An electromagnetic-analogy method of solving lifting-surface-theory problems


Material Information

An electromagnetic-analogy method of solving lifting-surface-theory problems
Alternate Title:
NACA wartime reports
Physical Description:
21, 11 p. : ill. ; 28 cm.
Swanson, Robert S
Crandall, Stewart M
Langley Aeronautical Laboratory
United States -- National Advisory Committee for Aeronautics
Langley Memorial Aeronautical Laboratory
Place of Publication:
Langley Field, VA
Publication Date:


Subjects / Keywords:
Compressibility   ( lcsh )
Aerodynamics -- Research   ( lcsh )
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )


Summary: A method is suggested for making lifting-surface calculations by means of magnetic measurements of an electromagnetic-analogy model. The method is based on the perfect analogy between the strength of the magnetic field around a conductor and the strength of the induced-velocity field around a vortex. Electric conductors are arranged to represent the vortex sheet. The magnetic-field strength is determined by measuring, with an electronic voltmeter, the voltage induced in a small search coil by the alternating current in the wires representing the vortex sheet. Solutions of nonlinear lifting-surface problems may be obtained by placing the conductors representing the trailing vortices along the fluid lines (Helmholtz condition). A potential-flow solution for the distortion and rolling up of the trailing-vortex sheet may be obtained. By use of the Prandtl-Glauert rule, the lifting-surface theory may be adapted to include first-order compressibility effects.
Includes bibliographic references (p. 21).
Statement of Responsibility:
by Robert S. Swanson and Stewart M. Crandall.
General Note:
"Report no. L-120."
General Note:
"Originally issued May 1945 as Advance Restricted Report L5D23."
General Note:
"Report date May 1945."
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."

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University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003613188
oclc - 71204231
sobekcm - AA00006278_00001
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Full Text

ARR No. L5D23



May 1945 as
Advance Restricted Report L5D23



By Robert S. Swanson and Stewart M. Crandall

Langley Memorial Aeronautical Laboratory
Langley Field, Va.


*-.-A "'.:'.". .'', k .. .
." .

~' '*'


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 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 120


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

71 L, I

1 315'





By Robert S. Swanson and Stewart i.I. Crandall


A method is suggested for maikine- liftin;j-surface
calculations by merns of magnetic rieesurements of an
electrompgnetic-anrlogy model. The method is based on
the pelrf,'ct ?nalocy t3ec':een the strl"_nt.h of the rmannetic
field around a conductor and the strenratl, of the induced-
velocity field around 9 vortex. L:lectl-ic conductors are
arranged to represent Uche vortex sheet. The magnetic-
field stren.tth is determined by mieecsiur-n, with an tlec-
tronic voltmeter, the voltage induced in a si:call sear-ch
coil by the alternrtin. current in the wires reorecenting
the vortex sheet.

Solutions of nonlinear liftini-surfPce ;problems mayT
be obtained by ,lacing the- conductors re-pres ntin-; the
trailing vortices along the fluid lines (Helmhoitz con-
dition). A potential-flov. solution for the distortion
and rolling up of the trailin-g-vort.e-: sheet r,,ay be obtained.
By use of the Prandtl-,Glsuert rule, the lifting-surfaca
theory may be adapted to include first-order ccmpr-esisioility

A comparison wTs made of the dovnmi-esh determined by
means of e preliminary electrcrnisnetic-analocy model with
the dowrnwash obtained by calculation for an .lliptic wing
having an aspect r-tio of 7. The sccurecy of the magnetic
measurements compared satisfactorily with the accuracy of
the downwash calculations.


There are many important aerod:neamic problems for
which solutions by lifting-line theory aire inadequate.
These problems can be solved much more satisfactorily by

2 NACA ATR No. L5D23

a lifting-surface theory; that is, a theory in which the
lift is assumed to be distributed over a surface instead
of along a line. The calculations necessary to determine
solutions by lifting-surface theory, however, are rEther
laborious even for the simplified case in which the vari-
ation in incremental pressures with the effective camber or
the angle of attack of the surface is linear. A more exact
nonlinear solution is very nearly impossible to calculate
except for a few special cases. A few of the aerodynamic
problems for which solutions by lifting-surface theory are
desired are: the plan-form corrections necessary for the
prediction of finite-span hinge-moment characteristics
from section data; the determination of spanwise and chord-
wise load distributions of wings with low aspect ratio,
wings with sweep, wings in sideslip, wings in roll, and
wings in turning flight; more exact solutions for the
unsteady lift of finite wings; and an improved theory of
the field of flow near propellers.

In reference 1 it was shown that the plan-form correc-
tions determined from lifting-line theory are inadequate
for hinge-moment predictions. The plan-form corrections
determined by a linear lifting-surface theory (reference 2),
however, were shown to be quite satisfactory for the pre-
diction of hinge moments at small angles of attack. For
wings at larger angles of attack, especially for wings with
square tips, a nonlinear lifting-surface theory is required.

The electromagnetic-analogy method was developed in
an attempt to make calculations by both linear and non-
linear lifting-surface theories practical. The time and
expense required to build and test an electromagnetic-
analogy model of a wing and wake were expected to be small
compared with the cost of applying other methods available
at present, even for the linear case. The electromagnetic-
analogy method is based on the fact that the magnetic
field around a wire carrying electric current is perfectly
analogous to the velocity field around a vortex. It has
also been shown (reference 5) that the lifting surface and
wake may be represented by a vortex sheet and may therefore
be replaced by conductors arranged in the configuration
of the equivalent vortex sheet. Simple measurements of
the magnetic-field strength then replace the difficult
induced-velocity calculations.

