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SECTION 1. AERODYNAMICS OF LIFTING SURFACES TOPIC 5. THE AERODYNAMIC CHARACTERISTICS OF WINGS IN GAS SUBSONIC FLOW As it was spoken earlier, aerodynamics of subsonic speeds is limited by Machnumbers 0 .4 ≤ M ∞ ≤ M* . Potential of speeds of disturb flow satisfies the linearized equation of gasdynamics in a linearized subsonic flow (1 − M )ϕ2 ∞ xx + ϕ yy + ϕ zz = 0 . (5.1) Lets pass to new variables, as to receive the Laplas equation for speed potentialϕ in incompressible fluid. Then, assuming flow parameters and aerodynamiccharacteristics in incompressible fluid as known, we shall connect them to the requiredcharacteristics in a subsonic gas flow. The passage to incompressible flow is performed by replacement of coordinates: ⎧x = x 1 − M 2 ⎪ n ∞ ⎪ ⎨ y = yn (5.2) ⎪z = z ⎪ ⎩ nwhere variables with an index n correspond to coordinates in incompressible gas flow.At that ϕ ( x , y , z ) → ϕ ( x n , y n , z n ) and ϕ x n x n + ϕ y n y n + ϕ z n z n = 0 . 2 ∂ϕ For the pressure factor in incompressible flow we have С pn = − and V∞ ∂ x n 2 ∂ϕ 2 ∂ ϕ d xn 1Сp = − =− = С pn . Finally we receive: V∞ ∂ x V∞ ∂ x n d x 2 1 − M∞ С p incompr Сp = . (5.3) 2 1 − M∞ Therefore, the aerodynamic characteristics connected to pressure forces aredetermined analogously: 48
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С ya n mz n С xi n С ya = , mz = , С xi = . (5.4) 2 2 2 1− M∞ 1− M∞ 1− M∞ It is necessary to emphasize that the characteristics С ya n , m z n and С xi ncorrespond to the transformed wing (Fig. 5.1) at passage to incompressible flow, i.e. toa wing with geometry λ n , χ n and η n . Lets define these parameters: n b0 n bw .t . l n = l , b0 = , bw .t . = , 2 2 1− M∞ 1− M∞ S n x0 Sn = , x0 = , ηn =η, 2 2 1− M∞ 1− M∞ 2 λ n = λ 1 − M ∞ , - reduced aspect ratio, n tgχ 0 n tgχ 0 = , λtgχ 0 = λ n tgχ 0 2 1 − M∞ Now it is possible to obtain the aerodynamic characteristics of the specified wing in subsonic flow, knowing the aerodynamic characteristics of the transformed wing in a flow of incompressible fluid. Lets notice, that the we consider linear Fig. 5.1. a) - initial wing; components of C yа and mz . Therefore it is possible to write C α and mα instead of C yа b) - transformed wing. yа zand mz in the formulae (5.4). 5.1. Wing lift coefficient In general case C yа = C yа лин + ΔC yа = C α (α − α0 ) + ΔC yа . yа At small angles of attack α and M ∞ < M∗ the lift coefficient of a wing isdetermined by equality 49
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C yа = C α (α − α 0 ) . yа (5.5)In compliance with the linear theory the lift coefficient C yа in compressed gas is С ya ndetermined by the formula С ya = 2 yа ( , where C yа n = C α n α − α 0 n is the lift ) 1 − M∞coefficient of the deformed wing in incompressible gas. It follows from the last ratios,that α Cα n yа C yа = , α0 = α0 n . (5.6) 2 1 − M∞ 5.1.1. High-aspect-ratio wings π λn For incompressible fluid we have C α n = yа , (1 − τ n ) + (1 − τ n ) 2 + (π mn ) 2 λn 2where mn = , (refer to section 4.1.1). As λ n = λ 1 − M ∞ and C α ∞ cos χ 0 .5 yа n 1 λ 1 − M ∞ + tg 2 χ 0 .5 2 ncos χ 0 .5 = , then mn = . Now C α should be yа 1 + tg 2 χ 0 .5 n Cα n∞ yа πλdetermined as Cα = yа , where (1 − τ n ) + (1 − τ n ) 2 + (π mn ) 2 λ 1 − M ∞ + tg 2 χ 0 .5 2mn = . Cα n∞ yа If one assumes that χ 0 .5 = 0 and λ → ∞ , then we get a result of a thin airfoil α Cα , where C α n = C α ∞ = 2π − 1,69 4 c . yаntheory C yа = yа yа 2 1− M∞ ( ) Parameter τ n = f λ n , χ n ,η n = f (λ , χ ,η ,M ∞ ) . 50
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mn Approximately τ n = τ 1 n ( mn ) ⋅ τ 2 n (η ) , where τ 1n ( mn ) = 0 .17 and π 1τ 2 n (η ) = η 2 + . (5η + 1) 3 As non-linear lift component is absent for a wing of high aspect ratio ΔC yа = 0 ,then C yа = C α (α − α0 ) . yа 5.1.2. Low aspect ratio wings It is possible to use the formula for determination of a derivative of a liftcoefficient by angle of attack C α : yа α C α ∞ λn ya C α ∞ λ 1 − M∞ ya 2 C yа n = = , (5.7) pn λ n + 2 pn λ 1 − 2 M∞ +2where pn is the parameter determined for a transformed wing. Particularly, we have for a tapered wing: 2 pn = 0 ,5 ⎛ 1 + tg 2 χ lne . + 1 + tg 2 χ tn.e . ⎞ + ⎜ ⎟ = ⎝ ⎠ λ n (η n + 1) . ⎧ ⎪ ⎜ 2 ⎫ ⎪ 1 = ⎨0 ,5 ⎛ 1 − M ∞ + tg 2 χ l .e . + 1 − M ∞ + tg 2 χ t .e . ⎞ + 2 2 ⎟ ⎬ . ⎪ ⎝ ⎠ λ (η + 1) ⎪ 2 ⎩ ⎭ 1 − M∞ α Cα ∞ λ yа Finally we shall write down the expression C yа = ~ , pλ + 2where ~ = pn 1 − M ∞ , p 2 ~ = 0 ,5 p ( 1 − M ∞ + tg 2 χ l .e . + 1 − M ∞ + tg 2 χ t .e . + 2 2 ) 2 λ ( η + 1) . The non-linear additive is calculated under the formula: 8 1 − M ∞ cos 2 χ l .e . (α − α 0 ) . 2 2 ΔC yа = (5.8) pλ + 2 51
