5. thin airfoil

CAIRO UNIVERSITY

FACULTY OF ENGINEERING

AEROSPACE DEPARTMENT

THIRD YEAR STUDENTS

FIRST TERM

Course Title: AERODYNAMICS (A)

Course Code: AER 301 A

PROF. Dr. MOHAMED MADBOULI ABDELRAHMAN

1

2

Thin Airfoil

Theory

Assume steady, 2-D, non viscous, irrotational, incompressible flow The solution of the airfoil problem may be obtained as

the superposition of the solutions of the following two simple problems:

Thin Airfoil Theory

1) Cambered airfoil with angle of attack

2) Symmetric airfoil at zero angle of attack y

y(x)

3

Assume steady, 2-D, non viscous, irrotational, incompressible flow The solution of the airfoil problem may be obtained as

the superposition of the solutions of the following two simple problems:

Thin Airfoil Theory

1) Cambered airfoil with angle of attack

2) Symmetric airfoil at zero angle of attack y

y(x)

4

The flow field may be represented by a distribution of vortices γ(x) along the x-axis on the chord line

The γ(x) distribution can be determine by applying the boundary condition (velocity must be tangent to the camber)

For small angle of attack and small camber the boundary condition can be written as “dy/dx ≈ α + v/V∞” where “v” is the velocity component in y-direction

Thin Airfoil Theory (Cambered airfoil with angle of attack)

c

0

1

1

dx)xx

(V2

1

dx

dy

y y

y(x) 5

Thin Airfoil Theory

c

0

1

1

dx)xx

(V2

1

dx

dy

To solve this integral-differential equation

Change the variables x and x1 to θ and θ1 with

Assume γ can be written in the following series form as

After substitution, the governing equation will be

)cos1(2

cx

)]nsin(Asin

)cos1(A[V2 1n

11

10

0

1

1

11n

0

1

1

10 dcoscos

sin)nsin(A1d)

coscos

)cos1(A1

dx

dy

0

1

1

11 )ncos(dcoscos

sin)nsin(

0

1

1

1

sin

)nsin(d

coscos

)ncos(

Using the following identities

6

Thin Airfoil Theory we can prove that

)nsin(AAdx

dyn

10

To calculate the coefficients A0 and An for n = 1 to ∞

we integrate this equation as

0

n1

0

0

d)nsin(AAddx

dy

0

n1

0

0

d)mcos()nsin(AAd)mcos(dx

dy

7

Thin Airfoil Theory

0

)mn(when0d)mcos()ncos(

and using the following identities

0

)mn(when)2

(d)mcos()ncos(

we can prove that

0

0 ddx

dy)

1(A

0

n d)ncos(dx

dy)

2(A&

8

Thin Airfoil Theory As a conclusion, if we have the camber line equation of an airfoil section given by “y = f(x)” then we can calculate the distribution of vortices γ(x) along the x-axis by

where

0

0 ddx

dy)

1(A

0

n d)ncos(dx

dy)

2(A&

9

)]nsin(Asin

)cos1(A[V2 n

10

with A0 and An are calculated by )cos1(2

cx

Thin Airfoil Theory

The lift force

c

0

dxVVL

where

)(2)2/AA(2C 010L

0

0 d)cos1)(dx

dy()

1(

)]nsin(Asin

)cos1(A[V2 n

10

)cos1(2

cx and

We can

prove that

Then

10

1cCV2

1

2

AAcVL L

210

2

where

Thin Airfoil Theory

The pitching moment about the leading edge point

c

0

LE xdxVM

)2/AAA)(2/(C 210MLE then

11

where

)]nsin(Asin

)cos1(A[V2 n

10

)cos1(2

cx and

We can prove that

c1cCV2

1

2

AAAcV

4M

LEM

2210

22

LE

Thin Airfoil Theory The pitching moment about any arbitrary reference point

c

xCCC LMM LEx

Neglecting the moment contribution due to drag, the pitching moment about any arbitrary reference point can be related by the pitching moment about the leading edge point by the following equation

