A discharge of 7 m³/(s m) width flows in a rectangular channel. A vertical sluice gate situated in the channel has an opening of 1.5 m. Cc = 0.62; Cv = 0.95. Assuming that downstream conditions permit free flow under the gate, draw the streamline pattern.

Question

A discharge of 7 m³/(s m) width flows in a rectangular channel. A vertical sluice gate situated in the channel has an opening of 1.5 m. [latex]C_{c } [/latex] = 0.62; [latex]C_{v } [/latex] = 0.95. Assuming that downstream conditions permit free flow under the gate, draw the streamline pattern.

Accepted Answer

See Figure 10.11.

[latex]y_{2} = C_{c } imes y_{0} = 0.62 × 1.5[/latex] = 0.93 m

[latex]y_{1} + \frac{V^{2}_{1} }{2g} = y_{2} + \frac{V^{2}_{2} }{2g} + b_{L}[/latex] (i)

and [latex]b_{L} = \left[\left(\frac{1}{C_{v }} ight)^{2} − 1 ight] \frac{V^{2}_{2} }{2g}[/latex]; [latex]V_{2} = \frac{7}{0.93}[/latex] = 7.527 m ∕ s

[latex]b_{L}[/latex] = 0.31 m

whence from Equation (i), y = 3.97 m (say 4.0 m).
Divide the field into a 1.0 m square grid (see Figure 10.11) and allocate boundary values and interior grid values of Ψ.
Numbering rows and columns from the top left-hand corner, note that at point (4, 7) two of the adjacent grid points do not coincide with the boundary. Using the notation of Figure 10.8, [latex]\lambda _{2}[/latex] = 0.6 and [latex]\lambda _{3}[/latex] = 0.5.
The results of successive corrections using Equation 10.17 in the form

[latex]\Psi_{0} = \frac{\Psi _{1} + (\Psi_{2}/\lambda _{2}) + \Psi _{3} + \Psi _{4 }}{3 + (1/\lambda _{2})}[/latex] [10.17]

[latex]\Psi_{0} = \frac{\Psi _{1} + \Psi _{2} + \Psi _{3} + \Psi _{4 }}{4}[/latex]

at all interior grid points with the exception of point (4, 7) where Equation 10.17

[latex]\Psi_{0} = \frac{\Psi _{1} + (\Psi _{2}/\lambda _{2}) + (\Psi _{3}/\lambda _{3}) + \Psi _{4 }}{2 + (1/\lambda _{2}) + (1/\lambda _{3})} [/latex]

is used are shown in the following table:

The above is probably sufficiently accurate for most purposes, but computations can be continued if required.

The discrete streamline can now be drawn (see Figure 10.12).

Question 10.2: A discharge of 7 m³/(s m) width flows in a rectangular chann...

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