aeronautical engineering AERODYNAMICS - Lecture 3.pdf

Este documento trata sobre aerodinámica y fuerzas aerodinámicas en el contexto de la ingeniería aeronáutica. Aquí tienes un resumen de los puntos clave:

Fuerzas Aerodinámicas: Se generan a partir de la distribución de presión y el esfuerzo cortante en la superficie de un objeto.
Flujo Incompresible sobre Perfiles Aerodinámicos: Describe cómo la presión y el esfuerzo cortante se correlacionan con las fuerzas aerodinámicas.
Resolución de Fuerzas Aerodinámicas: Explica cómo la fuerza aerodinámica total se puede descomponer en componentes de sustentación y resistencia.
Coeficientes de Sustentación, Resistencia y Momento: Analiza cómo estos coeficientes varían con el ángulo de ataque, la velocidad y la altitud.

1. AERODYNAMICS
Aeronautical Engineering
School of Engineering
UPB

2. •Aerodynamic forces exerted by airflow comes from only 2 sources:
1.Pressure, p, distribution on the surface.
•Acts normal to surface.
2.Shear stress, 
w, (friction) on surface.
•Acts tangentially to surface.
•Pressure and shear are in units of force per unit area (N/m
2
).
•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”.
WHAT CREATES
AERODYNAMIC FORCES?

3. •Pressure and Shear Stress – correlation with aerodynamic forces.
SV
∞
V
∞
Pressure – Acts perpendicular to the surface
Shear stress – Acts parallel to the surface)(Spp= )(S=
INCOMPRESSIBLE FLOW
OVER AIRFOILS

4. V
∞
N →p
u(S
u)
A →
u(S
u)
�=ඵ
??????
����+ඵ
??????
????????????��
INCOMPRESSIBLE FLOW
OVER AIRFOILS
•Pressure and Shear Stress – correlation with aerodynamic forces.

5. Relative Wind: Direction of V
∞
•We use the subscript ∞ to indicate far
upstream conditions.
Angle of Attack (AoA),  Angle between
relative wind (V
∞) and the chord line.
Total aerodynamic force, R, can be
resolved into two force components.
Lift, L: Component of aerodynamic force
perpendicular to the relative wind.
Drag, D: Component of aerodynamic
force parallel to the relative wind.
RESOLVING THE
AERODYNAMIC FORCE
V
∞

6. •Aerodynamic force, R, may also be
resolved into components perpendicular
and parallel to the chord line.
•Normal Force, N: Perpendicular to the
chord line.
•Axial Force, A: Parallel to the chord line.
•L and D are easily related to N and A.
•For airfoils and wings, L and D are the
most common.
•For rockets, missiles, bullets, etc. N and A
more useful.
RESOLVING THE
AERODYNAMIC FORCE
V
∞

7. •Aerodynamic forces and moments are a direct consequence of
pressure distribution.
V
∞
V
∞
L
L
D
D
R
R
M
M

N
AsincosANL −= cossinAND +=
INCOMPRESSIBLE FLOW
OVER AIRFOILS

8. AoA = 2°

9. AoA = 3°

10. AoA = 6°

11. AoA = 9°

12. AoA = 12°

13. AoA = 20°

14. AoA = 60°
AoA = 90°

16. •Behavior of L, D, and M depends on
, but also on velocity and altitude.
•V
∞, 
∞, Wing Reference Area (S), Wing
Shape, 
∞, compressibility.
•Characterize the behavior of L, D, M
with coefficients (c
l, c
d, c
m).( )Re,,
2
1
2
1
2
2




=
=
=
Mfc
Sq
L
SV
L
c
ScVL
l
l
l



Matching Mach and Reynolds
(called similarity parameters)
M
∞, Re
M
∞, Re
c
l, c
d, c
m identical
LIFT, DRAG AND MOMENT
COEFFICIENTS

17. •Behavior of L, D, and M depend on , but also on velocity and altitude.
•V
∞, 
∞, Wing Reference Area (S), Wing Shape, 
∞, compressibility.
•Characterize the behavior of L, D, M with coefficients (c
l, c
d, c
m).( )Re,,
2
1
2
1
3
2
2




