aerodynamics chapter 5

Copyright by Dr. Zheyan Jin

Zheyan

JinSchool of Aerospace Engineering and Applied Mechanics

Tongji

University

Shanghai, China, 200092

AerodynamicsAerodynamics


Chapter 5 Incompressible Flow Over Finite WingsChapter 5 Incompressible Flow Over Finite Wings

5.1 Introduction: Downwash and Induced Drag

Infinite wings versus finite wings

Airfoil shapes are two-dimensional. We consider them to stretch indefinitely along the span direction, and therefore the flow around them must also be two-dimensional.

Airfoil data comes from airfoil shapes that completely span the wind tunnel section, and this is a reasonable approximation to a 2D airfoil.





Real wings are not infinite in span. They have a finite wing-span. As we shall see, because real wings do not go on forever, the flow over them is not two-dimensional. This has very important effects.





Wing span bW

ing

root

rc

V

tc

Win

g tip

Streamline over the top surface Streamline over the bottom surface

Top view (planform)

Low pressure

High pressure

Front view





The physical mechanism for generating lift on the wing is the existence of a high pressure on the bottom surface and a low pressure on the top surface. As a by-product of this pressure imbalance, the flow near the wing tips tends to curl around the tips.

On the top surface, there is generally a spanwise component of the flow from the tip toward the wing root. On the contrary, there is generally a spanwise component of the flow from the root toward the tip on the bottom surface.

View from above





The tendency of the flow to “leak” around the wing tips has another important effect on the aerodynamics of the wing. This flow establishes a circulatory motion that trails downstream of the wing; that is, a trailing vortex is created at each wing tip.

These wing-tip vortices downstream of the wing induce a small downward component of air velocity in the neighborhood of the wing itself. This downward component is called downwash, denoted by the symbol w.

Condensation in the cores of wingtip vortices from an F-15E as it disengages from a KC-10 Extender following midair refueling.




Downwash

One reason that geese and pelicans fly in neat V- formation is that by maintaining a very precise spacing, birds can get a slight upward boost form the trailing vortex of the bird in front: the outer side of the trailing vortex of the bird in front has an upward component.

On course, the leader at the point of the V gets no such benefit, which is undoubtedly why birds exchange the lead position relatively often in such flocks.

Canada Geese in V formation




Downwash

Wingtip vortices can be harmful.

When small airplanes are thrown out of control by trailing of very large ones.

The vortex system of a fixed wing is quite simple, in practice consisting of the bound vortex on the wing and a trailing vortex off each wing tip that gradually dissipates far behind the wing.

Wake turbulence and tip vortices




Downwash

The downwash combines with the free stream velocity V ͚ to produce a local relative wind which is canted downward in the vicinity of each airfoil section of the wing.

The angle between the chord line and the direction of V ͚ is the angle of attack α. We now precisely define α as the geometric angle of attack.

The local relative wind is inclined below the direction of V ͚ by the angle αi, called the induced angle of attack.




Downwash

The presence of downwash has two important effects on the local airfoil section, as follows:

1. The angle of attack actually seen by the local airfoil section is the angle between the chord line and the local relative wind. This angle is defined as the effective angle of attack.

ieff




Downwash

2. The local lift vector is aligned perpendicular to the local relative wind, and hence is inclined behind the vertical by the angle αi.

Consequently, there is a component of the local lift vector in the direction of V ͚ ; this drag is created by the presence of downwash.

This drag is defined as induced drag.




Downwash

The tilting backward of the lift vector shown in the right picture is one way visualizing the physical generation of induced drag. Two alternative ways are as follows:

1. The three-dimensional flow induced by the wing-tip vortices alters the pressure distribution on the finite wing in such a fashion that a net pressure imbalance exists in the direction of V ͚.




Downwash

2. The wing-tip vortices contain a large amount of translational and rotational kinetic energy. This energy is provided by the aircraft engine.

In effect, the extra power provided by the engine that goes into the vortices is the extra power required from the engine to overcome the induced drag.




