How Rischio calculates an open channel

The units, conventions and assumptions of Rischio's open channel calculation, and its method in order, from the section to the bank-full capacity.

Rischio calculates an open channel from one cross-section, its longitudinal slope $S$ and the discharge $Q$ ; the overview says what it reports and what it checks.

Units and conventions

SI units: m, m², m³/s and m/s. The slope is in m/m: 0.002 is 0.2 %. $g = 9.81$ m/s².
The section is a polyline of vertices $(x, z)$ , drawn from bank to bank, left to right. $x$ never decreases; two vertices with the same $x$ make a vertical wall.
Each segment, from one vertex to the next, has its own Manning $n$ : $N$ vertices take $N - 1$ values of $n$ .
Water levels are elevations on the datum of $z$ . Depths are measured from the bed, the lowest vertex.

Assumptions

Steady, uniform flow in a prismatic channel: nothing changes along the reach, and the friction slope equals $S$ .
Where the roughness changes, the section is divided into parts, and each carries its own share: the divided channel method, as HEC-RAS does. A side steeper than 1 vertical to 5 horizontal joins the part it runs down to, so a lined channel with grassed sides stays one part.
Every part of the section below the water level carries flow, even a low area cut off by higher ground.

Method

At a given water level, the wet segments make the section, divided into parts where its roughness changes. Manning’s equation gives each part’s discharge, and the normal depth is the depth at which their sum equals $Q$ ; then come the Froude number, the critical depth and the specific energy. Last, Manning’s discharge with the water at the lower bank is the bank-full capacity $Q_{bank}$ , the limit of the one check, and the rating curve gives the discharge from the bed up to that bank. The equations go through every step, in the order of the calculation report.

References

Chow, V. T. (1959). Open-Channel Hydraulics. McGraw-Hill.
Horton, R. E. (1933). Separate roughness coefficients for channel bottom and sides. Engineering News-Record, 111(22), 652–653.
Einstein, H. A. (1934). Der hydraulische oder Profil-Radius. Schweizerische Bauzeitung, 103(8), 89–91.
Brent, R. P. (1973). Algorithms for Minimization without Derivatives. Prentice-Hall.
U.S. Army Corps of Engineers. HEC-RAS Hydraulic Reference Manual: cross section subdivision for conveyance calculations, and composite Manning’s n for the main channel.