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Chapter 6 Open channel flow.

Open channel flow occurs when a free water surface in a channel is at atmospheric pressure. Common examples of open channel flow are rivers, streams, drainage ditches, and irrigation canals. Open channel flow may also occur in pipes if the pipe is not flowing full and the water surface is at atmospheric pressure.

Open channel design is common in many applications of soil and water conservation, including drainage and irrigation ditches, grassed waterways, reservoir spillways, and large culverts. In all these applications, the designer must consider channel shape, slope, hydraulic roughness or resistance to flow, and in many cases, channel resistance to erosion.

Channels may be earth or concrete lined, vegetated, lined with impervious material such as rubber or fabric, or lined with erosion-resistant material such as large rocks or high-strength geotextile materials. Channels may be left in a natural condition, shaped to achieve a desired capacity, or designed to minimize bed erosion. In some cases, channels may be confined by vertical sides made from materials that are resistant to erosion or sloughing. A properly designed, earth-lined open channel should provide (1) velocity of flow such that neither serious scouring nor sedimentation will result, (2) sufficient capacity to carry the design flow, (3) hydraulic grade at the proper depth for good water management, (4) sideslopes that are stable, and (5) minimum initial and maintenance costs. Additional requirements must be met for carrying irrigation water, such as low seepage loss. Details on open channel flow and design can be found in Chow (1959), French (1985), and Henderson (1966).

States of Flow

The states of flow vary with viscous, inertial, and gravitational forces. The Reynolds and Froude numbers define the states of flow.

6.1 Reynolds Number

The state of flow in open channels may be laminar or turbulent. In laminar flow, viscous forces predominate over inertial forces. In turbulent flow, inertial forces predominate over viscous forces. The state of flow is determined by the Reynolds number, which is the ratio of the inertial forces and viscous forces, and for open channels is given by

[R.sub.e] = inertial forces/viscous forces = [rho]vR/[mu] [6.1]

where [R.sub.e] = Reynolds number,

[rho] = fluid density (M/[L.sup.3]),

v = velocity (L/T)

R = hydraulic radius (L),

[mu] = dynamic viscosity (M [L.sup.-1] [T.sup.-1]).

The hydraulic radius is the cross-sectional area of the flow divided by the channel wetted perimeter P (Figure 6-1). The wetted perimeter is the length of the channel cross section in contact with water.

R = cross-sectional area of flow/wetted perimeter of the channel = A/P [6.2]

Note that the Reynolds number uses the hydraulic radius for open channel flow, whereas the diameter is used for pipe flow. Reynolds number less than 500 are laminar flow and numbers over 1000 are turbulent flow in channels. A transition range exists between 500 to 1000 where the state of flow is uncertain; however, flows in the transition region are not common in engineering applications.

6.2 Froude Number

The state of flow in channels may be subcritical, critical, or supercritical depending on the ratio of inertial to gravitational forces. The determination is made by the Froude number [F.sub.r], which is

[F.sub.r] = inertial forces/gravitational forces = v[(gy).sup.1/2] [6.3]

where g = the gravitational acceleration (L/[T.sup.2]),

y = the depth of flow (L).

The Froude number describes the following states of flow

[F.sub.r] < the flow is subcritical (quiet),

[F.sub.r] = 1 the flow is critical,

[F.sub.r] > 1 the flow is supercritical (rapid).

Most flow in open channels is subcritical. The most common exceptions are spillways and flow over weirs and through measuring flumes where the flow is critical.

[FIGURE 6-1 OMITTED]

Equations of Flow

6.3 Continuity Equation

The continuity equation is based on the conservation of mass and in its simplest form for an incompressible fluid is given by

Q = Av [6.4]

where Q is the flow rate ([L.sup.3]/T), A is the area of flow perpendicular to the average velocity ([L.sup.2]), and v is the average velocity (L/T).

6.4 The Bernoulli Equation

The Bernoulli equation is based on the conservation of energy and for open channels is described by

[y.sub.1] + [z.sub.1] + [v.sup.2.sub.1]/28 = [y.sub.2] + [z.sub.2] + [v.sup.2.sub.2]/28 + [([h.sub.f]).sub.1-2] [6.5]

where y = depth of flow above the channel bottom (L),

z = distance of channel bottom above a datum (L),

v = average flow velocity (L/T),

[h.sub.f] = head loss due to friction (L),

1, 2 = subscripts denoting two channel locations.

Figure 6-2 illustrates these variables for uniform flow in an open channel.

6.5 Specific Energy

A given discharge of water in an open conduit may flow at two depths (called alternate depths) having the same total energy head. For convenience in design of control structures, specific energy is often used rather than total energy (Henderson, 1966). Specific energy (or specific head) is the energy (or head) of flow with respect to the channel bottom. It disregards elevation potential, since elevation changes are negligible in short reaches of mildly sloped conduits. The specific energy of flow in a channel having a rectangular cross section may be expressed as

[H.sub.e] = y + [v.sup.2]/28 = y + [q.sup.2]/2[a.sup.2]g = y + [q.sup.2]/2[b.sup.2][y.sup.2]g [6.6]

where [H.sub.e] = specific energy (L),

y = depth of flow (L),

v = average velocity of flow (L/T),

g = gravitational acceleration (L/[T.sup.2]),

q = discharge ([L.sup.3]/T),

a = cross-sectional area of flow ([L.sup.2]),

b = width of flow (L).

[FIGURE 6-2 OMITTED]

[FIGURE 6-3 OMITTED]

Figure 6-3 shows the relationship of depth of flow to specific energy in a rectangular channel for two discharges. For any discharge, specific energy has a minimum value that corresponds to a unique depth of flow, which is called the critical depth. Most control structures include a section at which flow passes through the critical depth. Being a unique value, it is useful for determination of discharge capacities and is therefore frequently used in the design equations for certain components of control structures.

