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Unformatted text preview: 650:460 Aerodynamics Chapter 3 Prof. Doyle Knight Tel: 732 445 4464, Email: [email protected] Office hours: Tues and Thur, 4:00 pm  6:00 pm and by appointment Fall 2008 1 Modern Commercial Aircraft Boeing 787 Airbus A380 2 Modern Military Aircraft F22 Raptor Sukhoi 35 3 History Orville Wright (p) (18711948) Wilbur Wright (18671912) 4 Course Objectives I Develop familiarity with the terminology of aeronautics I Understand the principles of lift and drag I Develop mathematical models for prediction of lift and drag I Compare results of mathematical models with experiment 5 Inviscid Flows The governing equations for an inviscid compressible flow are ρ du dt = ρ f x ∂ p ∂ x ρ dv dt = ρ f y ∂ p ∂ y ρ dw dt = ρ f z ∂ p ∂ z or in vector form ρ dv dt = ρ f ∇ p where dv dt = ∂ v dt + ( v · ∇ ) v These are the Euler equations 6 Bernoulli’s Equation Assume that the density is constant, and the body force is conservative, f =∇ F Furthermore, assume that the flow is steady. Then using the vector identity ( v · ∇ ) v = ∇ ( 1 2 U 2 ) v × ( ∇ × v ) where U =  v  , the Euler equations become ∇ ( 1 2 U 2 ) + ∇ F + 1 ρ ∇ p v × ( ∇ × v ) = 0 Consider the inner product of the Euler equations with an infinitesimal distance ds • ∇ ( 1 2 U 2 ) + ∇ F + 1 ρ ∇ p v × ( ∇ × v ) ‚ · ds = 0 7 Bernoulli’s Equation The individual term [ v × ( ∇ × v )] · ds = 0 if ∇ × v = 0 everywhere (known as irrotational flow ) or if ds is tangent to a streamline (prove for yourself). Under either of these conditions, the inner product of the Euler equations with ds yields ∇ • 1 2 U 2 + F + p ρ ‚ · ds = 0 which is d • 1 2 U 2 + F + p ρ ‚ = 0 Integrating 1 2 U 2 + F + p ρ = constant Bernoulli’s Equation 8 Bernoulli’s Equation The most common force potential is gravity. Assuming the z axis is positive upwards F = gz and thus Bernoulli’s equation becomes 1 2 U 2 + gz + p ρ = constant For aerodynamics problems, the change in potential energy is negligible, and Bernoulli’s equation may be approximated as 1 2 U 2 + p ρ = constant 9 Bernoulli’s Equation Example: An airfoil moves at 75 m/s in air at sealevel. What is the pressure measured at the stagnation point on the airfoil ? Solution: Bernoulli’s equation 1 2 U 2 1 + p 1 ρ = 1 2 U 2 2 + p 2 ρ where 1 and 2 are two points on the same streamline. Let point 1 be located far upstream, and point 2 be the stagnation point where U 2 = 0 by definition. Therefore p 2 = p ∞ + 1 2 ρ U 2 ∞ At sea level, p = 101 kPa and ρ = 1 . 22 kg/m 3 . Thus p 2 = 1 . 01 · 10 5 + (1 . 22)(75) 2 Pa = 1 . 078 · 10 5 Pa 10 Determination of Air Speed Pitotstatic tube is a device to measure airspeed Applying Bernoulli’s equation 1 2 U 2 ∞ + p ∞ ρ ∞ = p t ρ ∞ where p t is the total (“stagnation”) pressure. Thus U ∞ = s 2 ( p t p ∞ ) ρ ∞ Fig. 3.2 Pitotstatic tubes 11 Determination of Airspeed There are many flight conditions where the airspeed computed...
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This note was uploaded on 01/24/2012 for the course MECHE 343 taught by Professor Professor during the Spring '11 term at Carnegie Mellon.
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