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Aviation History
1915
1915 - 0216.PDF
the same. On looking into the matter more closely it will also be observed how some of the conclusions given in the previous work as the outcome of lengthy investigations arise immediately and directly as the natural outcome of the new line of treatment. Thus the con ception of the peripteral area and peripteral zone introduced into the author's "Aerodynamics" in connection with a discussion on the screw propeller, § 210 (Ch. IX.), and allowed to remain rather abstract in character, may now be identified as an area representing by its content the equivalent mass of the trailing vortices; it is here that we find the augmented mass of fluid in downward motion, which, according to the regular application of the Newtonian method, represents by its downward momentum the reaction of the sustained load. We know that in mathematical theory any free vortex ring or (in two-dimensional motion) vortex pair may be regarded as carrying the momentum communicated by the impulse by which it is generated,* and so is the equivalent of some appropriate mass of fluid in motion ; it is, strictly speaking, the vertical cross section of this hypothetical mass which, in the author's former work, constitutes the peripteral area. It will now be shown that the results given by the present line of reasoning are in substance identical with thoie obtained in the author's previous work. In both methods of treatment we have the aerofoil dealing directly with a cyclic component in the motion of the air, and ultimately leaving in the wake downward momentum whose value per second is the equivalent of the load sustained. Thus the direct action of the aerofoil is measured by the sum of the momenta of an ;// current received and a down current emitted ; it is in fact the downward momentum represented by the continuous reversal in the vertical component of the flow of a mass of fluid, just as, for example, in the Pelton wheel. The energy spent or work dose by the aerofoil is thus the difference between that received from the up current and given or imparted to the down current, or, in the symbols employed in the author's "Aerodynamics," W = momentum/sec. = m(fi + a) V and energy/sec. = m(0- - o*j K-'/a. If the mass per second representing the trailing vortex pair, that passing through the peripteral area, be denoted by Mand v be its downward velocity, we also have W = momentum/sec. = Mv and energy/sec. — Mv*lz or, Mv = m[B + a)V (1) and 2=2 (2 by (1) M=m fi + a)Vlr (3) m[p I a) Vv w(/3* - as) V* or (3) - - = — (4) or, v = B-a)V (5 8 + a also by (1) M - „ 111 (6) p — a In the author's "Aerodynamics" o//3 is represented by a con stant < ; substituting we have M=\^)m (7) Now the mass per second m is that coming within the " sweep " of the aerofoil as defined by an area KA (A being the area of the foil), hence the peripteral area is :::- « The above is the result and in substance the reasoning given in §210 of "Aerodynamics." Since our present basis is that of an aerofoil of fixed span and variable chord, we will take the span / as DAMS, and express the peripteral area in terms of I- ; thus, employing the values of the constants K and e as given in the author's " Aerodynamics," we obtain the results given in Table I, Referring to this table, we see that the side of the square representing the peripteral area varies from being approximately equal to the span / in the case of an aerofoil of aspect ratio = 3, to 0-83 in the case of aspect ratio = 12. Now the side of this square may be taken as roughly representing the base of the vortex pair, since the equivalent mass of a vortex pair is approximately equal to that of a mass of fluid represented by the content of the square on its base,t so that we have definitely a quantitative confirmation of the theory presented in the present communication based upon the constants determined by totally different and independent methods. Moreover, we find that so minute a matter as the influence of the * That any such vortex ring or pair dx:s not as a whole contain momentum the author has drfinitely proved ; however, we may nevertheless legitimately regard it as carrying a definite quantity of momentum just as though it were existent. The auth >r sug esis that this quantity should be termed the "pseudomotnentum" of the vortex ; it is ever equal to that of the impulse by which it is generated. t In tde case of a simple vortex pair generated by a uniform impulse as in Fig. to, the area representing the equivalent mass is n-, 3 times the square on the base x x', or approximately x '05. end effect or eddy, already noted, as tending to widen the base in the case of aerofoils of low aspect ratio, is accurately reflected in the figures obtained from the equation. TABLE I. . , Aspect Ratio. n 3 4 5 6 7 8 Constants. K I'OO 1 03 1-064 no i*ii 1*14 e 0-48 o-54 0-59 0-62 065 o-68 1 + « 1 — « 2-8S 3'4S 4'i3 470 5'3o 6-o A/" x W n 0"975 093 O-0I o-88 0-87 086 1-175 0-72 7-2 0-85 12 1-195 0-75 8-4 0-83 Again, we find complete harmony between the two liaes of treatment in the interpretation of equation (5). We have already seen that in accordance with our present hypothesis the velocity v is (with reference to Fig. 11) given by the expression V tan 0. When, as in Fig. 13, we add the arched section representing the path of flow of the cyclic component as a superposed system we have the angle of dip, the o of the previous investigation ("Aero dynamics") and an equal angle superposed on the angle f), making a total trail angle /8 ^ 9 + a, and we have (B-a)V=[(6+ a) -a]V=6V, since we are working on the " small single hypothesis "J the »j of the present investigation is identical with the v of equation (5), and we see that the present regime and that of the author's previous work give identical results. The agreement between the results of the author's previous work and the present investigation is not to be regarded so much as an independent confirmation, but rather as a justification of the original theory, and as a development directed to elucidate much that might otherwise be regarded as obscure in the regime. Put concisely, the earlier investigation and the present deal with the same main problem, the mode of support, on the same founda tion theory, but they begin at opposite ends : in the earlier work the cyclic motion around the aerofoil was taken as a basis and the remainder of the system was deduced as a corollary, the present line of argument begins with the ultimate or final step in the communication of momentum to the air in the trailing vortices, and works backwards to the motions and behaviour of the air in the more immediate vicinity of the aerofoil itself. $ 10. Quantitative Theoretical Treatment. — The cyclic or vortex theory of the aerofoil is capable of yielding quantitative results quite apart from any experimentally determined pressure values or constants whatever. There are difficulties at present, due to the limitations of mathematical analysis, but these will be without doubt overcome : if the mathematician fails us we can fall back upon graphic methods. Reverting to the hypothesis of § 6, we have seen that the simple "bifocal" vortex pair is impossible owing to the high velocity in the vicinity of the foci, or vortex filaments ; this would betoken an aerofoil whose camber and angle fi increased towards / / 1 A t ro 'a 11 Sn -- -H Fig. 16. Fig. 17. the lateral extremities to an indefinite degree, an altogether absurd and impracticable feature even from a theoretical standpoint. It has further been pointed out with reference to Fig. 11 that the more usual condition is that of uniform camber from end to end, but no attempt has been made so far either to carry the study of this X The author's method, as employed in his " Aerodynamics," is based on the assumption that the angles concerned in the sectional form of the aerofoil (relatively to the time of fl grit) come within tbe definition of a .small angle, i.e., tf»(in circular measure) = sin <£ as tan fj> within permissible limits of error. 2l6
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