thin airfoil.ppt

8/13/2019 thin airfoil.ppt

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Chap.4

Incompressible Flow overAirfoils


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OUTLINE

Airfoil nomenclature and characteristics

The vortex sheet

The Kutta condition

Kelvins circulation theorem

Classical thin airfoil theory

The cambered airfoil

The vortex panel numerical method


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Airfoil nomenclature and characteristics

Nomenclature


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Characteristics


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The vortex sheet

Vortex sheet withstrength =(s) Velocity at Pinduced by

a small section of vortexsheet of strength ds

For velocity potential (toavoid vector addition asfor velocity)

r

dsdV

2

2

dsd


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The velocity potential at Pdue to entire vortex sheet

The circulation around thevortex sheet

The local jump in tangentialvelocity across the vortexsheet is equal to .

b

ads

2

1

b

a

ds

0,21 dnuu


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Calculate (s) such that the induced velocity field

when added to Vwill make the vortex sheet(hence the airfoil surface) a streamline of the flow.

The resulting lift is given by Kutta-Joukowskitheorem

Thin airfoil approximation

VL


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The Kutta condition

Statement of the Kutta condition

The value of around the airfoil is such that theflow leaves the trailing edge smoothly.

If the trailing edge angle is finite, then the trailingedge is a stagnationpoint.

If the trailing edge is cusped, then the velocityleaving the top and bottom surface at the trailingedge are finite and equal.

Expression in terms of

0)TE(


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Kelvins circulation theorem

Statement of Kelvins circulation theorem

The time rate of change of circulationaround aclosed curve consisting of the same fluid elementsis zero.


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Classical thin airfoil theory

Goal

To calculate (s) such that the camber linebecomes a streamline.

Kutta condition (TE)=0 is satisfied. Calculate around the airfoil.

Calculate the lift via the Kutta-Joukowski theorem.


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Approach Place the vortex sheet on

the chord line, whereasdetermine =(x) to make

camber line be astreamline.

Condition for camber linetobe a streamline

where w'(s)is thecomponent of velocitynormal to the camber line.

0)(, swV n


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Expression of V,n

For small

)(tansin1

,dx

dzVV n

)(

)()(,tansin

,dx

dzVV

xwsw

n


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Expression for w(x)

Fundamental equation ofthin airfoil theory

)()(

2

1

0 dx

dzV

x

dc

c

x

dxw

0 )(2

)()(


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For symmetric airfoil (dz/dx=0)

Fundamental equation for ()

Transformation of , xinto

Solution

Vx

dc

0

)(

2

1

)cos1(2

,)cos1(2

0 c

xc

sin

)cos1(2)( V


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Check on Kutta condition by LHospitals rule

Total circulation around the airfoil

Lift per unit span

0cos

sin2)(

V

cVdc

0)(

2

VcVL


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Lift coefficient and lift slope

Moment about leading edge and moment coefficient

2,2d

dc

cq

Lc ll

42

2

2,

2

0

lLE

lem

c

LE

c

cq

M

c

cqLdM


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Moment coefficient about quarter-chord

For symmetric airfoil, the quarter-chord pointisboth the center of pressureand the aerodynamiccenter.

0

4

4/,

,4/,

cm

llemcm

c

ccc


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The cambered airfoil

Approach

Fundamental equation

Solution

CoefficientsA0andAn

)(

coscos

sin)(

2

1

00 dx

dzV

d

1

0 sinsin

cos12)(

n

n nAAV

0

000

00 cos2

,1

dndx

dzAd

dx

dzA n


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Aerodynamic coefficients Lift coefficient and slope

Form thin airfoil theory, the lift slopeis always 2for any shape airfoil.

Thin airfoil theory also provides a means to

predict the angle of zero lift.

2,)1(cos1

20

00

d

dcd

dx

dzc ll

000

0 )1(cos1

ddx

dzL


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Moment coefficients

For cambered airfoil, the quarter-chord point isnotthe center of pressure, but still is the

theoretical location of the aerodynamic center.

)(4

)(44

124/,

21,

AAc

AAc

c

cm

llem


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The location of the center of pressure

Since

the center of pressure is not convenient fordrawing the force system. Rather, theaerodynamic center is more convenient.

The location of aerodynamic center

)(1

421 AA

c

cx

l

cp

0as lcp cx

0

4/,

0

0

0 ,where,25.0 md

dca

d

dc

a

mx

cmlac


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The vortex panel numerical method

Why to use this method

For airfoil thickness larger than 12%, or high angle ofattack, results from thin airfoil theory are notgoodenough to agree with the experimental data.

Approach

Approximate the

airfoil surface by

a series of straightpanels with strength

which is to be

determined.

j


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The velocity potential induced at P due to thej thpanel is

The total potential at P

Put P at the control point of ith panel

j

j

pjjjj

pjjxx

yyds

1tan,2

1

2

)(11

jj

pj

n

j

jn

j

j dsP

2

),(1

jj

ij

n

j

j

ii dsyx


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The normal component of the velocity is zero atthe control points, i.e.

We then have n linear algebraic equation with nunknowns.

nidsn

V

dsnVVV

VV

jj

i

ijn

j

j

i

jj

i

ijn

j

j

nin

nn

,,1,02

cos

2,coswhere

0

1

1

,

,


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Kutta condition

To impose the Kutta condition,we choose to ignore one ofthe control points.

The need to ignore one of the

control points introducessome arbitrariness in thenumerical solution.

10)TE( ii

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