1. SECTION 2. AERODYNAMICS OF BODY OF REVOLUTION
THEME 13. DRAG COEFFICIENT OF A FUSELAGE
Drag of body of revolution can be presented as several components. On the one
hand, the drag can be presented as pressure drag and friction drag. The pressure drag is
caused by forces of pressure which act along perpendicular to the body surface. The
friction drag is caused by forces which act along tangent to the body surface.
At flow about the body of revolution with a blunt base, behind which there is no
jet stream from the engine, the drag from pressure on the blunt base occurs in addition.
On the other hand, the drag is divided into drag which is not connected with
lifting force, and induced drag connected with presence of lift. In this case, the drag
coefficient at small angle of attack can be presented as
C xa = C x 0 + C xi , (13.1)
where C x0 is the drag coefficient at zero lift, C xi is the induced drag, which is
determined similarly to low-aspect-ratio wing
2 1
C xi = AC ya , A = − CF (13.2)
Cα
ya
where C F is the relative factor of sucking force, if to neglect it, then the factor of a
1
polar pull-off will be equal A = .
Cα
ya
Generally drag coefficient at zero lift is equal
C x 0 = C xp + C xw + C x base , (13.3)
where C xp is the factor of profile drag; C xw is the factor of a wave drag; C x base is the
factor of base drag.
At subsonic speeds the drag consists of drag of friction in 75 ...80% and drag of
pressure in 25 ...15% . At transonic and supersonic speeds M∗Ф < M ∞ < 3 the drag of
pressure in 1.5 ... 2 times exceeds drag of friction (due to occurrence of wave drag).
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2. At M∞ < M* ф the wave drag is equal to zero C xw = 0 .
The critical Mach number of the body of revolution depends on its aspect ratio
and aspect ratio of its nose
1
M* ф = 1 − . (13.4)
λ f + 2λ nose
13.1. Wave drag
At first we shall consider a wave drag. This is drag of pressure, therefore it is
determined by the known distribution of the factor of pressure along lateral surface of
the body of revolution
lf
2π
C xw =
Sm. f . ∫ С p r r& dx . (13.5)
0
It is convenient to present the factor of wave drag as a sum of drag coefficients of
pressure of the nose C x nose and rear C x rear parts (the cylindrical part does not create
the wave drag)
C xw = C x nose + C x rear (13.6)
If considered body differs from a body of revolution then the factor of wave drag
includes an additional addend
C xw = C x nose + C x rear + ∑ ΔC xw , (13.7)
where ∑ ΔC xw is the` sum of wave drags of various sources. Such sources of an
additional wave drag of the fuselage can be: canopy, lateral and ventral air intakes,
coupled nozzles in the rear part and so on.
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3. 13.1.1. Wave drag of a nose part
The nose drag of pressure C x nose of the body of revolution substantially depends
on flow mode.
At subsonic speeds there is a reduced
pressure on some sites of a surface, owing to
that the sucking force can appear, and the
drag can be negative. At supersonic speeds
pressure is increased on the nose surface, due
to that drag of pressure appears (Fig. 13.1).
Fig. 13.1. Drag of pressure of a nose of For calculation of the wave drag
the body of revolution C x nose at M ∞ ≥ 1.20 . ..1.25 simple
engineering method exists it is the method of local cones (Fig. 13.2).
Fig. 13.2.
Let's write down ratios
( )
C p i ( M ∞ , ϑ i ) = C pcone M ∞ , β cone , β cone = ϑ i , tgϑ i = r( x ) .
&
The factor of pressure on the ray which is going out from cone top has constant
value and is determined by the ratio
0 .19 sin 2β cone
C pcone = 2 .09 sin 2 β cone + , (13.8)
2
M∞ −1
from here
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4. .2 .
r ( x) 0 .38 r( x )
C p = 2 .09 .2 + .2 . (13.9)
1 + r ( x) M∞ − 1 1 + r ( x)
2
It is possible to define size of C x nose by known value of the factor of pressure
С p integrating its analytically or numerically. In particular, for the conical shape of
nose:
lcone
2π .
C xcone = C
Sm . f . pcone ∫ r r dx = C pcone ;
0
0 .19 sin 2 β0
C x nose = 2 .09 sin 2 β0 + . (13.10)
2
M∞ − 1
For the parabolic nose:
⎛ ⎞
C x nose =
1 ⎜ 2 .30 + 0 .83λ nose ⎟ . (13.11)
2
4 λnose + 1⎜
⎝ M∞ − 1 ⎟
2
⎠
Generally C x nose will depend on the nose shape. If the head fuselage part has
bluntness or air intake, its drag coefficient C x nose varies in comparison with fuselage
without bluntness or air intake. At M ∞ > M∗ bluntness, as a rule, increases drag of the
head part. Besides, flow rate coefficient through the air intake provides the essential
influence onto character of flow about nose of the body of revolution with a channel. In
this case, drag of the fuselage is increased to some size called additional drag of the air
intake.
