Question 14.1: A trapezoidal channel having a bottom width of 6.1 m and sid......
A trapezoidal channel having a bottom width of 6.1 m and side slopes of 1.5H : 1V is carrying a flow of 126 m³ /s at a flow depth of 5.79 m. The bottom slope is 0.00008, Manning n = 0.013 and the channel length is 5 km. There is a constant-level reservoir at the upstream end of the channel. If a sluice gate at the downstream end is suddenly closed at time t = 0:
1. Compute the transient conditions until t = 2000 s by using:
\quad\quada. Method of specified intervals;
\quad\quadb. Lax diffusive scheme;
\quad\quadc. MacCormack scheme;
\quad\quadd. Preissmann scheme.
2. Plot the computed flow depth in the channel at t = 0, 500, 1000, 1500 and 2000 s.
3. Plot the variation of the flow depth with time at distance of 1.5, 2.5, 3 and 5 km from the reseoir.
A computer program was written based on these finite difference schemes for the interior grid points. In the first four schemes, the positive characteristic equation and the condition that flow velocity at the downstream end is always zero for t > 0 were utilized to simulate the downstream end condition. At the upstream end, the negative characteristic equation was used along with the condition that the flow depth is equal to the reservoir depth. In the Preissmann scheme, the upstream- and downstream-boundary conditions were directly incorporated into the solution. The downstream boundary condition specified the flow velocity at the downstream end to be always zero following the gate closure, and the upstream end condition specified the flow depth to be constant and equal to the reservoir depth at the channel entrance.
\quad\quadThe channel length was divided into 50 equal-length reaches and the computational time interval, Δt, was selected so that the stability condition was always satisfied in the explicit schemes at every grid point. If this was not the case, then the computational time interval was reduced by 20 percent and the flow conditions were recalculated. However, if the time step was considerably smaller than that required by the Courant condition, then the time step for the next interval was increased by 15 percent.
\quad\quadThe computed flow depths in the channel at different times by using these schemes are shown in Fig. 14-6. The variation of flow depth with respect to time at different locations is plotted in Fig. 14-7.
