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Theme 6

1. 1. SECTION 1. AERODYNAMICS OF LIFTING SURFACES THEME 6. THE AERODYNAMIC CHARACTERISTICS OF WINGS IN A SUPERSONIC GAS FLOW. WING IN TRANSONIC RANGE OF SPEEDS The wing aerodynamic characteristics in the supersonic gas flow( 1,20 ...1,25 ≤ M∞ ≤ 4 ...5 ) depend on edges type: subsonic or supersonic. In thebeginning we shall consider the aerodynamic characteristics of wings of the individualplan forms and on their example we shall reveal some common properties characteristicfor wings of derived plan forms. Then we shall consider features of the aerodynamiccharacteristics of wings with various plan forms. 6.1. Rectangular wings. The leading edge 1 − 2 and trailing edge 3 − 4 are supersonic, lateral edges 1 − 4 and 2 − 3 are subsonic on a wing (Fig. 6.1). The influence of lateral edges (pressure equalization between the lower and upper surface) has an effect only in areas 1 − 5 − 4 and 2 − 6 − 3 , limited by Mach cones, in contrast to the subsonic flow where the pressure equalization (influence of lateral edges) occurs all spanwise. Outside the area of influence (surface of a wing 1 − 2 − 6 − 5 ) the flow is similar to the flow about flat plate Fig. 6.1. Pressure distribution in two-dimensional flow. The pressure factor On a surface of a wing in any point of wing surface outside the areas 59
2. 2. of influence of lateral edges is equal 2α C р up .lo . = C р ∞ = ±2α tgμ 0 = ± . (6.1) 2 M∞ −1 In the field of influence of lateral edges the factor of pressure is determined 2 tgυ C р up .lo . = C р ∞ arcsin . (6.2) π tgμ ∞ (Here the wing is considered as a flat plate). The last formula is recorded for case when there is no mutual influence of the right-hand and left-hand lateral edges (Fig. 6.2). This influence appears at such Mach numbers, when the Mach cone of the left- hand (or the right-hand) lateral edge crosses the right-hand (left-hand) lateral edge. Fig. 6.2. The limit case of mutual influence absence is determined 2by a condition λ M∞ − 1 < 1 . Fig. 6.3. shows probable cases of lateral edges mutualinfluence. 2 2 2 2 2 λ M∞ − 1 < 1 λ M∞ − 1 = 1 λ M∞ − 1 < 2 λ M∞ − 1 = 2 λ M∞ − 1 > 2 Fig. 6.3. The factor of pressure in the field of mutual influence of lateral edges isdetermined by more complex formulae; they are shown in special literature. Summarizing distributed pressure we shall define the aerodynamic characteristics. 2 The following formulas are received for the case when λ M∞ − 1 ≥ 1 : C ya = C α ⋅ α ; ya (6.3) 4 ⎛ 1 ⎞ α ⎜1 − ⎟; C ya = (6.4) 2 ⎜ M∞ − 1 ⎝ 2 λ M∞ − 1 ⎟ 2 ⎠ 60
3. 3. C xa = C xw + C xi ; (6.5) Bc 2⎛ 1 ⎞ C xw = ⎜1 − ⎟; (6.6) 2 ⎜ M∞ − 1 ⎝ 2λ M ∞ − 1 ⎟ 2 ⎠ 2 C xi = C ya ⋅ α = AC ya ; (6.7) 2 λ M∞ − 1 − 2 3 x F = 0 .5 . (6.8) 2 λ M∞ −1−1 2 2 At λ M∞ − 1 < 1 equations for the aerodynamic characteristics for rectangularwings become considerably complicated. Lets analyze the aerodynamic characteristics. 6.1.1. Lift. 1. The dependence C ya = f ( α ) is linear for any aspect ratio wing ( λ ≥ 3 ). 2. With increasing of λ the value of a derivative C α grows and at λ → ∞ tends ya 4to the airfoil characteristic C α → C α ∞ = ya ya (Fig. 6.4). This tendency takes 2 M∞ −1place faster than in subsonic flow, that is explained by the limited area of the lateraledges influence at M∞ > 1 . 3. The value of derivative C α decreases and tends to C α ∞ with increasing of ya yaMach numbers M ∞ , that is connected with narrowing of Mach cones at growth of M ∞and reduction of lateral edges influence (Fig. 6.5). It is possible to assume, thatCα ≈ Cα ∞ ya ya already at 2 λ M∞ − 1 ≥ 7 ...8 (difference is less than 7% at 2 2λ M∞ − 1 = 7 , at λ M∞ − 1 = 10 - less than 5% ). 61
