Direct Step Method

This method has limited application, since it is used to solve for a distance given a known depth or for a change in depth (dx/dy rather than the typical dy/dx solution).

Normally, the engineer is interested in the depth at a specific location (dy/dx), such as at a bridge, rather than how far upstream is it necessary to go to find a selected change in depth (dx/dy). Also, this technique is generally used only for prismatic channel shapes because it is cumbersome to apply to nonprismatic sections. Multiplying Equation 2.37 by dx/dy, rearranging terms, recognizing that E = y + V²/2g, and assum-ing that α = 1 yields

(2.40)

where ∆x = the distance from location 1 to location 2 for the depth selected (ft, m) E₂ = the specific energy at upstream location 2 (ft, m)

E₁ = the specific energy at downstream location 1 (ft, m) s_o = the channel invert slope (ft/ft, m/m)

s_f = the average friction slope between the two locations (ft/ft, m/m)

Figure 2.24 may be used as an aid in understanding Equation 2.40. Figure 2.21 may also be used to derive Equation 2.40, rather than Equation 2.37.

∆x E₂–E₁ s_o–s_f

---=

Figure 2.24 Variables used in the direct step method.

Section 2.6 Computational Methods 53

The direct step method requires that the starting boundary conditions (depth or water surface elevation at the downstream-most cross section, assuming subcritical flow), cross-section geometry, s_o, Manning’s n, and discharge are all known. With the assign-ment of location 2 as upstream and location 1 as downstream, the velocity, specific energy, and friction slope at Section 1 can be computed. With the depth at location 2 selected (the difference from the depth at location 1 should be small and the depths must both be in the same classification zone), the velocity, specific energy, and friction slope at Section 2 can be computed. With specific energy at both locations computed, the value of the numerator of Equation 2.40 is now known, as well as s_o. The next step is to develop a value for s_f. This value is obtained from the Manning equation for velocity, which is rearranged to solve for s_f as

(2.41)

where s_f = the energy grade line slope (ft/ft, m/m)

n = the Manning roughness coefficient (dimensionless) V = the average velocity (ft/s, m/s)

k = 1.486 for English units and 1.0 for SI units (constant) R = the hydraulic radius (ft, m)

The average friction slope can be computed by averaging the friction slopes at loca-tions one and two, or by computing s_f directly with Equation 2.41, using the average velocity and hydraulic radius terms for the two locations. If V and R are carried to at least three significant figures following the decimal point, the average value of s_f is essentially the same as found by computing and averaging the two friction slopes sep-arately. No significant difference in the computed distance between the two locations is found using either technique. Using an average velocity and hydraulic radius assumes that Manning’s roughness value is the same between the two cross sections, normally the case for a prismatic channel. Knowing all the values on the right-hand side of Equation 2.40 allows the computation of the incremental distance between two cross sections. Example 2.12 illustrates the computational procedure for performing a direct step profile analysis. This technique can be employed to compute a water sur-face profile in a prismatic channel, such as a storm sewer, culvert, or small drainage channel of constant slope. The HEC-RAS program uses the direct step method to com-pute an open channel flow profile through a culvert. Since channels are normally non-prismatic, the standard step method, discussed in the following section, has a much wider application.

Example 2.12 Applying the direct step method.

A concrete (n = 0.013) trapezoidal channel has a bottom width of 12 ft and 1V:2H side slopes. The channel slope (s_o) is 0.00023 and the discharge is 600 ft³/s. The channel ends at a free overfall, with an invert elevation of 100 ft NGVD. Assume that critical depth occurs at the brink of the overfall. Using the direct step method, compute the profile from the location of critical depth to a point upstream where the depth is within 0.1 ft of normal depth.

Solution

Solving for normal depth (Equation 2.27) and critical depth (Equation 2.29) gives y_n = 6 ft and y_c = 3.48 ft.

s_f n²V² k²R^{4 3⁄}

---=

A water surface profile computed using the direct step method is performed by pre-paring a table, such as the table included with this example. Computations to deter-mine the distance from the previous depth for another selected depth proceed across each row, one row at a time. Critical depth was computed as 3.48 ft and is the initial depth used at the brink of the free overfall. Since it is known that the profile approaches normal depth (6 ft) at some upstream location, the resulting profile classi-fication is M2 and all selected depths are between 3.48 and 6 ft. Examining the M2 pro-file shape on Figure 2.23 shows that depth changes at a faster rate near the critical depth, compared to approaching normal depth. Therefore, the incremental depth changes are somewhat larger near critical depth compared to near normal depth. Also, the profile for a short distance upstream of the location of critical depth is actually rap-idly varied flow. Therefore, the computations immediately upstream of critical depth are acceptable for plotting the profile, but cannot be considered “highly accurate.”