=
=
=
Mfc
Scq
M
ScV
M
c
SccVM
m
m
m


 ( )Re,,
2
1
2
1
2
2
2




=
=
=
Mfc
Sq
D
SV
D
c
ScVD
d
d
d


 ( )Re,,
2
1
2
1
1
2
2




=
=
=
Mfc
Sq
L
SV
L
c
ScVL
l
l
l



Important note: Comment on Notation for the coefficients:
•Lowercase, c
l, c
d, and c
m for infinite wings (airfoils).
•Uppercase, C
L, C
D, and C
M for finite wings (actual 3D objects).
LIFT, DRAG AND MOMENT
COEFFICIENTS

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

19. •Lift, Drag and
Moment
Coefficients
INCOMPRESSIBLE
FLOW OVER
AIRFOILS
c
l
c
l, max

20. Lift Coefficient
Angle of Attack, 
A symmetric airfoil
generates zero lift at
zero 
SAMPLE DATA:
SYMMETRIC AIRFOIL

21. A cambered airfoil
generates positive
lift at zero 
SAMPLE DATA:
CAMBERED AIRFOIL
Lift Coefficient
Angle of Attack, 

22. •Lift slope
�
0=�
�??????
=
��
�
�??????
=
�
�−�
�0
??????−0
1
��??????
�
�=�
�0
+�
0??????=�
0??????−??????
??????=0
�
�0
=−�
0??????
??????=0
INCOMPRESSIBLE FLOW
OVER AIRFOILS
�
0=�
�??????
=
��
�
�??????
=
�
�−0
??????−??????
??????=0
1
��??????
c
l
c
l, max

23. •Lift, Drag, and Moment on an airfoil or wing will change as  changes.
•Variations of these quantities are some of the most essential information that
an airplane designer needs to know.
•Aerodynamic Center
•Point about which moments essentially do not vary with .
•M
ac = constant (independent of ).
•For low-speed airfoils, the aerodynamic center is near the quarter-chord point.
VARIATION OF L, D AND
M WITH 

24. •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!
•Called d’Alembert’s paradox.
• Friction (viscosity, m) must be included in theory.
•Flow adheres to the 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 the vicinity of the surface, called a ‘Boundary Layer’.
•At the outer edge of B.L., V
∞
•At the 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”.
REAL EFFECTS:
VISCOSITY ()

25. 
xV
forcesviscuos
forcesinercial 
==Re  = Density [kg/m
3
]
V = Velocity [m/s]
x = Length (characteristic object length) [m]
 = Dynamic viscosity [kg/ms]
Sometimes, the Reynolds number can be expressed
in terms of the kinematic viscosity (v [s/m
2
]):v
xV
=Re
Osborne Reynolds
(1842 –1912)
REYNOLDS NUMBER

26. Lift Coefficient
c
l = L/(½V
2
S)
Moment Coefficient
c
m, c/4

Flow
separation
Stall
SAMPLE
DATA: NACA
23012
AIRFOIL

27. c
l
c
m,c/4
Re dependence at high 
Separation and Stall
 c
l
c
d
c
m,a.c.
c
l vs. 
Independent of Re
c
d vs. 
Dependent on Re
c
m,a.c. vs. c
l very flat
R = Re
AIRFOIL
DATA: NACA
23012 WING
SECTION

28. 28
A constant chord length model wing is
placed in a low-speed subsonic wind
tunnel spanning the test section. The
wing has a NACA 2412 airfoil and a
chord length of 1.3 m. The flow in the
test section is at a velocity of 50 m/s at
standard sea-level conditions. If the
wing is at a 4° angle of attack, calculate
the following:
a.c
l, c
d and c
m_c/4 and
b.the lift, drag, and moments about
the quarter chord per unit span
EXAMPLE NACA 2412
��=
??????
∞??????
∞�
??????
∞
=
1.225ൗ
????????????
�
350Τ
�
�1.3�
1.789×10
−5
ൗ
????????????
��
��=4450810.51

