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SECTION 2. AERODYNAMICS OF BODYS OF REVOLUTION THEME 12. THE AERODYNAMIC CHARACTERISTICS OF BODYS OF REVOLUTION, FUSELAGES AND THEIR ANALYSIS 12.1. Lifting force of a body of revolution. According to the theory of an elongated body the factor of pressure on surface ofa body of revolution at flow about it at an angle of attack is determined by the formula . C p = −4α r( x ) cos ϕ , (12.1) .where r( x ) = dr . dx With taking that into account for a lift coefficient (11.9) we obtain 1 lf π lf π lf 8α 4πα ∫ ∫ ∫ ∫ ∫ 2 . .C ya = − r( x ) dx C p cos ϕ dϕ = r r dx cos 2 ϕ dϕ = r r dx . Sm. f . Sm. f . Sm. f . 0 0 0 0 0 Lets consider distribution of a lift coefficient along the length of a body ofrevolution dC ya 4πα 2α dS ( x ) = r ( x ) r( x ) = & , (12.2) dx Sm. f . S m . f . dxwhere S ( x ) = π r 2 ( x ) is the cross-sectional area of a body of revolution. The last formula is more general and fair for any shapes of cross sections. From the obtained expression it follows, that on a fuselage lift occurs only onsites with the variable area of cross sections S ( x ) , at that, the sign of lift is determinedby the sign of derivative dS ( x ) dx . Therefore, on extending nose part positive lifting π 1 cos 2 ϕ dϕ = π ∫ 2 Is used. 0 102
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force occurs, since here dS ( x ) dx > 0 , on the tapering rear part - negative lift, and onthe cylindrical part lift will be absent. Experiments and more precise calculations show, that the above mentionedqualitative analyses remain fair and for not thin fuselages. The quantitative resultsaccording to this theory are satisfactory only for nose and cylindrical parts at M∞ ≤ 1 .For the rear part the theory does not take into account the influence of a boundary layerand flow stall, due to this influence the absolute value of lifting force decreases. Atsupersonic speeds ( M∞ > 1 ) the theory does not take into account influence of noseshape and numbers M ∞ , for cylindrical part - occurrence of lift due to “carry” from thenose part. The lift coefficient C ya of a body of revolution can be presented as a sum of thefactors of lifts of its parts. The lift coefficient is calculated separately for nose,cylindrical and rear parts. For thin fuselages close to body of revolutions, the calculationshould be performed using the theory of an elongated body with consequent refinementof influence of the various factors which are not taken into account by this theory. So, generally it is possible to write down for the fuselage lift coefficient C ya = C α (α − α 0 ) + ΔC ya , yawhere C α = C α nose + C α cil + C α rear . ya ya ya ya (12.3) The size of the derivative C α depends on the shape of the body of revolution and yafirst of all on its nose part, angle of attack α , structure of a boundary layer, numberM ∞ and other factors. 12.1.1. Lift of a nose part. In accordance with the theory the lifting force is distributed according to the lawdC ya 2α dS ( x ) = in subsonic range of speeds ( M ∞ < 1 ). So dx S m . f . dx 102
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l nose Sm. f . dS ( x ) ∫ ∫ dS = 2(1 − ηnose ) ; 2 2 C α nose = ya dx = 2 Sm. f . dx Sm. f . 0 S nose 12 ⎛S ⎞ α C ya nose = 2 ( 1 − ηnose 2 ), η nose d = nose = ⎜ nose ⎟ d m. f . ⎜ Sm. f . ⎟ ⎝ ⎠ . (12.4) At absence of the air intake in the nose part ( η nose = 0 ) C α nose = 2 . ya It has to be noted, that at working engine, when air is sucked through the airintake, an additional air intake lift occurs which should be taken into account inC α nose . ya Approximately this force can be estimated by the formula ya ( C α a .i . = 2ϕ ( 1 − S c .b . ) 1 − S c .b . ηnose , 2 ) (12.5) where S c .b . = Sc .b . S nose is the relative area of the body central part in input cross-section of the air intake ( S c .b . = d c .b . d nose ) (Fig. 12.1), ϕ 2 2 is the flow coefficient of air flow rate (on computational Fig. 12.1. operational mode of the air intake ϕ = 1 ). The value of a derivative C α nose of the lift yacoefficient of the air intake C α a .i . is added to a derivative of the nose part. ya At supersonic speeds of flight the size of the derivative C α nose depends on the yashape of the nose part and aspect ratio (parameter x n = M ∞ − 1 λ nose ) (Fig. 12.2). 2 102
