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Chapter 5
Airfoils
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•Introduction
In this chapter the following will be studied:
1- Geometric characteristics of the airfoils.
2- Aerodynamic characteristics of the airfoils.
3- Flow similarity ( Dynamic similarity )
■ Airfoil Geometric Characteristics
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Airfoil geometric characteristics include:
1- Mean camber line : The locus of points halfway between
the upper and lower surfaces as measured perpendicular
to the mean camber line.
2- Leading & trailing edges: The most forward and rearward
points of the mean camber line.
3- Chord line: The straight line connecting the leading and
trailing edges.
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4- Chord C : The distance from the leading to trailing edge
measured along the chord line.
5- Camber : The maximum distance between the mean
camber line and the chord line.
6- Leading edge radius and its shape through the leading
edge.
7- The thickness distribution: The distance from the upper
surface to the lower surface, measured perpendicular
to chord line
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►Airfoil Families (Series)
# NACA (National Advisory Committee for Aeronautics) or
NASA (National Aeronautics and Space administration)
identified different airfoil shapes with a logical numbering
system.
# Abbott & Von Doenhoff “ Theory of Wing Sections”includes a summary of airfoil data ( geometric andaerodynamic data )
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■NACA Airfoil Series
1- NACA 4-digit series
2- NACA 5-digit series
3- NACA 1-series or 16-series
4- NACA 6- series
5- NACA 7- series
6- NACA 8- series
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►NACA Four-Digit Series
Example: NACA 2412
NACA 2 4 12
Camber inpercentage of chord
yc = 0.02 C
Position of camberin tenths of chordxc = 0.4 C
Maximum thickness (t )in percentage of chord(t/c)max = 0.12
xc
yc
C
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►NACA Five-Digit Series
Example: NACA 23012
NACA 2 30 12
When multiplied by 3/2yields the design liftcoefficient Cl in tenths.
Cl = 0.3
When divided by 2, givesthe position of thecamber in percent ofchord xc = 0.15 C
Maximum thickness(t ) in percentage ofchord (t/c)max = 0.12
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►NACA Six- Series
Example: NACA 64-212
NACA 6 4 - 2 12
Seriesdesignation 6
Location of minimumpressure in tenths ofchord (0.4 C)
Design liftcoefficient intenths (0.2)
Maximum thickness (t )in percentage of chord(t/c)max = 0.12
►Note that this is the series of laminar airfoils .
Comparison of conventional and laminar flow airfoils
is shown in the following Figure.
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Conventional Airfoil
Pressure distribution
On upper surface
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Laminar Airfoil
Pressure distribution
On upper surface
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•The Handbook “Theory of Wing Sections” gives theshape of airfoils in terms of upper and lower surfacesstation and ordinate as given in the following Tables.
•Airfoils can be drawn using these Tables.
•From airfoil drawing we can extract its geometric data:
- camber line
- maximum camber ratio and its position
- maximum thickness ratio and its position
-leading edge radius
-trailing edge angle
Assignment 1 : Meaning of numbering system for NACA 1-series,NACA 7-Series, and NACA 8- Series.
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•Tabe for NACA 2410, 2412, 2415
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■Center of Pressure and Aerodynamic Center
# Center or pressure : The point of intersection betweenthe chord line and the line of action of the resultantaerodynamic force R.
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# In addition to lift and drag, the surface pressure andshear stress distribution create a moment M which tendsto rotate the wing.
# Moment on Airfoil
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•Neglect shear stress
•F1 is the resultant pressure force on the upper surface.
•F2 is the resultant pressure force on the lower surface.
•Points 1 & 2 are the points of action of F1 & F2 .
•R is resultant force of F1 & F2 .
•F1 ≠ F2 because the pressure distribution on the uppersurface differs from the pressure distribution on the lowersurface.
•Thus, F1 & F2 will create an aerodynamic moment Mwhich will tend to rotate the airfoil.
•The value of M depends on the point about which wechoose to take moment.
•For subsonic airfoils it is common to take moments aboutthe quarter-chord point. It is denoted by Mc/4 .
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Mc/4 is function of angle of attack α, i.e. its value dependson α .
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■ Aerodynamic Center
≠ Aerodynamic center: The point on the chord line aboutwhich moments does not vary with α.
●The moment about the aerodynamic center (ac) isdesignated Mac .
● By definition, Mac = constant
● For low-speed and subsonic airfoils, ac is generally veryclose to the quarter-chord point
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■Lift, Drag, and Moment Coefficients
For an airplane in flight, L, D, and M depend on:
1- Angle of attack α
2- Free-stream velocity V∞
3- Free-stream density ρ∞ , that is, altitude
4- Viscosity coefficient µ∞
5- Compressibility of the airflow which is governed by
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Mach number M∞ = V∞/a∞. Since V∞ is listed above, wecan designate a∞ as our index for compressibility.
6- Size of the aerodynamic surface. For airplane we usethe plan form wing area S to indicate size.
7- Shape of the airfoil.
● Hence, for a given shape of airfoil, we can write:
L = f1( α, V∞, ρ∞, µ∞ , a∞, S )
D = f2( α, V∞, ρ∞, µ∞ , a∞, S )
M = f3( α, V∞, ρ∞, µ∞ , a∞, S )
●The variation of L with (α, V∞, ρ∞, µ∞ , a∞, S) taking one ata time with the others constant could be obtained byexperiment in a wind tunnel .
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L
α
(V∞, ρ∞, µ∞ , a∞, S)
= const
1
L
V∞
2
3
ρ∞
4
5
6
L
L
L
L
µ∞
a∞
S
Therefore, 6 experiments are required for each dependent variable.