29. Incompressible Flow over Aerofoils
•Lift, Drag and Moment Coefficients
For a 4°
angle of
attack
4
0.63
0.63
0.007
-0.035

30. 30
A constant chord length model
wing is placed in a low-speed
subsonic wind tunnel spanning
the test section. The wing has a
NACA 2412 airfoil and a chord
length of 1.3 m. The flow in the
test section is at a velocity of 50
m/s at standard sea-level
conditions. If the wing is at a 4°
angle of attack, calculate the
following:
a.c
l, c
d. and c
m_c/4 and
b.the lift, drag, and moments
about the quarter chord per
unit span
EXAMPLE NACA 2412
�
∞=
1
2
??????
∞??????
∞
2
=
1
2
1.225ൗ
????????????
�
350Τ
�
�
2
=1531.25??????�
�=�∙�=1�=1.3�
2
�=�
∞��
�=1531.25ൗ
�
�
21.3�
2
0.63=1254.09�
??????=�
∞��
�=1531.25ൗ
�
�
21.3�
2
0.007=13.93�
�
ൗ
�
4
=�
∞��
�
ൗ
??????
4
�=1531.25ൗ
�
�
21.3�
2
−0.0351.3�=−90.57�∙�
V
∞
c
b = 1
S

31. 31
The NACA 2412 wing, in the same
flow as in the previous example, is
pitched to an angle of attack such
that the lift per unit span is 700 N
(157 lb).
a.What is the angle of attack?
b.To what angle of attack must the
infinite wing be pitched to obtain
zero lift?
EXERCISE 1
�
∞=
1
2
??????
∞??????
∞
2
=
1
2
1.225ൗ
????????????
�
350Τ
�
�
2
=1531.25??????�
�=�∙�=1�=1.3�
2

32. Consider a NACA 23012 airfoil at 8 degrees of angle of attack. Calculate
the normal and axial force coefficients. Assume that Re = 8.8 x 10
6
.
EXERCISE 2
V
∞


33. •NACA 2415 flying upside down at certain AoA will generate positive lift but less than the same airfoil right side up at the
same AoA.
•Here is a way to understand this:
•If we take the airfoil on the left and turn it upside down, it is the same as the airfoil right side up but with a negative AoA.
•Therefore, the lift coefficient for upside down airfoil at a positive angle of attack is given by data for negative AoA.
•The negative c
l connotes a downward lift on the ordinary right-side-up airfoil when pitched to a negative Aoa.
•In an upside-down orientation (airfoil on the right), the lift is directed upward.
 
CAN AN AIRFOIL PRODUCE LIFT
WHEN IT IS FLYING UPSIDE DOWN?

34. •NACA 2415 flying right side up.
•Zero angle of attack.
•Lift in the positive vertical direction.
•NACA 2415 upside down.
•Zero angle of attack.
•Lift in the negative vertical direction.
•Positive angle of attack.
•Say  =10º.
•Lift in the positive vertical direction.
•Positive angle of attack.
•Say  =10º.
•Lift in the positive vertical direction, but less
than the right side up airfoil.
 
CAN AN AIRFOIL PRODUCE LIFT
WHEN IT IS FLYING UPSIDE DOWN?

35. •NACA 2415 AIRFOIL
•Zero-lift , 
L=0 = -2°
•The airfoil will generate positive lift
(when right side up) for  > -2º
•Now turn the airfoil upside down
•If  = 0º, negative lift
•If  = 2º, zero lift
•If  is greater than 2º (but reading –
 range), the airfoil will generate
lift in the positive vertical direction
•Upside-down airfoil at same  generates
less lift
•Example:
•Right side up:  = 10º, c
l = 1.2
•Upside down:  = 10º, c
l = - 0.8
CAN AN AIRFOIL PRODUCE
LIFT WHEN IT IS FLYING
UPSIDE DOWN?

