1. MAE 3241: AERODYNAMICS AND FLIGHT MECHANICS REVIEW OF INCOMPRESSIBLE FLOW OVER AIRFOILS AND WINGS
Mechanical and Aerospace Engineering Department
Florida Institute of Technology
D. R. Kirk

2. 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

3. 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

4. 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)

5. AIRFOILS VERSUS WINGS Why do airfoils have such a shape?
How are lift and drag produced?
NACA airfoil performance data
How do we design?
What is limit of behavior?

6. FINAL EXAM SAMPLE QUESTIONS
“Tell me everything can you about the design of these wings…” d
dihedral

7. AIRFOIL NOMENCLATURE
Mean Chamber Line: Set of points halfway between upper and lower surfaces
Measured perpendicular to mean chamber line itself
Leading Edge: Most forward point of mean chamber line
Trailing Edge: Most reward point of mean chamber line
Chord Line: Straight line connecting the leading and trailing edges
Chord, c: Distance along the chord line from leading to trailing edge
Chamber: Maximum distance between mean chamber line and chord line
Measured perpendicular to chord line

8. NACA AIRFOIL NAMING CONVENTION
NACA (National Advisory Committee for Aeronautics) precursor to NASA
Excellent historical discussion in Section 2.8
Early NACA series, 4-, 5-, modified 4-/5-digit generated with analytical equations
Later families, including 6-Series, are more complicated shapes derived using theoretical rather than geometrical methods
Before NACA series, airfoil design was rather arbitrary with nothing to guide designer’s except experience with known shapes and experimentation with modified shapes

9. EXAMPLE: NACA FOUR-DIGIT SERIES
First digit specifies maximum camber in percentage of chord
Second digit indicates position of maximum camber in tenths of chord
Last two digits provide maximum thickness of airfoil in percentage of chord
Example: NACA 2415
Airfoil has maximum thickness of 15%
of chord (0.15c)
Camber of 2% (0.02c) located 40%
back from airfoil leading edge (0.4c)
NACA 2415

10. WHAT CREATES AERODYNAMIC FORCES? (2.2)
Aerodynamic forces exerted by airflow comes from only 2 sources
Pressure, p, distribution on surface
Acts normal to surface
Shear stress, tw, (friction) on surface
Acts tangentially to surface
Pressure and shear are in units of force per unit area (N/m2)
Net unbalance creates an aerodynamic force
“No matter how complex the flow field, and no matter how complex the shape of the body, the only way nature has of communicating an aerodynamic force to a solid object or surface is through the pressure and shear stress distributions that exist on the surface.”
“The pressure and shear stress distributions are the two hands of nature that reach out and grab the body, exerting a force on the body – the aerodynamic force”

11. RESOLVING THE AERODYNAMIC FORCE: WING
Relative Wind: Direction of V∞
We used subscript ∞ to indicate far upstream conditions
Angle of Attack, a: Angle between relative wind (V∞) and chord line
Total aerodynamic force, R, can be resolved into two force components
Lift, L: Component of aerodynamic force perpendicular to relative wind
Drag, D: Component of aerodynamic force parallel to relative wind

12. RESOLVING THE AERODYNAMIC FORCE: ROCKET
Aerodynamic force, R, may also be resolved into components perpendicular and parallel to chord line
Normal Force, N: Perpendicular to chord line
Axial Force, A: Parallel to chord line
L and D are easily related to N and A
For airfoils and wings, L and D most common
For rockets, missiles, bullets, etc. N and A more useful

13. MORE DEFINITIONS
Total aerodynamic force on airfoil is summation of F1 and F2
Lift is obtained when F2 > F1
Misalignment of F1 and F2 creates Moments, M, which tend to rotate airfoil/wing
A moment (torque) is a force times a distance
Value of induced moment depends on point about which moments are taken
Moments about leading edge, MLE, or quarter-chord point, c/4, Mc/4
In general MLE ≠ Mc/4

14. VARIATION OF L, D, AND M WITH a
Lift, Drag and M on a airfoil or wing will change as a changes
Variations of these quantities are some of most important information that an airplane designer needs to know
Aerodynamic Center
Point about which moments essentially do not vary with a
Mac=constant (independent of a)
For low speed airfoils aerodynamic center is near quarter-chord point

15. SAMPLE DATA: SYMMETRIC AIRFOIL
Lift Coefficient
Angle of Attack, a
A symmetric airfoil generates zero lift at zero a

16. SAMPLE DATA: CAMBERED AIRFOIL
Lift Coefficient
Angle of Attack, a
A cambered airfoil generates positive lift at zero a