Road Map General features of finite-wing aerodynamics: downwash, effective angle of attack, and induced drag

Additional tools needed for finite wing analysis:

1. Curved vortex filament

2. Biot-Savart Law

3. Helmholz’s vortex theorems

Method of analysis

Prandtl’s classical lifting-line theory

Modern numerical lifting-line method

Lifting surface theory

Modern vortex lattice numerical method




Differences in nomenclature

For the two-dimensional bodies:

Lift, drag, and moments per unit span have been denoted with primes. For example, L’, D’, and M’.

The corresponding lift, drag, and moment coefficients have been denoted by lower case letters, for example, cl, cd, and cm.

For the three-dimensional bodies:

Lift, drag, and moments on a complete three-dimensional body are given without primes. For example, L, D, and M.

The corresponding lift, drag, and moment coefficients have been denoted by capital letters, for example, CL, CD, and CM.



DragTotal drag on a subsonic finite wing in real life

Pressure drag DpSkin friction drag DfInduced drag Di

Profile drag

Profile drag coefficient:

SqDC i

iD

,

SqDD

c pfd

iDdD CcC ,

Induced drag coefficient:

Total drag coefficient:




5.2 The vortex filament, the Biot-Savart law, and Helmholtzˈs theorems

The Biot-Savart law

In general, a vortex filament can be curved. The filament induces a flow field in the surrounding space.

Vortex filament of strength Г

dV

dlrThe velocity at point P, dV, induced by

a small directed segment dl of a curved filament with strength

is

34 rrdldV




The Biot-Savart law

Now apply the Biot-Savart law to a straight vortex filament of infinite length.

The direction of the velocity is downward.

The magnitude of the velocity is given by:

h2

VV-

P

r

h

dl

x

yz

34 r

rdl

V

dlr

2

sin4

V

The velocity induced at P by the entire vortex filament is:




The Biot-Savart law

Let h be the perpendicular distance form point P to the vortex filament. Then,

Thus, the velocity induced at a given point P by an infinite, straight vortex filament at a perpendicular distance h from P is simply:

h2

VV-

P

r

h

dl

x

yz

h2sin

h4sin

4V

0

2

ddl

r

dhdl

anhl

h

2sin

t

sinr




Helmholtz’s vortex theorems:

The great German mathematician, physicist, and physician Hermann von Helmholtz was the first to make use of the vortex filament concept in the analysis of inviscid, incompressible flow.

Helmholtz’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.




Lift distribution:

Most finite wings have a variable chord, with the exception of a simple rectangular wing.

Also, many wings are geometrically twisted so that α is different at different spanwise locations- so-called geometric twist. If the tip is at a lower α than the root the wing is said to have washout; if the tip is at a higher α than the root, the wing has washin.

The wings on a number of modern airplanes have different airfoil sections along the span, with different values of αL=0; this is called aerodynamic twist.

Front view of wing

y-b/2 b/2

)()('' yVyLL



5.3 Prandtlˈs Classical Lifting-Line Theory

Prandtl reasoned as follows.

A vortex filament of strength Г that is somehow bound to a fixed location in a flow -a so-called bound vortex- will experience a force Lˈ=ρV͚Г from the Kutta-Joukowski theorem.

This bound vortex is in contrast to a free vortex, which moves with the same fluid elements throughout a flow.




Prandtl reasoned as follows.

Let us replace a finite wing of span b with a bound vortex, extending from y=-b/2 to y=b/2. Since a vortex filament cannot end in the fluid, we assume the vortex filament continues as two free vortices trailing downstream from the wing tips to infinity.

This vortex is in the shape of horseshoe, and therefore is called a horseshoe vortex.

http://www.google.com.hk/imgres?imgurl=http://petticoatsandpistols.com/wp-content/uploads/2008/10/lucky-horseshoe.jpg&imgrefurl=http://petticoatsandpistols.com/2008/10/24/horseshoes-lucky-ups-downs/&usg=__tBa_0cECqjP_HHe28fC7VdalUW0=&h=376&w=400&sz=62&hl=zh-CN&start=9&zoom=1&tbnid=j-YmPQ8TUR_21M:&tbnh=117&tbnw=124&ei=-jWKT8vAPO_HmQXA3tGyCQ&prev=/search%3Fq%3Dhorseshoe%26um%3D1%26hl%3Dzh-CN%26newwindow%3D1%26safe%3Dstrict%26sa%3DN%26biw%3D1336%26bih%3D635%26tbm%3Disch&um=1&itbs=1



A single horseshoe vortex.

