16100lectre12_cg

# 16100lectre12_cg - Quick Visit to Bernoulli Land Although...

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Unformatted text preview: Quick Visit to Bernoulli Land Although we have seen the Bernoulli equation and seen it derived before, this next note shows its derivation for an uncompressible & inviscid flow. The derivation follows that of Kuethe &Chow most closely (I like it better than Anderson). 1 Start from inviscid, incompressible momentum equation 1 u u u p t ρ ∂ + ⋅ ∇ = − ∇ ∂ K K K There is a vector calculus identity: ( ) 2 , 1 2 vorticity u u u u u ω • ∇ = ∇ − × ∇ × K K K K K K ¡¢£ 2 1 1 2 u u p u t ω ρ ∂ ⇒ + ∇ + ∇ = × ∂ K K K K From here, we can make the final re-arrangement: 2 1 2 u p u u t ρ ρ ω ρ ∂ ∇ + = × − ∂ K K K K Two common applications: 1. Steady irrotational flow N Irrotational Steady u t ω ∂ = = ∂ K K ¡¢£ 2 2 1 2 1 . 2 p u p u const for entire flow ρ ρ ⇒ ∇ + = ⇒ + = K K 1 Kuethe and Chow, 5 th Ed. Sec 3.3-3.5 Quick Visit to Bernoulli Land 16.100 2002 2 2. Steady but rotational flow N Rotational Steady u t ω ∂ = ≠ ∂ K K ¡¢£ 2 1 2 p u u ρ ρ ω ⇒ ∇ + = × K K K This is a vector equation. If we dot product this into the streamwise direction: ← ≡ u u s K K K streamwise direction ( ) ( ) 2 0, 2 2 1 2 1 2 1 . 2 u u s p u s u d p u ds p u const along streamline ω ρ ρ ω ρ ρ = × ⊥ ⇒ • ∇ + = • × ⇒ + = ⇒ + = K K K K K K K K ¡¤¢¤£ K K Vortex Panel Methods 2 Step#1: Replace airfoil surface with panels Step #2: Distribute singularities on each panel with unknown strengths In our case we will use vortices distributed such that their strength varies linearly from node to node: Recall a point vortex at the origin is: 1 tan 2 2 y x φ θ π π − Γ Γ = − = − 2 Kuethe and Chow, 5 th Ed. Sec. 5.10 1 2 3 4 m m+1 i-1 i i+1...
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16100lectre12_cg - Quick Visit to Bernoulli Land Although...

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