ch01s(1)

MAE3303

Aerodynamics of

Compressible Flow

(Incomplete)

Introduction to

Aerodynamics

Compressible flow

Incompressible flow

Type of Flows

low-speed flow,

Incompressible flow…

where the fluid velocity is much less that its speed of sound.

Density is constant!

Compressible flow…

high-speed flow,

The density changes in response to changes in

pressure and/or temperature

where the fluid speed is comparable to its speed of sound.

Compressible flows are difficult to obtain in liquids.

is the fractional change in volume of the fluid element

per unit change in pressure.

Compressibility of a fluid…

p

P+dp

v

V-dv In general, the compressibility of gases is

several orders of magnitude larger than that

of liquid.

Bulk Modulus

Density change versus pressure

change…

If the fluid is liquid, density changes will be small.

For a low-speed gas flow, pressure changes are small.

If the fluid is a gas, density changes can be large.

Incompressible Mach No. < 0.3

Compressible Mach No. > 0.3

Gas flows…

can be classified with respect to the flow Mach number

Density changes will be more than 5% if M>0.3.

Mach number

Choking

Shock waves

Effects of compressibility…

---wherein the duct flow rate is shapely limited

by the sonic condition

---which are nearly discontinuous

property changes in a supersonic

flow

Flow Regions

• Subsonic flow (M<1 everywhere)

• Transonic flow (mixed regions

where M < 1 and M> 1)

• Supersonic flow (M>1 everywhere)

• Hypersonic flow (M>5)

Aerodynamic Characteristic of Airfoil and Wings

The following sections develop some of the terminology

and basic aerodynamic fundamentals of airfoil and wings.

What is an airfoil?

Airfoil Nomenclature

Mean camber line: the locus of points halfway between the upper and lower

surfaces as measured perpendicular to the mean camber itself.

An airfoil can be defined with mean camber line and thickness distribution

Thickness is the distance between the upper and lower surfaces measured

perpendicular to the camber line.


Chord Line: the straight line connecting the leading and trailing edges.

Camber (Maximum Camber): the maximum distance between the mean camber

line and the chord line, measured perpendicular to the chord line. Cambered vs.

symmetric.

Geometric Angle of attack (Angle of Attack): angle between the chord

and the direction of the undisturbed, “free-stream” flow.

Leading and trailing edges: the most forward and rearward points of the

mean camber line.

NACA Airfoils

There are a variety of classifications, including NACA four-digit wing

sections, NACA five-digit wing sections, and NACA six-digit wing sections.

The first integer indicates the maximum camber in percent of the chord.

NACA four-digit wing section: NACA 0012, NACA 4412

The second integer indicates the distance from the leading edge to the

maximum camber in tenths of the chord.

The last two integers indicate the maximum section thickness in percent of

the chord.

The first integer when multiplied by 3/2 gives the design lift coefficient in

tenths.

NACA five-digit wing section: NACA 23012

The next two integers when divided by 2 give the location of maximum

camber along the chord from LE in percent of the chord.


the chord.


The first integer simply identifies the series.

NACA ‘6-series’ wing section: NACA 65-218

The second integer gives the location of the minimum pressure in tenths of

the chord from the leading edge (for the basic symmetric thickness

distribution at zero lift).

The third integer is the design lift coefficient in tenths.


the chord.

University of Illinois at Urbana-Champaign (UIUC) Airfoil Coordinates

Database:

http://www.ae.illinois.edu/m-selig/ads/coord_database.html

NACA Four Digit Airfoils

Thickness distribution along the chord

First derived by Abbott and von Doenhoff in 1932.

The mean camber line is defined by two parabolic arcs tangent at the

maximum camber ordinate.

The leading-edge radius is

where, t is the maximum thickness as a fraction of the chord c and .

Where m is the maximum camber as a fraction of c, and p is the value of

x/c corresponding to this maximum.

Aerodynamic Forces and Moments

No matter how complex the body shape may be, the aerodynamic forces and

moments on the body are due entirely to two basic sources:

Pressure distribution over the body surface

Shear stress distribution over the body surface


Resultant force and moment

Body-oriented force components: Normal force N and Axial force A.