Example 6.1

Determine the specific energy of a channel with y = 0.5 m, b = 1.0, and q = 0.7 [m.sup.3]/s. Solution. Use Equation 6.6 to calculate [H.sub.e].

[H.sub.e] = y + [q.sup.2]/2[b.sup.2][y.sup.2]g = 0.5 + [0.7.sup.2]/2 x [1.0.sup.2] x [0.5.sup.2] x 9.81 = 0.6 m

6.6 Critical Depth

To determine the critical depth, differentiate Equation 6.6 with respect to y and set d[H.sub.e]/dy = 0. The value for y where [H.sub.e] is a minimum is the critical depth, [d.sub.c].

d[H.sub.e]/dy = 1 - [q.sup.2]/[b.sup.2][y.sup.3]g = 0

[d.sub.c] = [([q.sup.2]/[b.sup.2]g).sup.1/3] = 0 [6.7]

Equation 6.6 may also be used to determine the depth at which maximum discharge will occur for a given specific energy. Solve for q in Equation 6.6.

q = [[2[y.sup.2][b.sup.2]g ([H.sub.e] - y)].sup.1/2] 6.8

Assume [H.sub.e] is constant and set dq/dy = 0 to get

dq/dy = 2[b.sup.2]g (2y [H.sub.e] - 3[y.sup.2])/2[[2[y.sup.2][b.sup.2]g([H.sub.e] - y)].sup.1/2] = 0 [6.9]

If y = [d.sub.c], then solving Equation 6.9 gives

[d.sub.c] = 2/3 [H.sub.e] [6.10]

For a given specific energy, the maximum discharge occurs at the critical depth, that is, where [H.sub.e] Equation 6.8 is plotted in Figure 6-4 for [H.sub.e] = 1.0 m. The critical depth is 0.67 m, which corresponds to the maximum discharge of 1.83 [m.sup.3]/s per meter of channel width. The plot shows how the discharge drops off for any depth greater or less than the critical depth.

6.7 Hydraulic Jump as an Energy Dissipater

A change in flow depth from a depth less than the critical depth (supercritical) to a depth greater than the critical depth (subcritical) is called a hydraulic jump. Where the change in depth is great and the transition occurs very quickly, the jump is called a direct jump. Direct jumps are very turbulent and dissipate an appreciable fraction of the energy of the flow (Chow, 1959). The profile and depth-energy relationships of a hydraulic jump are shown in Figure 6-5. Note that the specific energy of flow after the jump is less than the specific energy of the flow entering the jump. Many control structures are designed so that a hydraulic jump forms within the structure, dissipating some of the energy of flow and reducing the flow velocity to a nonerosive value before exiting the downstream portion of the structure. The incoming depth of flow in Figure 6-5 must be in the supercritical range for a hydraulic jump to occur.

Types of Flow

The flow in open channels can be categorized according to the type of flow. The simplest case is steady, uniform flow (Figure 6-2). Uniform flow occurs in channels with uniform cross section, slope, flow depth, and flow rate. The flow is varied if any of the channel characteristics change along the channel length. Gradually varied flow occurs when the change in characteristics with length is small (Figure 6-6). Rapidly varied flow occurs when the change with length is rapid, such as at a sharp change in slope, over a weir, or a free overfall. With unsteady flow the depth and/or flow rate vary with time along the channel. Unsteady flow is nearly always nonuniform and much more difficult to analyze. Fortunately, the majority of varied flows are steady and uniform in a piece-wise manner. This chapter emphasizes steady, uniform flow.

[FIGURE 6-4 OMITTED]

Uniform Flow

In uniform flow the slope of the energy grade line, the water surface slope and the bottom slope are equal. Uniform flow occurs in long channels of uniform shape, constant flow rate, constant slope, and no obstructions (Figure 6-2). The depth of flow in such channels is known as the normal depth. If the normal depth is greater than the critical depth, the flow is subcritical ([F.sub.r] < 1). If the normal flow depth is less than the critical depth, the flow is supercritical ([F.sub.r] > 1).

Special equations have been developed for uniform flow conditions. These equations simplify the design and analysis of channels with uniform flow.

6.8 The Manning Equation

[FIGURE 6-5 OMITTED]

The Manning equation is widely used because of its simplicity and accuracy and is expressed as

v = [C.sub.u][R.sup.2/3][S.sup.1/2.sub.f]/n [6.11]

where v = average flow velocity (L/T),

[C.sub.u] = 1.0 [m.sup.1/2]/s for units in meters and seconds,

R = hydraulic radius (flow area divided by wetted perimeter) (L) (see Figure 6-1 for equations),

[S.sub.f] = friction slope (L/L),

n = Manning roughness coefficient, including soil and vegetative effects ([m.sup.1/6]) (see Appendix B for recommended values).

The Manning equation is also combined with the continuity equation (Equation 6.6) as follows

[FIGURE 6.6 OMITTED]

Q = [C.sub.u][A.sup.5/3[S.sup.1/2.sub.f]/n[P.sup.2/3] 6.12

where Q is the flow rate (L/[T.sup.3]).

Example 6.2

What is the normal flow depth in a rectangular concrete-lined channel, with b = 2 m wide, slope = 0.001 m/m, flow rate is 1 [m.sup.3]/s? What is the Froude number? Is the flow subcritical, critical, or supercritical?

Solution. Assume a depth, calculate Q, and compare to the required discharge. Substitute the equations for A and P into Equation 6.12 and solve for y by iteration. From Appendix B, read n for concrete = 0.015 and take [C.sub.u] = 1 for SI units.

