Determination of induced velocity in front of an inclined propeller by a magnetic-analogy method

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Title:
Determination of induced velocity in front of an inclined propeller by a magnetic-analogy method
Alternate Title:
NACA wartime reports
Physical Description:
20, 50 p. : ill. ; 28 cm.
Language:
English
Creator:
Gardner, Clifford S
LaHatte, James A
Langley Aeronautical Laboratory
United States -- National Advisory Committee for Aeronautics
Publisher:
Langley Memorial Aeronautical Laboratory
Place of Publication:
Langley Field, VA
Publication Date:

Subjects

Subjects / Keywords:
Propellers, Aerial   ( lcsh )
Aeronautics -- Research   ( lcsh )
Genre:
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Summary:
Summary: The horizontal and vertical components of the induced velocity in front of an inclined propeller in a horizontal stream were obtained by a magnetic-analogy method. The problem was formulated in terms of the linear theory of the acceleration potential of an incompressible nonviscous fluid. The propeller was assumed to be an actuator disk. The horizontal component of the induced velocity was found by a numerical calculation. Numerical calculation of the vertical component, however, was not practicable; therefore the vertical component was obtained from electrical measurements by use of the analogy between the acceleration potential of an incompressible nonviscous fluid and the potential of a magnetic field. An alternative formulation of the problem in terms of the trailing-vortex sheet is shown to be equivalent to the acceleration-potential formulation if the thrust coefficient is assumed so small that the slipstream is not deflected and undergoes no contraction. From the results presented, induced velocities of greater accuracy are shown to be obtainable from a modification of the vortex theory based on the assumption of a constant finite down-wash angle of the slipstream.
Statement of Responsibility:
Clifford S. Gardner and James A. LaHatte, Jr.
General Note:
"Report no. L-154."
General Note:
"Originally issued February 1946 Advance Restricted Report L6A05b."
General Note:
"Report date February 1946."
General Note:
"NACA WARTIME REPORTS are reprints of papers originally issued to provide rapid distribution of advance research results to an authorized group requiring them for the war effort. They were previously held under a security status but are now unclassified. Some of these reports were not technically edited. All have been reproduced without change in order to expedite general distribution."

Record Information

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University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003596963
oclc - 71012535
System ID:
AA00009355:00001


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Full Text

?JAcR L-


ARR No. L6A05b


NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS





WARTIME REPORT
ORIGINALLY ISSUED
February 1946 as
Advance Restricted Report L6AO5b

DETERMINATION OF INDUCED VELOCITY IN FRONT OF AN INCLINED
PROPELLER BY A MAGNETIC-ANALOGY METHOD
By Clifford S. Gardner and James A. LaHatte, Jr.

Langley Memorial Aeronautical Laboratory
Langley Field, Va.


'-N A CA- -.. ''. ,. -
NACA...


WASHINGTON
NACA WARTIME REPORTS are reprints of papers originally Issued tu provide rapid distribution of
advance research results to an authorized group requiring them for the war effort. They were pre-
viously held under a security status but are now unclassified. Some of these reports were not tech-
nically edited. All have been reproduced without change in order to expedite general distribution.


L 154


DOCUMENTS DEPARI M Nf


..-' hi...






































Digitized by the Internet Archive
in 2011 with funding from
University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundation


http://www.archive.org/details/determinationofi001ang






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NACA ARFR ?'o. L6AO5b

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A:' ACE PESTRaT. .EPOGT


E'-TI T 0'- UT F r1DUCED :---)CITY 11T rT AT rCUED

PFOP-LLER BY A AG7"CT-C-A. A7 v `1 D

By Clifford S. Gardner and James A. LaHatte, Jr.





Th:- horizontal and vertical cor- onents of the 'iniced
velocity in front of an inclined propeller in a ori zontal
stream were obtained by a magnntic-analoy method. The
problem was formulated in terrs of tie linear theory of
the acceleration potential of an incrc ressible ronvisc.as
flui d. The propeller wa3 assu'od to be an actuator dis h.
The horizoi.tal cororent of the induced velocity was found
by a numerical calculation. unerical calculation of the
vertical comoonent, however, was not praticable; th:ere-
fore the vertical c rroonent was ootained from electrical
measurements by' use of the analogy between the accelera-
tion potential of an incompressible nonviscous fluid and
the potential of a magnetic field.

An alternative formulation of the protlen in terms
of the trailing-vortex sheet is sho,?. to be equivalent to
the acceleration--cotential formiulation if the thrust coef-
ficient is assumed so small that the sliostrea-: is not
deflected and undergoes no contrsctior. From the results
presented, induced velocities of greater accuracy are
shown to be obtainable from a modification of the vortex
theory based on the assumption of a constant finite down-
vwash -

I1 .,"'DUCTIOT


The recent dI.-velc: rent of airplane designs with
pusher-propeller installations has occasioned several
inquiries regarding the nature of the flow in front of
an inclined propeller and the correspor.i., aerody7na-ic
effects on the ''.*!... cause of the difficulty of the
calculations, little effort has heretofore been made to









2 :-AC: ARR bh. L6A05b


compute the flow. Some exper imental work, however, has
been cone in connection with the pr olem of the lift
increment on the wing (fir exa:rle, references 1 and 2).
Further development of the theory is considered desirable
to serve as a 'asis for correlation of these and similar
da b.