Consider the downwash w induced along the bound vortex from –b/2 to b/2 by the horseshoe vortex.

The two trailing vortices both contribute to the induced velocity along the bound vortex, and both contributions are in the downward direction.

The bound vortex induces no velocity along itself.

b/2

-b/2

zx

y

Trailing vortex

Trailing vortexBound vortex

A single horseshoe vortex

)2/(4)2/(4)(

ybybyw

The first term on the right-hand side is the contribution from the left trailing vortex. The second term is the contribution from the right trailing vortex.




Prandtl’s Classical Lifting-Line Theory:

A single horseshoe vortex.

We can reduce the above equation to:

Note that w approaches -∞ as y approaches –b/2 or b/2.

A single horseshoe vortex22)2/(4

)(yb

byw

The downwash distribution due to the single horseshoe vortex shown in above figure does not realistically simulate that of a finite wing.

The downwash approaching an infinite value at the tips is especially disconcerting.


b/2

-b/2

zx

y

Trailing vortex

Trailing vortexBound vortex



Let us superimpose a large number of horseshoe vortices, each with a different length of bound vortex, but with all the bound vortices coincident along a single line, called the lifting line.

The circulation varies along the line of bound vortices.

321 ddd

Along AB and EF, the circulation is: 1d

21 dd

Lifting line

1d

b/2

A ∞

321 ddd1d

∞

∞

∞

∞

∞

1d

2d

2d

3d

3d

BC

DE

F

-b/2

21 dd

Along BC and DE, the circulation is:

Along CD, the circulation is:

Note that the strength of each trailing vortex is equal to the change in circulation along the lifting line.




Let us assume there are infinite number of horseshoe vortices along the lifting line.

The vortex sheet is parallel to the direction of V ͚.

There is continuous distribution of Г(y) along the lifting line. The value of the circulation at the origin is Г0。

The total strength of the sheet integrated across the span is zero, because it consists of pairs of trailing vortices of equal strength but in opposite direction.

z

x

y

Lifting line

0

b/2

dГV͚

-b/2

)(y

θ

y0




The velocity dw at y0 induced by the entire semi-infinite trailing vortex located at y is given by:

The minus sign in the above equation is needed for consistency with the right picture.

For the trailing vortex shown, the direction of dw at y0 is upward and hence is a positive value, whereas Г is decreasing in the y direction, making dГ/dy a negative quantity. The minus sign in the above equation makes the positive dw consistent with the negative dГ/dy.

)(4)/(

0 yydydyddw

dxz

x

y

Lifting line

0

b/2

dГV͚

-b/2

)(y

θ

y0

dw




The total velocity w induced at y0 by the entire trailing vortex sheet is the summation of the last equation over all the vortex filaments:

The value of the downwash at y0 due to all the trailing vortices.

2/

2/0

0)/(

41)(

b

b yydydydyw

Keep in mind that although we label w as downwash, w is treated as positive in the upward direction in order to be consistent with the normal convention in an xyz rectangular coordinate system.

dxz

x

y

Lifting line

0

b/2

dГV͚

-b/2

)(y

θ

y0

dw




Assume this section is located at the arbitrary spanwise station y0. The induced angle of attack αi is given by:

Generally, w is much smaller than V ͚, and hence αi is a small angle, on the order of a few degrees at most.

))((tan)( 010

Vywyi

Vywyi

)()( 00




By substituting the above equation into the downwash equation, we obtain

that is, an expression for the induced angle of attack in terms of the circulation distribution Г(y) along the wing.

2/

2/0

0)/(

41)(

b

bi yydydyd

Vy




Consider again the effective angle of attack αeff.

Since the downwash varies across the span, then αeff is also variable; αeff= αeff(y0). The lift coefficient for the airfoil section located at y=y0 is:

])([2])([ 0L00L00 yyac effeffl

)()(21

002 yVcycVL l

From the Kutta-Joukowski theorem, lift for the local airfoil section located at y0 is




Finally, we obtain

Thus，

0L0

0

0L0

)()(

])([2

ycVy

yc

eff

effl

)()(2

0

0

ycVycl

ieff Since

2/

2/0

000

00

)/(4

1)()(

)()(b

bL yydydyd

Vy

ycVyy




It simply states that the geometric angle of attack is equal to the sum of the effective angle plus the induced angle of attack.

The fundamental equation of Prandtl’s lifting-line theory.

2/

2/0

000

00

)/(4

1)()(

)()(b

bL yydydyd

Vy

ycVyy

αeff is expressed in terms of Г, and αi is expressed in terms of an integral containing dГ/dy. Hence, the above equation is an integro- differential equation, in which the only unknown is Г;

all the other quantities, α ,c, V ͚, and αL=0, are known for a finite wing of given design at a given geometric angle of attack in a freestream with given velocity.




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

The solution Г=Г(y0) obtained from the above equation gives us the three main aerodynamic characteristics of a finite wing, as follows:

2/

2/

2/

2/

2/

2/

'

)(2

)(

)(

b

bL

b

b

b

b

dyySVSq

LC

dyyVL

dyyLL

)()( 00' yVyL

2/

2/0

000

00

)/(4

1)()(

)()(b

bL yydydyd

Vy

ycVyy

2. The total lift is obtained by integrating the above equation over the span：




3. The induced drag per unit span is:

2/

2/,

2/

2/

2/

2/

'

)()(2

)()(

)()(

b

b ii

iD

b

b ii

b

b ii

dyyySVSq

DC

dyyyVD

dyyyLD

ii sinLD ''

Since αi is small, this relation becomes

ii '' LD

The total induced drag is obtained by integrating the above equation over the span




1. Г0 is the circulation at the origin.

Consider a circulation distribution given by:

)()(' yVyL 20

' )2(1)(byVyL

20 )

by2(1)( y

2. The circulation varies elliptically with distance y along the span; hence, it is designated as an elliptical circulation distribution. Since

Note:

0)2/()2/( bb

we also have

Hence, we are dealing with an elliptical lift distribution.

3.

Thus, the circulation, hence lift, properly goes to zero at the wing tips.




Elliptical Lift Distribution:

First, let us calculate the downwash.

What are the aerodynamic properties of a finite wing with such an elliptic lift distribution?

2/

2/0

2/12220

0 )()/41()(

b

bdy

yybyy

byw

2/12220

)/41(4

byy

bdyd

Substitute this into the down wash equation, we obtain,





The integral can be evaluated easily by making the substitution


00

00

0

0

00

coscoscos

2)(

coscoscos

2)(

db

w

db

w

dbdyby sin2

cos2

bw

2)( 0

0

Hence,

This states that the downwash is constant over the span for an elliptical lift distribution.





The induced angle of attack


bVVw

i 20

Note that both the downwash and induced angle of attack go to zero as the wing span becomes infinite- which is consistent with our previous discussions on airfoil theory.

The induced angle of attack is also constant over the span for an elliptical lift distribution.





A more useful expression for αi can be obtained as follows.


bVL

bVdbVL

4

4sin

2

0

00

20

dybyVL

b

b 2/

2/

2/12

2

0 )41(

Again use the transformation cos)2/(by





However,


2

0

212

2

bSC

bVbSCV

bSCV

LLi

L

Sb /AR 2

LSCV 2

21L

An important geometric property of a finite wing is the aspect ratio, denoted by AR and defined as

AR L

iC

The induced angle of attack:





Or,

The induced drag coefficient

bSCV

ARC

SVb LL

iD

2)(

2C ,

0

0202/

2/, 2sin

22)(2C

SVbdb

SVdyy

SViib

bi

iD

The above equation states that the induced drag coefficient is directly proportional to the square of the lift coefficient.

