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Aerodynamics
• Geometric properties of a wing
- Finite thickness much smaller than the span and
the chord
Finite WingsSteady, incompressible flow
Masters of Mechanical Engineering
the chord
- Definition of wing geometry:
a) Planform (variation of chord and sweep angle)
b) Section/Airfoil (thickness and camber)
c) Twist angle
d) Dihedral angle
Aerodynamics
• Geometric properties of a wing
- Cartesian coordinate system
Finite WingsSteady, incompressible flow
Masters of Mechanical Engineering
- Ox, longitudinal wing axis, positive
direction pointing to the rear
- Oy, lateral wing axis, perpendicular
to the symmetry plane
- Oz, vertical wing axis, positive upwards
Aerodynamics
• Geometric properties of a wing
a) Planform (variation of chord and sweep angle)
Finite WingsSteady, incompressible flow
Masters of Mechanical Engineering
Rectangular wing
Span, b
Chord, c
Sweep
Area,c
b
S
b==Λ
2
Aspect ratio,
Mean chord,b
Sc =
( )∫−=2
2
b
bdyycS
Aerodynamics
• Geometric properties of a wing
b) Section/airfoil (thickness and camber)
Finite WingsSteady, incompressible flow
Masters of Mechanical Engineering
Aerodynamics
• Geometric properties of a wing
c) Twist angle
Finite WingsSteady, incompressible flow
Masters of Mechanical Engineering
Twist angle
Aerodynamics
• Geometric properties of a wing
d) Dihedral angle
Finite WingsSteady, incompressible flow
Masters of Mechanical Engineering
Dihedral Dihedral
Dihedral Dihedral
Aerodynamics
• The (positive) lift force is a consequence of the
surface pressure distribution. The average
pressure on the lower surface is larger than the
upper surface average pressure
Finite WingsSteady, incompressible flow
Masters of Mechanical Engineering
upper surface average pressure
• This pressure difference causes a secondary flow
around the tip (from the lower to the upper surface)
that guarantees equal pressure at the tip
Aerodynamics
• The upper surface streamlines are displaced to the
symmetry plane of the wing and the lower surface
streamlines to the tip, generating longitudinal
vorticity in the wake (free vortex sheet)
Finite WingsSteady, incompressible flow
Masters of Mechanical Engineering
vorticity in the wake (free vortex sheet)
Aerodynamics
Finite WingsSteady, incompressible flow
• The free vortex sheet rolls-up around two
concentrated vortices close to the tip of the
wing (tip vortices)
Masters of Mechanical Engineering
Aerodynamics
Finite WingsSteady, incompressible flow
• The free vortex sheet rolls-up around two
concentrated vortices close to the tip of the
wing (tip vortices)
Masters of Mechanical Engineering
Aerodynamics
Finite WingsSteady, incompressible flow
• The free vortex sheet rolls-up around two
concentrated vortices close to the tip of the
wing (tip vortices)
Masters of Mechanical Engineering
Aerodynamics
Top view of the wing
Transversal flow around the tip←
←
Finite WingsSteady, incompressible flow
Masters of Mechanical Engineering
Transversal flow at the trailing
edge
Roll-up of the vortex sheet
Tip vortices←
←
←
←
Aerodynamics
• Ideal fluid models to simulate tip effect.
Simplest alternative: Horseshoe vortex
Finite WingsSteady, incompressible flow
Masters of Mechanical Engineering
• Qualitative representation of tip effect
Aerodynamics
• The vortex wake generates a vertical velocity
component : downwash between the two tips
and upwash outside the wing tips
Finite WingsSteady, incompressible flow
Masters of Mechanical Engineering
Aerodynamics
Finite WingsSteady, incompressible flow
Masters of Mechanical Engineering
Aerodynamics
Finite WingsSteady, incompressible flow
Masters of Mechanical Engineering
Aerodynamics
• The circulation around the wing sections, Γ,
is related to the lift force per unit width by
the Joukowski equation (positive lift corresponds
to negative Γ)
Finite WingsLifting line theory
Masters of Mechanical Engineering
to negative Γ)
• The circulation around the wing may be modelled
by a vortex line (along the spanwise direction)
between the two wing tips with an intensity, Γ(y),
defined by the lift/circulation of each wing section
(airfoil). Such vortex is called bound vortex or
lifting line
Aerodynamics
• Conservation of circulation in space (Helmohtz
theorem) implies that the change of intensity
of the bound vortex (Γ(y) must be zero at the wing
Finite WingsLifting line theory
Masters of Mechanical Engineering
of the bound vortex (Γ(y) must be zero at the wing
tip) must be linked to a system of trailing vortices.
