Water flows in a sloping drainage channel because of the force of gravity. Flow is resisted by the friction between the water and wetted surface of the channel. The quantity of water flowing (Q), the depth of flow (y), and the velocity of flow (V) depend upon the channel shape, roughness (n), and slope (S). Various equations have been devised to determine the velocity and discharge in open channels. A useful equation is the one that is named for Robert Manning, an Irish engineer.
Manning’s equation (HDS-4 Section 4.3.1) is valid for steady, uniform and turbulent flow, has the following form:
Q = (k/n)⋅A⋅R2/3⋅S½,
- k = 1.49 (US Customary units) or 1 (metric units),
- n is Manning’s roughness coefficient,
- A is the flow cross-section area (square feet or square meters),
- R is the hydraulic radius (feet or meters), and
- S is the longitudinal slope (feet/feet or meters/meters).
The hydraulic radius R is found by dividing the cross sectional area A by the wetted perimeter P. These cross-section properties are shown below for an open channel.
Over many decades, typical Manning’s n values have been compiled allowing an engineer to estimate the appropriate value by knowing the general nature of the channel boundaries. Most hydraulics textbooks and drainage design manuals provide tables of typical Manning’s n values. An abbreviated list of such Manning’s roughness coefficients is given in Appendix B, Table B.2 of HDS-4. Several pictorial guides are also available showing the Manning’s n value for different types of channels and floodplains (Barnes 1967 and Acrement and Schneider 1984). Special considerations exist for very steep channels (Jarrett 1985).
A numerical approach for n value estimates consists of the selection of a base roughness value for a straight, uniform, and smooth channel in the materials involved, and then adding values for the channel under consideration:
|n = (n0 + n1 + n2 + n3 + n4)•m5||(4.2)|
- no = Base value for straight uniform channels
- n1 = Additive value due to cross-section irregularity
- n2 = Additive value due to variations of the channel
- n3 = Additive value due to obstructions
- n4 = Additive value due to vegetation
- m5 = Multiplication factor due to sinuosity
A discussion of this method and coefficients can be found in Cowan (1956) and Chow (1959). This method may be useful for natural channels, but has limited application for most roadway drainage design work.
For rock riprap channels the Manning’s n is often described as some function of the rock size. Several equations are provided in HEC-15.
Roughness characteristics on the floodplain are complicated by the presence of vegetation, natural and artificial irregularities, buildings, undefined direction of flow, varying slopes, and other complexities. Resistance factors reflecting these effects must be selected largely on the basis of past experience with similar conditions. In general, resistance to flow is large on the floodplains. In some instances, conditions are further complicated by deposition of sediment and development of dunes and bars which affect resistance to flow and direction of flow.
The presence of ice affects channel roughness and resistance to flow in various ways. When an ice cover occurs, the open channel is more nearly comparable to a closed conduit. There is an added shear stress developed between the flowing water and ice cover. This surface shear is much larger than the normal shear stresses developed at the air-water interface. The ice-water interface is not always smooth. In many instances, the underside of the ice is deformed so that it resembles ripples or dunes observed on the bed of sand-bed channels. This may cause overall resistance to flow in the channel to be further increased. With total or partial ice cover, the drag of ice retards flow, decreasing the average velocity and increasing the depth.
When a channel cross section is irregular in shape such as one with a relatively narrow deep main channel and wide shallow overbank area, the cross section must be subdivided and the flow computed separately for the main channel and overbank area. The same procedure is used when different parts of the cross section have different roughness coefficients. In computing the hydraulic radius of the subsections, the water depth common to the two adjacent subsections is not counted as wetted perimeter (see Example Problem 4.3).
Conveyance can be computed and a curve drawn for any channel cross section. The area and hydraulic radius are computed for various assumed depths and the corresponding value of K is computed from the equation. Values of conveyance are plotted against the depths of flow and a smooth curve connecting the plotted points is the conveyance curve. If the section was subdivided, the conveyance of each subsection (Ka, Kb,…Kn) is computed and the total conveyance of the channel is the sum of the conveyances of the subsections. Discharge can then be computed using Equation 4.7
Example Problem 4.3 illustrates a conveyance curve for a compound cross section. The concept of channel conveyance is useful when computing the distribution of overbank flood flows in the stream cross section and the distribution through the openings in a proposed stream crossing. The discharge through each opening can be assumed to have the same ratio to the total discharge as the ratio of conveyance of the opening bears to the total conveyance of the channel.
4.3.2 Aids in the Solution of Manning’s Equation
Equations for the computation of Area, A, wetted perimeter, P, and hydraulic radius, R, in rectangular and trapezoidal channels (Figure 4.2) are:
|A = B•y + Z•y2||(4.8)|
|P = B + 2•y•*√(1 + Z2)||(4.9)|
|R = (B•y + Z•y2)/(B + 2•y•*√(1 + Z2))||(4.10)|
Variables are defined in Figure 4.2.
Figure 4.2. Trapezoidal channel.
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