17. WHY DOES LIFT CURVE BEND OVER?
Low a a
Moderate a a
High a a

19. REAL EFFECTS: VISCOSITY (m)
To understand drag and actual airfoil/wing behavior we need an understanding of viscous flows (all real flows have friction)
Inviscid (frictionless) flow around a body will result in zero drag!
Called d’Alembert’s paradox
Must include friction (viscosity, m) in theory
See §
Flow adheres to surface because of friction between gas and solid boundary
At surface flow velocity is zero, called ‘No-Slip Condition’
Thin region of retarded flow in vicinity of surface, called a ‘Boundary Layer’
At outer edge of B.L., V∞
At solid boundary, V=0
“The presence of friction in the flow causes a shear stress at the surface of a body, which, in turn contributes to the aerodynamic drag of the body: skin friction drag”

20. COMMENTS ON VISCOUS FLOWS

21. TYPES OF FLOWS: FRICTION VS. NO-FRICTION
Flow very close to surface of airfoil is
Influenced by friction and is viscous
(boundary layer flow)
Stall (separation) is a viscous phenomena
Flow away from airfoil is not influenced
by friction and is wholly inviscid

22. THE REYNOLDS NUMBER, Re Within B.L. flow Outside B.L. flow
One of most important dimensionless numbers in fluid mechanics/ aerodynamics
Reynolds number is ratio of two forces:
Inertial Forces
Viscous Forces
c is length scale (chord)
Reynolds number tells you when viscous forces are important and when viscosity may be neglected
Outside B.L. flow
Inviscid (high Re)
Within B.L. flow
highly viscous
(low Re)

23. LAMINAR VERSUS TURBULENT FLOW
Two types of viscous flows
Laminar: streamlines are smooth and regular and a fluid element moves smoothly along a streamline
Turbulent: streamlines break up and fluid elements move in a random, irregular, and chaotic fashion

24. LAMINAR VERSUS TURBULENT FLOW
All B.L.’s transition from laminar to turbulent
Turbulent velocity
profiles are ‘fuller’
cf,turb > cf,lam

25. FLOW SEPARATION
Key to understanding: Friction causes flow separation within boundary layer
Separation then creates another form of drag called pressure drag due to separation

26. COMPARISON OF DRAG FORCES
Same total drag as airfoil

27. OVERVIEW: AIRFOIL STALL
Key to understanding: Friction causes flow separation within boundary layer
B.L. either laminar or turbulent
All laminar B.L. → turbulent B.L.
Turbulent B.L. ‘fuller’ than laminar B.L., more resistant to separation
Separation creates another form of drag called pressure drag due to separation
Dramatic loss of lift and increase in drag

28. WHY DOES BOUNDARY LAYER SEPARATE?
Adverse pressure gradient interacting with velocity profile through B.L.
High speed flow near upper edge of B.L. has enough speed to keep moving through adverse pressure gradient
Lower speed fluid (which has been retarded by friction) is exposed to same adverse pressure gradient is stopped and direction of flow can be reversed
This reversal of flow direction causes flow to separate
Turbulent B.L. more resistance to flow separation than laminar B.L. because of fuller velocity profile
To help prevent flow separation we desire a turbulent B.L.

29. WHY DOES AN AIRFOIL STALL?
Two major consequences of separated flow over airfoil
Dramatic loss of lift (stalling)
Separated flow causes higher pressure on upper surface of airfoil
Major increase in drag
Separation causes lower pressure on trailing edge
Unbalance of pressure force causes pressure drag due to separation

30. SAMPLE DATA: CAMBERED AIRFOIL STALL
Lift or Lift Coefficient
Angle of Attack, a

31. SUMMARY OF VISCOUS EFFECTS ON DRAG
Friction has two effects:
Skin friction due to shear stress at wall
Pressure drag due to flow separation
Total drag due to
viscous effects
Called Profile Drag
Drag due to
skin friction
Drag due to
separation = +
Less for laminar
More for turbulent
More for laminar
Less for turbulent
So how do you design?
Depends on case by case basis, no definitive answer

32. LIFT, DRAG, AND MOMENT COEFFICIENTS
Behavior of L, D, and M depend on a, but also on velocity and altitude
V∞, r ∞, Wing Area (S), Wing Shape, m ∞, compressibility
Characterize behavior of L, D, M with coefficients (cl, cd, cm)
Matching Mach and Reynolds
(called similarity parameters)
M∞, Re
M∞, Re
cl, cd, cm identical

33. LIFT, DRAG, AND MOMENT COEFFICIENTS
Behavior of L, D, and M depend on a, but also on velocity and altitude
V∞, r ∞, Wing Area (S), Wing Shape, m ∞, compressibility
Characterize behavior of L, D, M with coefficients (cl, cd, cm)
Comment on Notation:
Lower case, cl, cd, and cm for infinite wings (airfoils)
Upper case, CL, CD, and CM for finite wings (real wings)

