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Flow over spillway

  1. Jan 17, 2017 #1
    1. The problem statement, all variables and given/known data

    When the level of water in a reservoir is too high, the water spills out over a spillway,
    as illustrated in the figure below

    jxI5TK0.jpg

    Neglecting viscosity, show that the water flow ##Q## over the spillway is given by
    $$Q=\frac{2}{3}w\sqrt{2gy^3}$$

    2. Relevant equations

    Bernoulli's Equation: ##P_1+\rho gh_1+\frac{1}{2}\rho v_1^2=P_2+\rho gh_2+\frac{1}{2}\rho v_2^2##
    Torricelli's Equation: ##v=\sqrt{2gh}##

    3. The attempt at a solution

    ##Q=Av=(wy)(\sqrt{2gy})=w\sqrt{2gy^3}##

    I am not sure how to get ##\frac{2}{3}## in this equation as indicated by the answer. I feel that I am oversimplifying this by assuming the height in Torricelli's equation is just ##y## when it should account for flow over an area as opposed to a point. Would this mean the height is ##\frac{y}{2}## to get the average velocity of the fluid?
     
  2. jcsd
  3. Jan 17, 2017 #2

    haruspex

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    Yes. The consequence for the total flow varies as the height above the spillway increases from0 to y.
    No, that's still too simplistic. Consider a thin horizontal slice from x to x+dx above the spillway. What is the flow due to that?
     
  4. Jan 17, 2017 #3
    Oh, that makes sense.

    ##Q=Av=(wy)\int_0^y (\sqrt{2gy})=(wy\sqrt{2g})\int_0^y \sqrt{y}=(wy\sqrt{2g})(\frac{\sqrt{y^3}}{\frac{3}{2}})=\frac{2}{3}w\sqrt{2gy^3}##

    Thank you!
     
  5. Jan 17, 2017 #4

    haruspex

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    Good job.
     
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