Lift and Drag Review and Renew Correlating 50 Years of NACA / NASA Test Data for the Effects of Wing Planform and Thickness 21 April 2013 Update J. Philip Barnes Pelican Aero Group cN cF cT vo a u This presentation revisits two classical problems of aerodynamics - those of determining the lift and drag of a wing, given its planform and thickness. Although wing thickness has only a minor effect on lift, we will show that reducing it yields a substantial increase of induced drag. To support our "review and renew" of lift and drag, limited for now to the low-speed and linear behavior of planar wings, we will process the test data of 114 wing or wing-body models tested over the last several decades in the NACA and NASA wind tunnels. In parallel, and in relation to the test data, we will review, clarify, and apply the theories of Ludwig Prandtl and Robert T. Jones, as well as those of perhaps less-well-known, but significant, contributors: Hienrich Helmbold, Frederick Diederich, and Edward Polhamus. The charts, representing an update to the 02 Sept. 2012 presentation to the Experimental Soaring Association in Tehachapi, CA, are perhaps best viewed in PowerPoint slide-show mode, but are best printed as "notes pages" to reveal the explanatory notes accompanying each chart. Lift and Drag Review and Renew - Correlations of 50 Years of NACA and NASA Test Data on the Effects of Wing Planform and Thickness www.HowFliesTheAlbatross.com J. Philip Barnes April 2013

Presentation Purpose and Contents Review & renew: wing / body lift & induced drag Aspect ratio, sweep, & thickness Subsonic, linear range (moderate incidence) Elliptical wing and Prandtl's formula for lift ~ 1918 Helmbold's enhancement for low aspect ratio ~ 1942 Diederich's enhancement for sweep ~ 1951 Polhamus' enhancement for sweep ~ 1957 Prandtl-Jones: "thick" wing or body induced-drag ~ 1918/1946 The thin-wing induced-drag surprise ~ 1950 Polhamus: "thin" wing or body induced drag ~ 1950 Transition, Prandtl-Jones to Polhamus ~ 2012 New: Synergy of airfoil & wing data thereof Summary and sample application of new method The purpose of the presentation is to review and renew the effects of "flat" wing planform and thickness on lift and induced drag at low-subsonic speed. By "flat" we mean that the wing has no dihedral, camber, or twist. However, we point out that whereas camber and/or twist will vertically shift the lift curve, neither will change the lift slope. Furthermore, we anticipate that follow-on studies (recommended) will reveal the general applicability of the trends herein to "non-flat" wings as well. Our "review" will study and correlate NACA and NASA wind-tunnel test data in parallel with the well-known theories of Prandtl and Jones, and lesser-known theories or studies of Helmbold, Diedrich, and Polhamus. Our "renew" then introduces a new way of thinking about induced drag. We will show that all wings exhibit induced drag which lies between upper and lower limits set by the theories of Polhamus and Prandtl-Jones, respectively. Indeed, our study shows that only if the wing is "very thick" (or forebody well rounded) can its induced drag be entirely characterized by the Prandtl-Jones formula. Conversely, only if a wing is "very thin" can its induced drag be entirely characterized by the theory of Polhamus. Essential features of our "renew" are (1) clarifying, condensing, and validating the theories with wide-ranging test data, (2) introducing an empirical correlation to determine where the induced drag of a given wing lies between the upper and lower limits thereof and (3) showing the synergy of wing and airfoil data therein. Lift and Drag Review and Renew - Correlations of 50 Years of NACA and NASA Test Data on the Effects of Wing Planform and Thickness www.HowFliesTheAlbatross.com J. Philip Barnes April 2013

Configurations studied ~ Data and theory references www.NTRS.NASA.gov www.AERADE.Cranfield.ac.uk www.Google.com In the sketch above, we show the planforms of the 114 models tested in the NACA and NASA wind tunnels at or below 0.40 Mach number. The models, some including a body, exhibited wing aspect ratios from 1 to 12 and wing streamwise thicknesses from 2 to 20%. Several models were tested with multiple thicknesses. Many of the models were defined with airfoil sections taken normal to the quarter-chord line, in which case such thickness was converted to a streamwise basis. Also shown are the NACA/NASA reports used for the study, together with the various websites from which the reports can be downloaded. In particular, the NTRS web site makes freely available almost a century of comprehensive aerodynamic test data and theory. For our present purpose, the essential information from each report consists of the report number, wing planform, thickness and orientation thereof, plots of lift versus angle of attack and drag versus lift, and references which point to additional test data and/or related theory. In processing the data herein, only the range exhibiting (a) linear lift with angle of attack and (b) drag below the "drag break" was used. For the induced drag, the total drag at zero lift was subtracted from the total drag at 0.408 lift coefficient to compute the change of drag coefficient with the square of lift coefficient. In some cases, this was instead done at 0.316 lift coefficient (0.408 and 0.316 are the square roots of 1/6 and 1/10 respectively, for convenient data reduction). This procedure then "lumps" any changes in section drag into the overall "induced drag" or "drag due to lift." 114 configurations, thickness: 02 - 20% Lift and Drag Review and Renew - Correlations of 50 Years of NACA and NASA Test Data on the Effects of Wing Planform and Thickness www.HowFliesTheAlbatross.com J. Philip Barnes April 2013

