1) The document describes using the direct step method to determine water surface profiles for trapezoidal, rectangular, and triangular channels. 2) The direct step method is an iterative process that tests different water depths to classify a channel's flow type as subcritical, critical, or supercritical based on comparisons to the normal and critical depths. 3) Two examples are provided demonstrating the use of the direct step method to calculate normal depth, critical depth, and classify the water surface profile for different channel geometries.

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Using the Direct Step Method to Profile Water Surfaces
The objective of this computer project is to profile water surfaces for trapezoidal,
rectangular, and triangular channels using an Excel spreadsheet. A water surface profile is a
description of the behavior of how water flows through a channel. To reveal the profile type
requires implementation of the direct step method. The direct step method is an iterative process
that incrementally tests increasing or decreasing depths in a channel in order to be able to
evaluate the flow type of a channel. The general flow type zones are one, two and three, which
are defined by the relationship between the actual flow depth and computed depth parameters.
The channel type is discovered by comparing the computed flow depth parameters, resulting in a
description of channel’s slope (mild, steep, critical, horizontal, or adverse).
The direct step method relies heavily on the following equation.
This describes a change in position a long a channel (delta X), such that change in
upstream and downstream conditions can yield an amount to move from one depth to the next.
To define some variables, Y is the water depth, with the subscripts necessitating upstream or
downstream. V describes velocity. G is a gravitational constant. The S’s describe channel slope
(So) and that which is simulated by friction (Sfm). This is operating under an assumed Y value,
either upstream or downstream.
To begin, the values of critical (Yc) and normal (Yn) depth must be computed. The
relationship between these two depth values is essential in classifying channel type. After this
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has been established, the direct step method must be used to determine the Y value that is being
considered for flow type. Y is determined by identifying boundary conditions. Boundary
conditions are known from whether a channel is critical or supercritical, which relies on the
status of Yc and Yn yet again. The relationships between Y, Yc, & Yn solely determine flow
type. The channel type and flow type make up the water surface profile.
It is important to note here that while the direct step method is iterative, is it not a trial
and error method. It is an analytical method to determine the flow depth based on boundary
conditions, which are determined by channel geometry and rate of discharge.
To go back to boundary conditions, it is worth considering how they are determined from
geometric parameters. If a channel is subcritical, its boundary conditions will be on the
downstream end of the channel section. If supercritical, conditions will be on the upstream end.
A channel flow is subcritical if the flow depth exceeds the critical depth, supercritical if it does
not.
In order to identify the water surface profile, there are many necessary computations that
must be performed. Calculation for slop values and relative changes (Sf, Sfm – So, etc.) are not
worth really investigating since they are products of later calculations. The velocity V is
calculated by dividing the flow by the area. Change in x is found from change in E over change
in (Sfm – So). Water depth is determined by a known boundary, and work towards the desired
flow depth.
For computing the normal depth, it is necessary to use Manning’s equation.
Q = (kn/n)AR^2/3)So^(1/2)= (kn/n)*{[(byn)^(5/3)]/[(b+2yn)^(2/3)]}So^(1/2)
(Eq. 3.26 from textbook).

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The only parameter which has not yet been defined here is R, hydraulic radius. Hydraulic
radius is the area divided by the wetted perimeter. Wetter perimeter (P) is wherever water
touches a surface of the channel. Here, Yn can be seen as the one unknown in the equation. It is
useful to use Excel’s “Solver” function to quickly find this Yn value, since all other values in the
equation are known.
For Yc, it is almost most useful to utilize the “Solver” function, unless the channel in
question is rectangular. The analytical solution for Yc for a rectangular channel is the following
equation.
Yc = [(Q^2/(gb^2)]^1/3) (Eq 2.3)
B in this equation is the bottom width of the channel. If the channel is not rectangular, Yc
can be computed with “Solver” by applying the Froude number (Fr) equation. D is hydraulic
diameter, usually in consideration of the top width of a channel (T).
Fr= Q/[A*sqrt(gD)] (Eq 2.27)
In my computer project, I utilized in-cell Excel coding, as well as some Visual basic
programming to solve and compute the necessary values, as well as determine the water surface
profile. For all basic computations, I entered known values into the sheet, and set up statements
within the sheet to determine system of units, as well as channel shape. This information shaped

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what Yc equation was used, as well as numerical outcomes considering the difference in units
from SI to US.
The programming portion is two-fold. One button labeled “Solve!” Computes Yc for
trapezoidal/triangular channels (if necessary) as well as figuring out what change in channel
depth accompanied by normal depth value would produce the least change in produced Q (from
the Manning’s equation). A second macro labeled “Graph!” takes the computed data, and
produces a position versus depth chart of the channel bottom, water surface, and depth values.
Here are the results of the macros and in-cell coding to give water surface profile. The
first one is for example 4.1, whose parameters are given in the attached Excel file.
Table 1: Yn & Yc for 4.1

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Figure 1: Position vs. Depth Graph for 4.1
Here, the critical depth and normal depth are not very far apart, which suggests it may be
easy for a type 3 flow type if the flow depth is able to drop under both values, which it does in
Example 4.1.
Table 2: Water Surface Profile for 4.1
The calculated values for Yn & Yc gave subcritical flow type, which set up downstream
boundary conditions. The eventually calculated depth was less than both, so the flow type is M3.
For 4.2, the situation is completely different.
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
0.0 5.0 10.0 15.0 20.0 25.0 30.0
x (ft/m)
Position vs. Depth
Zb (ft/m)
WS (ft/m)
y (ft/m)

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The water surface and depth are divergent because of the steep slope of the channel, as
seen in the channel bottom line’s (blue) very steady increase.
Table 3: Yn & Yc for 4.2
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0
x (ft)
Position vs. Depth
Zb (ft) WS (ft) y (ft)

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Table 4: Water Surface Profile for 4.2
The Yn is now less than the Yc, making it an S channel type. And the flow type is
supercritical, so the boundary is upstream. Since the computed depth was less than Yc & Yn, it is
in zone 1. Therefore, the water surface profile for 4.2 is S1.