For nonlinear solutions of lifting-surface problems,
the trailing-vortex sheet represented by the wires is
rolled up and distorted instead of lying in a plane as it

EACA ARR lio. L5D25 '

is usually assumed to do. In figure 1 is shown a sir lified
picture of a rectcni glar '; in of low asp..ect ratio at a l1rge
ancle of attack with a rolled-uo ?nd distorted trailin&.-
vortex sieet. Of tie various features of the distorted
vcrtex sheet that contribute to the nonlineari ty, the moist
inm'ortnrt -s the vertical sn-cifnj of the trailin:. vortices.
The increase in vertical spacin; as the anTle of' ttck'.. is
increased results in a decrease in uh: vertical component
of induced velocity at the surface, especi-illj near the
wing tips; thu3 the sloce of the lift curve is increased
as the an7le of attacil: incre, (s-r rsfer-nc.c 4) arnd theU
slones of hin-urnortent curv-s '-re imoi; nec-ative Treff-r-
enc:- 1).

The present re-:,ort describes thie basic theory of tbe
electroma.-netic-snaloy method ind the general procedure
by which various aerod/ynamic problem;. mna be solved by this
Penlogy. A few preliminlsrr results for the linear case
are presented for en ellintic rin. hav:in: n as-. ct ratio
of 3, as well as a coIImpIPiscn of th;- results obtained by
the present method and the results calculate-d by the method
of refsre-nce 2.


r vortex stren-th

'maX< mri;ax;imjm vortex strength

\p pressure difference across lifting sirf'c.2e

V free-stream velocity

V. mach number, ratio of free-stre.a velocity to sonic

p fluid. dens ity

x distance along free-stream direction from leading
ed-e of winz

y spanwise distance

z vertical distance above pine of vortex sheet

H msagnetic-field strength


i current in conductor

e induced voltage

1 length of conductor or vortex

r distance from element of conductor or vortex to point
in question

v induced velocity

w vertical component of induced velocity (downwash)

u horizontal component of induced velocity (free-stream

K constant

t time

b span

c chord

Bar above symbol indicates a vector, as h, T, r, and v.


Solution of Aerodynamic Problems by Available

Lifting-Surface Theories

The distribution of lift over a lifting surface cannot,
in general, be expressed in any simple mathematical form
such as can be obtained by lifting-line theory. This state-
ment is especially true for nonlinear lifting-surface
problems. Except for a few special Dlan forms (references 5
and 6), the method of determining the induced downwash for
a given lift distribution also is too complex for expression
in mathematical form. In order to obtain an exact, complete
analytical solution, however, such expressions must be known.

The determination of the surface to sustain an arbi-
trary lift distribution may be accomplished by means of
the electromagnetic-analogy method described herein or,
for the linear case, by the semigraphical method of refer-
ence 2. The inverse problem, determining the lift distri-


button over an arbitrary surface, msy then be solved by a
process of success: ive 3 n rpro-:imtti'r.s. A reason eole distri-
butzon of vortici O t is 0is0newa or' calculated from the
simr:le li-ftin:--line and thin:-sirfo*il theories, nd the
induce.- vel.:cities corrs-condlin, to that vortex distribution
aere sIeccrniin.d by i,,ikin- an electLromiagnetic-an1 lo' y mod.-.l
of the vortex sheet and measurin,- the Imr-netic-fi-id strength.
If the induced velocities do not satisfy the -.cundiry7 cor.-
ditions that is, the sl-p=s of the liftin-g urf,.ce the
vortex sheet is sutLbl, elt.i'red la-n the process Irepest.Ed
until the boundasry conditions are s&ti-fied. For the non-
linear problem, not only nmu i: tha induce velocities satisfy
the boundary conditions of the wink shane but Piso the
trailing vortices must satisfy the Helrrdmhiltz condition,
namely, that the vertices must tr-ail long Cllid lines,
(See reference 7.) In nrfrctice, satisfying these simple
conditions ;nqy, require a consid-erable a ounit of work unless
the first sop:ro:imt-tion is fairly accuraPe. In order to
obtain a somewhat Irore general solution, for the lin5?r
case qt lesst, the surfaces required to su::,oort s3'veral
different li't distributions aiy be determined so tiht the
shepe for s'. .Erticualar lift distribution may be estimated
by a process of interpolapti n or superposition.

There are, however, several nrobleris for which a com-
plete ootential-flow solution of the inverse problem: is not
necessary. For exeimple, in order -c. include the i. iin
effects of viscosity, the estimation of the hin,-e-rioment
parameter for fini e-soan win-s should be made by applying
theoretical aspect-ratio correction- to experimental section
hinge-moment parameters. For such problems the additional
aspect-rgtio corrections may be de Lc rmineid simply and
accurately from the surface required to support a .iven
lift distribution (reference 1) as found by lifting-surface

The results of the Qlectror.sanetic-nallory solution
of the liftin.-surf ce tlieorv may be cor-ected for first-
order compressiti litty, effects by a sir:le ap::licstion of
the Prandtl-Glauert rule (reference 8). Th m.nethod consists
of deter-mznin- t the incomjnressible-flvo characteristics of
an equivalent wins the chords of which are increased by the
factor It is therefore necessary only to build
'41 ,2
an electromagnztic-analogy model of a Wfinl: of this slightly
1. o t1 \
lower asoect ratio (lower by thi factor -' or to
~\test moel s of sverl i2t
test models of several aspect ratios and inter'olato. The


pressures (or vorticity) acting upon this incompressible
equivalent of lower aspect ratio, however, must be increased
by the factor In order to find the lift, these
increased pressures are referred to the original wing and