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Fig. 12.2. Influence of the shape of the nose part onto the derivative C α nose ya Examples:- conical nose part without the air intake (w/o a.i.) C α nose ≡ C α nose ya ya w / o a .i . ( = 8 1 − 0 .2 x n exp − xn ) λ2 nose 4 λ nose + 1 2 ; (12.6)- shape of the nose part with curvilinear generative line without the air intake (w/o a.i.) C α nose ≡ C α nose w / o a .i . = 1.65 + 0 .35(1 + 2 x n ) exp − 2 xn ; 2 ya ya (12.7)- at presence of the air intake ( ) + 2ϕ (1 − S )(1 − ηnose ) 1 + 0 .46 x 2 , (12.8) 2 α α C ya nose = C ya nose w / o a .i . 1 − ηnose 2 c .b . S c .b . nwhere x n = M ∞ − 1 λ nose . 2 12.1.2. Lift of the cylindrical part. In subsonic flow ( M ∞ < 1 ) the cylindrical part of a fuselage does not create lift atsmall angles of attack. According to the theory, as on the cylindrical part dS = 0 , thenC α cil = 0 . ya In the supersonic flow ( M ∞ > 1 ) there is a lift on the cylindrical part. It happensbecause of influence of the nose part. At presence of lift on the nose part pressure theyhave various values on its upper and lower parts. These pressures are propagated to the 102
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cylindrical part as disturbances after reflection from a head shock wave. As a result,there is a reduced pressure on the upper surface in comparison with the lower surface ofthe cylindrical part, that causes occurrence of lift on the cylindrical part C α cil ya(Fig. 12.3). (In the subsonic flow disturbances are spread in all directions, therefore theupper surface of the nose part effects both the upper and the lower parts of the cylindersurface. The influence of the lower surface of the nose part is similar. As a result ofmutual influence at M∞ < 1 C α cil = 0 ). ya In general, the size of the derivative C α cil depends on the Mach number, aspect yaratio of the nose part and type of coupling of nose and cylindrical parts (Fig. 12.4, 12.5)C α cil = f ya ( M ∞ − 1 λnose , λcil λnose , type of coupling . 2 ) Intersecting coupling Tangent coupling Fig. 12.3. Distribution of lift along length Fig. 12.4. Types of coupling of nose of the cylindrical part and cylindrical parts Approximately it is possible to estimate size of C α cil by the formula ya ya b ( C α cil = ax n exp − cx n 1 − exp − d xc ), (12.9)where xc = M ∞ − 1 λ cil . 2 102
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The values of factors a , b , c and d can also be adopted as the following:- for conical nose part a = 1.3 , b = 0 .5 , c = 0 .05 , d = 1.29 ;- for the nose with curvilinear generative line and tangent coupling a = 4 .5 , b = 3 .0 ,c = 1.5 , d = 0 .88 . It has to be noted, that the values C α cil are a little bit larger at presence of ya nose cone in comparison with other shapes of noses (Fig. 12.5). Fig. 12.5. 12.1.3. Lift of the rear part. The derivative of the lift coefficient of the rear part of the body of revolution doesnot depend on the shape of the rear part and is determined by the following ratios. In the subsonic flow ( M ∞ < 1 ) distribution of lift along body length according to dC ya 2α dSthe theory of an elongated body = , so dx S m . f . dx S rear C α rear = ya 2 Sm. f . ∫ dS dx dx = −2(1 − S base ) = −2(1 − ηrear ) . 2 (12.10) Sm. f . In real flow (Fig. 12.6) boundary layer δ ∗ rising happens in the rear part due to ∗influence of viscosity, that results in the body thickening S base and decreasing of angleof declination of generative line. 102
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Fig. 12.6. Thickening of the rear part due to the boundary layer As a result, the size of parameter C α rear should decrease on an absolute value. yaThe account of viscosity influence results in the following computational formula ya ( C α rear = −0 .4 1 − η rear . 2 ) (12.11) In the supersonic flow ( M∞ > 1 ) the Mach numbers M ∞ effect the amount of thederivative C α rear and determination of C α rear is performed by the formula ya ya α 1 − η rear 2 M∞ − 1 2 C ya rear = −0 .4 , xr = . (12.12) 1 + 0 .4 x r η rear 2 2 λ rear With increasing of numbers M ∞ the amount of C α rear decreases in an absolute yavalue. It is necessary to note one more effect, which is not taken into account in thetheory of the elongated body. This is an occurrence of the non-linear component on thefuselage due to formation of vortical structures on the upper surface (it is similar to thewing). The values ΔC ya are essential in general size C ya for thin body of revolutions atlarge angles of attack. For fuselages of airplanes the occurrence of the non-linearcomponent, as a rule, is not considered. Also it is necessary to remember, that in thesystem of an airplane the non-linear components from a wing and fuselage decreases. The size of zero lift angle of the fuselage α 0 is determined by chamber of its axiswhich is caused by nose deflection and splayed rear part. The value of α 0 is calculatedby the formula [ ( ) α 0 = 1.25 β nose λ nose λ f + 0 .1β rear λ rear λ f ( )] , (12.13)where β nose is the angle of nose deflection; β rear is the angle of taper of the rear part. The angles β nose and β rear also are taken with positive sign, if the nose isdeflected downwards, and the rear part is tapered upwards. 102