ARC

2

, L

iDC





The induced drag is a consequence of the presence of the wing-tip vortices, which in turn are produced by the difference in pressure between the lower and upper wing surfaces.

The lift is produced by this same pressure difference.

Hence, induced drag is intimately related to the production of lift on a finite wing; indeed, induced drag is frequently called drag due to lift.


The dependence of induced drag coefficient on the lift is not surprising, for the following reason.

ARC

2

, L

iDC

First property:





To reduce the induced drag, we want a finite wing with the highest possible aspect ratio.

Unfortunately, the design of very high aspect ratio wings with sufficient structural strength is difficult.

Therefore, the aspect ratio of a conventional aircraft is a compromise between conflicting aerodynamic and structural requirements.


CDi is inversely proportional to aspect ratio.

ARC

2

, L

iDC

Second property:





Assuming that a0 is the same for each section, cl must be constant along the span. The lift per unit span is given by

Consider a wing with no geometric twist (i.e., α is constant along the span) and no aerodynamic twist (i.e., αL=0 is constant along the span).

The local section lift coefficient cl is given by:

00ac Leffl

ll cq

yycccqy

)(L)()(L

''

Third property:

For an elliptic lift distribution, the chord must vary elliptically along the span; that is, for the condition given above, the wing planform is elliptical.





Illustration of the related quantities: an elliptic lift distribution, elliptic planform, and constant downwash


Although an elliptical lift distribution may appear to be a restricted, isolated case, in reality it gives a reasonable approximation for the induced drag coefficient for an arbitrary finite wing.



Elliptical lift distribution:


Supermarine Spitfire Supermarine Spitfire



General Lift Distribution:

where the coordinate in the spanwise direction is now given by θ, with 0<= θ

<=π.

In terms of θ, the elliptic lift distribution

Consider the transformation:cos

2by

sin)( 0


20 )

by2(1)( y

can be written as




This equation hints that a Fourier sine series would be an appropriate expression for the general circulation distribution along an arbitrary finite wing.


N

n nAbV1

sin2)(

where as many terms N in the series can be taken as we desire for accuracy.

An must satisfy the fundamental equation of Prandtl’s lifting-line theory.

dydnnAbV

dyd

dd

dyd N

n

1

cos2




Substituting the above equations into the angle of attack equation, we obtain


N

nL

N

nnnAnA

cb

1 0

000

10

00 sin

sin)(sin)(

2)(

This equation is evaluated at a given spanwise location; hence, θ0 is specified. In turn, b, c(θ0), and αL=0(θ0 ), are known quantities from the geometry and airfoil section of the finite wing. The only unknowns in the above equation are the An’s.

Hence, written at a given spanwise location, the above equation is one algebraic equation with N unknowns, A1,A2,…, AN.

However, let us choose N different spanwise stations, and let us evaluate the above equation at each of these N stations. We then obtain a system of N independent algebraic equations with N unknowns, namely, A1, A2,…, AN.




Now that Г(θ) is known.


ARASbAC

nn

dn

dnASbdyy

SVC

L

N

n

b

bL

1

2

1

0

10

22/

2/

)1(1(

02/

sinsin

sinsin2)(2

）

Note that CL depends only on the leading coefficient of the Fourier series expansion.




The induced drag is obtained by


001

2/

2/0

0 coscoscos1

y-y)/(

41)( dnAdydydV

yN

n

b

bi

The induced angle of attack is obtained by

01

22/

2/,D sin)(sin2)()(2 dnASbdyyy

SVC i

N

n

b

b ii

0

00

0 sinsin

coscoscos

ndn

0

0

10 sin

sin)( nnA

N

ni Thus,




In the above equation, θ0 is simply a dummy variable which ranges from 0 to π

across the span of the wing; it can therefore be replaced by θ, and the above equation can be written as:


dnnAnASb N

n

N

ni )sin()sin(2C1

01

2

,D

Substitute this equation to the drag coefficient equation, we have

sinsinn)(

1

nAN

ni

0

0

10 sin

sin)( nnA

N

ni




In the drag coefficient equation, the mixed product terms involving unequal subscripts are equal to zero. Hence,


