HYDRAULICS,] HYDROMECHANICS 499 bend. Hence the scouring at the outer side and the deposit at the inner side of the bend are not due to mere difference of velocity of flow in the general direction of the stream ; but, in virtue of the centrifugal force, the water passing round the bend presses outwards, and the free surface in a radial cross section has a slope from the inner side upwards to the outer side (fig. 118). For the greater part of the water flowing in curved paths, this difference of pressure produces no tendency to transverse motion. But the water imme diately in contact with the rough bottom and sides of the channel is retarded, and its centrifugal force is Inner Hank Outer Bank insufficient to bal ance the pressure due to the greater depth at the out side of the bend. It therefore flows Section at MN. inwards towards the inner side of Fig. 118. the bend, carrying with it detritus which is deposited at the inner bank. Conjointly with this flow inwards along the bottom and sides, the general mass of water must flow outwards to take its place. Fig. 117 shows the directions of flow as observed in a small artificial stream, by means of light seeds and specks of aniline dye. The lines CO show the directions of flow immediately in con tact with the sides and bottom. The dotted line AB shows the direction of motion of floating particles on the surface of the stream. STEADT MOTION OF WATER IN OPEN CHANNELS OF VARYING CROSS SECTION AND SLOPE. 104. 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 altera tion 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 uni form 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. 119 represent a longitudinal section of the stream, A^ being the water surface, B^ the stream bed. Let A B , AjBj be At C, Fig. 119. cross sections normal to e direction of flow. Suppose the mass of water AoBoAjBj comes in a short time to C D C,D,, and let the work done on the mass be equated to its change of kinetic energy during that period. Let I be the length A A, 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 there are as many particles possessing the same velocities in the space CoDoAjBj as at the beginning. The r ~> change of kinetic energy C: g, is therefore the difference of the kinetic energies of A B C D () and AjB^D,. Let fig. 120 represent the cross section A B n , and let <a be a small ele ment of its area at a F g- 120. point where the velocity is v. Let fl be the whole area of the cross action and U Q the mean velocity for the whole cross section. From the definition of mean velocity we have Let V = UQ + W, where w is the difference between the velocity at the small element and the mean velocity. For the whole cross section, 2,iaiv = Q, The mass of fluid passing through the element of section , in seconds, is uvO, and its kinetic energy is <av 3 . For the whole section, the kinetic energy of the mass A B C D passing in e seconds IS _G*J 2g 1 The factor Bu + w is equal to 2i/ + r, a quantity necessarily positive. Consequently 2&>?; ;t > n V, and consequently the kinetic energy of A B C D is greater than G0 fi u 3 Qr G0 2 2# 2# " which would be its value if all the particles passing the section had the same velocity . Let the kinetic energy be taken at where a is a corrective factor, the value of which has been esti mated by Belanger at I l. 1 Its precise value is not of great importance. In a similar way we should obtain for the kinetic energy of AjBjCjDj the expression a G0 fl u 3 = GO_Q U s where flj, u are the section and mean velocity at AjB,, and where a may be taken to have the same value as before without any import ant error. Hence the change of kinetic energy in the whole mass A B A 1 B 1 in seconds is a -^Q(V-O (i). Motive Work of the Weight and Pressures. Consider a small filament a^ which comes in seconds to c Cj. The work done by gravity during that movement is the same as if the portion a c n were carried to a x Cj. Let dQ0 be the volume of a c or a^, and y ot T/ X the depths of a , a l from the surface of the stream. Then the volume dQe or GdQ0 pounds falls through a vertical height z + y 1 - >j , and the work done by gravity is Putting ^7 fl for atmospheric pressure, the whole pressure per unit of area at a is Gy + p a> and that at a-^ is - (Gy 1 +2)a). The work of these pressures is Adding this to the work of gravity, the -whole work is GzdQ0 ; or, for the whole cross section, GiQ0 ........ (2). Work expended in Overcoming the Friction of the Stream Bed. Let A B , A"B" be two cross sections at distances s and s + ds from A B . Between these sections the velocity may be treated as uniform, because by hypothesis the changes of velocity from section to section are gradual. Hence, to this short length of stream the equation for uniform motion is applicable. But in that case the work in overcoming the friction of the stream bed between A B and A"B" is where u, x, H are the mean velocity, wetted perimeter, and section at A B . Hence the whole work lost in friction from A B to AjBj will be GQ0 (3). Equating the work given in (2) and (3) to the change of kinetic energy given in (1), / u * v <.? - 2 ) = GQc0 - GQ0/ C - ds ; 1 Poussinesq lias shown that this mode of determining the corrective factor <
is not satisfactory.
