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Zheyan JinSchool of Aerospace Engineering and Applied MechanicsTongji
University
Shanghai, China, 200092
AerodynamicsAerodynamics
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Chapter 4 Incompressible Flow Over AirfoilsChapter 4 Incompressible Flow Over Airfoils
4.1 Introduction
The aerodynamic consideration of
wings could be split into two parts:
(1). The study of the section of
a wing- an airfoi l.
(2). The modification of such
airfoi l properties to account
for the complete, finite wing.
Ludwig Prandtl and Göttingen (1912-1918):
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4.2 Airfoil Nomenclature
Mean camber line: is the locus of the points midway between upper and
lower surfaces of an airfoil as measured perpendicular to the mean camber
line.
Leading and trailing edges: the most forward and rearward points of the
man camber line.
Chord line: the straight l ine connecting the leading and trailing edges.
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4.2 Airfoil Nomenclature
Thickness: is the height of profile measured normal to the mean camber line.
Leading-edge radius: is the radius of a circ le, tangent to the upper and
lower surface, with i ts center located on a tangent to the mean camber line
drawn through the leading edge of this l ine.
Camber: is the maximum distance between the mean camber line and the
chord measured normal to the chord.
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4.2 Airfoil Nomenclature
Four-digit series: (for example, NACA 2412)
1. The first digit is the maximum camber in hundredths of chord.
2. The second digit is the location of maximum camber along the chord from the
leading edge in tenths of chord.
3. The last two digits give the maximum thickness in hundredths of chord.
Five-digit series: (for example, NACA 23012)
1. The first digit when multiplied by 3/2 gives the design lift coefficient in tenths.
2. The next two digit when divided by 2 give the location of maximum camber along
the chord from the leading edge in hundredths of chord.
3. The final two digits give the maximum thickness in hundredths of chord.
NACA airfoils (National Advisory Committee for Aeronautics):
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4.2 Airfoil Nomenclature
6-series: (for example, NACA 65-218)
1. The first digit simply identifies the series.
2. The second gives the location of minimum pressure in tenths of chord from the
leading edge.
3. The third digit is the design lift coefficient in tenths.
4. The last two digits give the maximum thickness in hundredths of chord.
Many of the large aircraft companies today design their own special purpose
airfoil; for example the Boeing 727,737,747, 757, 767, and 777 all have specially
designed Boeing airfoils.
Such capability is made possible by modern airfoil design computer programs
utilizing either panel techniques or direct numerical finite-difference solutionsof the governing partial differential equations for the flow field.
NACA airfoils:
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4.3 Airfoil Characteristics
At low-to-moderate angles of attack,
cl varies linearly withα.
The slope of this straight line is
denoted by a0 and is called lift slope.
The value ofα when lift equals zero
is called the zero-lift angle of attack.
Lift coefficient:
d
dcl0a
lc
0L
max,lc
Stall due to flow
separation
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4.3 Airfoil Characteristics
Drag coefficient:
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4.3 Airfoil Characteristics
At low to moderate angles of attack Cl-α curve is linear. The flow moves
slowly over the airfoil and is attached over most of the surface. At highangles of attack, the flow trends to separate from the top surface.
●
Cl,max occurs prior to stall
●
Cl,max is dependent on Re=ρvc/µ●
Cm,c/4 is independent of Re except for largeα
●
Cd is dependent on Re
●
The linear portion of the Cl-α curve is independent of Re and can be
predicted using analytical methods.
Aerodynamic center. There is one point on the airfoil about which the
moment is independent of angle of attack.
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4.4 Vortex Filament
Consider 2-D/point vortices of same strength duplicated in every planeparallel to z-x plane along the y-axis from to .
The flow is 2-D and is irrotational everywhere except the y-axis.
Y-axis is the straight vortex filament and may be defined as a line.
x
z
y
●
Definition: A vortex filament is a straight or
curved line in a fluid which coincides with the
axis of rotation of successive fluid elements.
