The fundamental relationships for hydraulic flow are the same for channels that are physically open at the top, such as roadway channels and curbs and gutters, and for pipes and culverts that have a free water surface. In both cases, hydraulic design is based on open-channel flow. An understanding of these relationships is important for comprehending various design aids subsequently presented.
Open-channel flow may be categorized by three characteristics: the flow may be (1) steady or unsteady, (2) uniform or nonuniform, and (3) either subcritical, critical, or supercritical. This discussion will begin with the first two categories, and the third will be discussed later.
Steady flow means that at a particular point, there is no change in depth with respect to time. By extension, this means that there is no change in the quantity of flow. Unsteady flow means that the depth does change with time.
Uniform flow assumes that there is no change in depth or quantity of water at any section along the length of the channel (or culvert) under investigation. This requires that there be no change in velocity of the flow, and it is possible only if the slope, roughness, and cross-section all remain constant along the length of the channel. This state is evidenced by the fact that the water surface is parallel to the channel bottom. Nonuniform flow assumes a change in depth or velocity along the length of the channel. This type of flow may be further classified as rapidly varying or gradually varying flow.
For most highway applications, the flow is steady and the changes in the section are so gradual that the flow may be considered uniform. The equations for open-channel flow are based on that assumption. Where the change in the cross-section of the channel is dramatic, nonuniform flow should be assumed. (For analysis of nonuniform flow, see E. F. Brater and H. W. King, Handbook of Hydraulics, McGraw-Hill, 1996.)
The continuity equation is based on the basic and fundamental concept that the quantity of flow passing any cross-section remains constant throughout the length of the stream flow:
Q = AV (5.10)
where Q = discharge, ft3/s (m3/s)
A = area, ft2 (m2)
V = velocity, ft/s (m/s)
Manning’s equation assumes uniform, turbulent flow conditions and computes the mean flow velocity for an open channel:
V = ^ 1,486 ^R2/3S1/2 in U. S. Customary units (5.11a)
n2/3o1/2
V = for SI units (5.11b)
n
where V = mean velocity, ft/s (m/s)
n = Manning coefficient of roughness R = hydraulic radius = A/WP, ft (m)
A = cross-sectional flow area, ft2 (m2)
WP = wetted perimeter = total perimeter of cross-sectional area of flow minus free surface width, ft (m)
S = channel slope
Manning’s equation may be solved directly or obtained from the nomograph in Fig. 5.3. Typical Manning’s n values are given in Table 5.6. For shallow flows, the effective n values should generally be increased, because the wetted perimeter will have a greater effect on the flow.
The continuity equation and Manning’s equation may be used in conjunction to directly compute channel discharges. Substitute Eq. (5.11) into Eq. (5.10) and rearrange terms to obtain
AR2/3 = 14Q—1/2 in U. S. Customary units (5.12a)
AR2/3 = —— for SI units (5.12b)
R is a function of A. Thus, for a given slope, flow quantity, and n value, AR2/3 may be determined and the normal depth of flow calculated by trial and error.
