Post on 14-Apr-2018
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AERODYNAMIC COEFFICIENTS The aerodynamic characteristics of a body are more fundamentally
described by the force and moment coefficients than by the actual forces
and moments themselves. aerodynamic force on a body depends on:
- velocity of the body through the air the density of the ambient air
- size of the body
- orientation of the body relative to the free-stream direction, (angle ofattack) (Clearly, if we change the velocity, the aerodynamic force shouldchange. Also, the force on a body moving at 100 feet per second through airis going to be smaller than the force on the same body moving at the samevelocity through water, which is nearly a thousand times denser than air.
Also, the aerodynamic force on a sphere of 1-inch diameter is going to besmaller than that for a sphere of 1-ft diameter, everything else being equal.Finally, the force on a wing will clearly depend on how much the wing is
inclined to the flow.- Moreover, since friction accounts for part of the aerodynamic force, theforce should depend on the ambient coefficient of viscosity.
- Also important is the compressibility of the medium through which thebody moves. A measure of the compressibility of a fluid is the speed of
sound in the fluid the higher the compressibility, the lower the speed ofsound.
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Hence we have
If we want to study how L,D,M depend on these variables we have vary one and
keep the others constant. With 6 unknowns it could be very time-consuming, and
moreover, the large amount of wind tunnel time could be quite costly. But the
amount of unknowns can be reduced using the non dimensional groups:
Reynolds number
Mach number
Dynamic pressure
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Imagine that we have a given body at a given angle of attack in a given
flow, where p, V, density, and a, are certain values. Let us call this the"green" flow. Consider another body of the same geometric shape (but not
the same size) in another flow where p, V, density and a, are all different;
let us call this flow the "red" flow. Dimensional analysis, tells us that even
though the green flows and the red flow are two different flows, if the
Reynolds number and the Mach number are the same for these twodifferent flows, then the lift coefficient will be the same for the two
geometrically similar bodies at the same angle of attack. The two flows, the
green flow and the red flow, are called dynamically similar.
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Variation of Cl with the angle of
attack and Reynolds
0=L
The slope of this linear portion is
called the lift slope and isdesignated by a0. For thin airfoils,
a theoretical value for the lift slope
is 2pi per radiant, or 0.11 per
degree.
there is a finite value of Cl at zero angle of
attack, and that the airfoil must be pitched
down to some negative angle of attack for
the lift to be zero. This angle of attack isdenoted by
If positively cambered airfoils have
negative zero-lift angles of attack. In
contrast, symmetric airfoil has
a negatively cambered airfoil has a positive
0=L
00==L
0=L
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At the other extreme, at high angles of attack, the lift coefficient becomes non-
linear, reaches a maximum value denoted by
then drops as a further increased. maxl
C
This is because a
separation occurs over the
top surface of the airfoil
and the lift decreases
(sometimes precipitously).
In this condition, the airfoil
is said to be stalled. In
contrast, over the linear
portion of the lift curve, the
flow is attached over most
of the airfoil surface.
the linear portion of the lift curve is essentially insensitive to variations in Re.By increasing Reynolds number Clmax increases
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Variation of Cm with the angle of
attack and Reynolds
over most of the practical range of the angle of attack the slope of the moment
coefficient curve is essentially constant.
This slope is positive for some airfoils (as shown here), but can be negative for
other airfoils. The variation becomes nonlinear at high angle of attack, when the
flow separates from the top surface of the airfoil, and at low, highly negative
angles of attack, when the flow separates from the bottom surface of the airfoil.
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Variation of Cd
with the angle of
attack and ReynoldsFor a cambered airfoil, the minimum
value Cd does not necessarily occur
at zero angle of attack, but rather atsome finite but small angle of attack.
For this angle-of-attack range, the
drag is due to friction drag and
pressure drag. In contrast, the rapid
increase in cd which occurs at highervalues of alpha, is due to the
increasing region of separated flow
over the airfoil, which creates a large
pressure drag.
The friction decreases by increasing the Reynolds number. Moreover, theReynolds number influences the extent and characteristics of the separated flow
region, and hence it is no surprise that Cd at the larger values of alpha is also
sensitive to the Reynolds number.
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NACA AIRFOIL
NOMENCLATURE
The major design feature of an airfoil is the mean camber line, which is the locus of
points halfway between the upper and lower surfaces, as measured perpendicular to the
mean camber line itself. The most forward and rearward points of the mean camber line
are the leading and trailing edges, respectively. The straight line connecting the leading
and trailing edges is the chord line of the airfoil, and the precise distance from the
leading to the trailing edge measured along the chord line is simply designated the chord
of the airfoil, denoted by c. The camber is the maximum distance between the mean
camber line and the chord line, measured perpendicular to the chord line. The camber,
the shape of the mean camber line, and, to a lesser extent, the thickness distribution of
the airfoil essentially control the lift and moment characteristics of the airfoil.