Г
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Chapter 4 Incompressible Flow Over AirfoilsChapter 4 Incompressible Flow Over Airfoils
4.4 Vortex Filament
●
Helmholt’s vortex theorems:
1. The strength of a vortex filament is constant along its length.
Proof: A vortex filament induces a velocity field that
is irrotational at every point excluding the filament.
Enclose a vortex filament with a sheath from which
a slit has been removed. The vorticity at every point
on the surface=0. Evaluate the circulation for the
sheath.
Circulation Ad V sd V S
C
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Chapter 4 Incompressible Flow Over AirfoilsChapter 4 Incompressible Flow Over Airfoils
4.4 Vortex Filament
●
Helmholt’s vortex theorems:
1. The strength of a vortex filament is constant along its length.
CC
0sdVor0sdV-0V
,
Sheath is irrotational. Thus
ad cb
dc ba
0sdVsdVsdVsdV
or
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Chapter 4 Incompressible Flow Over AirfoilsChapter 4 Incompressible Flow Over Airfoils
4.4 Vortex Filament
●
Helmholt’s vortex theorems:
1. The strength of a vortex filament is constant along its length.
c
d
d
c
b
a
d
c
b
a
sdVsdVsdV
0sdVsdV
However,
ac
d b
0sdVsdV
as it constitutes the integral across the slit.
Thus,
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Chapter 4 Incompressible Flow Over AirfoilsChapter 4 Incompressible Flow Over Airfoils
4.4 Vortex Filament
●
Helmholt’s vortex theorems:
1. The strength of a vortex filament is constant along its length.
2. A vortex filament cannot end in a fluid; it must extend to the
boundaries of the fluid or form a closed path.
3. In the absence of rotational external flow, a fluid that is irrotational
remains irrotational.
4. In the absence of rotational external force, if the circulationaround a path enclosing a definite group of particles is initially zero,
it will remain zero.
5. In the absence of rotational external force, the circulation around
a path that encloses a tagged group of elements is invariant.
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4.5 Vortex Sheet
An infinite number of straight vortex filamentplaced side by side from a vortex sheet. Each
vortex filament has an infinitesimal strength
γ(s):
γ(s) is the strength of vortex
sheet per unit length along s.
r 2v
for 2-D (point vortex)
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4.5 Vortex Sheet
A small portion of the vortex sheet of strength
γds induces an infinitesimally small velocity dV
at a field point P(r, θ).
Thus
r
ds
2mentvortexfilav
so
r
dsdv
2
P
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Chapter 4 Incompressible Flow Over AirfoilsChapter 4 Incompressible Flow Over Airfoils
4.5 Vortex Sheet
CirculationГ around a point vortexis equal to the strength of the vortex.
Similarly, the circulation around the
vortex sheet is the sum of the
strengths of the elemental vortices.
Therefore, the circulationГ for a
finite length from point ‘a’ to point ‘b’
on the vortex sheet is given by:
Across a vortex sheet, there is a discontinuous change in the tangential
component of velocity and the normal component of velocity is preserved.
b
)(a
dss
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Chapter 4 Incompressible Flow Over AirfoilsChapter 4 Incompressible Flow Over Airfoils
4.5 Vortex Sheet
AsΔn→0, we get
n()(
][
2121
2112
Box
)wwsuus
sunwsunwld v
ld v
)()(
21
21
uusuus
Γ =(u1-u2) states that the local jump in tangential velocity across
the vortex sheet is equal to the local sheet strength.
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Chapter 4 Incompressible Flow Over AirfoilsChapter 4 Incompressible Flow Over Airfoils
4.6 Kutta Conditions:
1. For a given airfoil at a given angle of attack, the value of Г
around the airfoil is such that the flow leaves the trailing edge
smoothly.
2. If the trailing-edge angle is finite, then the trailing edge is a
stagnation point.
3. If the trailing edge is cusped, then the velocities leaving the top
and bottom surfaces at the trailing edge are finite and equal inmagnitude and direction.
