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 5.3m3/s5.3m^{3}/s, and n=0.011. Find yny_{n} and ycy_{c}. If the actual depth of flow is 1.7m, what type of profile exists?
    (Answer: yny_{n} = 0.4m, ycy_{c} = 0.683m, S1 Curve)

  2. E3.2

    A rectangular channel with a bottom width of 4.0m4.0m and a bottom slope of 0.00080.0008 has a discharge of 1.50m3/s1.50m^{3}/s. In a gradually varied flow in this channel, the depth at a certain location is found to be 0.30m0.30m assuming n=0.016n=0.016, determining the type of GVF profile, the critical depth, and the normal depth.
    (Answer: M2, yc=0.24my_{c}=0.24m, yn=0.43my_{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=10m3/sQ=10m^{3}/s, b=3mb=3m, n=0.022n=0.022, and a bed slope of 0.0010.001. Take the depth just upstream of the dam as the control point equal to 5m5m. At what distance is the water level not affected by the dam? perform your integration using a) 2-steps and b) 10-steps
    (Answer: yn=2.437my_{n}=2.437m.
    Using y0y_{0} (Euler) 2-step x=3490mx=-3490m, 10-step x=4724mx=-4724m.
    Using y1/2y_{1/2} 2-step x=4480mx=-4480m, 10-step x=5906mx=-5906m.
    Using a forth-order Runge Kutta method gave 2-step x=3893mx=-3893m, 10-step x=5461mx=-5461m)

  4. E3.4

    A trapezoidal, concrete-lined, channel has a constant bed slope of 0.00150.0015, a bed width of 3m3m, and side slopes of 1:11:1. A control gate increased the depth immediately upstream to 4m4m. When the discharge is 19m3/s19m^{3}/s compute the water surface profile upstream and identify the distance when the water depth is 1.8m1.8m. (n=0.017n=0.017)
    (Answer: Using y0y_{0} (Euler) 2-step x=1540mx=-1540m, 10-step x=1695mx=-1695m.
    Using y1/2y_{1/2} 2-step x=1670mx=-1670m, 10-step x=1809mx=-1809m.
    Using mean GVF function: 2-step x=2729mx=-2729m, 10-step x=1933mx=-1933m.
    Using a forth-order Runge Kutta method gave 2-step x=2023mx=-2023m, 10-step x=1850mx=-1850m)

  5. E3.5

    Using the figure below, determine the profile for the channel conditions using a step length of Δx=100m\Delta x=100m. Q=600m3/sQ=600m^{3}/s, n=0.04n=0.04, the bed slope of the rectangular channel is So=0.002S_{o}=0.002 and has a width of B=50mB=50m. The sill height of the weir is 2.5m2.5m and the water depth over the weir is 4m4m. Compare results from each method.

    Refer to caption
    Figure 4: The backwater curve for question E3.5