11 YDBAULICS,] HYDROMECHANICS 515 Jet Propeller. In the case of vessels propelled by a jet of water (fig. 166), driven stern wards from orifices at the side of the vessel, the water, originally at rest outside the vessel, is drawn into the ship and caused to move with the forward velocity V of the ship. Afterwards it is projected stcrnwards from the jets with a velocity v relatively to the ship, or -v - V relatively to the earth. If O is the total sectional area of the jets, $2r is the quantity of water dis- - charged per second. The momentum generated per second in a sternward direction is Slv(v - Y), and this is equal to the forward acting reaction P which propels the ship. The energy carried away by the water The useful work done on the ship 9 Adding (1) and (2), we get the whole work expended on the water, neglecting friction : V = Slv V ^f . 9 2 Hence the efficiency of the jet propeller is PV 2V ...... (3). This increases towards unity as v approaches V. In other words, the less the velocity of the jets exceeds that of the ship, and there fore the greater the area of the orifice of discharge, the greater is the efficiency of the propeller. In the " Waterwitch " v was about twice V. Hence in this case the theoretical efficiency of the propeller, friction neglected, was. about f . 151. Pressure of a Steady Stream in a Uniform Pipe on a Plane normal to the Direction of Motion. Let CD (fig. 167) be a plane Ai Fig. 167. placed normally to the stream which, for simplicity, may be sup posed to flow horizontally. The fluid filaments are deviated in front of the plane, form a contraction at AjAj, and converge again, leaving- a mass of eddying water behind the plane. Suppose the section A A taken at a point where the parallel motion has not begun to be disturbed, and A 2 A 2 where the parallel motion is re established. Then, since the same quantity of water with the same velocity passes A A , A 2 A a in any given time, the external forces produce no change of momentum on the mass A A A A.,, and must therefore be in equilibrium. If fi is the section of "the stream at A A or A 2 A a , and u the area of the plate CD, the area of the contracted section of the stream at A l A l will be c e (fi-w), where c c is the coefficient of contraction. Hence, if v is the velo city at A A or A t A 2 , and t j the velocity at A^, Let;? , PJ, p 2 be the pressures at the three sections. Applying Bernoulli s theorem to the sections A A and AjAj, Also, for the sections AjAj and A^A 2 , allowing that the head due to the relative velocity v^ -v is lost in shock : _lhv*(vi-v?. . or, introducing the value in (1), 2-7 (2); Now the external forces in the direction of motion acting on the mass A A A 2 A 2 are the pressures Q, - ,fl at the ends, and the reaction - 11 of the plane on the water, which is equal and opposite to the pressure of the water on the plane. As these are in equilibrium, an expression like that for the pressure of an isolated jet on an in definitely extended plane, with the addition of the term in brackets which depends only on the areas of the stream and the plane For a given plane, the expression in brackets diminishes as 12 increases. If = p, the equation (4) becomes which is of the form (*), >- 2? where K depends only on the ratio of the sections of the stream and plane. For example, let c c =0 85, a value which is probable, if we allow that the sides of the pipe act as internal borders to an orifice. Then = p( 1-176 P = 1 2 3 4 5 10 50 100 oo 3-66 175 1-29 1-10 94 2-00 3-50 The assumption that the coefficient of contraction c c is constant for different values of p is probably only true when p is not very large. Further, the increase of K for large values of p is contrary to experience, and hence it may be inferred that the assumption that all the filaments have a common velocity v l at the section AjA, and a common velocity v at the section A 2 A 2 is not true when the stream is very much larger than the plane. Hence, in the expression K must be determined by experiment in each special case. 152. Pressure on a Cylindrical Body of a Length about three times its Diameter. A. contraction of the stream is formed at A : Aj A A A Fig. 168. (fig. 168). Let the same notation be used, the subscript figures indi cating the section to which the quantities belong. For sections AA, AA, for sections AjAj and A 2 A 2 , allowing for the abrupt enlargement of the stream, _ G + 2g G^2g + 2g and for sections A 2 A 2 , A 3 A 3 , allowing for another abrupt enlarge ment, a . = 3 . . 2gr G^V Zy Adding the three equations,
From the principle of momentum,
