2.1 Terminology
An open channel is any flow path with a free surface, which means that the flow path is open to the atmosphere. Open channels can be classified as prismatic or nonprismatic. A prismatic channel has a constant cross section and often has a constant bed slope for long lengths of the channel. Man-made channels (such as storm sewers, drainage ditches, and irrigation canals) are typically assumed to be prismatic, although they do have occasional changes in cross sections or slope to accommodate topographic conditions or changes in their discharge rate, as illustrated in Figure 2.1a. A nonprismatic channel varies in both the cross-sectional shape and bed slope between any two selected points along the channel length. Natural channels (rivers and creeks, such as the one shown in Figure 2.1b) are nonprismatic. Unless indicated otherwise, prismatic channels are assumed for examples in this book. Figure 2.2 shows cross sections of several classifications of channels that are operating under open channel flow.
Figure 2.1 (a) Prismatic and (b) nonprismatic channels.
Figure 2.2 Cross sections for open channel flow.
The theory and procedures of open channel hydraulic analysis were originally developed from experiments on fluid flow in pipes or conduits. Flow in a pressurized pipe, however, is not representative of open channel hydraulics. In open channel flow, atmospheric pressure acts continuously and constantly on the water surface and, unlike in a pressurized pipe, there is no constant internal pressure on the fluid boundaries. Consequently, a depth term, rather than a pressure term, is used in open channel analysis.
Figure 2.3 compares the pressure head terms in open channel and pressure flow. Atmospheric pressure is neglected because it acts on the water surface at every location. As shown in the figure, the pressure head term in open channel flow is the depth of flow (y). The same term in pressure flow analysis is indicated by the internal pressure of the pipe (p in lb/in2 or kg/cm2) divided by the unit weight of water (g in lb/ft3 or kg/m3). The pressure head term (p/g) is equal to the height to which the water would rise in a vertical tube attached to the pressurized system. In open channel hydraulics, a common assumption is that the pressure in the fluid is hydrostatic, meaning that pressure varies linearly with depth (p = gy). Thus, the pressure at any point in a column of water in open channel flow is equal to the vertical height above the selected location multiplied by the unit weight of water. Only under certain conditions, such as rapidly varying flow (see Section 2.2), is the assumption of hydrostatic pressure inappropriate.
Figure 2.3 Comparison of pressure heads between open channel and pressure flow.
Because of the presence of a free surface, open channel flow problems can be more challenging than closed conduit flow problems. In pressure conduits, the conduit flows full and the water exerts a pressure on the container's walls in all directions. The amount of discharge through the pipe is a function of the pressure differential over the length of the pipe. If the discharge doubles, the pipe cross-sectional area does not change, but the upstream pressure head must greatly increase to force this additional flow through the same pipe area. In open channel flow, boundaries are not fixed by the physical boundaries of a closed conduit; the free surface adjusts itself to accommodate the geometry of the channel. When the free surface adjusts itself, other geometric properties, such as the cross-sectional area, wetted perimeter, and top width, adjust accordingly.
In addition, the physical properties of open channels can vary widely, especially for natural channels. Cross-sectional geometry, roughness, and longitudinal slopes can change greatly even over short distances. Moreover, roughness can be difficult to quantify and, in fact, can vary vertically and horizontally over the depth of flow. Open channel hydraulic computations require several iterations to solve for flow depth or water surface elevation at a desired location. In pressure conduit analysis, however, the friction coefficient may be assumed constant, which leads to a direct solution. If the pressure conduit computations are adjusted for changing friction coefficient with the changes in other pressure conduit parameters, a successful solution is often obtained with only a single iteration. Typically, open channel flow computations for a natural channel cross section may require three or more iterations.
Consequently, open channel hydraulic analysis is more data intensive and empirical than closed conduit flow. In fact, much experimental effort has been invested in developing mathematical relationships that describe various open channel flow scenarios with sufficient accuracy.