Vortex Sheet
Inasmuch as the equivalence of a lifting wing and wake
to a vortex sheet may be considered to be well established
(reference 5), only the important characteristics of the
equivalent vortex sheet and the relations between the lifting
wing and the vortex sheet will be given.
The part of the vortex sheet representing the lifting
wing consists of a sheet of bound vortices. The strength
of the vortices is directly associated with the lift distri-
bution of the wing. The product of the air density, the
free-stream velocity, the vortex length perpendicular to
the free-stream velocity, and the vortex strength of each
elementary vortex equals the lift contribution by that
elementary vortex (Kutta-Joukowski law). If the lift
distribution of the wing is known or assumed, therefore,
the equivalent vortex distribution may be easily obtained.
A continuous lift distribution (as measured by pressure
distribution Ap) may be integrated to give a continuous
vortex distribution. The integration formula (reference 2)
for obtaining the vortex distribution is
r = p dx (1)
0J P
where pV is the product of the density and the free-
stream velocity and the integration is made in the free-
stream direction. Equation (1) sives the chordwise
r-function at each section. The values of F at the
trailing edge of the wing at each section also give the
spanwise vortex distribution of the wake. The bound
vortices may be assumed to lie along a mean surface, half-
way between the upper and lower surfaces of the wing.

The part of the vortex sheet representing the wake
consists of the so-called trailing vortices. As the name
implies, these vortices originate at the trailing edge of
the wiing and merely trail behind the wing. These vortices
are free to move and thus lie along the local stream lines,
or fluid lines. This simple kinematic condition, the
Helmholtz condition (reference 7), determines the configu-
ration of the trailing-vortex sheet.


The treiline ve:rtices for lirhtli leaded wings usually
lie very near a plane; that is, these vertices ter.vel
almost straight back: fror their origin at the trailin.
edge of the wvinj. For highly loaded wings, however, the
trsiling-vortex iheet is known to be considerably distorted,
rolled up, and inclined with respect to the free-stream
direction. (See fig. 1.) The ch-arscteristics of the air
flow behind wings are described in mior' detail in r-efer-
ences Q and 10.

ElIc tromagneti c An lo"y

The perfect nnaloy that exists bztw:en the strsinth
of the magnetic fieli round conductors and the strenalth
of the induced-velocity field around columnrer (finite-
diameter) vortices is Lxell ';nowrn. In fact, the rnhenomena
of the induced ve locities Laroiund vortiCeS 9re usually
explained in aerodynamiic textbookls by the anr:lojy with
electromagnetic phenomena. Both hain ormens are potential

The vector form of the diiferentitl equation for the
ma.gnetic-field strength dH at any point caused by the,
current i flowing in an infinitesimal len.-th dT of
wire is (from p. 2)2 of reference 11)
.jl x r
dH = I (2)

where r is the vector from the current element to the
ooint in question. This equation is u ually called the
Biot-Sivsrt law in aerodynamic te::tboo-!s.

The sarme forr. of e.uation (2) but '.iitii different
constants arplies to the induced velocity dv at any
point caused by an infinite i!nal length dT, of a vortex
of strength re (reference 9); that i,
P ,td x r
V= -= ()
.Tr ir13

The units in which the various quantities in
equations (2) and (5.) are usuiily meia.ured are widely
different. In equation. (2), for exasr.ple, H is usually
given in gauss, i in aba:srperes, and I and r in
centimeters. In equation (?), v is usually in feet per
second, d in feet squared per second, and 1 and r
in feet.

o NACA ARR No. L5D23

Small search coils are used to measure the strength
of the magnetic field. These search coils must be cali-
brated in magnetic fields of known strength for example,
in a Helmholtz coil (fig. 2). If the vortex equation (3)
and the usual vortex units are used to commute the induced
velocity in the Helmholtz coil (p. 269, reference 11) -
that is, are used as the calibrating unit and if the
ammeter measuring the current in the Helmholtz coil is
considered to read vortex strength, conversions of electro-
magnetic to aerodynamic units will not be necessary. All
conversions and constants become merely a part of the over-
all calibration constant of the search coils.


Approximate Representation of Continuous Vortex Sheet

The replacement of a continuous vortex distribution
by a finite number of conductors must, of course, involve
some approximation. Two procedures for constructing models
have been tried. For the preliminary electromagnetic-
analogy model, a set of 50 circular electric wires carrying
the same current (connected in series) representing
50 columnar vortices of equal strength were distributed
over the wing and wake, CSee fig. 5.) The arrangement
of the wires was determined as follows: Fifty contour
lines of P were calculated from equation (1). (See
reference 2.) Each wire was placed halfway between two
adjacent P-contour lines and thus represented
APF/max = 0.02. In order to illustrate the degree of
the approximation involved, the continuous chordwise
r-function at the plane of-symmetry and the stepwise
distribution of wires used to represent the continuous
distribution are given in figure 4..