12

Thin Airfoil Theory Determination of the center of pressure point

The center of pressure point is defined as that point about which the pitching moment is zero. Neglecting the moment contribution due to drag, it can be seen that

13

)AA(

LCC

C

c

cpx

L

M LE

2144

1

0

c

xCCC

cp

LMM LEcp

then

Thin Airfoil Theory Determination of the aerodynamic center point

14

c

xCCC ac

LMM LEac

A very important reference point on an airfoil is the aerodynamic center. The aerodynamic center is defined as that point about which the variation of the pitching moment with angle of attack is zero. Neglecting the moment contribution due to drag, it can be seen that

4

1

LC

C

c

acx

LEM

Then

0

c

xCCC acLMMac LE

Thin Airfoil Theory

Determination of the aerodynamic center point

c

xxCCC ac

LMM xac

The aerodynamic center (point about which the pitching moment is independent on the angle of attack) can be calculated using any arbitrary reference point as

0c

xxCCC acLMMac x

LC

C

c

x

c

acx

xM

&

15

Then

Using experimental data of CM versus CL it is possible to compute the location of the aerodynamic center xac. For low subsonic flow it is found that the aerodynamic center is at the quarter chord point.

Thin Airfoil Theory The pitching moment coefficients and center of pressure

)2/AAA)(2/(C 210MLE

and )AA)(4/(C 21M 4/c

)2

A1

A(

LC44

1

c

cpx

and 16

Thin Airfoil Theory Using the small perturbation for thin airfoil the

velocity on the airfoil surface can be approximated by

17

uVVanduVV lu

V

u

V

u

V

u

V

Vu 21

21

22

V

u

V

u

V

u

V

Vl 21

21

22

then

2

ll

22

uu V2

1pV

2

1pV

2

1p

If

and

using B.E.

then VuVpp ul 2

u2and

Thin Airfoil Theory

The velocity distribution over thin airfoil

18

The velocity distribution over the upper and lower surface of a thin airfoil can be calculated by

22

VVandVV lowerupper

where )]nsin(Asin

)cos1(A[V2 n

10

)cos1(2

cx and

Thin Airfoil Theory

The pressure coefficient can be calculated using

Bernoulli equation

19

The pressure distribution over the upper and lower surface of a thin airfoil can be calculated by

222

2

1

2

1

2

1lluu VpVpVp

where

)]nsin(Asin

)cos1(A[V2 n

10

)cos1(2

cx

and

22

VVandVV lu

with

22

11

V

VCand

V

VC l

pu

p luor

Thin Airfoil Theory

The chord-wise velocity distribution

2VV l,u

where )]nsin(Asin

)cos1(A[V2 n

10

)cos1(2

cx and

The chord-wise pressure distribution

V)VV(Vpp luul

)V/2()CC(Cullu ppp

20

Thin Airfoil Theory For symmetric airfoil (dy/dx=0), the vortex strength is

21 )V/2()CC(Cul ppp

where

then

and

where

Thin Airfoil Theory

22

)]nsin(Asin

)cos1(A[V2 n

10

)ncos(AAdx

dyn

10

)cos1(2

cx

Thin Airfoil Theory

By applying the boundary condition (velocity tangent to the camber line)

with x = 0 at the L.E. point and x = c at the T.E. point.

where

For a zero thickness cambered airfoil "y(x)" placed in a free stream velocity "V" with an angle of attack "α" the flow-field may be presented by a the following distribution of vortices "(x)" along the chord line "c" as

23

)ncos(AAdx

dyn

10

Thin Airfoil Theory

to1nforI2

Aand)I

(A nn

00

to1nford)ncos()dx

dy(Iandd)

dx

dy(I

0

n

0

0

The coefficients A0 & An for n = 1, 2, 3, …. are given by

where

The aerodynamic characteristics for this airfoil can be determined by:

]AA[4

)C(

]2

AAA[

2)C(

]2

AA[2C

214/cM

210LEM

10L

4

1)

c

x(

)C4(

)AA(

4

1)

c

x(

]2

AA[

ac

L

21cp

100

24

)ncos(AAdx

dyn

10

Thin Airfoil Theory

&

]II[2

1)C(

]II2I[2

1)C(

]II[2C

214/cM

210LEM

10L

4

1)

c

x(

)II(4

)II(

4

1)

c

x(

)II(

ac

10

21cp

100

V)VV(Vpp&2

VV&2

VV luullu

The aerodynamic characteristics for this airfoil can be determined in terms of “Io“ & ”In ” by:

where

25

Thin Airfoil Theory

mc

x

c

xa

c

xa

c

xaa

c

y )(0)()()( 3

3

2

2101

1)()()()( 3

3

2

2102

c

xm

c

xb

c

xb

c

xbb

c

y

])[2

)2sin((])[(sin])[(][ 221221112010200 tttttttI m

mm

]][6

)3sin([]][

4

)2sin([]

22)[(sin])[

2()

2( 22122111

221220102111211 tttt

tttttttI mm

mm

]][8

)4sin([]][

6

)3sin([]][

2

)2sin([])[

2

sin(])[

2(])[

2( 22122111201021112212222 tttttttttttI mmmmm

3123211321108

3

2

3

8

921cos ataataaatmm

3223221321208

3

2

3

8

9btbbtbbbt

For camber line airfoil given by the following 2 equations:-

The integrations I0, I1 and I2 can be calculated by

with

27

28

For a flat plate wing section airfoil where the mean camber line equation is

(y/c) = 0 for 0 =< (x/c) =< 1

Using thin airfoil theory we can calculate the following

the zero lift angle of attack αo = 0 rad = 0o,

the pitching moment coefficient at the aerodynamic center (Cm)ac = 0,

When the angle of attack is “α=8o ”,

the lift coefficient CL = 0.8773,

the pitching moment coefficient at the leading edge (Cm)LE = −0.2193

the center of pressure (xc.p/c) = 0.25

the aerodynamic center (xac/c) = 0.25

Thin Airfoil Theory

Flat plate

29

The NACA 25012 wing section airfoil has a mean camber line given by

(y/c) = 0.5383 (x/c)3 - 0.6315 (x/c)2 + 0.2147 (x/c) for 0 =< (x/c) =< 0.391

(y/c) = 0.0322 (1 - x/c ) for 0.391 =< (x/c) =< 1

Using thin airfoil theory we can calculate the following

the zero lift angle of attack αo = − 0.0259 rad = −1.483o,

the pitching moment coefficient at the aerodynamic center (Cm)ac = −0.0244,

When the angle of attack is “α=10o ”,

the lift coefficient CL = 1.26,

the pitching moment coefficient at the leading edge (Cm)LE = −0.339

the pitching moment coefficient at the half chord point (Cm)C/2= 0.29

the center of pressure (xc.p/c) = 0.27

the aerodynamic center (xac/c) = 0.25

Thin Airfoil Theory

NACA 25012

30

Thin Airfoil Theory

Flat plate with a trailing edge flap

31

Consider a flat plate airfoil with a trailing edge flap

as shown in figure, the flap chord to total chord

are "λ= cf/c" and the flap deflection angle are δ.

The expressions for the lift coefficient CL, the

pitching moment coefficient about the leading

edge point CM(LE), the pitching moment coefficient

about the aerodynamic center Cm(ac), the zero lift

angle of attack αo as functions of λ, δ, and the

angle of attack "α" using the thin airfoil theory can

be written as

Thin Airfoil Theory

Flat plate with a trailing edge flap

λ, δ

V∞, α After deflection

V∞, α c

Before deflection

cf

ffL

L

LLL

sin2C

and2Cwith

CCC

2

2sinsin2

2

1C

and2

Cwith

CCC

fff)LE(M

)LE(M

)LE(M)LE(MM )LE(

)12(cos 1

f

where

32

Thin Airfoil Theory

Flat plate with a trailing edge flap (cont.)