The purpose of the present paper is to give detailed
theoretical data on the induced velocities in front of an
inclined oro eller. Only the components important to the
problem have been obtained; namely,the component parallel
to the free-stream direction and the component non-al to
the free stream and in a vertical plane, which will be
designated horizontal and vertical components, respec-
tivelv. T.: determination of these components is based
on the linear theory of the acceleration potential of an
incom-oressbole nonviscous fluid, and the propeller is
assumed to be an actuator disk. Because the theory is
valid only for small perturbations, the results, which
are presented in dimensionless form independent of the
thrust coefficient, are valid only if used for propellers
operating at low thrust coefficients.

The horizontal component of the induced velocity was
determined by numerical computation. The computation of
this comioonent was practicable because. of certain simpli-
fications due to symmetry. erical calculation of the
vertical com:.ponent, Io :ever, is excessively laborious and
tirme-consru'riina: consequently, the vertical component was
obtained from electrical measuree. ents b- use of the
analo) bet',een the acceleration potential of an incom-
pressible nonviscous fluid and the potential of a magnetic
field.

An alternative formulation of the oroolem in terms
of the tralling-vortex sheet is shown to be equivalent to
the acceleration-potential formulation if the thrust coef-
ficient is assume so small that t e slipstream is not
deflected anc undergoes no contraction. er. the results
presented, induced velocities of greater accuracy are
shown to be obtainable from a modification of the vortex
theory based on the assumption of a constant finite down-
wash angle of the slipstream-.









lNACA ARR -o. LoAO5b


13 laT TE D17j


p
4-






U, V, VZ
u, v, vi V


u vT', w


local static pressure

static pressure at ov:::strea. face of pr-:Opeller
disk
air density

tie ,e

rectangular coordinates (fig. 1)

free-stre.n.T velocity

cornMonents of ~ erucurbation velaoity in -,Y-,
,:,' Z-directCins respectively

electric current, c;,s electr ... ;nstic units

radius of proplcler

diameter of proneller

thrust of propeller

thrust coefficient 2 --\


/ u
cimensionlcss veocities --

respectively )


V N
Vi ,.


y, y z'





-x, Ey, HZ


dimensionless pressure 1- -


dimensionless coordinates (x/R, y/R,
respectively)

:---nitic scalar potential, cgs elect
notic units

components ,of n 'netic-f).elc strength,
electroi:agnetic units

-:.e-ri:..entally measured voltage, volts


z/r,


cgs








L 'A ARR io. LA05b


ratio of w' to na1o oo3 seasurec. volt.-- e E

a a:ile of inclinat r n )f pr oeller disk to Z-axis,
degrees

as ;eued constc.nt dov:nwiash an I e of slipstreaT,
ra l ans

: mrass rate of ('clw ucross propeller disk

Subscripts:

2 in ultiLmate. wal:e
1 at downstream face of Droneller dis)

T H E,'1 F. I .L

Linear t@ ter of the acceleration potent a 1.- The
E ler ,1. 7. ... -. _.- le nin-
visc;us fluid ::a e ,v;riltten in .he following, form:

pD(V + u) -
DC x

Dv o



-t ~ p


These eq.t io:.s are in g. ner.l nonlinear in the velo-
cities. TiLe equations r. ay be 1do linear if the o: .v.nents
of the perturbation veloci by are assu 'o to be s:ail :.-
Sn. ed. i th bhe frpe-streami v(locity-. (See reference 3.)
If t. S assur'otio)n is valid and if terrs of the second
order are neglected,

r(V + u) Du .c :
v -V


Pv CV v



Dt 7 x








NACA ARR No. L6A05b 5


By virtue of equations (2), equations (1) become

6u p
x (3a)



6x by


pv = -- (3o)
ox 6z

If equations (3) are differentiated successively
with respect to x, y, and z and are added, the result

2 66 u IV ow
-v p = pv- + + o= (4)
6x Tx 6y 6z/

M -"netlc an 1 vy.- Since by equation (4) p satisfies
Laplace's equation, and since the scalar potential of a
magnetic field also satisfies Laplace's equation, it
follows that for similar boundary conditions p is
directly analogous to the scalar potential of a magnetic
field. This fact is the theoretical basis of the magnetic
analo.:, of the present oaoer.

For low thrust coefficients the three boundary condi-
tions for the pressure p in the problem of the actuator
disk are:

(1) i:.e pressure has some constant value -pl uni-
formly over the upstream face of the disk and a value pl
uniformly over the downstream face.

(2) ihe pressure has no singularities other than the
jump discontinuity at the disk.

(5) At great distances from the disk, the pressure
is uniform and without loss of generality may be assumed
to be zero.

For the magnetic potential the first condition is sat-
isfied by using as the source of the magnetic field a
circular wire loop carrying a current. The use of this








6 'NACA ARR :To. LbAC,


current-carryin, loon p actually ensure. the appropriate
behavior o' the magneticc field at the disk since the
.magnetic .'. entfa2l 1 such a 1)op has the value -2rTI
uniforr.ly over olie face of the loc nnc the value 2IT
over ,e othr face if cas electron ., -tic units are
used (referer.ce ). L hbe second condition is satisfied if
D obher mina;rtic fielc's and no ma.anetic materials are in
the nc.. crhod of che loop. .e third condition is
satisfied .c.ot atically, since tle maonetic potential of
the lo-o approaches z ro at great distances from the loop.

