02 Feb 2016 draft status

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 (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 computationally- efficient method for aerodynamic forces and moments. We then apply the method to study the top-level Messerschmitt 262 and Jumo 004 of seven German jets, each noted for its unique configuration. Also, renowned computer-graphic The rocket-powered Me 163 was test flown beyond its artists Mario Merino and Gery Gueville share some of critical speed by pilot Heini Dittmar, to Mach 0.84 where their work herein copyright-free. Thus our title: it exhibited instability about all three axes, in part “Aerodynamic and Artistic Study of the German Jets.” because the forebody, in effect a low-aspect-ratio wing, develops 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 Blohm und Voss P.209 at high speed, both lift coefficient and isolated-wing yaw stability vanish. Thus misbehaved the Me 163.

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 Lloyd S. Jones maximum speed of 870 km/hr to the Jumo 004 engine. Messerschmitt 163B

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SOME OF THE FOUNDING FATHERS THE VERSATILE HORSESHOE VORTEX Our aerodynamic study takes strong advantage of the Our “Wing-body Linear Longitudinal-Lateral Lifting Line” pioneering work of many of the “Founding Fathers” of (WBL5) method gives the lift (normal force) distribution, aerodynamics. With the horseshoe vortex as modeling as well as profile and induced drag over all clay, we’ll apply ’s lifting line and induced significant aerodynamic surfaces including the forebody drag concepts and shift the companion downwash line planform, forebody profile, wing, and empennage. The with the compressibility rule independently developed by analysis, limited to subsonic, linear aerodynamics below Ludwig Prandtl and Hermann Glauert, including the critical Mach, aligns “horseshoe vortices” along lifting effects of wing sweep. Although supersonic sweep was lines, nominally at ¼-chord, with empirical modifications. first theorized by Adolf Busemann, and later independently by Robert T. Jones, Albert Betz was first A corresponding downwash line, nominally at ¾-chord, to suggest and test the benefits of subsonic sweep to but shifted aft via the Prandtl-Glauert compressibility delay transonic effects (4,5). Finally, to include forebody “rule,” connects the points where the equivalent flat-plate effects, we integrate Max Munk’s theory of airships with (EFP) airfoil flow-tangency boundary condition is Robert T. Jones’ theory of low-aspect-ratio wings. 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 MaryEvans.com Rather than adjust aircraft and vortex geometry for angle CONFIGURATION AERODYNAMICS of attack and sideslip, instead the flight velocity vector is With the advent of the jet engine, the aircraft forebody tilted by these angles, with only selected adjustments of stretched forward as the wings swept aft. These wing coordinates applied to account for differential geometric changes decreased both the pitch and yaw sweep and downwash-node position in sideslip. stability of the aircraft. Accordingly, our method of Simultaneous equations, typically less than 100 per aerodynamic analysis includes the forebody as a aircraft, are then solved to yield the distribution of significant longitudinal and lateral “lifting surface.” In horsehoe-vortex strength, from which local normal forces keeping with the basic method, it will be modeled as a can be computed. The “apparent downwash” method “cruciform” consisting of two low-aspect-ratio wings. The then determines the local induced drag by comparing the forebody planform, wing and empennage together set observed local lift to that expected based on local the “aerodynamic center” (a.c.) of the aircraft, where the incidence and local sweep, together with Mach number. total lift is concentrated for the purposes of analysis. The German Jets would have been statically stable, with the We postulate that the well-known “Lifting-line” methods center of gravity (c.g.) residing forward of the a.c. and cannot obtain aerodynamic loads in sideslip without with a nose-up pitching moment balancing the offset of empirical modeling thereof. Thus, in the appendix we the lift and weight vectors. Such pitch-up moment can be characterize NACA-measured yaw and roll moments for provided by any combination of airfoil reflex, wing sweep planar wings of various sweep, aspect ratio, and taper with twist, and “decalage” of a canard or horizontal ratio. We then add or subtract “bumps” to or from the stabilizer, the latter designated herein as a “tail,” with lifting-line-computed lift and drag distributions to yield the vertical stabilizing surfaces designated herein as “fins.” overall yaw or roll moment. Such modeling is founded on Albert Betz’ observation that both yaw and roll moments Lift @ vehicle ac 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 Weight cMac vanishes. Thus, an aircraft configured largely as an isolated wing will go either marginally stable, or unstable, Vehicle Aerodynamic Center in yaw at “high” speeds approached by the German jets.

