Relation between Chezy, Manning and Darcy coefficients

Uniform flow: force balance

For uniform flow in a channel of hydraulic radius $R$ and slope $S$, the boundary shear stress is

$$\tau_0=\gamma RS=\rho gRS$$

Darcy-Weisbach

In terms of the Darcy friction factor $f$ and mean velocity $V$:

$$\tau_0=\frac{f}{8}\rho V^2$$

Equating the two:

$$\rho gRS=\frac f8\rho V^2\;\Rightarrow\;V=\sqrt{\frac{8g}{f}}\sqrt{RS}$$

Chezy

Chezy's equation is $V=C\sqrt{RS}$. Comparing:

$$\boxed{C=\sqrt{\frac{8g}{f}}}\qquad\left(f=\frac{8g}{C^2}\right)$$

Manning

Manning's equation is $V=\dfrac1nR^{2/3}S^{1/2}=\dfrac{R^{1/6}}{n}\sqrt{RS}$. Comparing with Chezy:

$$\boxed{C=\frac{R^{1/6}}{n}}$$

Combined relation

$$C=\sqrt{\frac{8g}{f}}=\frac{R^{1/6}}{n}\;\Rightarrow\;\boxed{f=\frac{8gn^2}{R^{1/3}}\quad\text{or}\quad n=R^{1/6}\sqrt{\frac{f}{8g}}}$$

For a pipe flowing full, $R=D/4$, so the same relation gives $h_f=\dfrac{fL}{D}\dfrac{V^2}{2g}$.