Topic 1
Topic 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
Pictures are from “An Album of Fluid Motion” by Van Dyke
Mach = 0.98
Flow Past a Sphere
Mach = 1.53
Mach = 4.01
Pictures are from “An Album of Fluid Motion” by Van Dyke
Shock wave: Very strong, thin wave, propagating supersonically, producing almost instantaneous compression of the flow, and increase in pressure and temperature.
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).
Hypersonic vehicle re‐entry
NASA Image Library
Cone‐cylinder in supersonic free flight, Mach = 1.84.
Picture from “An Album of Fluid Motion” by Van Dyke.
Compressible Aerodynamics
p
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?
2
What is compressibility? p
y
• Consider a volume of pressurized air p
V
p+dp
V+
dV
Compressibility   (Proportionate change in volume)/(Change in pressure)
Will this definition give a single answer?
3
• To be precise we can define the p
• Isothermal compressibility Isothermal?
• Isentropic compressibility Isentropic?
Compressibility   (Proportionate change in volume)/(Change in pressure), so…
Pressure change that resists a given proportionate change in volume ~ 1/
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When is Compressibility Important in Fluid Flow ? • When …
• Pressure changes occur primarily in response to…
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• The degree to which compressibility
h d
h h
b l is important in a flow is characterized by

Scale of Inertial Forces
Scale of Forces Resisting Compression
Scale of the inertial forces ~
Force resisting a given compression of the fluid ~
• So… 6
• Compressibility and speed of sound are related
a
1
 s
•
Ai a=340
Air:
340 m/s
/ @ STP
•
Water: a=1400 m/s @ STP
• So
M  V 2 s 
In Summary: the magnitude of compressibility effects depends
on the velocity
elocit of the flo
flow relati
relative
e to the speed of so
sound
nd
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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
Subsonic
V , a
M =V/a<1
Shock wave: Very strong, thin wave, propagating supersonically, producing almost
instantaneous compression
p
of the flow,, and increase in p
pressure and temperature.
p
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RAE 2822 Transonic Airfoil
2.3 degrees angle of attack
Pressure contours
M =0.729
http://www.grc.nasa.gov/WWW/wind/valid/raetaf/raetaf01/raetaf01.html
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Transonic flow over an airfoil
M very close to 1
Schlieren photograph
history.nasa.gov/SP-440/ch7-2.htm
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Supersonic free stream flow
V , a
M =V/a>1
•
Bad design for
f supersonic
flow (lots of drag), better
to have sharp leading
edge:
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Supersonic Flow Past an Airfoil
p
Visualization of constant density lines by Laser Holography. Image produced by
Laboratoire de Thermique Appliquee et de Turbomachines, EPF - Lausanne
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Flow Regimes
Incompressible
Cessna 150
Subsonic
P51 Mustang
Transonic
Global Express
Supersonic
SR‐71
Hypersonic
X‐43 14
4. How Can We Analyze Compressible Flows ?
Flows ?
• Like for incompressible flows we use
• Conservation of Mass
C
ti
fM
• Conservation of Momentum • However for compressible flows we must consider Thermodynamics (since the fluid properties are variables)
(since the fluid properties are variables)
• Conservation of energy ( 1st law of thermodynamics)
• Heat, Work, internal energy • 2nd law of thermodynamics
y
• 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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