aerodynamic coeff.pdf

7/27/2019 aerodynamic coeff.pdf

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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'

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aerodynamic coeff.pdf