Subsonic Airfoils

W.H. MasonConfiguration Aerodynamics Class

Typical Subsonic Methods: Panel Methods• For subsonic inviscid flow, the flowfield can be found by

solving an integral equation for the potential on the surface• This is done assuming a distribution of singularities along

the surface, and finding the “strengths” of the singularities• The airfoil is represented by a series of (typically) straight

line segments between “nodes”, and the nonpenetrationboundary condition is typically satisfied at control points

• Some version of a Kutta condition is required to close thesystem of equations.

1234

N + 1NN - 1

node

panel

Comparison of Panel Method Pressure Distributionwith Exact Conformal Transformation Results

-2.50

-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

PANELExact Conformal Mapping

Cp

x/c

Convergence with increasing numbers of panels

0.950

0.955

0.960

0.965

0.970

0.975

0.980

0 20 40 60 80 100 120

CL

No. of Panels

NACA 0012 Airfoil, α = 8°

A better way to examine convergence: Lift

0.9500.9550.9600.9650.9700.9750.980

0 0.01 0.02 0.03 0.04 0.05 0.06

CL

1/n

NACA 0012 Airfoil, α = 8°

Convergence with Panels: Moment

-0.250

-0.248

-0.246

-0.244

-0.242

-0.240

0 0.01 0.02 0.03 0.04 0.05 0.06

Cm

1/n

NACA 0012 Airfoil, α = 8°

Convergence with Panels: Drag

0.0000.0020.0040.0060.0080.0100.012

0 0.01 0.02 0.03 0.04 0.05 0.06

CD

1/n

NACA 0012 Airfoil, α = 8°

Pressures: 20 and 60 panels-5.00

-4.00

-3.00

-2.00

-1.00

0.00

1.000.0 0.2 0.4 0.6 0.8 1.0

20 panels60 panels

CP

x/c

NACA 0012 airfoil, α = 8°

Pressures: 60 and 100 panels-5.00

-4.00

-3.00

-2.00

-1.00

0.00

1.000.0 0.2 0.4 0.6 0.8 1.0

60 panels100 panels

CP

x/c

NACA 0012 airfoil, α = 8°

Comparison with WT Data: Lift- recall: panel methods are inviscid! -

-0.50

0.00

0.50

1.00

1.50

2.00

2.50

-5.0° 0.0° 5.0° 10.0° 15.0° 20.0° 25.0°

CL, NACA 0012 - PANEL

CL, NACA 0012 - exp. data

CL, NACA 4412 - PANEL

CL, NACA 4412 - exp. data

CL

α

NACA 0012

NACA 4412

Comparison with Data: Pitching Moment- about the quarter chord -

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

-0.00

0.05

0.10

-5.0 0.0 5.0 10.0 15.0 20.0 25.0

Cm, NACA 0012 - PANELCm, NACA 4412 - PANELCm, NACA 0012 - exp. dataCm, NACA 4412 - exp. data

Cm

α

c/4

NACA 0012

NACA 4412

For Completeness: Drag DataEffect of Camber

-0.50

0.00

0.50

1.00

1.50

2.00

0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018

CL

CD

Re = 6 million

NACA 4412

NACA 0012

data from Abbott and von Doehhoff

Comparison with WT Pressure Distribution-1.2

-0.8

-0.4

-0.0

0.4

0.8

1.20.0 0.2 0.4 0.6 0.8 1.0 1.2

α = 1.875°M = .191Re = 720,000transition free

Cp

x/c

NACA 4412 airfoil

Predictions from PANEL

data from NACA R-646

XFOIL: the code for subsonic airfoils

• Panel Methods: Inviscid!• Couple with a BL analysis to include

viscous effects• The single element viscous subsonic airfoil

analysis method of choice: XFOIL– by Prof. Mark Drela at MIT

• Link available from my software site

Airfoil pressures: What to look for-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00-0.1 0.1 0.3 0.5 0.7 0.9 1.1