The possibility of errors resulting from the use of
an incr-emental distribution is, however, more serious than
simply the possibility of not.obtaining a good represen-
tation of the distribution of vorticity in the continuous
wake. The induced velocity resulting from-an incremental
vortex pattern may vary greatly about the mean value that
would be obtained. by the-continuous sheet, because the
ma gnitude of the induced velocity varies inversely with
-distance..from. the voctex core and becomes higher than the
mean value on one side of each steopwise vortex increment
and lower than the mean value on the other side. The

I]ACA A7-R lFo. L5D23

effect is illustrated in figure 5. It is thus nece-ssry:
to us3e as :rsrny vi .es ,.s prac tic 9ale to rp-}:ro ach as nfe a1l
as pc-'sitle a c.on inuo:us sheet and to measure the iniuced
velocities at a. .reac number of' ,points so that the m,::an
!.vlue of the induced velocicy: may be obtained nim:r 1 easily
and :nore accurate ly, from f aired curves.

The pri 1 iminriary mLodel, co.ns trictcd of ,, wiree-, w'ive
satisfactory results -ird it is believed trit ',0 is slboi.t
the ootiTmuji n-iWnber of wires. Cojs truth ion difficulties
are too great if mnoie wires are used,, &rd the accuracy is
not great eini.ouh if fe.wer wires s:-e used. Another, minor
complex method of construction was adopted for a few
models out the results obtained were not much mnor- scc.urate
than those co'tsined with the siimoi.r ir cors truction
met.iod. The other cons tru.ccion met.1ho was to use t ii
alLuminum sr:trips resultingg in "flat 'ipr'.es") rather th-an
circulsa copoer wires (-_io. 6), the recs:n for th,, typ
of construction bein- thT t a m:ore n': rl continuous distri-
bution of vorticit:' Fnd f c:,res':or:in mnoothr induced-
velocity:; field could be obtained. The" ffec t cf oddy
currents in the a.uninum s tri.s proved to be more im.ort nt
thln w:as originally expected, hol.e'-e and the induc,:-d
:narrietic field was only sli,-t.:tly simoother th n with
circul ar wires and wvs nert so revul-ar" thus difficulties
in fairing the measured induced .na nli.Lic fieli proved to
be about the sarre for both t-:,es of model.

Correction for Finite Thic,'ness of ires

calculations by the lifting-surface theory are usually
made for points in the plne of the ';rtex sheet. Eec -:use
the wires raeoresentin- the vortices n-re of finite thicl:ness,
the mnagnetic-field :..ust tb me asre al at severals i
vertical heights and extra~,olated to cr., that is, to the
center of the wires. Z:cept ne:r t[li.: a ina, tips rnd the
leading z edge of the wini, tnis et:.r;.;:olation is u'-ually
lin-etr and can thus be rrm-de quite 9ccur:ely. The f'ct
that ,,'eesurements rce mauc above thi wires sir'.:lifies the
fasiinn problem, because th: vari: ion, in tne meanitude
of the vertical co:niponent of induced :elocicy about the
m sin vsluu (fig. 5) aecre- iaes with .nccrease in vertical


Correction for Finite Length of Trailing-Vortex Sheet

For the steady-state condition of a finite-span lifting
surface, the trailing-vortex sheet extends from the trailing
edge of the wing infinitely far downstream. It is necessary
to determine corrections for the finite length of trailing-
vortex sheet of the electromagnetic-analogy models. The
wires representing the incremental vortices were connected
in series; the closing loops were about 0.7 span behind the
wing for one model and about 2 spans behind the wing for
another model. (See fig. 3.) Because the correction for
the approximation of the infinite length of the trailing-
vortex sheet is fairly small about a 5-percent correction
for a 0.7-span wake and less than 1-percent correction for
a 1.5- to 2.0-span wake it appears that wakes need not
be longer than 1.5 spans. More dovwnwash is contributed by
the closing loops than by the missing trailing vortices.
The correction is simply the difference between the down-
wash contribution of the closing loops and the contribution
of the missing part of the trailing-vortex sheet. The
corrections are small and can usually be estimated by
assuming that the span loading is a simple rectangular
loading or the sum of two rectangular loadings.

The corrections may also be determined from measure-
ments of the induced field ct a distance behind the closing
wires equal to the length of wake represented. That is,
if a measurement is made at the corresponding spanwise
point I wake length behind the model, the effect will be
the same as if a measurement were made on the wing of the
downwash due to a mirror image of the model reflected from
the closing loops. Such an image would cancel the effect
of the closing loops and would double the length of the
wake. The remaining error is relatively small.


Several methods of measuring the magnetic-field
strength were investigated. The fundamental principle of
the method selected as being the simplest and as requiring
the least special equipment and the least development work
is given herein. This method consists in passing alternating
currents through the wires representing the vortex sheet
and measuring the magnetic-field strength by means of the

IHAC1. ARR lo. L5D23 I

voltage induced in a small search coil. A discussion or
other possible methods of rCeasuringf the rrnanet.c-field
stren..Lh is p-iven in the sppen i.::.

B3sic Principles

According toi the nrincoln-:-s of _electrxomPqnetLic
induction (refe.-cnce 11), the electrc-riotijve force inluced
in a fixed circui t (the- search, in tlhis case) by
chaning. the irqenetic flu.< throu h cho ccil is equel to
the time oate o1 chan;,e of ux linkaces ,'.ich the c:,il.
If ai altern rtinc c.irr-ent i2 crassed Lhrouh the wij.res
represen in 1 t-he I iftin p w nr c-id t.h- '. !k.e, the _ne-:i-ic-
field strengtth will 1iso :b alte'nristinc sat thz car.e
frequency and with the saie wave Sr;ac Thiis -nct may be
seen from equti-.n (2), bercuse tkL j ]stantFneous value
of d7 is ':roor orti.:n-- to th3 itnstsnt. ne.ous' v lue o i.

