E3 Questions on Gradually varied flow - including integration of the backwater curve.

1. E3.1

A Rectangular channel is 3.0m wide, has a 0.01 slope, discharge of $5.3m^{3}/s$, and n=0.011. Find $y_{n}$ and $y_{c}$. If the actual depth of flow is 1.7m, what type of profile exists?
(Answer: $y_{n}$ = 0.4m, $y_{c}$ = 0.683m, S1 Curve)

2. E3.2

A rectangular channel with a bottom width of $4.0m$ and a bottom slope of $0.0008$ has a discharge of $1.50m^{3}/s$. In a gradually varied flow in this channel, the depth at a certain location is found to be $0.30m$ assuming $n=0.016$, determining the type of GVF profile, the critical depth, and the normal depth.
(Answer: M2, $y_{c}=0.24m$, $y_{n}=0.43m$)

3. E3.3

The figure below shows a backwater curve in a long rectangular channel. Determine using a numerical integration method, the profile for the following high flow conditions: $Q=10m^{3}/s$, $b=3m$, $n=0.022$, and a bed slope of $0.001$. Take the depth just upstream of the dam as the control point equal to $5m$. At what distance is the water level not affected by the dam? perform your integration using a) 2-steps and b) 10-steps
(Answer: $y_{n}=2.437m$.
Using $y_{0}$ (Euler) 2-step $x=-3490m$, 10-step $x=-4724m$.
Using $y_{1/2}$ 2-step $x=-4480m$, 10-step $x=-5906m$.
Using a forth-order Runge Kutta method gave 2-step $x=-3893m$, 10-step $x=-5461m$)

4. E3.4

A trapezoidal, concrete-lined, channel has a constant bed slope of $0.0015$, a bed width of $3m$, and side slopes of $1:1$. A control gate increased the depth immediately upstream to $4m$. When the discharge is $19m^{3}/s$ compute the water surface profile upstream and identify the distance when the water depth is $1.8m$. ($n=0.017$)
(Answer: Using $y_{0}$ (Euler) 2-step $x=-1540m$, 10-step $x=-1695m$.
Using $y_{1/2}$ 2-step $x=-1670m$, 10-step $x=-1809m$.
Using mean GVF function: 2-step $x=-2729m$, 10-step $x=-1933m$.
Using a forth-order Runge Kutta method gave 2-step $x=-2023m$, 10-step $x=-1850m$)

5. E3.5

Using the figure below, determine the profile for the channel conditions using a step length of $\Delta x=100m$. $Q=600m^{3}/s$, $n=0.04$, the bed slope of the rectangular channel is $S_{o}=0.002$ and has a width of $B=50m$. The sill height of the weir is $2.5m$ and the water depth over the weir is $4m$. Compare results from each method.