13.1.2. Wave drag of the rear part
The fuselage rear parts have tapering in many cases. The reduced pressure is
established on tapered rear parts at supersonic speeds. The factor of wave drag of the
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5. fuselage rear part depends on the shapes of outlines, its tapering and aspect ratio,
number M ∞ , and also on the aspect ratio of the fuselage cylindrical part λ cil . It is more
less, when its aspect ratio and number M ∞ are more.
We shall mark, that wave drag C x rear will depend on aspect ratio of a cylindrical
part λ cil : for the fuselage with a short cylindrical part ( λ cil ≤ 3 ), because, in this case,
flow before the rear part will not have time to become uniform and to accept values of
undisturbed flow.
At λ cil > 3 it is possible to consider, that the rear part is streamlined by
undisturbed flow and calculation of the wave drag C x rear can be made irrespectively of
on what body it is located.
Calculation of the wave drag factor for conical rear part is performed by the
formula
⎡ 0 .76 λrear ⎤ (1 − ηrear ) 1 − ηrear
2
C x rear = ⎢ 2 .09(1 − ηrear ) + ⎥ . (13.12)
⎢ 2
M∞ − 1 ⎦ ⎥ 4 λrear + (1 − ηrear ) 2
2
⎣
For the rear part with any generative lines (close by shape to parabola)
(1 − η )
2 2
2
rear M∞ − 1
C x rear = , xr = . (13.13)
λ2
rear [1 + 0 .5(1 + η rear ) xr ] + ( xrη rear )
2 2 λ rear
In case of the pointed rear part it is necessary to accept η rear = 0 in the formulae
(13.12) and (13.13).
If the engine is installed in the fuselage rear part, the factors C x rear will depend
on the shape and parameters of outflowing jet. The jet extending at M ∞ > 1 causes
pressure increase near the rear part due to flow deceleration in rear shock waves. It
promotes decreasing of C x rear .
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6. 13.2. Fuselage profile drag
Fuselage profile drag is considered as drag of an equivalent body of revolution.
The amendments are entered for the account of fuselage design features which
distinguish it from the body of revolution.
Factor of fuselage profile drag
Cx р = Cx р
b .r .
+ ∑ ΔC x р
(13.14)
where C x р is the profile drag of an equivalent body of revolution, which is
b .r .
determined as follows:
⎛F ⎞
Cx р = C f η λ η м ⎜ l .s . ⎟, (13.15)
b .r . ⎝ S m . f .⎠
where C f is the friction drag coefficient of one side of a flat plate in an incompressible
fluid flow at identical with the specified fuselage Reynolds number Re and coordinate
of point in which laminar boundary layer becomes turbulent x t .
If we accept, that the fuselage is streamlined by completely turbulent flow
( x t = 0 ), that a little bit overestimates the drag, then
0 .087
Cf = . (13.16)
( lg Re − 1.6 ) 2
The number Re is calculated on fuselage length l f and flight parameters V∞
and H :
[ ]
V∞ l f ⎛ H ⎞
Re = = M ∞ l f f ( H ) , f ( H ) = 2 .33⎜ 1 − + 0 .00187 H 2 ⎟ 107 , 1 , H [ km] .
ϑ∞ ⎝ 12 ⎠ m
The factors η λ and η м in the formula (13.15) define the contribution of pressure
forces and compressibility effect in fuselage profile drag:
1 1.5 1 1
ηλ = 1 + + or ηλ = 1 + , ηм = . (13.17)
λf λ2
f 2λ f 2
1 + 0 .2 M ∞
The ratio of the area of the lateral (wetted) fuselage surface to the area of
midsection can be approximately calculated by the formula
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7. Fl .s . ⎡ λ λ ⎤
≈ 4 λ ⎢ 1 − 0 .2 nose − 0 .3 rear ⎥ . (13.18)
Sm. f . ⎢
⎣ λf λf ⎥ ⎦
The account of fuselage design features is carried out by summing of additional
drag factors ∑ ΔC x р
.
The increment of the factor of fuselage profile drag caused by tail fairing or
tapered rear part at subsonic speeds of flight is calculated by the formula
⎡ ⎛ ⎞ ⎤
ΔC x р =
0 .029
Cx р
⎢0 .2⎜ 1 +
⎢ ⎝
⎜
4
2 ⎟
1 + kλrear ⎠
3
( 3
⎟ 1 − ηrear + ξηrear ⎥ ,
⎥
) (13.19)
b .r . ⎣ ⎦
If the jet stream outflows from the blunt base then ξ = 0 , at absence of a jet stream
ξ = 1 . The factor k depends on the shape of rear part generative line: for an ellipse
k = 7 , for other curves (particular case - hemisphere) k = 3 . If tapering of the rear part
η rear = 0 then the value of the factor k = 7 .
