SECTION 3. AERODYNAMICS OF THE AIRCRAFT

THEME 15. MUTUAL INFLUENCE OF LIFTING SURFACES AND FUSELAGE

Practically all elements of the aircraft except wing either do not create lift at all or

create it at some flight angles of attack, but this lift is insignificant in comparison with

wing lift. The aircraft drag consists of drag of its separate parts with taking into account

their mutual influence. Let's consider the problem of wing and fuselage interference in details.

15.1. Drag of the wing - fuselage system

The drag of the wing - fuselage system is much more than the sum of separately

taken drags of wing and fuselage. Jointing the wing and the fuselage into the single

whole causes an additional drag. This drag is called the hazard interference. Let's

consider separately profile and wave drag of the wing-fuselage system.

15.1.1. Profile drag

The force of profile drag consists of three items:

137

X p X X Xp p pw f is w iz f+ = + +. . . . , (15.1) where is the profile drag force of an isolated wing composed

of consoles; is the profile drag force of the fuselage;

is the additional drag force caused by an interference.

X q Sp x wis w pis w. . . .= wm f

w

X q Sp xis f pis f. . . . . .= X q Sp x wp=

If the area of a wing with ventral part S is accepted as the characteristic area for

the wing - fuselage system and X q Sp xw f pw f+ += , we shall receive

k S S k Sx d w x w x f d w x wpw f pis w pis f p+ = + +. . . . , (15.2)

where k q qw w= is the coefficient of flow deceleration before the wing (influence of the fuselage), S S S

138

w w= , S S Sf m f= . . is the ratio of the areas of an isolated wing and of fuselage midsection to the characteristic area.

One should notice that profile drag of an isolated wing is calculated at x pis w. .

M M kw w= . In places of the wing and fuselage

jointing their boundary layers are combined

and the thickness of combined boundary layer

will increase. At subsonic speeds it promotes

a flow stalling in the place of wing and

fuselage jointing where "diffuser" effect -the

expansion of jets is observed (Fig. 15.1, a)

and there is positive pressure gradient. The

premature flow stalling is promoted also by

increasing of local angles of attack of wing

cross-sections and also decreasing of critical

numbers M in these cross-sections. All this

results in drag increasing due to friction forces

and pressure forces. To decrease the hazard

interference the so-called fillets (fairings) are

used (Fig. 15.1, b). It is possible to reduced

greatly the hazard interference at correct

selection of fillets.

Fig. 15.1. Diffuser effect in a place of

wing and fuselage jointing:

a) - area of diffuser flow;

b) - fillets at wing and fuselage jointing

For the factor of additional drag we have

k SSx xpww

wp = int , (15.3)

where is the interference factor; generally depends on lifting surface location

(wing, horizontal tail) on a fuselage and shape of fuselage cross section;

kint

Sw is the area of a wing occupied by a fuselage (S b dw b f or S b cw b b 2 ).

The point of view of drag the interference between a wing and fuselage (tail unit

and fuselage) is negative. The researches show, that such interference is the most

unfavorable for low-wing airplane, least unfavorable - for high-wing plane.

It is tentatively possible to assume interference factors which are listed in

table 15.1 irrespectively from numbers .

kint

MTable 15.1. Interference factors

Wing location kint Wing location kint

0 75.

r

0 5.

0 15.

0 4.

0 075.

0

It is possible to consider (taking into account fairings installation), that at thin

wing ( 5% ) installation on a fuselage cylindrical surface by the mid-wing scheme . It is necessary to note, that in some configurations the diffuser effect can be

more at wing installation by the high-wing scheme. It is possible to consider the diffuser

effect as equal to zero at the horizontal tail installation onto vertical according

to Tee-tail unit scheme, but at that it is necessary to displace the mutual position of

maximum thickness of vertical and horizontal tail airfoils.

kint = 0

kint = 0

139

15.1.2. Wave drag.

The interference problem gains special sharpness at configuration of a high-speed

aircraft. At high flight speeds there can appear so-called wave interference, i.e. the

additional wave drag, which is caused by occurrence of shock waves in a place of the

wing and fuselage joint. The unsuccessful joint of wing with fuselage can result in

substantial drag growth, decreasing of critical number and more intense growth of

drag after occurrence of wave crisis.

