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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 onehand, the drag can be presented as pressure drag and friction drag. The pressure drag iscaused by forces of pressure which act along perpendicular to the body surface. Thefriction 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 nojet 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 withlifting force, and induced drag connected with presence of lift. In this case, the dragcoefficient 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 isdetermined similarly to low-aspect-ratio wing 2 1 C xi = AC ya , A = − CF (13.2) Cα yawhere C F is the relative factor of sucking force, if to neglect it, then the factor of a 1polar 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 thefactor of base drag. At subsonic speeds the drag consists of drag of friction in 75 ...80% and drag ofpressure in 25 ...15% . At transonic and supersonic speeds M∗Ф < M ∞ < 3 the drag ofpressure in 1.5 ... 2 times exceeds drag of friction (due to occurrence of wave drag). 114
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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 ratioand 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 isdetermined by the known distribution of the factor of pressure along lateral surface ofthe 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 ofpressure of the nose C x nose and rear C x rear parts (the cylindrical part does not createthe 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 dragincludes 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 anadditional wave drag of the fuselage can be: canopy, lateral and ventral air intakes,coupled nozzles in the rear part and so on. 115
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13.1.1. Wave drag of a nose part The nose drag of pressure C x nose of the body of revolution substantially dependson 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 simpleengineering method exists it is the method of local cones (Fig. 13.2). Fig. 13.2. Lets 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 constantvalue and is determined by the ratio 0 .19 sin 2β cone C pcone = 2 .09 sin 2 β cone + , (13.8) 2 M∞ −1from here 116
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.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 pressureС p integrating its analytically or numerically. In particular, for the conical shape ofnose: 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∞ − 1For 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 hasbluntness or air intake, its drag coefficient C x nose varies in comparison with fuselagewithout bluntness or air intake. At M ∞ > M∗ bluntness, as a rule, increases drag of thehead part. Besides, flow rate coefficient through the air intake provides the essentialinfluence onto character of flow about nose of the body of revolution with a channel. Inthis case, drag of the fuselage is increased to some size called additional drag of the airintake. 13.1.2. Wave drag of the rear part The fuselage rear parts have tapering in many cases. The reduced pressure isestablished on tapered rear parts at supersonic speeds. The factor of wave drag of the 117
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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 moreless, 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 cylindricalpart 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 ofundisturbed flow. At λ cil > 3 it is possible to consider, that the rear part is streamlined byundisturbed flow and calculation of the wave drag C x rear can be made irrespectively ofon what body it is located. Calculation of the wave drag factor for conical rear part is performed by theformula ⎡ 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 dependon the shape and parameters of outflowing jet. The jet extending at M ∞ > 1 causespressure increase near the rear part due to flow deceleration in rear shock waves. Itpromotes decreasing of C x rear . 118
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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 whichdistinguish 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 incompressiblefluid flow at identical with the specified fuselage Reynolds number Re and coordinateof 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 pressureforces 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 ofmidsection can be approximately calculated by the formula 119
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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 additionaldrag factors ∑ ΔC x р . The increment of the factor of fuselage profile drag caused by tail fairing ortapered 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 streamξ = 1 . The factor k depends on the shape of rear part generative line: for an ellipsek = 7 , for other curves (particular case - hemisphere) k = 3 . If tapering of the rear partη 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* , 2a = 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 dragto 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. 120
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13.3. Base drag The base drag is caused by flow stall behind the blunt base. Thus the value ofrarefaction in the stagnant zone behind the blunt base depends on some factors: the rearpart 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 bluntbase causes pressure reducing. The level of pressure reducing depends on structure andthickness of the boundary layer. Increase of the boundary layer thickness reduces gasejection 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 propertiesstreamlining the blunt base, in the supersonic flow ( M ∞ > 1 ) the additional rarefactiontakes 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 thefactor 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 whichdiffers from C x base , the following formula is offered max 121
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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 theformula 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 itsthickness δ . Obviously, the more the thickness of the boundary layer δ is, then the lessthe 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 oftransported freight, requirements of aerodynamics and operation etc. The body of revolution of the perfectly streamlined shape should be chosen forfuselage. The changes of surface chamber should be small and smooth, as the fracturesincrease drag. The smoothness of the shape should not be broken by design juts, as thedrag is increased also due to mutual influence of body parts. 122
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It is possible to decrease fuselage drag and relay appearance of shock waveshaving given the laminar shape to fuselage, at which the maximum thickness displacesto 0 .4 K0 .5 shares of chord lengths, having created smooth contours, ideal smoothsurface, at that M ∗ increases up to values 0 .8 K0 .9 . The main part of fuselages drag of subsonic planes is the friction drag, thereforedesigners 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. Attransonic speeds ( M ∞ ≈ 0 .9 ) there is a wave drag on a fuselage, therefore it is moreexpedient 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 provisionof required angles of attack of an airplane while takeoff and landing. For transportairplanes 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 areused for drag decreasing of the rear part. These ribs allow to reduce fuselage drag to10 K15% and to increase lift-to-drag ratio approximately by unit during cruising modeof flight. 123
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