1. 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

2. READING AND HOMEWORK ASSIGNMENTS
Reading: Introduction to Flight, by John D. Anderson, Jr.
For this week’s lecture: Chapter 4, Sections
For next week’s lecture: Chapter 4, Sections , 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

3. ANSWERS TO LECTURE HOMEWORK
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 = m/s
Note: it takes a pressure difference of only 0.02 atm to produce such a high velocity
4.6: V2 = 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 = 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

4. 3 FUNDAMENTAL PRINCIPLES
Mass is neither created nor destroyed (mass is conserved)
Conservation of Mass
Often called Continuity
Force = Mass x Acceleration (F = ma)
Newton’s Second Law
Momentum Equation
Bernoulli’s Equation, Euler Equation, Navier-Stokes Equation
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?

5. 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
continuity
Bernoulli
continuity
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
isentropic
energy
equation of state
at any point

6. CONSERVATION OF MASS (4.1)
Physical Principle: Mass can be neither created nor destroyed
Stream tube
Funnel wall 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”

7. 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)

8. 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)
Conservation of mass could also give velocity, A2, if V2 was known
Conservation of mass tells us nothing about p2, T2, etc.

9. 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

10. APPLYING NEWTON’S SECOND LAW FOR FLOWSx y z
Consider a small fluid element moving along a streamline
Element is moving in x-direction V dy dz dx
What forces act on this element?
Pressure (force x area) acting in normal direction on all six faces
Frictional shear acting tangentially on all six faces (neglect for now)
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

11. APPLYING NEWTON’S SECOND LAW FOR FLOWSx y 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

12. APPLYING NEWTON’S SECOND LAW FOR FLOWSx y z
Pressure varies from point to point in a flow
There is a change in pressure per unit length, dp/dx
p+(dp/dx)dx
(N/m2) p
(N/m2) dy dz dx
Area of left face: dydz
Force on left face: p(dydz)
Forces is in positive x-direction
Change in pressure per length: dp/dx
Change in pressure along dx is (dp/dx)dx
Force on right face: [p+(dp/dx)dx](dydz)
Forces acts in negative x-direction

13. APPLYING NEWTON’S SECOND LAW FOR FLOWSx y z p
(N/m2)
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

14. 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)

15. SUMMARY: 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

16. 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 ↓

17. 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

18. INVISCID FLOW ALONG STREAMLINES2 1
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

19. Constant along a streamline
BERNOULLI’S EQUATION
Constant along a streamline
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

20. WHEN AND WHEN NOT TO APPLY BERNOULLI
YES NO

21. SIMPLE EXAMPLE
Given air flow through converging nozzle, what is exit pressure, p2?
p1 = 1.2x105 N/m2
T1 = 330 K
V1 = 10 m/s
A1 = 5 m2
p2 = ?
T2 = 330 K
V2 = 30 m/s
A2 = 1.67 m2
Since flow speed < 100 m/s assume flow is incompressible (r1=r2)
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

22. 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

23. HOW DOES AN AIRFOIL GENERATE LIFT?
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

24. HOW DOES AN AIRFOIL GENERATE LIFT?
As V ↑ p↓
Incompressible: Bernoulli’s Equation
Compressible: Euler’s Equation
Called Bernoulli Effect
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?

25. 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
continuity
Bernoulli
What if flow is high speed, M > 0.3?
What if there are temperature effects?
How does density change?

26. ONLINE REFERENCES http://www.aircraftenginedesign.com/enginepics.html