0)( lu V V TE
2V
021 V V
1V
a
At point a:
Finite angleCusp
2V
021 V V
1V a
At point a:
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4.7 Bound Vortex and Starting Vortex:
The question might arise: Does a real airfoil flying in a real fluid
give rise to a circulation about itself?
The answer is yes.
When a wing section with a sharp T.E is put into motion, the fluid
has a tendency to go around the sharp T.E from the lower to the
upper surface. As the airfoil moves along vortices are shed from
the T.E which from a vortex sheet.
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4.7 Bound Vortex and Starting Vortex:
●
Helmholt’s theorem:
IfГ=0 originally in a flow it remains zero.
●
Kelvin’s theorem:
Circulation around a closed curve
formed by a set of continuous fluid
elements remains constant as the fluid
elements move through the flow,
DГ/Dt=0.
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4.7 Bound Vortex and Starting Vortex:
Both the theorems are satisfied by the
starting vortex and bound vortex
system.
In the beginning,Г1 =0 when the flow
is started within the contour C1.
When the flow over the airfoil is
developed,Г2 within C2 is still zero
includes the starting vortexГ3 and the
bound vortexГ4 which are equal and
opposite.
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Chapter 4 Incompressible Flow Over AirfoilsChapter 4 Incompressible Flow Over Airfoils
4.8 Fundamental Equation of Thin Airfoil Theory
Thin airfoil theory is based on the assumption that under certain conditionsan airfoil section may be replaced by its mean camber line (mcl).
Experimental observation:
If airfoil sections of the same mcl but different thickness functions are tested
experimentally at the same angle of attack, it is found that the lift Lˈ
and thepoint application of the lift for the different airfoil sections are practically the
same provided that
(1) maximum airfoil thickness (t/c) is small;
(2) Camber distribution (z/c)max = m is small;
(3) Angle of attack (α) is small.
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This observation permitted the formulation of thin airfoil theory because it
allowed the airfoil to be replaced by the mcl.
V
α
mcl
V
α
mcl
Thin airfoi l theory
The problem now is to find, theoretically the flow of an ideal fluid around
this infinitely thin sheet (mcl) flying through the air at the velocity V ͚ at an
angle of attackα.
4.8 Fundamental Equation of Thin Airfoil Theory
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Any solution must satisfy:
(1) Equation of continuity
(2) Irrotational condition
(3) Outer b.c.- Flow at infinity must be undisturbed(4) Inner b.c. – mcl must be a streamline
(5) In addition, since the thin airfoil is being supported in level flight there
must be a lift Lˈ
acting on the airfoil.
(6) Since Lˈ=ρV ͚Г (Kutta-Joukowski Theorem), any theoretical
analysis must introduce a circulationГ around the airfoil section of
sufficient magnitude to satisfy the Kutta condition that the flow leave
the TE smoothly.
4.8 Fundamental Equation of Thin Airfoil Theory
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Summary:
Therefore in thin airfoil theory the mcl is replaced by a vortex sheet
of varying strengthγ(s) such that the above conditions aresatisfied and our aim is to determine this ‘γ’ distribution.
Thin airfoil theory stated as a problem says for a vortex sheet
placed on the mcl in a uniform flow of V ͚ determineγ(s) such that
the mcl is a streamline subject to the conditionγ(TE)=0.
4.8 Fundamental Equation of Thin Airfoil Theory
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Principle: Mean camber line is a streamline of the flow.
Velocity induced by a 2-D vortex is whereГ is the
strength of the 2-D vortex. Similarly the velocity induced by the vortex
sheet of infinitesimal length ds is given by
ê2
)(
Vd P r
dss
ê2
êvVr
4.8 Fundamental Equation of Thin Airfoil Theory
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To force the mean camber line to be a streamline, the sum of all
velocity components normal to the mcl must be equal to zero.Consider the flow induced by an elemental vortex sheet ds at point
P on the vortex sheet.
