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. g1,Re1, M1 g2 ,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 dydx 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 gdx 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 dydx dz dp dy pdx dz p dydx dz dy dz dp dy dx dy dz dx dy Gravity force dx dy dzg p dx dz gdx 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 dydx dz dp dx dy dz gdx dy dz 0 dy dy dp g dy dz dy → Hydrostatic equation dx p dx dz gdx 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 dydx dz dy p h 2 2 p ,h dp g dy 1 1 p h 1 1 dz p2 p1 gh2 h1 dy dx h h h p1 gh1 p2 gh2 1 2 p gh const. p dx dz p2 ,h2 gdx dy dz 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 gh1 h pa p1 pa ,h1 p pa gh1 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 -.