(α, ρ∞, µ∞ , a∞, S)
(α, V∞, µ∞ , a∞, S)
= const
= const
(α, V∞, ρ∞, a∞, S)
= const
(α, V∞, ρ∞, µ∞, S)
= const
(α, V∞, ρ∞, µ∞, a∞)
= const
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● Then by cross plotting that data obtained, we could beable to get a precise functional relation for L, D, and M.
● This is the hard way which could be very time consumingand costly.
●Instead, we can use the theory of dimensional analysis.
●This theory can reduce time, effort, and cost by groupingα, V∞, ρ∞, µ∞ , a∞, S , and L or D or M into a fewernumber of non-dimensional parameters.
●The results of this theory are:
CL= f1 (α,, M∞, Re)
CD = f2 (α,, M∞, Re)
CM = f3 (α,, M∞, Re)
-where CL = L/ q∞S = Lift coefficient
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- CD = D/ q∞S = Drag coefficient
- CM= M/ q∞S C = Moment coefficient
and q∞ = ½ ρ∞ V2∞ , C = Airfoil chord
- Re = ρ∞ V∞ C/μ∞ = Reynolds number
- M∞ = V∞ / a∞ = Mach number
►Note :
1- For airfoil ( 2D flow ) S = C x 1
2- CL cl , L l
3- CD cd , D d
4- CM cm , M m
Dynamicsimilarityparameters
Perunitspan
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■ Airfoil Data
● A goal of theoretical aerodynamics is to predictvalues of cl, cd, and cm from the basic equationsand concepts of physical science.
● However, simplifying assumptions are usuallynecessary to make the mathematics tractable.
● Therefore, when theoretical results areobtained, they are generally not “exact”.
● As a result we have to rely on experimentalmeasurements.
● cl, cd, and cm were measured by NACA for largenumber of airfoils in low-speed wind tunnels.
● At low-speed the effect of M∞ is cancelled.
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●These measurements were carried out on straight,constant-chord wings completely spanned the tunnel testsection from one side to the other.
● In this fashion, the flow essentially “ saw” a wing with nowing tips, and the experimental airfoil data were obtainedfor “infinite wings”
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● Results of airfoil measurements include cl, cd, cm,c/4, andcm,ac.
● The results are given in the form of graphs as follows:
- The 1st page of graph gives data for cl and cm,c/4 versus
angle of attack for the NACA airfoil.
- The 2nd page of graph gives cd and cm,ac versus cl for
the same airfoil.
Note: Some results of these airfoil data are given inAppendix D ( “Introduction to Flight”, Anderson )
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Example: Airfoil data for NACA 2415
cl and cm,c/4versus α
NACA 2415
Mach number is notincluded
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cd and cm,acversus cl
NACA 2415
Mach number is notincluded
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Mach number is notincluded
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Mach number is notincluded
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►Variation of cl with α
● This variation is shown in the following sketch.
* cl varies linearly with α over a large range of α.
* At α = 0 cl ≠ 0 due to the positive camber.
* cl = 0 at αL=0 ( zero lift direction/zero lift angle of attack)
* For large values of α,the linearity breaks down.
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* As α is increased beyond a certain value; cl reaches toclmax and then drops as α is further increased.
* When cl is rapidly decreasing at high α , the airfoil isstalled.
Flow mechanismassociated withstalling
Separated flow
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►Comparison of Lift Curves for Cambered and Symmetric Airfoils
* For symmetric airfoil the lift curve goes through the origin.
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►The Phenomenon of Airfoil Stall
Separated flow
*It is of critical importance
in airplane design.
*It is caused by flow
separation on the upper
surface of the airfoil due to
high adverse pressure
gradient.
*When separation occurs,
the lift decreases
drastically, and the drag
increases suddenly.
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■Compressibility Correction For Lift & Moment Coefficient
For 0.3 < M∞ ≤ 0.7 , the corrections for cl and cm , using<Prandtl-Glauert rule , are given as:
- cl = cl,0 / √ [1- M∞2]
- cm = cm,0 / √ [1- M∞2]
Where cl,0 is the low-speed value of the lift coefficient,
cm,0 is the low-speed value of the moment coefficient.
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■ Flow Similarity (Dynamic Similarity)
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•Consider two different flow fields over two differentbodies, as shown in figure.
•By definition, different flows are dynamically similar if:
1- The bodies and any other solid boundaries are
geometrically similar for the flow.
2- The dynamic similarity parameters are the same for
flows ( i.e.Re and M∞ are the same for the flows).
# If different flows are dynamically similar, the followingresults are satisfied:
1- The streamline patterns are geometrically similar.
2- The distribution of v/v∞ , p/p ∞,T/T ∞ ,..etc throughout
the flow field are the same when plotted against
common non-dimensional coordinates.
3- The force and moment coefficients are the same (i.e.
cl, cd, and cm are the same.
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v/v∞
s1/d1 , s2/d2
* Thus we can say that flows over geometrically similar bodies
at the same Mach and Reynolds numbers are dynamically
similar.
• Hence, the lift, drag, and moment coefficients will be identical
for the bodies.
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►This is the key point in the validity of wind-tunnel testing:
“ If a scale model of a flight vehicle is tested in a windtunnel, the measured lift, drag, and moment coefficientswill be the same as for free flight as long as the Machand Reynolds numbers of the wind-tunnel test-sectionflow are the same as for the free-flight case”
# This means that:
[ M∞1 ]model = [ M∞2 ]prototype
[ Re1 ]model = [ Re2 ]prototype
and [ cl1 ]model = [ cl2 ]prototype
[ cd1]model = [ cd2 ]prototype
[ cm1 ]model = [ cm2 ]prototype