Q = 1 [(b y).sup.5/3]/n[(b + 2y).sup.2/3] [S.sup.1/2.sub.f] = 1/0.015 [(2y).sup.5/3]/[(2 + 2y).sup.2/3] [0.001.sup.1/2]

Assume y = 0.3 m

Q = 1 [(2 x 0.3).sup.5/3]/0.015 [(2 + 2 x 0.3).sup.2/3] [0.001.sup.1/2] = 0.48 m/s

This is below the actual flow of 1.0 [m.sup.3]/s. Increase y and iterate to find that y = 0.496 m gives Q = 1.0 [m.sup.3]/s. The velocity is 1.0/(2 x 0.496) = 1.01 m/s. The Froude number is

[F.sub.r] = v/[(gy).sup.1/2] = 1.01/[(9.81 x 0.496).sup.1/2] = 0.46

The normal depth is 0.496 m and the flow is subcritical because [F.sub.r] < 1.

Channel Design

The major objective of open channel design is to provide a channel that has adequate capacity and is stable. Generally, one of the first steps in any channel design is a field survey to determine the limitations the grade places on the channel design. Wherever practical, the channel should be designed for a high hydraulic efficiency.

6.9 Channel Grade

The engineer frequently has little choice in the selection of grades for channels, because the grade is determined largely by the outlet elevation, elevation and distance to the beginning of the channel, and the channel depth. An allowance may be made for accumulation of sediment, depending on channel velocities and soil conditions. Structures may be required to limit grade, or specialized designs may be necessary to elevate the channel to maintain a uniform grade in uneven terrain.

In earth channels the stability of the soil places limitations on channel grade and sideslopes. The topography and desired water levels may limit the design grade and the velocity of flow.

6.10 Channel Roughness

The Manning roughness coefficient n for open channels varies with the height, density, and type of any vegetation; physical roughness of the bottom and sides of the channel; and variation in channel size and shape of the cross section, alignment, and hydraulic radius. Primarily because of differences in vegetation, the roughness coefficient may vary from season to season. In general, conditions that increase turbulence increase the roughness coefficient. Values of the Manning roughness n can be obtained from Appendix B or hydraulics books.

Experience and knowledge of local conditions are helpful in the selection of the roughness coefficient, but it is one of the most difficult values to define in channel design. In general, a value of 0.035 is satisfactory for medium-sized earth ditches with bottom widths of 1.5 to 3 m, and 0.04 is suitable for smaller bottom widths.

6.11 Channel Cross Sections

Natural channels tend to be parabolic in cross section. Constructed channels may be rectangular, trapezoidal, triangular, or parabolic. Under the normal action of channel flow, deposition, and bank erosion, unlined trapezoidal and triangular channels tend to become parabolic. The geometric characteristics of the trapezoidal, triangular, and parabolic channels are given in Figure 6-1. The figure defines the cross sections and presents the formulas to calculate area, wetted perimeter, hydraulic radius, and top width for each of the three.

Earth channels and lined canals are normally designed with trapezoidal cross sections (Figure 6-1). Waterways and streams are frequently designed with parabolic cross sections. The size of the cross section will vary with the velocity and quantity of water to be removed. Designers are often only able to vary the sideslopes and bottom width of a channel to meet the capacity requirements.

Channel dimensions are generally determined by an iterative analysis with a programmable calculator, spreadsheet, computer equation solver, or the NRCS (NRCS, 2002) computer programs to determine channel dimensions. The depth of the constructed channel should provide a freeboard allowance (D 2 d in Figure 6-1). Freeboard increases the depth to provide a safety factor to prevent overtopping and to allow a reasonable depth for sediment accumulation. Freeboard is typically 20 percent of the total depth, D. On large earthen irrigation canals the freeboard generally varies from 30 to 35 percent, and for lined canals it ranges from 15 to 20 percent. The freeboard for lined canals refers only to the top of the lining. The total freeboard is the same for lined and unlined channels. The freeboard should be increased on the outside edge of curves.

Sideslopes. Channel sideslopes are determined principally by soil texture and stability. The most critical condition for sloughing occurs after a rapid drop in the flow level that leaves the banks saturated, causing a positive pore water pressure that makes the banks highly unstable. For the same sideslopes, the deeper the ditch the more likely it is to slough. Sideslopes should be designed to suit soil conditions and not the limitations of construction equipment. Suggested sideslopes are shown in Table 6-1. Whenever possible, these slopes should be verified by experience and local practices. Narrow ditches should have slightly flatter sideslopes than wider ditches because of greater reduction in capacity in the narrow ditches if sloughing occurs.

Bottom Width. The bottom width for the most hydraulic-efficient cross section and minimum volume of excavation can be computed from the formula

b = 2d/z + [square root of [z.sup.2] + 1] [6.13]

where b = bottom width (L),

d = design depth (L),

z = sideslope ratio (L/L) (horizontal/vertical expressed as z:1).

For any sideslope it can be shown mathematically that, for a bottom width computed from Equation 6.13, the hydraulic radius is equal to one half the depth. The minimum bottom width of larger channels should be 1.2 m. It is not always possible to design for the most efficient cross section because of construction equipment limitations, site restrictions, allowable velocities, or increased maintenance.

Example 6.3

Determine the most hydraulically efficient cross section for a channel with Q = 4.4 [m.sup.3]/s, So = 0.2 percent, z = 2:1, and n = 0.02.

Solution. Perform an iterative solution where b is a function of d using Equations 6-12 and 6-13, and Figure 6-1. Assume an initial flow depth of 1.5 m with the variables in Figure 6-1.
Estimate    Equation     Figure 6-1
              6.13

   d            b             A

  1.5         0.71          5.56
  1.1         0.52          2.99

Figure 6-1  Equation
              6.12

   R            Q        Comment

  0.75        10.27      Q too high
  0.55         4.49      Q satisfactory

The most efficient cross section has a bottom width less than 1.2 m,
which may be difficult to construct with some machinery.


6.12 Channel Velocity and Tractive Force

In earth or vegetated channels, a channel shape is generally selected that will minimize the risk of channel erosion. The desired shape may be based on maximum velocities or on the shear strength of the channel material.

