A, A, A, Curve described by pivoting point. C, C, C, Curve described by outer edge of stern. D, Position of ship's centre of gravity when helm commenced to move over. E, Position of ship's centre of gravity when helm had reached 32°. F, Position of ship's centre of gravity when vessel had turned through 90°. Time from D, 49 sec. G, Position of ship's centre of gravity when vessel had turned through 180°. Time from D, I min. 20 sec. Fig. 80.-Gantry at Messrs Cramps' Shipbuilding Yard, Philadelphia. Fig. 81. Gantry at Messrs Harland & Wolff's Shipbuilding Yard, Belfast. The variation of the rate of growth of I.H.P. (or E.H.P.) with the speed is a result of the interference of the bow and stern wave systems, and is dependent upon the speed-length ratio (vide "Wave Resistance," above). A good illustration is afforded by taking the case of a vessel such as a torpedo-boat destroyer, which is run over a considerable range of speed. Fig. 37, Plate II. shows, for such a vessel, three curves plotted to a base of speed, the ordinates being respectivelyI.H.P. I.H.P. I.H.P. I.H.P., (speed)' (speed)** The second of these is of course a curve of resistance, and the rapid rise and fall of the rate of growth of resistance manifests itself in this resistance-curve by a very marked hump between 15 and 25 knots speed. The third curve, that of (speed) is interesting as affording, by its slope at different points, a very good indication of this rate of growth. Up to about 13 knots this curve is not far from being horizontal, indicating that till then the resistance is varying about as the square of the speed. The rate of growth increases from this point till it reaches a maximum of 15 knots, and then falls off till at about 20 knots the resistance once more varies as the square of the speed. From this point onward the resistance increases at a less rate than the square of the speed. It has been previously noted that the skin friction part of the E.H.P. does not obey the law of comparison; this is on account of variation of ƒ with length, and the index of the speed being different from 2. The coefficient f varies much more rapidly at the smaller lengths, and hence for these the skin friction correction is more important for a given change in length. For such lengths as are dealt with in ships, e.g. 100 ft. and upwards, and such lengths as we should deal with in applying the data that are now given, it has been found possible to express the correction for skin friction very accurately by the curves in fig. 38, Plate II. These indicate the absolute correction that must be applied to the E.H.P. deduced for the given displacement from the standard curves when interpreted by the law of comparison, and are drawn for a series of displacements on a base of speed; the correction for any odd displacement can be easily interpolated. An addition must be made for displacements under, and a deduction for displacements over, the standard 1000 tons. The following example illustrates this point and the method of using the standard curves: Beam A vessel 320'X35X13X2135 tons is being designed; to construct an E.H.P. curve, for speeds 11-22 knots. The proportions Draught ratio and block coefficient) of the design are most closely approximated to by type 2, group A (320' being the immersed length). First find the length for a similar vessel of 1000 tons displacement; 1= ordinates representing E.H.P. for the given speeds of the 1000-ton standard ship. These figures are converted into those appropriate for the design, by the law of comparison. If v and e are the speed and E.H.P. for the 1000-ton ship, and V and E corresponding quantities V (2-135) = =2.424: ν (2.135)=248.5 ft., and then from fig. 41 read off for the design, then = (2·135) = 1·135; and using these ratios we get a table thus: Shallow water. In the results hitherto recorded the depth of water has been supposed sufficient to prevent the disturbance attending the motion of a vessel on the surface from extending to the bottom; in these circumstances the resistance is unaffected by a moderate change in the depth. Conditions, however, frequently arise in which vessels are run at high speeds in comparatively shallow water; and a marked alteration is then observed in the resistance and power corresponding to a particular speed. An investigation of the effect of shallow water on resistance is therefore of importance and interest; and a brief account of this part of the subject is here appended. The change from deep to shallow water modifies the shape of the planes normal to the surface of the hull; those in shoal water tend stream lines, many of which in deep water are approximately in to lie more nearly in horizontal planes, owing to the reduced space under the bottom of the ship. In consequence, the velocity in the stream tubes in the vicinity of the ship is increased, and the changes of pressure and the "statical" wave heights are exaggerated. This becomes less; but the effect on the residuary resistance is more causes an increase in the frictional resistance as the depth of water complicated.. Firstly, the length of the waves corresponding to a speed is increased from that expressed by 28 to be in accordance with the formula which applies to shallow-water waves for a depth k. D2 termed a When the depth h is equal to the length of wave is infinite, and the wave becomes of the type investigated by Scott Russell in canals, and "solitary wave or a wave of translation." When the depth of water is less than no permanent wave system of speed ફર v can exist. These changes in the wave length considerably affect the wave pattern and alter the speeds at which interference between the bow and stern systems has a favourable or unfavourable effect on the efficiency of propulsion. In the second place the amount by which the speed of travel of the energy of the wave falls short of the speed of the ship is expressed by 4h/l sinh 4Th In deep water this difference