8.46 HANDBOOK OF HYDRAULICS

the position of the jump. Trial values of D2 should be computed for three sections a, b, and c, which will locate three points a', b', and c' to which the water must jump at the respective sections. One of the points should be on the opposite side of the jump from the other two, and, preferably, the middle point should be near the jump. The in-tersection of the line a'b'c' with the higher water-surface profile gives the position of the jump. See Fig. 8.23d for another example of non-uniform flow occurring both before and after the jump.

When several determinations of depth after jump in a channel are required for the same discharge, as in the preceding case, it may be less laborious to obtain the values from a graph of the momentum-pressure diagram in Fig. 8.12 than from a solution of Eq. (8.85). For rectangular channels, or for channels that are wide in comparison with the depth, Table 8.12 gives depths after the jump, and for trap-ezoidal channels approximate values can be taken from Table 8.11.

Minor Losses

The losses caused by rapid local changes in magnitude or direction of velocity are called minor losses. (For minor losses in pipes, see Sec. 6.) Such losses would occur at bends, contractions, enlargements, or obstructions in channels.

Channel Bends. When a fluid flows around a bend, the centrifugal force tends to develop a water surface which is higher at the outside of the channel. If the velocity in the channel were everywhere equal to the average velocity V, the amount that the water surface would rise at the outside wall and the amount that it would fall at the inner wall would be given approximately for subcritical flow by

V2b D = -T— (8.107) 2grc

where V is the average velocity, b the width of the channel, and rc the radius of curvature of the centerline. This equation is derived by noting that the water surface will be perpendicular to the resultant of the radial and gravitational forces on a particle of fluid. However, because the radial force is proportional to the square of the velocity, this force will be greatest on the high-velocity water near the center of the channel, thus developing crosscurrents, eddies, and spiral mo-tion. Also, there may be a tendency toward separation along the inner wall. Furthermore, for supercritical flow, a standing-wave pattern

OPEN CHANNELS WITH NONUNIFORM FLOW 8.47

complicates the flow pattern. Supercritical flow in bends is discussed in Sec. 9.

Information on losses at bends in rectangular channels has been presented by various investigators. Yen and Howe5 reported that Kb in the following expression was 0.38 for a 90° bend having a radius of curvature of 1.5 m and a width of 28 cm,

hb = Kb ^ (8.108)

Shukry6 reported test results in which the variables were the angle 6 through which the water was turned, the ratio of radius of curva-ture to width rjb, the ratio of depth to width D/b, and the Reynolds number. He used the Reynolds number in the following form:

rV R = — (8.109) V

where r is the hydraulic radius. These tests indicate that Kb is af-fected very little by D/b except when rjb is very small. When 0 <t 45°, the loss is negligible. The loss increases as 6 is increased from 45 to 90°, and for values of 6 ranging from 90 to 180° the loss is about constant. When rjb 5 3.0, losses were found to be negligible. (It should be noted that this does not agree with the results of Yen and Howe reported previously.) For values of rjb < 3, values of Kb are given in the following tabulation for R = 31,500. Also shown are ex-perimental values of Kb for various values of R, with rjb = 1.0.

Bend Loss Coefficients*

R = 31,500 rjb = 1.0 rjb Kb R Kb

2.5 0.02 1 X 104 0.59 2.0 0.07 3 X 104 0.27 1.5 0.12 5 X 104 0.25 1.0 0.25 7 X 104 0.35

•From Shukry.6

Tests on large canals7 showed that losses due to bends could be estimated from the following equation in which (2A°) is the summa-tion of deflection angles in the reach,

8.48 HANDBOOK OF HYDRAULICS

hb = 0.00K2A0) — Yl 2g

In sinuous natural rivers, the bend losses are included in the friction losses.

Contractions and Enlargements. The energy losses for contrac-tions have been expressed by Hinds8 in terms of the difference in kinetic energy at the two ends,

(VI V*\ hc = KA-1 - - 1 ) (8.110) 2g

and for enlargements,

, Vi Vï\ he = KA-1 - -1) (8.111) ,2g 2g

Values of Kc and Ke are given in the following table:

Form of transition Kc Ke

Sudden change in area, sharp corners 0.5 1.0 "Well designed":

Best 0.05 0.10 Design value 0.10 0.20

Additional information on entrance losses is given in Sec. 4. A "well-designed" transition is one in which all plane surfaces are con-nected by tangent curves and a straight line connecting flow lines at the two ends does not make an angle greater than 121/2° with the axis of the channel.

Contracting and enlarging sections are used at channel entrances or to form transitions between channels of different size. Hinds has summarized the art of designing transitions for subcritical flow as practiced by the U.S. Bureau of Reclamation as follows (transitions with supercritical flow are discussed in Sec. 9):

1. Sufficient fall must be allowed at all inlet structures to accel-erate the flow and to overcome frictional and entrance losses.

2. The theoretical recovery at an outlet structure is reduced by frictional and outlet losses.

OPEN CHANNELS WITH NONUNIFORM FLOW 8.49

3. At open-channel outlets a small factor of safety may be obtained by setting the transition for less than its maximum recovering ca-pacity, but erosion below the structure may be slightly increased.

4. At siphon outlets a small factor of safety may be obtained and erosion avoided by setting the transition for more than its assumed recovering capacity.

5. Simple designs may be prepared by adapting the details of pre-vious designs known to be satisfactory, if proper allowance is made for loss of head.

6. Important structures, where velocities are high, must be care-fully designed to conform to a smooth theoretical water surface. Sharp angles must be avoided.