Rapidly varied flow might reflect the reach for 25–50 ft (8–16 m) upstream of critical depth and is normally not a significant issue in profile plotting.

Column (1): The depths shown in Column (1) are arbitrarily selected by the engineer.

More or fewer values could be selected; more values result in greater precision in the profile calculation, while fewer values lead to less precision. As long as the changes in depth (and velocity) are small with respect to distance, the computed profile will be satisfactory.

Column (2): The cross-sectional flow area of the trapezoidal channel is computed.

Column (3): The wetted perimeter of the trapezoidal channel is computed.

Column (4): Column (2) is divided by column (3) to obtain hydraulic radius.

Column (5): The discharge (600 ft³/s) is divided by column (2) to obtain average veloc-ity.

Column (6): The velocity head (V²/2g) is computed for the velocity value in column (5).

Column (7): The friction slope is computed with Equation 2.41 for the values in col-umns (4) and (5) and the known n.

Column (8): The friction slope for this depth is averaged with the friction slope for the depth at the previous location to obtain the average friction slope between the two computation points.

Column (9): The average friction slope is subtracted from the invert slope. This value represents the denominator of Equation 2.40.

Column (10): The values in columns (1) and (6) are added to obtain the specific energy head at the location under analysis.

Column (11): The downstream specific energy (location 1) is subtracted from the upstream specific energy (location 2). This value represents the numerator in Equation 2.40.

Column (12): Column (11) is divided by column (9). This value could be positive or negative; however, the sign is not especially important as long as the engineer under-stands that the numerical value represents the distance between the two computation points.

Column (13): The incremental distance in column (12) is added to all the earlier com-puted distances to obtain the total distance from the start of computations to the loca-tion under analysis. This column represents the distance from the start of computations to each computation location for plotting the profile.

Column (14): Column (1) is added to the elevation of the channel invert elevation at the location of the start of computations and the product of the channel invert slope multi-plied by the value in column (13). The values in column (14) are given by

CWSEL = z₀+y±s₀Σx

Section 2.6Computational Methods55

3.48 65.98 27.56 2.394 9.094 1.284 0.001966 4.764 0 103.48

0.001846 –0.001616 0.006 –3.7

3.6 69.12 28.09 2.461 8.680 1.170 0.001726 4.770 3.7 103.60

0.001498 –0.001268 0.067 –52.8

3.9 77.22 29.43 2.624 7.770 0.937 0.00127 4.837 56.5 103.91

0.001112 –0.0008816 0.124 –140.6

4.2 85.68 30.774 2.784 7.003 0.761 0.0009533 4.961 197.1 104.25

0.0008405 –0.0006105 0.165 –270.3

4.5 94.50 32.115 2.943 6.347 0.626 0.0007276 5.126 467.4 104.61

0.0006459 –0.0004159 0.194 –466.4

4.8 103.68 33.456 3.099 5.787 0.520 0.0005643 5.32 933.8 105.01

0.0005038 –0.0002738 0.216 –788.9

5.1 113.22 34.797 3.254 5.299 0.436 0.0004493 5.536 1722.7 105.50

0.0004118 –0.0001817 0.154 –845.4

5.3 119.78 35.691 3.356 5.009 0.3896 0.0003801 5.689 2568 105.89

0.0003538 –0.0001238 0.16 –1294.8

5.5 126.50 36.585 3.458 4.743 0.349 0.0003275 5.849 3863 106.39

0.0003055 –0.00007598 0.165 –2188.9

5.7 133.38 37.479 2.559 4.500 0.314 0.0002834 6.014 6052 107.09

0.0002738 –0.00004379 0.084 –1927

5.8 136.88 37.926 3.609 4.383 0.298 0.0002642 6.098 7879 107.64

0.0002554 –0.00002536 0.086 –3372

5.9 140.42 38.373 3.659 4.273 0.284 0.0002465 6.184 11350 108.51

where CWSEL = the computed water surface elevation (ft, m)

z₀ = the elevation of channel invert at the start of computations (ft, m) y = the depth at the location under analysis (ft, m)

s₀ = the channel invert slope (ft/ft, m/m)

Σx = cumulative distance from the beginning of computations to the computation point (ft, m)

The ± sign is applied as follows: positive for subcritical profile analysis (proceeding upstream) and negative for supercritical profile analysis (proceeding downstream).

For Example 2.12, the sign is positive for all the computations. The equation is only applicable where the channel invert slope is constant.

Column (14) represents the water surface elevation at each computation point. The data in columns (13) and (14) are used to prepare the water surface profile shown in the figure below.