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TOP-LEVEL VALIDATION OF METHOD of attack, the lift distribution remains largely positive and A thorough validation of the WBL5 method would involve “half-sinusoidal,” with only a small region of negative lift a wide range of configurations and conditions far at the tips. exceeding the scope of this article. Also, at this time the Chord-weighted Normal-force Coef. method is still in development with validation on-going. 0.40 0.30 cN C/Cav

Nevertheless, in this section we validate the method for 0.20 selected cases applicable to the German jets to be 0.10 studied subsequently. 0.00 p/h -0.10 -1.25 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25 First and foremost is the distribution lift for a wing and body combination. NASA TN D-712 represents a rare, Plan View but immensely useful, characterization of wing-body aerodynamic loads by adding pressure taps to the portion of the body coinciding with the “submerged” Looking Fwd portion of the wing. As seen in the figure below, where h only the forebody has been retained for modeling purposes, the wing lift distribution exhibits a pronounced p

“dip” which is essentially replaced by forebody lift, Looking Starboard 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 Flat-plate Incidence, deg Chord-weighted Drag Coefficient 10 0.030 the method. Nevertheless, computed loads reasonably cD C/Cav match the measurements, even though 0.9 flight Mach 5 0.020 number is outside the applicability of the Prandtl-Glauert 0 0.010 p/h p/h -5 0.000 correction for compressibility used by the method. -1.25 -1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00 1.25 -1.25 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25 Method Validation ~ Wing-body lift with body pressure taps Chord-weighted Normal-force Coef. 0.40 cN C/Cav Our third and final top-level validation of the method 0.30 Data applies to a swept wing with dihedral in sideslip. As seen 0.20 Calc. in the figure below, the computed lift distribution 0.10 p/h reasonably matches the test data. Here, the model has 0.00 -1.25 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25 applied an empirical differential lift “bump” to yield the correlated yaw and roll moments of the appendix. Plan View Chord-weighted Normal-force Coef. 0.75 cN C/Cav Looking Fwd Calc. h 0.50 Data p 0.25 p/h 0.00 -1.25 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25 Looking Starboard

Plan View

Flat-plate Incidence, deg Chord-weighted Drag Coefficient 5.0 0.030 Looking Fwd 4.0 cD C/Cav h 0.020 3.0 p 2.0 0.010 1.0 p/h p/h 0.0 0.000 -1.25 -1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00 1.25 -1.25 -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 Flat-plate Incidence, deg Chord-weighted Drag Coefficient 8.2 o 0.050 8.0 wing with 10 twist, again tested with a body, but with 0.040 7.8 cD C/Cav 0.030 only “exposed wing” loads measured. Again, the 7.6 7.4 0.020 calculation agrees well with the test data. A key point of 7.2 0.010 7.0 p/h 0.000 p/h 6.8 -0.010 interest is the outboard upwash, whereby with the wing -1.25 -1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00 1.25 -1.25 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25 root at 3o angle of attack, and the wingtips at -7o angle Method Validation ~ Wing-body lift with body pressure taps

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GERMAN JET One ~ Focke-Wulf Ta183 local upwash (negative), profile drag, and induced drag. The FW Ta183, although radical for its day, is the The overall distribution of chord-weighted drag is shown closest we will come herein to a “conventional” at the bottom center. Although it may be difficult to configuration. Conceived by Hans Multhopp with distinguish the various curves with the chosen format, development well underway by Kurt Tank at F-W, the we note that the forebody planform drag is greater than Ta183 was noted by its swept, constant-chord wing. The the forebody profile drag because without sideslip only figures below describe the aerodynamics of the Ta183 at the forebody planform is lifting. 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 low- quality 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 Focke-Wulf Ta183 show the distributions of lift (normal force) coefficient,

Chord-weighted Normal-force Coef. Section Normal Force, cn 0.3 0.30 cN C/Cav 0.2 0.20 Wing Wing

0.10 Forebody 0.1 Forebody Tail Tail 0.00 0 Fin p/h Fin p/h -0.10 -0.1 -1.25 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25 -1.2 -0.8 -0.4 0 0.4 0.8 1.2 Upwash (Normal to section), deg 1 Fin Plan View 0 Wing -1 Tail -2 Forebody

-3 Looking Fwd c.g. p/h h -4 -1.2 -0.8 -0.4 0 0.4 0.8 1.2 p p Section Profile Drag, cDp 0.015

0.010 Looking Starboard a.c. 0.005

0.000 -1.2 -0.8 -0.4 0 0.4 0.8 1.2

Section Vortex Drag, cDv Flat-plate Incidence, deg Chord-weighted Drag Coefficient 0.008 5 0.025 0.006 4 Wing 0.020 cD C/Cav 3 0.004 Forebody 0.015 2 Tail 0.002 1 0.010 0.005 0 0 Fin p/h p/h p/h -1 0.000 -0.002 -1.25 -1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00 1.25 -1.25 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25 -1.2 -0.8 -0.4 0 0.4 0.8 1.2

Focke-Wulf Ta183, Cruise ~ Distribution of Lift and Drag Over the Forebody, Wing, and Empennage

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GERMAN JET Two ~ Horten iX / Gotha 229 “Wing-only” statically-stable configurations, such as the Next we visit an even more radical configuration, that of Ho-iX, typically suffer the need for ballast (550-lb, Ho-iX) the Horten-iX /Go-229. In the end, this configuration to shift the c.g. well forward from its inherent location, set proved too radical. Offered by the Hortens as a by structure and equipment, to the required location for contender for the last-ditch Volksjäger (People’s Fighter) pitch stability. But of course in aerial combat every competition, the Ho-iX was a favorite of Air Marshal pound counts and every round counts. Hermann Goering. He directed Gothaer Wagonfabrik Contrary to widespread popular literature, the Ho-iX was (Gotha) to carry out detailed design and construction. o Given the perseverance and skill of the Gotha engineers built with only 1.5 of chord-line twist, as indicated by the facing major setbacks and a near-impossible deadline, Arthur Bentley drawings. With a reflexed airfoil at the wing root, the local zero-lift-line incidence was 0.5o the Ho-iX is appropriately given the alternate designation o “Go-229.” 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.

Chord-weighted Normal-force Coef. 0.40 0.30 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

Plan View

c.g. Looking Fwd h

p Ho-iX / Go-229 a.c. Looking Starboard

Flat-plate Incidence, deg Chord-weighted Drag Coefficient 5 0.060 4 cD C/Cav 0.040 3 2 0.020 1 p/h p/h 0 0.000 -1.20 -0.80 -0.40 0.00 0.40 0.80 1.20 -1.25 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25 Aerodynamic Loads in Cruise ~ Ho-iX / Go-229

If instead the aircraft had incorporated bell-shaped lift, twist near 10o 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 Ho-iX / Go-229 in Action as left-right differences in induced drag. Therefore, since bell-shaped lift would by definition “unload” the wingtips, To stabilize the aircraft in yaw, the Hortens specified it appears that the Ho-iX would have been near-neutral drag rudders, in lieu of a swept vertical fin, which we or unstable in yaw at the lift coefficient corresponding to suggest would have offered superior handling and safety bell-shaped lift. with a net reduction of drag. The drag rudders, which opened both top and bottom, alternately left and right, With or without bell-shaped lift loading for the Ho-iX, the could not balance the yaw moment of single-engine consistent deployment of drag rudders, having wake flameout. This shortcoming proved fatal for the first test thickness and drag forces likely exceeding those of a fin, pilot (Lt. Irwin Ziller) to encounter this condition. defeated “reduction of drag by empennage removal.”

