This method is applicable for both prismatic and nonprismatic channels, including the adjacent floodplain. The technique is used by most computer programs that com-pute steady, gradually varied flow profiles and can be used for both subcritical and supercritical flow. The method uses the continuity, energy, and Manning equations to solve for depth or water surface elevations at selected locations along the stream.
Standard Step Equation. The basic equation for the standard step solution is a slight restatement of the terms of the energy equation (Equation 2.14). The resulting energy equation for water surface profile analysis is
(2.42) where WSEL1,2 = the water surface elevation (z + y) at the indicated location (ft, m)
= the friction loss plus expansion or contraction loss between the two points (ft, m)
Figure 2.25 illustrates the variables used in standard step computations.
WSEL2 α2V22 ---2g
+ WSEL1 α1V12 --- h2g L + + 1-2
=
hL
1-2
Section 2.6 Computational Methods 57
As mentioned previously, the head loss term is a combination of friction and other (expansion or contraction) losses between locations one and two. The head loss equa-tion is
(2.43) where hf = the energy loss due to friction between the two locations (ft, m)
ho = the energy loss due to expansion or contraction between the two locations (ft, m)
Bend losses also could be added as a separate loss component and are included as such in some programs. In HEC-RAS, however, bend losses are assumed to be incor-porated in the Manningʹs n used to compute friction losses. In Chapter 5, which cov-ers Cowan’s equation, the use of Manning’s n to include bend losses is further illustrated.
The friction loss is found from Equation 2.41 and the distance between the two loca-tions as
(2.44) where L = the length of the flow path between the two locations (ft, m)
sf = the average energy slope between the two locations (ft/ft, m/m)
The length term could represent an average flow length, because there could be as many as three different flow lengths between two cross-section locations for complex cross sections (one each for the left and right overbanks and one for the channel).
The other losses are sometimes referred to as eddy losses and are due to the expan-sion or contraction of cross-section flow area between the two locations. These losses, which are similar to minor losses in pipeline systems, are included by multiplying the absolute difference in velocity head between the two points by an appropriate coeffi-cient:
Figure 2.25 Variables used in the standard step method.
hL
1-2 = hf+ho
hf = Lsf
(2.45)
where Cc = the coefficient of contraction (dimensionless) Ce= the coefficient of expansion (dimensionless)
V1 = the average velocity at the downstream section (ft/s, m/s) V2 = the average velocity at the upstream section (ft/s, m/s)
When the difference in velocity head is positive (the downstream velocity head is less than the upstream velocity head), the flow area is expanding and the absolute value of the difference in velocity heads is multiplied by the coefficient of expansion (Ce).
When the difference in velocity head is negative (upstream velocity head is less than downstream velocity head), the flow area is contracting and the absolute value is mul-tiplied by the coefficient of contraction (Cc).
For subcritical flow, Cc and Ce are often taken as 0.1 and 0.3, respectively. For super-critical flow, the coefficients are much smaller. Values are left to the discretion of the user, but Cc and Ce for supercritical flow in natural or man-made channels are usually no more than 0.05 for contraction and 0.1 for expansion. Contraction and expansion coefficients in prismatic channels are often assumed to be zero for either flow regime.
Chapter 5 further discusses expansion and contraction coefficients for both subcritical and supercritical flow. Incorporating Equation 2.43 through Equation 2.45 into Equa-tion 2.42 and rearranging terms yields
(2.46)
Equation 2.46 is the form of the energy equation used by HEC-RAS and most other steady, gradually varied flow programs to solve for the unknown water surface eleva-tion at locaeleva-tion 2.
Application of the Standard Step Method. To use the standard step method, the discharge, geometry, roughness values, and expansion and contraction coeffi-cients must be known at each desired computation location. In addition, the discharge and boundary conditions (flow regime and starting water surface elevation) must be specified. Chapter 5 describes how to develop these data. An iterative procedure is required to compute the depth or water surface elevation at selected points. In Equa-tion 2.46, the velocity head and the fricEqua-tion slope at locaEqua-tion 2 cannot be computed until the water surface elevation at location 2 is determined. Thus, the procedure is to estimate the water surface elevation at location 2, calculate the velocity head and aver-age friction slope, and solve Equation 2.46 for WSEL2.