When designing for maximum velocity, no definite optimum velocity can be prescribed, but minimum and maximum limits can be approximated. Velocities should be low enough to prevent scour but high enough to prevent sedimentation. Usually an average velocity of 0.5 to 1.0 m/s for shallow channels is sufficient to prevent sedimentation. In channels that flow intermittently, vegetation may retard the flow to such an extent that adequate velocities at low discharges are difficult to maintain, if not impossible. For this reason maximum or even higher velocities may be desired, since scouring for short periods at high flows may aid in maintaining the required cross section.

A typical velocity distribution in an open channel is shown in Figure 6-7. The maximum velocity occurs near the center of the channel and slightly below the surface. The average velocity is generally about 70 percent of the maximum velocity near the surface. The maximum velocity and the velocity contours move toward the outer bank on channel bends.

Limiting Velocity Method. Maximum recommended velocities for earth-lined channels are presented in Table 6-2. These velocities may be exceeded for some soils where the flow contains sediment because deposition may produce a channel bed resistant to erosion. Where an abrasive material, such as coarse sand, is carried in the water, these velocities should be reduced by 0.15 m/s. For depths over 0.9 m, velocities can be increased by 0.15 m/s. If the channel is winding or curved, the limiting velocity should be reduced about 25 percent. Limiting velocities for grassed waterway designs are a specialized case and are discussed in Chapter 8.

Tractive Force Method. Tractive force is based on the hydraulic shear stress from the flowing water on the periphery of the channel. The method can be applied to earthen channels, or used to evaluate various channel liners (MoDOT, 2003). The critical tractive force is the maximum hydraulic shear that the channel lining can withstand before eroding. In a uniform channel of constant slope and constant flow, the water flow is steady state and uniform (without acceleration) because the force tending to prevent particle motion is equal to the force causing motion. For small channel slopes, hydraulic shear stress can be calculated from

[tau] = [rho]gR[S.sub.f] [6.14]

where [tau] = hydraulic shear stress (Pa),

[rho] = density of water (1000 kg/[m.sup.3]),

g = gravitational acceleration (9.81 m/[s.sup.2]),

R = hydraulic radius (Equation 6.3) (m),

[S.sub.f] = hydraulic gradient, generally assumed to be the channel slope (m/m).

When designing channels with the tractive force method, the channel is designed to ensure that the tractive force does not exceed a critical shear value for the bed of the channel. Critical shear depends on lining material properties. For cohesive soils, it has been correlated with the plasticity index, dispersion ratio, particle size, clay content, and void ratio. In noncohesive coarse soils, it approaches a value approximately equal to the diameter of the bed material in millimeters. For example, a gravel channel with bed material about 10 mm dia. has an estimated critical tractive force of about 10 Pa. Typical critical shear values for natural and artificial channel linings are presented in Table 6-2. Local design practices and critical shears for channel lining materials should be followed if they are available.

Example 6.4

Using the same conditions as described in Example 6.3, use the Tractive Force method to design an open channel with an 8-mm-dia. gravel lining to carry 4.4 [m.sup.3]/s with a slope of 0.2 percent. Assume a trapezoidal cross section with sideslopes of 2:1 and n = 0.020.

Solution. For 8-mm-dia. gravel, the critical shear value can be estimated as 8 Pa. Solving Equation 6.14 for the hydraulic radius where the critical shear is 8 Pa gives

R = 8.0/1000 x 9.8 x 0.002 = 0.41 m

Set up a spreadsheet or equation solver for an iterative solution. Starting with the solution to Example 6.2, decrease the depth and increase the width until a solution is obtained that provides the desired flow rate for the least width, to minimize excavation.
d      b      A      R      v       Q       Comment

1.1    0.52   2.99   0.55    1.50    4.49   R too high in
                                              Example 6.2
0.46    7.0   3.64   0.40    1.22    4.44   R and Q satisfactory

The depth of 0.46 m and bottom width of 7.0 m meet the design
requirements. The velocity of 1.22 m/s is also within the range of
small gravel as recommended in Table 6-2.


Gradually Varied Flow

Thus far, a straight, uniform-sloped channel has been assumed. For simple design problems the depth of flow and the velocity can be assumed constant. For this uniform flow condition the energy grade line, hydraulic grade line, and the channel bottom are all parallel (Figure 6-2). For gradually varied flow the channel characteristics change slowly with distance and the bottom slope, hydraulic grade line, and energy grade line are not parallel (Figure 6-8).

For gradually varied flow, special design considerations are needed. For example, if an obstruction, such as a culvert or a weir, is placed in the channel with subcritical flow, a concave water surface called a "backwater curve" develops upstream of the structure. A change from a gentle to a flatter grade could produce the same effect. If the channel slope changed from a flat to a steep grade, a convex water surface would develop. These examples illustrate two common types of gradually varied flow conditions.

[FIGURE 6-7 OMITTED]

The equation for gradually varied flow in a rectangular channel can be obtained from the following form of the Bernoulli equation (Henderson, 1966)

H = z + y + [v.sup.2]/2g [6.15]

where H is the total energy. Differentiating Equation 6.15 with respect to distance x and writing in finite difference form yields

[DELTA]H/[DELTA]x = [DELTA]z/[DELTA]x + [DELTA]/[DELTA]x (y + [v.sup.2]/2g) = [DELTA]z/[DELTA]x + 1/[DELTA]x ([y.sub.2] + [v.sup.2.sub.2]/2g - [y.sub.1] - [v.sup.2.sub.1]/2g) [6.16]

Substituting [DELTA]H/[DELTA]x = - [S.sub.f] and [DELTA]z/[DELTA]x = - [S.sub.o] into Equation 6.16 and solving for

[DELTA]x gives the equation for gradually varied flow

[DELTA]x = ([y.sub.2] + [v.sup.2.sub.2]/2g) - (y.sub.1] + [v.sup.2.sub.1]/2g)/[S.sub.o] - [S.sub.f] = [H.sub.e2] - [H.sub.e1]/[S.sub.o] - [S.sub.f] [6.17]

where [S.sub.o] is the bottom slope of the channel, [S.sub.f] is the friction slope calculated from the Manning equation, assuming uniform flow over the channel segments, and the specific energy is determined from Equation 6.6.

