JOURNAL OF AIRCRAFT

Vol. 39, No. 4, July– August 2002

Estimating the Low-Speed Downwash Angle on an Aft Tail

W. F. Phillips,¤ E. A. Anderson,† J. C. Jenkins,‡ and S. Sunouchi§

Utah State University, Logan, Utah 84322-4130

A closed-form analytic solution for the downwash induced on an aft tail by the main wing of an airplane is
presented. This inŽ nite series solution is based on Prandtl’s classical lifting-line theory and accounts for the effects
of wing planform shape, as well as tail position and vortex rollup. The results obtained from the present analysis
reduce to the known solution for an elliptic wing. Results for a tapered wing are also presented. This inŽ nite series
solution applies only to a main wing with no sweep or dihedral. However, an approximate correction is presented,
which can be used with some caution for swept wings. Results obtained from this analytic solution are compared
with other methods and experimental data.
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Nomenclature of the fuselage and nacelles. To obtain an accurate estimate for this
Bn = coefŽ cients in the inŽ nite series interaction between the main wing and the tail, we must make use of
b = wingspan computer simulations and/or wind-tunnel tests. Such methods are
b0 = wingtip vortex spacing always employed in the later phases of the airplane design process.
CLw = wing lift coefŽ cient However, for the purpose of preliminary design, it is useful to have
cw = wing chord an approximate closed-form solution for estimating the downwash
R Aw = wing aspect ratio induced on the tail by the main wing. Such an approximate solution
Vy = upwash velocity can be developed by assuming that the  ow-Ž eld in the vicinity of
V1 = magnitude of the freestream velocity the airplane is affected only by the main wing.
x = axial coordinate Prandtl’s classical lifting-line theory1;2 provides a closed-form
xN = dimensionless axial coordinate, 2x =b solution for the spanwise distribution of vorticity generated by a
y = normal coordinate Ž nite wing. If the circulation about any section of the wing is 0.z/,
yN = dimensionless normal coordinate, 2y =b and the strength of the shed vortex sheet per unit span is °t .z/, as
z = spanwise coordinate shown in Fig. 1, then Helmholtz’s vortex theorem requires that the
zN = dimensionless spanwise coordinate, 2z =b shed vorticity is related to the bound vorticity according to
® = angle of attack
0 = spanwise section circulation distribution d0
°t .z/ D ¡ (1)
0wt = wingtip vortex strength dz
°t = strength of shed vortex sheet per unit span
For the special case of an elliptic wing, Prandtl’s lifting-line theory
"d = downwash angle
gives a very simple closed-form solution for the spanwise distribu-
µ = change of variables, spanwise coordinate
tion of bound vorticity,
·b = wingtip vortex span factor
·p = tail position factor s
³ ´2
·s = wing sweep factor 2bV1 C L w 2z
·v = wingtip vortex strength factor 0.z/ D 1¡ (2)
¼ R Aw b
3 = quarter-chord wing sweep angle
By the use of Eq. (2) in Eq. (1), the spanwise distribution of shed
Introduction vorticity for an elliptic wing is

T HE downwash induced on an aft tail by the main wing has a

signiŽ cant effect on the trim and static stability of an airplane.
This downwash decreases the effective angle of attack for the hor-
°t .z/ D
4V1 C L w
¼ R Aw
p
.2z =b/
1 ¡ .2z =b/2
(3)
izontal tail and reduces its effectiveness in stabilizing the airplane.
The downwash varies along the span of the horizontal tail and is When this vorticity distribution is used and the rollup of the shed
affected by the planform shape of the wing, as well as the presence vortex sheet ignored, it is readily shown that the downwash angle
"d along the x axis far behind an elliptic wing, reduces to
Received 2 November 2001; presented as Paper 2002-0830 at the AIAA
40th Aerospace Sciences Meeting and Exhibit, Reno, NV, 14– 17 January "d .x; 0; 0/ D 2C L w =¼ R Aw (4)
2002; revision received 1 April 2002; accepted for publication 4 April 2002. x!1
Copyright ° c 2002 by the authors. Published by the American Institute of
Aeronautics and Astronautics, Inc., with permission. Copies of this paper The downwash angle computed from Eq. (4) has long been recom-
may be made for personal or internal use, on condition that the copier pay mended as a Ž rst approximation for predicting the downwash on an
the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rose- aft tail (See, for example, Perkins and Hage3 or Nelson4 ). This ap-
wood Drive, Danvers, MA 01923; include the code 0021-8669/02 $10.00 in proximation has three shortcomings. First of all, it does not account
correspondence with the CCC. for variations in the position of the tail relative to the main wing.
¤
Professor, Mechanical and Aerospace Engineering Department. Member In addition, it does not account for rollup of the shed vortex sheet.
AIAA.
† Assistant Professor, Mechanical and Aerospace Engineering Depart- Finally, it does not account for variations in the planform shape of
ment. Senior Member AIAA. the main wing.
‡ Graduate Student, Mechanical and Aerospace Engineering Department. The rollup of the shed vortex sheet affects the downwash on an aft
Member AIAA. tail because it affects the proximity of this vorticity to the tail. The
§
Research Assistant, Mechanical and Aerospace Engineering downwash angle computed from Eq. (4) is based on the assumption
Department. that the vortex Ž laments trailing downstream from the wing are all
600
 PHILLIPS ET AL. 601

