Introduction to Aerodynamics

< 1.7. Dimensional analysis > Physical parameters

 Q: What physical quantities influence forces and moments?

V A

l

Aerodynamics 2015 fall - 1 - Introduction to Aerodynamics

< 1.7. Dimensional analysis > Physical parameters  Physical quantities to be considered Parameter Symbol units Lift L' MLT-2 Angle of attack α -

-1 Freestream velocity V∞ LT -3 Freestream density ρ∞ ML -1 -1 Freestream viscosity μ∞ ML T -1 Freestream speed of sound a∞ LT Size of body (e.g. chord) c L  Generally, resultant aerodynamic force:

R = f(ρ∞, V∞, c, μ∞, a∞) (1)

Aerodynamics 2015 fall - 2 - 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 - 3 - < 1.7. Dimensional analysis > The Buckingham PI Theorem Example)

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

Aerodynamics 2015 fall - 6 - < 1.7. Dimensional analysis > The Buckingham PI Theorem

Through similar procedure

Aerodynamics 2015 fall - 7 - Introduction to Aerodynamics

< 1.7. Dimensional analysis > Dimensionless form

Aerodynamics 2015 fall - 8 - 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 - 9 - 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 - 10 - 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 3 3 ρ1=1.2 kg/m ρ2=3.0 kg/m

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 - 11 - Introduction to Aerodynamics

< 1.8. Flow similarity > Dynamic similarity  The PI products evaluate to the following values.

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

α1=5° α2=5°

RE1=1.4e7 RE2=1.4e7

M1=0.7 M2=0.7

 Two airfoil flows are dynamically similar.

g1,Re1, M1   g2 ,Re2 , M 2  c  c l1 l2

Aerodynamics 2015 fall - 12 - Introduction to Aerodynamics

< 1.9. Fluid statics : Buoyancy force > Fluid statics  The force on an element of fluid itself

• Pressure forces from the  dp   p  dydx dz surrounding fluid exerted on the  dy  surface on the element

dz dy • Gravity force due to the weight of dx the fluid inside the element

p dx dz gdx dy dz

Aerodynamics 2015 fall - 13 - Introduction to Aerodynamics

< 1.9. Fluid statics : Buoyancy force > Fluid statics  Net pressure force & Gravity force

Net pressure force  dp   p  dydx dz  dp   dy   pdx dz  p  dydx dz  dy  dz dp dy   dx dy dz dx dy

Gravity force dx dy dzg p dx dz gdx dy dz

Aerodynamics 2015 fall - 14 - Introduction to Aerodynamics

< 1.9. Fluid statics : Buoyancy force > Fluid statics  The fluid element is in equilibrium,

 dp   p  dydx dz dp    dx dy dz gdx dy dz  0  dy  dy dp  g dy dz dy → Hydrostatic equation dx

p dx dz gdx dy dz

Aerodynamics 2015 fall - 15 - 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.

 dp  p h  p  dydx dz 2 2  dy  p ,h dp  g dy   1 1 p h 1 1

p2  p1  gh2  h1  dz dy dx h  h  h p1  gh1  p2  gh2 1 2 p  gh  const. p dx dz p ,h gdx dy dz 2 2

Aerodynamics 2015 fall - 16 - Introduction to Aerodynamics

< 1.9. Fluid statics : Buoyancy force > Fluid statics  Applications : Walls of container

p  gh  p1  gh1  pa  gh1

p  pa  gh1  h

pa p1  pa ,h1 p  pa

 gh1  h p,h

p2 ,h2  0

pa  gh1

Aerodynamics 2015 fall - 17 - Introduction to Aerodynamics

< 1.9. Fluid statics : Buoyancy force > Fluid statics  Applications : U-tube manometer

Aerodynamics 2015 fall - 18 - Introduction to Aerodynamics

< 1.9. Fluid statics : Buoyancy force > Buoyancy force  Archimedes principle

Buoyancy force Weight of fluid on body = displaced by body

Aerodynamics 2015 fall - 19 -