6.13 Backwater Curve

Equation 6.17 can be numerically solved to determine the shape of the water surface for gradually varied flow in channels. The most common example is a backwater curve formed when a flow restriction (Figure 6-8) is placed in channel with a mild slope [F.sub.r] < 1. The solution process for subcritical flow begins at the obstacle and proceeds upstream.

[FIGURE 6-8 OMITTED]

Example 6.5

Assume a weir is to be placed in the channel in Example 6.2. If the weir will raise the flow depth 0.1 m, how far upstream will the flow depth be 0.5 m?

Solution. (1) Calculate area, wetted perimeter, and velocity of flow at y = 0.596.

A = by = 2.0 x 0.596 = 1.192 [m.sup.2]

P = b + 2y = 2.0 + 2 x 0.596 = 3.192 m

v = Q/A = 1.0/1.192 = 0.8389 m/s

(2) Calculate [S.sub.f] from Equation 6.12.

[S.sub.f] = [(Qn[P.sup.2/3]/A 5/3).sup.2] = [(1.0 x 0.015 x [3.192.sup.2/3]/1.192.sup.5/3]/[1.192.sup.5/3]).sup.2] = 0.000589

(3) Determine [H.sub.e] from Equation 6.6.

[H.sub.e] = y + [v.sup.2]/2g = 0.596 + [0.8389.sup.2]/2 x 9.81 = 0.6319 m

(4) Reduce y by 0.01 m to 0.586 m and repeat steps 1, 2, and 3 to obtain

A = 1.171 [m.sup.2], P = 3.172 m, v = 0.8532 m/s, [S.sub.f] = 0.000618, [H.sub.e] = 0.6231 m

(5) [DELTA]x is now determined from Equation 6.17.

[DELTA]x = 0.6231 - 0.6319/0.001 - (0.000589 + 0.000618)/2 = - 22.2 m

Note that [S.sub.f] is taken as the mean over the interval.
(6) Results from a spreadsheet for these calculations are

        A
y (m)   ([m.sup.2])  P (m)      v (m/s)  [S.sub.f]

0.596   1.192        3.192      0.8389   0.000589
0.586   1.172        3.172      0.8532   0.000618
0.576   1.152        3.152      0.8681   0.000649
0.566   1.132        3.132      0.8834   0.000682
0.556   1.112        3.112      0.8993   0.000718
0.546   1.092        3.092      0.9158   0.000756
0.536   1.072        3.072      0.9328   0.000798
0.526   1.052        3.052      0.9506   0.000842
0.516   1.032        3.032      0.9690   0.000890
0.506   1.012        3.012      0.9881   0.000941
0.5     1            3          1.0000   0.000974

        [H.sub.e]    [DELTA]x
y (m)    (m)         (m)        x (m)

0.596   0.6319                  0
0.586   0.6231       -22.2      -22
0.576   0.6144       -23.7      -46
0.566   0.6058       -25.8      -72
0.556   0.5972       -28.5      -100
0.546   0.5887       -32.2      -133
0.536   0.5804       -37.6      -170
0.526   0.5721       -46.1      -216
0.516   0.5639       -61.2      -277
0.506   0.5558       -95.8      -373
0.5     0.5510       -112.8     -486

The depth reaches 0.5 m at a distance of 486 m upstream
from the weir. This demonstrates that in channels with
small slopes, a partial obstruction will affect the flow
depth for large distances upstream.


Water Measurement

Good water management requires that flow rates and volumes be quantified. The most common measurements are flow velocity and stage. These are used in conjunction with knowledge of the shape of a channel or impoundment to calculate the flow rate or storage volume. Determination of a volume of flow requires a series of flow rate measurements that are integrated over the time of interest.

6.14 Units of Measurement

For volume flow rates, units in common use are liters per second or cubic meters per second. Units for volume are liters, cubic meters, or particularly for irrigation purposes, hectare-millimeters or hectare-meters. Flow velocity is typically in meters per second.

6.15 Float Method

A crude estimate of the velocity of water in an open channel can be made by determining the velocity of an object floating with the current. Select and mark a reasonably straight and uniform channel section about 100 meters long. Measure the time required for an object floating on the surface to travel the selected distance and calculate its velocity. Repeat this several times and calculate an average surface velocity. The mean velocity of the stream can be estimated at 0.8 to 0.9 times the average surface velocity.

[FIGURE 6-9 OMITTED]

6.16 Impeller Meters

Many instruments employ an impeller that rotates at speeds proportional to the velocity of the water. The relationship between rotational speed and velocity of water is usually not linear, so a calibration curve must be employed. Most modern impeller meters integrate this calibration into the electronic readout signal. Some impeller meters are designed for use in open channels (often called current meters), whereas others are designed for installation in pipes. Impeller meters function well in clear or sediment-laden water, but are not suitable for applications with large suspended solids or fibrous materials that could jam or tangle in the mechanism. Figure 6-9 shows examples of current meters.

Current meters measure the flow velocity at a specific point in the stream. It is not possible, however, to predetermine a single point where a measurement would represent the average velocity of the stream. To obtain a good estimate of the total stream flow, the cross section can be divided into a number of subsections in which measurements are taken to represent the velocity in each subsection (Figure 6-10). The average of readings taken at 0.2 and 0.8 times the depth provides a good estimate of the average velocity in a deep subsection. Since impeller-type current meters should not be used within 2-3 impeller diameters of the stream boundary (USBR, 2001), where the stream is too shallow to allow the reading at 0.8 times the depth, the velocity at 0.6 times the depth from the water surface may be taken as the average velocity. Summing the products of average velocity and cross-sectional area for each subsection gives the total stream flow. This method is demonstrated below.

