E1 Questions on Uniform Flow and Critical Flow. - Calculation of normal and critical depth.

  1. E1.1

    A rectangular channel is 3.0m3.0m wide, has a 0.01 slope, flow rate of 5.3m3/s5.3m^{3}/s, and n=0.011n=0.011. Find its normal depth yny_{n} and critical depth ycy_{c}.

    (Answer: yn=0.41my_{n}=0.41m, yc=0.683my_{c}=0.683m)

    Refer to caption
    Figure 1: A rectangular channel section
  2. E1.2

    Water flows in a long rectangular channel at a depth of 1.22m1.22m and discharge of Q=5.66m3/sQ=5.66m^{3}/s. Determine the minimum channel width if the channel is to be subcritical.

    (Answer: 1.34m1.34m)

  3. E1.3

    A rectangular channel has a bottom width of B=8mB=8m and Manning’s n=0.025n=0.025

    1. (a)

      Determine the slope to give a normal depth of yn=2my_{n}=2m when the discharge is 12m3/s12m^{3}/s

    2. (b)

      Determine the critical slope and the critical depth when the discharge is 12m3/s12m^{3}/s

    3. (c)

      Determine the critical slope to give a the critical depth of yc=1.5my_{c}=1.5m and compute the corresponding discharge.

    (Answer:(a) So=0.00024S_{o}=0.00024, (b) Soc=0.0087So_{c}=0.0087, yc=0.61my_{c}=0.61m, (c) Soc=46.03m3/sSo_{c}=46.03m^{3}/s, Q=0.00818Q=0.00818)

  4. E1.4

    For a trapezoidal channel with a base width b=3.0mb=3.0m, Manning’s n=0.025n=0.025 and side slope s=2s=2 (i.e. 1 vertical: 2 horizontal), calculate the critical depth, critical velocity, and critical slope if its discharge Q=10m3/sQ=10m^{3}/s.

    (Answer:yc=0.855my_{c}=0.855m, vc=2.483m/v_{c}=2.483m/s, Soc=0.00777So_{c}=0.00777)

  5. E1.5

    A rectangular channel 9m9m wide carries 7.6m3/s7.6m^{3}/s of water when flowing 1.0m1.0m deep. Work out the flow’s specific energy. Is the flow sub-critical or super-critical?

    (Answer: 1.0361.036m and flow is sub-critical)

  6. E1.6

    Two engineers observed two rivers and recorded the following flow parameters: River 1: flow discharge Q=130m3/sQ=130m^{3}/s, flow velocity V=1.6m/sV=1.6m/s, water surface width B=80mB=80m; River 2: flow discharge Q=1530m3/sQ=1530m^{3}/s, flow velocity V=5.6m/sV=5.6m/s, water surface width B=90mB=90m. Decide the flow regime of two rivers, i.e. sub-critical or super-critical.

    (Answer: River 1 is sub-critical and River 2 is super-critical)

  7. E1.7

    A concrete, trapezoidal channel has a bottom slope of So=0.0009S_{o}=0.0009 and a Manning roughness factor of n=0.013n=0.013. The bottom width of the channel is b=2.5mb=2.5m, and the side slopes are 1 in 2. Determine the velocity and discharge when the flow is normal at a depth of 1.8m1.8m.
    (Answer: v=2.37m/sv=2.37m/s, Q=26.01m3/sQ=26.01m^{3}/s)

  8. E1.8

    A trapezoidal channel has a bottom slope of So=1S_{o}=1 in 4040 and a Manning roughness factor of n=0.016n=0.016. The bottom width of the channel is b=6.0mb=6.0m, and the side slopes are 1 in 3. Determine the normal depth in this channel for Q=42.3m3/sQ=42.3m^{3}/s.
    (Answer: yn=0.75my_{n}=0.75m).

  9. E1.9

    The flow discharge in uniform flow in a rectangular channel 4.6m4.6m wide is 11.3m3/s11.3m^{3}/s when the slope is 1:100. Is the flow sub-critical or super-critical? Calculate the slope, ScS_{c}, that would give critical depth. The Manning roughness coefficient is n=0.012n=0.012.
    (Answer: super-critical, Fr=2.1Fr=2.1, Sc=0.002268S_{c}=0.002268).