If the original estimate does not approximate the calculated value, a new estimate of the water surface elevation at location 2 is made and the process repeated until the two values are within a specified tolerance. Two to four iterations are usually suffi-cient for most cross sections to meet the assigned tolerance. For hand computations, the tolerance may be 0.05–0.1 ft (0.015–0.03 m). The selected tolerance directly affects the profile accuracy. The engineer could experiment with various tolerances to deter-mine what tolerance provides acceptable accuracy. Errors in profile computations can accumulate with a loose tolerance and become excessive. If a profile computational accuracy of 0.1 ft (0.03 m) is desired, the tolerance should likely be 0.05 ft (0.015 m) or
ho Cc e, α2V22
Section 2.6 Computational Methods 59
less. Example 2.13 shows the steps for computing a water surface profile using the standard step method for a simple prismatic channel.
Example 2.13 Standard step method for a prismatic channel.
Building on Example 2.12 for the direct step method, compute the water surface pro-file starting at a point 5000 ft upstream of the free overfall and at 1000 ft increments thereafter to a point 9000 ft upstream of the free overfall. A 600 ft3/s discharge is car-ried in a trapezoidal channel (n = 0.013) of 12 ft bottom width and 1V:2H side slopes.
The channel invert slope is 0.00023 and the channel invert elevation at the free overfall is 100 ft NGVD. The expansion and contraction coefficients for this prismatic channel may be assumed to be zero and α = 1. The tolerance between the energy heads for the estimated and computed water surface elevations is 0.05 ft.
Solution
From the figure, which shows the water surface profile computed with the direct step method, the water surface elevation 5000 ft upstream of the free overfall is estimated as 106.8 ft NGVD, which becomes the elevation at the boundary for this example.
The water surface profile computed for a prismatic channel with the standard step method is determined by setting up a table, such as the one in this example. Following are descriptions of the contents of the table.
Column (1): The distance from the selected starting point to each cross section is input.
Since the first cross section is at 5000 ft upstream from the brink of the free overfall and the problem statement directs the computations to be made at 1000-ft intervals, all val-ues in this column can be included prior to the start of computations. However, ade-quate space on the table between each cross section should be included for the additional iterations required to achieve the desired tolerance.
Column (2): For the first cross section (5000 ft), the known water surface elevation (WSEL) is inserted. For all other cross sections, a water surface elevation will be esti-mated by the modeler when computations reach each location.
Column (3): The maximum depth for each cross section is inserted, obtained by sub-tracting the channel invert elevation at the location under analysis from the WSEL of column (2).
Column (4): The cross-section flow area is computed.
Column (5): The average velocity is computed from the known discharge (600 ft3/s) divided by the area in column (4).
Column (6): The velocity head is computed from the velocity of column (5).
Column (7): The total energy head is obtained by adding column (2) and column (6).
Column (8): After computing the wetted perimeter, the hydraulic radius is obtained by dividing the column (4) value by the wetted perimeter.
Column (9): The hydraulic radius is raised to the 4/3 power.
Column (10): The friction slope is computed using Equation 2.41 and the values in col-umns (5) and (9), using Manning’s n for the reach.
Column (11): The average friction slope between the two computation points is calcu-lated, normally by a simple average of sf at each section.
Column (12): The incremental distance between the two cross sections is input. For this example, a single value of distance is applicable because all flow is confined to the trapezoidal channel.
Column (13): The average friction slope in column (11) is multiplied by the incremental distance in column (12) to obtain the friction loss between the two cross sections.
Channel HydraulicsChapter 2 5000 106.8 5.65 131.645 4.558 0.322 107.12 3.528 5.371 0.0002944
0.0002868 1000 0.287 107.41 0
6000 107.1 5.72 134.08 4.475 0.311 107.41 3.569 5.454 0.0002793
0.0002719 1000 0.272 107.68 0.02
7000 107.4 5.80 136.84 4.385 0.299 107.70 3.608 5.534 0.0002645
0.000259 1000 0.259 107.96 0.03
8000 107.7 5.86 139 4.317 0.289 107.99 3.639 5.598 0.0002534
0.0002499 1000 0.250 108.24 0.06
9000 107.9 5.90 140.42 4.273 0.284 108.18 3.659 5.639 0.0002465
9000 107.93 5.93 141.49 4.231 0.279 108.21 3.674 5.670 0.0002415 0.0002474 1000 0.247 108.23 0.02
Section 2.6 Computational Methods 61
Column (14): The computed energy head is the result of the energy head at the previ-ous cross section (column 7) plus the friction loss between the two cross sections (col-umn 13).
