Compressibility

•  Bulk modulus •  Incompressible k=∞ •  Water: k = 2.9 . 109 N/m2

•  Transmission of sound: pressure waves through fluid

•  Speed of sound c depends on k

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k =ΔpdVV

=Δpdρρ

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c =kρ

Speed of Sound

•  Water •  Air •  Ideal Gases: where •  Moving Sources and Shocks

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c = γRT

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γ =Cp

Cv

Speed of Sound

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c = γRT

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γ =Cp

Cv

subsonic

sonic supersonic

See Fig.12-2 in Fox and McDonald

sin µ = c/V = 1/M

Classification

•  M∞<.3 incompressible •  .3 <M∞<.7 subsonic •  .7 <M∞<1.2 transonic •  1.2 <M∞<5 supersonic •  M∞>5 hypersonic

Classification

•  M∞<.3 incompressible •  .3 <M∞<.7 subsonic •  .7 <M∞<1.2 transonic •  1.2 <M∞<5 supersonic •  M∞>5 hypersonic

Classification

•  M∞<.3 incompressible •  .3 <M∞<.7 subsonic •  .7 <M∞<1.2 transonic •  1.2 <M∞<5 supersonic •  M∞>5 hypersonic

Classification

•  M∞<.3 incompressible •  .3 <M∞<.7 subsonic •  .7 <M∞<1.2 transonic •  1.2 <M∞<5 supersonic •  M∞>5 hypersonic

Classification

•  M∞<.3 incompressible •  .3 <M∞<.7 subsonic •  .7 <M∞<1.2 transonic •  1.2 <M∞<5 supersonic •  M∞>5 hypersonic

Shock Relations •  http://www.grc.nasa.gov/WWW/k-12/airplane/normal.html

Steady Quasi-1D Compressible Flow

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ρ1V1A1 = ρ2V2A2

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⇒dρρ

+dVV

+dAA

= 0

Or

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ρVA = const

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dp = −ρVdV = −ρd(V2

2)Using cons. momentum

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⇒dpρV 2 = −

dVV

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⇒dAA

=dpρV 2 −

dρρ

=dpρV 2 (1−

V 2

dpdρ

) =dpρV 2 (1−M

2)⇒ dVV

= −dAA

1(1−M 2)

Conservation of mass:

Internal Aerodynamics

•  Nozzles and diffusers – see Fig 13.3 in Fox and McD

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dVV

= −dAA

11−M 2

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dV < 0

Internal Aerodynamics

•  Wind Tunnels – see also Fig. 9.9 K&C