Question

Water discharging into a 11m wide rectangular channel from a sluice gate is observed to have undergone a hydraulic jump. The flow depth and velocity before the jump are 0.8 m and 7.5 m/s, respectively. Determine a) the flow depth after the hydraulic jump b) the Froude number after the jump c) the energy loss through the jump d) the flow condition upstream and downstream of the jump (subcritical or supercritical

Answer 1

a)
`y_(2) \approx 2.656\,m`, b)
`Fr_(2) \approx 0.442`, c) 20.5 % of initial energy is lost through the jump, d) Before the jump: Supercritical (Fr > 1), After the jump: Subcritical (Fr < 1).

Step-by-step explanation:

a) The Froude number before the jump is:

`Fr_(1) = \frac{v_(1)}{\sqrt{g\cdot y_(1)} }`

`Fr_(1) = \frac{7.5\,(m)/(s) }{\sqrt{(9.807\,(m)/(s^(2)) )\cdot (0.8\,m)} }`

`Fr_(1) \approx 2.678`

The depth after the jump is:

`y_(2) = 0.5\cdot y_(1)\cdot \left(-1 + \sqrt{1 + 8\cdot Fr_(1)^(2)} \right)`

`y_(2) = 0.5\cdot (0.8\,m)\cdot \left[-1 +\sqrt{1 +8\cdot (2.678)^(2)} \right]`

`y_(2) \approx 2.656\,m`

b) The speed after the hydraulic jump is derived from the continuity equation:

`v_(1)\cdot y_(1) = v_(2)\cdot y_(2)`

`v_(2) = (y_(1))/(y_(2))\cdot v_(1)`

`v_(2) = (0.8\,m)/(2.656\,m) \cdot (7.5\,(m)/(s) )`

`v_(2) = 2.259\,(m)/(s)`

The Froude number after the hydraulic jump is:

`Fr_(2) = \frac{v_(2)}{\sqrt{g\cdot y_(2)} }`

`Fr_(2) = \frac{2.259\,(m)/(s) }{\sqrt{(9.807\,(m)/(s^(2)) )\cdot (2.656\,m)} }`

`Fr_(2) \approx 0.442`

c) The head loss is:

`h_(L) = y_(1) - y_(2) + (v_(1)^(2)-v_(2)^(2))/(2\cdot g)`

`h_(L) = 0.8\,m - 2.656\,m + ((7.5\,(m)/(s) )^(2)-(2.259\,(m)/(s))^(2))/(2\cdot (9.807\,(m)/(s^(2)) ))`

`h_(L) = 0.752\,m`

The dissipation ratio is obtained afterwards:

`DR = (h_(L))/(y_(1)\cdot \left(1 + (Fr_(1)^(2))/(2) \right))`

`DR = (0.752\,m)/((0.8\,m)\cdot \left(1 + (2.678^(2))/(2) \right))`

`DR = 0.205`

20.5 % of initial energy is lost through the jump.

d) The flow conditions are described below:

Before the jump: Supercritical (Fr > 1)

After the jump: Subcritical (Fr < 1)