Uniform flow

1   What is uniform flow?

In uniform flow, the flow depth is the same at any section in the channel. It has no spatial variations in the flow depth.

2   Normal flow

Steady uniform flow is referred to as normal flow. Normal flow is a special case of uniform flow that is also temporally invariant.

In open channel design, we’re concerned about the determination of normal flow and normal depth.

Hydraulic resistance or surface roughness plays an important role in normal flow.

3   Momentum analysis

See Figure 4.1.

From $\sum\vec{F}$, \begin{equation} \gamma A\Delta L\sin\theta=\tau_0 P\Delta L \end{equation}

\begin{equation} \tau_0 =\gamma\frac{A}{P}\sin\theta=\gamma R\sin\theta\approx\gamma RS \end{equation} where $S=\tan\theta\approx\sin\theta$ for small values of $\theta$.

Considering $h_f=L\sin\theta\approx L\tan\theta=LS$ is the energy loss along the channel length $L$ and $\tau_0=\frac{f\rho V^2}{8}$ in pipe flow, \begin{equation} S=\frac{h_f}{L}=\frac{\tau_0}{\gamma R}=\frac{f\rho V^2/8}{\gamma R}=\frac{f}{4R}\frac{V^2}{2g} \label{eq:S} \end{equation} and we can see $4R$ replaces the pipe diameter in the Darcy-Weisbach equation $\frac{\Delta P}{L}=\frac{f}{D}\frac{\rho V^2}{2}$.

4   Chezy’s formula

Solving Eq. (\ref{eq:S}) for $V$ yields \begin{equation} V=\left(\frac{8g}{f}\right)^{1/2}\sqrt{RS}=C\sqrt{RS} \end{equation} where $C$ is the Chezy $C$.

5   Manning’s formula

Manning experimentally found \begin{equation} V=C_1R^{2/3}S^{1/2}=\frac{K_n}{n}R^{2/3}S^{1/2} \end{equation} where $n$ is Manning’s roughness coefficient and $K_n$ is 1 for $R$ in m and $V$ in m/s (SI units), and 1.49 for $R$ in ft and $V$ in ft/s (English units).

Note that the dimension of $\frac{K_n}{n}$ must be $L^{1/3}T^{-1}$, so for a value of $n$, $\text{m}^{1/3}\text{s}^{-1}=(3.28\text{ ft})^{1/3}\text{s}^{-1}=1.49\text{ ft}^{1/3}\text{s}^{-1}$, and hence $K_n=1.49$ for English units.

6   Exercise 4.1

Determine the normal depth and critical depth in a trapezoidal channel with a bottom width of 40 ft, side slopes of 3:1, and a bed slope of 0.002 ft/ft. The Manning’s n value is 0.025 and the discharge is 3000 cfs. Is the slope steep or mild? Repeat for n=0.012. Did the critical depth change? Why or why not?

7   Homework: Exercise 4.2

Please submit your full work.

8   Uniform flow computations

• Example 4.1
• Example 4.2
• Exercise 4.3

9   Slope classification

From Manning’s equation, we can obtain \begin{equation} S=\frac{n^2Q^2}{K_n^2A^2R^{4/3}}. \label{eq:S2} \end{equation}

For the rectangular channel, Eq. (\ref{eq:S2}) becomes \begin{equation} S=\frac{n^2Q^2}{K_n^2(by)^2R^{4/3}} =\frac{n^2q^2}{K_n^2y^2R^{4/3}} \label{eq:S3} \end{equation} and the Froude number is \begin{equation} \mathbf{F}^2=\frac{V^2}{gy} =\frac{Q^2}{gy(by)^2} =\frac{q^2}{gy^3}. \label{eq:F2} \end{equation}

Combining Eqs. (\ref{eq:S3}) and (\ref{eq:F2}) yields \begin{equation} S=\frac{n^2\mathbf{F}^2gyA^2}{K_n^2A^2R^{4/3}}=\mathbf{F}^2\frac{n^2gy}{K_n^2R^{4/3}} \label{eq:S4} \end{equation}

10   Flood control channels

Flow becomes unstable near the critical depth, so we want to limit the design flow depth such that it is outside the range of 0.9 to 1.1 times the critical depth.

A freeboard is recommended to allow for uncertainties in channel designs:

• Concrete-lined rectangular channels: 2 ft (0.6 m)
• Concrete-lined trapezoidal channels: 2.5 ft (0.8 m)
• Riprap-lined channels: 2.5 ft (0.8 m)
• Earthen channels: 3 ft (0.9 m)