496 HYDROMECHANICS [HYDRAULICS cot cot B fl b (1). (2); (3). From (1) and (2), This will be a minimum for an- - -TTT = 2 cosec. /3 - cot j8 a- (4). From (3) and (4), b _2(1- cos 0) rf sin 18 = 2 tan Proportions of Channels of Maximum Discharge for given Area and Side Slopes. Depth of channel = d ; Hydraulic mean depth = d; Area of section = ft . Inclination of Sides to Horizon. Ratio of Side Slopes Area of Section n. Bottom Width. Top width= twice length of each Side Slope. Semicircle Semi-hexagon... 60 ; " 3 : 5 1-571(Z 2 1-732J2
l-155d 2</ 2-310(7 Semi-square 90 75 58
- 1
1 :4 2(/2 l-812(/2 2d 1-562(Z 2d 2-062(7 63 26 1 : 2 1-736J2 l 236d 2-236(7 53 8 3 : 4 l-750(i2 d 2-500(7 45 1 : 1 l-828(/2 82S(f 2-828(7 38 40 H =1 1 -952^2 0-702<Z 3 202d 33 42 H : 1 2-106*2 0-60Gd 3-606(7 29 44 1| :1 2-28W2 532rf 4-032(7 26" 34 2 : 1 2 472(/2 0-472(2 4-472(7 23 58 2J :1 2 674</2 0-424(7 4-924(7 21 48 2jll 2 885t<2 0-385d 5-385(7 19 58 2| : 1 3-104d2 354(Z 5-854(7 18 2G 3 : 1 3 320(/2 0-325d 6-325(7 Half the top width is the length of each side slope. The wetted perimeter is the sum of the top and bottom widths. 97. Form of Cross Section of Channel in which lhe.Mean Velocity is Constant ivith 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. 112 represent the cross section of the channel. From the symmetry of the section, only half the channel need be considered. Scale id IncK = 1 Foot. Fig. 112. Let oboe be any section suitable for the minimum flow, and let it be required to find the curve beg for the upper part of the channel so that the mean velocity shall be constant. Take o as origin ol coordinates, and let de, fg be two levels of the water above ob. c = ds. The condition to be satisfied is that should be constant, whether the water level is at ob, de, orfg. Con sequently m= constant = k or all three sections, and can be found from the section obac. lence also Increment of section _ydx Increment of perimeter ds -k. kdy = Wds* == k-(dxt + dy"-) ; and dx = ._-. integrating, x = k log e (y + y 2 - k 2 ) + constant ; and, since y = - when x--=Q , Assuming values for i/, the values of x can be found and the curve drawn. The figure has been drawn for a channel the minimum section of which is a half hexagon of 4 feet depth. Hence ^ = 2 ; b= 9 2 ; the rapid flattening of the side slopes is remarkable. Variation of Velocity in Different Parts of the Cross Section of a Uniform Stream. Vertical Velocity Curve in a Stream. If it is assumed that the resistance to the relative sliding of the layers of water in a stream is of the nature of a viscous resistance, then the law of the distri bution of velocity in a vertical longitudinal section of the stream can be determined theoretically. For simplicity, suppose the stream of uniform depth and indefinite width. Let fig. 113 show a por tion of a vertical longi tudinal section of the stream, and let OA, O A be the intersections with this of two transverse sections at a distance apart 1. Let ab, cd be the traces of two planes parallel to the free surface or to the
d
A
equilibrium of a layer abed
of width unity. LetOa = y,
ac = dy, and let v be the I*" 1
velocity of the particles comprised in aied, v being a function of y
which is to be determined. Taking the components of the forces
acting on abed, parallel to 00, the pressures on ac, bd, being pro
portional to the depth from the free surface, are equal and opposite;
also, the frictions or viscous resistances on the lateral faces of the
prism are zero, since in a wide stream there is no relative sliding
between abed and the layers on each side. There remain only the
resistances on the upper and lower surface, and the component of the
weight.
The weight of the layer is Gl dy, and if i is the slope of the stream
the component of the weight parallel to 00 is Gli dy. The friction
or viscous resistance on the face ab is proportional to its area and
to the differential coefficient ( 3). The resistance is, therefore,
dy
kl~, the negative sign being used because, if v increases with y,
dy
is positive, while the action of the layers above ab is a retarding
dy
dv d*v
action. The resistance on the face cd is similarly Tel - kid -j- .
dy dy
The resultant of the action of the layers above and below is,
therefore, kl(
When the
Glidy
dy
integrating,
dv
dy~
i av
dy
motion is uniform,
1 J.i d dv o- 0^ -TV H
dy
Gi IT
3 "~ y; *L
A Vrr ^ x
"^ 1
k
- v ^/
t 1, f J7 "/ > HlKKSfOKIfK*rml~t an equation which gives the Fig. 114. velocity v at any depth y. If on a vertical line OA (fig. 114), representing the depth of the stream, the values of v are set off horizontally, a parabolic curve is obtained, termed the vertical velocity curve for the section considered. The constant v is evidently the surface velocity, being the value of v for 2/ = 0. The parabola has a horizontal axis corresponding to the position of the filament of maximum velocity. If there is no resist
ance at the surface of the stream like that at the bottom and sides,