A sinimle method cf der"mnirC. c.he :nitic-fi li
strength there 'ore, is to ireasure i'. ith en el3ctrcric
voltmeter the, v;lte, :- indrc.d in a sm-ll su,-r-ch coil (fig. 7)
when altern.-tino cui-rent is through the wires
representing the vortic-l c (1Ji 5). ''Thk sEerch .:cil is
direction il; thlai is, thi- coil measures only the componr-rnt
of maneritic-fiela intensity alonr thl Sxis of the ccil.

Practical Problems

Upp*r-fr limit.- E5ca.use the volts a:-- i.duce-d
in the sc.rch coil is proportions] to the :t-. ,.f char..e
of flux, it would sa'=: desir.bla to uise a v-ry ~: hig-h-
frequency curr"-nt so th.t the volts ,: inrd'cd in th.e
search coil would be very large 5nd thus sromrewhiL lesser
to measure e. The rn exini-mu frequency tnst :ir,': te uszd,
however, is cocut 50-j cycles, because the cC';acitance
between the closely s.-sced wires in the rmouil re3:'r'sent ng
the vortex s:.ae3 ;i'crt-t '-ove this frequency.
If greater frequencies -re ued, csnacitstive re-ctence
becomes sufficiently 1i 0 to have ,rcc, t ibli eff'oct. Vhen
the copecitative rea ct-nce between r't res becom::es smali,
the curr-nt is not thle s5e ,i in all e-ven tl.ou.h they/
are connected in series. The :ne.:-imun allo'vble frequency
wIE d-t,,rmined from mee- aurrimnts of tlhe cs:pecitntive
reactance of the el?ctrrm-nane tic-enaloey model Lsed for
the pr-:-liminsry tests, to b.-: described in the section
"Pr-l ri:in.sry. Tests w -,th Electrama.- netic-An12 1lcy olod3l."
The cDoscitative, r-ect-nce-, end thus the iea ,ags' currPnt
of this model, became- reasu'aible abovZe bout 5300 cycles.

12 NACA ARR No. L5D23

Several other models tested have had about the same or a
higher limiting frequency; therefore, a 300-cycle limit
is believed to be conservative.

Wave shape.- The fact that the frequency is limited
to ab- at ,- cycles also requires that the wave shape of
the current in the vortex sheet be very nearly sinusoidal;
that is, any departure from a simple sine wave means that
higher harmonic frequencies are also present. Usually the
most nearly sinusoidal wave shp.e is obtained by putting
enough condensers in series with the model to balance the
inductance of the model; in this way a pure resistance
load is put on the generator.

The easiest way to determine the wave shape is to look
at the voltage output of the search coil by means of an
oscilloscope. This wave shape is not that of the current
in the wires but depends on the rate of change of current i
with time t that is, the induced voltage e is equal to
K -. The amolitude of the third harmonic as seen on the
oscilloscope is, then, three times the amplitude of the
actual third harmonic of the current. In other words, the
wave shape of the voltage output of the search coil looks
much more irregular than the wave shape of the current in
the wires actually is. If the filters that are added to
the circuit make the voltage output of the search coil
appear satisfactory, no further test is necessary.

Practically all alternators have wave sh'.pes that are
not sinusoidal and that change with load. The wave form
of the current in the Helmholtz coil, used to calibrate
the search coil, must be identical with that of the current
in the electromagnetic-analogy model. Probably the easiest
way to make these wave forms identical is to connect the
Helmholtz coil and the electromagnetic-analogy model in
series. The Helmholtz coil and the electror-mnetic-
analogy model should be placed as far apart as possible
and should be oriented in such a way that downwash measure-
ments on the analogy model are unaffected by the field of
the Helmholtz coil and that the Helmholtz field is unaltered
by the analogy-model field while the search coil is being
calibrated. It would also be advantageous to effect an
arrangement such that the search-coil calibration could
be checked frequently. A fairly satisfactory arrangr.iernt
was employed for the preliminary tests. (See fig. 8.)
The leads from the search coil were long enough to go to
either the Helmholtz coil or the analogy model.

11ACA ARR No. L5D25

Search coils.- The optimum size of the search coil
depends u-on two factors. First, the search coil must be
small enough to make "point" measurements possible. If
the coil is too ler-,' and the inmnettic-field strength varies
nor.1 nearly v'ith ;osj tion, the effective center of the
coil muy be too far from the -gometric center. If the
coil dimensions :re kept small relative to the win-, little
error due to this cause will result The size cf the
search coil t-erefmrc devo-ndis upcn the size of the
electromae netic-analc.ry model. The e'conid cotnsider'StionL
is that, for accurte- v:ltvCe menasur-menit, crn voltPs
outout of the ser.rch ccil should be at least 0.0001 volt
and a minimuLTr value of 0.0O1. volt 3i crreftrsble.

Several sizes of coil tri:i with the pr-elimiinary
modil, and it is felt tht the iaximur.-size s-arch coil
that gives measurements fairly close to point rmeasurer..nts
is one with a Jiemater- about 1 percent of' the Viin ei:.i-
span and a height about J.5 or.rcent of th- wini-. semispen.
In order to get the desired search-cail voltage output,
the model semispan must be at leasL 5 to f-.-.t and the
search coil should have sbout 1000 Liurns nrid be wound with
about No. 13 wire (0.002 -in. diam.). If a larg-eri model
is used, larger wire may be ued to wind the c..ji, although
wire as small as 0.001 inch was wound satisfactorily for
the coil used for the p.elimrinr.ry tests. It is probably
desirable to have s number of search coils (fig. 7) of
various sizes, however, to meet saiy s.eciFl conditions .
Smaller coils are desirable near the leaiing ed'e rnd the
tios of the wing and in other places v'hore the flux field
varies ra-idly.

