Question 6.1: A trapezoidal channel having a bottom slope of 0.001 is carr......

The normal depth, y_{n}, for this channel was computed in Example 4-1 as 1.16 m. The flow depth approaches the normal depth asymptotically at an infinite distance. Therefore, the computation of the surface profile may be stopped when the flow depth is within about five percent of the normal depth. We will continue the calculations in this example until y = 1.05y_{n} = 1.05 × 1.16 = 1.21, say 1.20 m.
\quad\quadWe start the computations with a known depth of 5.0 m at the control structure and proceed in the upstream direction. Let us call the location at the control structure as x = 0. Since we are considering the distance in the downstream flow direction as positive, the values of x we determine from Eq. 6-17 are negative.
\quad\quad\quad x_{2}=x_{1}+\frac{E_{2}-E_{1}}{S_{o}-\frac{1}{2}(S_{f1}+S_{f2})} (6-17)
The calculations are done in a systematic manner, as shown in Table 6-1.
The following explanatory remarks should be helpful to understand these calculations. In this discussion, the depth for the step under consideration is the current depth and the depth for the previous step as the previous depth.

Column 1, y

We first use large increments of change in y, i.e., 0.5 m and then decrease their size, i.e., 0.1 m, as the rate of variation of y with x becomes small.

Column 2, A

This is the flow area for the depth of column 1.

Column 3, R

Hydraulic radius, R = A/P, where P = wetted perimeter for the flow depth of column 1.

Column 4, V

Flow velocity, V is computed by dividing the specified rate of discharge, Q, by the flow area, A, of column 2.

Column 5, S_{f}

By using the specified value of Manning n, and the computed values of V of column 4 and R of column 3, this column is computed from the equation, S_{f} = n²V²/(C_{o}^{2}R^{1.33}).

Column 6, \overline{S}_{f}

This is the average of S_{f} for the current depth and for the depth in previous step. This column is left blank for the first line since there is no previous depth when we start the computations. To indicate that this is an average slope, we list it between the lines corresponding to the current and the previous depths.

Column 7, S_{o}-\overline{S}_{f}

This is obtained by subtracting \overline{S}_{f} of column 6 from the specified value of S_{o}.

Column 8, E

The specific energy, E, is computed for the selected value of y of column 1 and corresponding computed value of V of column 4, i.e., E = y + αV²/(2g).

Column 9, ΔE = E_{2}  –  E_{1}

This column is obtained by subtracting E for the current depth from E for the previous depth. Again, since this column is the difference of E values corresponding to the current and the previous depths, we list its value between the lines for these depths.

Column 10, Δx = x_{2}  –  x_{1}

The distance increment is computed from the equation, Δx = (E_{2} – E_{1})/(S_{o}- \overline{S}_{f}), i.e., dividing column 9 by column 7.

Column 11, x_{2}

This is the distance where depth y will occur. It is obtained by algebraically adding Δx of column 10 to the x_{2} value for the previous depth.