pelican aero group 1 lift and drag review and renew - correlations of 50 years of naca and nasa test...
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Pelican Aero Group
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t
Streamwise thickness
Induced drag transition ~ Prandtl-Jones to Polhamus
K ≡ dcD/dcL2 ; t ≡ ( K-KPJ ) /( KPo-KPJ )
Prandtl-JonesdcD/dcL
2 = 1/(pA)
Polhamus cD ≈ a cL
dcD/dcL2 ≈ 1/(dcL/da)
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t
Streamwise thickness
Induced drag transition ~ Prandtl-Jones to Polhamus
K ≡ dcD/dcL2 ; t ≡ ( K-KPJ ) /( KPo-KPJ )
Prandtl-JonesdcD/dcL
2 = 1/(pA)
Polhamus cD ≈ a cL
dcD/dcL2 ≈ 1/(dcL/da)
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 and Drag Review and RenewCorrelating 50 Years of NACA / NASA Test Datafor the Effects of Wing Planform and Thickness
21 April 2013 Update J. Philip Barnes Pelican Aero Group
cN
cF
cT
cT
voau
au
Pelican Aero Group
2Lift 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
Pelican Aero Group
3Lift 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
• 114 configurations, thickness: 02 - 20%
Pelican Aero Group
4Lift 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 sweepSweep conversion (given quarter-chord sweep)
tanLn = tanLc/4 + (4/A) (n-¼) (r-1) / (r+1)
b
c
t
Lo
Pelican Aero Group
5Lift 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
JonesLift slope (low-A, any-L)
dcL/da = p A/2Induced drag: cDv = cL
2/(pA)
PrandtlLift slope (any-A, low-L)
dcL/da ≈ 2pA/(A+2)Induced drag: cDv ≈ cL
2/(pA)
Prandtl-JonesInduced drag: cDv ≈ cL
2/(pA)But what about thickness?
Pelican Aero Group
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(dcL /da) / 2ph
Aspect Ratio, A
Normalized Lift-slope ~ Test Data Vs. Theory
-10 TO 1527 to 3438 to 4659 to 60
Loc/2
to
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
Pelican Aero Group
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Equivalent Lift Slope,
dcL /da
2phcosLc/2
Equivalent Aspect Ratio, A / (hcosLc/2)
Helmbold-Diederich Low-speed Lift-slope Condensation(NACA TN 2335)
-10 to 1522 to 3438 to 4659 to 60Theory
Loc/2
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
42cos2
/
)cos/(
22/
2/
L
L
F
Fddc
AF
c
L
c
phah
Pelican Aero Group
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Equivalent Lift Slope,
dcL /daAp/2
Equivalent Aspect Ratio, A / cosLc/2
Helmbold-Polhamus Low-speed Lift-slope Condensation(NACA TN 3911)
-10 to 1527 to 3438 to 4659 to 60Theo (eta=0.95)
Loc/2
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
222/ 4)cos/(2
4
)2/(
/
hhh
pa
L
c
L
AA
ddc
Pelican Aero Group
10Lift 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
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Lift Coefficient, cL
Angle of attack, ao
Delta-wing-body Lift (A=2) ~ Effect of ThicknessNACA RM A50K20, A50K21, A51K28
0.080.050.03
8%
5%
3%
Test data ~ Delta wing-body lift ~ effect of thickness
Minor effect of thickness on lift
Pelican Aero Group
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0.00 0.10 0.20 0.30 0.40 0.50 0.60
Induced Drag Coefficient, cD
Square of Lift Coefficient ~ cL2
Delta-wing-body Induced Drag (A=2) ~ Effect of ThicknessNACA RM A50K20, A50K21, A51K28
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
8%
5%
3%
Prandtl-Jones: cDv = cL
2 /(pA)
The Thin-wing Induced-drag Surprise ~ Circa 1950
Polham
us: c Dv
≈ a
c L ≈
c L2 /(d
c L/da
)
Induced dragcoefficient, cDv
Delta wing-body linearized drag polarA=2, M 0.25, NACA RM A50K20, A50K21, A51K28
Pelican Aero Group
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Induced Drag Coefficient, cD
Square of Lift Coefficient ~ cL2
Rectangular Wing Induced Drag (A=4) ~ Effect of ThicknessNACA TN 3501
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
Prandtl-Jones: cDv = cL
2 /(pA)
10%
6%
4%
Polha
mus
: cDv
≈ a
c L ≈
c L2 /(d
c L/da
)
Rectangular wing linearized drag polarA=4, Effect of thickness, NACA TN 3501
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t
Streamwise thickness
Induced drag transition ~ Prandtl-Jones to Polhamus
K ≡ dcD/dcL2 ; t ≡ ( K-KPJ ) /( KPo-KPJ )
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
PreliminaryEmpirical
Correlation
t = e-a(t/c)-b(t/c)2
Prandtl-JonesdcD/dcL
2 = 1/(pA)
Polhamus cD ≈ a cL dcD/dcL
2 ≈ 1/(dcL/da)
t/c
t ≡ [dcD/dcL2 - 1/(pA)] / [1/(dcL/da) - 1/(pA)]
Pelican Aero Group
14Lift 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
Assume elliptical loadingAssume small anglescL ≈ cN ≈ 2 ( + )p 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]:
NomenclatureA aspect ratiovo flight velocity
a angle of attacku 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
cN
cF
cT
cT
voau
au
cD ≈ cF + (cN2) / (pA)
+ (1-k) (cN2) / (2p)
"very thin": k → 0
"thick" : k→1
No sweepNo twist
@ k = 0: cD ≈ cF + cNa
consistent with Polhamus
Pelican Aero Group
15Lift 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 = cL
2 / (pA)
• Polhamus: induced drag upper limit– zero thickness, symmetrical section– formula: cDv ≈ acL ≈ cL
2 / (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
Pelican Aero Group
16Lift 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/d a (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/D12. Plot all results versus a or cL
Pelican Aero Group
17Lift 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
b
c
t
Lo
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:
Pelican Aero Group
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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."
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."
About the Author
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