In terms of the geometry, cross sections for modeling purposes are described as a series of x and y coordinates of ground points, where x is the distance or stationing (in feet or meters) from the beginning of the cross section and y is the elevation (in feet or meters) above a datum. The datum is normally referenced to sea level and is expressed as National Geodetic Vertical Datum or NGVD in the United States. A cross section is typically taken from left to right looking downstream and describes the geometry of the channel and the left and right overbank (floodplain) areas, as shown in Figure 2.4. A characteristic that is important in modeling is the channel bank stations, which represent the breakpoints between the channel and overbank portions of the cross section. A cross section is oriented on a topographic map and surveyed in the field at right angles to the estimated flow path. Determining the cross-sectional geometry for every location for which a water surface elevation is desired is generally a required part of developing open channel hydraulic information, although HEC-RAS does have cross-section interpolation tools that can be helpful. Cross-section data are further discussed in Chapter 5.
Figure 2.4 A typical cross section consisting of a natural channel and floodplain.
Depth of Flow
Perhaps the key variable in floodplain modeling is the depth of flow, the elevation difference between the water-surface elevation and the deepest part of the channel. Depth is typically expressed by the variable y and represents the maximum vertical depth. However, to determine the cross-sectional area of flow (A) below the water surface, the area must be determined perpendicular to the channel-bottom slope (so). Consequently, depth perpendicular to the slope, and not in the vertical direction, must be determined. Depth perpendicular to the channel bottom slope is shown by the variable d, as shown in Figure 2.5. In most applications, y and d are used interchangeably, since the difference between the two values is negligibly small. A third term for depth (h) in the vertical is included for those infrequent situations in which d and y cannot be considered equal. The relationships among y, d, and h are as follows:
(2.1)
(2.2)
(2.3)
Figure 2.5 Channel depths and relations among variables.
Note: All three values of depth are essentially equal until the slope of the channel bottom becomes quite steep. A slope is considered steep when there is at least a one-percent difference between y and h. This difference occurs at an angle q of 5.7 degrees, which represents a 1 vertical to 10 horizontal (10-percent) slope.
Channel Top Width and Wetted Perimeter
The top width (T) of a channel is the horizontal width of the channel section at the water surface. The wetted perimeter (P) is the length of the channel boundary, typically the sides and bottom, that is in contact with the fluid; it is always larger than the top width, as shown in Figure 2.6. Both variables are used with the channel area term to develop two other variables that are important to open channel hydraulics: hydraulic depth and hydraulic radius.
Figure 2.6 Cross-section geometry.
Hydraulic Depth and Hydraulic Radius.
The hydraulic depth can be visualized as the average depth across the channel, whereas the depth (y) is the maximum depth at a cross-section location for the channel shapes shown in Figure 2.2. The equation for hydraulic depth is
| where |
D |
= |
the hydraulic depth (ft, m) |
| A |
= |
the cross-sectional flow area (ft2, m2) |
|
| T |
= |
the top width (ft, m) |
(2.4)
Another term that is critical in open channel flow problems is the hydraulic radius, given by
(2.5)
The hydraulic radius can also be thought of as a different measure of average channel depth. The two terms give values that are similar when the top-width-to-depth ratio for the flow area of any channel is greater than approximately 5. Hydraulic depth is most often used to determine the appropriate flow regime (subcritical or supercritical), while the hydraulic radius is most often applied in estimating channel velocity or discharge.
Expressions for A, P, R, T, and D for the channel shapes shown in Figure 2.2 are given in Table 2.1.
Discharge
The amount of water moving in a channel or stream system is characterized by the discharge (Q) or flow rate. The unit of discharge used in open channel flow is ft3/s for U.S. Standard units and m3/s for the SI system.
|
Channel Shape
|
Area, A
|
Wetted Perimeter, P
|
Hydraulic Radius, R
|
Top Width, T
|
Hydraulic Depth, D
|
|---|---|---|---|---|---|
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Rectangular
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Trapezoidal
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Triangular
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Circular1
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| 1f measured in radians. |
Velocity
The velocity is the speed at which the water moves in an open channel. The units for velocity are feet (meters) per second. Water movement adds kinetic energy to the system, which is computed using the stream velocity. The kinetic energy term is added to the water surface elevation to calculate the total energy head at a cross section. If the total energy head at several cross sections is connected by an imaginary line, the line is referred to as the energy grade line. These terms are further discussed in Section 2.3 and Section 2.6.
