Thin Aerofoil Theory notes
Transcript of Thin Aerofoil Theory notes
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To fnd correct combination o elementry ows overa specifed body
1. Source panel method
2. Vortex panel method
•. It become standard aerodynamics tool in industryand a research laboratories.
•. These are the numerical method appropriate orsolutions or a computers.
limitation to non!litin" ows
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Source sheet:
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Vortex filament:
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Vortex sheet:
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Vortex sheet over the airoilsurace#
Vortex sheet over the thin airfoil surface:dsdV 2rγ = − π
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Thin Airfoil theory:
Placement of the vortex sheet for thin airfoil analysis.
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Determination of the component of freestream velocity normal to
the camber line.
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Thin $iroil theory#
• Our purpose is to calculate the variation of (s) such
that the camber line becomes a streamline of the flow
and such that the utta condition is satisfied at the
trailin! ed!e" that is# γ $T%& ' (.• Once we have found the particular γ $s& that satisfies
these conditions# then the total circulation around
the airfoil is found by inte!ratin! γ $s& from the
leadin! ed!e to the trailin! ed!e.
• )n turn# the lift is calculated from Γ via the utta*
+ou,ows,i theorem.
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• The velocity at any point in the ow is the sum othe o the uniorm reestream velocity and thevelocity induced by the vortex sheet.
•
%et V∝&n be the component o the reestreamvelocity normal to the camber line.
• 'or a thin airoil at small an"le o attac(& both aresmall values. )sin" the approximation that orsmall θ& where θ& where θ is in radians& *+uation
reduces to
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-alculation o the induced velocity at the chordline.
( ) ( )
( )
c
0
dw x
2 x
γ ξ ξ= −
π − ξ∫
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)n this section# we treat the case of a symmetric airfoil. As state in section# a symmetri
airfoil has no camber" the camber line is coincident with the chord line.
-ence# for this case# d/d ' (# and %0uation becomes
( )
( )
c
0
ddzV 0
dx 2 x∞
γ ξ ξ α − − = ÷
π − ξ
∫
( )c
0
d1 dzV
2 x dx∞
γ ξ ξ = α − ÷π − ξ ∫
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• The help deal with the inte"ral in *+uations and&let us transorm into θ via the ollowin"transormation#
• Since x is a fxed point in *+uation and& itcorresponds to a particular value o θ& namely& θ&such that
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Substitutin" *+uations into& and notin" that the limits o inte"rationbecomes at the leadin" ed"e /where 0 and θ0π at the trailin" ed"e/where 0c& we obtain
cd sind
2ξ = θ θ
( )0
0
sind1V
2 cos cos
π
∞
γ θ θ θ= α
π θ − θ∫
( ) ( )1 cos2V
sin∞
+ θγ θ = α
θ
( ) ( )0 0
0 0
sind 1 cos d1 V
2 cos cos cos cos
π π∞γ θ θ θ + θ θθ=
π θ − θ π θ − θ∫ ∫
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0
00 0
sinncosnd
cos cos sin
π π θθ θ=θ − θ∫
( )
( )
0 0 00 0 0
1 cos dV V d cosd
cos cos cos cos cos cos
V 0 V
π π π∞ ∞
∞∞
+ θ θα α θ θ θ= + ÷π θ − θ π θ − θ θ −
α= + π = απ
∫ ∫ ∫
( )0
0
sin d1V
2 cos cos
π
∞γ θ θ θ = α
π θ −∫
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e are now n a pos t on to ca cu ate t e t coe c ent or a t n symmetr c a r o
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e are now n a pos t on to ca cu ate t e t coe c ent or a t n# symmetr c a r o .The total circulation around the airfoil is
4sin! %0uation and# e0uation transforms to
Substitutin! %0uation into# we obtain
Substitutin! %0uation into the utta*+ou,ows,i theorem# we find that the lift per unit span is
Substitutin! %0uation into# we have
( )c
0dΓ = γ ξ ξ∫
( )0
csind
2
πΓ = γ θ θ θ
∫ ( )
0cV 1 cos d cV
π
∞ ∞Γ = α + θ θ = πα∫
2L' V c V∞ ∞ ∞ ∞= ρ Γ = πα ρ
l
L'cqs∞
=
( )S c1=
( )
2
12
c Vc1Vc1
2
∞ ∞
∞ ∞
πα ρ=ρ
l
c 2= πα ldcLift slop= 2d
= π
α
( ) ( )1 cos2V
sin∞
+ θγ θ = α
θ
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-ence#
-owever# from %0uation#
5ombinin! e0uation and# we obtain
7rom %0uation# the moment coefficient about the 0uarter*chord point is
5ombinin! %0uation and# we have
'LE
m,le 2
Mc
qc 2∞
πα= = −
lc
2πα =
lm,le cc
4= −
l
m,c/4 m,le
c
c c 4= +
m,c/4c 0
=
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Important result#
Theoretical results or a symmetricairoil#
• -l 02πα.
• %it slope 0 2π.
•
The center o pressure and theaerodynamic center are both locatedat the +uarter!chord point.
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