16100lectre43_cj

# 16100lectre43_cj - Linearized Compressible Potential Flow...

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Unformatted text preview: Linearized Compressible Potential Flow Governing Equation Recall the 2-D full potential eqn is: 2 ) ( 1 1 ) ( 1 1 2 2 2 2 2 = − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − xy y x yy y xx x a a a φ φ φ φ φ φ φ Where [ ] 2 2 2 2 ) ( ) ( 1 y x o a a φ φ γ γ + − − = As you saw, for small perturbations to a uniform flow, the linearized form of the equation was: Where y x J y j x i v u u u u u φ φ φ φ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ = = ⇒ + = + = ∞ v v v v v v This equation is valid for both 1 &amp;lt; ∞ M and 1 &amp;gt; ∞ M . Note, though, it is not correct for 1 → ∞ M or for ∞ M large, say greater than about 2 or so. So what happens to the linearized potential equation for : 1 &amp;gt; ∞ M It also turns out that ˆ ˆ ) 1 ( 2 = + − ∞ yy xx M φ φ is much easier to solve when 1 &amp;gt; ∞ M . Define ˆ ˆ 1 2 2 = − ⇒ − = ∞ yy xx M φ φ λ λ Then, ) ( ˆ ) , ( ˆ η φ φ = y x where y x λ η − = is a solution to the linearized potential. To see this: ˆ ˆ...
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## This note was uploaded on 11/06/2011 for the course AERO 100 taught by Professor Willcox during the Fall '03 term at MIT.

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16100lectre43_cj - Linearized Compressible Potential Flow...

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