The equation for average velocity at any location is
(2.6)
Even though working with average velocities is convenient, the channel velocity is not constant at any location, regardless of whether the channel is prismatic or nonprismatic. An example is a small stream, where one can easily observe that the velocity near the channel bank is less than the velocity in the center of the channel. In fact, the velocity varies both horizontally and vertically for any given channel cross section. Figure 2.7 illustrates this phenomenon for different channel shapes.
Chow, 1959
Figure 2.7 Typical lines of equal velocity in various channel cross sections.
The difficulty of applying an average velocity is evident in the simple channel and floodplain section of Figure 2.8. The kinetic energy term is represented by the velocity head (V2/2g), where g is the gravitational constant (32.2 ft/s2 for English units, 9.81 m/s2 for SI). Analysis of open channel hydraulics problems typically requires the assumption that the water surface elevation and total energy elevation each have a constant value from one side of the cross section to the other side (defined as one-dimensional flow). Figure 2.8 shows different energy grade lines (elevation of the total energy head) for each of the three segments of the cross section having different average velocities.
Figure 2.8 Variation in velocity head at a cross section.
For the assumption of a constant energy grade line elevation for a section to be valid, a weighted velocity head must be developed that essentially collapses the three different values of the velocity head term for the channel and left and right floodplain areas into a single value. This modification incorporates a velocity distribution coefficient (a), thus making the kinetic energy head at a section equal to aV2/2g. The velocity distribution coefficient is given by
| where |
a |
= |
the velocity distribution coefficient (dimensionless) |
| Q1,2,3 |
= |
the discharges in the appropriate segments of the cross section of Figure 2.8 (ft3/s, m3/s) |
|
| V1,2,3 |
= |
the average velocities in the appropriate segments of the cross section (ft/s, m/s) |
|
| QTOT |
= |
the total discharge of the cross section (ft3/s, m3/s) |
|
| V |
= |
the average velocity in the full cross section (ft/s, m/s) |
(2.7)
Alpha (a) is always greater than or equal to one and generally ranges from 1.0 to 2.5. It is typically small for flows within the channel (1.0 < a < 1.5), but can be higher for floods occupying the channel and overbank areas or during the presence of ice jams (1.5 < a < 2.5).
Example 2.1 Computing the velocity distribution coefficient.
A certain discharge occurs in the complex channel for the dimensions and velocities shown in the figure. Compute the total discharge for the section and the velocity distribution coefficient.
The channel segment is separated from the left and right overbank segments by the imaginary vertical dashed line shown in the figure. This line is for illustration only and would not be included in hydraulic computations for the wetted perimeter.
Solution
The parameters for the left overbank area are
Left overbank area = 5 × 50 = 250 ft2
Velocity = 2.29 ft/s
Therefore, the flow rate is
left overbank discharge = AV = 250 × 2.29 = 572.5 ft3/s
Similarly, the area and discharge in the channel and right overbank area are
Channel area = 50 × 15 = 750 ft2
Channel discharge = 750 × 6.36 = 4770 ft3/s
Right overbank area = 50 × 3 = 150 ft2
Right overbank discharge = 150 × 1.33 = 199.5 ft3/s
Total discharge = 572.5 + 4770 + 199.5 = 5542 ft3/s
Average velocity = Q/A = 5542/(250 + 750 + 150) = 4.82 ft/s
The velocity distribution coefficient is computed with Equation 2.7 for the distribution of discharge and velocity in the section. As seen, the computed a results in a greater than 50-percent increase to the velocity head found using the average cross-section velocity:
