17 May 2011
Aerodynamic and Artistic Study of the German Jets
J. Philip Barnes
Pelican Aero Group
For more information, visit www.HowFliesTheAlbatross.com
INTRODUCTION
In April 1937, the world’s first turbojets independently
made their first runs in Britain and Germany (1). Just
four days before the outbreak of WWII, the world’s first
jet aircraft took flight (2). By the last year of the war, the
Germans were perhaps five years ahead of the allies in
aerodynamic technology. At the time of this article, some
65-years on, the conceptual and real designs of the early
“German Jets” continue capturing our imagination, as
evidenced by widespread literature and artwork thereof.
Our presentation, after reviewing some fundamental
aerodynamic principles, introduces a computationallyefficient method for aerodynamic forces and moments.
We then apply the method to study the top-level
aerodynamics of seven German jets, each noted for its
unique configuration. Also, renowned computer-graphic
artists Mario Merino and Gery Gueville share some of
their work herein copyright-free. Thus our title:
“Aerodynamic and Artistic Study of the German Jets.”
Messerschmitt 262 and Jumo 004
The rocket-powered Me 163 was test flown beyond its
critical speed by pilot Heini Dittmar, to Mach 0.84 where
it exhibited instability about all three axes, in part
because the forebody, in effect a low-aspect-ratio wing,
develops lift and side force coefficients which are
unaffected by Mach number (per Robert T. Jones),
whereas a fin with a relatively “thick” airfoil suffers a
reduction of lift coefficient (or side force coefficient)
above its critical transonic Mach number. In addition, just
as the wing is immersed in downwash due to forebody
lift, the fin is immersed in sidewash due to forebody side
forces. For both surfaces, such “downwash” degrades
aerodynamic effectiveness. Although the fin has higher
aspect ratio than the forebody and enjoys an “end-plate”
benefit in its attachment to the afterbody, the forebody
and fin of the Me 163 are in close competition with
nearly equal and opposite effects on yaw stability.
Although an isolated planar wing is marginally stable in
yaw, Albert Betz showed that such depends on lift. For
twisted wings, we interpret this to mean “tip-local” lift. But
at high speed, both lift coefficient and isolated-wing yaw
stability vanish. Thus misbehaved the Me 163.
Blohm und Voss P.209
THE FIRST JETS
Although the Allied aircraft Gloster Meteor entered
operational service just days before the Me 262, the Me
262 was the world’s first jet aircraft to be used in aerial
combat (3). Fortunately for the Allies, the contribution of
the Me 262 to the German war effort was delayed by
Hitler’s insistence on its conversion into a fighter-bomber
to hold off the coming Allied invasion. Nevertheless, the
Me 262 took its toll on Allied bombers in the last months
of the war. Its wing was swept to manage the center of
gravity (c.g.). Not taking advantage of available German
high-speed aerodynamic research, the Me 262 owed its
maximum speed of 870 km/hr to the Jumo 004 engine.
German Jets J. Philip Barnes www.HowFliesTheAlbatross.com
Lloyd S. Jones
Messerschmitt 163B
1
SOME OF THE FOUNDING FATHERS
Our aerodynamic study takes strong advantage of the
pioneering work of many of the “Founding Fathers” of
aerodynamics. With the horseshoe vortex as modeling
clay, we’ll apply Ludwig Prandtl’s lifting line and induced
drag concepts and shift the companion downwash line
with the compressibility rule independently developed by
Ludwig Prandtl and Hermann Glauert, including the
effects of wing sweep. Although supersonic sweep was
first theorized by Adolf Busemann, and later
independently by Robert T. Jones, Albert Betz was first
to suggest and test the benefits of subsonic sweep to
delay transonic effects (4,5). Finally, to include forebody
effects, we integrate Max Munk’s theory of airships with
Robert T. Jones’ theory of low-aspect-ratio wings.
THE VERSATILE HORSESHOE VORTEX
Our “Wing-body Linear Longitudinal-Lateral Lifting Line”
(WBL5) method gives the lift (normal force) distribution,
as well as profile drag and induced drag over all
significant aerodynamic surfaces including the forebody
planform, forebody profile, wing, and empennage. The
analysis, limited to subsonic, linear aerodynamics below
critical Mach, aligns “horseshoe vortices” along lifting
lines, nominally at ¼-chord, with empirical modifications.
A corresponding downwash line, nominally at ¾-chord,
but shifted aft via the Prandtl-Glauert compressibility
“rule,” connects the points where the equivalent flat-plate
(EFP) airfoil flow-tangency boundary condition is
applied. The local EFP incidence is determined from the
cross products of three vectors representing the chord,
flight velocity, and local dihedral.
Cambered airfoil
Chord line, Zero-lift line
ζ
α
Ludwig Prandtl
Max Munk
Albert Betz
Aero equivalent
Thin, flat airfoil
α+ζ
Adolf Busemann
Hermann Glauert
Robert T. Jones
© Godfrey Argent Studio
Rather than adjust aircraft and vortex geometry for angle
of attack and sideslip, instead the flight velocity vector is
tilted by these angles, with only selected adjustments of
wing coordinates applied to account for differential
sweep and downwash-node position in sideslip.
Simultaneous equations, typically less than 100 per
aircraft, are then solved to yield the distribution of
horsehoe-vortex strength, from which local normal forces
can be computed. The “apparent downwash” method
then determines the local induced drag by comparing the
observed local lift to that expected based on local
incidence and local sweep, together with Mach number.
CONFIGURATION AERODYNAMICS
With the advent of the jet engine, the aircraft forebody
stretched forward as the wings swept aft. These
geometric changes decreased both the pitch and yaw
stability of the aircraft. Accordingly, our method of
aerodynamic analysis includes the forebody as a
significant longitudinal and lateral “lifting surface.” In
keeping with the basic method, it will be modeled as a
“cruciform” consisting of two low-aspect-ratio wings. The
forebody planform, wing and empennage together set
the “aerodynamic center” (a.c.) of the aircraft, where the
total lift is concentrated for the purposes of analysis. The
German Jets would have been statically stable, with the
center of gravity (c.g.) residing forward of the a.c. and
with a nose-up pitching moment balancing the offset of
the lift and weight vectors. Such pitch-up moment can be
provided by any combination of airfoil reflex, wing sweep
with twist, and “decalage” of a canard or horizontal
stabilizer, the latter designated herein as a “tail,” with
vertical stabilizing surfaces designated herein as “fins.”
We postulate that the well-known “Lifting-line” methods
cannot obtain aerodynamic loads in sideslip without
empirical modeling thereof. Thus, in the appendix we
characterize NACA-measured yaw and roll moments for
planar wings of various sweep, aspect ratio, and taper
ratio. We then add or subtract “bumps” to or from the
lifting-line-computed lift and drag distributions to yield the
overall yaw or roll moment. Such modeling is founded on
Albert Betz’ observation that both yaw and roll moments
of an isolated wing arise from differences in local
induced drag, whereby such moments vanish as planar
(untwisted) wing lift coefficient (cL) becomes small, or in
our interpretation for twisted wings, as “tip-local” lift
vanishes. Thus, an aircraft configured largely as an
isolated wing will go either marginally stable, or unstable,
in yaw at “high” speeds approached by the German jets.
Lift @ vehicle ac
Weight
cMac
Vehicle Aerodynamic Center
German Jets J. Philip Barnes www.HowFliesTheAlbatross.com
2
TOP-LEVEL VALIDATION OF METHOD
A thorough validation of the WBL5 method would involve
a wide range of configurations and conditions far
exceeding the scope of this article. Also, at this time the
method is still in development with validation on-going.
Nevertheless, in this section we validate the method for
selected cases applicable to the German jets to be
studied subsequently.
of attack, the lift distribution remains largely positive and
“half-sinusoidal,” with only a small region of negative lift
at the tips.
Chord-weighted Normal-force Coef.
0.40
cN C/Cav
0.30
0.20
0.10
0.00
p/h
-0.10
-1
-1.25
First and foremost is the distribution lift for a wing and
body combination. NASA TN D-712 represents a rare,
but immensely useful, characterization of wing-body
aerodynamic loads by adding pressure taps to the
portion of the body coinciding with the “submerged”
portion of the wing. As seen in the figure below, where
only the forebody has been retained for modeling
purposes, the wing lift distribution exhibits a pronounced
“dip” which is essentially replaced by forebody lift,
whereby the total wing-body lift is nearly elliptical. Here,
the forebody wake has induced downwash on the wing
to an extent somewhat greater than that calculated by
the method. Nevertheless, computed loads reasonably
match the measurements, even though 0.9 flight Mach
number is outside the applicability of the Prandtl-Glauert
correction for compressibility used by the method.
-0.75 -0.5 -0.25
0
0.25
0.5
0.75
1
1.25
1
1.25
Plan View
Looking Fwd
h
p
Looking Starboard
10
Flat-plate Incidence, deg
Chord-weighted Drag Coefficient
0.030
cD C/Cav
0.020
5
0.010
0
p/h
-5
-1.25 -1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00 1.25
p/h
0.000
-1.25
-1
-0.75 -0.5 -0.25
0
0.25
0.5
0.75
Method Validation ~ Wing-body lift with body pressure taps
Chord-weighted Normal-force Coef.
0.40
0.30
cN C/Cav
Our third and final top-level validation of the method
applies to a swept wing with dihedral in sideslip. As seen
in the figure below, the computed lift distribution
reasonably matches the test data. Here, the model has
applied an empirical differential lift “bump” to yield the
correlated yaw and roll moments of the appendix.
Data
Calc.
0.20
0.10
p/h
0.00
-1.25
-1
-0.75 -0.5 -0.25
0
0.25
0.5
0.75
1
1.25
Plan View
Chord-weighted Normal-force Coef.
0.75
Looking Fwd
h
cN C/Cav
Calc.
0.50
Data
p
0.25
p/h
0.00
-1.25
Looking Starboard
-1
-0.75 -0.5 -0.25
0
0.25
0.5
0.75
1
1.25
1
1.25
Plan View
5.0
Flat-plate Incidence, deg
Chord-weighted Drag Coefficient
0.030
4.0
0.020
3.0
2.0
1.0
Looking Fwd
cD C/Cav
h
p
0.010
p/h
0.0
-1.25 -1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00 1.25
0.000
-1.25
p/h
-1
-0.75 -0.5 -0.25
0
0.25
0.5
0.75
1
1.25
Looking Starboard
Method Validation ~ Wing-body lift with body pressure taps
We now progress from a planar wing to a cambered
o
wing with 10 twist, again tested with a body, but with
only “exposed wing” loads measured. Again, the
calculation agrees well with the test data. A key point of
interest is the outboard upwash, whereby with the wing
o
o
root at 3 angle of attack, and the wingtips at -7 angle
Flat-plate Incidence, deg
8.2
8.0
7.8
7.6
7.4
7.2
7.0
p/h
6.8
-1.25 -1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00 1.25
Chord-weighted Drag Coefficient
0.050
0.040
D
0.030
0.020
0.010
0.000
-0.010
-1.25 -1
c C/Cav
p/h
-0.75 -0.5 -0.25
0
0.25
0.5
0.75
Method Validation ~ Wing-body lift with body pressure taps
German Jets J. Philip Barnes www.HowFliesTheAlbatross.com
3
GERMAN JET One ~ Focke-Wulf Ta183
The FW Ta183, although radical for its day, is the
closest we will come herein to a “conventional”
configuration. Conceived by Hans Multhopp with
development well underway by Kurt Tank at F-W, the
Ta183 was noted by its swept, constant-chord wing. The
figures below describe the aerodynamics of the Ta183 at
a representative “cruise” condition of 0.6 flight Mach
number. Along the left and in the center, we show the
aircraft with the lifting and downwash lines on each
aerodynamic surface The influence of the afterbody can
be ignored, as it is immersed in both downwash and lowquality flow. Notice the cruciform models of the forebody
planform and profile as thin, low-aspect-ratio wings. At
top center, we show the chord-weighted distribution of lift
as a function of non-dimensional position “p/h,” where
“h” designates halfspan and “p” the “screen-projected”
distance along the spar. The central dip is caused by
downwash imposed on the wing by the forebody wake
and lift which, per the theory of Robert T. Jones, is
distributed elliptically over the forebody. We assume a
zero-pitching-moment airfoil (but cambered forward). It is
interesting to see that the tail load is slightly positive.
With the chosen c.g. position, the aircraft has 8% static
margin (normalized c.g-to-a.c. distance). At the right, we
show the distributions of lift (normal force) coefficient,
local upwash (negative), profile drag, and induced drag.
