1. MAE 1202: AEROSPACE PRACTICUM
Lecture 7: Compressible Flow Review and Overview of Airfoils
March 11, 2013
Mechanical and Aerospace Engineering Department
Florida Institute of Technology
D. R. Kirk

2. UDPATES Mid-term grades
Team project: Introduced in Laboratory this week
Mid-Term Exam: Monday, March 18, 2013 in class
Covers Chapter 4 and Chapter 5.1 – 5.7
Open book / open notes… but no time to study during the exam
No computers, cell phones, etc.
Sample Mid-Term with Solution on line
Review Session: Thursday, March 14, 2013, Crawford Science Tower, Room 112, 8 – 10 pm
AIAA Meeting/Fund Raiser Friday, March 15, 2013, 7:00 pm – 11:00 pm, Buffalo Wild Wings (Palm Bay Road)

3. READING AND HOMEWORK ASSIGNMENTS
Reading: Introduction to Flight, by John D. Anderson, Jr.
For March 25, 2013 lecture: Chapter 5, Sections
Read these sections carefully, most interesting portions of Ch. 5
Lecture-Based Homework Assignment:
Problems: 5.7, 5.11, 5.13, 5.15, 5.17, 5.19
DUE: Friday, March 29, 2013 by 5pm
Turn in hard copy of homework
Also be sure to review and be familiar with textbook examples in Chapter 5

4. ANSWERS TO LECTURE HOMEWORK
5.7: Cp = -3.91
5.11: Cp =
Be careful here, if you check the Mach number it is around 0.71, so the flow is compressible and the formula for Cp based on Bernoulli’s equation is not valid. To calculate the pressure coefficient, first calculate r∞ from the equation of state and find the temperature from the energy equation. Finally make use of the isentropic relations and the definition of Cp given in Equation 5.27
5.13: cl = 0.97
Make use of Prandtl-Glauert rule
5.15: Mcr = 0.62
Use graphical technique of Section 5.9
Verify using Excel or Matlab
5.17: m = 30°
5.19: D = 366 lb
Remember that in steady, level flight the airplane’s lift must balance its weight
You may also assume that all lift is derived from the wings (this is not really true because the fuselage and horizontal tail also contribute to the airplane lift). Also assume that the wings can be approximated by a thin flat plate
Remember that Equation 5.50 gives a in radians

5. HOMEWORK EXAMPLES
Can you read this?
Who would you hire?

6. MODERN ART?

7. Compressible Flow Applications

8. 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

9. MEASUREMENT OF AIRSPEED: SUBSONIC COMRESSIBLE FLOW
If M > 0.3, flow is compressible (density changes are important)
Need to introduce energy equation and isentropic relations
cp: specific heat at constant pressure
M1=V1/a1
gair=1.4

10. EXAMPLE: TOTAL TEMPERATURE
Static temperature
Vehicle flight
Mach number
A rocket is flying at Mach 6 through a portion of the atmosphere where the static temperature is 200 K
What temperature does the nose of the rocket ‘feel’?
T0 = 200(1+ 0.2(36)) = 1,640 K!

11. MEASUREMENT OF AIRSPEED: SUBSONIC COMRESSIBLE FLOW
So, how do we use these results to measure airspeed
p0 and p1 give
Flight Mach number
Mach meter
M1=V1/a1
Actual Flight Speed
Actual Flight Speed
using pressure difference
What is T1 and a1?
Again use sea-level conditions Ts, as, ps (a1=340.3 m/s)

12. MEASUREMENT OF AIRSPEED: SUPERSONIC FLOW
What can happen in supersonic flows?
Supersonic flows (M > 1) are qualitatively and quantitatively different from subsonic flows (M < 1)

13. HOW AND WHY DOES A SHOCK WAVE FORM?
Think of a as ‘information speed’ and M=V/a as ratio of flow speed to information speed
If M < 1 information available throughout flow field
If M > 1 information confined to some region of flow field