The search coil must be wound rather carefully so
that all looos are Ps nearly ,erotndiculs-r e possible to
the axis of the coil in order to m,7.c, tain the directional
properties of the coil. The coil must then be carefully
mounted on the survey apparatus. A test fc- the correct
mounting (or alinement) cf the coil is to read the voltrge
output of the sn:mll coil when rrmount-d with its axis vertical
at a position on the electroiriencneTSic-n i, ogy model where
the horizontal component of the magnetic field is strcnger
than rth vertical component. aegzadles. of how the search
coil is turned about its vertical axis, the voltage out-
put should be the same if the search coil is kept &.t the
same lace on the model.

The leads from th3 search coil to the -lectronic
voltmet-r must be twisted so that no 1'.-g:e loops sre present.

i..l'A ARR No. L5D23

Otherwise, the voltage measured by the electronic volt-
meter will not only be the voltage induced in the search
coil but will also consist of the voltage induced in these
leads. The errors resulting from pickup in the leads may
be made negligible by tightly twisting the leads, by
making the number of turns in the search coil large so
that the lead pickup gives a small percentage error, and
by bringing the leads in perpendicular to the vortex sheet
so that the leads will be in the region of low magnetic-
field strength.

Extraneous fields.- One of the most important problems
in measuring the magnetic-field strength is to filter out
all extraneous fields. It is desirable to make the tests
in a wooden building fairly far from electrical distur-
bance.s such as electric motors and computing machines.
Reasonably satisfactory results may be obtained in spite
of these disturbances,if necessary, by the use of an
electrical filter in the circuit of the search coil and
electronic voltmeter. Suc. a filter suppresses any voltage
of frequencies other than the one passing through 'the
vortex sheet. Because the filter may act as a search coil
itself, it must be located some distance from the model.

Teasuring equioment.- No current should be drawn. in
measuring search-coil voltage, which is of the order of
only 1 millivolt. Commercial electronic voltmeters combine
the high sensitivity and high resistance demanded. Small
amounts of power-supply ripple usually exist in these
voltmeters. This power-supply ripple interferes with
measurements made at integral multiples of the line fre-
quency, and care-should be taken to avoid use of these
frequencies in testing.


In order to check the accuracy of the electromagnetic-
analogy method of solving lifting-surface problems, a
vortex pattern for which the induced velocities had already
been calculated by the method of reference 2 was inves-
tigated. Calculations were made of the vertical component
of the velocities induced by a plane vortex sheet repre-
senting the lift distribution estimated from the two-
dimensional theories for an elliptic wing having an aspect
ratio of 5.


Equi rentt

A small model of th plane vortex sheet was constructed
of 50 wires rooresenting 50 incremental vortices. This
model (fic. 5) reTresents about the srt-.lles t model (winT
scan of 5.5 ft) that will yield satisfactory results. A
srr.gll search coil (fig. 7(7)) of 1j30 turns was used to
.measure the mifgnetic-field strength. A -l-emlnholtz coil
(fi;. 2) ws used to celebrate the search ceil. Thi ;.ower
supply was a 500C-cycle alternator dJiven by a -l re t-
current cnotor, the s,.e-d of ~iaich 'was controlledc so lhat
the frequency out-'ut vi.s iheld nat 270 cycles. The itrect-
current field of tn altern-tor u\s r-- .ud L- to give a
current o it':ut of ea:. '*. The .va,' shcipe wcs ver.-
nearly sinuscidal (third harmoni c, cles then 4 percent of
the first harmoniic).

Some trial tests v.nee n:de in the w'-or :sho': of the
Lsnley Atmncao'so:ric W'ind T'unnel SecLion. The results
proved uns9tisfactory however, bee -9e or excessive elec-
tric and m r-netic interference. The acunertus .a.s then
moved to a lr-e vcen-Dier. .i.din', in which there was very
little electric i a .iac-netic interference. The setup is
shown in figure The smnll snmouiint of oC-cycle electric
interference that was still present was elinineaed by
using a 130-cycle hi-1rh-p5ss filter in the electrornic-
voltmeter circuit.


A comn lets survey was made of the induced-velocity
field (both the horizontal and vertical comr:ronents) st
from 50 to 100 chcrd,;ise points at each of 1; spcanjis
locations on the rodal and at several vertical heights.
Near the leading 6edg- and the tips of the .:o'el, the
effect of v4rticai height was vy l=r a en, at all
locations, the effect &was lrige cnoizh to r.-quire surveys
at several heights in order that t-,he r::sults could be
extrapolated to zerc vertical hs~lnt.

The most imnortanc co.-.7onrent of induced velocity
computed by liftingr-surf-ce theory is the vertical con:.onent.
The only reason for r.eqsuring the hcrizontal comionent is
to check the arrancemren of the wires reoresentin1: the
vortices. According to the assunmpticns of thin-airfoil
theory, the horizontal component of induced velocity is
proportional to the pressure disLribution. -,-lues of the


horizontal component u of the induced velocity
measured at a relative vertical height of 0.008
are shown in figure 9 along with the theoretical velocity
distribution that the wing was built to represent. The
agreement is satisfactory and indicates that the model
was constructed with sufficient accuracy.

