Lift and Drag Review and Renew Correlating 50 Years of NACA / NASA Test Data

Pelican Aero Group

1

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22

t

Streamwise thickness

Induced drag transition ~ Prandtl-Jones to PolhamusK ≡ dcD/dcL

2 ; t ≡ ( K-KPJ ) /( KPo-KPJ )

Prandtl-JonesdcD/dcL

2 = 1/(pA)

Polhamus cD ≈ a cLdcD/dcL

2 ≈ 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

cFcT

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



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



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



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

6

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 1 2 3 4 5 6 7 8 9 10 11 12 13

(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 slope data and validation of theory

UnsweptPrandtl

Helmbold

SweptHelmbold-Polhamus

Helmbold-Diederich

Pelican Aero Group

7

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 1 2 3 4 5 6 7 8 9 10 11 12 13

Equivalent Lift Slope,

dcL /da2phcosLc/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


Helmbold-Diederich ~ Low-speed lift slope of any wing

42cos2/

)cos/(

22/

2/

L

L

F

FddcAF

c

L

c

phah

Pelican Aero Group

8

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

0 1 2 3 4 5 6 7 8 9 10 11 12 13

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


Helmbold-Polhamus ~ Low-speed lift slope of any wing

222/ 4)cos/(2

4)2/(

/

hh

hp

a

L

c

L

AAddc

Pelican Aero Group



0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 2 4 6 8 10 12 14 16 18 20

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

11

0.00

0.05

0.10

0.15

0.20

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


8%

5%

3%

Prandtl-Jones: cDv = cL2 /(pA)

The Thin-wing Induced-drag Surprise ~ Circa 1950

Polham

us: c Dv

≈ ac

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

12

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

Induced Drag Coefficient, cD

Square of Lift Coefficient ~ cL2

Rectangular Wing Induced Drag (A=4) ~ Effect of ThicknessNACA TN 3501


The Thin-wing Induced-drag Surprise ~ Circa 1950

Prandtl-Jones: cDv = cL2 /(pA)

10%

6%

4%

Polham

us: c Dv

≈ a

c L ≈

c L2 /(d

c L/da)

Rectangular wing linearized drag polarA=4, Effect of thickness, NACA TN 3501

Pelican Aero Group

13

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22

t

Streamwise thickness

Induced drag transition ~ Prandtl-Jones to PolhamusK ≡ dcD/dcL

2 ; t ≡ ( K-KPJ ) /( KPo-KPJ )


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



Effect of thickness on induced drag ~ symmetrical section

Assume elliptical loadingAssume small anglescL ≈ cN ≈ 2p (au) [1]u ≈ -cN /(pA) [2]cD ≈ cN a - cT cF [3]

Define thrust recovery: k ≡ cT / [cN tan(au)] ≈ cT / [cN (au)] [4]Combine [1,2,3,4]:

NomenclatureA aspect ratiovo flight velocitya 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

cFcT

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 + cNaconsistent with Polhamus

Pelican Aero Group



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 wings2) Suggested correlation for thick-to-thin drag transition3) New formula for induced drag with symmetrical sections

Pelican Aero Group



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/D12. Plot all results versus a or cL

Pelican Aero Group



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:[email protected]

Pelican Aero Group

18

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 Correlating 50 Years of NACA / NASA Test Data

Documents

Transcript of Lift and Drag Review and Renew Correlating 50 Years of NACA / NASA Test Data