approximate correction is also presented that can be used with some

caution to account for the effects of sweep in the main wing.

Downwash Aft of an Unswept Wing

The well-known inŽ nite series solution to Prandtl’s classical
lifting-line equation applies to a single Ž nite wing with no sweep
or dihedral, having an arbitrary spanwise variation in chord length.
This solution is based on the change of variables
z D ¡ 12 b cos.µ / (5)
The variation in section circulation along the span of the wing, as
predicted by this solution, is

2bV1 C L w X Bn
1
0.µ / D sin.nµ / (6)
¼ R Aw B1
Fig. 1 Prandtl’s model for the bound vorticity and the trailing vortex nD1
sheet generated by a Ž nite wing.
For a complete presentation of Prandtl’s lifting-line theory, see, for
example, Anderson,20 Katz and Plotkin,21 McCormick,22 or Bertin
and Smith.23
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Historically, the coefŽ cients in this inŽ nite series solution have
usually been evaluated from collocation methods. Typically, the se-
ries is truncated to a Ž nite number of terms, and the coefŽ cients in
the Ž nite series are evaluated by requiring the lifting-line equation
to be satisŽ ed at a number of spanwise locationsequal to the number
of terms in the series. A very straightforward method was Ž rst pre-
sented by Glauert.24 The most popular method, based on Gaussian
quadrature,was originally presented by Multhopp.25 Most recently,
Rasmussen and Smith26 have presented a more rigorous and rapidly
converging method, based on a Fourier series expansion similar to
that Ž rst used by Lotz27 and Karamcheti.28
By the use of Eqs. (5) and (6) in Eq. (1), the spanwise variation
of shed vorticity is

4V1 C L w X Bn n cos.nµ /
1
°t .µ / D ¡ (7)
¼ R Aw n D 1 B1 sin.µ /

The downwash distribution that is predicted directly from Eq. (7)

is not accurate in the region behind the wing. This is because the
development of Eq. (7) is based on the assumption that the vortex
Ž laments trailing downstream from the wing are all straight and
parallel to the freestream  ow, as was shown in Fig. 1. In reality, the
vorticity trailing from each side of the wing will roll up around an
axis trailing slightly inboard from the wingtip, as is shown schemat-
ically for an elliptic wing in Fig. 2.
Fig. 2 Schematic of the vorticity rollup behind a Ž nite wing with The rollup of the vortex sheet trailing behind each semispan of
elliptic planform shape. the wing can be viewed as a result of the vortex lifting law (see
Saffman29 ). This vortex lifting law requires that, in any potential
 ow containing vortex Ž laments, a force is exerted on the surround-
straight and parallel to the freestream  ow, as shown in Fig. 1. In ings that is proportionalto the cross product of the local  uid veloc-
reality, the vorticity trailing from each side of the wing will roll ity with the local Ž lament vorticity. Because a free vortex Ž lament
up around an axis trailing somewhat inboard from the wingtip, as cannot support a force, the cross product of the local  uid velocity
shown in Fig. 2. Within two to three chord lengths behind the wing, with the local Ž lament vorticity must be zero at every point along a
the vortex sheet becomes completely rolled up to form the wingtip free vortex Ž lament. This means that all free vortex Ž laments must
vortices.5¡8 everywhere follow the streamlines of the  ow. Thus, the free vortex
Early experimental and theoretical investigations of wing Ž laments trailing behind each semispan of the wing will follow the
downwash9¡14 eventually led to the development of a widely used streamlines and rollup about the center of vorticity shed from that
empirical relation15 for the downwash induced on an aft tail by a semispan. Within a few chord lengths behind the wing, the vortex
tapered wing. The result accounts for the effects of tail position, as sheet becomes completely rolled up to form the wingtip vortices.
well as variations in both aspect ratio and taper ratio of the main This rollup has a signiŽ cant effect on the downwash.
wing. This relation is still frequently used for preliminary calcu- Each wingtip vortex is generated from the trailing vortex sheet
lations (for example, Roskam,16 Etkin and Reid,17 or Pamadi18 ). produced by one-half of the wing. Therefore, a wingtip vortex, a
Recently, McCormick19 proposed an analytical vortex model that few chord lengths or more behind the wing, can be approximated
accounts for the rollup of shed vorticity, as well as variations in the as a single vortex of strength 0wt which is given by
position of the tail relative to the main wing. However, this model Z b =2
is based on an elliptic main wing and does not account for other 0wt D °t .z/ dz (8)
spanwise variations in chord length. z D0
The present paper presents an analyticalmodel for the downwash Using Eqs. (5) and (7) in Eq. (8) gives
induced on an aft tail by a main wing having arbitrary spanwise
Z ¼
variation in chord length. This model is based on the inŽ nite series 2bV1 C Lw X Bn
1