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Chord-weighted Normal-force Coef. 0.60 GERMAN JET Four ~ Blohm und Voss P.209 c C/C 0.40 N av Characteristic of design by Dr. Richard Vogt of B&V 0.20 were the outboard stabilizers or “taillets” of the P.209. p/h 0.00 -1.25 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25 Most fascinating is the upwash and positive lift on the taillets, in spite of their 2.5o net decalage needed for the Plan View nose-up trim pitching moment.

With neither 3D-aero analysis tools available only after

c.g. the war, nor wind-tunnel data for the P.209, Dr. Vogt Looking Fwd would have been unaware that the “gull-wing” dihedral h as originally drawn for the P.209 was insufficient to overcome the yaw-destabilizing effect of the forebody. p For study purposes, such dihedral has been increased in a.c. Looking Starboard the figure below, with thus our designation “P.209A.”

Chord-weighted Normal-force Coef. Flat-plate Incidence, deg Chord-weighted Drag Coefficient 0.40 8 0.160 0.30 cN C/Cav 6 0.120 cD C/Cav 0.20 4 0.10 0.080 0.00 2 p/h -0.10 0 0.040 p/h p/h -1.25 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25 -2 0.000 -1.20 -0.80 -0.40 0.00 0.40 0.80 1.20 -1.25 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 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 Looking Fwd implementation of the “essentially-all-wing” concept by h (1) incorporating a forebody to reduce or eliminate nose ballast, while providing superior visibility, and by (2) p a.c. incorporating inverted winglets to stabilize the aircraft in Looking Starboard yaw. Whereas a modern winglet, whether above or below the wing, would have a high aspect ratio to Flat-plate Incidence, deg Chord-weighted Drag Coefficient generate local aerodynamic thrust with local induced 6 0.030 4 cD C/Cav drag (negative) exceeding in magnitude the local 0.020 2 parasitic drag, the winglets of the Schwanzloser were 0 0.010 -2 too “stubby” for that role. Nevertheless, they provided p/h p/h -4 0.000 -1.20 -0.80 -0.40 0.00 0.40 0.80 1.20 -1.25 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25 the necessary yaw stability. Readers familiar with the British Avro Vulcan, the design of which followed in time Aerodynamic Loads in Cruise ~ Blohm & Voss P.209A the Schwanzloser by only two years, will immediately note strong similarities in the configuration overall.

Mario Merino

Focke-Wulf Schwanzloser in Action Blohm & Voss P.209 in Action

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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.

Chord-weighted Normal-force Coef. 0.35 0.30 c C/C 0.25 N av 0.20 0.15 0.10 0.05 0.00 p/h -0.05 -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

GERMAN JET Five ~ Junkers EF.128 p

The Junkers EF128 was characterized by a prominent Looking Starboard pair of “finlets” and ventral fin, together ensuring a stable gun platform. Also of interest is the inlet boundary layer Flat-plate Incidence, deg Chord-weighted Drag Coefficient bleed, discharged from the aft step behind the canopy. 6 0.030 4 cD C/Cav 0.020 2 0 0.010 -2 p/h p/h -4 0.000 -1.25 -1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00 1.25 -1.25 -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

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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. Aero Loads in Sideslip ~ Me Ente