Chordwise surveys of the measured value of the ver-
tical component w of the induced velocity are presented
in figure 10 for several spanwise locations and vertical
heights. (Not all of the data are presented.) These and
similar data were extrapolated to zero vertical height
(fig. 11) and corrected for the finite length of the
trailing-vortex sheet (correction, about one-half of 1 per-
cent) and are summarized in figure 12. Included in fig-
ure 12 are the calculated values. The agreement between
the two sets of results may be seen to be satisfactory.
The time and labor involved in obtaining the solution were
considerably less (approximately one-third the man-hours)
by the analogy method than by calculation, after the proper
experimental technique had been determined.


A method for making lifting-surface calculations by
means of magnetic measurements of an electromagnetic-
analogy model has been developed. The method is based on
the perfect analogy between the strength of the magnetic
field around a conductor and the strength of the induced-
velocity field around a vortex. Electric conductors are
arranged to represent the vortex sheet. The magnetic-
field strength is determined by measuring, with an elec-
tronic voltmeter, the voltage induced in a small search
coil by the alternating current in the wires representing
the vortex sheet.

A comparison was made of the downwash determined by
means of a preliminary electromagnetic-analogy model with
the downwash obtained by calculation for an elliptic wing
having an aspect ratio of 5. The accuracy of the magnetic
measurements compared satisfactorily with the accuracy of
the downwash calculations.

Other applications of the method include solutions
of nonlinear lifting-surface problems obtained by placing

NACA ARR No. L5D25 17

the conductors representing the trailin, vertices alon;
thi fluid lines (hnelrmholtz condiiticnr). A potent l9-flow
solution for the diEtortion and rolling up of the trailing-
vortex sheet nmy be obtained. By use of the Prendtl-
GlIuert rules the liftit-,'.-sur' fce the-Iry Q aT y be a.: 'pted
to include fi rst-or:der -coipressib, lity effects.

Langley ,emori l Aer'on sliical Labor~tory
Ni&tiorin l Advisor-' Comnr'ittee fo-,r Acr.-,r,'autics
Ln;ley Field, Ve.




By R. A. Gardiner

Several methods could be used to make the measurements
described in this report. The methods that were considered

Pickup Device

1. With d-c. field on wing

a. Search coil (flip
coil or collapse
of field)

b. Rotating search coil

With slip rings

With commentator

c. Saturated-core

d. Torsion-type

2. With a-c. field on wing

a. Saturated-core

b. Bismuth bridge

c. Search coil

Measuring Instrument

Ballistic galvanometer

A-c. electronic voltmeter

D-c. amplifier and voltmeter

Suitable electronic
e equipment

Suitable optical equipment

Suitable electronic

vheatstone bridge

A-c. electronic voltmeter

The considerations that led to the choice of the
method used (method 2c in the foregoing list) were the
simplicity, the sturdiness and availability of the equipment,
the development work required, the probable success and
accuracy, the magnitude of field strength required, the
minimum possible size of the pickup device, and the free-
dom from interference of stray fields. The outstanding


advaiite es- or dissiavantages of the various methods mayo be
surmmrarized as follow;:

Dtet:od l.- The use of a search coil and ballistic
galvanonmeter is the established method of maanetic-field
measuren.ent. In ord.r to secure the necessary sensitivity/ ,
however, a rthner delicate galvsnorieter would hlve to be
used 2nd would probably require r se-,cial vibration-free
support. The flip coil would require te-r readinfgs of the
earth's m!a.retic field. ith a stationa.rv coil end
collapsing field, the inuictance of the hingi would prevent
the desired instantanreous colls cse, the galvPnometer would
not be used in a true ballistic manner, ndri errors would

Method lb.- A rotstini search coil and, associated
equipment have been used to ma.:a mari-etic messuren.ents;
however, the induced voltpees to ba ;.eesured are lower than
those usually measured by this Irh: t"d. The necessary
sliding contacts woulG prob-bl' introduce th-;rmal
electromotive forces ar v:ariabl r.s isitan.ce and would be
subject to corrosicn. P'cision rr -hire worl would be
necessa-ry to minimize difficulties from the r.otor and

Metihos Ic end 2a.- It is ka-.o'wn that a coil corntlSninig
a high--asrmeability .:netal (Per::alloy or M lu.etl) will have
a larsce induce volt.oa-e across its rermirinais as the core
becomes mnsanetic l lly saturate. This induced voltage is
due to the great change in induct .nce that occurs at the
saturation point. This principle has been used in the
measurement of magnetic fields; the field to be measured
is superimposed upon s. fisld set up in the ccre and the
chasne in voltage due to saturaticn is measured. Although
this method can be made very sensi ti' e, a large amount of
electronic equipment is necessary and the nres-ssne of a
ferromarnnetic substance might cause distortion of the field.

ivMethod 13.- The use .of torsio-n mri:.-netometer is -n accurate
method of mneasurin2 the eearh's fiel'.. A sall -manet
suspended by a suitable fiber is deflected by the earth's
field. This deflection is measured and, from th; known
magnetic moment of this ma r et, the ear-th's field ray be
determined. The mreasuring eleiient is very sensitive and
the time necessary to take one reading is rather long.
In addition, such an instrument would probably be of
delicate construction.

20 NACA ARR No. L5D25

6-thod 2b.- Resistance change due to the presence of
magnetic lines takes place in bismuth. Measurement of the
resistance change by use of a bridge is a possible method
of measuring the magnetic-field strength. At present a
powerful magnetic field is necessary in order to make
practical measurements. Considerations of the available
power supply eliminated this method.

