Dimensions of the Channel

Hydraulic Radius

The cross sectional area and wetted perimeter of the channel are given by

     A = (25 + y/tan30^o)y = (25 + 1.732y)y
     P = 25 + 2y/sin30^o = 25 + 4y

The hydraulic radius thus can be obtained from R = A/P.

The normal depth y_o can be determined from the Manning equation as follows:

     

Note that the subscript "o" represents the normal depth condition.

Substituting R_o = A_o/P_o, and rearranging to yield

By trial and error, it is found that the normal depth, y_o, is 4.42 ft.

Next, to find the critical depth y_c, set the Froude number equal to unity:

     Fr = 1 = V_c/(gL_c)^0.5

where the hydraulic depth is given by L_c = A_c/B_c as shown in the figure. Note that the subscript c represents the critical depth condition.

Substituting the definition of the hydraulic depth and V_c = Q/A_c into the above equation yields:

By trial and error, it is determined that the critical depth y_c = 2.83 ft.

Since y_o > y_c, from the water surface profiles table, it can be concluded that the slope is mild, and the water surface has the profile of M2.

The direct step method is then used to compute the actual water surface profile. Computations are done through a spreadsheet, as summarized in the following table:

Water Surface Profile

The computations start from the critical depth (y_c = 2.8 ft), and proceed upstream to the normal depth (y_o = 4.4 ft).

The velocity and the specific energy are determined from V = Q/A and E = y + V²/2g, respectively.

The slope of the energy grade line is given by

     

while and ΔE = E_i+1 - E_i. The subscript i refers to the point of the section (i.e., i = 1 for y = 2.8 ft; i = 2 for y = 2.9 ft, etc.).

The distance to the assumed water surface elevation is computed by

     ,

and the last column (Sum Δx) is simply the cumulative distance along the channel.

The water surface profile is shown in the figure.