Wing geometry and aerodynamic terms S ≡ plan area b ≡ span c ≡ chord r ≡ tip chord / root chord t ≡ streamwise thickness t/c ≡ thickness ratio A ≡ aspect ratio = b2/S = b/cav a ≡ angle of attack cL ≡ lift coefficient h ≡ lift slope / (2p) cDv ≡ vortex drag coefficient Lo ≡ leading-edge sweep Lc/2 ≡ mid-chord sweep Lc/4 ≡ quarter-chord sweep b c t Lo Sweep conversion (given quarter-chord sweep) tanLn = tanLc/4 + (4/A) (n-¼) (r-1) / (r+1) Lift and Drag Review and Renew - Correlations of 50 Years of NACA and NASA Test Data on the Effects of Wing Planform and Thickness www.HowFliesTheAlbatross.com J. Philip Barnes April 2013

Prandtl and Jones Theories Ludwig Prandtl Robert T. Jones Prandtl Lift slope (any-A, low-L) dcL/da ≈ 2pA/(A+2) Induced drag: cDv ≈ cL2/(pA) Jones Lift slope (low-A, any-L) dcL/da = p A/2 Induced drag: cDv = cL2/(pA) Prandtl-Jones Induced drag: cDv ≈ cL2/(pA) But what about thickness? Lift and Drag Review and Renew - Correlations of 50 Years of NACA and NASA Test Data on the Effects of Wing Planform and Thickness www.HowFliesTheAlbatross.com J. Philip Barnes April 2013

Lift slope data and validation of theory Unswept Prandtl Helmbold Swept Helmbold-Polhamus Helmbold-Diederich Lift and Drag Review and Renew - Correlations of 50 Years of NACA and NASA Test Data on the Effects of Wing Planform and Thickness www.HowFliesTheAlbatross.com J. Philip Barnes April 2013

Helmbold-Diederich ~ Low-speed lift slope of any wing Lift and Drag Review and Renew - Correlations of 50 Years of NACA and NASA Test Data on the Effects of Wing Planform and Thickness www.HowFliesTheAlbatross.com J. Philip Barnes April 2013

Helmbold-Polhamus ~ Low-speed lift slope of any wing Lift and Drag Review and Renew - Correlations of 50 Years of NACA and NASA Test Data on the Effects of Wing Planform and Thickness www.HowFliesTheAlbatross.com J. Philip Barnes April 2013

Test data ~ rectangular wing lift ~ effect of thickness Theory, Helmbold (h=0.95) Lift and Drag Review and Renew - Correlations of 50 Years of NACA and NASA Test Data on the Effects of Wing Planform and Thickness www.HowFliesTheAlbatross.com J. Philip Barnes April 2013

Test data ~ Delta wing-body lift ~ effect of thickness 5% 3% 8% Minor effect of thickness on lift Lift and Drag Review and Renew - Correlations of 50 Years of NACA and NASA Test Data on the Effects of Wing Planform and Thickness www.HowFliesTheAlbatross.com J. Philip Barnes April 2013

The Thin-wing Induced-drag Surprise ~ Circa 1950 Delta wing-body linearized drag polar A=2, M 0.25, NACA RM A50K20, A50K21, A51K28 3% 5% Polhamus: cDv ≈ acL ≈ cL2/(dcL/da) 8% Induced drag coefficient, cDv Prandtl-Jones: cDv = cL2/(pA) Lift and Drag Review and Renew - Correlations of 50 Years of NACA and NASA Test Data on the Effects of Wing Planform and Thickness www.HowFliesTheAlbatross.com J. Philip Barnes April 2013

The Thin-wing Induced-drag Surprise ~ Circa 1950 Rectangular wing linearized drag polar A=4, Effect of thickness, NACA TN 3501 4% 6% Polhamus: cDv ≈ acL ≈ cL2/(dcL/da) 10% Prandtl-Jones: cDv = cL2/(pA) Lift and Drag Review and Renew - Correlations of 50 Years of NACA and NASA Test Data on the Effects of Wing Planform and Thickness www.HowFliesTheAlbatross.com J. Philip Barnes April 2013

Induced-drag Transition ~ Prandtl-Jones to Polhamus t ≡ [dcD/dcL2 - 1/(pA)] / [1/(dcL/da) - 1/(pA)] Preliminary Empirical Correlation t = e-a(t/c)-b(t/c)2 Polhamus cD ≈ a cL dcD/dcL2 ≈ 1/(dcL/da) Prandtl-Jones dcD/dcL2 = 1/(pA) t/c Lift and Drag Review and Renew - Correlations of 50 Years of NACA and NASA Test Data on the Effects of Wing Planform and Thickness www.HowFliesTheAlbatross.com J. Philip Barnes April 2013