Flight path-oriented force components: Lift L and Drag D.


Two-dimensional body ---Airfoil

Sign convention for θ

pl and pu

and


Two-dimensional body ---Airfoil

pl, and pu

and


The aerodynamic moment (pitching moment), M’,

depends on the moment center.

- Moment about the leading edge

Dimensionless Aerodynamic Forces and Moments

Let ρ∞ and U∞ be the density and velocity, respectively in the free-stream.

Freestream dynamic pressure

Lift coefficient:

Drag coefficient:

Normal force coefficient:

Axial force coefficient:

Moment coefficient:

Reference Quantities:

Also, define S as a reference area and as a reference length.


For a two dimensional body, such as an airfoil section, the forces and moments

are for unit span section, S = c(1) = c,

The dimensionless pressure and shear stress are defined as follows,

Pressure coefficient

Skin friction coefficient

Coefficients are denoted by lowercase letters.


NACA0012, M∞=0.345, α=3.93, Re=3.245x105




Neglecting the shear stress contribution, at small angle of attack

Load distribution

Example Problem: Calculation of aerodynamic coefficients

Consider an airfoil with chord length c and the running distance x measured along

the chord. The leading edge is located at x/c = 0 and the trailing edge at x/c =1.

The pressure coefficient variation over the upper and lower surfaces are given,

respectively, as

Calculate the normal force coefficient.

Numerical Integrations

Consider pressure only Given: x, y, and p (or Cp) at nodes



Example Problem: Calculation of aerodynamic coefficients

For the airfoil section shown, compute the lift, drag and pitching moment about

the leading edge coefficients for angle of attack of 100

Pitching Moments

The pitching moment is measured about some definite point on the airfoil chord.

For some particular purpose, it may be desirable to know what it is about other

point.

Known: Ma To Know: Mx

Taking moment for each case

about the leading edge

Then

Converting to coefficient form gives

In terms of , (a = 0)

Center of Pressure

Force-and-Moment Single Force

or

For small α,

Aerodynamic Center

There is one point on the airfoil about which the moment is independent of

angle of attack; such a point is defined as the aerodynamic center (AC). It is

close to, but not generally on, the chord line, between 23% ~ 25 % of the chord

from the L.E.

(x=xac)

For small α,

a = c/4

Pitching Moment about AC

For small α,

Let

If is made zero,

That is, the pitching moment coefficient about an axis at zero lift is equal to the

constant pitching moment coefficient about the aerodynamic center.

AC vs CP

For small α,

cm,ac is almost invariably negative, so the center of pressure is behind the

aerodynamic center.

Let x = xac and a = xcp. Then


Airfoil Characteristics

During the 1930s and 1940s, the NACA carried out numerous measurements of

the lift, drag, and moment coefficients on the standard NACA airfoils.

Airfoil data are frequently called infinite wing data.

As α becomes large, the flow tends to

separate from the top surface.

At a certain angle 15 to 20°, the flow is separated completely from the upper

surface. The airfoil is said to be stalled: Lift drops off markedly, drag increases

markedly, and the airfoil is no longer flyable.

At low-to-moderate angle of attack,

cl varies linearly with α; the flow

moves smoothly and is attached to

the surface.

Cl vs. α


The maximum lift coefficient, cl,max occurs just prior to the stall.

The value of α when lift equals zero is

called the zero-lift angle of attack, αL=0.

For symmetric airfoils, αL=0 = 0.

For all airfoils with positive camber, αL=0 is a negative value, usually on the order of

-2 or -30.


The lift slope α0 is not affected by Re.

cl,max is dependent upon Re.

The moment coefficient is also

insensitive to Re except at large α.

When Re= 3.1x106, αL=0 ≈ -2.10,

cl,max ≈ 1.6, and the stall occurs at

α ≈ 160.

Experimental results for lift and moment coefficients for the NACA 2412 airfoil:

Viscous Effects:


Drag coefficient for the NACA 2412 airfoil.

The physical source of this drag coefficient is

both skin friction drag and pressure drag.