[FIGURE 6-10 OMITTED]

Example 6.5 Determine the discharge for the stream shown in Figure 6-10. The measured velocities are given in the following table.

Solution. Begin by dividing the cross section into sections that can be approximated by triangles or trapezoids, as shown by the dashed lines in the figure. For each section, find its width and average depth. For a shallow section, calculate the measurement distance from the water surface as 0.6 times the average depth for that section. For deeper sections, calculate the measurement depths as 0.2 and 0.8 times the average depth of each section. Measure the velocities at those points. Then find the average velocity for each of the deeper sections as the average of the two measurements. Finally, multiply the area of each section by its average velocity and add them to obtain total discharge. The measurements and calculations for the stream are given in tabular form below.
                           Avg.          Meas.       Measured
             Width         Depth         Depths      Velocity
Section       (m)           (m)         (m) (a)        (m/s)

1             2.0           0.9           0.54         0.09
                                          0.40         0.29
2             3.2           2.0
                                          1.60         0.25
                                          0.60         0.33
3             2.4           3.0
                                          2.40         0.27

                                          0.62         0.32
4             2.6           3.1
                                          2.48         0.28
                                          0.42         0.28
5             1.4           2.1
                                          1.68         0.26
6             2.6           0.7           0.42         0.11
Total

            Average
            Velocity       Area        Discharge
Section      (m/s)      ([m.sup.2])   ([m.sup.3]/s)

1             0.09         1.80          0.162

2             0.27         6.40          1.728

3             0.30         7.20          2.160

4             0.30         8.06          2.418

5             0.27         2.94          0.794

6             0.11         1.82          0.200
Total                                     7.46

(a) Distance from the water surface where velocities
were measured.

The number of sections to use for a particular stream
is a matter of judgment.

Where cross sections have more complex shapes, use
more sections.


A variation of the impeller meter is the paddle-wheel meter. A small paddle wheel with a shield that protects half of it from exposure to the flow is positioned inside a pipe. The speed of rotation is proportional to the flow velocity.

6.17 Doppler Meters

Doppler meters emit acoustic signals into a stream and measure the apparent frequencies of the reflections. Although the concept is simple, the actual workings are not, and will not be detailed here. Doppler meters require something in the water to reflect the signal, such as suspended solids or bubbles. Some are designed for use in open water as current meters. Others are intended for use in well-defined sections such as culverts. The meter lies on the bottom of the channel, has an integrated pressure sensor for flow depth measurement, and can be programmed to calculate discharges directly. Such a configuration is not suitable for a stream with a high sediment load that could bury the meter.

6.18 Magnetic Meters

Magnetic meters are based on the principle of Faraday's law, that the voltage induced in a conductor moving at right angles to a magnetic field is proportional to the velocity of the conductor. Since natural waters contain some dissolved salts, they are conductive and work with this method. The moving water (conductor) causes a distortion in an induced magnetic field that is sensed by the meter and converted to a velocity readout.

Magnetic meters are used with pipe flow. Various models are available that either clamp onto the outside of the pipe or are fabricated as short in-line sections.

6.19 Pitot Meters

Pitot tubes are used in a variety of meters for flow velocity. Pitot tubes are probes with openings positioned to sense the velocity head of the flow. Modern pitot meters couple pressure transducers with microprocessors to sense the differential pressures and calculate flow velocities and/or discharges.

6.20 Slope Area

The principles of open channel flow can be used to estimate velocity and discharge. The reach of channel should be uniform in cross section and, if possible, up to 300 meters long. Given data for the shape of the cross section, the gradient, and the channel roughness, the Manning equation can be applied. This method gives relatively crude estimates with natural channels because of the difficulty of selecting an appropriate value for the roughness coefficient.

6.21 Orifice Meters

Orifice meters are usually circular openings in a plate placed perpendicular to the flow stream to measure the flow rate. The orifice flow equation is based on the conservation of energy with an orifice discharge coefficient accounting for the head loss through the orifice and the velocity of the upstream flow. The equation for flow through a sharp-edged orifice is

Q = CA[(2gh).sup.1/2] [6.18]

where Q = flow rate ([L.sup.3]/T),

C = discharge coefficient,

A = area of the orifice ([L.sup.2]),

g = acceleration due to gravity (L/[T.sup.2]),

h = head on the orifice (L).

[FIGURE 6-11 OMITTED]

The value of C is 0.61 to 0.63 if the orifice area is small relative to the upstream flow area. The head is measured from the center of the orifice to the upstream water surface, or the head difference if the orifice outlet is submerged. As the orifice size becomes larger relative to the upstream flow area, the value of C increases.

If an orifice plate is installed in a pipe as shown in Figure 6-11a, the differential head measurements (upstream one pipe diameter and downstream one half the pipe diameter) are used to calculate the flow rate. If an orifice is configured as an end-cap on a horizontal pipe (Figure 6-11b), a head measurement taken about 0.6 m upstream can be used to calculate the discharge. The orifice diameter should be 0.5-0.8 times the pipe diameter. The orifice coefficient must be determined for each configuration and can be obtained from hydraulic handbooks.

6.22 Coordinate Method

The coordinate method can be used where water discharges freely from the end of a horizontal pipe. Measurements of the distance the stream has fallen at a given distance from the end of the pipe are used to calculate the exit velocity (Figure 6-11c). Discharge is estimated as the exit velocity times the cross-sectional area of the pipe.

6.23 Level Sensors

Many applications require measurement of water levels or depths. Methods range from something as simple as a staff gauge to ultrasonic-level sensors. There are more methods available than can be discussed here. The best choice of sensor will depend on the application, data requirements, power available at the site, temperature, consequences of failure, etc. A few methods for sensing water levels are mentioned below.

Float. A number of devices use a float to ride the water surface, providing a mechanical link to some type of position sensor. Floats linked to pens that traced lines on moving paper charts were the standard for much of the 1900s. Floats have also been used to carry small magnets that activate reed switches or otherwise influence accessory circuits.