7. Horizontal curvature in the conduit before an outlet appears to reduce its efficiency and to produce objectionable cutting velocities in the canal beyond.

8. Kc [Eq. (8.110)] for a well-designed inlet is likely to be less than 0.05. A value of 0.1 is safe for use in design.

9. Ke [Eq. (8.111)] for a well-designed outlet is likely to be less than 0.2, unless the conduit before the structure is curved. A value of 0.2 is safe for use in design.

10. No definite data as to the best form of water-surface profile, best form of structure, or most efficient length of transition are avail-able.

11. Special care is required where critical depth is approached or where hydraulic jump is involved.

12. The disturbances often observed in long, uncontrolled siphons, at part capacity, are not caused by entrained air but by the hydraulic jump in the pipe.

Losses at Obstructions. Water passing through a constriction in an open channel at subcritical velocity decreases in depth, as shown in Fig. 8.3. The depth downstream from the constriction must be the uniform flow depth or normal depth for this discharge because no other water-surface profile can exist (see Equations of Gradually Var-ied Flow). The Bernoulli equation, written from a point just upstream from the obstruction to a point just downstream, is

z« + Du + au P = zd + Dd + ad + ht (8.112)

8.50 HANDBOOK OF HYDRAULICS

Datum

FIGURE 8.16 Flow past obstructions.

The symbols used in Eq. (8.112) are defined in Fig. 8.16. The amount of backwater caused by the obstruction D can then be obtained from Eq. (8.112), using Fig. 8.16 as a reference,

AD = (zu+ Du) - (zd + Dd) = ad + h, (8.113)

The losses at obstructions in open channels consist of the loss due to a constriction and an enlargement and, if the obstruction has con-siderable length in the direction of flow, of a friction loss. Usually the principal loss is that due to the enlargement at the downstream end of the obstruction because losses are invariably larger when velocities are decreased than when flow is speeded up. This is illustrated by the coefficients for losses in the previous subsection, the coefficients for enlargements being twice those for contractions under similar con-ditions. Energy losses at piers can be reduced to a minimum by rounding the upstream corners and tapering, or "streamlining," the downstream end. The losses could be estimated by treating them as combinations of constriction and enlargements and using the coeffi-cients given in the previous subsection.

Flow through bridge openings has been investigated by means of model studies.9 The results are presented in a series of curves which are useful in designing bridge openings. The procedure for expressing the losses is based on the equation

(8.114)

where h¡ is the total loss, Kb the loss coefficient, and Vn the average

OPEN CHANNELS WITH NONUNIFORM FLOW 8.51

velocity that would occur in the bridge opening if the entire discharge were to pass through the bridge opening at the normal depth in the river for this discharge. Values of Kb are related to the bridge-opening ratio M. The value of M is obtained by dividing the portion of dis-charge that would normally flow through the bridge opening if no piers were present by the total discharge. Figure 8.17 shows two curves relating Kb to M. One curve applies to abutments with vertical walls and 90° corners, as well as to abutments with sloped embank-ments on the upstream and downstream sides held in place at the ends by wing walls making an angle of 90° with the piers, as illus-trated in Fig. 8.176. For wing walls having angles other than 90°, as shown in Fig. 8.17c, the values of Kb are smaller than those shown in the graph, the reduction being, on the average, about 12 percent for a wing-wall angle of 30° and approximately 30 percent for angles of 45 and 60°.

The second curve applies to piers, referred to as spill-through abut-ments, which have the sloped embankment extending around the ends of the piers, as illustrated in Fig. 8.17d. The curve shown is for an embankment slope of 1.5:1, horizontal to vertical. Values of Kb for an embankment slope of 2:1 are 5 to 10 percent larger than those shown, and for a 1:1 slope, the values are 4 to 9 percent lower than those shown by the curve.

A U.S. Bureau of Public Roads publication9 also provided coeffi-cients AK to be added to Kb to take care of minor effects on the losses at the bridge opening. One such coefficient takes care of the increase in loss which occurs when the bridge opening is not in the center of the river. Another includes the additional losses caused by obstruc-

FIGURE 8.17 Losses at bridge piers.

8.52 HANDBOOK OF HYDRAULICS

tions in the opening. A third one introduces the effect of having the bridge cross the river at an angle differing from 90°.

Transition through Critical Depth without Jump

If water flowing at less than critical depth enters a channel having less than critical slope, change to a higher stage will normally occur in a jump (see Hydraulic Jump) unless special means are provided for making velocity changes gradually A transition designed to pre-vent a jump, for the specific data indicated, is illustrated in Fig. 8.18. The raised bottom has a smooth surface, the elevation at the crest C being such that the minimum energy gradient is tangent to the en-ergy gradient of the stream. For this condition a jump is impossible. A similar design could be prepared for channels having other sec-tional forms.

In Fig. 8.18 lower-stage flow is indicated up to section C, where critical depth occurs and then follows higher-stage flow. On both sides of C the other stages which could be computed are not shown. The force curves (QV/g + ay) [see Eq. (8.86a)] for the two stages of flow are tangent to each other. If the crest C is lower than that indicated in the figure, the curves will intersect to the right of C at the section

Min. energy gradient

\ 1 I —] [̂Energy gradient | r»i«Vr-l--.—r.'û I -

•Water surface

Reatangular c r o s s »act ion 4m wlda D l s c h s r g a 4.6m ,par sac

FIGURE 8.18 Transition through critical depth to higher stage without jump.