M

Analogously to profile drag it can be written as

140

Xw X X Xw w ww f is w is f+ = + +. . . . (15.4) and

k S S k Sx d w x w x f d w x www f wis w wis w w+ = + +. . . . . (15.5)

The factor should be determined at xwis w. . M M kw = d w . In particular

( ) f k M tgx d wwis w. . , , 2 2 1= . The additional wave drag occurs as a result of interaction of two flows about the

fuselage and the wing.

C x x xw ww f w= +0 ; ( ) x kw n= 0 25 3 1. exp ; = M MMw*1 , (15.6)

where the factors and also depend on the wing plan form:

, - fuselage - swept wing;

k n

k = 3 n = 1 k = 1 5. , n = 2 - fuselage - delta wing. Number is the critical number M* M of the fuselage - wing system. It will be

less than critical number of an isolated wing and fuselage. It is possible to assume

taking into account the effect of flows interaction.

M*

{M M w f* *. min ,= 0 95 }M*

Effective way to decrease an additional wave drag is using the '' area-rule ". Using

the " area-rule " results into wave drag drop first of all in the zone of transonic speeds

( ) (Fig. 15.2). M M* . ... . 1 15 1 20

Fig. 15.2. Wave drag of fuselage - wing system.

According to "area-rule" the wave drag of the wing - fuselage system is about to

drag of the equivalent body of revolution. Without using the area-rule, this body will

have a bulge in the place of jointing of the wing with the fuselage (Fig. 15.3, a). It is

necessary to make thinner the

fuselage cross section on the value of

wing cross-sections area in the place

of wing jointing to fuselage (Fig.

15.3, b) (or to increase fuselage

cross-section on its remaining part

outside the wing) with the purpose of

the equivalent body of revolution

should have smaller drag.

Fig. 15.3.

There is a generalization of subsonic area-rule to supersonic speeds (supersonic

area-rule).

141

15.2. Lift of the wing - fuselage system

Let's consider a wing - fuselage system put into flow under the angle of attack . For simplification we shall assume, that the fuselage is a body of revolution close to

cylindrical, and the wing is installed on it by the mid-wing scheme with angle . Within borders of the linear theory the general configuration wing - fuselage can

be presented as a sum of two schemes:

Fig. 15.4.

Each scheme contribution in lift is

represented as follows:

142

ya ya ya= + 0 0 where is the lift coefficient of the

fuselage - wing system with a symmetrical

airfoil at a zero setting angle (

ya 0

= 0 , fuselage with a straight-line axis), is the lift

coefficient of the fuselage - wing system with

aerodynamic and geometric twist and with

setting angle (

ya0

0 , fuselage with curved axis).

Fig. 15.5.

Let's consider lift of these schemes.

Scheme" 0 ". Let's write down lift as a sum Y Y Y Y Ya a a a ais f is w w f f w 0 0 0 0 0= + + +. . . . ( ) ( ) , (15.7) where first two items Y and Y refer to an isolated fuselage with a

straight-line axis (in horizontal plane of symmetry) and flat wing with a symmetrical

airfoil; is the additional lift arising on the wing because of fuselage

ais f. .0 ais w. .0

Yaw f( )0

influence; is the additional lift arising on a fuselage because of wing

influence.

Ya f w( )

Obviously, the sum Y Ya ais w w f. . ( ) 0 0+ represents lift of a wing set on the fuselage.

We accept

( )( )

Y Y YY

YY Ka a a

a

aais w w f is w

w f

is wis w. . . .

. .. .