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12.2. Aerodynamic moment of a body of revolution. Coordinate of aerodynamic center. According to the theory of a thin (elongated) body the longitudinal moment isdetermined under the formula l π ∫ x r dx ∫ C p cos ϕ dϕ 2 mz = (12.14) SL 0 0 As the lifting (normal) force was determined for separate parts of a body ofrevolution (for nose, cylindrical and rear parts), and moment characteristics should bealso calculated for parts of body of revolution. 12.2.1. Aerodynamic moment of a nose and coordinate of an aerodynamic center. Lets use the results of the theory of an elongated body, according to which thefactor of pressure on surface of the body of revolution at streamlining under the angle of .attack is determined by the formula (12.1) C p = −4α r cos ϕ . We have l nose π l nose π 8α ∫ ∫ ∫ ∫ 2 .m z nose = x r dx C p cos ϕ dϕ = − x r r dx cos 2 ϕ dϕ , S m . f .l nose S m . f .l nose 0 0 0 0 l nose 4πα ∫ . m z nose =− x r r dx . (12.15) S m . f .l nose 0 l nose ∫ . Lets consider the integral function x r r dx : 0 102
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l nose l nose l nose 2 ∫ ∫ ∫ r dx = . 1 1 x r r dx = x r dr = xr 2 − 2 2 0 2 0 0 1 1 1 ⎛ Wnose ⎞ = l nose rm . f . − 2 Wnose = l nose rm . f . ⎜ 1 − 2 ⎜ ⎟ . 2 2π 2 ⎝ l nose S m . f . ⎟ ⎠ Having accounted it an aerodynamic moment of the nose part m z nose = −2α (1 − W nose ) , (12.16)where W nose = Wnose S m . f .l nose - relative volume of the nose part. Coordinate of the nose aerodynamic center relatively to nose of thebody of revolution in shares of length of the nose partx F nose = x F nose l nose = m z C ya ( = mα C α z ya ) nose :- at absence of the air intake in the nose part x F nose = 1 − W nose . 1 − W nose- at presence of the air intake in the nose part x F nose = . 1 − η nose 2 The obtained formulae can be used at any Mach numbers M ∞ (despite of the factthat the theory of an elongated body was applied which is fair for calculation of thederivative C α nose only at subsonic speeds M ∞ < 1 ). ya For conical nose part 1 2 + η nose x F nose = . 3 1 + η nose For chambered nose part 1 7 + 3η nose x F nose = . 15 1 + η nose l nose ∫ r 2dx = 2 Wnose Is used π 0 102
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It is necessary to note, that for bodies with the parabolic nose part coordinate ofthe aerodynamic center practically does not vary with the increase of Mach numbersM∞ . 12.2.2. Coordinate of the aerodynamic center of the cylindrical part. In the subsonic flow ( M ∞ < 1 ) the lift of the cylindrical part C ya cil = 0 ,therefore the moment characteristics of the cylindrical part are not calculated. In the supersonic flow ( M∞ > 1 ) coordinate of the aerodynamic center x F cil , aswell as the derivative C α cil , depends on Mach number, aspect ratio of the nose and yatype of coupling of nose and cylindrical parts (Fig. 12.7): ( ) x F cil x F cil = = f M ∞ − 1 λnose , λcil λnose , type of coupling . 2 l nose Lets express a coordinate of the aerodynamic center of the cylindrical part x F cil x F cil = in shares of fuselage nose l nose length −1 ⎛ d λ cil ⎞ xn λcil ⎜ x n λ nose ⎟ M∞ − 1 2 x F cil = 1+ − exp − 1⎟ , xn = , (12.17) d λnose ⎜ ⎜ ⎟ λ nose ⎝ ⎠where the factor d value can be accepted as the following ones:- for conical nose part d = 1.29 ;- for the nose with chambered generative line and tangent coupling d = 0 .88 . 102
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M∞ − 1 2 Lets note, that at → ∞ the λ nose coordinate of the aerodynamic center depends only on the attitude of aspect ratios of cylindrical and nose parts of the body of revolution x Fcil → 1 + 0 .5( λ cil λ nose ) . At presence of smooth coupling of the nose and cylindrical parts the aerodynamic center xFcil is located a little bit distant, than Fig. 12.7. Coordinate of the aero- in the case of conical nose and intersecting dynamic center of the cylindrical part coupling. 12.2.3. Coordinate of the aerodynamic center of the rear part. Irrespectively of the shape of the rear part, for any Mach numbers M ∞ coordinateof the aerodynamic center of rear part can be calculated by the formula xFrear = l f − 0 .5 l rear = ( ) (12.18) = l f 1 − 0 .5 λrear λ f , i.e. we accept, that the rear part aerodynamic center is located in its middle. 12.2.4. Coordinate of the aerodynamic center of body of revolution in a whole. Lets consider the configuration of a body of revolution (Fig. 12.8). In this casethe coordinate of the aerodynamic center relatively to the nose is determined asxF = − mα C α : z ya 102
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Fig. 12.8. C α nose xFnose + C α cil xFcil + C α rear xFrear ya ya ya xF = , (12.19) Cα yawhere C α = C α nose + C α cil + C α rear . ya ya ya ya It is necessary to note, that at subsonic speeds ( M ∞ < 1 ) and small angles ofattack, at which C α cil = 0 the aerodynamic center of the body of revolution can be yaplaced ahead of a nose, i.e. xF < 0 . 102
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