This vortex sheet along the wake has an intensity,
γ, defined by the change of circulation along the
lifting line
Aerodynamics
• The vortex sheet along the wake tends to be
aligned with the flow streamlines and to roll-up
around the tip vortices. This is a non-linear
problem (unknown location of the wake vortex sheet)
Finite WingsLifting line theory
Masters of Mechanical Engineering
problem (unknown location of the wake vortex sheet)
Aerodynamics
• Linearized theory (Prandtl)
For small angles of attack, thickness and camber
(small perturbations) the wake trailing vortices
are approximately aligned with the undisturbed
Finite WingsLifting line theory
Masters of Mechanical Engineering
are approximately aligned with the undisturbed
uniform incoming flow
Lifting line
trailing vortices
Aerodynamics
• Linearized theory (Prandtl)
For this simplified model, the trailing vortices
are lines with a known location and so the
problem becomes linear
Finite WingsLifting line theory
Masters of Mechanical Engineering
problem becomes linear
trailing vortices
Lifting line
Aerodynamics
• Linearized theory (Prandtl)
The wing is modelled by a vortex system including
the bound vortex and a plane vortex sheet along
the wake aligned with the undisturbed incoming flow
Finite WingsLifting line theory
Masters of Mechanical Engineering
the wake aligned with the undisturbed incoming flow
Lifting line
trailing vortices
Aerodynamics
• Linearized theory (Prandtl)
The trailing vortices intensity, γ, is related to the
bound vortex circulation, Γ(y), (Helmothz theorem)
dΓ
Finite WingsLifting line theory
Masters of Mechanical Engineering
( ) ( ) dydy
dydyyd
Γ=Γ−+Γ=γ
Aerodynamics
• Linearized theory (Prandtl)
A three-dimensional velocity field is induced
by the vortex system. The induced velocity of
a semi-infinite vortex line (by comparison to an
Finite WingsLifting line theory
Masters of Mechanical Engineering
a semi-infinite vortex line (by comparison to an
infinite vortex line) is given by , with a direction
perpendicular to the vector rπ
γ
4rr
Aerodynamics
Finite WingsLifting line theory
• Linearized theory (Prandtl)
Trailing vortices induced (descending) velocity
at a point y on the lifting line (without sweep)
Masters of Mechanical Engineering
'''
1
4
1 2
2dy
dy
d
yy
b
bi ∫−
Γ
−=
πω
trailing vortices
Aerodynamics
• Assuming two-dimensional flow (valid for large
aspect ratio) at each wing section (airfoil) and an
approximately uniform wake induced velocity, ωi, in
the vicinity of the wing (lifting line) leads to
Finite WingsLifting line theory
Masters of Mechanical Engineering
iω
iα
effα
α
iα
iα
iD
LR
∞Vr
RVr i
ωi
α
effα
α
iα
iα
idD
dLdR
∞Vr
RVr i
ωi
ωi
αi
α
effα
effα
α
iα
iα
iα
iα
iDiD
LR
∞Vr
∞Vr
RVr
RVr i
ωi
ωi
αi
α
effα
effα
α
iα
iα
iα
iα
idD
idD
dLdR
∞Vr
∞Vr
RVr
RVr
Aerodynamics
iα
iD
LR
iα
idD
dLdR
is the velocity of the undisturbed
flow
is the velocity induced by the
wake trailing vortices
∞Vr
iω
Finite WingsLifting line theory
Masters of Mechanical Engineering
iω
iα
effα
α
iα
iR
∞Vr
RVr i
ωi
α
effα
α
iα
idR
∞Vr
RVr