solution to Prandtl’s classical lifting-line theory and, thus, can be 0wt D ¡ n cos.nµ / dµ (9)
¼ R Aw B 1 µ D ¼ =2
directly applied only to a main wing with no sweep or dihedral. An nD1
 602 PHILLIPS ET AL.

When Eqs. (5), (7), and (10) are applied, this can be rewritten as
( Z ," ³ ´#)
X1 ¼ X
1
n¼
0
b Db n Bn cos.nµ / cos.µ / dµ Bn sin
nD1 µ D ¼ =2 nD1
2
(14)
The integration with respect to µ in Eq. (14) is readily carried out
to give
8
> ¼
Z ¼
>
< ; nD1
4
cos.nµ / cos.µ / dµ D (15)
>
> cos.n¼ =2/
µ D ¼ =2
: 2 ; n 6D 1
.n ¡ 1/
Using Eq. (15) in Eq. (14) results in
(" ³ ´#," ³ ´#)
Fig. 3 Vortex model used for estimating the downwash a few chord
¼ X X
1 1
lengths or more aft of an unswept wing. 0 n Bn n¼ Bn n¼
b Db C cos 1C sin
4 .n 2 ¡ 1/B1 2 B1 2
nD2 nD2
Performing the indicated integration, we have
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³ ´ (16)
2bV1 C L w X Bn
1
n¼
0wt D sin (10) Because the downwash is small compared to the freestream ve-
¼ R Aw n D 1 B1 2 locity, the downwash angle can be closely approximated as the
downwash velocity divided by the freestream velocity. Thus, when
When computing the downwash a few chord lengths or more
Eqs. (10) and (16) are applied to Eq. (11), the downwash angle a
downstream from a Ž nite wing, we can approximate the rolled-up
few chord lengths or more downstream from an unswept wing is
vortex sheet as a single horseshoe-shapedvortexŽ lament of strength
approximated as
0wt , as shown in Fig. 3. The distance between the trailing vortices
b 0 is less than the wingspan because the vortex sheet from each N yN ; zN / »
"d . x; D ¡Vy . x;
N y;
N zN /= V1 D .·v · p =·b /.C L w = R Aw / (17)
side of the wing rolls up around the center of vorticity, which is
somewhat inboard from the wingtip. The horseshoe Ž lament starts where
an inŽ nite distance downstream from a point slightly inboard of the X
1 ³ ´
left wingtip, .1; 0; b 0 =2/, and runs upstream along the left wingtip Bn n¼
·v D 1 C sin (18)
vortex to the left wing, .0; 0; b0 =2/. From there it runs across the B1 2
nD2
wing quarter-chord to a point slightly inboard of the right wingtip, ("
.0; 0; ¡b0 =2/, and then downstream along the right wingtip vortex ³ ´#," ³ ´#)
¼ X X
N N
n Bn n¼ Bn n¼
to inŽ nity, .1; 0; ¡b0 =2/. From the Biot – Savart law, the y-velocity ·b D C cos 1C sin
component induced at any point (x, y, z) by this entire horseshoe 4 n D 2 .n 2 ¡ 1/B1 2 B1 2
n D2
vortex is (19)
V y .x; y; z/
8 2 3 · p . x;
N yN ; zN /
>
< 1 0
( " #
0wt 2
b ¡z 6 x 7 1 ·b .·b ¡ zN / xN
D ¡ ¡1 ¢2 4 1 C q ¡ ¢ 5 D 2 1C p
4¼ > 2
: y C 2 b0 ¡ z 2 ¼ yN 2 C .·b ¡ zN /2
x 2 C y 2 C 12 b0 ¡ z xN C yN C .·b ¡ zN /2
2 2