The overall distribution of chord-weighted drag is shown
at the bottom center. Although it may be difficult to
distinguish the various curves with the chosen format,
we note that the forebody planform drag is greater than
the forebody profile drag because without sideslip only
the forebody planform is lifting.
Focke-Wulf Ta183
Chord-weighted Normal-force Coef.
0.30
0.20
cN C/Cav
0.2
Wing
Forebody
0.10
Fin
-1
-0.75 -0.5 -0.25
0
Forebody
0.25
0.5
0.75
Tail
0
p/h
-0.10
-1.25
Wing
0.1
Tail
0.00
Section Normal Force, cn
0.3
1
1.25
Fin
-1.2
1
Plan View
-0.8
p
0
0.4
0.8
1.2
0.4
0.8
1.2
0.4
0.8
1.2
0.8
1.2
Fin
0
Wing
Tail
-2
Forebody
-3
c.g.
h
-0.4
Upwash (Normal to section), deg
-1
Looking Fwd
p/h
-0.1
p/h
-4
-1.2
p
-0.8
-0.4
0
Section Profile Drag, cDp
0.015
0.010
Looking Starboard
a.c.
0.005
0.000
-1.2
5
4
1
0
-1
0.025
Wing
3
2
Chord-weighted Drag Coefficient
Flat-plate Incidence, deg
0.020
Forebody
Tail
Fin
0.008
-1.25 -1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00 1.25
0
0.004
0.015
0.002
0.005
0
p/h
0.000
-1.25
-0.4
0.006
cD C/Cav
0.010
p/h
-0.8
Section Vortex Drag, cDv
-1
-0.75 -0.5 -0.25
0
0.25
0.5
0.75
p/h
-0.002
1
1.25
-1.2
-0.8
-0.4
0
Focke-Wulf Ta183, Cruise ~ Distribution of Lift and Drag Over the Forebody, Wing, and Empennage
German Jets J. Philip Barnes www.HowFliesTheAlbatross.com
4
0.4
GERMAN JET Two ~ Horten iX / Gotha 229
Next we visit an even more radical configuration, that of
the Horten-iX /Go-229. In the end, this configuration
proved too radical. Offered by the Hortens as a
contender for the last-ditch Volksjäger (People’s Fighter)
competition, the Ho-iX was a favorite of Air Marshal
Hermann Goering. He directed Gothaer Wagonfabrik
(Gotha) to carry out detailed design and construction.
Given the perseverance and skill of the Gotha engineers
facing major setbacks and a near-impossible deadline,
the Ho-iX is appropriately given the alternate designation
“Go-229.”
“Wing-only” statically-stable configurations, such as the
Ho-iX, typically suffer the need for ballast (550-lb, Ho-iX)
to shift the c.g. well forward from its inherent location, set
by structure and equipment, to the required location for
pitch stability. But of course in aerial combat every
pound counts and every round counts.
Contrary to widespread popular literature, the Ho-iX was
o
built with only 1.5 of chord-line twist, as indicated by the
Arthur Bentley drawings. With a reflexed airfoil at the
o
wing root, the local zero-lift-line incidence was 0.5
o
below the chord line, whereby the wing had just 1 of
“equivalent-flat-plate” twist, far short of that needed to
obtain a “bell-shaped” lift load distribution. As built, the
Ho-iX/Go-229 had a “pseudo-elliptical” lift distribution.
0.40
0.30
Chord-weighted Normal-force Coef.
cN C/Cav
0.20
0.10
p/h
0.00
-1.25
-1
-0.75 -0.5 -0.25
0
0.25
0.5
0.75
1
1.25
0.75
1
1.25
Plan View
c.g.
Looking Fwd
h
p
Ho-iX / Go-229
a.c.