Basis of keter.nination of horizontal velocity.- If
equa-_ ~. '- '~ "".".. g t :-c '- : ,.d t 0 x, The result
is

pvu = -p (5)

Tnasmiuc as the iensionles s velocity u' and the dimen-
sonless pressure o' are defined by



VTc

ana

-)



equation (5) beco::es

u' = -p' (6)

- value f the dimr n onless n pressure T' at the :;n-
st.r e -i faoe Gf )he I1 1 L


1
C





S- -

= (7)








NAA -. 7o. L6A05b 7


Since this boundary value is a universal numerical con-
st-nt, the values of the :. ensionless velocities
th.r- ht space, which are determined oy the oounfdry
values )_ p', are also universal numerical constants.
-y means of the dimensionless coefficients u', v w',
and pr, therefore, the robler:. is stated in a nn-
dimensional forn: that is indepenient of all relevant
variables such as the dis: radius, density, thrust, and
free-stroami velocity.

In acccrd.-nce ':.ith tIle 2 ... tic enalogy, the ;- .--tic
potential is directly ancl~gous to the opr sure p' that
is,
= i (d)

where r is the ".. -ttic potential measure- at a ooi.nt
cf ';i ch the dirienslonless coordinates x', y', -nd z'
are the same as the dimensionless c or. xnates of the
-o at heree p' is :.easuec., and. _-ure w: is "o, con-
stant of roortonaity -hat Ie-enis on th!e dinerni oins
of the elect:- ..etic sste:n. --- alue of a a
hence, u' :as thus obtained r. c alating u a nd
u ltinly the constant k- .. ay be determined
by ".- .-...':.g. ".r;- the value of p' "at cae & isk, which h b
equation (7) is 2/7r, with the value of f at the disk,
which is 2TI S. calculation of > :as effected b-
nu.cerical inteGration of a form' la o v "- Slycte
(reference 5) for the -_..- tic field -f a circular loco.

potential j and, consequently, the i1orizontal
velocity ul at ar.- noint cepend on.l en the position of
this point relative to the disk; thus, he results f)r u
at positions given in terms of coordinates fixed in the
disk are the sane for all & : lea of inclination. C cause
of rotational sn- entry, moreover, the values of u' at
corresponding points in any two planes throi.;. the a.:is
rf the disk are the sa me. It was thus necessary to
calculate u' over only one axial plane.

The linear theo- of the -.'esent --oar gives values
of u' that are valid for low thrust coefficients;
VTc
that is, the results for u' are essentially the deriva-
tives of u/V with respect to Tc at Tc Th
momentura theory of the propeller (reference 6) gives the
inflow velocity at the disk as









ACO A-RR No. LCAO5b


u -- ,1 + -- -i
-a -
), a -'r d4 T 2rt / ee
..... e ..... .. u at t

/ .U \'

-' C
T=0
c


S : l ..... tin'. c.:.,._ ,,_., l .i! ... value -iven ty the
lin-p the..r; of the resent anr eu atcl'ns (6) and (7)).

tO sis o- te r.i i O 0 erticzal velocity.- If

to x, te result is


X
piw = I


--
5 Z


If eqs:tin (j) is divided br p'- an if dimensionless
Sor:; not -- are int4o ced, tre r..suit is


(10)


<..2\


In acc^ri nce :.vith th "'netic ...l", if euation ( )
iS used, eq.rat n (10) ':e cor.c C


; '~'-
t --I
)


= k f -- d i
=!<0z


(11)


But since / is the n: ten lai o:' a -, r.-.-tc field,








.:.A ;-..- No. LbA05b 9


i- 'z
6z

where Hz is the z-coimoonent of the .,.-:n tic-field
3tre'n h. '. -.tion (11) now becomes


w, = k Hz dx (12)


1:i order to find the value of w' at a point (xI, T', z')
t is therefore sufficient to measure the integral
f H. alo-i,' a path parallel to the X-E.xLs, extending
from minus infinity to the point (x, y, z).

An alternating 6.- netic fiell in air _-"t.ces in a
coil of wire a volta:e proportional to the total flux
linking the coil, whichh in turn is proportional to the
inte.al over the face of the coil of the normal cor no-
nent of the ri.. .-tic-field stren th. The voltage induced
in a long narrow search coil is proportional to the sur-
face integral of the normal field over the area of this
coil; since the coil is narrow and, consequently, the
field tr-e:.gth is almost constant across the width of
the coil, this surface integral is prooortional to the
line integral along the length of the coil. i-e line
integral in equation (12), therefore, is proportional to
the voltage induced in a long narrow search coil, the
plane of which is perpendicular to the S-axis, which
extends parallel to the X-axis from the point (-m, y, z)
the noint (x, v, z), as shown in figure 2.
Since the mr.gnetic field dies out rapidly with distance,
a search coil of practical length actually suffices to
obtain accurately enough the infinite integral.

Thus the follc.ving equation holds:

w = KE (15)

where E is a measured voltage proportional to the
voltage induced in the search coil and K is a constant
of proportionality to be determined by calibration.








NACA ARR No. L6AO5b


APPARATUS A.T. ':?T-TDS


Field coil.- A field coil of 61p turns of Brown
and Sharpe o,. 18 copper wire wound on a circular wooden
form was used to simulate the actuator disk. The mean
radius of the coil was 12 inches and the cross section
was a square 0.&7) inch by 0.575 inch. The coil ;ias
supported by pivots about its horizontal diameter in
such a way that its angle of inclination to the Z-axis
could be varied and the support could be moved up and
down. The arrangement is shown in figure 2.

