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Aerodynamics • Geometric properties of a wing - Finite thickness much smaller than the span and the chord Finite Wings Steady, 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 Wings Steady, 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

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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)

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

effα

α

iD

LR

∞Vr

RVr i

ωi

α

effα

α

idD

dLdR

∞Vr

RVr i

ωi

ωi

αi

α

effα

effα

α

iDiD

LR

∞Vr

∞Vr

RVr

RVr i

ωi

ωi

αi

α

effα

effα

α

idD

idD

dLdR

∞Vr

∞Vr

RVr

RVr

Aerodynamics

iD

LR

idD

dLdR

is the velocity of the undisturbed

flow

is the velocity induced by the

wake trailing vortices

∞Vr

Finite WingsLifting line theory

Masters of Mechanical Engineering

effα

α

iR

∞Vr

RVr i

ωi

α

effα

α

idR

∞Vr

RVr

wake trailing vortices

ieff ααα −=

γρ dVdR R

r=

Aerodynamics

iD

LR

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

effα

α

iR

∞Vr

RVr i

ωα

effα

α

idR

∞Vr

RVr

the chord direction

ieff ααα −=

γρ dVdR R

r=

Aerodynamics

iD

LR

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

effα

α

iR

∞Vr

RVr i

ωα

effα

α

idR

∞Vr

RVr

ieff ααα −=

γρ dVdR R

r=

Aerodynamics

iD

LR

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

effα

α

iR

∞Vr

RVr i

ωi

α

effα

α

idR

∞Vr

RVr

ieff ααα −=

γρ dVdR R

r=

Aerodynamics

iD

LR

idD

dLdR

( )∞

=V

ii r

ωαtan

≅ iωα

Finite WingsLifting line theory

For small values ofiα

Masters of Mechanical Engineering

effα

α

iR

∞Vr

RVr i

ωi

α

effα

α

idR

∞Vr

RVr

ieff ααα −=∞

≅V

ii r

ωα

γρ dVdR R

r=

Aerodynamics

iD

LR

idD

dLdR

( )i

R

VV

αcos

∞=

rr

≅ VVrr

Finite WingsLifting line theory

For small values ofiα

Masters of Mechanical Engineering

effα

α

iR

∞Vr

RVr i

ωi

α

effα

α

idR

∞Vr

RVr

ieff ααα −=∞≅ VVR

rr

γρ dVdR R

r=

Aerodynamics

iD

LR

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

effα

α

iR

∞Vr

RVr i

ωi

α

effα

α

idR

∞Vr

RVr

with the chord direction

ieff ααα −=

γρ dVdR R

r=

Aerodynamics

iD

LR

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

effα

α

iR

∞Vr

RVr i

ωi

α

effα

α

idR

∞Vr

RVr

( ) ( )iii dRdDdRdL αα sencos ==

ieff ααα −=

γρ dVdR R

r=

Aerodynamics

iD

LR

idD

dLdR

The wake induced descending

velocity, , originates a drag

force, , designated by induced

drag

iD

Finite WingsLifting line theory

Masters of Mechanical Engineering

effα

α

iR

∞Vr

RVr i

ωi

α

effα

α

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

( ) '''

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