A hydraulic engineer wants to analyze steady flow in a rectangular channel featuring a hydraulic yump immediately downstream from a sluice gate that is open to a vertical clearance of \(3 \mathrm{ft}\) The flow depth upstream from the sluice gate \(8.7 \mathrm{ft}\), and the flow velocity beyond the sluice gate and prior to the hydraulic jump is \(21.5 \mathrm{ft} / \mathrm{s}\). Assume that the flow upstrean from the sluice gate is subcritical. Find: (a) The discharge in the channel; (b) The flow depth before and after the hydraulic jump; (c) The flow velocities upstream from the sluice gate and beyond the hydraulic jump: (d) The energy loss rate in the hydraulic jump: (e) The force the sluice gate exerts on the fluid. How does this compare with the force computed assuming a hydrostatic pressure distribution?

Step by step solution

01

Discharge calculation

The discharge \(Q\) can be calculated using the formula \(Q = A \cdot V\), where \(A\) is the area of the flow and \(V\) is the velocity. The area before the jump can be assumed to be the clearance height multiplied by the width of the channel, and the velocity is given as \(21.5 \, ft/s\). Let's substitute these values in to calculate \(Q\).

02

Flow depth calculation

The flow depths before and after the hydraulic jump can be calculated using the energy equation \(H = h + V^2/2g\) where \(H\) is the total energy head, \(h\) is the elevation head (the flow depth), \(V\) is the flow velocity, and \(g\) is the acceleration due to gravity. We know the total energy head both before and after the sluice gate, so we can calculate the flow depths.

03

Flow velocity calculation

The flow velocity before the hydraulic jump can be calculated using the continuity equation \(Q = A \cdot V\), where \(Q\) is the discharge, \(A\) is the area of the flow, and \(V\) is the velocity. After the hydraulic jump, the velocity can be calculated using the energy equation as we did for the flow depth.

04

Energy loss calculation

The energy loss \(\Delta E\) in the hydraulic jump can be calculated using the equation \(\Delta E = H - h - V^2/2g\), where \(H\) is the total energy head before the jump, \(h\) is the elevation head (flow depth) after the jump, \(V\) is the flow speed after the jump, and \(g\) is the acceleration due to gravity.

05

Force calculation

The force \(F\) that the sluice gate exerts on the fluid can be calculated using the equation \(F = \rho \cdot g \cdot Q \cdot (V_1 - V_0)\), where \(\rho\) is the fluid density, \(g\) is the acceleration due to gravity, \(Q\) is the discharge, \(V_1\) is the flow velocity after the gate, and \(V_0\) is the flow velocity before the gate. The pressure distribution can be assumed to be hydrostatic, so the force can be computed using the equation \(F = p \cdot A\), where \(p\) is the pressure and \(A\) is the area of the gate.

06

Comparison of the forces

By comparing the forces calculated in step 5 using the dynamic and hydrostatic pressure distributions, differences and similarities can be identified.

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