Question 15.9: Find the critical depth and critical flow velocity in a 20 ft...

The geometry of the flow channel is shown in Figure 15.32. Using the formula for flow area given in Figure 15.6 for a trapezoidal channel, A(y) = y(w + y cot θ), and evaluating this formula for 45°, we have A(y) = y(w + y). We also know dA/dy = b, thus we find b = w + 2y. At the critical point the depth satisfies −(Q²/gA³)(dA/dy) + 1 = 0, so y_C is determined by solving

\frac{Q^2(w+2y_C)}{g[y_C(w+y_C)]^3 }=1                                (A)

Generally speaking, this equation will need to be solved iteratively or with a symbolic math code. To demonstrate the iterative approach, first note that (A) can be written as

\frac{Q^2(w+2y_C)}{g[y_C(w+y_C)]^3 }=\frac{Q^2w\left(1+2\frac{y_C}{w} \right) }{g\left[y_Cw\left(1+\frac{y_C}{w} \right)\right]^3}=1

Rearranging, and solving for y^3_C we have

y^3_C=\frac{Q^2\left(1+2\frac{y_C}{w} \right) }{gw^2\left(1+\frac{y_C}{w} \right)^3}=\frac{Q^2}{gw^2}\left(1+2\frac{y_C}{w} \right)\left(1+\frac{y_C}{w} \right)^{-3}                                  (B)

For y_C \ll w, this equation can be approximated as y^3_C ≈ Q²/gw². Thus we can obtain a first estimate for y_C by writing

y_{C,1}=\left(\frac{Q^2}{gw^2}\right)^{1/3}

The iteration converges when (A) or equivalently (B) is satisfied. In either case the result for the critical depth is y_C = 3.0 ft .

Since we know that A(y) = y(w + y) and b = w + 2y, the critical flow area and critical free surface width are found to be

A_C = y_C(w + y_C) = 3.0 ft (20 ft + 3.0 ft ) = 69 ft²

and

b_C = w + 2y_C = 20 ft + 2(3.0 ft ) = 26 ft

Also, since Q = AV, the critical velocity is given by V_C = Q/AC = 640 ft³/s/69 ft² = 9.28 ft/s. Finally, the critical hydraulic depth is found by inserting the data into Eq. 15.45: y_HC = A_C/b_C = 69 ft²/26 ft = 2.65 ft.