])A

n(1[)n(C

n2

)n(2C

2

2

1

21

2

221,D

1

2

1

22

,D

Nn

N

ni

N

n

N

ni

AARAAAAR

AARASb

0 )(2/)(0

sinsinkm

kmkm




Thus,


]1[CC2

,D

AR

Li

where2

1

n

2

An

A

N

eARCC L

iD

2

,

Note that δ ≥0;hence, the factor 1+ δ in the above equation is either greater than 1 or at least equal to 1. Let us define a span efficiency factor, e, as e=1/(1+ δ).

where e ≤1. Note that δ =0 and e =1 for the elliptical lift distribution. Hence, the lift distribution which yields minimum induced drag is the elliptical lift distribution.





However, elliptic planforms are more expensive to manufacture than, say, a simple rectangular wing.

On the other hand, a rectangular wing generates a lift distribution far from optimum. A compromise is the tapered wing.

Recall that for a wing with no aerodynamic twist and no geometric twist, an elliptical lift distribution is generated by a wing with an elliptical planform.

Rectangular wing

Elliptic wing

Tapered wing



Effect of Aspect Ratio:

Note that the induced drag coefficient for a finite wing with a general lift distribution is inversely proportional to the aspect ratio.

Note that AR, which typically varies from 6 to 22 for standard subsonic airplanes and sailplanes, has a much stronger effect on CD,i than the value of δ.


Induced drag factor δ as a function of taper ratio.

]1[CC2

,D

AR

Li




Hence, the primary design factor for minimizing induced drag is not the closeness to an elliptical lift distribution, but rather, the ability to make the aspect ratio as large as possible.


Induced drag factor δ as a function of taper ratio.

]1[CC2

,D

AR

Li




There are two primary differences between airfoil and finite-wing properties.

1. A finite wing generates induced drag.

2. The lift slope is not the same.


constARCC

constC

ddC

LL

iL

i

L

)(

)(

)(

0

0

0

ARaaa

ddCL

/1 0

0

0aLC Infinite

wing

0L

i eff

LC

a

Finite elliptic wing

For a finite wing of general planform, the left equation is slightly modified

)1(/1 0

0

ARaaa



Example 1:


055.0

1

0

0 97.4055.18/21

21/1

radARaaa

）（）（

Assume a0=2πfor thin airfoil theory

00789.08

055.014335.0)1(22

,

）（）（

ARCC L

iD

Consider a finite wing with an aspect ratio of 8 and a taper ratio of 0.8. The airfoil section is thin and symmetric. Calculate the lift and induced drag coefficients for the wing when its angle of attack is 5°. Assume that δ=τ.

Solution:

Since the airfoil is symmetric, aL=0=0°. Thus,

4335.0)5(deg0867.0 01 reeaCL

From a figure in the textbook, we can obtain



Example 2:


423.0LC

1787.0055.01001.06

1C ,2

L

iDARC

The lift slope of this wing is therefore

Consider a rectangular wing with an aspect ratio of 6, and induced drag factor δ=0.055, and a zero lift angle of attack of -2°. At an angle of attack of 3.4°, the induced drag coefficient for this wing is 0.01. Calculate the induced drag for a similar wing at the same angle of attack, but with an aspect ratio of 10. Assume that the induced factors for drag and the lift slope, δ and τ, respectively, are equal to each other. Also, for AR=10, δ=0.105.

Solution:

Hence,

radreeddCL /485.4deg/078.0

)2(4.3423.0

00

Firstly, let us calculate CL for the wing with aspect ratio 6.