2 "
·b xN ·b ¡ zN
x 6
1 0
b ¡z C 2 p
C 2 4q 2 xN C yN 2 xN 2 C yN 2 C .·b ¡ zN /2
x C y2 ¡1 ¢2
x2 C y2 C 2
b0 ¡z #
3 ·b C zN
Cp
1 0
b Cz 7 xN 2 C yN 2 C .·b C zN /2
2
C q ¡1 ¢2 5 " #)
x 2 C y2 C 2
b0 C z ·b .·b C zN / xN
C 2 1C p (20)
2 39 yN C .·b C zN /2 xN C yN C .·b C zN /2
2 2

1 0
>
=
2
b Cz 6 x 7 The dimensionless parameters ·v and ·b depend on the planform
C ¡ ¢2 4 1 C q ¡ ¢2 > (11)
5 shape of the wing. For an elliptic wing, all of the coefŽ cients Bn in
y 2 C 12 b0 C z x 2 C y 2 C 1 b0 C z ;
2 the inŽ nite series solution, except for the Ž rst, are zero. Using this
with Eq. (18), we Ž nd that ·v is 1.0 for an elliptic wing. Thus, from
Because the vortex sheet shed from each semispan of the wing rolls Eqs. (10) and (18), we see that the vortex strength factor ·v is the
up about the center of vorticity, we have ratio of the wingtip vortex strength to that generated by an elliptic
R b=2 wing having the same lift coefŽ cient and aspect ratio. The vortex
1 0 z°t .z/ dz
b D Rz Db=02 (12) span factor ·b is deŽ ned as the spacing between the wingtip vortices
2 °t .z/ dz divided by the wingspan. Both ·v and ·b were determined analyti-
zD0
cally from the series solution to Prandtl’s lifting-line equation. For
Using Eq. (8) in Eq. (12) gives an elliptic wing with no sweep, dihedral, or twist, ·v is 1.0 and ·b
Z b =2 is ¼ =4. For an unswept tapered wing with no dihedral or twist, ·v
2
b0 D z°t .z/ dz (13) and ·b are related to the aspect ratio and taper ratio as is shown in
0wt zD0 Figs. 4 and 5.
 PHILLIPS ET AL. 603

Fig. 4 Wingtip vortex strength factor as predicted from the series

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solution to Prandtl’s lifting-line theory. Fig. 6 Effect of tail position on the downwash angle in the plane of
symmetry aft of an unswept wing.