14. MEASUREMENT OF AIRSPEED: SUPERSONIC FLOW
Notice how different this expression is from previous expressions
You will learn a lot more about shock wave in compressible flow course

15. SUMMARY OF AIR SPEED MEASUREMENT
Subsonic, incompressible
Subsonic, compressible
Supersonic

16. HOW ARE ROCKET NOZZLES SHAPPED?

17. MORE ON SUPERSONIC FLOWS (4.13)
Isentropic flow in a streamtube
Differentiate
Euler’s Equation
Since flow is isentropic
a2=dp/dr
Area-Velocity Relation

18. CONSEQUENCES OF AREA-VELOCITY RELATION
IF Flow is Subsonic (M < 1)
For V to increase (dV positive) area must decrease (dA negative)
Note that this is consistent with Euler’s equation for dV and dp
IF Flow is Supersonic (M > 1)
For V to increase (dV positive) area must increase (dA positive)
IF Flow is Sonic (M = 1)
M = 1 occurs at a minimum area of cross-section
Minimum area is called a throat (dA/A = 0)

19. 2: OUTLET 1: INLET TRENDS: CONTRACTION M1 < 1 M1 > 1 V2 > V1

20. 2: OUTLET 1: INLET TRENDS: EXPANSION M1 < 1 M1 > 1 V2 < V1

21. PUT IT TOGETHER: C-D NOZZLE
1: INLET
2: OUTLET

22. MORE ON SUPERSONIC FLOWS (4.13)
A converging-diverging, with a minimum area throat, is necessary to produce a supersonic flow from rest
Supersonic wind tunnel section
Rocket nozzle

23. Chapter 5 Overview

24. 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

25. 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

26. 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?

27. 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?

28. 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

29. 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

30. 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

31. WHAT CREATES AERODYNAMIC FORCES? (2.2)
Aerodynamic forces exerted by airflow comes from only two 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”

32. RESOLVING THE AERODYNAMIC FORCE
Relative Wind: Direction of V∞
We use 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

33. 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 F1 F2

34. VARIATION OF L, D, AND M WITH a
Lift, Drag, and Moments 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, c/4

35. AOA = 2°

36. AOA = 3°

37. AOA = 6°

38. AOA = 9°

39. AOA = 12°

40. AOA = 20°

41. AOA = 60°

42. AOA = 90°

43. SAMPLE DATA: SYMMETRIC AIRFOIL
Lift (for now)
Angle of Attack, a
A symmetric airfoil generates zero lift at zero a

44. SAMPLE DATA: CAMBERED AIRFOIL
Lift (for now)
Angle of Attack, a
A cambered airfoil generates positive lift at zero a

45. SAMPLE DATA Lift (for now) Cambered airfoil has lift at a=0
Lift coefficient (or lift) linear variation with angle of attack, a
Cambered airfoils have positive lift when a = 0
Symmetric airfoils have zero lift when a = 0
At high enough angle of attack, the performance of the airfoil rapidly degrades → stall
Lift (for now)
Cambered airfoil has
lift at a=0
At negative a airfoil
will have zero lift

46. SAMPLE DATA: STALL BEHAVIOR
Lift (for now)
What is really going on here
What is stall?
Can we predict it?
Can we design for it?

47. 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!
This is called d’Alembert’s paradox
Must include friction (viscosity, m) in theory
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” p.219, Section 4.20

48. 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

49. COMMENTS ON VISCOUS FLOWS (4.15)

50. 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)

51. LAMINAR VS. 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

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

53. 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

54. REVIEW: AIRFOIL STALL (4.20, 5.4)
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

55. SUMMARY OF VISCOUS EFFECTS ON DRAG (4.21)
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!

56. COMPARISON OF DRAG FORCES
Same total drag as airfoil

57. TRUCK SPOILER EXAMPLE Note ‘messy’ or turbulent flow pattern High drag
Lower fuel efficiency
Spoiler angle increased by + 5°
Flow behavior more closely resembles a laminar flow
Tremendous savings (< $10,000/yr) on Miami-NYC route

