4 a- 2: unit a - 5: list of subjectsmercury.pr.erau.edu/~hayasd87/ae301/ae301_notes_a-5.pdfpitching...
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AE301 Aerodynamics I
UNIT A: Fundamental Concepts
ROAD MAP . . .
A-1: Engineering Fundamentals Review
A-2: Standard Atmosphere
A-3: Governing Equations of Aerodynamics
A-4: Airspeed Measurements
A-5: Aerodynamic Forces and Moments
AE301 Aerodynamics I
Unit A-5: List of Subjects
Pressure and Shear Force Distributions
Lift, Drag, and Moment
Aerodynamic Coefficients
Aerodynamic Forces and Moments per Unit Span
2-D Aerodynamic Coefficients
Center of Pressure
Aerodynamic Center
SOURCES OF AERODYNAMIC FORCES AND MOMENTS
There are two sources for aerodynamic forces and moments.
(1) Pressure distribution over the body surface
(2) Shear stress distribution over the body surface
The resultant aerodynamic force (R) can be resolved into 2 components of forces:
(i) In the direction of chord (“body-fitted” coordinate system)
(ii) In the direction of freestream (V)
ANGLE OF ATTACK ( )
Angle of attack is defined as an angle between chord line & freestream.
Unit A-5Page 1 of 11
Pressure and Shear Force Distributions
LIFT, DRAG, AND MOMENT
Lift (L) is defined as the component of the aerodynamic force perpendicular to the relative wind (called,
freestream: V∞). Non-dimensional lift coefficient (CL) is:
Sq
L
SV
LC
L
==2
2
1
Drag (D) is defined as the component of the aerodynamic force parallel to the relative wind (called,
freestream: V∞). Non-dimensional drag coefficient (CD) is:
Sq
D
SV
DC
D
==2
2
1
Pitching moment (M) is defined as a moment created by pressure differences between bottom and top
surfaces. Pitch up moment is positive and pitch down moment is negative. Non-dimensional moment
coefficient (CM) is:
Scq
M
ScV
MC
M
==2
2
1
Note: the moment depends on the location, about which we choose to take moments at:
MLE, Mc/4, and MTE, where: MLE, ≠ Mc/4 ≠ MTE.
(You MUST always specify the LOCATION (that you took the moment) with a subscript)
Unit A-5Page 2 of 11
Lift, Drag, and Moment
CL
CD CM
AERODYNAMIC COEFFICIENTS
There are two different aerodynamic coefficients:
• Upper case: CL, CD, CM = coefficients for 3-D finite wings
L
LC
q S
= D
DC
q S
= M
MC
q Sc
=
S: projected surface area (characteristic surface)
For aircraft wings (lift) = “projected planform” area
Note: for circular cylinder (drag) = “projected frontal” area
• Lower case: cl, cd, cm = coefficients for 2-D airfoils (“unit span” of an infinite wing)
( )
( )' '
1l
L b L Lc
q S q c q c
= = = ( )
( )' '
1d
D b D Dc
q S q c q c
= = = ( )
( ) 2
' '
1m
M b M Mc
q Sc q c c q c
= = =
c: chord length
b: wing span
L’: lift per unit span
D’: drag per unit span
M’: moment per unit span
Unit A-5Page 3 of 11
Aerodynamic Coefficients
S = frontal area
Unit D-1Page 1 of 18
Infinite v.s. Finite Wings
Unit A-5Page 4 of 11
Aerodynamic Forces and Moments per Unit Span (1)
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LIFT, DRAG, AND MOMENT PER UNIT SPAN (1)
Normal and axial forces (per unit span) as well as moment at the leading edge can be calculated by
integrating these equations over the entire surface of the 2-D body (airfoil): TE
LE
' 'N dN=
TE
LE
' 'A dA=
TE
LE LE
LE
' 'M dM=
LIFT, DRAG, AND MOMENT PER UNIT SPAN (2)
TE TE TE
LE LE LE
' ' ( cos sin ) ( cos sin )u u u l l lN dN p ds p ds = = − + + −
TE TE TE
LE LE LE
' ' ( sin cos ) ( sin cos )u u u l l lA dA p ds p ds = = − + + +
TE TE
LE LE
LE LE
' ' ( cos sin ) ( sin cos )u u u u uM dM p x p y ds = = + − −
TE
LE
( cos sin ) ( sin cos )l l l l lp x p y ds + − + + +
Lift and drag forces (per unit span) can be calculated from normal force (N’) and axial force (A’) and
angle of attack () as: ' 'cos 'sinL N A = − ' 'sin 'cosD N A = +
Unit A-5Page 5 of 11
Aerodynamic Forces and Moments per Unit Span (2)
2-D AERODYNAMIC COEFFICIENTS
Lower case cl, cd, cm: 2-D airfoil
L’, D’, and M’: aerodynamic forces and moment per unit span.