ê2
)(Vd Pr dss
r
dssd w PP
2
cos)(cosvd '
Thus, the velocity normal to the mcl is:
whereβ is the angle made by dvp to the normal at P, and r is the
distance from the center of ds to the point P.
4.8 Fundamental Equation of Thin Airfoil Theory
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The induced velocity due to the vortex sheet representing the entire
mcl is given by.
)sin(V n, V
TE
LE
' cos)(
2
1
r
dsswP
Now determine the component
of the free stream velocity
normal to the mcl.
whereα is the angle of attack and ε is the angle made by the tangentat point P to the x-axis.
4.8 Fundamental Equation of Thin Airfoil Theory
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The slope of the tangent line at point P is given by:
))tan(sin
)tan
1
,
1
dx
dzV V
dx
dz
n
(
(
0))(tan(sincos)(
2
1 1TE
LE
dxdz
V dsr
sr
tan)tan( dx
dz
or
In order that the mcl is a streamline. 0)( ,' nP V sw or
4.8 Fundamental Equation of Thin Airfoil Theory
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After changing these variables and making the small angle approximationfor sin and tan, and upon rearrangement we get:
)()(
2
1 c
00 dx
dzV dx
x x
x
within thin airfoil theory
approximation s→x,
ds→dx,cosβ =1 and
r →(x0-x), where x
varies from 0 to c, and
x0 refers to the point P.
4.8 Fundamental Equation of Thin Airfoil Theory
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In order to facilitate analytic solution, we do a variable transformation
such that:
)cos1(2
)cos1(2
00
c
x
c x
The following analysis is an exact solution to the flat plate or an
approximate solution to the symmetric airfoil. The mean camber linebecomes the chord and hence:
4.9 Flat Plate at an Angle of Attack
V dx x x x
dx
dz
c
00
)(
2
1
0
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Here we simply state a rigorous solution forγ(θ) as:
V d 00 )cos1()cos1(
2
c
sin2
c)(
2
1
sincos12)(
V
d V d
00
00 )cos(cos
)cos1()cos(cos
sin)(21
θ =0 at LE andθ =π at TE andθ increases in CW,
dx=(sinθdθ)c/2
4.9 Flat Plate at an Angle of Attack
We can verify this solution by substitution as follows:
V d
00 )cos(cos
sin)(
2
1
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Thus, it satisfies the equation:
0
0
00 sin
sin
)cos(cos
cos
nd
n
We now use the following result to evaluate the above integral.
4.9 Flat Plate at an Angle of Attack
V
V d d
V d
V )0(
)cos(cos
cos
)cos(cos
1
)cos(cos
)cos1(
00
00
00
V d
00 )cos(cos
sin)(
2
1
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Whenθ=π,0112)( V
In addition, the solution forγ also satisfies the Kutta condition.
4.9 Flat Plate at an Angle of Attack
0cos
sin
2)(
V
By using L’Hospital’s rule, we get
Thus it satisfies the Kutta condition.
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Where s is along the mcl.
By using thin airfoil approximation:
TE
LE dssV V )(L
'
TE
LE
c
dx xV dx xV L0
' )()(
d cV dx xV c
00' sin)(
2
1)(L
Using the transformation x= (1-cosθ) c/2
4.10 2-D lift coefficient for a thin/symmetrical airfoil
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Substituting the solution:
sin
)cos1(2)(
V
2,2'
d
dC and
S q
LorC ll
4.10 2-D lift coefficient for a thin/symmetrical airfoil
0
' )cos1(22
1d V cV L
0
2'
)cos1( d cV L
2'
V c L
)1(22
2'
c
V
L
S q L 2'
2d
dC lshows that lift curve is linearly proportional to the angle of attack.
