Topic 1Topic 1

Introduction and Basic Concepts

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Flow Past a Circular Cylinder

Re = 10,000 and Mach approximately zero

Mach = 0.45 Mach = 0.64

Pictures are from “An Album of Fluid Motion” by Van Dyke

Flow Past a Circular Cylinder

Mach = 0.80 Mach = 0.90 Mach = 0.95 Mach = 0.98

Pictures are from “An Album of Fluid Motion” by Van Dyke

Flow Past a Sphere

Mach = 1.53 Mach = 4.01

Pictures are from “An Album of Fluid Motion” by Van Dyke

Hypersonic vehicle re‐entryNASA Image Library

Shock wave: Very strong, thin wave, propagating supersonically, producing almost instantaneous compression of the flow, and increase in pressure and temperature.

Cone‐cylinder in supersonic free flight, Mach = 1.84.Picture from “An Album of Fluid Motion” by Van Dyke.

Expansion or isentropic compression wave: Finite wave moving at the sound speed, producing gradual compression or expansion of a flow  (and raising or lowering of the temperature and pressure).

Compressible Aerodynamicsp y• What is meant by compressibility?

• When is compressibility important  in fluid flows?

• What are the effects of compressibility in fluid flows?

• How can we analyze compressible flows?

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What is compressibility? p y• Consider a volume of pressurized air 

Vp

V+dV

p+dp

Compressibility (Proportionate change in volume)/(Change in pressure)

3Will this definition give a single answer?

• To be precise we can define the  p• Isothermal compressibility  Isothermal?

• Isentropic compressibility  Isentropic?

Compressibility (Proportionate change in volume)/(Change in pressure), so…

4Pressure change that resists a given proportionate change in volume ~ 1/

When is Compressibility Important in Fluid Flow ? 

• When …

• Pressure changes occur primarily in response to…

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h d h h b l• The degree to which compressibility is important in a flow is characterized by

nCompressioResistingForcesofScaleForcesInertialofScale

Scale of the inertial forces ~

Force resisting a given compression of the fluid ~

• So…  

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• Compressibility and speed of sound are related

Ai 340 / @ STP1 • Air: a=340 m/s @ STP

• Water: a=1400 m/s @ STPs

a1

• So

sVM 2

In Summary: the magnitude of compressibility effects depends on the elocit of the flo relati e to the speed of so nd

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on the velocity of the flow relative to the speed of sound

What Are the Effects of Compressibility on a Flow?Compressibility on a Flow?

E.g. Uniform flow past an airfoil vs Mach no.

Entirely subsonic flow

V , a

M =V/a <<1

• Smooth streamlines. Even far upstream of the airfoil the streamlines have begun to move under its influence 

• How does the airfoil exert this upstream influence?

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Mixed subsonic/supersonic flow

V , a

Subsonic

M =V/a<1

Shock wave: Very strong, thin wave, propagating supersonically, producing almost instantaneous compression of the flow, and increase in pressure and temperature.

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p , p p

RAE 2822 Transonic Airfoil2.3 degrees angle of attack

Pressure contoursPressure contours

M =0.729

10http://www.grc.nasa.gov/WWW/wind/valid/raetaf/raetaf01/raetaf01.html

Transonic flow over an airfoilM very close to 1

Schlieren photographhistory.nasa.gov/SP-440/ch7-2.htm

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Supersonic free stream flow

V , a

M =V/a>1

f• Bad design for supersonic flow (lots of drag), better to have sharp leading edge:

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edge:

Supersonic Flow Past an Airfoilp

13Visualization of constant density lines by Laser Holography. Image produced by Laboratoire de Thermique Appliquee et de Turbomachines, EPF - Lausanne

Flow Regimes

Incompressible Cessna 150

SubsonicP51 

Mustang

TransonicGlobal Express

Supersonic SR‐71

Hypersonic X‐43 

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4. How Can We Analyze Compressible Flows ?Flows ?

• Like for incompressible flows we useC ti f M• Conservation of Mass

• Conservation of Momentum • However for compressible flows we must consider 

Thermodynamics (since the fluid properties are variables)Thermodynamics (since the fluid properties are variables)• Conservation of energy ( 1st law of thermodynamics)

• Heat, Work, internal energy • 2nd law of thermodynamicsy

• Entropy • Equation of state (to relate variables)

• We will neglect viscosity• Can be ignored in many situations (just like for incompressible flows)• Viscous effects are generally localized 

in a thin region close to the wall  (Boundary Layer AOE 3044)

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