Transcript Governing Equations Aerodynamics II

MAE 1202: AEROSPACE PRACTICUM
Lecture 3: Introduction to Basic Aerodynamics 2
January 28, 2013
Mechanical and Aerospace Engineering Department
Florida Institute of Technology
D. R. Kirk
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READING AND HOMEWORK ASSIGNMENTS
• Reading: Introduction to Flight, by John D. Anderson, Jr.
– For this week’s lecture: Chapter 4, Sections 4.1 - 4.9
– For next week’s lecture: Chapter 4, Sections 4.10 - 4.21, 4.27
• Lecture-Based Homework Assignment:
– Problems: 4.1, 4.2, 4.4, 4.5, 4.6, 4.8, 4.11, 4.15, 4.16
• DUE: Friday, February 8, 2013 by 11:00 am
• Turn in hard copy of homework
– Also be sure to review and be familiar with textbook examples in
Chapter 4
• Lab this week:
– Machine shop (remember to dress appropriately, no ‘open-toe’ shoes)
– Team Challenge #1
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ANSWERS TO LECTURE HOMEWORK
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4.1: V2 = 1.25 ft/s
4.2: p2-p1 = 22.7 lb/ft2
4.4: V1 = 67 ft/s (or 46 MPH)
4.5: V2 = 102.22 m/s
– Note: it takes a pressure difference of only 0.02 atm to produce such a high
velocity
4.6: V2 = 216.8 ft/s
4.8: Te = 155 K and re = 2.26 kg/m3
– Note: you can also verify using equation of state
4.11: Ae = 0.0061 ft2 (or 0.88 in2)
4.15: M∞ = 0.847
4.16: V∞ = 2,283 MPH
• Notes:
– Outline problem/strategy clearly – rewrite question and discuss approach
– Include a brief comment on your answer, especially if different than above
– Write as neatly as you possibly can
– If you have any questions come to office hours or consult GSA’s
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3 FUNDAMENTAL PRINCIPLES
1. Mass is neither created nor destroyed (mass is conserved)
– Conservation of Mass
– Often called Continuity
2. Force = Mass x Acceleration (F = ma)
– Newton’s Second Law
– Momentum Equation
– Bernoulli’s Equation, Euler Equation, Navier-Stokes Equation
3. Energy Is Conserved
– Energy neither created nor destroyed; can only change physical form
– Energy Equation (1st Law of Thermodynamics)
How do we express these statements mathematically?
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SUMMARY OF GOVERNING EQUATIONS (4.8)
STEADY AND INVISCID FLOW
• Incompressible flow of fluid along a
streamline or in a stream tube of
varying area
• Most important variables: p and V
• T and r are constants throughout flow
A1V1  A2V2
continuity
1
1
2
p1  rV1  p2  rV22
2
2
Bernoulli
• Compressible, isentropic
(adiabatic and frictionless)
flow along a streamline or in a
stream tube of varying area
• T, p, r, and V are all variables
continuity
r1 A1V1  r 2 A2V2

