Critical depth

Calculate the critical depth.

critical_depth(Q, yopt, g, B, SS)

Arguments

Q

Flow rate [\(L^3 T^{-1}\)].

yopt

Initial guess for normal depth [\(L\)].

g

Gravitational acceleration [\(L T^{-2}\)].

B

Channel bottom width [\(L\)].

SS

Channel sideslope [\(L L^{-1}\)].

Value

The critical depth \(y_c\) [\(L\)].

Details

The critical depth is the water depth at which a channel flow regime will transition from supercritical to subcritical (or vice versa). Calculation of the critical depth is based on a specific energy formulation, i.e. $$E = y + z + \frac{Q^2}{2gB^2y^2}$$ where \(y\) is the flow depth, \(z\) is the elevation relative to some datum (assumed to be 0), and the last term represents kinetic energy. More specifically, the function operates by finding the point where the derivative of specific energy w.r.t. \(y\) is zero, i.e. \(y = y_c\) when $$\frac{dE}{dy} = 1 - \frac{Q^2}{gA^3}\frac{dA}{dy} = 0$$.

Examples

critical_depth(250, 2, 32.2, 100, 0) # rectangular channel
#> [1] 0.5789949
critical_depth(126, 1, 9.81, 6.1, 1.5) # trapezoidal channel with sideslope 3H:2V
#> [1] 2.783155