AE301 Aerodynamics I UNIT C: 2-D Airfoils ROAD MAP . . . C-1: Aerodynamics of Airfoils 1 C-2: Aerodynamics of Airfoils 2 C-3: Panel Methods C-4: Thin Airfoil Theory Unit C-2: List of Subjects Pressure Coefficient Obtaining Lift from C P Compressibility Effects Critical Mach Number Drag-Divergence Mach Number Supercritical Airfoil Wave Drag
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AE301 Aerodynamics I
UNIT C: 2-D Airfoils
ROAD MAP . . .
C-1: Aerodynamics of Airfoils 1
C-2: Aerodynamics of Airfoils 2
C-3: Panel Methods
C-4: Thin Airfoil Theory
AE301 Aerodynamics I
Unit C-2: List of Subjects
Pressure Coefficient
Obtaining Lift from CP
Compressibility Effects
Critical Mach Number
Drag-Divergence Mach Number
Supercritical Airfoil
Wave Drag
PRESSURE COEFFICIENT
Pressure coefficient (Cp) is a dimensionless quantity
q
ppC
p
Convention of Cp plot: opposed vertical axis (negative up positive down)
Cp = 1: means that the location where the velocity (V) is equal to zero (stagnation point)
Note that the “highest” positive value of pressure coefficient is: Cp = 1 (Cp cannot become more than
1: it is impossible, by definition).
Cp = 0: means that the location where the static pressure at that point (p) becomes equal to the
freestream static pressure (p∞), commonly called: the static port location.
Cp = negative: means that the surface pressure is lower than the freestream static pressure. The
surface of negative pressure coefficient is called “suction surface.”
Cp = positive (but less than 1): means that the surface pressure is higher than the freestream static
pressure. The surface of positive pressure coefficient is called “pressure surface.”
CP DISTRIBUTION OVER AN AIRFOIL (WITH A SMALL POSITIVE AOA)
Upper surface: Cp at the leading edge starts from 1 (stagnation point). Cp starts to decrease (favorable
pressure gradient) very rapidly (p < p∞) and reaches the minimum pressure point. After this minimum
pressure point, Cp increases (adverse pressure gradient) toward the trailing edge.
Lower surface: Cp at the leading edge starts from 1 (stagnation point). Cp starts to decrease (favorable
pressure gradient) and then slightly increase (adverse pressure gradient) toward the trailing edge.
Unit C-2Page 1 of 11
Pressure Coefficient
Cp = 0: pressure is the same as p
Cp = 1: stagnation point
V
At standard sea-level condition,
= 0.0023769 slug/ft3
p = 2,116.2 lb/ft2
The test section airspeed is: 70 mph = 88 ft/s
70 mph60 mph
= 102.667 ft/s
Thus, the dynamic pressure is:
2 21 1(0.0023769)(102.667)
2 2q V = 12.527 lb/ft2
2,100 2,116.2
12.527p
p pC
q
1.293
Unit C-2Page 2 of 11
Class Example Problem C-2-1
Related Subjects . . . “Pressure Coefficient”
Consider an airfoil model mounted in a subsonic wind tunnel. The test section
airspeed is 70 mph, and the condition is the standard sea-level. If the pressure
measured at a point on the airfoil (using a static pressure tap connected to a U-tube
manometer) is 2,100 lb/ft2, what is the corresponding pressure coefficient?
NASA-SC airfoil is relatively “thick” airfoil and not suitable for supersonic flight. The design purpose
is to delay formation of “shock-induced flow separation” to avoid drag divergence at the high-end of
transonic flight.
Boeing (MD) C-17 Globemaster III (US Patent
No: 4,858,853, McDonnell Douglas Corporation,
1987)
Unit C-2Page 9 of 11
Supercritical Airfoil
NASA-SC
Characteristics of NASA-SC Airfoil:• Flattened upper surface,• Highly cambered (curved) aft section, and• Greater leading edge radius as compared to traditional airfoils
WAVE DRAG
Wave drag is the pressure drag due to the formation of shock waves. At supersonic flight, the entire
vehicle is placed “behind” the shockwave. Under this condition, the flow behind the shockwave is not
the same condition of freestream: it is much higher pressure, density, and temperature.
Wave drag is usually the order of magnitude higher than other drag components (such as skin friction &
pressure drags). Super-cruising (cruising at supersonic) is technically very challenging, due to the
massive increase of drag (due to the formation of shock waves: wave drag), usually requires higher
thrust to compensate the higher drag for supersonic cruising.
AERODYNAMIC COEFFICIENTS AT SUPERSONIC FLIGHT
Lift and drag at supersonic flight is mainly dependent upon Mach number (as well as the angle of
attack). For a thin supersonic airfoil (close to a flat plate), the lift and wave drag coefficients can be
estimated as:
2
4 or
1l Lc C
M
2
, ,2
4 or
1d w D wc C
M
Unit C-2Page 10 of 11
Wave Drag
2
, ,2
4 or
1d w D wc C
M
2
4 or
1l Lc C
M
Lift and wave drag coefficients can be calculated as:
= 5 degrees = 5(/180) = 0.087266 radians
2 2
4 4(0.087266)
1 (3) 1lc
M
= 0.1234
2 2
,2 2
4 4(0.087266)
1 (3) 1d wc
M
= 0.01077
At 22,000 ft:
= 0.0011836 slug/ft3
a = 1,028.6 ft/s => M = 3 means that V = (1,028.6)(3) = 3,085.8 ft/s
2 21 1(0.0011836)(3,085.8)
2 2q V = 5,635.215 lb/ft2
Hence, lift and wave drag (per unit span) are:
' (5,635.215)(10)(0.1234)lL q cc = 6,953.85 lb
,' (5,635.215)(10)(0.01077)w d wD q cc = 606.91 lb
Unit C-2Page 11 of 11
Class Example Problem C-2-3
Related Subjects . . . “Wave Drag”
Consider a thin supersonic airfoil with chord length c = 10 ft in a Mach 3 freestream at
an altitude of 22,000 ft. The airfoil is at an angle of attack of 5 degrees. Calculate the
lift & wave drag coefficients and the lift & wave drag per unit span.
Figure: X-43 Hyper X Prototype(NASA Ames Research Center)