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Unformatted text preview: Fluids – Lecture 4 Notes 1. Dimensional Analysis – Buckingham Pi Theorem 2. Dynamic Similarity – Mach and Reynolds Numbers Reading: Anderson 1.7 Dimensional Analysis Physical parameters A large number of physical parameters determine aerodynamic forces and moments. Specif- ically, the following parameters are involved in the production of lift. Parameter Symbol Units Lift per span L ′ mt − 2 Angle of attack α — Freestream velocity V ∞ lt − 1 Freestream density ρ ∞ ml − 3 Freestream viscosity µ ∞ ml − 1 t − 1 Freestream speed of sound a ∞ lt − 1 Size of body (e.g. chord) c l For an airfoil of a given shape, the lift per span in general will be a function of the remaining parameters in the above list. ′ L = f ( α, ρ ∞ , V ∞ , c, µ ∞ , a ∞ ) (1) In this particular example, the functional statement has N = 7 parameters, expressed in a total of K = 3 units (mass m , length l , and time t ). Dimensionless Forms The Buckingham Pi Theorem states that this functional statement can be rescaled into an equivalent dimensionless statement ¯ Π 1 = f ( Π 2 , Π 3 . . . Π N − K ) having only N − K = 4 dimensionless parameters. These are called Pi products, since they are suitable products of the dimensional parameters. In the particular case of statement (1), suitable Pi products are: Π L ′ 1 = 1 = c ℓ lift coeﬃcient 2 ρ...
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- Fall '05
- Aerodynamics, parameters, Buckingham Pi Theorem, Pi products