Method 2c.- The method finally selected that using
an a-c. voltmeter, an a-c. field, and a small search coil
for the pickup device appeared to be the means which
would be least troublesome and which would use easily
available and simple components. Details of this method
are given in the report.



1. S-.wnsIon, Robert S., ant- Gl]lir, Clarence L.: Linita-
sion:'- of Lifting-Line.- Theory fo, Estimation of
Aile.:'on Hinge-Mionent Clharacterict-lcs. HIAC3i CB
uNo. 5L02, 15L?.

2. Cohen, Doris: A i.'othod for Determining the Camber
and 'lTist of a Surface to Suvport a Given Distri-
bution of Lift. 1IAC. T11 lio. u55, 194-.

3. von E.arian, Ph., ani Durgers, J. .i.: General Aero-
Jynarmp.c theory Perfect Fluids. Vol. II of Aero-
dynami.c Tijory, div. L, '. Durandi, b., Julius
Springer ( ierlin), 1955.

4. Bolls.y, jilliam A Non-Linear .in; Theo:r and Its
Application to Rectan ular '.ins of Small Aspect
Patio. Z.f.a...i.:., Be. 19, Ieft 1, Feb. 10.).,
:p. 21-75.

5. Kinner, J.: Die kreisformije Tragfl che auf potencial-
theoretischer Grundlage. Tnr,-Archiv B-i. VIII,
Heft 1, Feb. 1957, pp. .7-O0.

6. Krienes, Klaus: The Elliptic -Tin. Based on the
Potential Theory. UL: TM Ic. 971, 131.

7. Prandtl, L.: Applications of I'odern Hydrodynamics to
Aeronautics. 'NACA Rep. HIo. 116, 1921.

8. Goldstein, S., and Young, A. D.: The Linear Pertur-
bation Theory of Compressible Flow, with Apprlications
tc ,-iind-Tunnel Interference. Ri. & :,. ilo. 1909,
British A.ri.C., 19?43.

9. Silverstein, ."ie, Katzoff, S., and Eullivant, d. Kenrneth:
Downwash and vsake behind Plain and Flapped Aiirfoils.
.A'CA Rep. Nio. 651, 1959.

10. Kaden, H.: Aufwicklung einer tustabilen Unstetig-
keitsfl.che. Ing.-Archiv Ed. II, Heft 2, IIay 1951,
pp. 0-168.

11. Page, Leigh, and Adams, Iorman Ilsley, Jr.: Frinciples
of Electricity. D. Van Nostrand Co., Inc. (New
York), 1951.


Fig. 1



I 1f
th 6


O% .




Electrio wires wound around these hoops

Di5tance between co, I
equals rodius of coils


Fiqure Z.- A Helmholt coil. The, earch
coil to be calibrated is located in
the geome-tric, center of the f-elm holtz,
coi I

Fig. 2

NACA ARR No. L5D23 Fig. 3


0 :
.-I -


0 0

-r- Q.
o -4

cod o

+3 40
C ..-)

t a) t
..-.," .I ,

...0 -

: ,. ,, .; ,)t!. ,ol
.i : ... ... ;" '.] ;: ei '
"4 ... ...... ;: .-:;
..... :
,- .--.I
., ,


%. \9 4 N
't/so.^ t/uop /Ou o /suawpo

00 t W, ',3'eNvoad uqo/njjo

- s o t
J I I k-
:- o




-o s
^ "

Figs. 4, 5

NACA ARR No. L5D23 Fig. 6

E r



.- o o

0 .
ii 0

,-1 '

D 0 0

Ic i .-I

I AI c Q

.) w 0

l *



Fig. 7a,b


INACA ARP No. L5f)23



va vo/fmeter




Figure General/ rra/ngemenf of e/ecfromogneo/ -
ona/ogy experiment .


Fi g. 8


0 .- .4 .6 -8
Ch/ordvwise station, x/c

/'~/re 9. Chon/d'-es d/s/-rL /o4 of/he /7end//me,/7ona/
Yor0zot.a-/ re/a/y cmpv-e' t 6'// 'g,,; ao/,s & o
-=0.008. F/ot-p/ate Iwoe of /oooewy.

Fi g. 9


Fig. 10a, b, c

_ [ l *- __ _





0 >q ldlq f l)
'/ a /tfgo 7o3o//ftUOp\o



0.- I

i b


to.q o

c 0

4 b
o s


*d ^5
0 '0

P1 C
0 k

o P

9I c

0 Q
t) 0x
^ ^
E b ^

'* sg




(a) --0.
6h T -

7 =Q7c

5 1 4
(c) =0. 4 0O

0 .OZ .04
Re/lafve height,

(d) Y 0.60.

0 .02 .04
,Re/ofve height, -

)Y = 0.95.

Figure .//. -Extropo/a/on to zero vertfco/ he/ghf of
dofo obtained from on re/ecfromagnef/- analogy
model of on e//'pL/c wing having on aspect
ratio of 3 F/of-p/afe type of /oadng .


.n ..'- *- -

Fig. lla-f


C---"---- 1----
.8- -- ] ----

4 .4

x Ca/cu/ofed points

0 .2 .4 .. -8 /0
Span wise station, ;

Figure /2 A The nonod/meens.ona downwash
wb for an e/llphc wing having an aspect
ratfl of 3 with f/of-p/ate type of /oad/ng.
Conopar/son of results obtained by method
of reference 2 with data, extrapolated to
z = 0, obtained from tests of preliminary
e/ecromragnetOc ano/ogy model/

Fig. 12


3 1262 08104 968 5

P.O. BOX 117011