The Normal Component of the Induced Velocity in the Vicinity of a Lifting Rotor and Some Examples of Its Application Page: 4 of 18
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REPORT 1184 I--NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS
, radial component of velocity induced at a point
P hy a vortex ring
Saxial component. of velocity induced at a. point P
by a vortex ring
IV gross weight of elicol)ter
Sslope oflollngitudinal variation of nondimensional
induced velocity in plane of rotor
X, Y7, Z rotor axes (fig. 3)
x nondimensional radial distance of appoint P from
axis of a vortex ring, RpR (fig. 1)
x', y', z' nondimensional coordinatesof a point P wxitl
respect to rotor axes (fig. 2)
y slope of lateral variation of nondimensional
induced velocity in plane of rotor
Z. distance of a point P from plane of a vortex ring
z nondimensional distance of a point P from plane
of a vortex ring, positive in direction of r,
Z7)1 (fig. 1)
a angl t of plttac anek of plne of zero feathering
a, induced angle of attack
af fuselage angle of attack
a, angle of attack of tip-path plane
9r angle between radius vector from center of rotor
to a point P lying in XZ-plane and positive
X-axis, positive above rotor (fig. 3)
r vortex strength
X- (7 sin a - r) ,/R
1,- (C sin a,--c)/I-
P=-Vcos a,/iQR
g, V cos a,/2iH a
p density of airangle between fligllt )athl al nd orizontal, positive
below horizontal
angle between axis of the wake and normal to
tip-path plane (fig. 3)
azimuth angle mcasured in XIY-plane between
radius vector to a point and positive X-axis,
positive in going from positive X-axis to
positive Y-axis
angular velocity of rotor, radlians/ise
values of back rotor of two rotors in tandentl
vialtles of front, rotor of two rotors in tandem
values taken with respect to virtual axis of
rotation or to tip-path planevortex ring of strength P and radius R may be expressed as
I-(
=- (/I +2lwL)[(T- (T)1
2,T(1)
wl here R is tthe ra dius of the vortex ring, dR and (121? are the
least and greatest distances of the point P to the vortex ring,
Vortex-ring oxisd2R d, R Z=zR
_---R,=xR
zFrI;Ur 1.-Coordinates for vortex ring and table I.
d2-dl(2)
and K(r) and E(r) are the complete elliptic integrals of tlhe
first and second kinds, respectively.
The flow field of a vortex ring is axially symmetric and
thus the axial and radial velocity components v, and r, at a
point I', having an axial distance Z, from the plane of the
vortex ring and a radial distance R, from the axis of sym-
metry, are given byand
1 d(I
ve= E-- R dRy,
1 d4
'V'= dZ,It is shown in reference 3 that equat ions
by expressed as
v, G (AB1+CDF)
rj, 2and
whereANALYSIS
VELOCITY INDUCED BY A VORTEX RING
It is shown in reference 4 (ch. VII, see. 161, p. 237) that
tlhe st.ream0 functions at a point P (fig. 1) in the flow field of aT
rxI? (AB'+ ODF')
A= K(r)-E (r)
--1 x+l
= l d((3)
(4)
(3) and (4) may
(5)(6)
(7)
(8)C=d,+d, (9)
(12 -- (
r=c l2
Subscripts:
B
F
VI
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Castles, Walter, Jr. & De Leeuw, Jacob Henri. The Normal Component of the Induced Velocity in the Vicinity of a Lifting Rotor and Some Examples of Its Application, report, August 11, 1952; (https://digital.library.unt.edu/ark:/67531/metadc65544/m1/4/: accessed April 24, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.