0 0 00

00

1+ = +

=

,

( ) Y Ya aw f is w 0 0 K= . . ,

where and are interference factors for a flat wing with a symmetrical airfoil

and fuselage having a horizontal plane of symmetry (at

K K = 0 Yais w. .0 0=

). Yais f. .0 0=Taking it in account we obtain

( )Y Y Y K Ka a ais f is w 0 0 0= + +. . . . . (15.8) If one passes to lift coefficients

, YY q Sa ya 0 0= q Sa y m fis f ais f. . . . . . 0 0 q Sa y w wis w ais w. . . . 0 0=, Y , =

where S is the characteristic area; is the dynamic pressure before the wing, we shall

receive:

qw

( ) S k K K Sy y f d w y wa ais f ais w 0 0 0= + +. . . . , (15.9) where k q qd w w= is the factor of flow deceleration before a wing (fuselage influence), S Sw w= S , S Sf m f= . . S is the ratio of the isolated wing area and fuselage mid-section area to the characteristic area S . At it lift coefficient of an isolated

wing is determined at yais w. .0 M M kw d= w . Scheme "0 ". Let's write down lift as a sum

( ) ( )Y Y Y Y Ya a a a ais f is w w f f w0 0 0 0 0 = + + +. . . . , (15.10) 143

where first two items Y and Y - characteristic of an isolated fuselage and

wing at

ais f. .0 ais w. .0 = 0 ; lift occurs at the expense of camber of fuselage axis, camber of airfoil,

twist and angle of wing setting onto fuselage; ( )Yaw f 0 is the additional lift arising

on a wing because of fuselage influence; ( )Ya f w 0 is the additional lift arising on a

fuselage because of wing influence.

Here, as well as for scheme " 0 ", the sum ( )Y Ya ais w w f. .0 0 + represents lift

of a wing set on the fuselage.

Let's write down similarly to the previous case

( )( )

Y Y YY

YYa a a

a

aais w w f is w

w f

is wis w. . . .

. .. .0 0 0

0

00

1

+ = +

=

K , (15.11)

where and is the interference factor for the wing - fuselage system at K K = 0 . If, as well as in the previous scheme, we pass to lift coefficients, we shall receive

( ) S k K K Sy y f d w y wa ais f ais w0 0 0 = + +. . . . . (15.12) Finally, for the general scheme " " it is obtained:

( ) ( )

S

k K K S k K K S

y y y y y f

d w y w y w

a a a ais f ais f

ais w ais w

= + = + +

+ + + +

0 0 0 0

0 0

. . . .

. . . .* .

(15.13)

Let's consider isolated elements:

Fuselage: ( ) y y y yais f ais f is f ais f ais f is f. . . . . . . . . . . .= = 0 0 . Obviously y yais f ais f. . . .

0= , and y yais f ais f is f. . . . . .0 0

= .

Wing: ( ) ( y y y yais w ais w is w ais w ais w is w. . . . . . . . . . . .= + = ) 0 0 . Obviously y yais w ais w. . . .

0= ; ( ) y yais w ais w is w. . . . . .0 0 = .

144

Taking it into account, we obtain

( )

( ) ( )( )[ ] S

k K K K K S

y y f

d w y w

a ais f is f

ais w is w

= +

+ + + +. . . .

. . . .

0

0 . (15.14)

Let's consider the separate characteristic of a wing set on fuselage (w/f):

( ) ( )( )[ ]( ) ( )

( )

k K K K K S

k K KK KK K

S

S

y d w y w

d w y w

y w

aw f ais w is w

ais w is w

aw w

/ . . . .

. . . .

= + + +

= + + ++

=

=

0

0

0 ,

=

where yaw and 0w are characteristics of a wing in system with fuselage.

Now we can write in the same type form

( ) ( ) S Sy y f y wa f f aw w= + 0 0 (15.15) And finally for a wing - fuselage system

( ) y ya = 0 ; S Sy y f y w f aw = + ; 0 0 0

1= + S Sy y

f y w

f f w aw,

where a derivative of a fuselage lift coefficient and angle of zero lift are taken for the

isolated fuselage y y f ais f =

. . and 0 0f is f= . . ; the derivative of wing lift

coefficient and angle of zero lift are calculated under the formulas with taking into

account the interference factors ( k K Ky d w yaw ais w ) = . . + and ( ) 0 0w is w K KK K= ++. . .