isentropic
energy
equation of state
at any point

p1  r1   T1   1
     
p2  r 2   T2 
1 2
1 2
c pT1  V1  c pT2  V2
2
2
p1  r1 RT1
p2  r 2 RT2
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CONSERVATION OF MASS (4.1)
• Physical Principle: Mass can be neither created nor destroyed
Funnel wall
Stream tube
A2
A1
V1
V2
• As long as flow is steady, mass that flows through cross section at point 1
(at entrance) must be same as mass that flows through point 2 (at exit)
• Flow cannot enter or leave any other way (definition of a stream tube)
• Also applies to solid surfaces, pipe, funnel, wind tunnels, airplane engine
• “What goes in one side must come out the other side”
CONSERVATION OF MASS (4.1)
Stream tube
A1: cross-sectional area
of stream tube at 1
V1: flow velocity
normal (perpendicular) to A1
• Consider all fluid elements in plane A1
• During time dt, elements have moved V1dt and swept out volume A1V1dt
• Mass of fluid swept through A1 during dt: dm=r1(A1V1dt)
dm
 kg 
Mass Flow 
 m 1  r1 A1V1  
dt
 s 
m 2  r 2 A2V2
m 1  m 2
SIMPLE EXAMPLE
Given air flow through converging nozzle, what is exit velocity, V2?
p1 = 1.2x105 N/m2
T1 = 330 K
V1 = 10 m/s
A1 = 5 m2
p2 = ?
T2 = ?
V2 = ? m/s
A 2= 1.67 m2
IF flow speed < 100 m/s assume flow is incompressible (r1=r2)
m 1  m 2  r1 A1V1  r 2 A2V2
A1V1  A2V2
A1
m
 5 
V2  V1
 10
  30
A2
s
 1.67 
Conservation of mass could also give velocity, A2, if V2 was known
Conservation of mass tells us nothing about p2, T2, etc.
INVISCID MOMENTUM EQUATION (4.3)
• Physical Principle: Newton’s Second Law
• How to apply F = ma for air flows?
• Lots of derivation coming up…
• Derivation looks nasty… final result is very easy is use…
• What we will end up with is a relation between pressure and velocity
– Differences in pressure from one point to another in a flow create
forces
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APPLYING NEWTON’S SECOND LAW FOR FLOWS
y
Consider a small fluid element moving along a streamline
Element is moving in x-direction
x
z
V
dy
dz
dx
What forces act on this element?
1. Pressure (force x area) acting in normal direction on all six faces
2. Frictional shear acting tangentially on all six faces (neglect for now)
3. Gravity acting on all mass inside element (neglect for now)
Note on pressure:
Always acts inward and varies from point to point in a flow
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APPLYING NEWTON’S SECOND LAW FOR FLOWS
y
x
z
p
(N/m2)
dy
dz
dx
Area of left face: dydz
Force on left face: p(dydz)
Note that P(dydz) = N/m2(m2)=N
Forces is in positive x-direction
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APPLYING NEWTON’S SECOND LAW FOR FLOWS
y
Pressure varies from point to point in a flow
There is a change in pressure per unit length, dp/dx
x
z
p
(N/m2)
p+(dp/dx)dx
(N/m2)
dy
dz
dx
Area of left face: dydz
Change in pressure per length: dp/dx
Force on left face: p(dydz)
Change in pressure along dx is (dp/dx)dx
Forces is in positive x-direction Force on right face: [p+(dp/dx)dx](dydz)
Forces acts in negative x-direction
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APPLYING NEWTON’S SECOND LAW FOR FLOWS
y
x
p
(N/m2)
z
p+(dp/dx)dx
(N/m2)
dy
dz
dx
Net Force is sum of left and right sides
Net Force on element due to pressure
dp 

F  pdydz   p  dx dydz
dx 

dp
F   dxdydz
dx
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APPLYING NEWTON’S SECOND LAW FOR FLOWS
Now put this into F=ma
First, identify mass of element
Next, write acceleration, a, as
(to get rid of time variable)
mass
r
volume
volume  dxdydz
mass  r dxdydz
dV
a
dt
dx
V
dt
dV dx dV dx dV
a


V
dt dx dx dt dx
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SUMMARY: EULER’S EQUATION
F  ma
dp
dV
 dxdydz  r dxdydzV
dx
dx
dp   rVdV Euler’s Equation
• Euler’s Equation (Differential Equation)
– Relates changes in momentum to changes in force (momentum equation)
– Relates a change in pressure (dp) to a chance in velocity (dV)
• Assumptions we made:
– Neglected friction (inviscid flow)
– Neglected gravity
– Assumed that flow is steady
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WHAT DOES EULER’S EQUATION TELL US?
• Notice that dp and dV are of opposite sign: dp = -rVdV
• IF dp ↑
– Increased pressure on right side of element relative to left side
– dV ↓
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WHAT DOES EULER’S EQUATION TELL US?
• Notice that dp and dV are of opposite sign: dp = -rVdV
• IF dp ↑
– Increased pressure on right side of element relative to left side
– dV ↓ (flow slows down)
• IF dp ↓
– Decreased pressure on right side of element relative to left side
– dV ↑ (flow speeds up)
• Euler’s Equation is true for Incompressible and Compressible flows
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INVISCID FLOW ALONG STREAMLINES
1
2
Points 1 and 2 are on same streamline!
Relate p1 and V1 at point 1 to p2 and V2 at point 2
Integrate Euler’s equation from point 1 to point 2 taking r = constant
dp  rVdV  0
p2
V2
p1
V1
 dp  r  VdV  0
 V22 V12 
  0
p2  p1  r 