Search coil.- As previously explained, a long narrow
search coil was used to perform the integration of the
magnetic-field strength indicated in equation (12). The
search coil was made up of 110 turns of c-r:-wn and Sharpe
,. O copper wire wound lengthwise on a glass rod
72 inches by 0.225 inch by 1.2 inches. The coil rested
on Lucite supports at the two ends. The supports were
scribed with cross-hair linc,, to aid in setting the posi-
tion of the coil and were sunoolied with leveling screws
so that the face of the coil coul be turned exactly 900
to the flux being measured. voltage developed in the
search coil was fed through a filter eliminating O0-cycle
pickup to an electronic voltmeter by which the voltage
was measured.

PoTwer suoply.- Current was supplied to the field
coil from :a motor-generator set delivering 5.0 amperes
at 590 cycles per second. The Ward-Leonard speed control
system was used ss that the frequeue- and output voltage
could be adjusted by rheostats. The output volta ~e was
continuously adjusted to maintain through the field coil
a constant current of 5.0 amoeres, as measured on a
standard high-frequency ammeter. The output of the
generator was connected in parallel with the input of a
cathode-ray oscilloscope, and a 60-cycle line volt., .was
connected across the sweep circuit. The resulting
Lissajous pattern was held stationary by continuous
adjustment of the frequency control rheostat; thus the
frequency of the current was maintained constant at
390 cycles per second. The arruin ' fi .-.e 5 .

Test procedure.- In order to measure the int?;.r l
in equation (12) at the point (x, r, a) when the disk









NACA ARR "--!. L6A05b


was inclined to the Z-axis by an angle a, the field
coil was set so that its center was a distance z below
a horizontal tale; then it was set at the angle a Cit)
a protractor and the search coil was placed on the table
parallel to the X-axis with one end at the noint (x, y, z)
and the other end away from the field coil. Tne arrange-
ment is shown in figure 2. For each setting of a and z,
readings of the voltage were made at the 170 vertices of
a rectangular grid 6. inches by )0 inches that was made
up .f lines parallel to the X-axis and the Y-axis soaced
at intervals of )L inches. The arrangement is shown in
figure 1.

Zero hei -ht adiustment.- In order to locate the
height _-..... eac oll corresponding to a value
of z = 0, a was set at 00 and the height of t-e field
coil was then adjusted for zero voltage in the search
coil. The voltage in the search coil is zero when a
and z are zero, so that the search coil is on a plane
through the axis of tihe field coil, since the component
of the magnetic field normal to such a plane is zero.

Leveling: adjustment for the search coil.- The com-
nonen ,.et,.: 1 -.L -- i:. .:::ic about the
AZ-olane for all values of a; consequently if bhe
search coil actually measures the component --z of the
r,:-.:.etic field the voltage readings should be the same
for two positions of the search coil in which the values
of x and z are the same and in which the values ofi y
have equal magnitudes but opposite si i'. If, however,
the search coil is not level so that the component Hy
also contributes to the induced voltage, the voltage
readings will not be the same at symmetric points since H
has opposite signs at symmetric points. When the com-
ponent Hy is strong, the error in the voltage reading
may be lar if the search coil is not level; hence at
each setting of a and z the coil was leveled by
adjusting the leveling screws until the readings were the
sam:e for a oair of symiietric positions. Because of some
uneveness of the table top on which the search coil
rested, the readings were not exactly the same for other
pairs of -.... tric positions; hence average values were
used for the data at other positions.

(1iibrt1 '- of -'ten n, manti h -nl'_Y anartus v- In
,order "ci.-1 re ,' i L, t,.- nit 1 T-er by use









NACA ARR No. L6AO5b


of equation (13) determination of the value of the con-
stant K was necessary. This value was obtained by
calibrating the aooaratus; that is, by comparing values
of E Ieasured at a series of calibration points with
values of w' calculated for those points. The values
of w' were calculated from equation (12) by use of an
electromagnetic formula (reference 5). The method is
similar to the method previously discussed by which the
horizontal velocity u' was calculated.

The values of w' wert calculated for values of a =90
and z = 0 at a series of points along the X-axis. A
comparison of calculated values of w' and measured
values of E is given in table I to show how the cali-
bration constant K was obtained.

Accuracy.- In order to estimate the accuracy of the
experiment, values of w' were computed at several
points for a value of a = 00 and were compared with the
corresponding experimental values. The experimental
values were found to be low, some by as much as 8 percent.
This inaccuracy in the data can be attributed to errors
in the measurement of distances and angles and to the
fact that the search coil used vas of finite length. The
error due to the finite length of the search coil could
be calculated by means of the ass'i .ition that at great
distances from the field coil the magnetic field could be
approximated by the field of a magnetic dipole. This
error was found to amount to less than 5 percent at great
distances from the field coil, where the magnetic field
falls off slowly with distance. In general, aoout half
of the error may be attributed to the finite length of
the search coil and the other half, to inaccuracies in
measuring distances and alles.

Te error of the calibration reading ,s (table I) may
be seen to be less than the 8 percent error mentioned
previously. This greater precision is probably due to
the fact that for values of a = 900 and z = 0 the
voltage reading is insensitive to s:All errors in the
alinement of the field and search coils because the
m u.!tic field is symmetric about the origin and is a
maximum relative to both a and z.









NACA ARR c. L6A05b


-' mLTS


The horizontal veloc ty field of the actuator disk
is presented in figure L as a -ap of contours of constant
di:-ensionless horizontal velocit.-- u, in a lane ;thrIo.
the axis of .-.. etry of the disk. The horizontal velocity
at any point for any value of a is then the sar.e as the
horizontal velocity at the corresponding point that is in
this plane of .--.t ,try and has the same position in te:'.3
of coordinates fixed in the actuator disk.