Column (15): Compute the difference between the column 14 value (computed energy head) from that of column 7 (estimated energy head).
As seen, the value in Column (7) is identical to the value in Column (14) and well within the tolerance. A single iteration to achieve the desired tolerance is not normally the case. Because the WSEL at each location can be initially estimated closely from the existing profile for the direct step method, only one iteration at each location (except the last cross section) was needed for this simple example. As would be expected, the computed profile with the standard step method between the 5000 and 9000 ft dis-tances is essentially identical to that computed by the direct step method.
Standard Step Method Using Conveyance. When used to compute water sur-face elevations for normal cross sections having a left and right floodplain as well as a nonprismatic channel (Figure 2.3), the standard step method, described in the previ-ous section for prismatic channels, includes some modifications to facilitate the com-putations, especially in the use of conveyance. Conveyance is taken from the geometry and roughness terms in the Manning equation for discharge (Equation 2.27) and is computed with
(2.47) where K = the conveyance (ft3/s, m3/s)
k = 1.486 for English units and 1 for SI
Equation 2.7 was used earlier in this chapter to compute α from the velocity and dis-charge values for the left and right overbanks and for the channel. However, most computer programs use the conveyance to determine α. Similarly, using conveyance is computationally more efficient for determining the distribution of flow in the left and right overbanks and the channel of a cross section, and in calculating the friction
H6000 = H5000+hL5000 6000– = 107.12 0.287+ = 107.41 ft
K k
n---AR2 3⁄
=
slope. To compute the velocity distribution coefficient using conveyance, HEC-RAS, HEC-2, and other similar programs use
(2.48)
whereATOT = the total cross-sectional area (ft2, m2) Klob = the left overbank conveyance (ft3/s, m3/s) Alob = the left overbank cross-sectional area (ft2, m2)
Kch = the channel conveyance (ft3/s, m3/s) Ach = the channel cross-sectional area (ft2, m2) Krob = the right overbank conveyance (ft3/s, m3/s) Arob = the right overbank cross-sectional area (ft2, m2) KTOT = the total cross-sectional conveyance (ft3/s, m3/s)
The friction slope at a cross section may also be computed using the conveyance and the Manning equation for discharge (Equation 2.27):
(2.49)
After an initial estimate of the unknown water surface elevation, the conveyance in each of the three cross-section segments can be calculated. For a cross section consist-ing of the channel and the left and right overbank areas, Equation 2.49 is expanded to the form
(2.50) Rearranging the terms of Equation 2.50 gives the equation used to solve for the fric-tion slope at locafric-tion 2 after estimating the unknown water surface elevafric-tion and computing the conveyance:
(2.51)
The distribution of discharge for the left and right overbanks and channel sections is also based on the conveyance. For one-dimensional steady flow, the friction slope (sf) must be the same for each part of an individual section. Therefore, the discharge in each subsection is computed from the Manning equation for discharge, using the con-veyance. The discharges in the three main segments of the cross section (left and right overbanks and the channel) are found using the total discharge and the conveyance of each of the three segments:
(2.52)
The calculation of total energy head at the upstream location uses a calculated α2 for location 2 from Equation 2.48 and the assumption of subcritical flow. The average of
α
Section 2.6 Computational Methods 63
the friction slopes for locations 1 and 2 is used as the average friction slope between the two cross sections. The average friction loss is computed using Equation 2.44, with the known length between the two cross sections; a weighted length is used if the lengths vary among the channel and left and right overbank reaches. The equation used in HEC-RAS for a discharge-weighted reach length is
(2.53)
where LQ = the discharge-weighted reach length used to compute the friction loss (ft, m)
Llob= the average length of flow between the left overbanks of adjacent cross sections (ft, m)
Lch= the average length of flow between the channels of adjacent cross sections (ft, m)
Lrob= the average length of flow between the right overbanks of adjacent cross sections (ft, m)
The development of different lengths between two cross sections is further discussed in Chapter 5.
