1. MAE 3241: AERODYNAMICS AND FLIGHT MECHANICS
Overview of Compressible Flows:
Critical Mach Number and Wing Sweep
April 25, 2011
Mechanical and Aerospace Engineering Department
Florida Institute of Technology
D. R. Kirk

2. EXAMPLES: INFLUENCE OF COMPRESSIBILITY

3. WHEN IS FLOW COMPRESSIBLE?

4. WHEN IS FLOW COMPRESSIBLE?

5. EXAMPLES: COMPRESSIBLE INTERNAL FLOW

6. EXAMPLE: H2 VARIABLE SPECIFIC HEAT, CP

7. COMPRESSIBILITY SENSITIVITY WITH g

8. PRESSURE COEFFICIENT, CP
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 tunnel
Shows effect of M∞ on cl

9. EXAMPLE: CP CALCULATION
See §4.10

10. COMPRESSIBILITY CORRECTION: EFFECT OF M∞ ON CP
Cp at a point on an airfoil of fixed shape
and fixed angle of attack
For M∞ < 0.3, r ~ const
Cp = Cp,0 = 0.5 = const
Flight Mach Number, M∞

11. 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

12. EXAMPLE: SUPERSONIC WAVE DRAG
F-104 Starfighter

13. CRITICAL MACH NUMBER, MCR
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!

14. CRITICAL FLOW AND SHOCK WAVES
MCR
Sharp increase in cd is combined effect of shock waves and flow separation
Freestream Mach number at which cd begins to increase rapidly called Drag-Divergence Mach number

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

16. CRITICAL FLOW AND SHOCK WAVES
MCR

17. EXAMPLE: IMPACT ON AIRFOIL / WING DRAG
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

18. 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!

19. CAN WE PREDICT MCR? A
Pressure coefficient defined in terms of Mach number (instead of velocity)
PROVE THIS FOR CONCEPT QUIZ
In an isentropic flow total pressure, p0, is constant
May be related to freestream pressure, p∞, and static pressure at A, pA

20. CAN WE PREDICT MCR? Combined result
Relates local value of CP to local Mach number
Can think of this as compressible flow version of Bernoulli’s equation
Set MA = 1 (onset of supersonic flow)
Relates CP,CR to MCR

21. HOW DO WE USE THIS? 1 3 2 Plot curve of CP,CR vs. M∞
Obtain incompressible value of CP at minimum pressure point on given airfoil
Use any compressibility correction (such as P-G) and plot CP vs. M∞
Intersection of these two curves represents point corresponding to sonic flow at minimum pressure location on airfoil
Value of M∞ at this intersection is MCR 1 3 2

22. IMPLICATIONS: AIRFOIL THICKNESS
Note: thickness is relative
to chord in all cases
Ex. NACA 0012 → 12 %
Thick airfoils have a lower critical Mach number than thin airfoils
Desirable to have MCR as high as possible
Implication for design → high speed wings usually design with thin airfoils
Supercritical airfoil is somewhat thicker

23. THICKNESS-TO-CHORD RATIO TRENDS
Root: NACA 6716
TIP: NACA 6713
Thickness to chord ratio, %
F-15
Root: NACA 64A(.055)5.9
TIP: NACA 64A203
Flight Mach Number, M∞

24. ROOT TO TIP AIRFOIL THICKNESS TRENDS
Boeing 737
Root
Mid-Span
Tip

25. SWEPT WINGS
All modern high-speed aircraft have swept wings: WHY?

26. WHY WING SWEEP? V∞ V∞
Wing sees component of flow normal to leading edge

27. V∞ V∞,n V∞ V∞,n < V∞ WHY WING SWEEP? W W
Wing sees component of flow normal to leading edge

28. SWEPT WINGS: SUBSONIC FLIGHT
Recall MCR
If M∞ > MCR large increase in drag
Wing sees component of flow normal to leading edge
Can increase M∞
By sweeping wings of subsonic aircraft, drag divergence is delayed to higher Mach numbers

29. SWEPT WINGS: SUBSONIC FLIGHT
Alternate Explanation:
Airfoil has same thickness but longer effective chord
Effective airfoil section is thinner
Making airfoil thinner increases critical Mach number
Sweeping wing usually reduces lift for subsonic flight

30. SWEPT WINGS: SUPERSONIC FLIGHT
If leading edge of swept wing is outside Mach cone, component of Mach number normal to leading edge is supersonic → Large Wave Drag
If leading edge of swept wing is inside Mach cone, component of Mach number normal to leading edge is subsonic → Reduced Wave Drag
For supersonic flight, swept wings reduce wave drag

31. WING SWEEP COMPARISON
F-100D
English Lightning

32. SWEPT WINGS: SUPERSONIC FLIGHT
M∞ < 1
SU-27 q
M∞ > 1
~ 26º
m(M=1.2) ~ 56º
m(M=2.2) ~ 27º

33. WING SWEEP DISADVANTAGE
At M ~ 0.6, severely reduced L/D
Benefit of this design is at M > 1, to sweep wings inside Mach cone
Wing sweep beneficial in that it increases drag-divergences Mach number
Increasing wing sweep reduces the lift coefficient