36. 36
The question is sometimes asked: Can an airfoil product lift when
flying upside-down? This exercise will answer that question.
a.Consider, a NACA 2415 airfoil flying right side up at an angle of
attack of 6°. The airfoil has a chord length of 1.5 m and is flying at
a standard altitude of 2 km at a velocity of 150 m/s. Calculate the
lift per unit span.
b.Now, turn this airfoil upside-down at the same flight conditions at
an angle of attack of 6°. Calculate the lift per unit span.
c.Compare and discuss the results.
EXERCISE 3

37. •Aerodynamic forces and moments as a
consequence of pressure distribution
•Total aerodynamic force on the airfoil is the
summation of F
1 and F
2
•Lift is obtained when F
2 > F
1
•Misalignment of F
1 and F
2 creates
Moments, M, which tend to rotate
airfoil/wing
•Value of the induced moment depends on
the point about which it is taken
•Moments about leading edge, M
LE or
quarter-chord point, c/4, (M
c/4)
•In general, M
LE ≠ M
c/4
F
1
F
2
V
∞
INCOMPRESSIBLE FLOW
OVER AIRFOILS

38. INCOMPRESSIBLE FLOW
OVER AIRFOILS
•Aerodynamic Moments
•Essentially linear over a practical range
of AoA.
•Slope is positive for some airfoils and
negative for others.
•Variation becomes non-linear at high
AoA, when the flow is separated.
• The linear portion of the curve is
essentially independent of Re.

39. •Aerodynamic Moments
M
c/4
V
∞
D
c/4
L
c/4
x
c/4
c

INCOMPRESSIBLE FLOW
OVER AIRFOILS
•The aerodynamic moment exerted on
the body depends on the point about
which the moments are taken.
•Considering moments about the L.E.:
moments that tend to increase the AoA
(pitch up – positive).
(-)
L.E.
(+)
T.E.x
ac
L
ac
D
acM
ac

40. M
c/4
V
∞
D
c/4
L
c/4
x
c/4
c

L.E.
T.E.x
ac
L
ac
D
acM
ac
�
????????????=�
ൗ
�
4
−�
ൗ
�
4
??????
ൗ
�
4
cos??????−??????
ൗ
�
4
??????
ൗ
�
4
sin??????
�
????????????=�
??????�−�
??????�??????
??????�cos??????−??????
??????�??????
??????�sin??????
0=�
ൗ
�
4
−�
??????�−�
ൗ
�
4
??????
ൗ
�
4
cos??????+�
??????�??????
??????�cos??????−??????
ൗ
�
4
??????
ൗ
�
4
sin??????+??????
??????�??????
??????�sin??????
�
ൗ
�
4
=�
??????�=�
??????
ൗ
�
4
=??????
??????�=??????
0=�
ൗ
�
4
−�
??????�+�cos????????????
??????�−??????
ൗ
�
4
+??????sin????????????
??????�−??????
ൗ
�
4
�
??????�=�
ൗ
�
4
+�cos????????????
??????�−??????
ൗ
�
4
+??????sin????????????
??????�−??????
ൗ
�
4
01
�
??????�=�
ൗ
�
4
+�??????
??????�−??????
ൗ
�
4
�
�????????????
=�
�
ൗ
??????
4
+�
�
??????
??????�−??????
ൗ
�
4
�
�
�
????????????
=�
�
ൗ
??????
4
+�
�ℎ
??????�−ℎ
ൗ
�
4

41. •Aerodynamic center – The point of reference on
the airfoil about which aerodynamic moments
do not vary with AoA, assuming V
∞ = const.
•If L = 0, the moment is a pure couple equal to
the M
a.c.
•Simple airfoil theory places a.c.:
•At c/4 for low-speed airfoils
•At c/2 for supersonic airfoils
AERODYNAMIC CENTER