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Calculation of Moment Coefficient:
c
LE dL x M 0
'' )(
c
LE d V x M 0
')(
c
LE xdx xV M 0
' )(
4.10 2-D lift coefficient for a thin/symmetrical airfoil
c
LE dx xV x M 0
'
))((
0
' sin2
)cos1(2sin
)cos1(2d
ccV V M LE
)2
()2
(2
)cos1(2
22
2
0
22
2'
cqcV d
cV M LE
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Calculation of Moment Coefficient:
2c
2
''
,
cq
M
Scq
M LE LE LE m
2c l
4c ,
l LE m
C
4.10 2-D lift coefficient for a thin/symmetrical airfoil
'
4/
'
4/
'
cc LE L M M
4/ccc 4/,, lcm LE m
C i ht b D Zh Ji
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Chapter 4 Incompressible Flow Over AirfoilsChapter 4 Incompressible Flow Over Airfoils
Calculation of Moment Coefficient:
4/,c cm
4.10 2-D lift coefficient for a thin/symmetrical airfoil
4/, cc l LE m
is equal to zero for all values ofα.
c/4 is the aerodynamic center.
Aerodynamic center is that point on an airfoil where moments
are independent of angle of attack.
0c 4/m, c
C i ht b D Zh Ji
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(A)
4.11 Thin Airfoil Theory for Cambered Airfoil
d c
dx
c x
sin2
)cos1(2
where dz/dx is the slope of mcl at x0.
For symmetrical airfoil, mcl is a straight line and hence dz/dx=0 everywhere.
On the other hand, for a cambered airfoil dz/dx varies from point to point.
)(
)(
2
1 c
00 dx
dz
V dx x x
x
As before, we do a variable transformation given by:
C i ht b D Zh Ji
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(B)
4.11 Thin Airfoil Theory for Cambered Airfoil
Equation (A) becomes
Such a solution forγ(θ) will make the camber line astreamline of the flow.
However, as before a rigorous solution of Equation (B) for γ(θ)
is beyond the scope of this course:
)(coscos
sin)(
2
1 c
00 dx
dzV d
1
0 sin)sin
cos1(2)(
n
n n A AV
C i ht b D Zh Ji
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4.11 Thin Airfoil Theory for Cambered Airfoil
01 0 00 0
0
)cos(cos
sinsin1
)cos(cos
)cos1(1
xn
n
dx
dzd
n Ad
A
The first integral can be evaluated from the standard from given in
equation
Substitute this solution in equation (B)
0
0
00 sin
sin
)cos(cos
cos
nd
n
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4.11 Thin Airfoil Theory for Cambered Airfoil
00 0
cos)cos(cos
sinsin
nd n
The first term becomes:
The remaining integrals can be obtained from another standard form,
which is given below:
0
00
00
0
0
00
0
)coscos
cos(1)coscos
(1
)coscos
cos1(
1
A
d Ad A
d A
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4.11 Thin Airfoil Theory for Cambered Airfoil
1
00 cos)(n
n n A Adx
dz
The second term becomes:
Therefore Equation (B) becomes:
1
0
10
0
coscoscos
sinsin1
n
n
n
n n Ad n A
Upon rearrangement the slope at a point P on the mcl is given by:
01
00 cos xn
ndx
dzn A A
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4.11 Thin Airfoil Theory for Cambered Airfoil
nn B A
d dz
dz B A
000 )(
1)(
Where,
10 cos)(n
n n B B f
,,2,1
cos)(2
)(
1
0
00
n
d n f B
d f B
n
From Fourier series:
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4.11 Thin Airfoil Theory for Cambered Airfoil
cc
d c
dx x00
sin)(2
)(
]sinsin)sin([
sinsin22
)cos1(22
sin)(2
sinsin
)cos1(2)(
100
0
01
0 0
0
1
0
n
n
n
n
c
n
n
d n A AcV
d n AV c
d AV c
d c
n A AV
From thin airfoil theory:
Evaluation of Circulation :
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Chapter 4 Incompressible Flow Over AirfoilsChapter 4 Incompressible Flow Over Airfoils
4.11 Thin Airfoil Theory for Cambered Airfoil
]2
[2c
]2
[
]2
[
)1(0
)1(2
sinsin
10
''
10
2'
10
10
A Acq
L
sq
L
A AcV V L
A AcV
n
n
d n A
l
n
n
Using:
Cl is normalized by theα as seen by the chord connecting the LE and
TE of the mcl. c is the chord connecting the LE and TE of the mcl.