58. LIFT, DRAG, AND MOMENT COEFFICIENTS (5.3)
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

59. LIFT, DRAG, AND MOMENT COEFFICIENTS (5.3)
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)
Note on Notation:
We use lower case, cl, cd, and cm for infinite wings (airfoils)
We use upper case, CL, CD, and CM for finite wings

60. SAMPLE DATA: NACA 23012 AIRFOIL
Flow separation
Stall
Lift Coefficient cl
Moment Coefficient
cm, c/4 a

61. AIRFOIL DATA (5.4 AND APPENDIX D) 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
R=Re
cm,a.c. vs. cl very flat
cm,a.c.
cm,c/4 a cl

62. EXAMPLE: BOEING 727 FLAPS/SLATS

63. EXAMPLE: SLATS AND FLAPS

64. AIRFOIL DATA (5.4 AND APPENDIX D) NACA 1408 WING SECTION
Flaps shift lift curve
Effective increase in camber of airfoil
Flap extended
Flap retracted

65. PRESSURE DISTRIBUTION AND LIFT
Lift comes from pressure distribution over top (suction surface) and bottom (pressure surface)
Lift coefficient also result of pressure distribution

66. PRESSURE COEFFICIENT, CP (5.6)
Use non-dimensional description, instead of plotting actual values of pressure
Pressure distribution in aerodynamic literature often given as Cp
So why do we care?
Distribution of Cp leads to value of cl
Easy to get pressure data in wind tunnels
Shows effect of M∞ on cl

67. EXAMPLE: CP CALCULATION

68. COMPRESSIBILITY CORRECTION: EFFECT OF M∞ ON CP
For M∞ < 0.3, r ~ const
Cp = Cp,0 = 0.5 = const M∞

69. COMPRESSIBILITY CORRECTION: EFFECT OF M∞ ON CP
Effect of compressibility
(M∞ > 0.3) is to increase
absolute magnitude of Cp as M∞ increases
Called: Prandtl-Glauert Rule
For M∞ < 0.3, r ~ const
Cp = Cp,0 = 0.5 = const M∞
Prandtl-Glauert rule applies for 0.3 < M∞ < 0.7

70. OBTAINING LIFT COEFFICIENT FROM CP (5.7)

71. COMPRESSIBILITY CORRECTION SUMMARY
If M0 > 0.3, use a compressibility correction for Cp, and cl
Compressibility corrections gets poor above M0 ~ 0.7
This is because shock waves may start to form over parts of airfoil
Many proposed correction methods, but a very good on is: Prandtl-Glauert Rule
Cp,0 and cl,0 are the low-speed (uncorrected) pressure and lift coefficients
This is lift coefficient from Appendix D in Anderson
Cp and cl are the actual pressure and lift coefficients at M∞

72. CRITICAL MACH NUMBER, MCR (5.9)
As air expands around top surface near leading edge, velocity and M will increase
Local M > M∞
Flow over airfoil may have
sonic regions even though
freestream M∞ < 1
INCREASED DRAG!

73. CRITICAL FLOW AND SHOCK WAVES
MCR

74. CRITICAL FLOW AND SHOCK WAVES
‘bubble’ of supersonic flow

75. AIRFOIL THICKNESS SUMMARY
Note: thickness is relative
to chord in all cases
Ex. NACA 0012 → 12 %
Which creates most lift?
Thicker airfoil
Which has higher critical Mach number?
Thinner airfoil
Which is better?
Application dependent!

76. 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

77. THICKNESS-TO-CHORD RATIO TRENDS
Root: NACA 6716
TIP: NACA 6713
F-15
Root: NACA 64A(.055)5.9
TIP: NACA 64A203

78. MODERN AIRFOIL SHAPES Boeing 737 Root Mid-Span Tip

79. SUMMARY OF AIRFOIL DRAG (5.12)
Only at transonic and
supersonic speeds
Dwave=0 for subsonic speeds
below Mdrag-divergence
Profile Drag
Profile Drag coefficient relatively constant with M∞ at subsonic speeds