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NACA airfoils are indicated by a series of 4 digits. The numbers in the designationmean the following:
The first digit gives the maximum camber in percentage of chord.
The second digit is the location of the maximum camber in tenths of chord,
measured from the leading edge.
The last two digits give the maximum thickness in percentage of chord.For example, the NACA 2412 airfoil has a maximum camber of 2% of the chord (or
0.02c), located at 0.4c from the leading edge. The maximum thickness is 12% of the
chord (or 0.12c)
First family of airfoils
The numbers mean the following:
The first digit, when multiplied by 3/2, gives the design lift coefficient in tenths.
The second and third digits together are a number which, when multiplied by 1/2,
gives the location of maximum camber relative to the leading edge in percentage of
chord.The last two digits give the maximum thickness in percentage of chord.
For example, the NACA 23012 airfoil has a design lift coefficient of 0.3, the location
of maximum camber at 15% of the chord (or 0.15c) from the leading edge, and a
maximum thickness of 12% of the chord (or 0.12c).
Second family of airfoils
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THE AERODYNAMIC CENTER
The aerodynamic center is the point on a
body about which the moments areindependent of the angle of attack.
Differentiating with respect to angle of attack a gives
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If the aerodynamic center is the point about which moments areindependent of the angle of attack.
0. =d
dcca
m
for a body with linear lift and moment curves, where m0 and
a0 are the values, the aerodynamic center does exist as a
fixed point on the airfoil.
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Variation of Cl with Ma
Hence, CI increases as Ma, increases. The Prandtl-Glauert rule, the first and simplest(and also the least accurate) of the several formulas for subsonic "compressibility
corrections," predicts that Cl will rise inversely proportional to (1-Ma2)0.5.
In the supersonic region, the dashed curve shows the theoretical supersonic variation for
a thin airfoil, where CI = 4/(1-Ma2)0.5-. The oscillatory variation of Cl near Mach=1 is
typical of the transonic regime, and is due to the shock wave-boundary layer interactionthat is prominent for transonic Mach numbers.
At subsonic speeds, the
"compressibility effects"
associated with increasing Ma,
result in a progressive increasein CI. The reason for this is that
the lift is mainly due to the
pressure distribution on the
surface. As Ma, increases, the
differences in pressure fromone point to another on the
surface become more
pronounced.
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Dependence of Cd with MaCd stays relatively constant with Ma,up to, and slightly beyond the critical
Mach number (that free-stream Mach
number at which sonic flow is first
encountered at some location on the
airfoil). The drag in the subsonic
region is mainly due to friction, and
the "compressibility effect" on friction
in the subsonic regime is small. The
flow over the airfoil in this regime is
smooth and attached, with no shock
waves present.
As Ma increases above Ma critical, a large pocket of locally supersonic flow forms
above, and sometimes also below, the airfoil. These pockets of supersonic flow are
terminated at the downstream end by shock waves. The presence of these Shocks will
affect the pressure distribution in such a fashion as to cause an increase in pressure
drag (this drag increase is related to the loss of total pressure across the shock waves).
However, the dominant effect is that the shock wave interacts with the boundary layer
on the surface, causing the boundary layer to separate. Finally, in the supersonic
regime, Cd gradually decreases,
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Incompressible Flow
about Wings of Finite Span For a wing of finite span, the high-pressure air beneath
the wing spills out around the wing tips toward the low-pressure regions above the wing. As a consequence ofthe tendency of the pressures acting on the top surfacenear the tip of the wing to equalize with those on thebottom surface, the lift force per unit span decreasestoward the tips.
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Variation of lift along the span
The resultant lift force acting on a section,
obtained by integrating the pressuredistribution over the chord length, has a
spanwise variation:
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As a result of the spanwise pressure
variation, the air on the upper surfaceflows inboard toward the root. On the
lower surface, air will tend to flow outward
toward the tips. The resultant flow around
a wing of finite span is three dimensional,
having both chordwise and spanwisevelocity components.
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Trailing vortices the difference in
spanwise velocitycomponents will cause
the air to roll up into a
number of streamwisevortices, distributed
along the span. These
small vortices roll upinto two large vortices
just inboard of the wing
tips
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Visualization of tip vortices Very high velocities and low pressures exist at the
core of the wing-tip vortices. In many instances,water vapor condenses as the air is drawn into thelow-pressure flow field of the tip vortices.Condensation clearly defines the tip vortices
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LIFTING-LINE THEORY FOR
UNSWEPT WINGS
We assume that the lift acting on an element of
the wing is related to the local circulationthrough the Kutta-Joukowski theorem
we represent the spanwise lift distribution by a system of vortex filaments the
axis of which is normal to the plane of symmetry and which passes through
the aerodynamic center of the lifting surface The strength of the bound-vortex
system at any spanwise location is proportional to the local lift acting at thatlocation
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Trailing vortices the vortex theorems of Helmholtz state that a vortex
filament cannot end in a fluid. Therefore, we model thelifting character of the wing by a large number of vortexfilaments (infinitesimal strength filaments) that lie alongthe quarter chord of the wing.
