Question

At a hydraulic jump, the flow depths are 0.4 m and 5 m at the upstream and downstream, respectively. The channel is wide rectangular. The discharge per unit width is nearly
1. 5.8 m²/s
2. 6.4 m²/s
3. 7.3 m²/s
4. 8.3 m²/s

Answer 1

Correct Answer - Option 3 : 7.3 m²/s

Concept:

Hydraulic Jump: It is a phenomena associated with sudden rise in water depth when a flow with high velocity and low depth (supercritical flow) strikes another flow with low velocity and high depth (subcritical flow).

For any hydraulic jump, the following relation always holds good:

\({{\rm{y}}_1}{{\rm{y}}_2}\left( {{{\rm{y}}_1} + {{\rm{y}}_2}} \right) = \frac{{2{{\rm{q}}^2}}}{{\rm{g}}}\)

Where, y₁ and y₂ are the depths before and after the hydraulic jump respectively.

q is discharge per unit width and g is acceleration due to gravity.

Calculation:

Given, upstream depth which is depth before hydraulic jump = y₁ = 0.4 m.

Downstream depth which is depth after hydraulic jump = y₂ = 5 m.

Acceleration due to gravity = g = 9.81 m/s².

\(\therefore {{\rm{q}}^2} = \frac{{{{\rm{y}}_1}{{\rm{y}}_2}\left( {{{\rm{y}}_1} + {{\rm{y}}_2}} \right) \times {\rm{g}}}}{2} = \frac{{0.4 \times 5 \times \left( {0.4 + 5} \right) \times 9.81}}{2} = 52.974{\rm{\;}}\therefore {\rm{q}} = 7.278\frac{{{{\rm{m}}^2}}}{{\rm{s}}} \approx 7.3{\rm{\;}}\frac{{{{\rm{m}}^2}}}{{\rm{s}}}{\rm{\;}}\)

∴ The discharge per unit width is nearly 7.3 m²/s.