unit c: 2 - d airfoils - embry–riddle aeronautical...

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


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?

Static Pressure Taps

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LIFT COEFFICIENT CALCULATION FROM CP DISTRIBUTION

Lift force (per unit span) can be found by integrating Cp over the surface: TE TE

LE LE

' cos cosl uL p ds p ds ,

where, lp = Pressure on the lower surface

up = Pressure on the lower surface

also, cosds dx

0 0

'

c c

l uL p dx p dx

Adding and subtracting p, we have:

0 0

'

c c

l uL p p dx p p dx

From the definition of lift coefficient:

0 0

' 1 1c c

l ul

p p p pLc dx dx

q c c q c q

, ,

0

1( )

c

p l p uC C dxc

NORMALIZED COORDINATE LOCATION

Often, it is convenient to specify the coordinate “along the chord”: x

xc

1

, , , ,

0 0

1( ) ( )

x cc

p l p u p l p u

x c

xC C dx C C d

c c

1

, ,

0

( )p l p uC C dx


Obtaining Lift from CP



Related Subjects . . . “Obtaining Lift from Cp”

Consider an airfoil with chord length c and the running distance x measured along the

chord line. The leading edge is located at x/c = 0 and the trailing edge at x/c = 1.

Calculate the lift coefficient for this airfoil. The pressure coefficient variations over the

upper and lower surfaces are given respectively as:

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COMPRESSIBILITY CORRECTION FOR PRESSURE COEFFICIENT

As the freestream Mach number is increased (M > 0.3), the compressibility effects can no longer be

ignored. A correction for compressibility is called, the Prandtl-Glauert rule:

,0

21

p

p

CC

M

This is simply a “correction factor” and will only provide “estimated” pressure coefficient at a given

Mach number

In order to more accurately determine pressure coefficient, one needs to apply energy equation (if it

is reasonable to assume that the flow is still isentropic and ideal gas of air).

CRITICAL MACH NUMBER

Critical Mach number (Mcr): the freestream Mach number at which sonic flow is first obtained

somewhere on the airfoil surface.

Critical pressure coefficient (Cp,cr): the specific value of Cp that corresponds to the presence of sonic

flow (M = 1).


Compressibility Effects

Prandtl-Glauert Rule:

EFFECTS OF AIRFOIL THCKNESS

If the airfoil is thick, the pressure at the minimum pressure point on the surface of the airfoil becomes

lower. As a result, the sonic flow begins to appear at much lower Mach number. Therefore, the critical

Mach number is lower.

EQUATION OF CRITICAL MACH NUMBER (1)

Starting from the pressure coefficient: 1p

p p p pC

q q p

eqn. 1

Recall, the definition of dynamic pressure:

2

2 21 1 1

2 2 2

Vq V p V p

p p

eqn. 2

Also, recall, speed of sound is 2a RT p , and eqn. 2 becomes:

2 22 2

2

1 1 1

2 2 2 2

V Vq V p p p M

p a

eqn. 3


Critical Mach Number (1)

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EQUATION OF CRITICAL MACH NUMBER (2)

Recall, for isentropic flow (between freestream and stagnation point):

1

20 11

2

pM

p

eqn. 4

Applying eqn. 4 between freestream and stagnation point:

1

20 11

2

pM

p

eqn. 5

Combining eqns. 4 & 5 yields:

1

2

0

20

11 1

21

1 12

Mpp p

p p pM

eqn. 6

Now, substituting eqn. 6 into eqn. 1:

1 1

2 2

22 2 2

1 11 1 1 1

22 21 1 11 1 1

1 1 1 12 2 2

p

M Mp pp

Cq p M

p M M M

Cp = Cp,cr, where M = 1, therefore the Cp at critical Mach number is:

1

2

, 2

2 121

1p cr

MC

M


Critical Mach Number (2)

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DRAG DIVERGENCE MACH NUMBER

Drag divergence Mach number: the freestream Mach number at which drag coefficient begins to

increase rapidly.

Mcr < Mdrag divergence < 1.0

Drag divergence is based mainly on the formation of strong normal shock wave at the minimum

pressure point on the upper surface of the airfoil and associated flow separation behind the shock wave,

called the “shock-induced” flow separation.

Due to the drag divergence, the early attempts of supersonic flight (NASA / Bell X-1) faced

difficulties of achieving the successful supersonic flight: this is commonly known as a “SONIC

BARRIER.”


Drag-Divergence Mach Number

SUPERCRITICAL AIRFOIL

A supercritical (SC) airfoil (NASA-SC or NACA 8-series) is an airfoil designed, primarily, to delay the

onset of wave drag in the transonic speed range. The supercritical airfoil was created in the 1960s, by

NASA scientist Richard Whitcomb.

While the design was initially developed as part of the supersonic transport (SST) project at NASA, it

has since been mainly applied to increase the fuel efficiency of many high subsonic aircraft.

Supercritical airfoils have three main benefits:

Higher (or “delayed”) critical Mach number,

Develop shock waves further aft than traditional airfoils, and thus,

Greatly reduce shock-induced boundary layer separation.

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)


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


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



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)

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unit c: 2 - d airfoils - embry–riddle aeronautical...

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