16100lectre46_cj

16100lectre46_cj - ⇒ − = ⇒ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜...

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Prandtl-Meyer Expansion Waves When a supersonic flow is turned around a corner, an expansion fan occurs producing a higher speed, lower pressure, etc. in an isentropic process. Just as we saw with shock waves, if we apply conservation of mass and momentum across a single wave, the tangential velocity is unchanged. Unlike a shock wave, an expansion wave is isentropic. So let’s pick out a single wave: From law of sines: + = + θ µ π d V dV V 2 sin 2 sin Using M d 1 sin & 0 = , we can find : 1 1 > M Forward Mach line fan Rearward Mach line 1 2 waves V u du V w dv V + d Mach wave V dV M d 1 2 =
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Prandtl-Meyer Expansion Waves 16.100 2002 2 Next, a da M dM v dv M v a + = = Using adiabatic relationships, we can re-write:
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Unformatted text preview: ⇒ − + = ⇒ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − = ⇒ ∂ − ∂ + = = = − M dM M v dv dM M M a da M T T a a RT RT c o o 2 1 2 2 2 1 1 1 1 1 2 1 1 1 γ Finally, integrating θ d we find: ) ( ) ( 1 2 M v M v − = Where 1 tan ) 1 ( 1 1 tan 1 1 ) ( 2 1 2 1 − − − − − − + = − − M M M v Prandtl-Meyer function Problem: Estimate the rates of the pressure inside the pitot probe to the freestream static pressure, ∞ p p a . M dM M M d 2 2 2 1 1 1 − + − = 3 = ∞ M ° 15 Pressure in pitot probe a p =...
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16100lectre46_cj - ⇒ − = ⇒ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜...

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