42. •The a.c. is that reference point on a body about which the aerodynamically
generated moment is independent of the AoA ().
•For most airfoils, it is close to, but not exactly at x
c/4.
AERODYNAMIC CENTER
��
�
????????????
�??????
=0
��
�????????????
�??????
=
��
�
ൗ
??????
4
�??????
+
��
�
�??????
ℎ
??????�−ℎ
ൗ
�
4
m
0 a
0
��
�????????????
�??????
=0=�
0+�
0ℎ
??????�−ℎ
ൗ
�
4
ℎ
??????�=−
�
0
�
0
+ℎ
ൗ
�
4

43. •Example Aerodynamic Moments.
M
c/4V
∞ D
c/4
L
c/4
x
c/4
c

INCOMPRESSIBLE FLOW
OVER AIRFOILS
L.E.
T.E.x
ac
L
ac
D
acM
ac
 [deg]c
lc
m,c/4
-4 0-0.095
-3 0.1-0.095
-2 0.2-0.095
-1 0.3-0.095
0 0.4-0.090
1 0.5-0.087
2 0.6-0.086
3 0.7-0.080
4 0.8-0.079
5 0.9-0.077
6 1.0-0.075
NACA 4415

44. •Example Aerodynamic Moments
INCOMPRESSIBLE FLOW
OVER AIRFOILS
 [deg]c
lc
m,c/4
-4 0-0.095
-3 0.1-0.095
-2 0.2-0.095
-1 0.3-0.095
0 0.4-0.090
1 0.5-0.087
2 0.6-0.086
3 0.7-0.080
4 0.8-0.079
5 0.9-0.077
6 1.0-0.075
��
�
�??????
=�
0=
1−0.2
6−−2
=0.1ൗ
1
��??????
��
�
ൗ
??????
4
�??????
=�
0=
−0.075−−0.095
6−−2
=0.0025ൗ
1
��??????
��
�????????????
�??????
=0=�
0+�
0ℎ
??????�−ℎ
ൗ
�
4
ℎ
??????�=−
�
0
�
0
+ℎ
ൗ
�
4
=−
0.0025
0.1
+0.25=0.225=22.5%
NACA 4415

45. INCOMPRESSIBLE FLOW
OVER AIRFOILS
•Example Aerodynamic Moments
If
For this example:
�=0→�
�??????=0
�
�????????????
=�
�
ൗ
??????
4
+�
�ℎ
??????�−ℎ
ൗ
�
4
 [deg]c
lc
m,c/4
-4 0-0.095
-3 0.1-0.095
-2 0.2-0.095
-1 0.3-0.095
0 0.4-0.090
1 0.5-0.087
2 0.6-0.086
3 0.7-0.080
4 0.8-0.079
5 0.9-0.077
6 1.0-0.075
0
�
�????????????
=�
�
ൗ
??????
4
=�
�??????=0
=�����.
�
�
????????????
=−0.095
NACA 4415

46. •Center of pressure – location on the
airfoil where the pitching moment is
zero.
•The resultant forces (L and D) acting
at c.p. produce no moment (use
centroid rule to find R location).
•Since the pressure distribution over
the airfoil changes with AoA, the
location of c.p. varies with AoA.
CENTER OF PRESSURE
�
�
????????????
=0

47. CENTER OF PRESSURE
•Example
If
For
NACA 4415
�
�
????????????
=0
�
�
????????????
=�
�
????????????
+�
�ℎ
??????�−ℎ
�??????
 [deg]c
lc
m,c/4
-4 0-0.095
-3 0.1-0.095
-2 0.2-0.095
-1 0.3-0.095
0 0.4-0.090
1 0.5-0.087
2 0.6-0.086
3 0.7-0.080
4 0.8-0.079
5 0.9-0.077
6 1.0-0.075
0
ℎ
�??????=−
�
�????????????
�
�
+ℎ
??????�
�
�=0.2→ℎ
�??????
ℎ
�??????=−
−0.095
0.2
+0.225
ℎ
�??????=0.7=70%

48. EXERCISE
AERODYNAMIC
MOMENTS
Consider the NACA 63-210 airfoil at
6 [deg] angle of attack. Calculate the
moment coefficient about the
leading edge.
�
�????????????
=?