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lectures-chapter9

# lectures-chapter9 - 650:460 Aerodynamics Chapter 9 Prof...

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650:460 Aerodynamics Chapter 9 Prof. Doyle Knight Tel: 732 445 4464, Email: [email protected] Office hours: Tues and Thur, 4:30 pm - 6:00 pm and by appointment Fall 2009 1

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Compressible Flow Assumptions Inviscid Steady Irrotational Isentropic (and therefore barotropic) From Kelvin’s Theorem, the flow is irrotational if the flow upstream of any aerodynamic body is irrotational and p = p ( ρ ). This latter condition is satisfied if we assume the flow is isentropic. This implies that there are no shock waves. 2
Compressible Flow Bernoulli’s Equation Conservation of momentum ρ bracketleftbigg ∂vector v t + ( vector v · ∇ ) vector v bracketrightbigg = −∇ p where is the gradient operator p = p x vector i + p y vector j + p z vector k and vector i , vector j and vector k are the unit vectors in the x -, y - and z -directions. 3

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Compressible Flow Bernoulli’s Equation Vector identity ( vector v · ∇ ) vector v = ( 1 2 vector v 2 ) vector v × ω where ω = ∇ × vector v is the vorticity. Substitute into the conservation of momentum ∂vector v t + ( 1 2 vector v 2 ) + 1 ρ p = 0 since the vorticity is assumed zero everywhere. Since the fluid is assumed barotropic 1 ρ p = parenleftbiggintegraldisplay dp ρ parenrightbigg 4
Compressible Flow Bernoulli’s Equation Since the fluid is assumed irrotational vector v = Φ and thus ∂vector v t = Φ t = parenleftbigg Φ t parenrightbigg Therefore the conservation of momentum is bracketleftbigg Φ t + 1 2 vector v 2 + integraldisplay dp ρ bracketrightbigg = 0 and thus Φ t + 1 2 vector v 2 + integraldisplay dp ρ = f ( t ) where f ( t ) is a function of time only. 5

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Compressible Flow Bernoulli’s Equation For isentropic flow p p o = parenleftbigg ρ ρ o parenrightbigg γ and thus integraldisplay dp ρ = γ ( γ 1) p ρ Thus, the integral of the conservation of momentum is Φ t + 1 2 Φ · ∇ Φ + γ ( γ 1) p ρ = f ( t ) This is the compressible Bernoulli equation . 6
Compressible Flow Equation for Φ Take the inner product of vector v with the conservation of momentum ρ parenleftbigg vector v · ∂vector v t + vector v · ( vector v · ∇ ) vector v parenrightbigg = vector v · ∇ p Identity vector v · ( vector v · ∇ ) vector v = vector v · ( ( 1 2 vector v 2 ) vector v × ω ) = vector v · ∇ ( 1 2 vector v 2 ) since ω = ∇ × vector v = 0 by assumption. 7

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Compressible Flow Equation for Φ Now the convective derivative of the pressure is defined as Dp Dt = p t + vector v · ∇ p Since the flow is assumed isentropic Dp Dt = dp d ρ D ρ Dt = a 2 D ρ Dt where a = γ RT is the speed of sound.
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lectures-chapter9 - 650:460 Aerodynamics Chapter 9 Prof...

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