498 HYDROMECHANICS [HYDRAULICS. From a discussion of experiments in which the maximum velocity was at the surface, Bazin was led to take M=-36-3VAi (4); and for that case the equation to the vertical velocity curve is (a = 0) In the cases in which the maximum velocity was below the surface, Bazin found that the difference between the maximum velocity V and the bottom velocity r& remained constant. But, putting x = = 1 in the equation (2a), and v b for the bottom ill velocity, U=V-M(l-o) 2 (o); V - v b = M(l - a) 2 = constant, for different positions of the axis of the parabola. Let N where N is a constant ; then v-V-N 1-a for any position of the axis of the parabola. But this must agree with the equation (4) above, for a = ; hence, and the general equation for all cases becomes Bazin has shown that this equation agrees well with experiments on artificial channels by himself, and on the Saone, Seine, Garonne, and Rhine. In all these the ratio ranged from 1 10 to 1 13, ex cept in the case of the Rhine at Basel, for which the ratio was 117. The parameter 36 3V lies between 13 and 20, and the ratio of M" (2), an equation which gives the velocity v at any depth y from the sur face in the region below the filament of maximum velocity. For the region above the filament of maximum velocity Boileau assumes where v is the velocity at the depth y and V is the surface velo city. 102. Ratio of Mean to Greatest Surface Velocity, for the wlwla Cross Section in Trapezoidal Channels. It is often very important to be able to deduce the mean velocity, and thence the discharge, from observation of the greatest surface velocity. The simplest method of gauging small streams and channels is to observe the greatest surface velocity by floats, and thence to deduce the mean velocity. Now, for channels not widely differing from those experi mented on by Bazin, the expression obtained by him for the ratio of surface to mean velocity may be relied on as at least a good approximation to the truth. Let r be the greatest surface velo city, I m the mean velocity of the stream. Then, according to Bazin, Let OS = V + c, then CT = 2CS = 2c. -1 , and CM 2 = PC- CT ; 2 PC 1 CT c Put r/mi UUL VM v v //tt- ^ where c is a coefficient, the values of which have been already given in the table in 90. Hence c + 25 - 4 Values of Coefficient , ^ IC Formula v m = rr (1 - a) 2 the depth at which the maximum velocity is found to the whole depth, =a, ranges from zero to 2, except in some of the artificial channels, where it reached 35. The Mississippi experiments give different results, and Bazin inclines to believe that the method of experimenting was untrustworthy. The extreme difference V- n between the maximum and bottom velocity is found by Bazin to range from V to V in artificial channels, being greater the greater the roughness of the Asides. In natural streams it is more generally V, but in the Rhine at Basel it reached V, the bed being covered with boulders. Boileau s Formulae. Boileau also assumes the vertical velocity curve to be a parabola ; below the filament of greatest velocity the curve is expressed by the relation A "R 2 f~] That is, the velocity curve is a parabola having its axis at the free surface of the stream. Above the filament of greatest velocity this law fails, and , the velocities di- P MOST minish instead of increasing. The vertical ve locity curve is therefore such a curve as M MM" (fig. 116), where the part MM" is a parabola hav ing its vertex at S, and DM is the maximum velocity. The part M M does ,,. ,, fi not follow the parabolic law. Let Vbe the maximum velocity DM, CMits depth -= 6. Draw at M the tangent and normal to the parabola. Then PC is Hydraulic Mean Depth =m. Very Smooth Channels. Cement. Smooth Channels. Ashlar or Brickwork. Hough Channels. Rubble Masonry. Very Hough Channels. Canals in Earth. Channels encumbered with Detritus. 0-25 83 79 69 1 42 0-5 84 81 74 8 50 0-75 84 82 7G 63 55 1-0 85 77 65 58 2-0 83 79 -1 64 3-0 80 "3 67 4-0 81 "5 70 5-0 76 71 6-0 84 "7 72 7-0 -8 73 8-0 9-0 88 74 10-0 15-0 79 75 20 80 76 30-0 82 77 40-0 50-0 00 79 the half parameter . Boileau finds c = B0 2 to be nearly constant for very different streams. Thus from two experiments of his ow r n on streams 2 and 3 metres deep, c = 01070 and 01072. In Hennocque s experiments on the Rhine, 2 45 metres deep, c = 0107 ; and in tli" Mississippi experi ments with a depth of 32 metres, c = 0093 to 0113. Replacing A and B in (1 ) by the values now given 103. River Bends. In rivers flowing in alluvial plains, the wind ings which already exist tend to increase in curvature by the scouring away of material from the outer bank and the deposition of detritus along the inner bank. The sinuosities sometimes increase till a loop is formed with only a narrow strip of land between the tw r o encroaching branches of the river. Finally a "cut off" may occur, a waterway being opened through the strip of land and the loop left separated from the stream, forming a horse-shoe shaped lagoon or marsh. Professor James Thomson has pointed out (Proc. limjal Soc. 1877, p. 356 ; Proc. Inst. of Mech. Engineers, 1879, p. 456) that the usual sup- ,.,. position is that the water tending to go forwards in a straight line, rushes against the outer bank and scours it, at the same time creatingdeposits at the inner bank. That view is very far from a complete ac- ! count of the matter, 1 and Professor Thorn- 1 son has given a much 1 more ingenious ac count of- the action 1 at the bend, which he has completely confirmed by experi ment. When water moves Fig. 117. round a circular curve under the action of gravity only, it takes a motion like that in a free vortex.
to the axis of the stream at the inner than at the outer side of the