This is the bound-vortex system, which represents thespanwise loading distribution. When the lift changes atsome spanwise location, the total strength of the bound-vortex system changes proportionally. But vortexfilaments cannot end in the fluid. Thus, the change isrepresented in our model by having some of thefilaments from our bundle of filaments turn 90 degreeand continue in the streamwise direction.
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Trailing vortices
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Lanchester's own drawing
of the wing-tip vortex on a finitewing.
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Downwash velocity The strength of the trailing vortex is given
by
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Downwash velocity (2) The vortex at y induces a velocity at a general point y1
on the aerodynamic centerline which is one-half thevelocity that would be induced by an infinitely long vortexfilament of the same strength:
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Downwash velocity (3) the resultant induced velocity at any point
y1 due to the cumulative effect of all thetrailing vortices is
The resultant induced velocity at y1 is in a downward direction (i.e., negative)
and is called the downwash.
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High-Aspect-Ratio Straight Wing
d
dCa
L=
d
dca
l=0
The classic theory for such wings was worked out by Prandtl during World War I
and is called Prandtl's lifting line theory.
airfoil
wing
lift slope per radian and e1 is a factor that
depends on the geometric shape of the wing,
including the aspect ratio and taper ratio.
S
bAR
2
=
Prandtl's lifting line theory does not apply to
low-aspect-ratio wings. It holds for aspect
ratios of about 4 or larger.
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the lift slope for a finite wing decreases as the aspect ratio decreases.
The angle of attack for zero lift, denotedis the same for all the seven wings; at zero lift the induced effects
theoretically disappear. At any given angle of attack larger than
the value of CL becomes smaller as the aspect ratio is decreased.
0=CL
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Prandtl's lifting line theory, also holds for subsonic compressible flow,
where
Substituting we have
It gives a quick, but approximate correction to the lift slope; because it is
derived from linear subsonic flow theory it is not recommended for use for Ma
greater than 0.7.
For supersonic flow over a high-aspect-ratio straight wing, the lift slope
can be approximated from supersonic linear theory
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Low-Aspect-Ratio Straight Wings When applied to straight wings at AR < 4, the equations for high AR
do not apply because are derived from a theoretical model whichrepresents the finite wing with a single lifting line across the span of
the wing. However, when the aspect ratio is small, the same intuitionleads to some misgivings-how can a short, stubby wing be properlymodeled by a single lifting line? The fact is-it cannot.
Instead of a single spanwise lifting line, the low-aspect-ratio wingmust be modeled by a large number of spanwise vortices, each
located at a different chordwise station
Modern panel methods can quickly and
accurately calculate the inviscid flow
properties of low-aspect-ratio straight
wings,
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An approximate relation for the lift slope for
low-aspect-ratio straight wings wasobtained by H. B. Helmbold in Gemany in
1942
For subsonic compressible flow, is modified
as follows
In the case of supersonic flow over a low-
aspect-ratio straight wing,
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At subsonic speeds, a low-aspect-ratio wing is plagued by large induced drag,
and hence subsonic aircraft (since World War I) do not have low-aspect-ratio wings.
On the other hand, a low-aspect-ratio straight wing has low supersonic wave drag,and this is why such a wing was used on the F-104-the first military fighter designed
for sustained Mach 2 flight. At subsonic speeds, and especially for takeoff and
landing, the low-aspect-ratio wings were a major liability to the F-104.
F104
Fortunately, there are two other wing platforms that reduce wave drag
without suffering nearly as large a penalty at subsonic speeds, namely,
the swept wing and the delta wing.
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Swept WingsThe main function of a swept wing is to reduce wave drag at transonic and
supersonic speeds. Consider a straight wing and a swept wing in a flow with a
free-stream velocity V. Assume that the aspect ratio is high for both wings, so that
we can ignore tip effects. Let u and w be the components of V, perpendicular andparallel to the leading edge, respectively. The pressure distribution over the airfoil
section oriented perpendicular to the leading edge is mainly governed by the
chordwise component of velocity u; the spanwise component of velocity w has
little effect on the pressure distribution. For the straight wing the chordwise velocity
component u is the full V, for the swept wing the chordwise component of thevelocity u is smaller than V: = cosVu
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Since u for the swept wing is smaller than u for the straight wing, the difference
in pressure between the top and bottom surfaces of the swept wing will be less
than the difference in pressure between the top and bottom surfaces of thestraight wing. Since lift is generated by these differences in pressure, the lift on
the swept wing will be less than that on the straight wing.