Looking Starboard
Flat-plate Incidence, deg
5
0.060
4
0.040
3
2
Chord-weighted Drag Coefficient
cD C/Cav
0.020
1
0
-1.20
p/h
-0.80
-0.40
0.00
0.40
0.80
1.20
0.000
-1.25
p/h
-1
-0.75 -0.5 -0.25
0
0.25
0.5
Aerodynamic Loads in Cruise ~ Ho-iX / Go-229
If instead the aircraft had incorporated bell-shaped lift,
o
twist near 10 would be needed, as shown at lower left
of the figure on the next page. Such would develop a
strong pitchup moment, balanced by a forward c.g. shift
needing even more ballast. But, as noted earlier,
isolated-planar-wing lateral stability requires non-zero lift
near the tips, where the yaw-stabilizing forces originate
as left-right differences in induced drag. Therefore, since
bell-shaped lift would by definition “unload” the wingtips,
it appears that the Ho-iX would have been near-neutral
or unstable in yaw at the lift coefficient corresponding to
bell-shaped lift.
Ho-iX / Go-229 in Action
To stabilize the aircraft in yaw, the Hortens specified
drag rudders, in lieu of a swept vertical fin, which we
suggest would have offered superior handling and safety
with a net reduction of drag. The drag rudders, which
opened both top and bottom, alternately left and right,
could not balance the yaw moment of single-engine
flameout. This shortcoming proved fatal for the first test
pilot (Lt. Irwin Ziller) to encounter this condition.
German Jets J. Philip Barnes www.HowFliesTheAlbatross.com
With or without bell-shaped lift loading for the Ho-iX, the
consistent deployment of drag rudders, having wake
thickness and drag forces likely exceeding those of a fin,
defeated “reduction of drag by empennage removal.”
5
0.60
0.40
GERMAN JET Four ~ Blohm und Voss P.209
Characteristic of design by Dr. Richard Vogt of B&V
were the outboard stabilizers or “taillets” of the P.209.
Most fascinating is the upwash and positive lift on the
o
taillets, in spite of their 2.5 net decalage needed for the
nose-up trim pitching moment.
Chord-weighted Normal-force Coef.
cN C/Cav
0.20
p/h
0.00
-1.25
-1
-0.75 -0.5 -0.25
0
0.25
0.5
0.75
1
1.25
Plan View
With neither 3D-aero analysis tools available only after
the war, nor wind-tunnel data for the P.209, Dr. Vogt
would have been unaware that the “gull-wing” dihedral
as originally drawn for the P.209 was insufficient to
overcome the yaw-destabilizing effect of the forebody.
For study purposes, such dihedral has been increased in
the figure below, with thus our designation “P.209A.”
c.g.
Looking Fwd
h
p
a.c.
Looking Starboard
Flat-plate Incidence, deg
8
0.160
6
0.120
4
Chord-weighted Drag Coefficient
0.40
0.30
cD C/Cav
0.20
0.10
0.080
2
0
-2
-1.20
-0.80
-0.40
0.00
0.40
0.00
0.80
1.20
0.000
-1.25
p/h
-0.10
0.040
p/h
Chord-weighted Normal-force Coef.
cN C/Cav
-1.25
p/h
-1
-0.75 -0.5 -0.25
0
0.25
0.5
0.75
1
1.25
-1
-0.75 -0.5 -0.25
0
0.25
0.5
0.75
1
1.25
1
1.25
Plan View
Aerodynamic Loads in Cruise ~ Ho-iX / Go-229 ~ Bell-Lift Option
c.g.
GERMAN JET Three ~ Focke-Wulf Schwanzloser
The Schwanzloser represented a far-more practical
implementation of the “essentially-all-wing” concept by
(1) incorporating a forebody to reduce or eliminate nose
ballast, while providing superior visibility, and by (2)
incorporating inverted winglets to stabilize the aircraft in
yaw. Whereas a modern winglet, whether above or
below the wing, would have a high aspect ratio to
generate local aerodynamic thrust with local induced
drag (negative) exceeding in magnitude the local
parasitic drag, the winglets of the Schwanzloser were
too “stubby” for that role. Nevertheless, they provided
the necessary yaw stability. Readers familiar with the
British Avro Vulcan, the design of which followed in time
the Schwanzloser by only two years, will immediately
note strong similarities in the configuration overall.
Looking Fwd
h
p
a.c.