Example 2:


Solving for a0, we find that this yields a0=5.989/rad. Since the second wing (with AR=10) has the same airfoil section, then a0 is the same. The lift slope of the second wing is given by


Solution: The lift slope for the airfoil can be obtained by

0

0

0

0

0

0

056.01055.016/1485.4

1/1

aa

aa

ARaaa

ddCL

）（

）（



Example 2:


The lift coefficient for the second wing is therefore


Solution: For AR=10

radARaaa /95.4

105.0110/989.51989.5

1/1 0

0

）（）（

0076.010

105.01464.0)1(22

,

）（）（

ARCC L

iD

464.0]2(4.3[086.0)( 000L ）LaC



5.4 A numerical nonlinear lifting-line method

Consider the most general case of a finite wing of given planform and geometric twist, with different airfoil sections at different spanwise stations. Assume that we have experimental data for the lift curves of the airfoil sections, including the nonlinear regime. A numerical iterative solution for the finite-wing properties can be obtained as follows:

1: Divide the wing into a number of spanwise stations. Here k+1 stations are shown, with n designating any specific station. y

Δy

1 2 3 n k k+1

For the given wing at a given α, assume the lift distribution along the span; that is, assume values for Г at all the stations Г1, Г2,…., Гn,…, Гk+1 . An elliptical lift distribution is satisfactory for such an assumed distribution.

2:




3:With this assumed variation of Г，

calculate the induced angle of attack αi

y

Δy

1 2 3 n k k+1

Using αi from step 3, obtain the effective angle of attack αeff at each station form

4:

2/

2/

)/(4

1)(b

bn

ni yydydyd

Vy

1

1

6,4,2 1

1 )/()/(4

)/(34

1)(

jn

j

jn

jk

j jn

jni yy

dydyy

dydyy

dydyV

y

The integral is evaluated numerically. By using Simpson’s rule,

)()( nineff yy

where Δy is the distance between stations.




5: With the distribution of αeff calculated from step 4, obtain the section lift coefficient (cl)n at each station. These values are read from the known lift curve for the airfoil.

6:

nlnn

nlnnn

ccVy

ccVyVyL

)(21)(

)(21)()( 2'

where cn is the local section chord. Keep in mind that in all the above steps, n ranges from 1 to k+1.

From (cl)n obtained in step 5, a new circulation distribution is calculated from the Kutta-Joukowski theorem and the definition of lift coefficient:




7: The new distribution of Г obtained in step 6 is compared with the values that were initially fed into step 3. If the results from step 6 do not agree with the input to step 3, then a new input is generated. If the previous input to step 3 is designated as Гold and the result of step 6 is designated as Гnew, then the new input to step 3 is determined from

8:

)( oldnewoldinput D

where D is a damping factor for the iterations.

Steps 3 to 7 are repeated a sufficient number of cycles until Гnew and Гold

agree at each spanwise station to within acceptable accuracy.

From the converged Г(y), the lift and induced drag coefficients are obtained. The integrations in these equation can again be carried out by Simpson’s rule.

9:




Lift coefficient versus angle of attack

The numerical lifting-line solution at high angle of attack agrees with the experiment to within 20 percent, and much closer for many cases.

Therefore, such solutions give reasonable preliminary engineering results for the high-angle-of-attack poststall region.




where D is a damping factor for the iterations.

Surface oil flow pattern on a stalled, finite rectangular wing with a Clark Y-14 airfoil section.

At high angle of attack, there is a strong spanwise flow, in combination with mushroom-shaped flow separation regions.

Clearly, the basic assumptions of lifting-line theory, classical or numerical, cannot properly account for such three-dimensional flows.


5.5 The Delta Wing


Supersonic airplanes usually have highly swept wings.

A special case of swept wings is those aircraft with a triangular planform - called delta wings.


5.5 The Delta Wing


Question:

Since delta-winged aircraft are high- speed vehicles, why are we discussing this topic in the present chapter, which deals with the low-speed, incompressible flow over finite wings?

Answer:

All high-speed aircraft fly at low speeds for takeoff and landing;

Moreover, in most cases, these aircraft spend the vast majority of their flight time at subsonic speeds, using their supersonic capability for short “supersonic dashes,” depending on their mission.

Therefore, the low-speed aerodynamics of delta wings has been a subject of much serious study over the past years.


