Art. LIX.—On the Mechanical Principles involved in the Sailing Flight of the Albatros.

Transactions and Proceedings of the Royal Society of New Zealand, Volume 2, 1869, Page 233

Art. LIX.—On the Mechanical Principles involved in the Sailing Flight of the Albatros. By J. S. Webb. [Read before the Otago Institute, November 2, 1869.] In the first volume of the “Transactions of the New Zealand Institute,” there is an abstract of a paper by Captain Hutton on this subject, but for want of necessary type for the algebraic formulæ the paper could not be printed in full. A revised and modified copy of the more important portions of it has, however, been published in the “Philosophical Magazine” for August last. The subject dealt with by Captain Hutton is what has been, somewhat inappropriately, called the “Sailing flight of the Albatros”—that steady and continued motion, without any observable action of the wings, which has attracted the attention of every one who has made a voyage in the southern seas. The paper is ably written and very interesting, but in his mathematical treatment of the subject Captain Hutton has not been happy, having as it appears to me, made a mistake at the outset of his calculations. The object of the present communication is to supply what I consider to be the necessary corrections, and to show the effect which the new results thus arrived at have upon the general conclusions drawn by Captain Hutton from his own calculations. As I cannot ask this meeting to follow me through a dry, though by no means abstruse, process of mathematical reasoning, I have thrown that portion of the paper into form of an appendix, which I lay on the table for perusal of those members who may be desirous of examining it. In his paper* “Phil. Mag.,” Vol. xxxviii., p. 130.—(No. 253, August, 1869.) Captain Hutton proposes to himself “to determine approximately the probable resistance of the air in order to allow ‘the albatros’ to sail for half an hour without moving its wings.” He estimates the under surface of the wings, body, and tail, at 8 sq. ft., and the weight of the bird at 16 lbs. From these data he concludes that “if an albatros starts with a velocity of 115 feet per second, it could maintain a constant height above the sea until its velocity was reduced to 64 feet per second, by merely increasing the angle to the horizon at which it was flying from 0° to 7°.” He assumes that the wings are always inclined 15° more than the body of the bird. The corrections I offer to-day show that, on the data assumed, the velocity at starting must be 155 feet per second, instead of 115, and that it must not be reduced during the interval below 100 feet per second, instead of to 64 feet. A consideration of the following extract from Captain Hutton's paper will show the bearing of this correction on his general results. “The velocity of the air in a ‘fresh sailing breeze’ is about 30 feet per second, in a ‘moderate gale’ 60 feet, in a ‘strong gale’ 90 feet, and in a ‘great storm’ 120 feet per second. Now, an albatros can often be seen sailing, though slowly, directly against a strong gale: his velocity” (through the air) “must therefore often be more than 90 feet per second. He is however most at home in a strong breeze or moderate gale, when the velocity of the wind is 50 or 60 feet per second, and consequently when his velocity would have to be 70 or 80 feet per second to enable him to fly easily against it. In a calm or light air, when the wind has a velocity of only 10 feet per second, the albatros rarely sails for so long as a minute at a time,—the reason for this being that as, in order to sustain himself in the air he must move through it with a velocity not less than 64 feet per second, he would even when flying against it have to travel over the sea at the rate of not less than 54 feet per second, or 36 miles an hour, and so could not reach it for good, or stop himself quickly enough when he saw anything; so that the velocity and manner of flight observed in the albatros correspond closely enough with those calculated as necessary from theoretical

considerations.” It will readily be seen that the very much higher velocities which I derive from Captain Hutton's data, upset the conclusion he has here drawn. It is however in the choice of his assumed data and in his calculations based on them, and not on the principle by which he accounts for the power of the albatros to sail for a long time without moving its wings, that Captain Hutton is in error. I do not know whether the merit of the demonstration belongs to him (he appears to claim it), but if so, notwithstanding the criticisms I have ventured upon, I willingly bear testimony to his success in the primary object of his paper, viz, to “indicate the principles involved in the flight of the albatros when sailing along without moving its wings.” Captain Hutton proceeds to calculate from his first results “what the resistance of the air to the forward progress of the albatros ought to be, to enable him to start with a velocity of 115 feet per second, and sail for half an hour without flapping his wings, and at the end of that time to have reduced his velocity to 64 feet per second.” He arrives at a result for which he himself deems it necessary to offer excuses, viz, that the resistance to a body of the shape of this bird is only 1–300th of that to round shot. Had he used the figures which I have brought out, instead of his own, his estimate would have been only about half what it is,—a further proof, if any were needed, that the real details of the bird's flight are very different to those assumed in his calculations. I have not the necessary leisure to attempt to deduce these details from such physical data as are available by the aid of the undoubtedly true principle laid down by Captain Hutton. I repeat and endorse his own closing remark:—“the problem still remains to be solved; but until experiments have been made on the resistance offered to the air by the front and lower surfaces of birds, a tolerably accurate solution is not possible.” I may add, that some careful observations of the duration of the “sailing flight” of various birds, and of their ordinary position in the air, whilst flying without flapping the wings, are absolutely necessary before anything like approximately correct calculations on the subject can be made. Appendix. The references in what follows are to the annexed copy of Captain Hutton's diagram, to which I have added the are H A′ C′, and the dotted lines A′ T and C′ S. Captain Hutton assumes the under surface of the bird at 8 feet, its weight at 16 lbs., the surface of the wings at (about) three times that of the body and tail, and the upward current of air necessary to support the bird against gravity, at 30 feet per second acting upon the whole bearing surface. “Let A B” he says, “represent the axis of the body of the bird flying in the direction B A and at an angle A E H to the horizon. Let C D represent the wings making ∠ C E H with the horizon. Take the line H E to represent the velocity at which the bird is flying, or the number of feet it passes through the air in one second. From H draw the perpendicular H A, the line will represent the distance which the bird will rise (omitting for the present the force of gravity) by means of the angle at which he is flying to the horizon.” Here Captain Hutton first assumes that the number of feet the bird travels in one second = H E and then that the bird will pass in the same time through the longer distance A E. The mistake leads him to the further error of adopting H E tan A E H as the measure of the vertical component of the

