Normal depth

Calculate the normal (equilibrium) depth using Manning's equation.

normal_depth(So, n, Q, yopt, Cm, B, SS)

Arguments

So

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

n

Manning's roughness coefficient.

Q

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

yopt

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

Cm

Unit conversion coefficient for Manning's equation. For SI units, Cm = 1.

B

Channel bottom width [\(L\)].

SS

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

Value

The normal depth \(y_n\) [\(L\)].

Details

The normal depth is the equilibrium depth of a channel for a given flow rate, channel slope, geometry and roughness. Manning's equation is used to calculate the equilibrium depth. Manning's equation for normal flow is defined as $$Q = \frac{C_m}{n} AR^{2/3}S_0^{1/2}$$ where \(Q\) is the channel flow, \(S_0\) is the channel slope, \(A\) is the cross-sectional flow area, \(R\) is the hydraulic depth and \(C_m\) is a conversion factor based on the unit system used. This function uses a Newton-Raphson root-finding approach to calculate the normal depth, i.e. \(y = y_n\) when $$f(y) = \frac{A^{5/3}}{P^{2/3}} - \frac{nQ}{C_mS_0^{1/2}} = 0$$.

Examples

normal_depth(0.001, 0.045, 250, 3, 1.486, 100, 0) # rectangular channel
#> [1] 1.711301
normal_depth(0.0008, 0.013, 126, 5, 1, 6.1, 1.5) # trapezoidal channel with sideslope 3H:2V
#> [1] 3.2864