Looking Starboard
Flat-plate Incidence, deg
6
0.030
4
0.020
2
0
cD C/Cav
0.010
-2
-4
-1.20
p/h
-0.80
-0.40
0.00
0.40
0.80
1.20
0.000
-1.25
p/h
-1
-0.75 -0.5 -0.25
0
0.25
Aerodynamic Loads in Cruise ~ Blohm & Voss P.209A
Mario Merino
Blohm & Voss P.209 in Action
Focke-Wulf Schwanzloser in Action
German Jets J. Philip Barnes www.HowFliesTheAlbatross.com
Chord-weighted Drag Coefficient
6
0.5
0.75
Another interesting feature is the inboard-directed lift of
the finlets (assuming they were not “toed out”). These
forces arise from the inboard component of flow above
the wing, yielding step changes in lift distribution.
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
-0.05
Chord-weighted Normal-force Coef.
cN C/Cav
p/h
-1.25
-1
-0.75 -0.5 -0.25
0
0.25
0.5
0.75
1
1.25
Plan View
Looking Fwd
h
Blohm & Voss P.209 in Action
p
GERMAN JET Five ~ Junkers EF.128
The Junkers EF128 was characterized by a prominent
pair of “finlets” and ventral fin, together ensuring a stable
gun platform. Also of interest is the inlet boundary layer
bleed, discharged from the aft step behind the canopy.
Looking Starboard
6
Flat-plate Incidence, deg
4
0.020
2
0
-2
0.030
Chord-weighted Drag Coefficient
cD C/Cav
0.010
p/h
-4
-1.25 -1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00 1.25
0.000
-1.25
p/h
-1
-0.75 -0.5 -0.25
0
0.25
0.5
0.75
1
1.25
Aerodynamic loads in cruise ~ Junkers EF.128
GERMAN JET Six ~ Heinkel P.1078
The pronounced gull-wing dihedral of the P.1078 seems
to emphasize the overall simplicity of the configuration.
The sharp and sudden reversal of dihedral at the “wrist”
of each wing yields, working outboard, a sudden
reduction of incidence. However, the distribution of lift
normal to the spar remains continuous, as seen in the
corresponding figure showing cruise aerodynamic loads.
Junkers EF.128
Heinkel P.1078
Junkers EF.128 in Action
German Jets J. Philip Barnes www.HowFliesTheAlbatross.com
7
Messerschmitt Ente in Action
Aerodynamic Loads at Cruise ~ Heinkel P.1078
GERMAN JET Seven ~ Messerschmitt Ente
Our last German jet, the Ente, is certainly among the
most interesting. Its many design challenges would have
included (1) locating an appropriate position for the c.g.,
(2) calculating the lift distributions for the wing and
canard, (3) validating yaw stability, and (4) mitigating
the effects of canard wake ingestion at the engine inlets.
With the aid of computational methods and hardware
unavailable to the Messerschmitt engineers at the time,
we can today readily meet most of these challenges.
Aerodynamic Loads at Cruise ~ Messerschmitt Ente
Messerschmitt Ente
At right we show the distributions of lift and drag for the
Ente at cruise, with or without sideslip. The small “dome”
at upper right represents the distribution of forebody lift.
The forebody wake induces a depression in lift at the
canard, and the canard wake induces a depression of lift
at the wing. With 12% “static margin,” the c.g. resides
well forward of the mean-aero-chord leading edge. The
aircraft is stable in yaw at the stated conditions.
German Jets J. Philip Barnes www.HowFliesTheAlbatross.com
Aero Loads in Sideslip ~ Me Ente
8
SUMMARY and CONCLUSIONS
We have studied the real and conceptual “German Jets”
of WWII from both aerodynamic and artistic viewpoints.
We introduced the “wing-body longitudinal, lateral liftingline” method for computationally-efficient analysis of
linear aero loads on simple or complex configurations.
We showed that the forebody is in effect a low-aspectratio wing, with its planform wake inducing downwash on
the wing, and in sideslip, with its profile wake imposing
“sidewash” on the fin. An empirical correlation was
introduced to characterize NACA test data for the yawing
and rolling moments of isolated wings. The German jets
were the first to encounter the full force of “transonic
treachery.” But below their critical Mach numbers, most
would probably have flown well in spite of their radical
configurations. Today, some 65-years on, we continue to
learn form the German Jets.
