HYDKAULICS.] HYDROMECHANICS 49)3 This may be put in the form A 2 - /,/-^
- =V c Vmi=
(2); where c is a coefficient depending on the roughness and form of the channel. The coefficient of friction f varies greatly with the degree of roughness of the channel sides, and somewhat also with the velocity. It must also be made to depend on the absolute dimensions of the section, to eliminate the, error of neglecting the variations of velocity in the cross section. A common mean value assumed for is Q 00757. The range of values will be discussed presently. It is often convenient to estimate the fall of the stream in feet per mile, instead of in feet per foot. If/ is the fall in feet per mile /-528<H. Putting this and the above value of in (2a), we get the very simple and long-known approximate formula for the mean velocity of a stream = HV2^/ (3). The flow down the stream per second, or discharge of the stream, is 89. Coefficient of Friction for Open Channels. Various expressions have been proposed for the coefficient of friction for channels as for pipes. W eisbach, giving attention chiefly to the variation of the coefficient of friction with the velocity, proposed an expression of the form / n (5), and from 255 experiments obtained for the constants the values = 0-007409 ; = 1920. This gives the following values at different velocities : t =: ?= 03 0-01215 0-4 0-91907 0-5 0-01025 0-6 0-00978 0-7 0-00944 0-8 0-06918 0-9 0-00898 1 0-00883 1J 0-00836 2 0-00812 3 0-9C788 5 0-00769 7 0-00761 10 0-00755 15 0-00750 In using this value of when v is not known, it is best to proceed by approximation. Calculate a rough value of v by taking any mean value of for instance the one given in the preceding section. Then from this value of v calculate a revised value of and from this a new value of v. 90. Darcy < Bazin s Expression for the Coefficient of Friction. Darcy & Ba/in s researches have shown that f varies very greatly for different degrees of roughness of the channel bed, and that it also varies with the dimensions of the channel. They give for an empirical expression (similar to that for pipes) of the form where m is the hydraulic mean depth. For different kinds of chan nels they give the following values of the coefficient of friction : Kind of Channel. a. ft I. Very smooth channels, sides of smooth j cement or planed timber } 0-00316 o-i II. Smooth channels, sides of ashlar, brick- ) work, planks 0-00401 0-23 III. Rough channels, sides of rubble masonry j or pitched with stone ) 0-00507 0-82 IV. Very rou^h canals in earth . . 0-00592 4-10 V. Torrential streams encumbered with de- ) tritus ) 0-00846 8-2 The last values (Class V.) are not Darcy & Bazin s, but are taken from experiments by Ganguillet & Kutter on Swiss streams. The following table very much facilitates the calculation of the mean velocity and discharge of channels, when Darcy & Bazin s value of the coefficient of friction is used. Taking the general formula for the mean velocity already given in equation (2a) above, v = c/mi , where c= */ -?- > the following table gives values of c for channels of different degrees of roughness, and for such values of the hy draulic mean depths as are likely to occur in practical calculations: Values of c in v = cVrni, deduced from Darcy it- Bazin s Values. c
S* 2Z > H "* ? 5 c i
- i
1 - s^ a -~ " - ^ 5 2 - .t^ 5 5 i a
c j* II" 3? = " a y s o o> 5 5 ^ 77 " ?3 .%
- 3
^ S, ii go c* . 5 & ow S Q s ~ S o II S C ** ft {/) _ | H. |"i c o ^0 S "r, c ~~ 7!? If a o j- a C "" r - > 5^ 3 $3 s i i * i > 5
11 c - X
- 3
JB E2 C 3 bo ^ .C iJ H ^* cS A 3 KM bo W r- -j ^" 53
02 jo > ^ _g
M <j > 25 125 95 57 26 18-5 8-5 147 150 112 89 5 135 110 72 3fi 25-6 9-0 147 130 112 90 71 75 13!) 116 81 42 30-8 95 147 130 112 90 1-0 141 119 87 48 34-9 10-0 147 130 112 91 72 1-5 143 122 94 56 41-2 11 147 130 118 92 20 144 124 98 62 46-0 12 147 130 118 93 74 2 5 145 12G 101 67 13 147 130 113 94 3-0 145 126 104 70 53 14 147 130 113 95 3-5 14fi 127 105 73 15 147 130 114 96 77 40 146 1 128 106 76 58 16 147 130 114 97 4-5 HI: 128 107 78 17 147 130 114 97 6-0 146 128 108 80 62 18 147 130 114 98 5-5 146 12!) 109 82 20 147 131 114 98 80 6-0 117 !--".i 110 84 65 25 148 131 115 Ion 6-5 147 129 110 85 30 148 131 115 102 83 7-0 147 129 110 86 67 40 148 131 116 103 85 7-5 147 129 111 87 50 148 131 110 104 86 8-0 147 j 130 111 88 69 CO 148 131 117 108 91 91. Ganguillet <fc Kutter s modified Darcy Formula. Starting from the general expression v = c/mi, Messrs Ganguillet & Kutter have examined the variations of c for a wider variety of cases than those discussed by Darcy & Bazin. Darcy & Bazin s experiments are confined to channels of moderate section, and to a limited variation of slope. Ganguillet & Kutter brought into tlie discussion two very distinct and important additional series of results. The gaugings of the Mississippi by Messrs Humphreys & Abbot afford data of discharge for the case of a stream of exceptionally large section and of very low slope. On the other hand, their own measurements of the flow in the regulated channels of some Swiss torrents gave data for cases in which the inclination and roughness of the channels were exceptionally great. Darcy & Bazin s experiments alone were conclusive as to the dependence of the coefficient c on the dimensions of the channel and on its rough ness of surface. Plotting values of c for channels of different in clination indicated to Ganguillet & Kutter that it also depended on the slope of the stream. Taking the Mississippi data only, they found c = 256 for an inclination of - 0034 per thousand, = 154 ,, 0-02 so that for very low inclinations no constant value of c independent of the slope would furnish good values of the discharge, In small rivers, on the other hand, the values of c vary little with the slope. As regards the influence of roughness of the sides of the channel a different law holds. For very small channels differences of rough ness have a great influence on the discharge, but for very large channels different degrees of roughness have but little influence, and for indefinitely large channels the influence of different degrees of roughness must be assumed to vanish. The coefficients given by Darcy & Bazin are different for each of the classes of channels of dim-rent roughness, even when the dimensions of the channel are infinite. But, as it is much more probable that the influence of the nature of the sides diminishes indefinitely as the channel is larger, this must be regarded as a defect in their formula. Comparing their own measurements in torrential streams in Switzerland with those of Darcy & Bazin, Ganguillet & Kutter found that the four classes of coefficients proposed by Darcy & Bazin were insufficient to cover all cases. Some of the Swiss streams gave results which showed that the roughness of the bed was markedly greater than in any of the channels tried by the French engineers. It was necessary therefore in adopting the plan of arranging the different channels in classes of approximately similar roughness to increase the number of classes. Especially an addi tional class was required for channels obstructed by detritus. To obtain a new expression for the coefficient in the formula
- = A/ -^Vmi cVmi ,
in which Darcy & Bazin take ,_ / ^_ V /! + - Ganguillet & Kutter proceeded in a purely empirical way. found that an expression of the form They could be made to fit the experiments somewhat better than Darcy n
expression. Inverting this, we get