atmospheric resistance instead of H E sin A E H. If we take A′ E=H E and draw the perpendicular A′ T, then A′ T represents the height the body would rise (irrespective of gravity) in one second. Now A′ T=A E sin A E H =H E sin A E H. Again, Captain Hutton has unaccountably adopted a totally different method to arrive at the vertical component of resistance in the case of the wings, and has resolved the force represented by H E into (1) H K non-effective (2) L E, resisting gravity, and (3) K L retarding the motion of the bird. He has thus arrived at the strange conclusion that at one angle of inclination (and for the body of the bird) the upward pressure is in proportion to the tangent of the angle, i. e., to the ratio of the sine to the cosine (sin A E H/cos A E H) and that at another angle (for the wings) it is in proportion to the product of the sine and cosine of the angle of inclination (sin C E H. cos C E H). The error lies here. On his own assumption H E is the absolute velocity in one second, therefore the retarding force has been overcome in addition to the, production of so much motion. The whole force exerted by the bird is, in fact, H E + R where R : H E ::K E, ∴ it is not L E but K E (= H E sin C E H) which represents the vertical component of the force actually at work. Instead therefore of H A and L E as measures of the upward pressures on the body and wings respectively, we must take H E sin A E H and H E sin C E H. Captain Hutton goes on to say “the total amount the bird will rise per second will be L E + H A feet.” Introducing the corrections just made, this amounts to saying that the upward pressure on the whole area of the bird =H F (sin A E H + sin C E H). This is a grave error. Let P be the total pressure which supports the bird; p the average pressure on each square foot of sustaining surface; M the area of the lower surface of the body and tail; N the area of the under surface of the wings; Then it is evident that P=M × H E sin A E H + N × H E sin C E H. Now by the assumed data p= P/8 M=2 and N=6 therefore ∴p=H E sin A E H+3. H E sin C E H/4   (1) Also p=2 lbs. per square foot, and is assumed to represent the pressure of an upward current of air having a velocity of 30 feet per second. From this we obtain H E=120/sin A E H+3. sin C E H   (2) This equation gives, when ∠ A E H=0°, and C E H=15°, H E=120/3 × .258819 = 155(nearly)   (3) And for ∠ A E H=7°, and ∠ C E H=22° H E=120/.121869+3×.374607=96 (and a little more). (4) Captain Hutton closes this part of his calculations at this stage, and omits to consider that as the angle of flight is increased, the sustaining surface is reduced in the same proportion as the cosine of the ∠ of inclination to the horizon. In passing from the conditions of equation (3) to those of (4), this

cause is influential to the extent of about 4 per cent. Hence the bird will not preserve a horizontal flight, if the velocity falls below 100 feet per second, without increasing its angle of flight more than the assumed 7°. Another slight error occurs in Captain Hutton's calculations which is probably an oversight. Using his equation H E = 30/tan A E H + sin C E H. cos C E H he makes H E= 115, when A E H= 0, and C E H= 15°. The true result is 120. When proper data have been obtained, the solution of the problems connected with this “sailing flight” should, I think, be approached in an entirely different manner from that adopted by Captain Hutton. His deductions as to the resistance of the air to a projectile of the form of the albatros are of no value at all, and may, I think, be shown to be inconsistent with facts already ascertained. The principal portion of the resistance is that which is resolved into a sustaining, or upward bearing, force, and this is exerted against the obliquely exposed under surfaces of the bird. The formula for the resistance of a fluid to a plane, moving obliquely through it is— R= ½ Q v2 sin 3 I. A where Q is the density of the fluid, v the velocity of the stream plus that of the plane if it is moving against it, I the < of inclination to the stream, and A the area of the plane. The two latter coefficients will have to be determined from observations, which in the case of I it will be very difficult to make. If the part of R which is resolved into a force sustaining the bird against gravity be known, let this= C, then the retarding force of atmospheric resistance against the inclined surfaces of the body and wings,— = ½ Q v2 sin3 I. A—C We must deduct this quantity from the whole retardation observed to find what would be the resistance to the front surfaces of the bird when both body and wings were horizontal. It is only by this process that we can obtain a quantity which is comparable with the atmospheric resistance to round shot.