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It is necessary to note, that the conversion linear theory can not be used forconnection between compressed and incompressible flows while calculating the non-linear additive ΔC yа . Thus: C yа = C α (α − α0 ) + ΔC yа . yа 5.1.3. Extreme small-aspect-ratio wings ( λ < 1 ). α πλ n πλ 2 α Cα n yа πλ In such case we have C yа n = = 1− M∞ and C yа = = , 2 2 1− 2 M∞ 2 πλtherefore derivative C α = yа does not depend on Mach numbers M ∞ . 2 The non-linear additive ΔC yа is determined by the above mentioned formula(5.8). Note: it is possible to notice in the above mentioned formulae for C α , that the yаratio C α λ is a function of parameters λ 1 − M∞ , λ tgχ 0 .5 and η . yа 2 2 Parameter λ 1 − M∞ is the reduced aspect ratio. These parameters can be considered as parameters of similarity and used forcreation of the diagrams. So C α λ = f ⎛ λ 1 − M ∞ , λ tgχ 0 .5 , η ⎞ . ⎜ 2 ⎟ ya ⎝ ⎠ The analysis of wing lift in a subsonic gas flow: 1. The angle of zero lift α 0 does not depend on numbers M ∞ (Fig. 5.2). 2. The derivative C α grows with increasing of numbers M ∞ (Fig. 5.2). ya 3. The effect of a compressibility (influence of Mach numbers M ∞ onto thederivative C α ) decreases with reduction of λ (it is the reason of spatial flow of low yaaspect ratio wings) (Fig. 5.3). 4. The non-linear component ΔC ya decreases with increasing of numbers M ∞ . 52
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Fig. 5.2. Influence of number M ∞ Fig. 5.3. Influence of wing aspect ratio λ onto onto dependence C ya = f ( α ) compressibility effect 5. The effect of a compressibility decreases with increasing of sweep angle χ(Fig. 5.4). The reason of it is as follows: with rising up of sweep angle χ , the speedscomponent normal to the leading edge from which depends the characteristic C yabecomes less,. 6. The value of C ya max decreases with increasing of numbers M ∞ (the reason ismore earlier flow stalling) (Fig. 5.5). For example, for the airplane JaK − 40 : atM ∞ = 0 .2 - C ya max = 1.44 , and at M ∞ = 0 .6 - C ya max = 1.1 .Fig. 5.4. Influence of a sweep angle χ onto Fig. 5.5. Influence of number M ∞ effect of compressibility at λ = const onto dependence C ya = f ( α ) 53
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Using parameters of similarity the dependence ⎜ 2 ⎞C α λ = f ⎛ λ 1 − M ∞ , λ tgχ 0 .5 , η ⎟ looks like it is shown in fig. 5.6. ya ⎝ ⎠ Fig. 5.6. Dependence of a factor C α λ on parameters of similarity ya 5.2. Induced drag. The wing induced drag coefficient taking into account of compressibility is equal C xi n C xi = , (5.9) 2 1− M∞ 2Where C xi n = AnC ya n is the induced drag of the “deformed” wing in incompressible 2gas flow; C ya n = C ya 1 − M ∞ ; the polar pull-off factor is equal to: ⎧ 1 + δn 1 + δn ⎪ πλ = 2 − for high - aspect - ratio wing; ⎪ n π λ 1 − M∞ ⎪ 1 1 An = ⎨ = − for low - aspect - ratio wing. ⎪ Cα n Cα 1 − M 2 ⎪ ya ya ∞ ⎪. ⎩ 2 2 After substitution C xi n = AnC ya n , C ya n = C ya 1 − M ∞ and An into (5.9) wereceive again 54