λ, δ

V∞, α After deflection

V∞, α c

Before deflection

cf

)sin(

)cos1(sin

44

1

c

x

ff

ffcp

)12(cos 1

f where

2

2sinsin

2

1C

and0Cwith

CCC

ff)ac(M

)ac(M

)ac(M)ac(MM )ac(

ff

0

sin)1(

33

Thin Airfoil Theory


λ, δ

V∞, α After deflection V∞, α c

Before deflection

cf

)12(cos 1

f where

fff

2

ff2)H(M

ffff2)H(M

)H(M)H(MM

sin)(4cos21)(2)2cos(1(4

1C

and)2sin(sin4)1cos2)((24

1Cwith

CCC)H(

H

The hinge moment coefficient due to flap deflection

34

Thin Airfoil Theory

Flat plate with a trailing edge flap (cont.) If the flap chord to total chord “ λ = cf /c = 1/4”, the flap

deflection angle ” δ = 4o ”, and the angle of attack

“ α = 8o “. Using thin airfoil theory :-

Before deflection

the lift coefficient

“ CL = 0.877 ”

the pitching moment

coefficient about the

leading edge point

“ CM(LE) = - 0.219 ”

the center of pressure

point “ xcp/c = 0.25 ”

After deflection


“ CL = 1.144 ”

the pitching moment


leading edge point

“ CM(LE) = - 0.331 ”



35

Thin Airfoil Theory

Flat plate with a trailing edge flap (cont.) If the flap chord to total chord “ λ = cf /c = 0.2 ”, the flap



Before deflection


“ CL = 1.316 ”

the pitching moment


leading edge point

“ CM(LE) = - 0.329 ”



After deflection


“ CL = 1.678 ”

the pitching moment


leading edge point

“ CM(LE) = - 0.486 ”



36

Thin Airfoil Theory


If the flap chord to total chord “ λ = cf /c = 0.2 ”, the flap



After deflection

the hinge moment coefficient “ CM(H) = - 0.201 ”

923.0Cand5.0Cwith

CCC

)H(M)H(M

)H(M)H(MM )H(

37

λ1,δ1

λ2,δ2 Voo, α c

Voo, α Before deflection After deflection

Problem Consider a flat plate airfoil with double trailing edge flap as shown in

figure, the flap chord to total chord are "λ1= cf1/c" and "λ2= cf2/c". The flap

deflection angles are δ1 and δ2, respectively.

•Derive the lift coefficient CL as function of λ1, λ2, δ1, δ2 and the angle of attack

"α" using the thin airfoil theory.

•What is the change in the lift coefficient due to flap deflections δ1=5o and δ2 =

5o for λ1 = 0.2 and λ2 = 0.2 when α = 10o.

Thin Airfoil Theory Problem Consider the NACA 6412 wing section airfoil with a mean camber line

given by

(y/c) = 0.3 (x/c) - 0.375 (x/c)2 for 0 =< (x/c) =< 0.4

(y/c) = 0.0333 + 0.1332 (x/c) – 0.1665 (x/c) 2 for 0.4 =< (x/c) =< 1.0

Using thin airfoil theory calculate the zero lift angle of attack “αo” and the

pitching moment coefficient at the aerodynamic center (Cm)ac . When the angle

of attack is “α=10o ”, find the lift coefficient CL, the pitching moment coefficient

at the leading edge (Cm)LE, the pitching moment coefficient at the half chord

point (Cm)C/2, and the center of pressure (Xc.p/c).

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END OF THIN AIRFOIL THEORY

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5. thin airfoil

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