Assume the gate in the previous problem has been operating for a long time in which it produces a depth of 3.5 ft downstream from it, and now suddenly the gate is lowered to decrease the flow rate past it, so it produces a depth of 2 ft downstream from it. Solve the resulting flows based on the

Question

Assume the gate in the previous problem has been operating for a long time in which it produces a depth of 3.5 ft downstream from it, and now suddenly the gate is lowered to decrease the flow rate past it, so it produces a depth of 2 ft downstream from it. Solve the resulting flows based on the assumptions that allow the characteristic method to be used to solve the unsteady flow.

Accepted Answer

With the gate instantly lowered, both an upstream and a downstream wave will be created and shown in the sketch. Typically, as in this problem, the new depth downstream from the gate as well as the depth [latex]Y_{o1}[/latex] and the velocity [latex]V_{o1}[/latex], and the downstream depth [latex]Y_{o1}[/latex] and the velocity [latex]V_{o1}[/latex]are known, and the seven unknown variables are: [latex]Y_{1}, V_{1}, V_{2}, Y_{3}, V_{3}[/latex], [latex]w_{u}[/latex], and [latex]w_{d}[/latex]. The seven equations that allow for these to be solved are

1. Energy across the gate (stationary observer)

[latex]Y_{1}+\frac{V^{2}_{1}}{2g}=Y_{2}+\frac{V^{2}_{2}}{2g}[/latex]

2. Continuity across the gate (stationary observer)

[latex]V_{1}Y_{1}=V_{2}Y_{2}[/latex]

3. Continuity across the upstream wave (moving observer)

[latex]\left(w_{u}+V_{o1}\right)Y_{o1}=\left(w_{u}+V_{1}\right)Y_{1}[/latex]

4. Momentum across the upstream wave (moving observer)

[latex]\frac{Y^{2}_{o1}}{2}+\frac{\left(w_{u}+V_{o1}\right)^{2}Y_{o1}}{g}= \frac{Y^{2}_{1}}{2}+\frac{\left(w_{u}+V_{1}\right)^{2}Y_{1}}{g}[/latex]

5. Continuity across the downstream wave (moving observer)

[latex]\left(V_{2}-w_{d}\right)Y_{2}=\left(V_{3}-w_{d}\right)Y_{3}[/latex]

6. Momentum across the downstream wave (moving observer)

[latex]\frac{Y^{2}_{2}}{2}+\frac{\left(V_{2}-w_{d}\right)^{2}Y_{2}}{g}= \frac{Y^{2}_{3}}{2}+\frac{\left(V_{3}-w_{d}\right)^{2}Y_{3}}{g}[/latex]

7. Characteristic equation for unsteady flow

[latex]V_{3}-2c_{3}=V_{o2}-2c_{o2} or V_{3}-2 \left(gY_{3}\right)^{{1}/{2}}=V_{o2}-2\left(gY_{o2}\right)^{{1}/{2}}[/latex]

A TK-Solver model is given below. An alternative is to solve the upstream and the downstream portions of the problem separately. In so doing, the first four equations would be solved for the first four unknowns for the variables upstream of the gate including [latex]V_{2}[/latex], and then the last three equations for the last three unknowns.

Solving all 7 equations, model GATEDW.TK (see GATEDW1.TK to solve first 4 equations followed by

Question 6.22: Assume the gate in the previous problem has been operating f...

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