156 6 COMPUTATION OF GRADUALLY VARIED FLOW
6-3 Direct-Step Method
In the previous section, we discussed how we may compute from the known flow
depth at a section the location of an adjacent section where a specified depth
will occur. Let us discuss this in more detail to develop a systematic procedure
for the computations. Following Chow[1959], we call this procedure the direct-
step method.
Referring to Fig. 6-2, let us say that we know the flow depth at section 1 and
we want to determine the location of section 2, where a specified flow depth, y2
will occur in a given channel for a specified discharge, Q. In other words, the
statement of our problem is as follows: The flow depth, y1, at distance x1 (i.e.
section 1 in Fig. 6-2) is known; determine distance x2 where a specified flow
depth y2 will occur. The properties of the channel section, So, Q, and n are
known.
If So = slope of the channel bottom, then referring to Fig. 6-2,
z2 = z1 − So (x2 − x1 ) (6 − 11)
In addition, the specific energy
α1 V12
E1 = y1 +
2g
α2 V22
E2 = y2 + (6-12)
2g
The slope of the energy grade line (For simplicity we will refer to it as the
friction slope in the following discussion) in gradually varied flow may be
computed with negligible error by using the corresponding formulas for friction
slopes in uniform flow [Chow, 1959; Henderson, 1966]. However, since the flow
depth, y, varies with distance, x, the friction slope Sf is a function of x as well.
The following approximations have been used [U.S. Army Corps of Engineers,
1982] to select a representative value of Sf for the channel length between
Sections 1 and 2:
6-3 Direct-Step Method 157
Fig. 6-2. Computation of distance for specified depth
Average friction slope
1
S̄f = (Sf + Sf2 ) (6 − 13a)
2 1
Geometric mean friction slope
S̄f = Sf1 Sf2 (6 − 13b)
Harmonic mean friction slope
2Sf1 Sf2
S̄f = (6 − 13c)
Sf1 + Sf2
By expanding the right-hand side of the above approximations in a Taylor
series, we can prove (Problem 6-15) that these three formaulations for the ap-
proximation of the friction slope give identical results if the terms of the order
(ΔSf /Sf 1 )2 and higher are neglected. In this expression, ΔSf = Sf 2 − Sf 1 .
Laurenson [1986] showed that the average slope (Eq. 6-13a) gives the lowest
maximum error although it is not always the smallest error. If the distance
between sections 1 and 2 is short or the flow depths y1 and y2 are not signifi-
cantly different, then Eq. 6-13a yields satisfactory results, in addition to being
the simplest of the three approximations. Therefore, its use is recommended,
and we will use it herein. Hence, an expression for hf may be written as
1
hf = (Sf 1 + Sf 2 )(x2 − x1 ) (6 − 14)
2
158 6 COMPUTATION OF GRADUALLY VARIED FLOW
Substitution of Eqs. 6-12 and 6-14 into Eq. 6-2 yields
1
z1 + E1 = z2 + E2 + (Sf 1 + Sf 2 )(x2 − x1 ) (6 − 15)
2
By substituting the expression for z2 from Eq. 6-11 into Eq. 6-15, and can-
celling out z1 , we obtain
1
E2 − E1 = So (x2 − x1 ) − (Sf 1 + Sf 2 )(x2 − x1 ) (6 − 16)
2
This equation may be written as
E2 − E1
x2 = x1 + (6 − 17)
So − 12 (Sf 1 + Sf 2 )
Now, the location of section 2 is known. This is the starting value for the next
step. Then, by successively increasing or decreasing the flow depth and deter-
mining where these depths will occur, the water-surface profile in the desired
channel length may be computed. In Eq. 6-17, the direction of computations
is automatically taken care of if proper sign for the numerator and for the
denominator is used. Note that both the numerator and the denominator are
very small and extreme care should be excercised in using the proper number
of significant digits in the computations and rounding off the values.
There are two, main disadvantages of this method. (1) The flow depth is
not computed at the predetermined locations. Therefore, interpolations may
become necessary if the flow depths are required at specified locations. Simi-
larly, the cross-sectional information has to be estimated if such information
is available only at the given locations. This may not yield accurate results
in addition to requiring additional effort. (2) It is cumbersome to apply to
nonprismatic channels.
The following example should help in understanding this computational
procedure.
Example 6-1
A trapezoidal channel having a bottom slope of 0.001 is carrying a flow of 30
m3 /s. The bottom width is 10.0 m and the side slopes are 2H to 1V. A control
structure is built at the downstream end which raises the water depth at the
downstream end to 5.0 m. Compute the water surface profile. Manning n for
the flow surfaces is 0.013 and α = 1.
Given:
Bottom slope, So = 0.001
Discharge, Q = 30 m3 /s
Channel width, Bo = 10.0 m
Manning n = 0.013
Depth at the downstream end (i.e., at x = 0) = 5.0 m
α=1
6-3 Direct-Step Method 159
Determine:
Water-surface profile in the channel.
Solution:
The normal depth, yn , 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 per cent of the normal depth. We
will continue the calculations in this example until y = 1.05yn = 1.05 × 1.16
= 1.21, say 1.20 m.
We 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.
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.
Coulmn 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, Sf
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,
Sf = n2 V 2 /(Co2 R1.33 ).
160 6 COMPUTATION OF GRADUALLY VARIED FLOW
Column 6, S̄f
This is the average of Sf for the current depth and for the previous depth.
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, So − S̄f
This is obtained by subtracting S̄f of column 6 from the specified value of So .
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 + αV 2 /(2g).
Column 9, ΔE = E2 − E1
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 = x2 − x1
The distance increment is computed from the equation, Δx = (E2 −E1 )/(So −
S̄f ), i.e., dividing column 9 by column 7.
Column 11, x2
This is the distance where depth y will occur. It is obtained by algebraically
adding Δx of column 10 to the x2 value for the previous depth.