'l
Lc
q c
= '
d
Dc
q c
= 2
'm
Mc
q c
=
SKIN FRICTION & PRESSURE COEFFICIENT
Skin friction coefficient (per unit length): fcq
=
: Shear stress developed on the surface (due to viscosity)
Skin friction drag coefficient (for a given surface S): ( )f
f
f D
DC C
q S
= =
fD : Skin friction drag (due to viscosity)
Pressure coefficient: p
p pC
q
−=
Unit A-5Page 6 of 11
2-D Aerodynamic CoefficientsPressure coefficient:
Skin friction coefficient:
Unit A-5Page 7 of 11
Center of Pressure
LE
cp
'
'
Mx
L= − / 4
cp
'
4 '
cMcx
L= −
Center of pressure:
or
M’
= =
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Unit A-5Page 8 of 11
Class Example Problem A-5-1
Related Subjects . . . “Center of Pressure”
A set of experimental data for a series of conventional
airfoil is readily available. This is called “NACA
conventional airfoil data.” NACA 4412 is one of
these NACA conventional airfoils, and it shows the
following aerodynamic coefficients at an angle of
attack of 4 degrees: cl = 0.8 and cm,c/4 = −0.1.
Calculate the location of the center of pressure.
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Unit A-5Page 9 of 11
Aerodynamic Center
=
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Unit A-5Page 10 of 11
Class Example Problem A-5-2
Related Subjects . . . “Aerodynamics Center”
Determine the location of the aerodynamic center (in % chord) of NACA 23012 airfoil.
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Center of pressure: a location where moments become zero: Mcp = 0.
Center of pressure will vary as angle of attack is changed.
Aerodynamic center (also, often called “Neutral Point” or “NP”): a location where moments do not
change with angle of attack (): Mac = const.
Aerodynamic center is independent of angle of attack (usually very close to the quarter chord
location, especially for “thin” airfoils).
The aerodynamic center is also often called the “Neutral Point” (NP), where it is used as a reference
location for aircraft control moments in three axes (“yaw,” “pitch,” and “roll”).
The aerodynamic center is independent of angle of attack (), while the center of pressure depends on
the angle of attack (): this makes the aerodynamic center an important (useful) reference location for
flight mechanics.
Imagine, when you “pitch up” an aircraft, using a control surface. The aircraft will rotate around a
center of gravity (CG). As an aircraft rotates around CG, the increased angle of attack will result in
increased lift force. This lift force will be resolved into two components: (i) a moment generated at the
aerodynamic center (which is “constant”) and (ii) a resultant force acting through the aerodynamic
center that will, in turn, generate control moment (“pitch” either up or down) around a pivot point (CG).
The center of pressure location keeps changing as the angle of attack changes . . . hence, this cannot
be used as a reference location for flight mechanics!
Unit A-5Page 11 of 11
Class Example Problem A-5-3
Related Subjects . . . “Center of Pressure” & “Aerodynamic Center”
For designing an aircraft, the location of
“aerodynamic center” (AC), not “center of pressure”
(CP) is usually used. What is the difference between
the center of pressure and aerodynamic center?
Apparently, the aerodynamic center is more important
(or “useful”) in flight mechanics than the center of
pressure. Why?