Fig. 7 Vortex model used for estimating the downwash a few chord
lengths or more aft of a swept wing.
Fig. 5 Wingtip vortex span factor as predicted from the series solution
to Prandtl’s lifting-line theory.
aft tail. Similar results were observed empirically by Hoak,15 but
The dimensionless parameter · p is a position factor that accounts are not accounted for in the model proposed by McCormick.19
for spatial variations in downwash. As a Ž rst approximation, the
Approximate Correction for Swept Wings
variation in downwash along the span of the horizontaltail is usually
neglected. The downwash for the entire tail is typically taken to be Sweep in the main wing also has a signiŽ cant effect on the down-
that evaluated at the aerodynamic center. For a symmetric airplane, wash induced on an aft tail. Sweep affects this downwash in three
the aerodynamic center of the tail is in the plane of symmetry. The ways. Because sweep changes the spanwise vorticity distributionon
change in the downwash with respect to the spanwise coordinate the wing, it changes the strength of the wingtip vortices for a given
is zero at the aircraft plane of symmetry. Furthermore, the span of lift coefŽ cient and aspect ratio. This same changein the vorticitydis-
the horizontal tail is usually small compared to that of the wing. tribution will also change the location of the center of vorticityin the
Thus, the downwash is often fairly uniform over this span, and a vortex sheet shed from each semispan. Because each wingtip vortex
reasonable Ž rst approximation for the downwash on an aft tail is rolls up around the center of vorticity from one side of the wing,
found by setting the dimensionless spanwise coordinate zN equal to wing sweep affects the spacing of the wingtip vortices. Because
zero in Eq. (20). This gives the relatively simple relation wing sweep affects both the strength and spacing of the wingtip
" # vortices, sweep in the main wing will affect both ·v and ·b . More
¡ ¢ signiŽ cantly, sweep in the main wing affects the downwash on an
2·b2 xN xN 2 C 2 yN 2 C ·b2
· p . x;
N yN ; 0/ D ¡ ¢ 1C p aft tail through a simple change in proximity of the wing surface to
¼ 2 yN 2 C ·b2 . xN 2 C yN 2 / xN 2 C yN 2 C ·b2 the tail. As the wing is swept back, the outboardportions of the wing
(21) are brought closer to the tail, as is shown in Fig. 7. This places the
bound portion of the vortex system closer to the aft tail and, thus,
The tail position factor · p depends on the planform shape of the changes the downwash induced on the tail.
wing and the position of the tail relative to the wing. The variation Unfortunately,the series solution to Prandtl’s classicallifting-line
of · p with tail position in the plane of symmetry is shown in Fig. 6. equation does not apply to a swept wing. No closed-form solution
The planform shape of the wing affects the value of · p only through for the spanwise vorticity distribution on a swept wing has ever
its effect on ·b . Thus, for a main wing with no sweep or dihedral,the been obtained. In the absence of such a solution for this vorticity
value of · p in the plane of symmetry is a unique function of xN =·b distribution,it is not possible to obtain a closed-formsolution for the
and yN =·b , as shown in Fig. 6. variation of ·v and ·b with sweep. Neglecting the effects of sweep
Notice from Figs. 4 and 5 that the planform shape of the main on ·v and ·b is quite restrictive, and such results should be used
wing has a very signiŽ cant effect on the downwash induced on an with extreme caution for highly swept wings. Nevertheless, if we
 604 PHILLIPS ET AL.

are willing to neglect the effect of sweep on ·v and ·b , it is possible

to obtain a closed-form approximation for the proximity effect that
results from moving the bound vortex closer to the aft tail when the
wing is swept back.
This approximation is based on the vortex model suggested by
McCormick,19 which is shown here in Fig. 7. With this model, the
boundvorticityis approximatedas two straightvortexline segments,
one aligned with the quarter-chord of each semispan. Each wingtip
vortexis modeled as a semi-inŽ nite line vortextrailingfrom the wing
at the center of shed vorticity. In reality, the direction of the trailing
wingtip vortices is determined by the geometry of the airplane, the
angle of attack, and the downward de ection of the vortex system,
which is causedby the downwash that is inducedon one vortexby the
other. However, this level of sophistication is hardly justiŽ ed when
one considers the approximate nature of the other aspects of this
vortex model. Instead, we shall simply assume that wingtip vortices
trail downstream from the wing quarter-chord at the center of shed
vorticity in a direction parallel to the x axis. In the development of
Prandtl’s classical theory, this same approximation was used.
With these approximations, the vorticity generated by a lifting
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swept wing is modeled as four straight vortex line segments, all of

which fall in the x – z plane as shown in Fig. 7. By the application
of the Biot– Savart law to this vortex system, in a manner similar
to that used to obtain Eq. (11), the downwash in the aircraft plane
of symmetry can be evaluated. If we continue with the assumption
that the downwash is small compared to the freestream velocity, the
result can be rearrangedsuch that the downwash angle in the aircraft
plane of symmetry a few chord lengths or more downstream from
a swept wing is approximated as