The note: the shown dependence for the wing - fuselage system remains valid for

a system a horizontal tail-fuselage located on a fuselage ahead of a wing (canard

scheme). 145

146

For normal scheme, when horizontal tail is located behind the wing it is necessary

to take into account a wing influence on horizontal tail in addition to the interference of

horizontal tail and fuselage. In this case horizontal tail is streamlined at smaller angle of

attack equal to . ( ) h t. . = 1 0Let's consider the characteristic of horizontal tail yah t f. . located on the

fuselage similar to the wing characteristic : yaw f/

( ) ( ) k K K K KK K Sy d h t y h t h t h t h t h tah t f ais h t is h t. . . . . . . .. . . . . . . . . . . .= + + ++

0 .

Substituting here ( ) h t. . = 1 0 , we obtain for horizontal tail in the system of aircraft with normal scheme:

( ) Sy y h tah t f ah t h t. . . . . . . .= 0 , where the derivative of lift coefficient of horizontal tail and angle of zero lift are equal

to

( )( ) k K Ky d h t y h tah t ais h t. . . . .. . . . = +1 and

( ) 0 011h t is h th t h tK KK K. . . . .. . . .

= ++

0 ;

h t. . is the angle of the horizontal tail setting relatively to the fuselage axis; as a rule, the symmetrical airfoil is installed on horizontal tail and 0 0is h t. . . = . Obviously, the wing lift in the system of a canard aircraft should also be determined with accounting of

flow downwash located ahead of horizontal tail. Therefore, the last expressions remain

fair and for a wing in the canard scheme after replacement of parameters of horizontal

tail on wing parameters.

Let's consider interference factors , K K , and . Generally they depend on the ratio of fuselage midsection diameter d to wing span with ventral part

K Kf

l , wing shape and fuselage cross-section, wing setting on fuselage altitude and length,

number and influence of the boundary layer. These dependencies are complex and

systematic data about them are absent in the literature. The most essential ones, as

researches show, are the dependencies on

M

dl

f , shown in fig. 15.6.

For an approximate estimation it is

possible to use the following ratios:

( )K K K 1 , K 1, K K 1 ; K K K + = 2

K K K + =

( ) ( )K K K K K + + 1= . The last three equalities are more

exact than the first three. Fig. 15.6. Dependence of interference factors

from dl

f

As we see, calculation of

interference factors is reduced to

definition of the factor . KLet's pass to consideration of mutual influence of a wing and fuselage.

15.2.1. Fuselage influence onto wing (horizontal tail).

Let's consider the wing - fuselage system set in a flow under the angle of attack (a fig. 15.7). Let's assume, that the fuselage is a body of revolution close to cylindrical,

and the wing is located on it under the mid-wing scheme. Let's factor the incoming

undisturbed flow moving with speed V into two components: directed along fuselage

axis with speed V V

x = cos and normal to it with speed V Vy = sin . At flow

147

about fuselage cylindrical part by this transversal flow the speed V is increased in

comparison to V .

y

y

Fig. 15.7. The scheme of flow about wing - fuselage system

Supposing speeds V are small, from the theory of potential flow about cylinder

by transversal flow with speed V V

y

y = sin for local streamlining speeds we obtain

V Vd

zV

d

zy

f f= +

= +

sin 1 4 1 4

2

2

2

2 .

The influence of fuselage onto wing has an effect in changing of the wing angle

of attack, which is equal to w fd

z= +

1 4

2

2 .

At z d f= 2 the wing angle of attack increases twice. So it follows, that for a wing which span with ventral part differs a little from we shall have d f K = 2 . Approximately, for a wing set by the mid-wing scheme K +1 , ( )K +1 , K = where = d lf .

