Frederick William Lanchester

“ In the forms that succeed in practice the propeller is usually behind in the frictional wake, never in front; and the successful forms of propulsion are those in which a sufficient mass of water per second can be conveniently handled; thus, jet propulsion has become practically extinct. We are, therefore, led to appreciate the soundness of the Newtonian method.
There are many methods of mechanical propulsion, that is to say, there are several known mechanical devices for producing the reaction on the fluid which we have so far regarded as being accomplished by action at a distance. ”

“ We may therefore expect to find the blade form becoming of more compact outline as higher speeds come into vogue.
It is manifest that for marine work at high speeds it is impossible to construct a fine pitch propeller to give any reasonable economy, for at a comparatively low vessel speed the velocity of the blades will begin to exceed the limiting value, and it will be necessary to add so much extra surface to reduce the pressure that the efficiency will be poor. ”

“ If we make the blades too long, the efficiency at the extremities would be so low as to involve an extravagant expenditure of power; if, on the other hand, we confine the length of the blade to the region where the efficiency is about its maximum, in order to reap the benefit of the full value (as given in § 204) , we encounter practical disadvantages in the increased propeller diameter required to deal with a given quantity of fluid (the proportion of the “disc” area utilised becoming small) , and in the length (and consequent resistance) of the arm necessary to attach the blade to its boss. ”

“ The form of blade employed in the marine propeller is of necessity confined to low values of long slender forms such as may be well suited to an air propeller will be too weak (unless made of disproportionate thickness) to stand the pressure required. In practice the value of employed is less than 3, usually very much less; we will therefore confine our attention to blades of this proportion with the knowledge that the results as to pressures and efficiency ought in general to be higher than those that obtain in practice. ”

“ The resistance of the phantom ship is supposed, at its velocity to be equal to that of the real ship at its velocity so that the propeller designed for the phantom ship on the basis of simple theory will be correct for the real ship to work in its frictional wake. The important fact is thus rendered apparent, that the form of the propeller proper to highest efficiency is independent of the existence or otherwise of a frictional wake. ”

“ If we take the pitch as equal to 1 times the diameter, this velocity, at the extremity of the blade, corresponds to 39.7 feet per second for speed of vessel, that is, approximately, 27 miles per hour or 23 knots. In practice, the blade of a propeller is never brought to a point as we are now supposing; the end of the blade is always rounded and the pressure consequently less than contemplated by our theory. It is probable that from this cause cavitation does not commence to give trouble till a somewhat higher speed is reached. ”

“ We have an aerofoil whose aspect ratio is of considerable magnitude, and whose grading is specified, and we prove that the reaction due to each increment of length is proportional to the grading ordinate proper to that increment, irrespectively of the fore-and-aft dimension. The proof involves the tacit assumption that for the smooth curve form of grading specified, a geometrical similarity of section at all points involves a uniform pressure distribution. ”

“ Pressure Distribution.—It is evident that, according to the present theory, the propeller blade is amenable to precisely the same laws so far as its pressure-velocity relation is concerned as the ordinary aerofoil, and we presumably also have the two alternative types of fluid motion, the continuous and the discontinuous, according as the blade is given a pterygoid form (based on a helix) or whether a simple helical surface or sheet (the analogue of an aeroplane) is employed. ”

“ If, as an example, we suppose that 25 per cent, of the weight of the machine is taken up by the motor itself, and if the motor weigh only 2 lbs. per horse-power, it is improbable, taking everything into account, that seventy miles per hour can be exceeded. Flying is comparable to hill climbing on a road automobile where represents the gradient, this being about 1 in 5 or 1 in 6, and where the transmission efficiency is limited to about 66 per cent. The velocity limit rests entirely on the weight per horse-power, the aerodrome being presumed designed for least resistance ”

“ When we have to deal with the propeller blade on the assumption of a discontinuous system of flow, it is evident that the fluid in the dead-water region must be following the blade in its spiral path and must consequently be subject to centrifugal force. ”

“ Leaving all attendant difficulties on one side, it is evident that the conveyance of goods by flying machine would be comparable, so far as power expenditure is concerned, with drawing them on a sleigh over a common road, so that where any other method of transport is possible, flight may be regarded as out of the question. ”

“ Let us now re-calculate the efficiency as in § 198: we have work done usefully per second and energy taken out of the fluid, that is, energy received per second, is or, total energy expended per second , or,
efficiency
(5) This is greater than unity; the result being, as anticipated, that it is theoretically possible that a vessel should be propelled for a less expenditure of power than that by which it can be towed. ”