¡Vy . x;
N y;
N 0/ ·v · p ·s C L w
N yN ; 0/ »
"d . x; D D (22)
V1 ·b R Aw

where
" ¡ ¢#," ¡ ¢#
N r C tN/ tN02 ¡ xN 2
xN ¡ sN x.N xN rN 2 C tN02 ¡ xN 2
·s D 1 C C 1C
tN rN tN.rN tN C rN 2 ¡ xN sN / rN 2 tN0 Fig. 8 Effect of wing sweep on the downwash in the plane of symmetry
aft of the main wing.
(23)
p
rN ´ xN 2 C yN 2 (24) the 1.83-m-diam variable pitch propeller. Maximum speed capabil-
ity of the tunnel is 50 m/s with a correspondingturbulence intensity
sN ´ ·b tan 3 (25) of less that 0.5%. Experiments were performed at a tunnel velocity
p of 25 m/s corresponding to a chord Reynolds number of 1:7 £ 105 .
tN ´ . xN ¡ sN /2 C yN 2 C ·b2 (26) A rectangularplanform NACA 0015 wing with rounded end caps
was used in this study. The wing chord of 12.7 cm and span of
p 66.0 cm gave an aspect ratio of 5.2. The relative positioning ac-
tN0 ´ xN 2 C yN 2 C ·b2 (27)
curacy of wing incidence was §0.1 deg. Note that the accuracy
of the absolute incidence angle is not reported because the results
The wing sweep factor ·s depends on the planform shape of the are presented in terms of change in downwash angle with respect
wing and the position of the tail in addition to the quarter-chord to changes in wing incidence, which is independent of the abso-
sweep angle 3. However, as was the case for · p , the planform shape lute incidence angle. The wing was mounted at the vertical center
of the wing affects the value of ·s as predicted by Eq. (23) only of the test section on two vertical struts extending from a splitter
through its effect on ·b . Thus, in the aircraft plane of symmetry, ·s plate located 10 cm above the tunnel  oor. The struts were attached
is found to be a unique function of 3, xN =·b , and yN =·b . The variation to the wing at quarter-chord positions with a symmetric spanwise
in ·s with axial tail position is shown in Fig. 8, for several values separation of 18 cm.
of wing quarter-chord sweep and tail height. The results shown in Downwash angle and velocity measurements were obtained us-
Fig. 8, for the case y D 0, agree exactly with the results presented by ing a TSI 1240-20 x-type hot-Ž lm probe and a TSI IFA300 constant
McCormick19 for the special case of an elliptic planform shape, that temperature anemometer system. The mean velocity results pre-
is, ·b D ¼ =4. However, McCormick states that the sweep correction sented in this study are based on an average of 214 samples per data
“does not depend signiŽ cantly on the tail height,” and he suggests point acquired at a sample rate of 2000 Hz for a total sample pe-
that the zero height solution may be used in general. Figure 8 does riod of 8.192 s. The sample period was deŽ ned by the minimum
not support that statement. period beyond which the statistical quantities remain constant and
repeatable. Calibration of the hot-wire probes was accomplished
Experimental Procedure and Uncertainty with a TSI Model 1129 automatic air velocity calibrator using 11
As part of the process of validating the analytical solution pre- points to cover a velocity range of 0 – 30 m/s. The mean standard
sented here, experiments were conducted in the low-speed wind errors of the velocity calibration are as follows: less than 2% for
tunnel at Utah State University’s Aerodynamics Research Labora- velocities between 0 and 1 m/s, less than 1% for velocities between
tory. The tunnel is an in-draft type with a 1:2 £ 1:2 m test section 1 and 8 m/s, and less than 0.5% for velocities greater than 8 m/s.
and an inlet contraction ratio of 9:1. A 200-hp three-phase electric The probe was calibrated over a pitch angle range of §30 deg using
motor with a computer controlled variable frequency drive rotates 5-deg increments at a velocity of 23.0 m/s. Several recalibrations
 PHILLIPS ET AL. 605

were required to maintain the accuracy of the velocity results and

they occurred at least within 24 h of any test results reported in
this study. The freestream dynamic pressure was determined from a
pitot-static probe located upstream of the model. Dynamic pressure
was converted to tunnel velocity using real-time ambient pressure
and temperature measurements.
The streamwise origin of the coordinate system is located at the
aerodynamic center of the wing section, whereas the origin of the
transverse axis is deŽ ned by the location of the trailing vortex at
the streamwise measurement location. Hence, the transverse origin
is a functionof wing incidence.The transverseorigin at each stream-
wise measurement location was determined by x-sensor traverses
near the spanwise positions of the wingtips.
Downwash velocity components were obtained using traverses
from 2y =b D 0:00 to 0.46 with an increment of 0.019 at three span-
wise locations, 2z =b D ¡0:077, 0.0, and 0.077. The reported results
represent a spatial average over the spanwise measurement range.
The area based solid blockage of the wind tunnel, including support
struts, varied from 2.5 to 3.1% over the tested angle-of-attackrange
of 2.0– 8.0 deg. Because the solid blockage was small and varied lit-
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tle over the angle-of-attack range covered, the reported downwash

Fig. 10 Downwash in the plane of symmetry aft of a rectangular wing
results were not corrected for tunnel wall interference, and tunnel of aspect ratio 5.2, at xÅ = 0.75.
upwash corrections were not applied. This is justiŽ ed because the
results are presentedin terms of a relativechangein downwash angle
with respect to change in wing incidence at a Ž xed measurement lo-
cation. Because there was very little variationin solid blockageover
the angle-of-attackrange covered, any upwash due to imperfections
in the tunnel walls should not change signiŽ cantly with wing angle
of attack. Thus, any  ow angularity caused by the tunnel walls was
assumed to be independentof wing incidence.Although such tunnel
upwash has a signiŽ cant effect on the magnitude of the downwash
angle measured aft of the wing, it should not signiŽ cantly affect the
measured downwash gradient,which is determined from the change
in downwash angle measured at different angles of attack.