34. SAMPLE DATA: NACA 23012 AIRFOIL
Lift Coefficient
cl=L/(½rV2S)
Moment Coefficient
cm, c/4 a

35. AIRFOIL DATA: NACA 23012 WING SECTION
Re dependence at high a
Separation and Stall
cd vs. a
Dependent on Re cl
cl vs. a
Independent of Re cd
cm,a.c. vs. cl very flat
cm,a.c.
cm,c/4 a cl

36. AIRFOIL THICKNESS: WWI AIRPLANES
English Sopwith Camel
Thin wing, lower maximum CL
Bracing wires required – high drag
German Fokker Dr-1
Higher maximum CL
Internal wing structure
Higher rates of climb
Improved maneuverability

37. EXAMPLE: F-104 LOCKHEED STARFIGHTER
First airplane designed for sustained flight at Mach 2
Very sharp leading edge on wings (razor sharp leading edges, thickness 3.4 %)
Designed to minimize wave drag at supersonic speeds
Very poor low-speed aerodynamic performance
Such wings tend to stall at low angles of attack, CLmax is only about 1.15
Vstall (full) ~ 198 MPH, Vstall (empty) ~ 152 MPH (Vstall proportional to W1/2)

38. EXAMPLE: BOEING 727
Designed in 1960’s to operate out of airports with relatively short runways
Desire to minimize take-off and landing distances
Maximum CL = 3.0
Vstall ~ 113 MPH

39. OVERVIEW: KUTTA CONDITION
Suppose we model the flow around an airfoil using a potential flow approach:
How many theoretical potential flow solutions are exist?
The answer is that there are infinitely many.
Flow (1) and Flow (2) satisfy all conditions for potential flow theory, however G1 ≠ G2, so we know that L1’ ≠ L2’ (recall rotating cylinder example)
Experimental evidence (nature) tells us which G should be used → Kutta Condition (see §4.5)
Flow 1
Flow 2

40. EXAMPLE: NACA 65-006 SYMMETRIC AIRFOIL
dcl/da = 2p
cm,c/4 = 0
Bell X-1 used NACA (6% thickness) as horizontal tail
Thin airfoil theory lift slope:
dcl/da = 2p rad-1 = 0.11 deg-1
Compare with data
At a = -4º: cl ~ -0.45
At a = 6º: cl ~ 0.65
dcl/da = 0.11 deg-1
Thin airfoil theory:
cm,c/4 = 0

41. CAN AN AIRFOIL PRODUCE LIFT WHEN IT IS FLYING UPSIDE DOWN?
NACA 2415 flying right side up
Zero angle of attack
Lift in positive vertical direction
NACA 2415 upside down
Zero angle of attack
Lift in negative vertical direction a a
Positive angle of attack
Say a=10º
Lift in positive vertical direction
Positive angle of attack
Say a=10º
Lift in positive vertical direction, but less than the right side up airfoil

42. CAN AN AIRFOIL PRODUCE LIFT WHEN IT IS FLYING UPSIDE DOWN?
NACA 2415 flying upside down at certain angles of attack will generate positive lift, but less than same airfoil right side up at the same angle of attack
Here is a way to understand this:
If we take airfoil on left and turn it upside down it is same as airfoil right side up but with a negative angle of attack
Therefore, lift coefficient for upside down airfoil at positive angle of attack is given by data for negative angles of attack
The negative cl connotes a downward lift on the ordinary right side up airfoil when pitched to a negative angle of attack
In upside down orientation (airfoil on right), lift is directed upward

43. CAN AN AIRFOIL PRODUCE LIFT WHEN IT IS FLYING UPSIDE DOWN?
NACA 2415 AIRFOIL
Zero-lift a, aL=0 = -2º
So airfoil will generate positive lift (when right side up) for a > -2º
Now turn airfoil upside down
If a = 0º, negative lift
If a = 2º, zero lift
If a is greater than 2º (but reading –a range) airfoil will generate lift in positive vertical direction
Upside down airfoil at same a generates less lift
Example:
Right side up: a = 10º, cl = 1.2
Upside down: a = 10º, cl = - 0.8

44. VORTEX MOVING BLADE ROW MODEL
Thrust of engine largely
dependent on change in
total pressure across blade rows
Blade row interaction critical to
overall performance
Detail of multiple
compressor rows

45. FLIGHT MECHANICS EXAMPLE: GOSSAMER CONDOR
August 26, 1977: Gossamer Condor, designed by Dr. Paul MacCready (Caltech graduate) in United States and piloted by racing cyclist Bryan Allen, wins £50,000 Kremer prize for first 1 mile figure-of-eight flight by a human-powered aircraft
Wingspan, b ~ 29 m, average chord, c ~ 2.3 m, total mass ~ 95 kg, CD ~ 0.05
Pilot delivered ¼ HP to propel aircraft
For cruise at sea-level, estimate:
Cruise speed attained CL
HP required input by pilot to achieve a speed of 20 MPH