2 
 2
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BERNOULLI’S EQUATION
2
2
2
1
V
V
p2  r
 p1  r
2
2
2
V
pr
 Constant along a streamline
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• One of most fundamental and useful equations in aerospace engineering!
• Remember:
– Bernoulli’s equation holds only for inviscid (frictionless) and
incompressible (r = constant) flows
– Bernoulli’s equation relates properties between different points along a
streamline
– For a compressible flow Euler’s equation must be used (r is variable)
– Both Euler’s and Bernoulli’s equations are expressions of F = ma
expressed in a useful form for fluid flows and aerodynamics
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WHEN AND WHEN NOT TO APPLY BERNOULLI
YES
NO
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SIMPLE EXAMPLE
Given air flow through converging nozzle, what is exit pressure, p2?
p2 = ?
T2 = 330 K
V2 = 30 m/s
A2 = 1.67 m2
p1 = 1.2x105 N/m2
T1 = 330 K
V1 = 10 m/s
A1 = 5 m2
Since flow speed < 100 m/s assume flow is incompressible (r1=r2)
p1
1.2 x105
kg
r1 

 1.27 3
RT1 287 330
m




1
1
N
p2  p1  r V12  V22  1.2 x105  1.27  102  30 2  1.195 x105 2
2
2
m
Since velocity is increasing along flow, it is an accelerating flow
Notice that even with a 3-fold increase in velocity pressure decreases
by only about 0.8 %, which is characteristic of low velocity flow
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HOW DOES AN AIRFOIL GENERATE LIFT?
• Lift due to imbalance of pressure distribution over top and bottom surfaces of
airfoil (or wing)
– If pressure on top is lower than pressure on bottom surface, lift is generated
– Why is pressure lower on top surface?
• We can understand answer from basic physics:
– Continuity (Mass Conservation)
– Newton’s 2nd law (Euler or Bernoulli Equation)
Lift = PA
HOW DOES AN AIRFOIL GENERATE LIFT?
1. Flow velocity over top of airfoil is faster than over bottom surface
– Streamtube A senses upper portion of airfoil as an obstruction
– Streamtube A is squashed to smaller cross-sectional area
– Mass continuity rAV=constant: IF A↓ THEN V↑
Streamtube A is squashed
most in nose region
(ahead of maximum thickness)
A
B
HOW DOES AN AIRFOIL GENERATE LIFT?
2. As V ↑ p↓
1
p

rV 2  constant
– Incompressible: Bernoulli’s Equation
2
– Compressible: Euler’s Equation
dp   rVdV
– Called Bernoulli Effect
3. With lower pressure over upper surface and higher pressure over bottom surface,
airfoil feels a net force in upward direction → Lift
Most of lift is produced
in first 20-30% of wing
(just downstream of leading edge)
Can you express these ideas in your own words?
SUMMARY OF GOVERNING EQUATIONS (4.8)
STEADY AND INVISCID FLOW
• Incompressible flow of fluid along a
streamline or in a stream tube of
varying area
• Most important variables: p and V
• T and r are constants throughout flow
A1V1  A2V2
continuity
1
1
2
p1  rV1  p2  rV22
2
2
Bernoulli
What if flow is high speed, M > 0.3?
What if there are temperature effects?
How does density change?
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ONLINE REFERENCES
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http://www.aircraftenginedesign.com/enginepics.html
http://www.pratt-whitney.com/
http://www.geae.com/
http://www.geae.com/education/engines101/
http://www.ueet.nasa.gov/StudentSite/engines.html
http://www.aeromuseum.org/Education/Lessons/HowPlaneFly/HowPlaneFly.html
http://www.nasm.si.edu/exhibitions/gal109/NEWHTF/HTF532.HTM
http://www.aircav.com/histturb.html
http://inventors.about.com/library/inventors/bljjetenginehistory.htm
http://inventors.about.com/library/inventors/blenginegasturbine.htm
http://www.gas-turbines.com/primer/primer.htm
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