Results
To validate the spatial variation in downwash predicted by the
proposed analytical solution, results obtained from this model were
compared with the experimentalwind-tunneldata. The solution was
also compared with the empirical correlation of Hoak15 and the
analytical method proposed by McCormick.19 These comparisons
are shown in Figs. 9 – 13 for the unswept rectangular wing that was
described in the preceding section. The data collected in the present
study were restricted to a single wing of aspect ratio 5.2 and taper
ratio 1.0. Although this allows us to examine the accuracy of the
proposed method for predicting the spatial variation in downwash, Fig. 11 Downwash in the plane of symmetry aft of a rectangular wing
little can be inferred from these data about the ability of the model of aspect ratio 5.2, at xÅ = 1.0.
to predict the effects of wing planform.

Fig. 9 Downwash in the plane of symmetry aft of a rectangular wing Fig. 12 Downwash in the plane of symmetry aft of a rectangular wing
of aspect ratio 5.2, at xÅ = 0.5. of aspect ratio 5.2, at xÅ = 1.5.
 606 PHILLIPS ET AL.
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Fig. 15 Effects of aspect ratio on the downwash in the plane of sym-

Fig. 13 Downwash in the plane of symmetry aft of a rectangular wing
metry aft of a rectangular wing.
of aspect ratio 5.2, at xÅ = 2.0.

Fig. 16 Effects of sweep on the downwash in the plane of symmetry

Fig. 14 Effects of taper ratio on the downwash in the plane of symme- aft of a wing with no taper.
try aft of a tapered wing.
as shown in Fig. 3. This model should not be expected to give accu-
To examine how well the present model predicts the effects of rate results for locations very close to the wing, and it is tempting
wing planform shape, the proposed solution was compared with the to blame the errors that occur close to the wing on our failure to
empirical correlation of Hoak.15 Because the empirical correlation account for the transition region of partial vortex rollup. However,
was taken from experimental data, it should be reasonably accurate, as will be demonstrated, this is not the case.
at least over the range of parameters for which the correlation was To examine how vortex rollup affects the downwash angle aft of
obtained. Figure 14 shows a comparison between the empirical cor- the wing, we can compare results computed from the present model
relation and the present analytical solution for taper ratios from 0.0 with those obtained from the classical lifting-line model, which
to 1.0. This comparison is based on the downwash one semispan neglects all vortex rollup, as shown in Fig. 1. From the classical
directly aft of the center of a wing with aspect ratio 6.0. The result lifting-line solution, the downwash angle induced by the unrolled
predicted by the analytical method of McCormick19 is also shown vortex sheet at an arbitrary point in space is given by
in Fig. 14. Similar comparisons showing the effect of aspect ratio Z ¼ »
C L w X Bn
N
and sweep are presented in Figs. 15 and 16. n cos.nµ /[Nz C cos.µ /]
"d . x;
N y;
N zN / D
In the region very close behind the wing, both the present analyt- R Aw ¼ B1 µ D 0
2 yN 2 C [Nz C cos.µ /]2
nD1
ical solution and the empirical correlation15 begin to deviate from
the experimentaldata obtained in the current study. Because the em- n xN cos.nµ /[Nz C cos.µ /]
pirical correlation was developed for prediction of the downwash C 1
gradient on an aft tail, it is not likely that any attempt was made to f yN 2 C [Nz C cos.µ /]2 gf xN 2 C yN 2 C [Nz C cos.µ /]2 g 2
correlate data that were taken very close behind the wing. The devi- ¼
ation between the experimental data taken close to the wing and the xN sin.nµ / sin.µ /
present analyticalmodel is likely a result of the approximate manner C 3 dµ (28)
f xN 2 C yN 2 C [Nz C cos.µ /]2 g 2
in which lifting-line theory handles the bound vorticity. The present
analytical model neglects the streamwise distance required for the The integration required to evaluate this expression for an arbitrary
shed vorticity to roll up and form the wingtip vortices. The model point in space is quite complex and will not be addressed here.
assumes a single wingtip vortex shed from the wing quarter-chord However, for large distancesdirectlydownstreamfrom the spanwise
 PHILLIPS ET AL. 607