Beveled or bended rear part causes an additional drag
(β )
a 3
o
ΔC x р = tg 2
rear (13.20)
Cx р
b .r .
where β rear is the angle of deflection of the rear part mean line, a = 0 .04 at M ∞ ≤ M* ,
2
a = 0 .04 M∞ at M ∞ ≥ 1.1 .
The influence of canopy is estimated by such values:
- for a passenger or transport airplane ΔC x p = 0 .038 λf ;
- for a maneuverable airplane ΔC x p = 0 .042 Scocpit S m . f . ;( )
- the fairings of main landing gears located on the lateral fuselage surface increase drag
to size ΔC x p = 0 .08C x b .r . ;
- side or ventral air intakes increase drag to size ΔC x p = 0 .085 Sa .i . S m . f . , where ( )
S a .i . is the summarized area of all air intakes.
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8. 13.3. Base drag
The base drag is caused by flow stall behind the blunt base. Thus the value of
rarefaction in the stagnant zone behind the blunt base depends on some factors: the rear
part shape, presence or absence of a jet stream, geometric characteristics, flow mode,
boundary layer status etc. Friction between an external flow and flow behind the blunt
base causes pressure reducing. The level of pressure reducing depends on structure and
thickness of the boundary layer. Increase of the boundary layer thickness reduces gas
ejection in stagnant area, reduces rarefaction and increases the factor of base drag.
In the subsonic flow the base drag occurs as a result of air ejection properties
streamlining the blunt base, in the supersonic flow ( M ∞ > 1 ) the additional rarefaction
takes place from expansion of the supersonic flow.
Fig. 13.3. Subsonic and supersonic flow about the blunt base
The greatest size of base drag X base will be at pbase = 0 (vacuum). Then the
factor of pressure on the blunt base will be equal
pbase − p∞ 2 p∞ 2 a∞ 1.43
C p base = =− =− ≈− 2 (13.21)
q∞ ρ∞V∞ 2 γ V∞
2
M∞
and as C x base = −C pbase S base , then
1.43
C x base = S base . (13.22)
max 2
M∞
In the supersonic flow ( M ∞ > 1 ) for calculation of factor of base drag which
differs from C x base , the following formula is offered
max
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9. 1.43
C x base = ξ bξ η C x base = ξ bξ η 2
S base , (13.23)
max M∞
where the factor ξ b takes into account the influence of the boundary layer, ξη -
tapering of the rear part:
ξb =
(
3 − p base
*
) *
, − p base =
0 .029
S12 .
( )
2 base
* Cx p
1 + 10 − p base f
In the subsonic flow ( M ∞ < 1 ) the factor of base drag can be defined by the
formula
0 .029 32
C x base = S base . (13.24)
Cx p
f
The ejection effect depends on the boundary layer status, in particular on its
thickness δ . Obviously, the more the thickness of the boundary layer δ is, then the less
the suction and C x base are.
Fig. 13.4. Function of base drag on Mach numbers
The fuselage shape is determined by airplane assignment, type and weight of
transported freight, requirements of aerodynamics and operation etc.
The body of revolution of the perfectly streamlined shape should be chosen for
fuselage. The changes of surface chamber should be small and smooth, as the fractures
increase drag. The smoothness of the shape should not be broken by design juts, as the
drag is increased also due to mutual influence of body parts.
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10. It is possible to decrease fuselage drag and relay appearance of shock waves
having given the laminar shape to fuselage, at which the maximum thickness displaces
to 0 .4 K0 .5 shares of chord lengths, having created smooth contours, ideal smooth
surface, at that M ∗ increases up to values 0 .8 K0 .9 .
The main part of fuselages drag of subsonic planes is the friction drag, therefore
designers try to give them the shape with minimal surface. The modern subsonic planes
( M ∞ < 0 .7 ) have fuselages with optimum aspect ratio λ f = 7 K9 and rounded nose. At
transonic speeds ( M ∞ ≈ 0 .9 ) there is a wave drag on a fuselage, therefore it is more
expedient to use fuselages with high aspect ratio λ f = 10 K13 and more pointed nose.
The tail unit of passenger plane fuselages is usually a little elevated for provision
of required angles of attack of an airplane while takeoff and landing. For transport
airplanes the tail unit is beveled and is even more elevated for freights loading.
Therefore, the fuselage aerodynamic characteristics of transport plane are usually worse,
than of passenger one. Special ribs installed along fuselage near the back doorway are
used for drag decreasing of the rear part. These ribs allow to reduce fuselage drag to
10 K15% and to increase lift-to-drag ratio approximately by unit during cruising mode
of flight.
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