wake trailing vortices
ieff ααα −=
γρ dVdR R
r=
Aerodynamics
iα
iD
LR
iα
idD
dLdR
is the relative velocity for the flow
around the wing section (airfoil)
that makes an angle with
the chord directioneffα
RVr
Finite WingsLifting line theory
Masters of Mechanical Engineering
iω
iα
effα
α
iα
iR
∞Vr
RVr i
ωα
effα
α
iα
idR
∞Vr
RVr
the chord direction
ieff ααα −=
γρ dVdR R
r=
Aerodynamics
iα
iD
LR
iα
idD
dLdR
is the circulation around the
wing section (airfoil) that is
assumed to be negative
γd
Finite WingsLifting line theory
Masters of Mechanical Engineering
iω
iα
effα
α
iα
iR
∞Vr
RVr i
ωα
effα
α
iα
idR
∞Vr
RVr
ieff ααα −=
γρ dVdR R
r=
Aerodynamics
iα
iD
LR
iα
idD
dLdR
α is the geometrical angle of attack
αi is the induced angle of attack
αeff is the effective angle of attack
Finite WingsLifting line theory
Masters of Mechanical Engineering
iω
iα
effα
α
iα
iR
∞Vr
RVr i
ωi
α
effα
α
iα
idR
∞Vr
RVr
ieff ααα −=
γρ dVdR R
r=
Aerodynamics
iα
iD
LR
iα
idD
dLdR
( )∞
=V
ii r
ωαtan
≅ iωα
Finite WingsLifting line theory
For small values ofiα
Masters of Mechanical Engineering
iω
iα
effα
α
iα
iR
∞Vr
RVr i
ωi
α
effα
α
iα
idR
∞Vr
RVr
ieff ααα −=∞
≅V
ii r
ωα
γρ dVdR R
r=
Aerodynamics
iα
iD
LR
iα
idD
dLdR
( )i
R
VV
αcos
∞=
rr
≅ VVrr
Finite WingsLifting line theory
For small values ofiα
Masters of Mechanical Engineering
iω
iα
effα
α
iα
iR
∞Vr
RVr i
ωi
α
effα
α
iα
idR
∞Vr
RVr
ieff ααα −=∞≅ VVR
rr
γρ dVdR R
r=
Aerodynamics
iα
iD
LR
iα
idD
dLdR
dR is the force perpendicular to
the local flow around the airfoil, ,
that makes an angle αeff
with the chord direction
RVr
Finite WingsLifting line theory
Masters of Mechanical Engineering
iω
iα
effα
α
iα
iR
∞Vr
RVr i
ωi
α
effα
α
iα
idR
∞Vr
RVr
with the chord direction
ieff ααα −=
γρ dVdR R
r=
Aerodynamics
iα
iD
LR
iα
idD
dLdR
Projecting dR to the directions
parallel and perpendicular to the
undisturbed incoming flow, ∞Vr
( ) ( )dRdDdRdL αα sencos ==
Finite WingsLifting line theory
Masters of Mechanical Engineering
iω
iα
effα
α
iα
iR
∞Vr
RVr i
ωi
α
effα
α
iα
idR
∞Vr
RVr
( ) ( )iii dRdDdRdL αα sencos ==
ieff ααα −=
γρ dVdR R
r=
Aerodynamics
iα
iD
LR
iα
idD
dLdR
The wake induced descending
velocity, , originates a drag
force, , designated by induced
drag
iω
iD
Finite WingsLifting line theory
Masters of Mechanical Engineering
iω
iα
effα
α
iα
iR
∞Vr
RVr i
ωi
α
effα
α
iα
idR
∞Vr
RVr
drag
ieff ααα −=
γρ dVdR R
r=
Aerodynamics
( )dyyVdVdR RR Γ==rr
ργρ
( ) ( ) Γ= dyyVdL αρ cosr
Finite WingsLifting line theory
• Determination of forces for the finite wing
Masters of Mechanical Engineering
( )( ) ( )
( ) ( )
Γ=
Γ=⇒Γ=
dyyVdD
dyyVdLdyyVdR
iRi
iR
R
αρ
αρρ
sen
cos
r
r
r
( )( )
( ) ( )
Γ=
Γ=⇒Γ=
∞
∞
dyyVdD
dyyVdLdyyVdR
ii
R r
r
r
αρ
ρρ
tan
Aerodynamics
( )
( )dyydD
dyyVdL
ii Γ=
Γ= ∞
ρω
ρr
Finite WingsLifting line theory
• Determination of forces for the finite wing
Masters of Mechanical Engineering
- Integrating in the spanwise direction
( )dyydD ii Γ= ρω
( )
( )∫
∫
−
−∞
Γ=
Γ=
2
2
2
2
b
bii
b
b
dyyD
dyyVL
ωρ
ρr
Aerodynamics
( )
( )∫
∫−∞
Γ=
Γ=
2
2
2
b
b
b
dyyD
dyyVL
ωρ
ρr
Finite WingsLifting line theory