For a wing set on the fuselage of round cross-section by the low-wing or high-

wing configurations, wing distance from fuselage axis influences the interference factor

H

K hh

=

+ +

1 11

2

2 2 , hHd f

= 2 , = dlf .

For such configuration at h= 1 : K < 1 , K < 0 , K < 0 .

148

15.2.2. Wing (horizontal tail) influence onto fuselage

The wing influence on fuselage has double effect.

At first, the lift of the fuselage part is increased by "carry" of raised pressure

under the wing and reduced above the wing on fuselage (Fig. 15.8, a).

Secondly, the lift of fuselage part behind the wing decreases because of presence

flow downwash (Fig. 15.8, b).

Fig. 15.8. Wing influence onto fuselage

As a result the additional lift on the fuselage is equal to

Y Y Ya a af w f w f w( ) ( ) ( )= + . Let's mark, that the position Ya f w( ) depends on number at . With

increasing of applying point

M M > 1M Ya f w( ) displaces back. 15.3. Moment of pitch and position of aerodynamic center

of the wing - fuselage system.

Strictly speaking, the moment of pitch is created by full aerodynamic force or, in

most cases, by normal forces. Approximately it is possible to consider that the moment

of pitch is created by lifting forces of wing, fuselage and additional lifts caused by

interference. At that, we neglect moments of drag forces. Proceeding from mentioned

above we accept: Y , Yf a f ( ) ( )Y Yw f aw f , ( ) ( ) Y Yf w a f w , where Y is the lift of an isolated fuselage, is the lift of wing installed on the fuselage

a f

( )Yaw f

149

150

Yaw f a aw fis w= +. .(Y Y( ) ( ) ), is the additional lift on the fuselage from wing influence.

( )Ya f w

The applying point of wing lift generally places a little ahead of the aerodynamic

center of isolated wing ( ), however, it is possible to accept without large error, that

the applying point of lift coincides with aerodynamic center of isolated wing

.

xFw

( )Yaw fxFw

Using fig. 15.9, we shall make an equation for moment of pitch for wing -

fuselage system relatively to fuselage nose.

Mz

Fig. 15.9.

( ) ( ) ( ) ( )

( ) ( )( )( ) ( ) ( )M M Y x Y x x Y x x

M Y x Y Y x x Y x x

z z a f F aw f w F a f w w F

z a f F aw f a f w w F a f w F F

f w f w

f w f w w

= + + =

= + + 0

0

*

*

.

Here is the moment at zero lift of isolated parts and their interference. Mz0*

. M M M Mz z z zis f is w f w0 0 0 0*

. . . .= + + +The last item in equation for is small, because of small force value and small

arm (especially at ) and it can be neglected. The summarized value

represents wing lift in the system Y

Mz

M < 1( )Y Yaw f a f w+ ( ) aw and then

( )M M Y x Y x xz z a f F aw w Ff w= +0* . Let's pass to the factor of moments, for that we divide the right and the left parts

of equation on dynamic pressure, characteristic area and characteristic length . It

is possible to accept as characteristic length

q SLL , for example, the length of a fuselage:

( )

( ) ( )( )m m C x S C x x S

m C x S C x x S

z z ya F f ya w F w

z ya F f ya w F w

f f w w

f f f w w w

= + == +

0

0 0 0

*

* . (15.16)

Here

m k m S m Sz d w z w z fis w is f0 0 0*

. . . .= + , C Cya yaf is f

151

=. .

, 0 0f is f= . . ,

( )C k C K Kya d w yaw is w = +. . , ( ) 0 0w is ww K KK K= ++. . , location of the wing relatively to the fuselage nose, the position of aerodynamic centers

of the fuselage and wing are expressed in shares of fuselage length x xw w L= , x x LF Fw w= , x x LF Ff f= .

Let's differentiate the equation (15.16) for on mz : ( )m C x S C x x Sz ya F f ya w Ff f w w w = + .

