Darcy-Chezy-Manning link and composite roughness

Part 1: Relationship between Darcy, Chezy and Manning

Consider uniform flow in a channel length $L$ with area $A$ and wetted perimeter $P$, bed slope $S_0$. Force balance along the flow (gravity component = boundary shear):

$$\gamma AL\,S_0=\tau_0PL\;\Rightarrow\;\tau_0=\gamma RS_0$$

• Darcy-Weisbach: shear stress in terms of friction factor $f$: $\tau_0=\dfrac f8\rho V^2$.
• Equating: $\rho gRS_0=\dfrac f8\rho V^2\;\Rightarrow\;V=\sqrt{\dfrac{8g}{f}}\sqrt{RS_0}$.
• Chezy: $V=C\sqrt{RS_0}$, therefore $C=\sqrt{\dfrac{8g}{f}}$.
• Manning: $V=\dfrac1nR^{2/3}S_0^{1/2}=\dfrac{R^{1/6}}{n}\sqrt{RS_0}$, therefore $C=\dfrac{R^{1/6}}{n}$.

Combining:

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

Part 2: Manning's $n$ for a composite (mixed) boundary

When parts of the wetted perimeter have different roughness ($P_1,n_1$; $P_2,n_2$; ... $P_N,n_N$), the section is given an equivalent $n_e$, by assuming a rule about how the parts behave:

1. Horton-Einstein (equal velocity): every part has the same mean velocity as the whole section.

$$n_e=\left[\frac{\sum P_in_i^{3/2}}{P}\right]^{2/3}$$

2. Pavlovskii, Muhlhofer and Einstein-Banks (equal total resistance): the total resisting force is the sum of the resistance on the parts.

$$n_e=\left[\frac{\sum P_in_i^2}{P}\right]^{1/2}$$

3. Lotter (equal discharge sum): total discharge is the sum of the part discharges, each part with the same slope and with its own area and roughness.

$$n_e=\frac{PR^{5/3}}{\sum\dfrac{P_iR_i^{5/3}}{n_i}}$$

4. Cowan's method: when no data is available, $n=(n_0+n_1+n_2+n_3+n_4)m_5$ from the basic roughness, irregularity, variation of section, obstruction and vegetation, with a meandering factor.
5. Compound channels: the section is divided into main channel and flood plains by vertical lines; discharge is the sum of subsection discharges, and then the effective $n$ is found from $n_e=AR^{2/3}S^{1/2}/Q$.

Example: bed $n=0.015$, $P=3.6$ m; walls $n=0.03$, $P=2.4$ m. Horton-Einstein gives $n_e=0.0216$; Pavlovskii gives $0.0222$.