flow is subcritical, below it supercritical. In a circular pipe the flow reaches its maximum at about 0.94 of the diameter, so the normal depth is sought only up to that filling.
Best hydraulic section
The best hydraulic section carries a given flow with the smallest area. For a trapezoid with side slope m the ratio of width to depth is b/h = 2·(√(1 + m²) − m) and R = h/2. For a rectangle b = 2h. Practical dimensions often differ because of the soil, the excavation cost and velocity limits.
Sources: Chow V. T. Open-Channel Hydraulics, McGraw-Hill, 1959 (Table 5-6, values of n); Chugaev R. R. Hydraulics; Shterenlikht D. V. Hydraulics; Bogomolov A. I., Mikhailov K. A. Hydraulics.