The vertical velocity field of the actuator -isk is
presented in figures 5 to 9 as a series of r:.sa of contours
of constant dirmensionless vertical velocity w' for five
different values of a at nine verti cal sections that
are parallel to the free stream and spaced ot intervals
of 1/3 radius from values of = 0 to y = 2.-. radii.

In order to plot each contour nap, aer...e. of the volteD
readi. s on the t-o sides of he. ....-olane were used. )n
each section the contours of constant velocity are drawn
throui. -ut a rectan -ular area extening horizontally
5 radii unstream from the actuator disk and vertically
1 radius above and below the center of she disk. The
nine sections on which contr)r ra-ors are drawn are labeled
a to i fr.m the *1..- of syr3met-- out:ards, as sho'
in figure 1.


VORTEX TREk .i"I OF "T ,7CTA --D1T P70. L-


Equivalence at low thrust coefficients of vorte: anI
a cce l( -7'-- :-n -' : .t .-:n ,. '--: "~l" i -- ..:. .- ":

with an actuator disk at low. thrust :)fficieiLt a
cyli n r c a 1 sheet of circular vortex ri n-s hi h 1 ave
the disk an travel c ownstream in the ree-str3* direction.
T.:s vortex pattern is approximated ';a propeller
operating at low thrust coefficient and rotational speeds
high in cornarison w-ith the free-strean velocit- and
ha -.:.z blades alon:- which the ":-;und vortex strength is
uniform. Because the bound vortex strength is uniform,
tra. .n- vortices leave the blades only at the tip and
at the center of the propeller. The tip vortices travel
downstream in helical paths and the vortices from the
pr'.-ller center travel downstream in a straight line.









PACA ARR 'o. L6AO5b


At any instant the density o the bound vortices and of
the trailinC vortices fr-.. the pro Iller center is necr-
ligible compared with the density of the helical vortices,
since the density of these tip vortices is pro .-rtional
to the high speed of the blade tios. The velocity field
of the propeller is therefore the induced velocity field
of the infinite cylindrical vortex sheet shed from the
blade tips. Since the rotational velocity is high, the
pitch of a helical vortex is small and the sheet can
therefore be considered to consist of an infinite con-
inu-us r.-: 'f circular vortex lines.

It may also be shown that the induced velocity field
of a propeller operating at low rotational speed and low
thrust coefficient and having an infinite number of blades
along which the bounv: vortex strength is uniform is the
same outside the slipstream as the velocity field of an
actuator disk. The vortex pattern of such a propeller
may be considered to consist of a system of vortex rl. s
to vhich must be added another vortex system composed of
the radial bound vortices together with straight trailing
vortices from the nrooeller center and from the blade
tips. The Induced velocity field of the system of rings
is the sane as that of an actuator disk. -. induced
velocity field of the remaining vortex system.;, however,
may be shol-n to be zero outside the slipstream.
rotational induced velocity is zero because of rotational
*-..etr- and the fact that the total circulation around
a closed path exterior to the slipstream is zero (since
the total included vorticity is zer ). The radial and
axial components are zero, since only the radial bound
vortex elements in the plane of the disk could contribute
to such components and the contribution of these elements
vanishes because of synmetry.

The velocity field of an actuator disk may be cal-
culated by interatin the effect of the infinite row of
circular vortex lines. Since the velocity induced by a
vortex line is directly analo_-as to the ri- n'tic field
of a current filament, the induced velocity of the infinite
vortex sheet is analo.--us to the integral of the n1tic
field of an infinite row of circular current filam::ents.
This inte :r_, in :Iwhich the point at which the field is
evaluated is fixed and the position of the source is
variable, evidently has the sa e value as a related
integral of the ma ..-tic field at a variable roint due
to a fixed source. T related inte r--!, however, as








.::.AA nRR 10o. LbA05b 15


has been shown oreviov.sly, gives the induced velocity
according to the acceleration-ootential formulation of
the prUolem. Tncer the assumption of --1_ perturbations,
the -altenative treatment of the orobem of the act ator
disk us" thIe trai ling vortex sheet, Whic hay obe con-
sidered to be th treatment in terms of vhe v' locit
potential, -'elds the sa_-e results as the treat&.ent in
terms of t!e acc aeration ot zn-ia!