Capacitance. The dielectric constant of water is significantly different from that of air. As a water level rises and falls around a probe, it changes the capacitance of the probe, which can be detected and converted to a level readout.

Pressure transducer. Pressure transducers can be used in a variety of ways. The simplest is to submerge the transducer with its sensing element at a known position. A pressure signal can be easily converted to a depth of water above the sensor. Indirect linkage can be provided in a bubbler arrangement. A gas-filled tube is connected to the pressure transducer and positioned with the open end at a known elevation. As long as the water does not intrude into the tube, the pressure at the transducer will be the same as the pressure at the end of the tube. That value can be converted to a depth readout, as above.

A wide range of pressure transducers is available. Unit costs have declined as production methods improved to the point that many are both inexpensive and robust. Pressure transducers are probably the most widely used method for level sensing, especially for research or feedback/control applications.

Ultrasonic. In some applications (e.g., a tank or a manhole), an ultrasonic unit can be positioned some distance above the water surface. An acoustic signal is emitted and the time of return of the reflected signal is used to calculate the distance to the water surface. Ultrasonic sensors require a fairly clean and quiet surface. Foam or floating debris can absorb or scatter the signal.

6.24 Weirs and Flumes

For open channel flow measurements, the best accuracy requires structures having known hydraulic behavior. Weirs and flumes cause water to pass through a critical depth, which is unique for a given structure and flow rate.

Weirs. Weirs are the simplest, least expensive, and most common devices used for flow measurements in open channels (Figure 6-12). A weir is some type of barrier placed in the channel that restricts the flow, causing it to flow through a well-defined section. The most common are sharp-crested weirs, which are very thin in the direction of flow so that the flow will spring free of the weir crest. Weirs are further classified according to the shape of the opening or notch and whether the weir is contracted, i.e., narrower than the channel. Rectangular-notch weirs are most common. For accurate measurement of small flows, V-notch weirs are best. Trapezoidal notches provide a combination of the characteristics of the rectangular and V-notch designs. A number of more exotic shapes have been developed to provide specific stage-discharge relationships. For design details, rating curves, etc., see a hydraulics reference such as Brater et al. (1996).

[FIGURE 6-12 OMITTED]

Weirs can be designed for a wide range of flow rates. Equations for discharge for several common weirs are shown in Figure 6-12 for lengths in m and flow rate in [m.sup.3]/s. For rectangular and trapezoidal weirs, discharge is proportional to h3/2. For V-notch weirs, the exponent on h is 2.5. The weir coefficient C depends on the specific weir configuration. (To obtain SI coefficients from English values, multiply by 0.552.)

Head measurements should be taken upstream of the weir, at a distance at least 3-4 times the maximum height of flow over the weir. A limitation on the application of weirs is that they require significant upstream--downstream differences in water levels to work properly. Common problems with weirs are the tendencies for sediment deposition in front of the weir and for debris to snag on the weir crest. A detailed presentation on weirs can be found in USBR (2001).

Flumes. Flumes are stable, specially shaped channel sections that cause flow to pass through well-defined configurations. Compared to weirs, they pass debris much more easily and do not require as much head change. Parshall (1950) developed a common measuring flume (Figure 6-13). Designs are available for throat widths from 25 mm to 3.66 m, with flow rates from 0 to 7.67 [m.sup.3]/s. Design specifications and discharge relationships may be found in Parshall (1950) or hydraulics handbooks. Various flow calculators are available on the Internet.

Simpler flumes that are easier to construct have been developed. These include the cutthroat flume described by Skogerboe (1973) and the long-throated flume designed by Replogle (Clemmens et al., 2001) shown in Figure 6-14. The Bureau of Reclamation recommends the use of long-throated flumes for their many advantages (USBR, 2001) and distributes the design software WinFlume via the Internet (Wahl, 2001). For applications such as runoff measurement from small plots, the H-flume developed by the USDA Soil Conservation Service in the 1930s provides good accuracy at low flows as well as having high capacity. H-flumes (Figure 6-15) have flat bottoms and require very little head loss. They pass debris well.

[FIGURE 6-13 OMITTED]

[FIGURE 6-14 OMITTED]

There are three H-flume variants: HS for flows up to 0.023 [m.sup.3]/s, H for flows up to 0.88 [m.sup.3]/s, and HL for flows up to 3.3 [m.sup.3]/s. As with weirs, the water level must be measured a short distance upstream from a broad-crested or H-flume. The discharge can then be determined from a rating table. Computer programs are also available for discharge calculation. Clemmens et al. (2001) provide a thorough treatment of the use of flumes and weirs.

[FIGURE 6-15 OMITTED]

Internet Resources

U. S. Army Waterways Experiment Station library

http://libweb.wes.army.mil

U. S. Bureau of Reclamation library

http://www.usbr.gov/main/library/

Source of pictures and Manning n values for channels

http://manningsn.sdsu.edu

References

Brater, E. F., H. W. King, J. E. Lindell, & C. Y. Wei. (1996). Handbook of Hydraulics, 7th ed. New York: McGraw-Hill.

Clemmens, A. J., T. L. Wahl, M. G. Bos, & J. A. Replogle. (2001). Water Measurement with Flumes and Weirs. Wageningen, The Netherlands: International Institute for Land Reclamation and Improvement.

Chow, V. T. (1959). Open-Channel Hydraulics. New York: McGraw-Hill.

Fischenich, C. (2001). Stability thresholds for stream restoration materials. U.S. Army Corps of Engineers. Online at http://www.wes.army.mil. Accessed January 2004.

French, R. H. (1985). Open-Channel Hydraulics. New York: McGraw-Hill.

Henderson, F. M. (1966). Open Channel Flow. New York: Macmillan.

Lane, E. W. (1955). Design of stable channels. Transactions of the ASCE, 120, 1234-1260.