of speed is 2; in shallow water it diminishes, becoming zero at the critical depth producing a wave of translation. is increased in shallow water, theoretical investigation showing that, Thirdly, the local disturbance immediately surrounding the ship at the critical depth above referred to, it becomes indefinite or is only limited by its own viscosity and eddying resistance. In still shallower water, the amount of disturbance is reduced as the departure from the critical depth becomes greater. higher velocity of rubbing is further modified by the large dimen Finally, the increase of the frictional resistance due to the sions of the wave accompanying the ship; the particles of a wave in very shallow water are moving appreciably in the direction of travel, which might lead to a reduction in the frictional resistance. From these considerations it appears impossible to obtain, a priori, the net effect of shallow water on the resistance, owing to the divergent character of the component effects producing the final result. This difficulty is confirmed by the inconsistency of the readings frequently obtained during experiments in shallow water, pointing to instability in the conditions then existing A number of experiments have been carried out in shallow water with both ships and models; the most important are those by Constructor Paulus (SchleswigHolstein District Club, 1904), Captain Rasmussen, Mr Yarrow. Herr Popper and Major Rota, many of which are recorded in the I.N.A. Transactions. A summary of the conclusions drawn from them is appended: Col.4-Col.5 =E.H.P. Corrected. 348 Friction: read from Figure. 638 fore, that the increase of resistance due to the enhanced dimensions of the wave then accompanying the ship is more than sufficient to counteract the gain resulting from the diminished drain of energy from the wave system astern. 3. At high speeds, when a considerable portion of the resistance is due to wave-making, the total resistance diminishes at depths lower than the critical depth, and is frequently less in very shallow water than in deep water. 4. The humps" in the curves of resistance on a base of speed occur at lower speeds in shallow water, and are more pronounced; the resistance is occasionally reduced when the speed is increased. 5. The changes of resistance produced by shallowness are accompanied by corresponding changes in the speed of revolution of the engines and in the trim of the vessel. These are illustrated by the curves in fig. 52, Plate VI., which are taken from a paper read before the I.N.A. by the writer in 1909, giving the results of some trials on H.M. torpedo-boat destroyer "Cossack." The data obtained from the various shallow water experiments are capable of extension to ships of similar types by the applica-only a very low efficiency is obtained. A second type of propeller is tion of the law of comparison at corresponding depths (proportional to the linear dimensions) and at corresponding speeds. The influence of shallow water on the speed of a large number of ships can be thus obtained; but the data at present available are insufficient to enable a general law, if any exists, to be determined. Accelera When the speed of a ship is not uniform, the resistance is altered by an amount depending on the acceleration, the inertia of the ship and the motion of the surrounding water. In the ideal conditions of a vessel wholly submerged in a perfect fluid, tion. the force producing acceleration is the product of the acceleration with the "virtual mass," which is the mass of the vessel increased by a proportion of the displacement; e.g. for a sphere, one half the displacement added to the mass is equal to the virtual mass. The effect of acceleration on a ship under actual conditions is less simple; and the virtual mass, defined as the increase of resistance divided by the acceleration of the ship, varies considerably with the circumstances of the previous motion. The mean value of the virtual mass of the "Greyhound," obtained by Froude from the resistance experiments, was about 20% in excess of the displacement. This value is probably approximately correct for all ships of ordinary form, and is of use in estimating the time and distance required to make a moderate alteration in speed; the conditions during the stopping, starting and reversing of ships are generally, however, such as to make this method inapplicable. Propulsion. The action of a marine propeller consists fundamentally of the sternward projection of a column of water termed the propeller race; the change of momentum per unit time of this water is equal to the thrust of the propeller, which during steady motion is balanced by the resistance of the ship. a low efficiency. The foregoing considerations show that, with a given thrust, the larger m the quantity of water acted upon (and the smaller, therefore, the slip), the higher is the efficiency generally obtained. The type of propeller most nearly conforming to the fundamental assumption is the jet propeller in which water is drawn into the ship through a pipe, accelerated by a pump, and discharged aft. The "Waterwitch" and a few other vessels have been propelled in this manner; since, however, the quantity of water dealt with is limited for practical reasons, a considerable sternward velocity in the jet is required to produce the thrust, and the slip being necessarily large, the paddle, or stern-wheel which operates by means of floats mounted radially on a circular frame, and which project a race similar to that of the jet propeller. Certain practical difficulties inherent to this form of propulsion render it unsuitable or inefficient for general use, although it is of service in some ships of moderate speed which require large manœuvring powers, e.g. tugs and pleasure steamers, or in vessels that have to run in very shallow water. The screw, which is the staple form of steamship propeller, has an action similar in effect