The sum of skin friction and form drags

yields the profile drag coefficient, cd for

the airfoil.

cd is sensitive to Re as expected.

cm,ac does not change with α and Re.

Pressure drag has several distinct

contributions: form drag (BL), wave drag

(SW), and induced drag (3D Vortex).

General Thin Airfoil Theory

The essential assumptions of thin-airfoil theory are,

(3) Irrotational incompressible flow.

(1) that the airfoil is operating at a small angle of attack

(2) Ratios of camber to chord and maximum thickness to chord are small.

Circulation and the Generation of Lift

For a lifting airfoil, the pressure on the lower surface of the airfoil is, on the

average, greater than the pressure on the upper surface.

Thus, the flow around the airfoil can be represented by the combination of a

translational flow from left to right and a circulating flow in a clockwise

direction,

The rounded leading edge prevents flow separation there, but the sharp trailing

edge causes a tangential wake motion that generates the lift.

Kutta-Joukowski Law

For any two-dimensional object of any cross- sectional shape placed in a

uniform, inviscid stream, the lift per unit span is

The direction of the lift is 900 from the stream direction, rotating opposite to

the circulation.

The circulation is determined around any closed curve containing the body,

The Kutta Condition

Nonuniqueness of the potential flow theory solution.

The case (c) best simulates

a real airfoil flow.

The Kutta Condition: the

circulation around an airfoil

is just right value to ensure

that the flows from the

upper surface and the

lower surface join smoothly

at the trailing edge.

General Thin Airfoil Theory

In thin-airfoil theory, the airfoil is replaced with its mean camber line.

A vortex sheet is placed along the mean camber line to produce the

required velocity jump, and its strength is adjusted so that the camber line

becomes a stream line and the Kutta condition is satisfied.

The velocity pattern, then, is composed of a uniform stream plus the field

induced by the vortex sheet.

Thin, Flat-Plate Airfoil (Symmetric Airfoil)

The lift per unit span (from Kutta-Joukowski Law) is

The section lift coefficient is

Lift slope:

Circulation:


The section moment coefficient about the leading edge is given by

The center of pressure , xcp, is the x coordinate, where the resultant lift force

could be placed to produce the pitching moment about the leading edge, i.e.

The result is independent of the angle of attack and is therefore independent

of the section lift coefficient.

The quarter-chord point is both the CP and AC.

the quarter-chord

or


The following important theoretical results for a symmetric airfoil are

obtained:

1. The sectional lift coefficient is directly

proportional to the geometric angle of

attack and is equal to zero when the

angle of attack is zero.

2. Lift slope = 2π

3. The center of pressure (CP) is at

the quarter-chord point for all

values of the lift coefficients.

The quarter-chord point is both the CP and AC.

Thin, Cambered Airfoil

The method of determining the aerodynamic characteristics for a cambered

airfoil is similar to that followed for the symmetric airfoil.

and

The values of An depend on the shape of the mean camber line and ,

with the coordinate transformation:

It is a general result from thin airfoil theory that the lift slope is equal to 2π for

any shape of airfoil.

Setting , the angle of zero lift is obtained as

For a symmetric airfoil,

The more highly cambered the airfoil, the larger will be the absolute

magnitude of .

The lift coefficient can be rewritten as,


Following a similar process, the moment coefficient about the leading edge can

be obtained as

in terms of the lift coefficient,

The center of pressure position behind the leading edge is found by:

The position of the center of pressure will vary as the lift coefficient varies.


The quarter chord is the theoretical location of the aerodynamic center for a

cambered airfoil.

The moment coefficient about the quarter chord

independent of α


Experimental Results

indicate remarkable agreement with the foregoing formulas based on thin airfoil

theory.

Re = 9x106

Example Problem: Theoretical aerodynamic coefficients for a

cambered airfoil

Consider the airfoil NACA 2412. The equation for the mean camber line is defined

in terms of the maximum camber and its location. Forward of the maximum

camber position, the equation of the mean camber line is

while aft of the maximum camber position,

Calculate the aerodynamic properties of the airfoil section.