4 >-i : t rf< t f 1 t t-_
i- "" 1i .- i JL J e 1 )--s^ Us'tut o

results are valid cil-q at 1,-;: tl.2ust coefficienc:. In
the cGcelerati n- o:e liol a... I n s urc- of
tii s li-O it- atl o ccc b_-1 0.in c- in t` J L; 1 "/ -
turbation velocities are s"ull c.r ... .ith free-
streaD.x veloc ty; in the tr lin- vorte:. for.ulatin, Le
limitation occurs in the assur1'or-..ons thoat ;he slipsttieam
under es no contraction and that it travels dI:nstrea
in the free-strean direction. Ile t ery .' the vertex
.frmulation may ee T:o3 ified to _ive greater acc racy at
.'... ^- cU-1ust coefficienos :' as: : tLt ce sli str
is oelc.ted o.n;.;ard J o. u free scorer. :- a constant
finite angle E. T1e Cn znwash an!e i. te l .lti:tLte
...... c si pl4- c vcu aati cn '-d -ay
taken as the valuevilo of n. ince, however, ohe 'axi .
influence is exerted b the traili".- vertices just behind
the propeller, it bi.i.t be more ace-rate -) ise as tie
value of F the dKc:n:/ash an le L ...i. tely .ehind th
.Doeller, which is aoout one-half of e I. ".e integration
S- .1en :erformed in the section n of the trai v! vorte
t rather than in tn. free- st-ro d..irecton, e orc-
S- the -ans ('-7. 1). _, the 'i -xis and the
-..s are rotated through an ar!le about th -axi
S- a new' set of c rcin-te &>s ,), Y, a), Tn
i .:-ooration in the "Irection of t o slipstreamr. is i.te ra-
tl- alonen te -axis of the field of a coil .in a
.: Ie a 57-.xE -in I-" -)v:ith the a:i s. (?ee fi,. .,10
... -orionents of the o~,rturbatiorn velocity orall1e to
t.: .-a:is and the :-axis cuan be obtained fito. the results
.r :,nteda in _-.,-'es ; to 9 -.r t eI hori ontal and
v-rt: cal perturbation velocities -or th an le a 7 .3-s
F'r the x- and 5-corponcnts of- t e perturbation velocity,
. x- and z-cor-oonents :lay then be Iound by a si ole
2 !hulation.








16 NACA ARR :10. L6AO5b


Calculation of slipstream. down',aash an le.- The do;:n-
wash ... 1. ,i -. ;, -"..-: l, r :.. ,.- the co".,po-
nents of the porpeller thrust parallel and normal to the
free stream to the corresponding components of the rate
of change of -::o;:entur of the air flow. When the normal
thrust-m:oentum equation is set up, care must oe exercised
to inclun0 the -iomentum of the flow about the slipstrean;
Lhat is, the morentuu of its virtual mass (references 8
and 9). :I normal thrust-smomentum equation therefore is


T-- ;(V + u2) + V 2 ( .)
57.3 2

here is the m-ass rate of flow across the ropDeller
disk, and u2 is the velocity increment in the slipstream
in the ultimats ;:ake. ;- term X(V + u2)E of equa-
tion (1) is the vertical r;rentuem in the slipstreanm; the
term I'VE D is the vertical r:o:ientui of the virtual mass
of the slimstream. :..e thr'i.' is .iven by the equation

S = .7i (15)

',ubstit ut -n of equation (15) in equation (1L.) gives

u2a1
57.3" 2 = (10)
u2 + 2V

In order to ap-ly equation (16) the value


U2 = V + 1 -1 (17)

derived from the :-omentum theory of the pr'p'eller with
no inclination (reference 6) ';ay be used.

In practice, the correction angle e is very small.
For exar9ple, if a = 10' and T. = 0.2, equation (17)
becomes

r x 0.2
u2 V 1 + T 1!
iT/


u2 = 0.22,V








yACA ARR "3. LIA05b


and, consequently, equation (16) becomes

0.228V x 10
57.3C2 0.228V + 2V

ich ives for e a value of

1 o0.5
e = 5 E = 57- radian 0.50
2 57.5


-Iv ODF PUSHER PROPLL.J, 0 LIFT

AYD PITC .': hOV-"::-_ OF :'.--


Tha incremental horizontal and vertical velocities
induced b- a usher Droeller -ayv be expected -to cause an
increase in the lift of the wr.- (reference 1) and a
decrease in the pitching mor.ent, inasmuch as tnese incre-
mental velocities increase toward the trailing e(f- c The
results of the present n:xuer indicate that the induced
vertical velocities (figs. 5 to 9) in the region directly
ahea: of the orop-eller are s-iall in comparison with the
induced horizontal velocities (fig. '). The effect of
the induced vertical flow on the lift and pitching moment
of the wing may consequently be expected to be small in
co-.narison with the effect of the induced horizontal flow.
.lculation of the magnitudes of the increments of lift
.0i pitching moment due to the presence of the pusher
r-peller is however considered impracticable, inasmuch
S(1) the available liftiy surface theories require a
.hibitive amount of labor, especially for a flow field
7 nonuniform as that in front of a propeller, and
i' the c .-. --.s wrought by the pressure field and velocity
"iZld of the propeller in the boundary layer, vmhich cannot
-: taken into account in the lifting-surface theory, are
'..-'ected to cause increments in lift and moment comparable
i-h the total increments due to the presence of the
r-...:her propeller (reference 1). It nay be concluded,
-n, that further work both to clarify the physical
.--.nor'.ena and to improve the computational methods will








If8 ,'..CA ARR No. L6A05b


be required before the effect of the propeller on the
winrg can be accurately predicted.


Lang -. Memorial Aeronautical Laboratory
N.ltional Aivisory Co-m2ittee for Aeronautics
Langley Field, Va.








.. AR -. LcAC`b 19





1. ~.~lt, R., and 3mith, P.: "ote on Lift Cl 2nge Due to
an Airscrew mountedd behind a '.:... Rep. .:'. B. l1
.ritish R.A.E., Dec. 1953, and _cdencan, hep.
Do. 3.A. 151a, April 1939.

2. .. n, J. S., Selt, R., -;vison, B., and Smit 7-.
Co,'-.arison of '..sher a:. Tractor Airscrevws oun te
on a ;>l 3. 1R V ",. B. o. 1610 ., 'Jitish R.A.E.,
JTne 1949.

. rndtl, L.: ccnt .,ork on Airfoil -. ...
No. 962, -,Lo.