CP

x/c

NACA 0012 airfoil, α = 4°

Trailing edge pressure recovery

Expansion/recovery around leading edge(minimum pressure or max velocity, first appearance of sonic flow)

upper surface pressure recovery(adverse pressure gradient)

lower surface

Leading edge stagnation point

Rapidly accelerating flow,favorable pressure gradient

Effect of Angle of Attack-5.00

-4.00

-3.00

-2.00

-1.00

0.00

1.00-0.1 0.1 0.3 0.5 0.7 0.9 1.1

α = 0°α = 4α = 8CP

x/c

NACA 0012 airfoilInviscid calculation from PANEL

Comparison of NACA 4-Digit Airfoils0006, 0012, 0018

-0.30

-0.20

-0.10

-0.00

0.10

0.20

0.30-0.1 0.1 0.3 0.5 0.7 0.9 1.1

NACA 0006 (max t/c = 6%)NACA 0012 (max t/c = 12%)NACA 0018 (max t/c = 18%)

y/c

x/c

Thickness Effects on Airfoil PressuresZero Lift Case

-1.00

-0.50

0.00

0.50

1.00-0.1 0.1 0.3 0.5 0.7 0.9 1.1

NACA 0006, α = 0°NACA 0012, α = 0°NACA 0018, α = 0°

CP

x/c

Inviscid calculation from PANEL

Thickness Effects on Airfoil Pressures, CL = 0.48-3.00

-2.50

-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00-0.1 0.1 0.3 0.5 0.7 0.9 1.1

NACA 0006, α = 4°

NACA 0012, α = 4°

NACA 0018, α = 4°

CP

x/c

Inviscid calculation from PANEL

Comparison of NACA 4-Digit Airfoilsthe 0012 and 4412

-0.30

-0.20

-0.10

-0.00

0.10

0.20

0.30

-0.1 0.1 0.3 0.5 0.7 0.9 1.1

NACA 0012 (max t/c = 12%)NACA 4412 foil (max t/c = 12%)

y/c

x/c

Highly Cambered Airfoil Pressure Distribution- NACA 4412 -

-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00-0.1 0.1 0.3 0.5 0.7 0.9 1.1

NACA 4412, α = 0°NACA 4412, α = 4°

CP

x/c

Note: For a comparison of cambered and uncambered presuure distributions at the same lift, see Fig. 18.

Inviscid calculation from PANEL

Camber Effects on Airfoil Pressures, CL = 0.48-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00-0.1 0.1 0.3 0.5 0.7 0.9 1.1

NACA 0012, α = 4°

NACA 4412, α = 0°

CP

x/c

Inviscid calculation from PANEL

Camber Effects on Airfoil Pressures, CL = 0.96-4.00

-3.00

-2.00

-1.00

0.00

1.00-0.1 0.1 0.3 0.5 0.7 0.9 1.1

NACA 0012, α = 8°NACA 4412, α = 4°

CP

x/c

Inviscid calculations from PANEL

Camber Effects on Airfoil Pressures, CL = 1.43-6.00

-5.00

-4.00

-3.00

-2.00

-1.00

0.00

1.00-0.1 0.1 0.3 0.5 0.7 0.9 1.1

NACA 0012, α = 12°NACA 4412, α = 8°

CP

x/c

Inviscid calculations from PANEL

NACA 6712 Airfoil- Heavy Aft Camber Geometry -

-0.05

0.05

0.15

-0.1 0.1 0.3 0.5 0.7 0.9 1.1

y/c

x/c

NACA 6712 Airfoil- Heavy Aft Camber, Pressure Distribution -

-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00-0.1 0.1 0.3 0.5 0.7 0.9 1.1

α = -.6 (CL = 1.0)

CP

x/c

NACA 6712

Inviscid calculations from PANEL

Whitcomb GA(W)-1 Airfoil

-0.10-0.050.000.050.100.15

0.0 0.2 0.4 0.6 0.8 1.0

y/c

x/c-1.00

-0.50

0.00

0.50

1.000.0 0.2 0.4 0.6 0.8 1.0

Cp

x/c

GA(W)-1

α = 0°

Inviscid calculations from PANEL

Liebeck’s Hi-Lift Airfoil: Geometry and Lift- note shape of pressure recovery -

From R.T. Jones, Wing Theory

Liebeck’s Hi-Lift Airfoil: Drag

From Bertin,Aerodynamics for Engineers

Camberline Design: DesCam

0.00

0.02

0.04

0.06

0.08

0.10

0.12

-0.50

0.00

0.50

1.00

1.50

2.00

0.0 0.2 0.4 0.6 0.8 1.0

(Z-Z0)/C - DesCamZ/C - from Abbott & vonDoenhoff

Z/CΔCP

X/C

Design ChordLoading

Airfoil Selection

Issues:• Cruise CL, and CLmax, don’t forget Cm0

-large LE radius?-Near parallel trailing edge closure

• Profile Drag: Laminar flow? • Thickness for low weight and internal volume• Tails: often symmetric, 6 series foils picked

To Conclude

You have the tools to do single element airfoil design