The position of aerodynamic center of the wing - fuselage system relatively to the

fuselage nose in shares of fuselage length are determined

x x L m m CF F zC

z yya= = = a .

It is accepted for flight dynamics

problems to determine the position of

aerodynamic center relatively to mean

aerodynamic chord nose in shares of MAC

length (Fig. 15.10): bA

Fig. 15.10.

( )[ ]x xL

FF

AA= = x x x L

bF w A

A

+.

Substituting in expression for (15.16) parameter mz = 0 , we obtain the moment factor at zero lift

( ) ( )( )

( )m m C x S C x x S

m m C x S C x x S

z z ya F f ya w F w

z z ya F f ya w F w

f f f w w w

f f f w w w

= += + + + +

0

0

0 0

0 0 0

*

*

.=

As 0 0 01= + C C S C Sya ya

f ya wf f w w and m x Cz F ya

= , then

, m m m mz z z zf w0 0 0 0= + +*

where ( )m x x C S [

152

z F F ya ff f f f0 0= + , ( )]m x x x C Sz F w F ya ww w w w0 0= + .

15.4. Influence of aircraft configuration onto its aerodynamic

characteristics

The wing setting relatively to fuselage essentially influences the aerodynamic,

weight, operational and economic characteristics of the aircraft.

Let's consider the main virtues and shortages of all the schemes of mutual

arrangement of a wing and fuselage.

The aircraft made by the low-wing configuration has the following advantages: it

allows to receive the improved take-off and landing characteristics due to more effective

influence of ground proximity, and, also, capability of increasing of high-lift devices

area due to ventral wing part; the design of glider and service of aggregates installed on

the wing become simpler.

Imperfections of this configuration are the following: decreases due to

interference of wing and fuselage, C increases, the lift-to-drag ratio decreases, the

falling of foreign objects is possible at takeoff and landing at engines installation on the

wing.

Cya max

xa

The advantages of high-wing configuration are: lift-to-drag ratio is higher on

than for low-wing configuration with other conditions being equal due to

decreasing of aerodynamic interference; the characteristics of longitudinal stability at

4 5%K

high angles of attack are better in comparison with the characteristics of the low-wing

configuration; in case of engines installation on the wing, the probability of outside

objects falling from runway surface essentially decreases, that allows more successful

use of high-wing aircraft on unpaved runways.

The disadvantages of the configuration are worsening of lateral stability

characteristics at high angles of attack, when vertical tail falls into the wake jet from the

wing that demands necessity of vertical tail area increasing on in comparison

with vertical tail area of low-wing configuration.

30 50%K

Weight of high-wing construction is more, than low wing one. In particular,

weight of a heavy high-wing-airplane with the main landing gears fixed on the fuselage,

exceeds the weight of low-wing airplane on 15 . 20%KThe operating economy of passenger high-wing aircraft is a little bit lower, than

of low-wing one because of losses in weight.

The midwing airplane, from a point of view of an aerodynamic interference, has

the greatest advantages. However, this scheme is not applied to the aircraft of mean

seating capacity because the wing should pass through the passenger compartment,

forming ledges or cavities on the compartment floor. The midwing configuration is used

on some wide-fuselage aircraft (airbuses), which wing passes not through the passenger

compartment, but in space between the upper and lower decks.

Now the majority of passenger aircraft has the low-wing configuration. For cargo

and cargo-passenger aircraft the high-wing configuration is more preferable in many

cases.

153

SECTION 3. AERODYNAMICS OF THE AIRCRAFT THEME 15. MUTUAL INFLUENCE OF LIFTING SURFACES AND FUSELAGE 15.1. Drag of the wing - fuselage system 15.1.1. Profile drag 15.1.2. Wave drag. 15.2. Lift of the wing - fuselage system 15.2.1. Fuselage influence onto wing (horizontal tail). 15.2.2. Wing (horizontal tail) influence onto fuselage 15.3. Moment of pitch and position of aerodynamic center of the wing - fuselage system. 15.4. Influence of aircraft configuration onto its aerodynamic characteristics