34. TRANSONIC AREA RULE
Drag created related to change in cross-sectional area of vehicle from nose to tail
Shape itself is not as critical in creation of drag, but rate of change in shape
Wave drag related to 2nd derivative of volume distribution of vehicle

35. EXAMPLE: YF-102A vs. F-102A

36. EXAMPLE: YF-102A vs. F-102A

37. CURRENT EXAMPLES No longer as relevant today – more powerful engines
F-5 Fighter
Partial upper deck on 747 tapers off cross-sectional area of fuselage, smoothing transition in total cross-sectional area as wing starts adding in
Not as effective as true ‘waisting’ but does yield some benefit.
Full double-decker does not glean this wave drag benefit (no different than any single-deck airliner with a truly constant cross-section through entire cabin area)

38. EXAMPLE OF SUPERSONIC AIRFOILS

39. SUPERSONIC AIRFOIL MODELS
Supersonic airfoil modeled as a flat plate
Combination of oblique shock waves and expansion fans acting at leading and trailing edges
R’=(p3-p2)c
L’=(p3-p2)c(cosa)
D’=(p3-p2)c(sina)
Supersonic airfoil modeled as double diamond
Combination of oblique shock waves and expansion fans acting at leading and trailing edge, and at turning corner
D’=(p2-p3)t

40. APPROXIMATE RELATIONS FOR LIFT AND DRAG COEFFICIENTS

41. http://www. hasdeu. bz. edu
CASE 1: a=0°
Shock waves
Expansion

42. CASE 1: a=0°

43. CASE 2: a=4°
Aerodynamic Force Vector
Note large L/D=5.57 at a=4°

44. CASE 3: a=8°

45. CASE 5: a=20°
At around a=30°, a detached shock begins to form before bottom leading edge

46. CASE 6: a=30°

47. DESIGN OF ASYMMETRIC AIRFOILS

48. QUESTION 9.14 Compare with your solution
Consider a diamond-wedge airfoil as shown in Figure 9.36, with half angle e=10°
Airfoil is at an angle of attack a=15° in a Mach 3 flow.
Calculate the lift and wave-drag coefficients for the airfoil.
Compare with your solution

49. EXAMPLE: MEASUREMENT OF AIRSPEED
Pitot tubes are used on aircraft as speedometers (point measurement)
Subsonic
M < 0.3
Subsonic
M > 0.3
Supersonic
M > 1
M < 0.3 and M > 0.3: Flows are qualitatively
similar but quantitatively different
M < 1 and M > 1: Flows are
qualitatively and quantitatively different

50. MEASUREMENT OF AIRSPEED: INCOMPRESSIBLE FLOW (M < 0.3)
May apply Bernoulli Equation with relatively small error since compressibility effects may be neglected
To find velocity all that is needed is pressure sensed by Pitot tube (total or stagnation pressure) and static pressure
Comment: What is value of r?
If r is measured in actual air around airplane (difficult to do)
V is called true airspeed, Vtrue
Practically easier to use value at standard seal-level conditions, rs
V is called equivalent airspeed, Ve
Static
pressure
Dynamic
pressure
Total
pressure
Incompressible Flow

51. MEASUREMENT OF AIRSPEED: SUBSONIC COMRESSIBLE FLOW (0. 3 < M < 1
If M > 0.3, flow is compressible (density changes are important)
Need to introduce energy equation and isentropic relations

52. MEASUREMENT OF AIRSPEED: SUBSONIC COMRESSIBLE FLOW (0. 3 < M < 1
How do we use these results to measure airspeed?
p0 and p1 give flight Mach number
Instrument called Mach meter
M1 = V1/a1
V1 is actual flight speed
Actual flight speed using pressure difference
What are T1 and a1?
Again use sea-level conditions Ts, as, ps (a1 = (gRT)½ = m/s)
V is called Calibrated Velocity, Vcal

53. MEASUREMENT OF AIRSPEED: SUPERSONIC FLOW (M > 1)
Rayleigh Pitot Tube Formula

54. EXAMPLE: SUBSONIC AND SUPERSONIC FLIGHT
Flight at four different speeds, pitot measures p0 = 1.05, 1.2, 3 and 10 atm
What is flight speed if flying in 1 atm static pressure and Tambient = 288 K (a = 340 m/s)?
Determine which measurements are in subsonic or supersonic flow
p0/p = is boundary between subsonic and sonic flows
1.05 atm → p0/p = 1.05 → subsonic
Use compressible flow form, M = 0.265, V ~ 90 m/s ~ 200 MPH
Could use Bernoulli which will provide small error (~ 1%) and give V directly
Compressible form requires knowledge of speed of sound (temperature)
Apply Bernoulli safely? p0/p < 1.065
1.2 atm → p0/p = 1.2 → subsonic
M = 0.52, V ~ 177 m/s ~ 396 MPH
Use of compressible subsonic form justified (Bernoulli ~ 3% error)
3 atm → p02/p1 = 3 → supersonic
M1 = 1.39, V ~ 473 m/s ~ 1057 MPH (Bernoulli ~ 22% error)
10 atm → p02/p1 = 10 → supersonic
M1 = 2.73, V ~ 928 m/s ~ 2076 MPH (MCO → LAX in 1 hour 30 minutes)