Manning's equation, terms and slope S

Manning's equation

Manning's equation is an empirical formula for mean velocity in open channels, for uniform (turbulent, rough) flow:

$$V=\frac{1}{n}R^{2/3}S^{1/2}\qquad\Rightarrow\qquad Q=\frac{1}{n}AR^{2/3}S^{1/2}\quad(\text{SI units})$$

It is related to Chezy's formula $V=C\sqrt{RS}$ by $C=R^{1/6}/n$.

The slope $S$ and the energy diagram

$S$ is the energy slope $S_f$, the loss of total energy head per unit length:

$$S=S_f=-\frac{dH}{dx},\qquad H=z+y+\frac{\alpha V^2}{2g}$$

On the energy diagram it is the slope of the energy grade line (EGL), measured relative to the horizontal. It is not necessarily the bed slope.

 EGL -----\
 WSL  ------\  \  S = slope of EGL (S_f)
 Bed  -------\---\--

For uniform flow, $S_f=S_w=S_0$, since depth and velocity do not change, so the EGL, water surface and bed are parallel. Then $S$ may be taken as the bed slope. For non-uniform flow, $S_f$ is used at each section (the local friction slope), found from the Manning equation: $S_f=\dfrac{n^2V^2}{R^{4/3}}$.

Why uniform flow is assumed in most applications

1. Most canals, flumes and drains are prismatic with constant slope and roughness, and long reaches reach equilibrium (normal depth).
2. The bed slope can be measured, but the energy slope cannot be measured directly. With uniform flow, $S_f=S_0$ so only the bed slope is needed.
3. In gradually varied flow, depth changes slowly, so the uniform-flow formula is valid for local friction loss at each section.
4. It gives a simple design method: for a given $Q$, $n$ and $S_0$, find the normal depth.
5. Manning's equation was itself derived from data of (nearly) uniform flow.

Hence, Manning's equation gives acceptable accuracy when the flow is turbulent, steady and nearly uniform.