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2 C xi = AC ya , (5.10) 1 + δn 1where А = - for high-aspect-ratio wings, А = α - for low-aspect-ratio wings. πλ C ya Thus formula of induced drag in gas subsonic flow is kept in a prior form (atangles of attack, where the linear dependence C ya = f ( α ) . Polar does not vary either(at condition of C xp = const ). 5.3. Moment characteristics. Location of center of pressure and aerodynamic center. The factor of wing aerodynamic moment of pitch relatively to axis 0 z passingthrough center of forces reductions is determined by the formula mz = mz0 + ⎛ mz ya ⎞ C ya , C ⎜ ⎟ (5.11) ⎝ ⎠ at that factor of pitch moment for linear part of dependence is determined as m z0 nWhere m z0 = is the factor of pitch aerodynamic moment at C ya = 0 , 2 1− M∞ C dm z mαxF = − m z ya =− =− z is the relative coordinate of aerodynamic center dC ya Cα yaposition. The position of aerodynamic center x F = x F n and center of pressurex c . p . = x c . p . n does not depend on Mach numbers M ∞ for a wing in subsonic andincompressible gas flows! Received result is approximate for low-aspect-ratio wings with taking intoaccount the non-linear effects. (it is exact at α → 0 ). For tapered high-aspect-ratiowings it is possible to offer the following formula for definition of aerodynamic centerlocation relatively to wing top: 55
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xF ⎛ η − 1⎞ η + 1 xF = = x F∞ ⎜ 1 − 4 ⎟+ λ tg χ l .e . b0 ⎝ 3π η ⎠ 3π η ( x F∞ = 0 ,25 1 − 1,6 f 2 ) - airfoil. Fig. 5.7. It is noteworthy, that if concepts of bА and x Аare used, then position of aerodynamic center relatively to the leading edge MAC inshares of bА for wings of large aspect ratio is determined by the formula: xFA x FA = = 0 .25 ; bА xFA = xF − x A . It is possible to consider this ratio as fair for wings with curvilinear edges or with fracture (Fig. 5.8). The Fig. 5.8. position of aerodynamic centerrelatively to wing top is determined by the formula: l 2 2 xF = x A + 0 .25 bA = S ∫ [ xl .e .( z) + 0 .25 b( z)] b( z)dz . 0 There is no common formula for low-aspect-ratio wings. In particular cases: λ 1xF = is for rectangular wing, x F = is for triangular wing. 2 .2 + 3 .6 λ 1.52 + 0 .12λ It is noteworthy, that the aerodynamic center displaces forward with decreasingof λ for rectangular wing (at λ → 0 , x F → 0 and all aerodynamic load is concentratedon the leading edge), and the aerodynamic center with λ reduction displaces back fortriangular wing (at λ → 0 , x F → 0 .66 , more precisely 2 3 ). 56
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5.4. Wing critical Mach number M* . The critical Mach number M* determines the upper border of subsonic flows andthe above mentioned formulae are fair at condition of M∞ ≤ M* . Generally ( )M* = f η , λ , χ , c , C ya . Parameters χ , c and C ya have the greatest influence. The value M* can be defined by theoretical curve by S.A. Christianovich (Fig. 5.9), having the diagram of distribution of pressure factors Cp along wing surface in incompressible flow. It is also possible to use the following formula for assessment M* of wings with ordinary airfoils Fig. 5.9. Christianovich dependence (Fig. 5.10) at lift coefficientvalue C y = 0 : 0 ,7 λ 2 c M* = 1 − cos χ c , (5.12) 2 λ + 0 ,1Where χc is the sweep angle at a line of maximum thickness. Other formula M* = 1 − mλ 2 c* cos χ ; c = c (c + 17 f ) , * 2 c (5.13) λ 2 + 0 ,1 xcWhere m = 0 .35 is for classical airfoil, m = 0 .27 is for supercritical airfoil. As it is visible from the above mentioned formulae, the value of M* depends onrelative thickness c and airfoil camber f , on the airfoil shape (first of all on maximumthickness location x c ) and on the wing plan form λ , χ c . 57
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It is possible to increase M* by application of supercritical airfoils (Fig. 5.11).They are characterized by more uniform distribution of a pressure factor chordlengthwise. Fig. 5.10. Pressure distribution on the Fig. 5.11. Pressure distribution on the upper surface for ordinary airfoil upper surface for supercritical airfoil The account of C y influence can be done by the following formula ⎛ 3 2 cos 2 χ ⎞ λ2 M* = 1 − ⎜ 0 .7 c cos χ c + 3 .2 c C ya c⎟ 2 . (5.14) ⎝ ⎠ λ + 0 .1 It is possible to use dependence for supercritical airfoil and wings with suchairfoils: ( M* = 1 − 0 .55 c cos χ c + 3 c C ya 2 cos χ c )λ 2 λ2 + 0 ,1 . (5.15) At C ya = 0 the last formula gives 0 .55 λ 2 c M* = 1 − cos χ c for supercritical airfoils 2 λ + 0 .1 58
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