Missouri Department of Transport (MoDOT). (2003). Chapter 9, Hydraulics and drainage, Section 9-04. Online at http://www.modot.state.mo.us. Accessed January 2004.

Natural Resource Conservation Service (NRCS). (2001). Part 650: Engineering Field Handbook. National Engineering Handbook, Chapter 14. Online at http://www.info.usda.gov/CED/ftp/CED/EFH-Ch14.pdf. Accessed January 2004.

--. (2002). Ohio Engineering Software. Online at http://www.zert.info. Accessed December 2003.

Parshall, R. L. (1950). Measuring Water in Irrigation Channels with Parshall Flumes and Small Weirs. USDA Agriculture Cirular 843. Washington, DC: USDA.

Skogerboe, G. V. (1973). Selection and Installation of Cutthroat Flumes for Measuring Irrigation and Drainage Water. Colorado State University Engineering Experiment Station Technical Bulletin 120.

U.S. Bureau of Reclamation (USBR). (1987). Design of Small Dams, 3rd ed. Washington, DC: U.S. Government Printing Office.

--. (2001). Water Measurement Manual, 3rd ed., rev. Denver, CO: U.S. Government Printing Office.

Wahl, T. L. (2001). WinFlume User's Manual. U.S. Department of the Interior, Bureau of Reclamation. Online at http://www.usbr.gov/wrrl/winflume. Accessed January 2004.

Problems

6.1 What is the flow rate in a trapezoidal channel if the depth of flow is 0.5 m, slope is 0.002 m/m, bottom width is 1 m, and sideslope is 1:1? Is the flow subcritical?

6.2 What is the normal depth of flow in the channel in Problem 6.1 if the flow rate is 0.5 [m.sup.3]/s?

6.3 Determine the velocity of flow in a parabolic, a triangular, and a trapezoidal waterway, all having a cross-sectional area of 2 [m.sup.2], a depth of flow of 0.3 m, a channel slope of 4 percent, and a roughness coefficient of 0.04. Assume 4:1 side slopes for the trapezoidal cross section.

6.4 Compute the most efficient bottom width for a trapezoidal channel (z = 2) in a silty loam soil with a flow depth of 0.75 m. What are the velocity and the channel capacity if the hydraulic gradient is 0.1 percent and n = 0.04? Is the velocity below the critical velocity for this channel lining?

6.5 Determine the specific energy for the channel in Problem 6.4. Is the flow subcritical?

6.6 If the critical shear on the bottom of a channel in cohesive soil is 10 Pa, what is the maximum average velocity of flow at maximum slope for a trapezoidal channel where the depth of flow is 1.2 m, bottom width 1.5 m, n = 0.04, and sideslopes 1:1?

6.7 A rectangular channel must carry 5 m3/s of clear water. The slope of the channel is 1 percent. What width of channel is necessary to limit the shear to prevent channel erosion if the bottom is (a) noncolloidal silt and (b) fine gravel?

6.8 Specify a nonerodible channel lining for the channel in Example 6.3, using the tractive force method.

6.9 Determine the backwater curve for a concrete-lined rectangular channel 3 m wide with a flow rate of 1.5 [m.sup.3]/s and bottom slope is 0.002 if an obstruction is added that raises the water depth by 0.1 m.

6.10 Flow depth and velocity measurements, as in Example 6.5 were repeated for several stages of flow as follows: 2.5 m, 4 [m.sup.3]/s; 3.5 m, 10 [m.sup.3]/s; 4.4 m, 25 [m.sup.3]/s. Plot a stage-discharge curve for this channel. Explain the shape of the curve.

6.11 What is the flow in a channel if a contracted rectangular weir with a 1-m crest length is placed in the channel and has a head of 0.4 m?

6.12 Find the recommended dimensions for the height of the weir crest above the channel bottom and the width of the side contractions for the weir in Problem 6.11.
TABLE 6-1

Maximum Recommended Sideslopes for Open Channels

Material                                            Sideslope
                                                    (Horizontal/
                                                    Vertical)

Solid rock, cut section                             0.25:1
Loose rock or cemented gravel, cut section          0.75:1
Heavy clay, cut section                                1:1
Heavy clay, fill section                               2:1
Heavy clay in CH classification                        4:1
Sand or silt with clay binder, cut or fill section   1.5:1
Loam                                                   2:1
Peak, muck, and sand                                   1:1
Silts and sands with high water table                3.5:1

Source: NRCS (2001).

TABLE 6-2 Critical Shear and Velocity for Selected Channel
Lining Materials

                                              Maximum
                                 Critical     Velocity
Category                         Shear (Pa)    (m/s)

Soils
Fine sand (colloidal)               1.2        0.46
Sandy loam                          1.7        0.53
Alluvial silt (noncolloidal)        2.3        0.61
Alluvial silt (colloidal)           12.5       1.14
Silty loam (noncolloidal)           2.3        0.61
Firm loam                           3.6        0.76
Fine gravels                        3.6        0.76
Stiff clay                          12.5       1.16
Graded loam to cobbles              18.2       1.14
Graded silts to cobbles             20.6       1.22
Shales and hardpan                  32.1       1.83

Gravel/Cobble
25-mm                               15.8       1.14
50-mm                               32.1       1.37
150-mm                              95.8       1.75
300-mm                             191.5       2.67

Vegetation
Long native grasses                  70        1.52
Short native and bunchgrasses        40        1.07

Degradable Linings
Jute net                             22        0.53
Straw with net                       80        0.61
Coconut fiber with net              110        1.07
Fiberglass roving                    96        1.45

Soil Bioengineering
Live fascine                        104         2.1
Willow stakes                       125         2.0

Hard Surfacing
Gabions                             480         5.0
Concrete                            600         5.5

Source: Fischenich (2001).
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Publication:Soil and Water Conservation Engineering, 5th ed.
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Date:Jan 1, 2006
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