0.0020
Yawing Moment, Isolated Planar Wings
0.0018
cn β o/cL2
0.0016
β
0.0014
A>=5, Tapered or Rounded
0.0012
A>=5, Constant Chord
0.0010
n
A=3.5, Tapered
0.0008
A=2.6, Taper=1/4;1/2;1
0.0006
0.0004
0.0002
Quarter-chord Sweep, deg
0.0000
-40
-30
-20
-10
0
10
20
30
40
50
60
Figure A-1 Empirical Correlation, Isolated Wing Yawing Moment
REFERENCES
An aircraft which yaws nose-left (positive sideslip)
should naturally roll to the left if unfavorable handling
qualities are to be avoided. Whereas the yawing moment
is proportional to the square of planar wing (or twisted
wingtip-local) lift coefficient, the rolling moment
coefficient is proportional to the first power of lift
coefficient (Figure A-2). Again, sweep combined with low
aspect ratio affords the greatest “roll due to yaw.” As
with the yawing moment, a rectangular wing exhibits
marginally favorable characteristics, but for roll-due-toyaw, taper or rounded wingtips tend to degrade the
handling qualities.
1. Archer and Saarlas, Introduction to Aerospace Propulsion
2. A.C. Piccirillo, Heinkel and the Turbojet Engine:
Origin of the First Jet Fighter
3. F. Crosby, A Handbook of Fighter Aircraft
4. Theodore von Kármán, “Aerodynamics,” Dover, ‘57, p. 133
5. Meier, “German Development of the Swept Wing,” p. 42
6. Web site: www.wapedia.mobi/en/ho-229
RECOMMENDED READING
•
•
•
•
•
•
•
•
•
•
Jet Planes of the Third Reich, Vol.1&2, M. Griehl
Luftwaffe Secret Projects, Schick & Meyer
Secret Messerschmitt Projects, Radinger & Schick
Ho 229 Spirit of Thuringia, Shepelev & Ottens
A Handbook of Fighter Aircraft, F. Crosby
Jet and Turbine Aero Engines, B. Gunston
High-speed Wing Theory, R.T. Jones & D. Cohen
German Aircraft of WWII, D. Donald
Fluid Dynamic Lift, Fluid Dynamic Drag, S. Hoerner
www.SAE.org, search with quotes: “J.Philip Barnes”
0.004
0.000
cl β o/cL
-0.002
APPENDIX
Figure A-1, based on test data from NACA reports
(TN703, TN1468, TN1581, TN1671, TN 2445, RM
A6K15), correlates the yawing moment coefficient with
lift, sweep, and aspect ratio for isolated, planar wings
(untwisted with no dihedral). Interestingly, even a planar
rectangular wing is “somewhat” stable in yaw, but for any
wing without dihedral, yaw stability requires lift in the
vicinity of the wingtips. This was shown by Albert Betz,
among the first to investigate the yaw stability of isolated
wings. Here, yaw stability vanishes at zero lift because
the differential forces which provide yaw stabilization
originate from induced drag. Thus, if a planar wing
develops no lift, or if the tip regions of a twisted wing
develop no lift, then the wing will be “neutral” in yaw.
Indeed, the restoring yawing moment is proportional to
the square of wing (or tip-local) lift coefficient. Sweep,
particularly when combined with low aspect ratio,
provides a significant increase in the yaw stability of an
isolated wing. Based on limited data, it appears that
forward sweep yields near-neutral or slightly-negative
yaw stability, whereby dihedral becomes essential
unless active yaw stability is to be provided.
German Jets J. Philip Barnes www.HowFliesTheAlbatross.com
Rolling Moment, Isolated Planar Wings
0.002
-0.004
β
-0.006
l
A>=5, Tapered or Rounded
A>=5, Constant Chord
A=3.5, Tapered
A=2.6, Taper=1/4;1/2;1
A=1.6, Constant Chord
-0.008
-0.010
-0.012
Quarter-chord Sweep, deg
-0.014
-40
-30
-20
-10
0
10
20
30
40
50
60
Figure A-2 Empirical Correlation, Isolated Wing Rolling Moment
ABOUT THE AUTHOR
Phil Barnes has a Master of Engineering degree from
Cal Poly Pomona and has recently celebrated 30-years
with a major aircraft manufacturer where he is
responsible for air vehicle and subsystem performance
analysis. He is the author of landmark studies of
dynamic soaring and regenerative-electric flight, both
found at www.HowFliesTheAlbatross.com
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