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Chapter 4 Incompressible Flow Over AirfoilsChapter 4 Incompressible Flow Over Airfoils
4.11 Thin Airfoil Theory for Cambered Airfoil
d
dx
dz
d dx
dzd dx
dz
A A
)1(cos)(1
2c
cos)(2
))(1
(2c
2c
0
l
00l
10l
d
dx
dz
d
d
L
L Ll
l
00
00
)1(cos1
)(2)(c
c
Note that as in the case of symmetric airfoil, the theoretical lift slope
for a cambered airfoil is 2π. It is a general result from thin airfoil
theory that dcl/dα=2π for any shape airfoil.
Also,
zero lift angle of attack
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Chapter 4 Incompressible Flow Over AirfoilsChapter 4 Incompressible Flow Over Airfoils
4.11 Thin Airfoil Theory for Cambered Airfoil
c LE
c
dx x xcV scq
M
dx x xV
02
'
LEm,
0
'
LE
)(2
c
)(M
01
0
20,
1
0
sinsin)cos1()cos1(c
]sinsin
cos)1([2)(
sin2
)cos1(2
n
n LE m
n
n
d n Ad A
n A AV
d cdx
c x
As before we do a variable transformation from x toθ.
Thus,
Determination of moment coefficient:
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Chapter 4 Incompressible Flow Over AirfoilsChapter 4 Incompressible Flow Over Airfoils
4.11 Thin Airfoil Theory for Cambered Airfoil
0
2
0
2
2sin
2
cos
d
d
),,3(0
)2(4
)1(0
sincossin
),,2(0
)1(2
sinsin
0
0
n
n
n
d n
n
n
d n
Using the following definite integrals:
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Chapter 4 Incompressible Flow Over AirfoilsChapter 4 Incompressible Flow Over Airfoils
4.11 Thin Airfoil Theory for Cambered Airfoil
422c
422
c
sincossinsinsincosc
210,
2100,
10 00
2
00
0,
A A A
A A A A
d n And n And Ad A
LE m
LE m
n
LE m
)(44
c
2
22
2c
2c
21210
,
10
A A A A A
A A
l LE m
l
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Chapter 4 Incompressible Flow Over AirfoilsChapter 4 Incompressible Flow Over Airfoils
4.11 Thin Airfoil Theory for Cambered Airfoil
214
,
21
4,
,
''
4
'
4c
)(44
c
4
cc
4
A A
A Ac
c L M M
cm
llcm
LE m
c LE
)(
c4
1)(
c44
cc
2121
,
A Ac
c A Acc
x
c x
ll
cp
lcp LE m
The location of center of pressure:
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Chapter 4 Incompressible Flow Over AirfoilsChapter 4 Incompressible Flow Over Airfoils
4.11 Thin Airfoil Theory for Cambered Airfoil
)(c4
121 A A
c
c xl
cp
As the angle of attack changes, the center of pressure also changes.
Indeed, as the lift approaches zero, xcp moves toward infinity; that is, it
leaves the airfoil.
For this reason, the center of pressure is not always a convenient
point at which to draw the force system on an airfoil.
Rather, the force-and-moment system on an airfoil is more
conveniently considered at the aerodynamic center .