midpoint of the wing, integration of Eq. (28) yields a very simple thermore, agreement between the empirical correlation15 and the
result for the far-Ž eld downwash with no vortex rollup, proposed analytical model appears to be reasonably good for sweep
" ³ ´# angles less than 30 deg.
2 X n Bn N
n¼ C Lw
"d .1; 0; 0/ D 1C sin (29) Conclusions
¼ B1 2 R Aw
n D2 From the results presented here, it can be seen that the proposed
analytical model agrees very closely with both the empirical cor-
From Eqs. (17) and (20), the far-Ž eld downwash including vortex
relation of Hoak and with the present experimental data, over the
rollup is given by
range of spatial coordinates where an aft tail might typically be lo-
¡ ¯ ¢ cated. This should give some conŽ dence in the model, at least for
"d .1; 0; 0/ D 4·v ¼ 2 ·b .C L w = R Aw / (30)
the prediction of downwash on an aft tail.
For the special case of an elliptic wing, Eq. (29) reduces Because the analytical model proposed here agrees very closely
to Eq. (4) and the far-Ž eld downwash with no vortex rollup with the empirical correlation of Hoak, a question may naturally
would be "d .1; 0; 0/R Aw =C L w D 2=¼ . From Eq. (30), the far- arise in the mind of the reader. That is, what advantage does the
Ž eld downwash for an elliptic wing including vortex rollup present analytical model provide? The answer is quite simple. The
is "d .1; 0; 0/R Aw = C L w D16=¼ 3 . Similarly, for a rectangular analytical model proposed by McCormick is valid only for an el-
wing of aspect ratio 6.0, neglecting vortex rollup gives liptic wing and the empirical correlation of Hoak applies only to
"d .1; 0; 0/R Aw =C L w D 0:464, and including vortex rollup yields a tapered wing with no geometric or aerodynamic twist. The ana-
"d .1; 0; 0/R Aw =C L w D 0:417. For a tapered wing of aspectratio 6.0 lytical model proposed here, on the other hand, can be used for a
and taper ratio 0.5, Eq. (29) results in "d .1; 0; 0/R Aw = C L w D 0:743 wing with completely arbitrary spanwise variations in chord length,
Downloaded by UNIVERSITY OF MICHIGAN on February 3, 2015 | http://arc.aiaa.org | DOI: 10.2514/2.2998

and Eq. (30) predicts "d .1; 0; 0/R Aw =C Lw D 0:523. Thus, we see section geometric angle of attack, and section zero-lift angle of at-
that complete vortex rollup reduces the far-Ž eld downwash by tack. For example, the present model can be used to predict the
10– 30%, dependingon the planformshape of the wing. In the region effects of double taper, washout, and other spanwise variations in
close behind the wing, we would expect the partial vortex rollup to the wing section properties.Furthermore,the computationsrequired
reduce the downwash somewhat, but the reduction should be less for this model are simple enough to be carried out on a modern pro-
than that predicted under the assumption of complete vortex rollup. grammable calculator.
As a result, the rollup approximation used in the present analytical
model tends to underpredict the downwash induced by the trailing References
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in the region close behind the wing. Thus, the effects of these two NACA TR-116, June 1921.
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both the empirical correlation of Hoak15 and the present experi- mance Stability and Control, Wiley, New York, 1949, pp. 219– 223.
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wing. For example, the comparison shown in Fig. 10 corresponds matic Control, 2nd ed., McGraw – Hill, New York, 1998, pp. 47 – 52.
5 McCormick, B. W., Tangler, T. L., and Sherrieb, H. E., “Structure of
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6 Green, S. I., and Acosta, A. J., “Unsteady Flow in Trailing Vortices,”
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it is not likely that the empirical correlation15 is based on data that Diehl, W. S., “The Determination of Downwash,” NACA TN-42, Jan.
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10 Munk, M. M., and Cairo, G., “Downwash of Airplane Wings,” NACA
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14 Diederich, F. W., “Charts and Tables for use in Calculations of Down-
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low-speed aircraft. Because the analytical model of McCormick19 15 Hoak, D. E., “USAF Stability and Control Datcom,” U.S. Air Force
is based on an elliptic wing planform, Fig. 14 shows no variation in Wright Aeronautical Labs., AFWAL-TR-83-3048, Wright– Patterson AFB,
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17 Etkin, B., and Reid, L. D., “Downwash,” Dynamics of Flight: Stability
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