.F.e, LeT ., and odam, !oitan lisley, Jr. Principles
of Ilectrieity. i an ostr: .. Inc., 1 $ 1,
*', 7, 2-1 5 27 -257


5. 3 'he, 1illioa R.: Static and Dna--ic Electricity.
I;cr..--:ill ok Co., Inc., pp. 26-271.

6. Gl uert, H.: The Ele sents of Ae1ofoil and airscrew
Th-or". Cajbriage Univ. Press, 19)7, P. .

7. von -an, .., anrd aE ".ers, J. I.: --neral Aerac,'-
nar.ic --..ry Perfect ?luids. T.thematical
Far. .:-.tin o2f the T ,-- ry of ..- .with Finite : ,-..
Vol. i! of Aerod-'n. c Theory, riv. E, ch. ITI,
sec. 3, F.. F. Durand, ed., Juli-s .- ':-. er (.-rlin),
1355, 1-- 105-1'-

brer, Herbert S.: Pro ellers in Yaw. NACA .RR


..e., ': ". .:--ntals of Fluid Dynamaics for
Aircraft Designers. Te Ronald Press Ca., 1929.
p.-. 15--.







NACA ARiR Yo. L6AO5b


TABLE I

CO PRISON OF COPUTED VALUES OF w ~ N ME.-:"D VOLTAGE E
-a = 90; y = 0; z = 0


x wi v E = w'/E
(radii)

2.0 0.0: ll 0.04225 1.367

2.5 .55 .01891, .65

5.0 .01690 .01230 1.571

5.5 .0 q .00855 1.58
| .0 .00872 .o06l6o 1.565

4.5 .oo66 .oo90 1.5 35L

Average value -f I = 1.366



:'.ATIO 0:AL ADVISORY
COD 'TEE FOR AERONAUTICS









NACA ARR No. L6AO5b


c^H

C-)





s I

-l0
C.)"


Fig. 1





































0
r.

























0
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0
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,-4










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Fig. 2


I
'~1
4',


NACA ARR No. L6A05b












S














4
0-,-
>m-
4z






0



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to
-I
A:







NACA ARR No. L6A05b Fig. 3





S---- 4a 4 0





oo




0 40
A 0 q4








0*
00


rd

0

21o






I
.5.0
I






NACA ARR No. L6A05b


3.25 ----

3.00 ,_-_ __-

2.75 ,

2.50 O- / ____-____6

2.25 /_ _






S/75 //1 2 0-------
450 -_------_-

125 -_' -_1_





.50
S60 .. 06 \0\




0 2 .5 70 .75 /00 125 /50 175 2.00 2.25 2,0 2.75 300 3.2S
Ax1 c/ /is n7ce from center of prolpe//er, rad/l
NATIONAL ADVISORY
CONNITTEE FOI AEDMNAUTICS


Figure 4.~ Conifours ,'- ?orzorfo/ ve/ofIy.


Fig. 4




NACA ARR No. L6AO5b


Fig. 5a


0


0






N1_______ ______.


NI ~- + 4 -~---- 4 4 -4----- 4 + -~--


S------- I


I /1


I/


+


0,
T1
4


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I-.
Z
8


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0
0:


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0
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____________ ___________ ___in


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IIGV 'a3713dOEd 0O 831N30 W08-I 30ONV1SIO 1VOIIa3A


*1 -\I


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II


t 0 t


f/


l-:


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j-




NACA ARR No. L6A05b


q a




i///



































PROPELLER PROJECT ION


0n o i


0\O
-o0 0 Ln 0 1-
INL4


aIIOV '377113dO8d dO 831N30 i8OJd 30ONVISIG IVOiId3A


Fig. 5b


In

0
r'-.
cJ








cin
oJ
C'4

5
0c
0-
(m cr
w
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-J
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0.
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0
LL
Z
z
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Ir
00

0
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0



in
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in





NACA ARR No. L6AO5b


ro1












+ iO
















0















PROPELLER PROJ SECTION
o\ \ / X



r ">. \ ()DOD i
----- ^ ^ --- --- ^---------- --- --------- _


______________ \ ______ \ ______o 4___ __ ________


^"SlZ^

T^ \ / / ^Sto
q s. ------ v t -- -- f -r ------ ^ o CM


O O N30 3 0SI OI3A
IIGlV 8I'8377-3dOE~d JO 831N30 V108 4 3ONViSI(] --7VOIIi?3!


Fig. 5c


V)

4
4g.
2
- 8
< Z


Q I'
or





NACA ARR No. L6AO5b


0 0








O




ci
+0



















P- 0




















PROPELLER PROJEClION


-O 'I-


0 wt
WM


0,'


IIOVy '831-13dO8d -o0 I31N30 INO84 "4'WISIO -iVOIII3A


Fig. 5d


0 in
LP 04




NACA ARR No. L6A05b


r
0 /

0 0

?. 9 ____ _______ / 0
N.


0
Lc)



0 C4


0



_________ If)/ __ __






+, Ln

0)C
0 0
rv__\___ \__ /_


0o






P0
+
0 r






00




PROPELLER PROJECTION


_. ___ /)~


0 LOn 0 0
0 p-: L 0N


0 .0
I"


IIGVa '83-7713dOad jO I31N30 IAOId BO3NVISIG -1VOIW3A


Fig. 5e


0=

0



8
a-
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8


o
Qr










LL
(oK


0
01i




NACA ARR No. L6AO5b


\ /+
0i


0
c\J



0





U-)0
ui
0 0+
















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\ L+'0
q 0





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0* 0








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o
o. r,


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PROPE


0


LLER PROJECTION



0 U)


'1*
0
.4-.
II


0



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81
zI
li


a
aJ
UJ
Ll
Q.
0


L-
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IIOV'd3113dO8d -io0 3iN30 A0O8A 3ONVIJ-SI1 71VoI3A


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0.
I.