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SUMMARY and CONCLUSIONS 0.0020 We have studied the real and conceptual “German Jets” Yawing Moment, Isolated Planar Wings of WWII from both aerodynamic and artistic viewpoints. 0.0018 We introduced the “wing-body longitudinal, lateral lifting- 0.0016 o 2 cn β /cL line” method for computationally-efficient analysis of β linear aero loads on simple or complex configurations. 0.0014 A>=5, Tapered or Rounded We showed that the forebody is in effect a low-aspect- 0.0012 ratio wing, with its planform wake inducing downwash on A>=5, Constant Chord n 0.0010 the wing, and in sideslip, with its profile wake imposing A=3.5, Tapered “sidewash” on the fin. An empirical correlation was 0.0008 A=2.6, Taper=1/4;1/2;1 introduced to characterize NACA test data for the yawing 0.0006 and rolling moments of isolated wings. The German jets were the first to encounter the full force of “transonic 0.0004 treachery.” But below their critical Mach numbers, most 0.0002 would probably have flown well in spite of their radical Quarter-chord Sweep, deg configurations. Today, some 65-years on, we continue to 0.0000 -40 -30 -20 -10 0 10 20 30 40 50 60 learn form the German Jets. Figure A-1 Empirical Correlation, Isolated Wing Yawing Moment REFERENCES 1. Archer and Saarlas, Introduction to Aerospace Propulsion An aircraft which yaws nose-left (positive sideslip) 2. A.C. Piccirillo, Heinkel and the Turbojet Engine: should naturally roll to the left if unfavorable handling Origin of the First Jet Fighter qualities are to be avoided. Whereas the yawing moment 3. F. Crosby, A Handbook of Fighter Aircraft is proportional to the square of planar wing (or twisted 4. Theodore von Kármán, “Aerodynamics,” Dover, ‘57, p. 133 wingtip-local) lift coefficient, the rolling moment 5. Meier, “German Development of the Swept Wing,” p. 42 coefficient is proportional to the first power of lift 6. Web site: www.wapedia.mobi/en/ho-229 coefficient (Figure A-2). Again, sweep combined with low

aspect ratio affords the greatest “roll due to yaw.” As RECOMMENDED READING with the yawing moment, a rectangular wing exhibits • Jet Planes of the Third Reich, Vol.1&2, M. Griehl • Luftwaffe Secret Projects, Schick & Meyer marginally favorable characteristics, but for roll-due-to- • Secret Messerschmitt Projects, Radinger & Schick yaw, taper or rounded wingtips tend to degrade the • Ho 229 Spirit of Thuringia, Shepelev & Ottens handling qualities. • A Handbook of Fighter Aircraft, F. Crosby • Jet and Turbine Aero Engines, B. Gunston 0.004 • High-speed Wing Theory, R.T. Jones & D. Cohen Rolling Moment, Isolated Planar Wings • German Aircraft of WWII, D. Donald 0.002 • Fluid Dynamic Lift, Fluid Dynamic Drag, S. Hoerner • www.SAE.org, search with quotes: “J.Philip Barnes” 0.000

-0.002 c βo/c APPENDIX l L Figure A-1, based on test data from NACA reports -0.004 (TN703, TN1468, TN1581, TN1671, TN 2445, RM β -0.006 A6K15), correlates the yawing moment coefficient with l lift, sweep, and aspect ratio for isolated, planar wings -0.008 A>=5, Tapered or Rounded (untwisted with no dihedral). Interestingly, even a planar A>=5, Constant Chord rectangular wing is “somewhat” stable in yaw, but for any -0.010 A=3.5, Tapered wing without dihedral, yaw stability requires lift in the A=2.6, Taper=1/4;1/2;1 -0.012 vicinity of the wingtips. This was shown by Albert Betz, A=1.6, Constant Chord among the first to investigate the yaw stability of isolated Quarter-chord Sweep, deg -0.014 wings. Here, yaw stability vanishes at zero lift because -40 -30 -20 -10 0 10 20 30 40 50 60 the differential forces which provide yaw stabilization Figure A-2 Empirical Correlation, Isolated Wing Rolling Moment 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. ABOUT THE AUTHOR Indeed, the restoring yawing moment is proportional to Phil Barnes has a Master of Engineering degree from the square of wing (or tip-local) lift coefficient. Sweep, Cal Poly Pomona and has recently celebrated 30-years particularly when combined with low aspect ratio, with a major aircraft manufacturer where he is provides a significant increase in the yaw stability of an responsible for air vehicle and subsystem performance isolated wing. Based on limited data, it appears that analysis. He is the author of landmark studies of forward sweep yields near-neutral or slightly-negative dynamic soaring and regenerative-electric flight, both yaw stability, whereby dihedral becomes essential found at www.HowFliesTheAlbatross.com unless active yaw stability is to be provided.

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