An expansion or contraction is determined and the value for ho is found from Equa-tion 2.45. EquaEqua-tion 2.46 is then used to compute a value for the water surface elevaEqua-tion at location 2. The computed elevation is compared to the estimated water surface ele-vation and the difference between the two water surface eleele-vations is compared to the allowable tolerance. If the difference is not within the specified tolerance, the estimate of the water surface elevation is modified and the process is repeated until conver-gence is reached. Normally, two to five iterations are needed to achieve converconver-gence in hand computations of nonprismatic channels.
Example 2.14 Standard step method for complex cross sections using conveyance.
The following figure shows three nonprismatic cross sections, including the elevation-distance data for the geometry of each section, the bank stations (located at the vertical dotted lines), and the n values for each section. Use the standard step method, apply-ing conveyance, to compute a water surface profile for a discharge of 1000 ft3/s. Use Cc
= 0.1 and Ce = 0.3. The starting water surface elevation is 412 ft NGVD.
Solution
When computing a water surface profile by hand for nonprismatic cross sections with flow in the overbanks, a table similar to the following table should be prepared.
Sequential computations are performed at each cross section until a balance between the assumed and computed water surface elevation is achieved, within the defined tol-erance.
Column (1): Cross-section identification number. This value normally corresponds to the cross sectionʹs location (distance) on the stream, as measured from the mouth of the stream. For this simple example, the location number corresponds to the cross-section number, as surveyed.
Column (2): Assumed water surface elevation. At the first cross section, this value is specified by the engineer. The estimated normal depth, the critical depth, a known ele-vation, or simply a best estimate of the starting water surface elevation are all appro-priate. At all other sections, the elevation is an initial estimate to begin the
LQ LlobQlob+LchQch+LrobQrob QTOT
---=
computations. Chapter 5 describes methods for estimating a starting water surface ele-vation.
Column (3): The cross-sectional area below the water surface in the left overbank (floodplain) area is computed.
Column (4): The cross-sectional area of the channel below the water surface is com-puted.
Column (5): The cross-sectional area in the right overbank (floodplain) below the water surface is computed.
Column (6): The values in columns (3) through (5) are summed to obtain the total cross-sectional flow area.
Column (7): The wetted perimeter of the left overbank is determined. Note that the depth on the imaginary vertical dashed line separating the left overbank from the channel is not included in the wetted perimeter.
Column (8): The wetted perimeter of the channel is determined. The depths on both imaginary vertical dashed lines separating the channel from the left and right over-banks are not included in the wetted perimeter.
Section 2.6 Computational Methods 65
Column (9): The wetted perimeter of the right overbank is determined. The depth on the imaginary vertical dashed line separating the right overbank from the channel is not included in the wetted perimeter.
Column (10): The values in columns (7) through (9) are summed to determine the total wetted perimeter of the cross section.
Column (11): The value in column (3) is divided by the value in column (7) to give the hydraulic radius of the left overbank.
Column (12): The value in column (4) is divided by the value in column (8) to give the hydraulic radius of the channel.
Column (13): The value in column (5) is divided by the value in column (9) to give the hydraulic radius of the right overbank.
Column (14): The value in column (6) is divided by the value in column (10) to give the hydraulic radius of the full cross section.
Column (15): The conveyance of the left overbank is computed, using the values in col-umns (3) and (11), and knowing the value of Manning’s n for the left overbank.
Column (16): K3/A2 is computed for the left overbank, from the values in columns (3) and (15).
Column (17): The conveyance of the channel is computed, using the values in columns (4) and (12), and knowing the value of Manningʹs n for the channel.
Column (18): K3/A2 is computed for the channel, from the values in columns (4) and (17).
Column (19): The conveyance of the right overbank is computed, using the values in columns (5) and (13), and knowing the value of Manning’s n for the right overbank.
