Question 12.14: Water flows down a dam spillway and then forms a hydraulic j......

Fluid Mechanics [1012629]

Water flows down a dam spillway and then forms a hydraulic jump, Fig. 12–31. Just before the jump, the velocity is 8 m/s, and the water depth is 0.5 m. Determine the velocity downstream in the channel.

fig 12-31

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Fluid Description. Steady uniform flow occurs before and after the jump. The water is considered incompressible, and average velocity profiles will be used.
Analysis. The Froude number for the flow before the jump is

Fr_1  =  \frac{V_1}{\sqrt{gy_1}}  =  \frac{8  m/s}{\sqrt{(9.81  m/s^2)(0.5  m)}}  =  3.6122  >  1

Thus the flow is supercritical, as expected. After the jump the depth of the water is

\frac{y_2}{y_1}  =  \frac{1}{2} (\sqrt{1  +  8Fr_1^2}  –  1)

\frac{y_2}{0.5  m}  =  \frac{1}{2} (\sqrt{1  +  8(3.6122)^2}  –  1)

$y_2$ = 2.316 m
We can now obtain the velocity $V_2$ by applying the continuity equation, Eq. 12–27.

$V_1y_1 = V_2 y_2$
(8 m/s)(0.5 m) = $V_2$ (2.316 m)
$V_2$ = 1.727 m/s = 1.73 m/s

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