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py g y y
Chapter 4 Incompressible Flow Over AirfoilsChapter 4 Incompressible Flow Over Airfoils
4.11 Thin Airfoil Theory for Cambered Airfoil
TE
LE
TE
LE ul
ul
dssV L
ds p p L
ds p pdL
)(
cos)(
)1(cos)(
'
'
'
TE
LE
TE
LE)()( dssV ds p p ul
Equating (A) and (B) and setting cosɳ
≈ 1, we get
Relationship between pressure on mcl and :
(A)
(B)
or
)()( sV p pul
(1)
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py g y y
Chapter 4 Incompressible Flow Over AirfoilsChapter 4 Incompressible Flow Over Airfoils
4.11 Thin Airfoil Theory for Cambered Airfoil
))((2
)(2
1)(
2
1 2222
luluul
uull
uuuu p p
wu pwu p
2
lu uuV
Using Bernoulli’s equation:
(2)
(3)
From vortex sheet theory:
)(u sulu
From (1), (2) and (3)
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py g y y
Chapter 4 Incompressible Flow Over AirfoilsChapter 4 Incompressible Flow Over Airfoils
4.11 Thin Airfoil Theory for Cambered Airfoil
V s
V V sc
q
uuuu
c
q
p pc
q p p
q p pc
u pl p
luul
u pl p
ulu pl p
ulu pl p
)(22)(c
))((2
1
c
c
c
2,,
,,
,,
,,
i.e., within thin airfoil approximation, the average of top and bottom
surface velocities at any point on the mcl is equal to the freestreamvelocity.
Pressure coefficient difference between lower surface and upper surface:
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py g y y
Chapter 4 Incompressible Flow Over AirfoilsChapter 4 Incompressible Flow Over Airfoils
4.12 Summary
b
adss y x )(
2
1),(
b
adss)(
A vortex sheet can be used to synthesize the inviscid, incompressible
flow over an airfoil. If the distance along the sheet is given by s and
the strength of the sheet per unit length is γ(s), then the velocity
potential induced at point (x,y) by a vortex sheet that extends from
point a to point b is
The circulation associated with this vortex sheet is
21 uu
Across the vortex sheet, there is a tangential velocity discontinuity, where
Vortex Sheet:
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Chapter 4 Incompressible Flow Over AirfoilsChapter 4 Incompressible Flow Over Airfoils
4.12 Summary
0)TE(
The Kutta condition is an observation that for a airfoil of given shape
at a given angle of attack, nature adopts that particular value of
circulation around the airfoil which results in the flow leaving smoothly
at the trailing edge.
If the trailing-edge angle is finite, then the trailing edge is a stagnation
point.
If the trailing edge is cusped, then the velocities leaving the top and
bottom surfaces at the trailing edge are finite and equal in magnitudeand direction.
Kutta Condition:
In either case:
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Chapter 4 Incompressible Flow Over AirfoilsChapter 4 Incompressible Flow Over Airfoils
4.12 Summary
)()(
2
1 c
0 dx
dzV
x
d
Thin airfoil theory is predicated on the replacement of the airfoil by the
mean camber line.
A vortex sheet is placed along the chord line, and its strength adjusted
such that, in conjunction with the uniform freestream, the camber linebecomes a streamline of the flow while at the same time satisfying the
Kutta condition.
The strength of such a vortex sheet is obtained from the fundamental
equation of thin airfoil theory:
Thin airfoil theory:
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Chapter 4 Incompressible Flow Over AirfoilsChapter 4 Incompressible Flow Over Airfoils
4.12 Summary
Symmetrical airfoil
1. cl=2πα.
2. Lift slope = dcl
/dα
=2π
.3. The center of pressure and the aerodynamic center are
both at the quarter-chord point.
4. cm,c/4=cm,ac=0.
Results of thin airfoil theory:
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Chapter 4 Incompressible Flow Over AirfoilsChapter 4 Incompressible Flow Over Airfoils
4.12 Summary
Cambered airfoil
1.
2. Lift slope = dcl /dα=2π.
3. The aerodynamic center is at the quarter-chord pint.
4. The center of pressure varies with the lift coefficient.
Results of thin airfoil theory:
0
00 )1(cos1
2 d dx
dzcl
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