5.5 The Delta Wing


Variants:

There are several variants of the basic delta wing used on modern aircraft:

(a) Simple delta

(b) Cropped delta

(c) Notched delta

(d) Double delta


5.5 The Delta Wing



5.5 The Delta Wing


Leading-edge vortices over the top surface of a delta wing at angle of attack. The vortices are made visible by dye streaks in water flow.


5.5 The Delta Wing


The flow field in the crossflow plane above a delta wing at angle of attack, showing the two primary leading-edge vortices. The vortices are made visible by small air bubbles in water.


5.5 The Delta Wing


Schematic of the spanwise pressure coefficient distribution across a delta wing.

Pressure over the bottom surface:

The spanwise variation of pressure over the bottom surface is essentially constant and higher than the free stream pressure.

Pressure over the top surface:

The spanwise variation in the midsection of the wing is essentially constant and lower than the freestream pressure.

However, near the leading edge the static pressure drops considerably. The leading-edge vortices are literally creating a strong “suction” on the top surface near the leading edge.


5.5 The Delta Wing


The suction effect of the leading-edge vortices enhances the lift; for this reason, the lift coefficient curve for a delta wing exhibits an increase in CL for values of α at which conventional wing planforms would be stalled.

Note the following characteristics:

1. The lift slope is small, on the order of 0.05/degree.

2. The lift continues to increase to large values of α; the stalling angle of attack is on the order of 35°. The net result is a reasonable value of CL,max, on the order of 1.3.


5.5 The Delta Wing


Schematic of the spanwise pressure coefficient distribution over the top of a delta wing as

modified by leading-edge vortex flaps.

Leading edge vortex flap:

The direction of the suction due to the leading-edge vortices is now modified.

Since the pressure is low over this frontal area, the net drag can decrease.


5.5 The Delta Wing


Vortical flow over a 70 degree delta wing at an angle of attack of 30 degrees

Vortex breakdown

The primary vortices begin to fall apart somewhere along the length of the vortex when a delta wing is at a high enough angle of attack.

In summary, the delta wing is a common planform for supersonic aircraft. The low- speed aerodynamics of these wings are quite different from a conventional planform.


5.6 Ludwig Prandtl - Father of modern aerodynamics


Ludwig Prandtl (1875-1953)

Major contributions:1. Thin airfoil theory.

2. Finite-wing theory.

3. Boundary-layer concept.

4. Compressibility corrections.

5. Supersonic shock and expansion-wave theory.

Ludwig Prandtl was a German scientist. He was a pioneer in the development of rigorous systematic mathematical analyses which he used to underlay the science of aerodynamics, which have come to form the basis of the applied science of aeronautical engineering.

Hubert Ludwieg, Hermann Schlichting,

Theodore von Kármán, Reinhold Rudenberg

Notable students:



5.7 Summary

ieff

The wing-tip vortices from a finite wing induce a downwash which reduces the angle of attack effectively seen by a local airfoil section:

In turn, the presence of downwash results in a component of drag defined as the induced drag Di.

Downwash



5.7 Summary

34 rrdl

dV

Vortex sheets and vortex filaments are useful in modeling the aerodynamics of finite wings.

The velocity induced by a directed segment dl of a vortex filament is given by the Biot-Savart law:

Vortex Filament



5.7 Summary

2/

2/0

000

00

)/(4

1)()(

)()(b

bL yydydyd

Vy

ycVyy

In Prandtl’s classical lifting-line theory, the finite wing is replaced by a single spanwise lifting line along which the circulation Г(y) varies.

A system of vortices trails downstream from the lifting line, which induces a downwash at the lifting line.

The circulation distribution is determined from the fundamental equation




5.7 Summary

ARaaa

ARCC

ARC

bw

LiD

Li

/1

2

0

0

2

,

0

Results from classical lifting-line theory:


Elliptic wing:

Downwash is constant:

General wing:

)1)(/(1

1

0

0

22

,

ARaaa

eARC

ARCC LL

iD ）（

aerodynamics chapter 5

Documents