Applications of Modern Hydrodynamics to Aeronautics Part 1: Fundamental Concepts and the Most Important Theorems. Part 2: Applications Page: 28 of 56
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REPORT NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS.
according to the same method, we must seek for the theory of monoplanes another suitable
object of comparison. As such, the infinitely long wing will serve. Where the discussion
previously was about change of angle of attack, increase of drag, etc., we intend now to refer
these to the infinitely long wing as a starting point. Since in the theoretical nonviscous flow
the infinitely long wing experiences no drag, the total drag of such a wing in such a fluid must
be due to vortices amenable to our calculations, as the follo ing treatment will show. In a
viscous fluid drag will arise for both wings; infinitely long or not, which for those angles of
at tack for which the profile is said to be goodo" is, according to the results of experiment, of the
ordtlr of magnitude of the frictional resistance of a plane surface.
The carrying out of this problem is accompanied with greater difficulties than the calcula-
tion for a biplane as given. In order to obtain the necessary assistance for the solution of the
problem, we shall first be obliged to improve the accuracy of our picture of the vortex system.
The density of the lift (lift per unit length) is not constant over the whole span, but in
general falls off gradually from a maximum at the middle nearly to zero at the ends. In ac-
cordance with what has been proved, there. corresponds to this a circulation decreasing from
within outward. Therefore, according to the theorem that by the displacement of the closed
curve the circulation F can change only if a corresponding quantity of vortex filaments are cut,
we must assume that vortex filaments proceed off from the trailing edge wherever 1' changes.
dr
For a portion of this edge of length dx the vortex strength is therefore to be written d x d, and
hence per unit length of the
edge is dx" These vortex fila-
ments flowing off, closely side
by side, form, taken as a whole,
,- a surface-like figure, which we
shall call a" vortex ribbon."
For an understanding of
this vortex ribbon we can also
VIG. 42.--Change in shape of vortex ribbons at great distances behind the aving.
approach the subject from an
entirely different side. Let us consider the flow in the immediate neighborhood of the surface of
the wing. Since the excess in pressure below the wing and the depression above it must vanish
as one goes beyond the side edges of the wing in any manner, there must be a fall in pressure near
these edges, which is directed outward on the lower side of the wing and inward on the upper.
Tle oncoming flow, under the action of this pressure drop, whliile it passes along the wing, will
receive on the lower side an additional component outward, on the upper side, one inwar(l,
which does not vanish later. If we assume that at the trailing edge the flow is completely
closed again, as is the case in nonviscous flow, we will therefore have a difference in direction
between the upper and lower flow; the upper one has a relative velocity inward with reference
to the lower one, and this is perpendicular to the mean velocity, since on account of the Ber-
nouilli equation in the absence of a pressure difference between the two layers the numerical
values of their velocities must be the same. This relative velocity of the two flows is exactly
the result of the surface distribution of vortices mentioned above (as the vortex theory proves,
a surface distribution of vortices always means a discontinuity of velocity between the regions
lying on the two sides of the surface). The relative velocity is the greater, the greater the side-
wise pressure drop, i. e., the greater the sidewise change in lift. The picture thus obtained
agrees in all respects with the former one.
21. The strengths of our vortex ribbon remain unchanged during the whole flight, yet
the separate parts of the ribbon influence each other, and there takes place, somewhat as is
shown in figure 42, a gradual rolling up of the ribbon, as a closer examination proves. An
exact theoretical investigation of this phenomenon is not possible at this time; it can only be
said that the two halves of the vortex ribbon become concentrated more and more, and that
finally at great distances from the wing there remain a pair of vortices with rather weak cores.28
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Prandtl, L. Applications of Modern Hydrodynamics to Aeronautics Part 1: Fundamental Concepts and the Most Important Theorems. Part 2: Applications, report, 1979%; (https://digital.library.unt.edu/ark:/67531/metadc53396/m1/28/: accessed April 25, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.