Aerodynamics 2015 fall - 1 -

Introduction to Aerodynamics

< 1.7. Dimensional analysis >

Physical parameters

Q: What physical quantities influence forces and moments?

V

l

A

Aerodynamics 2015 fall - 2 -

Introduction to Aerodynamics

< 1.7. Dimensional analysis >

Physical parameters

Physical quantities to be considered

Generally, resultant aerodynamic force:

R = f(ρ∞, V∞, c, μ∞, a∞) (1)

Parameter Symbol units

Lift L' MLT-2

Angle of attack α -

Freestream velocity V∞ LT-1

Freestream density ρ∞ ML-3

Freestream viscosity μ∞ ML-1T-1

Freestream speed of sound a∞ LT-1

Size of body (e.g. chord) c L

Aerodynamics 2015 fall - 3 -

Introduction to Aerodynamics

< 1.7. Dimensional analysis >

The Buckingham PI Theorem

The relation with N physical variables

f1 ( p1 , p2 , p3 , … , pN ) = 0

can be expressed as

f 2( P1 , P2 , ... , PN-K ) = 0

where K is the No. of fundamental dimensions

Then P1 =f3 ( p1 , p2 , … , pK , pK+1 )

P2 =f4 ( p1 , p2 , … , pK , pK+2 )

….

PN-K =f ( p1 , p2 , … , pK , pN )

Aerodynamics 2015 fall - 4 -

< 1.7. Dimensional analysis >

The Buckingham PI Theorem

Example)

Aerodynamics 2015 fall - 5 -

< 1.7. Dimensional analysis >

The Buckingham PI Theorem

Example)

Aerodynamics 2015 fall - 6 -

< 1.7. Dimensional analysis >

The Buckingham PI Theorem

Aerodynamics 2015 fall - 7 -

< 1.7. Dimensional analysis >

The Buckingham PI Theorem

Through similar procedure

Aerodynamics 2015 fall - 8 -

Introduction to Aerodynamics

< 1.7. Dimensional analysis >

Dimensionless form

Aerodynamics 2015 fall - 9 -

Introduction to Aerodynamics

< 1.8. Flow similarity >

Dynamic similarity

Two different flows are dynamically similar if

• Streamline patterns are similar

• Velocity, pressure, temperature distributions are same

• Force coefficients are same

Criteria

• Geometrically similar

• Similarity parameters (Re, M) are same

Aerodynamics 2015 fall - 10 -

Introduction to Aerodynamics

< 1.8. Flow similarity >

Dynamic similarity

Airfoil flow example

• Consider two airfoils which have the same shape and angle of

attack, but have different sizes and are operating in two different

fluids.

Aerodynamics 2015 fall - 11 -

Introduction to Aerodynamics

< 1.8. Flow similarity >

Dynamic similarity

Airfoil 1 (sea level) Airfoil 2 (cryogenic tunnel)

α1=5° α2=5°

V1=210 m/s V2=140 m/s

ρ1=1.2 kg/m3 ρ2=3.0 kg/m3

μ1=1.8e-5 kg/m·s μ2=1.5e-5 kg/m·s

a1=300 m/s a2=200 m/s

c1=1.0 m c2=0.5 m

Aerodynamics 2015 fall - 12 -

Introduction to Aerodynamics

< 1.8. Flow similarity >

Dynamic similarity

The PI products evaluate to the following values.

Two airfoil flows are dynamically similar.

Airfoil 1 (sea level) Airfoil 2 (cryogenic tunnel)

α1=5° α2=5°

RE1=1.4e7 RE2=1.4e7

M1=0.7 M2=0.7

21

222111 ,Re,,Re,

ll cc

MgMg

Aerodynamics 2015 fall - 13 -

Introduction to Aerodynamics

< 1.9. Fluid statics : Buoyancy force >

Fluid statics

The force on an element of fluid itself

• Pressure forces from the

surrounding fluid exerted on the

surface on the element

• Gravity force due to the weight of

the fluid inside the element

dzdxdydy

dpp

dzdxp

dxdy

dz

dzdydxg

Aerodynamics 2015 fall - 14 -

Introduction to Aerodynamics

< 1.9. Fluid statics : Buoyancy force >

Fluid statics

Net pressure force & Gravity force

dzdxdydy

dpp

dzdxp

dxdy

dz

dzdydxg

dzdydxdy

dp

dzdxdydy

dppdzdxp

forcepressureNet

gdzdydxforceGravity

Aerodynamics 2015 fall - 15 -

Introduction to Aerodynamics

< 1.9. Fluid statics : Buoyancy force >

Fluid statics

The fluid element is in equilibrium,

→ Hydrostatic equation

dzdxdydy

dpp

dzdxp

dxdy

dz

dzdydxg

dygdp

dzdydxgdzdydxdy

dp

0

Aerodynamics 2015 fall - 16 -

Introduction to Aerodynamics

< 1.9. Fluid statics : Buoyancy force >

Fluid statics

Let the fluid be a liquid, for which we can assume ρ is

constant.

.

2211

1212

2

1

2

1

constghp

ghpghp

hhgpp

dygdph

h

p

p

11,hp

22 ,hp

21 hhh

dzdxdydy

dpp

dzdxp

dxdy

dz

dzdydxg

Aerodynamics 2015 fall - 17 -

Introduction to Aerodynamics

< 1.9. Fluid statics : Buoyancy force >

Fluid statics

Applications : Walls of container

ap

11 ,hpp a

hp,

0, 22 hp

hhg

pp a

1

ap

1ghpa

hhgpp

ghpghpghp

a

a

1

111

Aerodynamics 2015 fall - 18 -

Introduction to Aerodynamics

< 1.9. Fluid statics : Buoyancy force >

Fluid statics

Applications : U-tube manometer

Aerodynamics 2015 fall - 19 -

Introduction to Aerodynamics

< 1.9. Fluid statics : Buoyancy force >

Buoyancy force

Archimedes principle

=Buoyancy force

on body

Weight of fluid

displaced by body