• Determination of forces for the finite wing
Masters of Mechanical Engineering
- The lift force, L, depends on the circulation
distribution along the span
- The induced drag force, Di, depends directlty
and indirectly (ωi) on the circulation distribution
along the span
( )∫− Γ=2
2
b
bii dyyD ωρ
Aerodynamics
• Assuming two-dimensional flow around the
wing sections (airfoils) and ideal fluid (irrotational
flow) theory
( ) ( ) ( ) ( ) ( )( )yyyCy
yC βα +=Γ
= '2r
Finite WingsLifting line theory
Masters of Mechanical Engineering
effα
( ) ( )( ) ( )
( )yycVyC
yy
l
eff βα −Γ
=∞∞
r'
2
( ) ( )( )
( ) ( ) ( )( )yyyCycV
yyC effll βα +=
Γ=
∞
∞
'2r
• In such conditions, is related to the
circulation distribution, Γ(y), by
Aerodynamics
( ) ( )( ) ( )
( )yycVyC
yy
l
eff βα −Γ
=∞∞
r'
2
( )yC'
Slope of the change of C with α, that depends
Finite WingsLifting line theory
Masters of Mechanical Engineering
( )
( )
( )yc
y
yCl
β
'
∞Slope of the change of Cl with α, that depends
on the airfoil thickness
Symmetric of the zero lift angle that depends
on the airfoil camber
Airfoil chord dependent on the wing planform
Aerodynamics
• For small induced angles of attack
( ) ( )
∞
=V
yy i
i rω
α with ( ) '''
1
4
1 2
2dy
dy
d
yyy
b
bi ∫−
Γ
−=
πω
Finite WingsLifting line theory
Masters of Mechanical Engineering
iα
∞
( ) '''
1
4
1 2
2dy
dy
d
yyVy
b
bi ∫−
∞
Γ
−= r
πα
• In such conditions, is related to the
circulation distribution, Γ(y), by
Aerodynamics
relations between and the angles of attack
ieff ααα +=
( )yΓ
• Simple relation between geometric, induced
and effective angles of attack
Finite WingsLifting line theory
Masters of Mechanical Engineering
( ) ( )( ) ( )
( ) '''
1
4
12 2
2'dy
dy
d
yyVy
ycVyC
yy
b
bl
∫−∞∞
Γ
−+−
Γ=
∞
rrπ
βα
relations between and the angles of attack
lead toieff αα and
( )yΓ
Aerodynamics
( ) ( )( ) ( )
( ) '''
1
4
12 2
2'dy
dy
d
yyVy
ycVyC
yy
b
bl
∫−∞∞
Γ
−+−
Γ=
∞
rrπ
βα
Geometric parameters that define the wing
Finite WingsLifting line theory
Masters of Mechanical Engineering
( )
( ) ( )
( )( ) →Γ
→
→
→
∞
y
yc
yyC
y
l β
α
,'
Change with y depends on twist angle
Airfoils selected for the wing sections
Change with y depends on the wing planform
Spanwise circulation distribution
Geometric parameters that define the wing
Aerodynamics
• Analysis or direct problem:
( ) ( )( ) ( )
( ) '''
1
4
12 2
2'dy
dy
d
yyVy
ycVyC
yy
b
bl
∫−∞∞
Γ
−+−
Γ=
∞
rrπ
βα
Finite WingsLifting line theory
Masters of Mechanical Engineering
• Analysis or direct problem:
- Starting data
Wing geometry and angle of attack, α
- Unknowns
( ) ( ) ( ) ( )ycyyCy l e,,' βα∞
( ) iDLy and,Γ
Aerodynamics
• Project or inverse problem:
( ) ( )( ) ( )
( ) '''
1
4
12 2
2'dy
dy
d
yyVy
ycVyC
yy
b
bl
∫−∞∞
Γ
−+−
Γ=
∞
rrπ
βα
Finite WingsLifting line theory
Masters of Mechanical Engineering
• Project or inverse problem:
- Starting data
Lift and induced drag, i.e. spanwise circulation
distribution
- Unknowns
Wing geometry and/or angle of attack
( ) ( ) ( ) ( )ycyyCy l and,,' βα∞
( ) iDLy and,Γ
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