to the propellers already considered. Before proceeding to discuss the action of screw propellers, it is desirable to define some of the terms employed. The product of the revolutions and pitch is often called the speed of the propeller; it represents what the speed would be in the absence of slip. Speed of advance, on the other hand, is applied to the forward movement of the propeller without reference to its rotation; and is equal to the speed of the ship or body carrying the propeller. The difference between the speed of the propeller and the speed of advance is termed the slip; and if the two former speeds be denoted by and V respectively, the slip is -V and the slip ratio (or properly the apparent slip ratio) "V. This notation corresponds to that previously used, -V being then defined as the absolute velocity of the race; it is found with propellers of the usual type, that zero thrust is obtained when v=V, provided that the "conventional" pitch, which for large screws is approximately 1-02 times the pitch of the driving surface, is used in estimating v. The pitch divided by the diameter is termed the pitch ratio. The theories formulated to explain the action of the screw propeller are divisible into two classes-(i.) those in which the action of the screw as a whole is considered with reference to the change of motion produced in the water which it encounters, the blade friction being, however, deduced from experiments on planes; and (ii.) those in which the action of each elementary portion of the blade surface is separately estimated from the known forces on planes moved through water with various speeds and at different angles of obliquity; the force on any element being assumed uninfluenced by the surrounding elements, and being resolved axially and circumferentially, the thrust, turning moment, and efficiency are given by summation. Professor Rankine in Trans. Inst. Nav. Archs., 1865, assumed that the propeller impressed change of motion upon the water without change of pressure except such as is caused by the rotation of the race. In Sir George Greenhill's investigation (Trans. Inst. Nav. Archs., 1888) it is assumed conversely that the thrust is obtained by change of pressure, the only changes of motion being the necessary circumferential velocity due to the rotation of the screw, and a sufficient sternward momentum to equalize the radial and axial pressures. These two theories are both illustrative of class (i.); and this idea was further developed by Mr R. E. Froude in 1889, who concluded that the screw probably obtained its thrust by momentarily impressing an increase of pressure on the water which eventually resulted in an increase of velocity about one-half of which was obtained before and one-half abaft the screw. A lateral contraction of the race necessarily accompanies each process of acceleration. These general conclusions have been in some degree confirmed by experiments carried out by Mr D. W. Taylor, Proceedings of the (American) Society of Naval Architects, &c., 1906, and by Professor Flamm, who obtained photographs of a screw race in a glass tank, air being drawn in to show the spiral path of the wake. In Trans. Inst. Nav. Archs., 1878, Froude propounded a theory of the screw propeller illustrative of the second class above mentioned, the normal and tangential pressures on an elementary area being deduced from the results of his own previous experiments on obliquely moving planes. He was led to the following conclusions regarding maximum efficiency:-(1) The slip angle (obliquity of surface to the direction of its motion) should have a particular value (proportional to the square root of the coefficient of friction); and (2) when this is so, the pitch angle should be 45°. The maximum efficiency obtained from this investigation was 77%. This theoretical investigation, though of importance and interest, does not exactly represent the actual conditions, inasmuch as the deductions from a small element are applied to the whole blade, and, further, the considerable disturbance of the water when a blade reaches it, owing to the passage of the preceding blade, is ignored. Assuming in the first place that the passage of the ship does not affect and is uninfluenced by the working of the propeller, let V be the speed of the ship, v that of the propeller race relative to the ship, and m the mass of water added to the propeller race per second. The thrust T is then equal to m (v-V), and the rate at which useful work is done is TV or mV (v-V). Loss of energy is caused by (a) shock or disturbance at the propeller, (b) friction at the propeller surface, (c) rotational motions of the water in the race, and (d) the astern motion of the race. Of these (a), (b) and (c) are capable of variation and reduction by suitable propeller design; though unavoidable in practice, they may be disregarded for the purpose of obtaining the theoretical maximum efficiency of a perfect propeller. The remaining loss, due to the sternward race, is equal to m(-V); whence the whole energy supplied to the propeller in unit time is expressed by m(2-V2) and the efficiency by The quantity -V is commonly termed D-V the slip, and the slip ratio; the latter expression being denoted by s, the theoretical maximum efficiency obtained on this basis The most complete information respecting the properties of screw becomes. It appears, therefore, that the maximum efficiency propellers has been obtained from model experiments, the law of comparison which has been shown to hold for should be obtained with minimum slip; actually, however, with screw ship resistance being assumed to apply equally to screw propellers the losses here disregarded entirely modify this result, propellers. No frictional correction is made in obtain which is true only to the extent that very large slip is accompanied bying the values for large screws from the model ones; as stated by I-S 2V Experimental results. |