The wingspan b is the
straight-line distancebetween the wing tips, the
wing platform area is S, and
the aspect ratio and the
taper ratio are defined
AR = b^2/S and taperratio ct/cr.
an approximate calculation of the lift slope for a swept finite wing,Kuchemann suggests the following approach. The lift slope for an infinite
swept wing should be cos0a
therefore
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The subsonic compressibility effect is added by replacing
0a Maa 10with
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Supersonic Delta wingsFor a swept wing moving at
supersonic speeds, the
aerodynamic properties depend
on the location of the leadingedge relative to a Mach wave
emanating from the apex of the
wing.
The Mach angle is given by
)/1(cos
1Ma
=
If the wing leading edge is swept inside the Mach cone the component of Ma
perpendicular to the leading edge is subsonic; hence, the swept wing is said to have
a subsonic leading edge. For the wing in supersonic flight, there is a weak shock
that emanates from the apex, but there is no shock attached elsewhere along the
wing leading edge. In contrast, if the wing leading edge is swept outside the
Mach cone the component of Ma, perpendicular to the leading edge is supersonic;
hence the swept wing is said to have a supersonic leading edge. For this wing in
supersonic flight, there will be a shock wave attached along the entire leading edge.
A swept wing with a subsonic leading edge behaves somewhat as a wing at
subsonic speeds, although the actual free-stream Mach number is supersonic.
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Delta WingsSwept wings that have platforms such as shown in Fig are called delta wings.
dominant aspect of this flow
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Thus, the flow on the bottom surface in the vicinity of the leading edge tries to curl around the
leading edge from the bottom to the top. If the leading edge is relatively sharp, the flow willseparate along its entire length. This separated flow curls into a primary vortex above the wing
just inboard of each leading edge. The stream surface which has separated at the leading
edge loops above the wing and then reattaches along the primary attachment line. The primary
vortex is contained within this loop.A secondary vortex is formed underneath the primary
vortex, with its own separation line, and its own reattachment line. Unlike many separated
flows in aerodynamics, the vortex pattern over a delta wing is a friendly flow in regard to the
production of lift. The vortices are strong and generally stable. They are a source of high
energy, relatively high vorticity flow, and the local static pressure in the vicinity of the vortices is
small. Hence, the vortices create a lower pressure on the top surface than would exist if the
vortices were not there. This increases the lift compared to what it would be without the
vortices.
dominant aspect of this flow
is the two vortices that are
formed along the highly
swept leading edges, andthat trail downstream over
the top of the wing. This
vortex pattern is created by
the following mechanism.
The pressure on the bottomsurface of the wing is
higher than the pressure on
the top surface.
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The net result is a reasonable value of CLmax=1.35. The lift curve is nonlinear, in contrast
to the linear variation exhibited by conventional wings for subsonic aircraft. The vortex lift is
mainly responsible for this nonlinearity.
The next time you have an opportunity to watch a delta-wing aircraft take off or land, for
example, the televised landing of the space shuttle, note the large angle of attack of the
vehicle. Also, this is why the Concorde supersonic transport, with its low-aspect-ratio
deltalike wing, lands at a high angle of attack. In fact, the angle of attack is so high that the
front part of the fuselage must be mechanically drooped upon landing in order for the pilots tosee the runway.
The difference between the
experimental data and the potential
flow lift is the vortex lift . The vortexlift is a major contributor to the
overall lift; The lift slope is small, on
the order of 0.05 per degree. The lift,
however, continues to increase over
a large range of angle of attack (thestalling angle of attack is about 35).
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Static Aeroelasticity
Rigid flat plate mounted on a torsional spring
If the spring were very stiff or
airspeed were very slow, the
rotation would be rather small;
however, for flexible springs or
high flow velocities the rotationmay twist the spring beyond its
ultimate strength
and lead to structural failure.
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The equation of static equilibrium simply states that the sum of aerodynamic plus
elastic moments about any point on the airfoil is zero. By convention,
we take the point about which moments are summed as the point of springattachment, the so-called 'elastic center' or 'elastic axis' of the airfoil.
The total aerodynamic angle of attack, , is taken as the sum of some
initial angle of attack, 0 (with the spring untwisted), plus an additional increment
due to elastic twist of the spring e.
No changes with
For a symmetrical airfoil CL0=0
ke
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Ifgoes to infinity
This is the divergence condition
and the corresponding dynamic pressure is
termed the 'divergence dynamic pressure'