Effect of thickness on induced drag ~ symmetrical section Nomenclature A aspect ratio vo flight velocity a angle of attack u upwash angle * cL lift coefficient cD drag coefficient cN normal force coef. cF friction force coef. ** cT chord thrust coef. *** k thrust recovery (0-1) * Usually negative ** Upper + lower, chordwise *** Pressure integration, chordwise No sweep No twist Assume elliptical loading Assume small angles cL ≈ cN ≈ 2p (a+u) [1] u ≈ -cN /(pA) [2] cD ≈ cN a - cT + cF [3] Define thrust recovery: k ≡ cT / [cN tan(a+u)] ≈ cT / [cN (a+u)] [4] Combine [1,2,3,4]: cN cF cT vo a u cD ≈ cF + (cN2) / (pA) + (1-k) (cN2) / (2p) "very thin": k → 0 "thick" : k →1 @ k = 0: cD ≈ cF + cNa consistent with Polhamus Lift and Drag Review and Renew - Correlations of 50 Years of NACA and NASA Test Data on the Effects of Wing Planform and Thickness www.HowFliesTheAlbatross.com J. Philip Barnes April 2013

Summary ~ Lift and Drag Review and Renew Prandtl: Good prediction of unswept wing lift slope Helmbold: Excellent prediction thereof particularly at low aspect ratio Diederich & Polhamus: added effect of sweep different formulas ~ quite-different curve shapes essentially identical results, nonetheless Prandtl & Jones: thick-wing or body induced drag totally independent methods & purposes Prandtl: any aspect ratio ~ Jones: Low-A same formula: cDv = cL2 / (pA) Polhamus: induced drag upper limit zero thickness, symmetrical section formula: cDv ≈ acL ≈ cL2 / (dcL/da) Enhancements via our review & renew study: 1) Showed Prandtl-Jones drag is limited to thick wings 2) Suggested correlation for thick-to-thin drag transition 3) New formula for induced drag with symmetrical sections Lift and Drag Review and Renew - Correlations of 50 Years of NACA and NASA Test Data on the Effects of Wing Planform and Thickness www.HowFliesTheAlbatross.com J. Philip Barnes April 2013

Application of method ~ "Neutral-trimmed" drag polar 01. Set geom (aspect ratio, thickness, & sweep) {A, t/c, Lc/2} 02. Loop on specified angle of attack, a (say from 0o to 10o) 03. Compute the lift slope, dcL/da (Diederich or Polhamus) 04. Compute the lift coefficient, cL (given a and dcL/da) 05. Compute Prandtl-Jones induced drag coefficient, cDv_PJ 06. Compute Polhamus induced-drag coefficient, cDv_Po 07. Get induced-drag transition (t) at thickness ratio (t/c) 08. Compute induced drag coefficient (cDv) given (t) 09. Est. zero-lift drag (cDo) {1st mention ~ use 0.02} 10. Compute total drag coefficient, cD = cDo + cDv 11. Compute lift/drag ratio, L/D 12. Plot all results versus a or cL Lift and Drag Review and Renew - Correlations of 50 Years of NACA and NASA Test Data on the Effects of Wing Planform and Thickness www.HowFliesTheAlbatross.com J. Philip Barnes April 2013

Sample application of method ~ homework assignment Application: Me-163 Assume: a) no twist, low Mach number b) 9% thickness (t/c) c) section h = 0.95 Measure from sketch: a) Leading-edge sweep (Lo) b) Span (b) c) Root (centerline) & tip chords Tasks: 1) Get parameters S, A, r, Lc/2 2) Find L/D, a and cL at max L/D 3) e-mail results to: Phil@HowFliesTheAlbatross.com b c t Lo Lift and Drag Review and Renew - Correlations of 50 Years of NACA and NASA Test Data on the Effects of Wing Planform and Thickness www.HowFliesTheAlbatross.com J. Philip Barnes April 2013

About the Author Phil Barnes has a Master’s Degree in Aerospace Engineering from Cal Poly Pomona and a Bachelor’s Degree in Mechanical Engineering from the University of Arizona. He has 31-years of experience in the performance analysis and computer modeling of aerospace vehicles and subsystems at Northrop Grumman. Phil has authored diverse technical papers and studies of gears, computer graphics, orbital mechanics, aerodynamics, and propellers, including internationally-recognized studies of albatross dynamic soaring, regenerative-electric flight, and "German Jets." This chart has no footnotes Lift and Drag Review and Renew - Correlations of 50 Years of NACA and NASA Test Data on the Effects of Wing Planform and Thickness www.HowFliesTheAlbatross.com J. Philip Barnes April 2013