Computation of the critical depth

We previously observed that it was necessary to calculate the critical depth in each computational section in order to determine if a sub-critical solution exists. The critical depth in a section is defined by:

= 1

This equation can be transformed as follows:

f(Zci) = Log ( ) = 0
which is a decreasing function.

which can be solved by Newton’s method.

The initial value is determined by taking the critical water depth in the rectangle equivalent to the section corresponding to the overtopping water elevation (ZDi).

Zcio = ( )1/3 + ZFi

with: X =

Starting values for the bisection algorithm are:

Zmin = ZFi
Zmax = ZDM + 10

with: ZDM = elevation of the last point of the section.

These values are modified in the following way:

If f(Zcil) > 0 : Zmin = Zcil

If f(Zcil) < 0 : Zmax = Zcil

since f is a decreasing function.