The aerodynamic properties of airfoils are the same as the properties of a wing of

infinite span.

However, all airplanes have wings of finite span. And the flow over the finite wing

is 3D.

An airfoil is simply a section of a wing. And the flow over an airfoil is 2D.

Wings of Finite Span

By placing the airfoil sections discussed in the preceding section in span-wise

combinations, wings, horizontal tails, vertical tails, canards, and/ or other lifting

surfaces are formed.

WING GEOMETRY PARAMETERS

The planform of a wing is its shape seen on a plan (top) view of the aircraft.

Its area is called Wing Area (S).

Wingspan (b): the distance between two wingtips.

Average Chord ( ), is determined from the equation that the product of the

span and the average chord is the wing area ( ).

Mean Aerodynamic Chord (mac) is used together with S to non-

dimensionalize the pitching moments.


Aspect Ratio (AR), is a measure of the narrowness of the wing planform. It

is defined as

For a rectangular wing,

Typical aspect ratios vary from 35 for a high-performance sailplane to 2 for a

supersonic jet fighter.

Taper Ratio, is the ratio of the tip chord to the root

chord:

Root Chord, is the chord at the wing centerline, and the tip chord, is

measured at the tip.

A rectangular wing has a taper ratio of 1.0 while the pointed tip delta wing

has a taper ratio of 0.0.


Sweep Angle, is usually measured as the angle between the

line of 25% chord and a perpendicular to the root chord.

Dihedral Angle, is the angle between a horizontal

plane containing the root chord and a plane midway

between the upper and lower surfaces of the wing. If

the wing lies below the horizontal plane, it is termed

an Anhedral Angle.

Geometric twist defines the situation where the chord

lines for the spanwise distribution of airfoil sections do

not all lie in the same plane (AOA of all sections is not a

constant). Wash-in vs. Wash-out


Wing planforms



Wings of Finite Span: Downwash and Induced Drag

A trailing vortex is created at each wing tips.

These wing-tip vortices downstream of the wing induce a small downward

component of velocity in the neighborhood of the wing itself.

This downward component is called downwash.

The downwash combines with the free stream velocity to produce a local relative

wind which is canted downward in the vicinity of each airfoil section of the wing.

Wings of Finite Span: Downwash and Induced Drag

and creates a component of drag

– the induced drag Di

The presence of downwash over a finite wing reduces the angle of attack by αi

The induced drag coefficient,

Total drag coefficient for a wing

- Profile Drag

Wings of Finite Span: Lifting-Line Theory

Superposition of an infinite number of horseshoe vortices coincident along a

single line, called the lift line.

The strength of each trailing vortex is equal to the change in circulation along

the lifting line

Wings of Finite Span: Lifting-Line Theory

The three main aerodynamic characteristics of a finite wing:

1. The lift distribution is obtained from the Kutta-Joukowski theory:

2. The total lift of the wing

and the lift coefficient

3. The induced drag per unit span is

The total induced drag:

The coefficient of the total induced drag:

Wings of Finite Span: General Lift Distribution

For all wings in general,

The lift distribution which yields the minimum induced drag is the elliptical lift

distribution.

where, e is called span efficiency factor.

For elliptical lift plan forms, e=1; for all other planforms, e< 1. Typical

values for e are between 0.6 and 0.95.


The induced drag coefficient is directly proportional to the square of the lift

coefficient.

The induced drag coefficient is inversely proportional to aspect ratio.

Aspect ratio varies from about 6 to 22 for subsonic airplanes and sailplanes.

Clearly, an airplane cannot generate lift for free. The induced drag is also

called the drag due to lift.


The lift slope for an airfoil is defined as

The lift slope for a finite-wing is defined as

Clearly, the effect of a finite wing is to reduce the lift slope.


Also, note that at zero lift, there are no

induced effects; that is,

Thus, when CL = 0,

As a result, αL=0 is the same for the finite

wing and the infinite wing.


In summary, a finite wing introduces two major changes to the airfoil data:

1. Induced drag must be added to the finite wing:

2. The slope of the lift curve for a finite wing is less than that for an infinite wing,

Documents

ch01s(1)