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74
[FLOW IN RIVERS
HYDRAULICS


Proportions of Channels o Maximum Discharge for given Area and Side Slopes. Depth 0 channel=d; Hydraulic mean depth=§ d; Area of section =Q.

.

- ~ ° f I Top width =

?§§ ;§ ..;§ 
§ :.°§ . me;i§ .tsa*;

ope.

Semicircle . .. .. I'57Id2 0 2d

Semi-hexagon . 6o° o' 3:5 I°732d2 I'155d 2'3I0d Semi-square . 90° o' o; 1 zd' 2d 2d 75° 58' I:4 1-812112 1 56211 2-o62d 63° 26' 1: 2 1-736:12 1-23611 2'236d 53° 8' 3:4 I~75Od2 d 2'500d

45° o' I:I I-828122 0°828d 2-828d 38° 40' Ig; 1 I'952d2 0'702d 3-202d 33° 42' 1%: I 2-IO6d2 o~6o6d 3-6o6d 29° 44' 1%: 1 2-282d2 o-532d 4~032d 26° 34' 2; 1 2'472112 O'472d 4-472d 23° 58' 2%: 1 2-674d=' 0°424d 4~924d 21° 48' 2;: I 2-885112 o-385d 5'385d 19° 58' 2%:1 3'IO4d2 o-354d 5-854d 18° 26' 3: 1 3~325d2 o-325d 6-325d Half the top width is the length of each side slope. The wetted perimeter is the sum of the top and bottom widths § 1 14. Form of Cross Section of Channel in which the Mean Velocity is Constant with Varying Discharge.-In designing waste channels from canals, and in some other cases, it is desirable that the mean velocity should be restricted within narrow limits with very different volumes of discharge. In channels of trapezoidal form the velocity increases and diminishes with the discharge. Hence when the discharge is large there is danger of erosion, and when it is small of silting or obstruction by weeds. A theoretical form of section for which the mean velocity would be constant can be found, and, although this is not very suitable for practical purposes, it can be more or less approximated to in actual channels. Let fig. 117 represent the cross section of the channel. From the symmetry of the section, only half the channel need be considered. xi

fi:;:~':;~';°-°~~° °-'° —°~- - ~~°-~°- if """" i """'°"“"

ci ll

Scale 1% Inch =1Foot.

FIG. 117.

Let obac be any section suitable for the minimum How, and let it be required to find the curve beg for the upper art of the channel so that the mean velocity shall be constant. Take o as origin of coordinates, and let de, fg be two levels of the water above ab. Let ob=b/2; de=y, fg=y-l-dy, od=x, of=x+dx; eg=ds. The condition to be satisfied is that v=c / (mi)

should be constant, whether the water-level is at ob, de, or fg. Consequently m = constant = k

for all three sections, and can be found from the section obac. Hence also

Increment of section 3Lclic k

Increment of perimeter- ds "

y2dx2 = kldsf = k'(dx2+dy2) and dx =kdy/ / (y'-k'). Integrating,

x =k logf {y-l- V (y'-k')l-l-constant; and, since y=b/2 when x=o,

x=k logs lly+ w/ (yi-k')}/l%b+ w/ (tif-k')llglssuming values for y, the values of x can be found and the curve rawn.

The figure has been drawn for a channel the minimum section of which is a half hexagon of 4 ft. depth. Hence k=2; b=9-2; the rapid flattening of the side slopes is remarkable. Srmuv MOTION or WATER IN OPEN CHANNELS or VARYING Cnoss SECTION AND SLOPE

