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Aviation History
1917
1917 - 0838.PDF
[/TIGHT AUGUST 16, 1917. THEORY OF PRESSURE ON A C - PLANE SURFACE DUE TO RELATIVE WIND. By A. E. WATSON. . . Concluded from page 816.) - " To find the pressure on the back of the element K it should be noted that the plane is moving away from the incident streams. Thus streams which, when the plane is at rest, are incident with normal velocity components less than v, are unable to overtake the plane when it is moving with velocity v. These streams correspond to the zone DMNF. There is thus a corresponding decrease in the number of impacts in unit time, with corresponding decrease in effective density of incident streams, and therefore also of actual density in the immediate vicinity of the back of the element in the ratio surface MRN V-t,, ; surface DRF V A limit is reached when v — V. At this value the region in the immediate vicinity of the back of the element becomes vacuous, and remains vacuous for all values of v higher than V. It will be seen that the surface MRN has the same significance with respect to the pressure on the back of the element K as the surface MEN has with respect to the pressure en the front of the element. The pressure on the back is therefore obtained from (8) by integrating between limits corresponding to the surface MRN. It is convenient to work in terms of a. From the triangle GCK we have V2 — a2 + v2 — 2av sin <p (9)lav •--. Again, we have a* - V2 + n2 + 2*Vv sin 9 whence, by differentiating . .~ .:. . za-da — zVv cos 6.d6 a. da.•. cos 9.d6 — Substituting (9) and (10) in (8) "W (10) VV»)Y -J- 2av 1 Vv da. a? a4 6 - 2 This is the general integral to be evaluated between the necessary limits. Two cases must be considered, viz.:— (1) For values of v up to and including V. (2) For values of v from V upwards. Case 1.—When v is less than V, the plane MN cuts the sphere of reference. For the pressure on the front of the element, a varies between the limits EK and NK, or (V + v) and V Thus •4W 4V1/8 I "L (V 2 — v2 2 J (V + »)«_(V«-»»)3(V2 - v*) (12) which, by straightforward simplication, becomes V2 y g V* and the pressure is obtained by multiplying this expression by the density of the gas. If we put v = «V, the expression for the pressure on the front of the element becomes It should be remembered that this expression only holds for values of n iip to unity. The first term is constant and equal to the static pressure, the remaining terms giving the increase of pressure due to the relative wind. ..... r ... pV2 * • ' : •••-:• •••' -'• When n =• 0, p = L-r-, the static pressure. The interesting result is also shown that, when n is small, the third and fourth terms are negligible in comparison with the second, so that for low values of v the increase of pressure is approximately proportional to the velocity. As v increases the second and third powers become more important. When v = V, so that « — 1 8p - - pV* = eight times the static pressure. ' • - Expression (13) can be written in the easily remembered form * pV2 / * * \ pV2/ \3 ,".'- . '.::_' t*-*\ 3 \ 3 V / For the pressure on the back of the element, the limits of a are RK and NK, or (V -v) and */V2 - v*. Thus -^rrf^ - — (V2 - r8) + °2 (V2 - v"f ~ " 4V^ a|_6 2 ^ 2V W-v 1 H^ vll ~ <Y_T^8 _ (V8-*»*)'-(V~»)*,™ « ^*L 6 2 (V ~v) . which by simplification becomes - •--- Putting v = wV as before, the pressure on the back of the element is given by Here, again, the first term is the constant static pressure, the remaining terms giving the negative increase [i.e., decrease) Vin pressure. As before, when v — o, p «• Also, when v is small, the decrease of pressure is proportional to the velocity. When v = V, or n '— 1, p = 0. Expression (16) may be written in the form pV2- p -~ (i-»)8. (i7l The total resistance r due to relative wind is given by the difference between the pressures on the front and back of the element, i.e., by subtracting (17) from (14). Thus = (2»* + 6M). (18) Case 2.—When v>V, the pressure on the back of the element is always zero, the sphere of reference being entirely out of contact with the plane, values of a for the back vanishing. The pressure on the front of the element (which is equal to the total resistance r in this case) is found by integrating over the whole surface of the sphere between the limits of- a, (v + V) and (1/ -V). ., - (V - V)* -fo- V)* - V) which reduces to -V2 + 2f*. so that p r>- pV2 + 2Pv*. (19) Remembering that we are now dealing with a state of affairs in which the density of the gas in the immediate vicinity of the plane is 2p, as previously explained, we see that the first-term is the constant static pressure due to gas of this density, and the second term is the pressure due to a relative wind of velocity v calculated for gas of density 2p by the usually given application of the Newtonian method. This result shows clearly that the pressure due to the relative wind is only proportional to the square of the wind velocity 2v when the term _pV2 is negligible—that is, when V — 0, or v is infinite. Considered in another way, we see that the pressure can only be written in the form p •= pv2 + a constant when V is less than v, so that for such an expression to hold good for values of v down to zero, the constant must "be zero, fifearing in mind also the fact that (19) is estab- lished under conditions in which the pressure on the back of the element is zero, the reader will now be in a position to grasp more fully the significance of the comments with which we started. 838
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