HYDROMECHANICS [HYDRAULICS. Putting <a for the section of the body, c, for the coefficient of con traction. c c (tt - a>) for the area of the stream at A^, -77: - ; c c (0 - w) or, putting p = Then where Taking c c = 85 and p=4, K x = 0-467, a value less than before. Hence there is less pressure on the cylinder than on the thin plane. 153. Distribution of Pressure on a Surface on which a Jet impinges normally. The principle of momentum gives readily enough the total or resultant pressure of a jet impinging on a plane surface, but in some cases it is useful to know the distribution of the pressure. The problem in the case in which the plane is struck normally, and the jet spreads in all directions, is one of great complexity, but even in that case the maximum intensity of the pressure is easily assigned. Each layer of water flowing from an orifice is gradu ally deviated (fig. 169) by contact with the surface, and during deviation exercises a centrifugal pressure towards the axis of the jet. The force exerted by each small mass of water is normal to its path, and inversely as the radius of curvature of the path. Hence the greatest pressure on the plane must be at the axis ot the jet, and the pressure must de crease from the axis outwards, in some such way as is shown by the curve of pressure in fig. 170, the branches of the curve being probably asymptotic to the plane. For simplicity suppose the jet is a vertical one. Let h^ be the depth of the orifice from the free surface, and v r the velocity of dis- Fig. 170. charge. Then, if u is the area of the orifice, the quantity of water impinging on the plane is obviously that is, supposing the orifice rounded, and neglecting the coefficient of discharge. The velocity with which the fluid reaches the plane is, however, greater than this, and may reach the value v = V2J/A ; where h is the depth of the plane below the free surface. The external layers of fluid subjected throughout, after leaving the orifice, to the atmospheric pressure will attain the velocity v, and will How away with this velocity unchanged except by friction. The layers towards the interior of the jet, being subjected to a pressure greater than atmospheric pressure, will attain a less velocity, and so much less as they are nearer the centre of the jet. v i Hut the pressure can in no case exceed the pressure or h measured 2<7 in feet of water, or the direction of motion of the water would be reversed, and there would be reflux. Hence the maximum intensity of the pressure of the jet on the plane is h feet of water. If the pressure curve is drawn with pressures represented by feet of water, it will touch the free water surface at the centre of the jet. Suppose the pressure curve rotated so ar, to form a solid of revolu tion. The weight of water contained in that solid is the total pressure of the jet on the surface, which has already been deter mined. Let V = volume of this solid, then G V is its weight in pounds. Consequently GY = -wv,v g We have already, therefore, two conditions to be satisfied by the pressure curve. Some very interesting experiments on the distribution of pressure on a surface struck by a jet have been made by Mr J. S. Beresford (Prof. Papers on Indian Engineering, No. cccxxii.), with a view to afford imformation as to the forces acting on the aprons of weirs. Cylindrical jets inch to 2 inches diameter, issuing from a vessel in which the water level was constant, were allowed to fall vertically on a brass plate 9 inches in diti meter. A small hole in the brass plate communicated by a flexible tube with a vertical pressure column. Arrangements were made by which this aperture could be moved -^ inch at a time across the area struck by the jet. The height of the pressure column, for each position of the aperture, gave the pressure at that point of the area struck by the jet. When the aperture was exactly in the axis of the jet, the pressure column was very nearly level with the free surface in the reservoir supplying the v 2 jet ; that is, the pressure was very nearly-- . As the aperture moved away from the axis of the jet, the pressure diminished, and it became insensibly small at a distance from the axis of the jet about equal to the diameter of the jet. Hence, roughly, the pressure due to the jet extends over an area about four times the area of section of the jet. Fig 171 shows the pressure curves obtained in three experiments with three jets of the sizes shown, and with the free surface level in the reservoir at the heights marked. Experiment 1. Experiment 2. Experiment 3. Jet -475 in. diameter. Jet -988 in. diameter. Jet l - 95 in. diameter. , from Free
- e to Brass
in inches. ce from Axis
- in inches.
re in inches Water.
- from Free
ce to Brass in inches. cefrom Axis
- in inches.
re in inches Water.
- from Free
ce to Brass in inches. cefrom Axis
- in inches.
re in inches Water. 3>"2 C tf> O SfS C u
- c o
2 * js 5 5^
- o
a** 3 1 P" 5 5
JP E 5 = 1 43
40-5 42-15
42 27-15
26-9
- )
05 39-40 05 41-9 >} 08 26 9 1 37-5-39-5 fj 1 41-5-41-8 13 26-8 15 35 15 41 18 26-5-26-6
2 33-5-37 H 2 40-3 )( 23 26-4-26-5
25 31 25 39-2 28 26-3-26-6 , 3 21-27 3 37-5 27 33 26-2 ( 35 21 }) 35 34-8 }) 38 25-9 ( 4 14 M 45 27 43 25-5
45 8 42-25 5 23 , 48 25 5 3-5 55 18-5 53 24-5 t 55 1 6 13 58 24 ( 6 0-5 J( 65 8-3
63 23-3 65
7 5 ( 68 22 5 75 3 73 21-8 8 2 2 f 78 21 42-15 85 1-6 ( 83 20-3 95 1 ( 88 19 3 93 18 ( 98 17 26-5 1-13 13-5 1-18 12-5 ,, 1-23 10-8 ,, jl 28 9-5 ,, 1-33 8 ,, 1-38 7 ,, 1-43 6-3 il 48 5 ., 1-53 4-3 ,, 1-58 3-5 ,, 1-9 2 As the general form of the pressure curve has been already indicated, it may be assumed that its equation is of the form y = air** (1).
But it has already been shown that for x=0, y = ?i, hence ah.