§ 1 15. In every stream the discharge of which is constant, or may be regarded as constant for the time considered, the velocity at different places depends on the slope of the bed. Except at certain exceptional points the velocity will be greater as the slope of the bed is greater, and, as the velocity and cross section of the stream vary inversely, the section of the stream will be least where the velocity and slope are greatest. If in a stream of tolerably uniform slope an obstruction such as a weir is built, that will cause an alteration of flow similar to that of an alteration of the slope of the bed for a greater or less distance above the Weir, and the originally uniform cross section of the stream will become a varied one. In such cases it is often of much practical importance to determine the longitudinal section of the stream. The cases now considered will be those in which the changes of velocity and cross section are gradual and not abrupt, and in which the only internal work which needs to be taken into account is that due to the friction of the stream bed, as in cases of uniform motion. Further, the motion will be supposed to be steady, the mean velocity at-each given cross section remaining constant, though it Varies from section to section along the course of the stream. Let fig. 118 represent a longitudinal section of the stream, AeA, being the water surface, BOBI the stream bed. Let AOBU, AIB1 be 2 4' ffl" <'» Qs

"'~4"¢' al

T Y/'il l I . lar itil,

B0 W;/tar

B if W

FIG. 118.

cross sections normal to the direction of flow. Suppose the mass of water AUBOAIBI comes in a short time 6 to CQDOCIDI, and let the work done on the mass be equated to its change of kinetic energy during that period. Let l be the length ADA; of the portion of the stream considered, and z the fall of surface level in that distance. Let Q be the discharge of the stream per second. Change of Kinetic Energy.-At the end of the time 0 there are as many particles possessing the same velocities In the space CQDOAIB, as at the beginning. The

change of kinetic energy is

therefore the difference of

the kinetic energies of

AnB0C0Dg and A;B1CID1.

Let fig. 119 represent the

cross section AOBO, and let

w be a small element of its

area at a point where the

velocity is v. Let S20 be the

~ Wi.., - .f

We a-

ts. ... . — — - , f.g.,

- - ~~- -~ ~-'- - -*

”“ Elw ~'§ *“"

"4 .

ss 1 1

- 'Nw . <>°»gfgirs

0 Wg, - u' .

ss ;», agyféxo, La

FIG. 1 19.

whole area of the cross section and ug the mean velocity for the whole cross section. From the definition of mean velocity we have no =Ewv/90. s.

Let v=u0-I-'w, where w is the difference between the velocity at the small element w and the mean velocity. For the whole cross section, Eww = 0.

The mass of fluid passing through the element of section oi, in 0 seconds, is (G/g)wv0, and its kinetic energy is (G/2g)w1J50. For the whole section, the kinetic energy of the mass AOBOCQDO passing in 0 seconds is A

(GH/2g)Zwv3 = (G0/2g)Ew(u03+3u0”w+3u0w” -l-103), = (GH/2g){u03SZ ~l-Eww2(3u0 J, -ze) }. The factor 3u0-I-fw is equal to 2ii0-I-v, a quantity necessarily positive. Consequently ZIwz'3>S20u03, and consequently the kinetic energy of AOBQCDDU is greater than (GH/2g)S10u03 or (G0/2g)QufF.

which would be its value if all the particles passing the section had the same velocity uo. Let the kinetic energy be taken at a(G6/2g)S29ii03 = a(G0/2g)Qu02,

where a is a corrective factor, the value of which was estimated by ]. B. C. ]. Bélanger at I~I.' Its precise value is not of great importance. In a similar way we should obtain for the kinetic energy of A1B1C1D, the expression

a(G0/2g)S2, u13 = a(G6/'2g)Qu12,

where $21, ui are the section and mean velocity at AIB1, and where ui may be taken to have the same value as before without any important error.

Hence the change of kinetic energy in the whole mass A0B0A1B in 6 seconds is

a(G0, '2g)Q(n,2-nog). (I)

Motive Work of the Weight and Pressures.-Consider a small filament anal which comes in 0 seconds to cocl. The work done by gravity during that movement is the same as if the portion agco were carried to a1c1. Let dQ0 be the volume of aoco or a1c1, and yo, yi the depths of ao, al from the surface of the stream. Then the volume 1 Boussinesq has shown that this mode of determining the corrective

factor a is not satisfactory.