In this section, the equations for gradually varied open-channel flow will be discussed, and different possible profiles of the water surface will be presented. The direct step method is introduced to determine the water surface profile.

For nonuniform open-channel flow, the cross sectional area, depth, and velocity vary along the channel. The flow is classified as gradually varied flow when the change of the fluid depth along the channel dy/dx is much less than one. For gradually varied flow, the channel can be divided into different sections called reaches, and the uniform flow equations, such as the Chezy and Manning equations, are applicable for calculating head losses in such reaches.

For steady and one-dimensional flow, the equation for gradually varied flow, based on the energy equation, can be written as

     dE/dx = S_o - S_f                                                [1]

where E is the specific energy, S_o is the slope of the channel bed, and S_f is the slope of the energy grade line (EGL).

The equation above can also be rewritten in terms of the change of the water surface elevation and Froude number (Fr) as follows:

                                       [2]

For uniform flow, S_o is the same as S_f, hence the water surface elevation stays constant. On the other hand, for gradually varied flow, S_o does not equal S_f, and the water surface elevation can either increase or decrease in the direction of flow, depending on the type of profile. The slope of the energy grade line S_f can be determined from the Manning equation using the local value of fluid depth:

                                     [3]

Different Types of Water Surface Profiles
(NDL: Normal Depth Line and
CDL: Critical Depth Line)

According to Eqn. 2, different water surface profiles can be derived, and there are altogether 12 possible shapes, as shown in the table and figure. The surface profiles can be classified into 5 categories according to the channel bed slopes. They include mild (M), steep (S), critical (C), horizontal (H) and adverse (A) slopes. The surface profiles can be further divided into different types according to its depth relative to the channel's normal depth (y_o) and critical depth (y_c).

Note that equations 1 and 2 are nonlinear differential equations, and analytical solutions are possible only for a few problems. Hence numerical methods are necessary to approximate the solutions. One of the simplest numerical methods is the finite difference method. Eqn. 1 can be discretized as follows:
     
                                      [4]

where is the average slope of the energy grade line between the end points of a given reach.

The direct step method starts from a boundary condition or control point (i.e., a point where the water surface elevation is given), and proceeds by assuming a surface elevation (y) and computing the distance Δx to the assumed elevation:

                                       [5]

The computations then carry on to the next reach, and stop when a predetermined limit or criterion is reached. The case study provides detailed procedures for using the direct step method in determining the water surface elevation.