“ On the Marine Propeller.—The marine propeller, which has already been made the subject of comparison, is of especial interest as representing in a concentrated form the experience of over half a century. Beyond the advantages to be gained from an examination of established practice as a quantitative check on our investigation, there is some reasonable probability that we may arrive at facts having a bearing on the future evolution of screw propeller design. ”

“ But the total work done in propulsion is the same in both cases, therefore if is the efficiency under real conditions, we have—
The value of depends upon the lines of the vessel and position of the propeller, the essential point being the extent to which the frictional wake is led into the periptery of the propeller blades. It is manifest that is limited to a value less than the sternward component of the impressed velocity (compare § 200) . ”

“ If the conditions of least gliding angle are infringed, the present theory continues to apply, but the result will be the best efficiency possible under the adverse conditions imposed, and not the real maximum.
Now and therefore is constant in our expression; consequently we have to solve—
for maximum value where is the only variable. ”

“ If we presume that the propeller is of the customary proportions, based on a 95 per cent, discard as to diameter, pitch, etc., the length of the blade (in the sense here employed) is approximately twice the radius at the point chosen, so that, expressing the circumferential spacing of the blades in terms of the radius, we shall have for 3, radius 2.8, which, multiplied by gives 160 degrees apart, or two blades nearest whole number, and for 8 we get 2.1, which gives 120 degrees, or three blades almost exactly. ”

“ Now the energy that is left in the kinetic form takes some time to disappear, and in the meantime it constitutes a wake current with a corresponding counterwake whose momenta are equal and opposite, the wake current being situated in the immediate track of the vessel, and the counterwake further afield and extending theoretically to the confines of the fluid region, but only sensible for a limited distance. ”

“ The desirability of keeping the propeller circle as small as possible is probably responsible for the employment of a fourth blade in the marine propeller. Four blades doubtless give rise to some slight interference and loss of power, but not sufficient to be seriously detrimental. ”

“ It is possible that the constant ought to vary from point to point along the blade; if this is so it is a matter on which we have so far no information; for the present it should be taken as the aerofoil value of proper to the value of employed. In Fig. 137 the blade is supposed continued to connect to the boss. Such continuation is always necessary, unless a boss of very large diameter is employed, the continuation being of stream-line section symmetrically disposed about the pitch helix. ”

“ Fig. 131. the supporting reaction is continually derived from the virgin fluid, and the line of pressure reaction of any plane, or any component of it, only cuts the path of that plane once. In the case of the propeller, on the contrary, the component of the pressure reaction of any blade in the line of motion cuts the paths of that blade an in- definite number of times. We have here to deal with a fact new to our theory. ”

“ It is probable that, owing to the much lighter pressures required, it will be practicable to employ greater values of in the aerial propeller than in the marine propeller: an aspect ratio of 6 or 8 does not appear to present any difficulty. ”

“ Draw the line parallel to the axis of to represent the component of in the direction of motion of the vessel, cut off and draw perpendicular to draw perpendicular to and perpendicular to
Then being equal to , we have equal to the two triangles being equal in every respect. Let us denote and
Now while the blade moves from to h the energy lost will be that is to say, we regard the matter as a case of gliding, to which it is strictly analogous. The energy utilised in propulsion during the same period will be (These quantities are indicated by the shaded areas in the figure.) ”

“ Now, if we suppose the limits of the blade length be moved from place to place, so that, however, the annular disc area of the propeller remains contant (which is our fundamental condition) , it is a matter of simple geometry to show that the area of the thrust grading curve (of which is the ordinate) must remain constant; consequently, if we suppose any small variation to take place, and represent the various quantities as given on Fig. ”

“ Indicated Horse-power and Thrust Horse-power per 100 lbs. weight at different Velocities and for different Values of .
If we take account of body resistance, we know that the total resistance is greater the higher the velocity, for the body resistance increases as so that the angle is no longer constant in respect of We also know (§ 171) that the influence of the weight of the aerofoil, as additional to the load carried, is to place a lower limit on the velocity that may be usefully employed. ”

“ The calculation has been made for values of extending from 6 degrees to 12 degrees, as in the preceding Table; experiment would appear to show that 10 degrees is as low a value of as can be obtained under practical conditions; it is, however, possible that with increased experience lower values may be obtained. An aerodrome whose is greater than 12 degrees is certainly of inefficient design. ”