Fig. 5f




NACA ARR No. L6AO5b


o







OO
0









\ J_
















0 LO
N




























+0












o o q'
__ s^ \ ___ / / -^o


-^^ ^----^^--72







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NO19POI \ l 1 / /dO /d


0 \ \0 \ 'C 0 10 0
\ \o \ N / /


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2-
t-I-
S



0-
of









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0:


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i-
U-
0

a:





w
0cr
LiL


0
z
4
I-
(I


-j
4
z
0

0
m


Fig. 5g




NACA ARR No. L6AO5b


o o Um o to o0
Q i.-. 0 cq ,n rI.
IIGV8 '3-113dO8d dO 831N3O INOd8 3ONVISIOG VOiOU3A


Fig. 5h




NACA ARR No. L6AO5b


0 0

I_ \ 0



Si




0
(D






'




0 0
I00








00

















__ -PROPELLER PROJECTION 0


I _ I _


U) 1,)
0* 1c~


(M 03n


9'
O I"
_..


IIOVi'831-13dO8d O 831N30 VY08d 30NVISIO -1VOI1I3A


a













L-.
0 S
ZR








a:
-J



LJ
0
a-





w "
Z- c.


cUJ
o
z
I-
0


Fn

z
0
N
x:
0
T


Fig. 5i





Fig. 6a


NACA ARR No. L6A05b


0



















+/






































-------.------- -





E-PRoPELLLER PR 0 10.


SLO N0


N t)I>


IIaVy '31 13dO8d dO 831N30 IYOj- 30NVISIO IVOIUI3A


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in N
ir

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NACA ARR No. L6AO5b


I.. t I
I I
0


c'J
0
I.


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0
+


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-pRPEOLERCPI6 O1


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a
0
0


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AIIVU '83y1-d08d 4o 83iNJ3 iV zlQid 3ONVISIO IVI1I83A


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Fig. 6b


y

/





Fig. 6c


NACA ARR No. L6AO5b


PROPELLER PRUoJ-


0 0) 0 0 0)
O o ) o
IICV9 '83-713d08d JO 831N30 INIO- 30NV1SIG 7VOIUI3A


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0











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0

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0 u0















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0


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NACA ARR No. L6AO5b Fig. 6d

0 0
O
ri


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c,j



IN t
Lo\








2.:
0 w
---------- O

Q-


4: LL









-ww







---------- E RJE-- -- --T---------N 0
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-r-C- -^ ----o--- 0
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co^~ -~. ^ S \ \ \ \ f .'r+ r> N 0







_______________________________ 4 0^


IIOV8 '83173dOad 10O 831N30 OHj1 30ONVISIO 7VOI1a3A




Fig. 6e


0
0
C


NACA ARR No. L6A05b


0



\ o0


\ / /
5N
4+:U














. \ / /
C\j


0





\0
Cc~j
_______^____ U)/_____>



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0 L

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IIUV8'8d31-13dOd 40 831N30 IVOd 30NVISIG 1VOIU13A




NACA ARR No. L6AO5b


Fig. 6f


0 o





0 ------ _
0



























-- ------- ------ -- -----O------- _"L C
toJ


'C)













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I )






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w
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"




NACA ARR No. L6AO5b


to a
-0
0*


0 0 o
zO


oz
N


0

LiJ
-J
-J
w
Z
w
0
a::
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0
z


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1-





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1'
0Oz


II


PROELLER PROJECTIONS


to 0 to
N- C
un oin
r^ Q (71


I/I


/


/_


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Q/
4:
k


IJo
0 "


11IGV 'dl1l3doyId dO I31N30 V08-I 3ONVISIG 1VOIII3A


0


m
u


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Fig. 6g





NACA ARR No. L6AO5b


0
0 Ko





o 0 0 0
I I 4: + /



LO


C-







00

0 o







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-- pROPE-ER ---- ---




















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z
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-I
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0


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xr


Fig. 6h




NACA ARR No. L6AO5b


r0
0
I \


N


0
-0
I


a
-0
I .


I'I


/ ro /
0
_o0
+


0
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0

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Fig. 6i


4Q




NACA ARR No. L6AO5b Fig. 7a
0
0--10

00







l'




C.J..
o -



--- ||p-LW
/ +"Z








0 __ -.--- ---- --- __


0

>- _-_ _i" L"














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__________J 1> i


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Fig. 7b


IIVa '37l713dOUd JO 831N30 VYO8d 30t.'.lSl7IG 1VDII3A


NACA ARR No. L6AO5b
0 0
I 0


r d









/- -
/ 0 0















\ o _=
0 0 Lii












z-
0/




cr D










-o
_L _A ^ _y-- ^ z -O F ,

< \ 1 / / /+ o
-\ \ -/-- ^ -^^-q
\ \ / ^'^wz'

\ \ / / ^"^ <




NACA ARR No. L6AO5b


llV8 '83--13dOId 30 i31N30 V408oJ 3oNViSIG 7VOI183A


Fig. 7c




NACA ARR No. L6AO5b


II (-)



o rd

S / oC\j

+









o /o

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