    Compound Channels

  10. E1.10

    The cross-section of a stream can be approximated by the compound channel shown in figure 2. The bottom slope is So=0.0009S_{o}=0.0009. The Manning roughness factor n=0.025n=0.025 for the main channel and n=0.035n=0.035 for the overbank areas. Determine the normal depth for a discharge of 197m3/s197m^{3}/s. Also, calculate the energy coefficient α\alpha and the momentum coefficient β\beta for the channel with this flow condition.

    Refer to caption
    Figure 2: A Compound section

    (Answer: yn=5.507my_{n}=5.507m, α=1.23\alpha=1.23, β=1.09\beta=1.09.)

  11. E1.11

    The total width of the channel considered in Question E1.10 is to be decreased by reducing the overbank portions symmetrically; however, this reduction must not cause an increase of more than 0.15m0.15m in the flow depth for the discharge of 197m3/s197m^{3}/s. Assuming normal depth still is present in the channel, determine the minimum allowable channel total width, BB.
    (Answer: B=16.157mB=16.157m.)

  12. E1.12

    The cross-section of a river with flood plains flowing in uniform flow may be idealized as shown in Fig. 3. Determine the discharge carried by the river when its dimensions and roughness parameters are:

    Bed slope: So=2×104S_{o}=2\times 10^{-4}
    Manning’s ns: n1=n2=n3=0.02n_{1}=n_{2}=n_{3}=0.02
    Side slopes: s1=s2=s3=1s_{1}=s_{2}=s_{3}=1
    Bed widths: B1=3mB_{1}=3m, B2=5mB_{2}=5m, B3=4mB_{3}=4m
    Main channel depth: ymain=3.0my_{main}=3.0m


    Normal depth yn=4.5my_{n}=4.5m
    (Answer: Q=69.42m3/sQ=69.42m^{3}/s.)

    Refer to caption
    Figure 3: Idealized river channel with flood plains
  13. E1.13

    For the channel of question E1.12 calculate the flow, if all dimensions, including the normal depth were the same, but the slope of the channel is 0.002.
    (Answer Q=219.53m3/sQ=219.53m^{3}/s).

    Efficient Channels

  14. E1.14

    A trapezoidal channel has side slopes of 1:3/4 and the slope of the bed is 1 in 2000. Determine the optimum dimensions of the channel if it is to carry water at 0.5m3/s0.5m^{3}/s. Use the Chezy formula, assuming that C=80m1/2/sC=80m^{1/2}/s.
    (Answer: yn=0.552my_{n}=0.552m, b=0.552mb=0.552m).

  15. E1.15

    An open channel with n=0.011n=0.011 is to be designed to carry 1.0m3/s1.0m^{3}/s of water at a slope of 0.0065. Find the most efficient cross-section for a rectangular section.
    (Answer: b=2y=0.869mb=2y=0.869m).

  16. E1.16

    A rectangular channel has width B=3mB=3m and normal depth y=1my=1m. What is the diameter of a semicircular channel that will have the same discharge as in the rectangular channel, when flowing just full in uniform flow? Assume that nn and SoS_{o} are the same in the two cases. Compare the two wetted perimeters.
    (Answer: D=2.057mD=2.057m, Prectangular=5.0mP_{rectangular}=5.0m, Pcircular=6.463mP_{circular}=6.463m)

  17. E1.17

    What are the dimensions of the most efficient rectangular channel section to carry 5m3/s5m^{3}/s at a slope of 1 in 900. The surface of the channel is of concrete.
    (Answer: y=1.21my=1.21m, b=2y=2.42mb=2y=2.42m using n=0.012n=0.012)

  18. E1.18

    What is the most efficient depth for a brick channel of a trapezoidal section with sides sloping at 4545^{\circ} to the horizontal to carry 3m3/s3m^{3}/s. The bed slope is 0.0009.
    (Answer: y=1.104my=1.104m, using n=0.015n=0.015)