Applications of Modern Hydrodynamics to Aeronautics Part 1: Fundamental Concepts and the Most Important Theorems. Part 2: Applications Page: 35 of 56
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API''I'ATIONS 1OF Mo0 1ra:N IHYIrIciim YNAMI(' TO) A.;I)(oNALTI('S.
of this power series, ley the solution of linear equations. lBy this the lift distribution and everv-
thing else are known.
Another method for the solution of this "second roblelm" will be obtained hv the fol-
lowing consideration: Tihe velocities at a tlist.ance behind the wing, on account of the connect ion
mentioned so often between a vortex filament extending to infinity in ,one direction only andn
one extending to infinity in both directions, are twice as great as those in the cross section of
the lifting line, if we do not take into account te change i slalpe of tlice vortex ribbon. We
therefore lhave here, neglecting tlfis calilge in slaipe, 4Tn ill ustra tion of a. two-iimellnsional fluidl
flow (uniplanar flow), for which thle verticall velocityy components at thle point, where the wing
is reached are specliified. For tlie simple cese that tie vertical velocity u' is constant. as was
found to, be true for the elliptical lift distribution, the shape of the flow that arises has been known
for a long tirnie. It is given in figure 49a.. It is the same as that already considered, in another
connection, in section 15. 'h'l(e picture of thle streamlines show clearly the velocity disconti-
nuity between the upper anc lower sides of the vortex ribbon, indicated Lx the nick in the stream-
lines, and also the vortical motion around the two extreme points of the vortex ribbon, corre-
sponding tc t te ends iof the wing.
Any1. p"' oblems of this kind can therefore be solved by means of the methods provided by
the potential theory for tlihe cor'respmonding problem of two-dimensional fluid flow. We can
not go into these matters more closely at this time; by a later opportunity some special rela-
tions will be discussed, however.
A " t hird problem"' c insists in determinirig the lift (list rib)ution for a Ilefinite wing having a
given shape andl given angle of attack. This problem, as iaLy v e iningined, wiNs the first we
proposed; its solution n has taken the longest., sin ce it leails Int an integral w hicht is awkward( t
handle. )r. Bet.z in 1919 succeeded after very great eff crts in solving it for the cse of a sclquare-
cornlered wirg havil y evye rewtlee same profile alid te sniame angle of ittnack. The way the
solution was cl)tainedl may bie indli'ateld briefly here. \We start, as before, front the relation
1i" eqluatin (37) u, is expressed in terms of the circulation. The effective angle of attack
c' ln be ex prssedi in terms f I', sinrice, according tci thle assulmrptic) is made before the lift,
distribution, whichlI is proiportional toi I', dlependis directly trl)(o al'. The relation between a'
anl1 I' n be assumed to 1he give surf1cientlx exactly y ( i' l()I' prpop1ses yv a linear expression
F 1 i' (c, a' + c) (47)
ill which I is the length h of the chlord (nleasured in the direction of flig f li }). By tle introduction
of tile facti'r 1, c, and c2 are made prre numbers. The irieneical value of c ,, whiht is the more
important., can be expressed, if Ca, is the lift coefficient forn tIhe infinitely lon wing wing at the angle
cf attack a', )yv the relation
I dc.
C1= 2 da'
In fact
A p 1' I. _ 2I'
Ca - F .. t 2 (ca' 4 c,)
It . ., p lV
For a flat-plate theory Iprves that , = r, for cuirv(,i wings it lhas a slightly greater value.
If, according to what has gone before, we express a' by F and ' by <l and write
dI'
dx=f(x) and therefore I'= f(.r)dx
we obtain after a sinplle calculation thle integral equatii
o f(x)dx + fs- 2'= Vt (c,a + ce) =const. (48)35
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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/m1/35/: accessed April 19, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.