Gradually varied flow
1 Equation of gradually varied flow
See Figure 5.1.
\begin{equation} H=z+y+\alpha\frac{V^2}{2g} \end{equation}
\begin{equation} \frac{dH}{dx}=-S_e=-S_0+\frac{dE}{dx} \end{equation}
\begin{equation} \frac{dE}{dx}=\frac{dE}{dy}\frac{dy}{dx}=\left(1-\mathbf{F}^2\right)\frac{dy}{dx}=S_0-S_e \end{equation}
\begin{equation} \frac{dy}{dx}=\frac{S_0-S_e}{1-\mathbf{F}^2} \end{equation}
2 Local uniform flow assumption
Assuming that the local value of $S_e$ can be calculated from Manning’s equation using the local value of $y$, \begin{equation} Q=\frac{K_n}{n}AR^{2/3}S_e^{1/2}=\frac{K_n}{n}A_0R_0^{2/3}S_0^{1/2}. \end{equation}
For a wide rectangular channel ($b\gg y$ and $R=by/(b+2y)\approx y$), \begin{equation} Q=\frac{K_n}{n}by^{5/3}S_e^{1/2}=\frac{K_n}{n}by_0^{5/3}S_0^{1/2}. \end{equation}
If $y<y_0$, $S_e>S_0$. Otherwise, $S_e<S_0$.
3 Classification of water surface profiles
See Figure 5.2.
Mild slopes
- M1: $y_c<y_0<y$, $S_e<S_0$, $\mathbf{F}^2<1$, $\frac{dy}{dx}>0$
- M2: $y_c<y<y_0$, $S_e>S_0$, $\mathbf{F}^2<1$, $\frac{dy}{dx}<0$
- M3: $y<y_c<y_0$, $S_e>S_0$, $\mathbf{F}^2>1$, $\frac{dy}{dx}>0$
Steep slopes
- S1: $y_0<y_c<y$, $S_e<S_0$, $\mathbf{F}^2<1$, $\frac{dy}{dx}>0$
- S2: $y_0<y<y_c$, $S_e<S_0$, $\mathbf{F}^2>1$, $\frac{dy}{dx}<0$
- S3: $y<y_0<y_c$, $S_e>S_0$, $\mathbf{F}^2>1$, $\frac{dy}{dx}>0$
Critical slopes
- C1: $y_0=y_c<y$: $S_e<S_0$, $\mathbf{F}^2<1$, $\frac{dy}{dx}>0$
- C2: $y<y_0=y_c$: $S_e>S_0$, $\mathbf{F}^2>1$, $\frac{dy}{dx}>0$
4 Lake discharge problem
Example 5.1