Column (20): K3/A2 is computed for the right overbank, from the values in columns (5) and (19).
Column (21): The total cross-section conveyance is computed by adding the values in columns (15), (17), and (19).
Column (22): The friction slope (sf) is computed for the cross section using Equation 2.49 with the total discharge at the cross section and the total conveyance of column (21).
Column (23): The computed friction slope at each of the two cross sections is averaged from the values in column (22) for each location. A simple average is used for this example; there are four different methods to compute average friction slope in HEC-RAS.
Column (24): The average distance between the two cross sections is entered. Distances used in this example are the same between each pair of sections, but this distance is normally a discharge-weighted reach length computed with Equation 2.53, using the distribution of flow in each of the three cross-section segments and the distance between the two sections for each of the three segments.
Column (25): The friction loss between the two cross sections is computed with Equa-tion 2.44 and the values in columns (23) and (24).
Column (26): The velocity distribution coefficient (α) is computed with Equation 2.48, using the values in columns (6), (16), (18), (20), and (21).
Column (27): The average velocity for the full cross section is computed from the known discharge and the value in column (6).
Column (28): The adjusted velocity head is computed, using the values in columns (26) and (27).
Channel HydraulicsChapter 2
1 412.00 60.0 440.0 80.0 580.0 32.0 53.3 42.0 127.3 1.9 8.3 1.9 4.6 2260 3204472 66784
2, Trial 1 412.50 37.5 420.0 75.0 532.5 26.5 58.0 51.5 136.0 1.4 7.2 1.5 3.9 1171 1140787 58401
2, Trial 2 412.26 31.5 410.4 63.0 504.9 26.3 58.0 51.3 135.5 1.2 7.1 1.2 3.7 881 688565 56193
3, Trial 1 412.50 0.0 332.0 0.0 332.0 0.0 56.6 0.0 56.6 0.0 5.9 0.0 5.9 0 0 40115
1538530292 3045 4409345 72088 0.00019 1.39 1.72 0.064 412.00 0.00
0.00023 1200 0.273 0.011 0.3 0.003 0.277
112913419 2046 1521709 61617 0.00026 1.37 1.88 0.075 412.27 –-0.23
0.00024 1200 0.290 0.017 0.3 0.005 0.295
1053519366 1535 910388 58609 0.00029 1.34 1.98 0.081 412.28 0.02
0.00046 1500 0.684 0.059 0.3 0.018 0.702
585676560 0 0 40115 0.00062 1.00 3.01 0.141 412.92 0.42
0.00042 1500 0.624 0.047 0.3 0.014 0.638
655828675 0 0 42985 0.00054 1.00 2.87 0.128 412.87 –0.03
0.00042 1500 0.630 0.048 0.3 0.014 0.644
648631059 0 0 42696 0.00055 1.00 2.89 0.130 412.86 0.00
Section 2.6 Computational Methods 67
Column (29): The velocity head at the section of known water surface elevation (down-stream in this example) is subtracted from the velocity head at the section under anal-ysis. Include the sign of the value.
Column (30): If the value in column (29) is positive, insert the coefficient of expansion (often 0.3). If the value in column (29) is negative, insert the coefficient of contraction (often 0.1).
Column (31): Compute the other losses (ho) by multiplying the absolute value in col-umn (29) by the value in colcol-umn (30).
Column (32): Compute the total losses between the two cross sections by adding the values in columns (25) and (31).
Column (33): Compute the water surface elevation for the cross section under analysis using Equation 2.46 and the values for the previous cross section in columns (2) and (28) and the values for the cross section under analysis in columns (28) and (32).
Column (34): Compare the value in column (33) to the assumed value in column (2) for the cross section under analysis. If the difference is equal to or less than the specified tolerance, accept the value in column (2) as correct and move to the next cross section to be analyzed. If the difference is greater than the specified tolerance, select a new value for column (2) and repeat steps (3) through (34) until the tolerance is met.
Column (35): Add any comments necessary for documentation.
The profile is plotted in the following figure, based on the final water surface values for each section in column (2) and the cumulative distance from the start of computa-tions to each cross section (1200 ft from Section 1 to Section 2 and 2700 ft from Section 1 to Section 3).