Transcript Lift and Drag Review and Renew

Induced drag transition ~ Prandtl-Jones to Polhamus
Lift and Drag
Review
and
Renew
2
K ≡ dcD/dc
; t ≡of( NACA
K-KPJ )/ /(
KPo-KTest
Correlating
50LYears
NASA
PJ ) Data
for the Effects of Wing Planform and Thickness
1.2
t
21 April 2013 Update
J. Philip Barnes
Pelican Aero Group
1.0
0.8
Polhamus cD ≈ a cL
c
2
dcD/dcL ≈ 1/(dcL/da)
T
u
a
cN
0.6
0.4
vo
a
u
cT
cF
0.2
0.0
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
Pelican
Aero Group
1
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
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
Pelican
Aero Group
2
Configurations studied ~ Data and theory references
• www.NTRS.NASA.gov
• www.AERADE.Cranfield.ac.uk
• www.Google.com
• 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
Pelican
Aero Group
3
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
Lo
b
t
c
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
Pelican
Aero Group
4
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
Pelican
Aero Group
5
Lift slope data and validation of theory
Normalized Lift-slope ~ Test Data Vs. Theory
1.0
0.9
(dcL /da) / 2ph
0.8
0.7
0.6
0.5
0.4
0.3
to 15
-10 TO
27 to 34
38 to 46
59 to 60
0.2
Loc/2
0.1
Aspect Ratio, A
0.0
0
1
2
3
4
5
6
7
8
9
10
11
12
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
13
Pelican
Aero Group
6
Helmbold-Diederich ~ Low-speed lift slope of any wing
Helmbold-Diederich Low-speed Lift-slope Condensation
(NACA TN 2335)
1.0
Equivalent Lift
Slope,
dcL /da
0.9
0.8
2phcosL c/2
0.7
Loc/2
0.6
-10 to 15
22 to 34
38 to 46
59 to 60
Theory
0.5
0.4
F  A /(h cos L c / 2 )
0.3
dcL / da
F

2ph cos L c / 2 2  F 2  4
0.2
0.1
Equivalent Aspect Ratio, A / (hcosL c/2)
0.0
0
1
2
3
4
5
6
7
8
9
10
11
12
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
13
Pelican
Aero Group
7
Helmbold-Polhamus ~ Low-speed lift slope of any wing
Helmbold-Polhamus Low-speed Lift-slope Condensation
(NACA TN 3911)
1.2
1.1
Loc/2
1.0
-10 to 15
27 to 34
38 to 46
59 to 60
Theo (eta=0.95)
0.9
0.8
0.7
0.6
Equivalent Lift
Slope,
dcL /da
Ap/2
0.5
0.4
dcL / da
4h

A(p / 2) 2h  ( A / cos L c / 2 ) 2  4h 2
0.3
0.2
0.1
Equivalent Aspect Ratio, A / cosL c/2
0.0
0
1
2
3
4
5
6
7
8
9
10
11
12
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
13
Pelican
Aero Group
8
Test data ~ Delta wing-body lift ~ effect of thickness
Delta-wing-body Lift (A=2) ~ Effect of Thickness
NACA RM A50K20, A50K21, A51K28
0.8
5%
3%
8%
0.7
0.6
0.5
0.4
Lift Coefficient,
0.08
0.05
0.03
cL
0.3
0.2
Minor effect of thickness on lift
0.1
Angle of attack, a o
0.0
0
2
4
6
8
10
12
14
16
18
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
20
Pelican
Aero Group
10
The Thin-wing Induced-drag Surprise ~ Circa 1950
0.20
Delta wing-body
linearized
dragofpolar
Delta-wing-body
Induced Drag
(A=2) ~ Effect
Thickness
A=2,
M 0.25,
NACA RMA50K21,
A50K20, A51K28
A50K21, A51K28
NACA
RM A50K20,
3%
0.15
5%
8%
Induced drag
coefficient, cDv
Induced Drag
Coefficient, cD
0.10
0.05
Square of Lift Coefficient ~ cL2
0.00
0.00
0.10
0.20
0.30
0.40
0.50
0.60
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
Pelican
Aero Group
11
The Thin-wing Induced-drag Surprise ~ Circa 1950
0.07
Rectangular
wing
linearized
dragof
polar
Rectangular
Wing Induced
Drag
(A=4) ~ Effect
Thickness
A=4, Effect NACA
of thickness,
TN 3501NACA TN 3501
4%
0.06
Induced Drag
Coefficient, cD
0.05
6%
0.04
0.03
10%
0.02
0.01
Square of Lift Coefficient ~ cL2
0.00
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
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
0.40
Pelican
Aero Group
12
Induced drag transition ~ Prandtl-Jones to Polhamus
Induced-drag KTransition
≡ dcD/dcL2 ; ~t Prandtl-Jones
≡ ( K-KPJ ) /( KPo-KPJ to
) Polhamus
1.2
t
t ≡ [dcD/dcL2 - 1/(pA)] / [1/(dcL/da) - 1/(pA)]
1.0
Polhamus cD ≈ a cL
dcD/dcL2 ≈ 1/(dcL/da)
0.8
Preliminary
Empirical
Correlation
0.6
t = e-a(t/c)-b(t/c)
0.4
2
0.2
0.0
Prandtl-Jones
dcD/dcL2 = 1/(pA)
-0.2
0.00
0.02
0.04
0.06
Streamwise thickness t/c
0.08
0.10
0.12
0.14
0.16
0.18
0.20
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
0.22
Pelican
Aero Group
13
Effect of thickness on induced drag ~ symmetrical section
No sweep
No twist
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
cT
k
vo
cT
Define thrust recovery:
u
friction force coef. **
k ≡ cT / [cN tan(au)]
≈ cT / [cN (au)]
[4]
a
Combine [1,2,3,4]:
chord thrust coef. ***
cN
thrust recovery (0-1)
a
u
cT
* Usually negative
** Upper + lower, chordwise
*** Pressure integration,
chordwise
Assume elliptical loading
Assume small angles
cL ≈ cN ≈ 2p (au)
[1]
u ≈ -cN /(pA)
[2]
cD ≈ cN a - cT  cF [3]
@
cD ≈ cF + (cN2) / (pA)
+ (1-k) (cN2) / (2p)
"very thin": k → 0
cF
k = 0: cD ≈ cF + cNa
"thick" : k →1
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
Pelican
Aero Group
14
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
Pelican
Aero Group
15
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
Pelican
Aero Group
16
Sample application of method ~ homework assignment
Application: Me-163
Assume:
a) no twist, low Mach number
b) 9% thickness (t/c)
Lo
c) section h = 0.95
Measure from sketch:
b
a) Leading-edge sweep (Lo)
b) Span (b)
c) Root (centerline) & tip chords
Tasks:
t
c
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]
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
Pelican
Aero Group
17
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."
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
Pelican
Aero Group
18