4. 4. Fig. 6.4. Dependence C ya = f ( α ) Fig. 6.5. Dependence C ya = f ( α ) at M ∞ = const at λ = const Cα ya 4. The ratio is the universal dependence on the reduced aspect ratio λ 2λ M∞ − 1 . 6.1.2. Drag. There is only pressure drag which determines wave drag and induced dragC xa = C xw + C xi in the inviscid supersonic flow. As the wing leading edge issupersonic, then there is no sucking force and C xi = C ya ⋅ α (for flat wing). As 1 1C ya = C α ⋅ α and α = ya 2 C ya , then C xi = C ya ⋅ A ; A = α . So influence of λ onto Cα ya C yaC xi at M ∞ > 1 is weaker than in subsonic flow (for example at λ ≥ 4 2 1+δ 2 BcC xi = C ya ). The wave drag C xw is defined by airfoil drag C xw ∞ = and πλ 2 M∞ −1 ⎛ 1 ⎞multiplier ⎜1− ⎟ which is taking into account span finite. Lets notice, that ⎜ 2 λ M∞ − 1 ⎟ 2 ⎝ ⎠ 62
5. 5. in case of unswept wing, the same multiplier is included in the formula for C α (6.4). yaAs well as C ya , the value of parameter C xw → C xw ∞ with increasing of M ∞ and λ 2(more precisely λ M∞ − 1 ). It is explained by narrowing of Mach cone and reduction 2of lateral edges influence. It is possible to consider that at λ M∞ − 1 ≥ 7 ...8 C xwC xw ≈ C xw∞ (error ≈ 7% ). The ratio is dependent only on reduced aspect ratio 2 λcand factor of the airfoil plan form B i.e. C xw λc 2 ( 2 ) = f λ M ∞ − 1 , airfoil planform . 6.1.3. Location of aerodynamic center. "Loss" of lift in the wing areas falling inside the Mach cones will cause displacement of pressure center and aerodynamic center forward, to the leading edge, in comparison with location of the airfoil aerodynamic center x F∞ = 0 .5 (Fig. 6.6). The location of aerodynamic center is a function of aspect Fig. 6.6 ratio x F = f ⎛ λ M ∞ − 1 ⎞ . ⎜ 2 ⎟ ⎝ ⎠ 2 2 At λ M∞ − 1 → ∞ x F → x F∞ . Approximately x F ≈ x F∞ at λ M∞ − 1 ≥ 7 2with an error less than 3% . At λ M∞ − 1 = 5 x F = 0 .96 ⋅ 0 .5 = 0 .48 difference fromx F∞ is 4% . 63
6. 6. 6.2. Triangular wings with subsonic leading edges. Lets consider a triangular wing with subsonic leading edges. With the help of fig. 6.7. we get an internal sweep angle 90 o − χ l .e . < μ ∞ and ctgχ l .e . 2 1 2 n= = M ∞ − 1 ctgχ l .e . = λ M∞ − 1 < 1 . tg μ ∞ 4 Thus triangular wing with subsonic edges has 2 reduced aspect ratio less than 4 ( λ M∞ − 1 < 4 ). In this case there is an overflow from the lower surface to Fig. 6.7. A triangular wing the upper surface. The sucking force is realized on the with subsonic edges leading edge and reduces induced drag. There is also non-linear additive to the lift coefficient ΔC ya . Pressure distribution along wing surface (fig. 6.8) submits to the law within the linear theory ( α << 1 ) 2α tgγ l .e . C р up .l . = ± , .9) E 1 − t2 tgϑ where t = and γ l .e . = 90 o − χ l .e . (refer to fig. tgγ l .e . 6.7); it is possible to define approximately E - the elliptical integral of II type dependent on 1 2Fig. 6.8. Pressure distribution parameter n = λ M∞ − 1 , by the formula 4 in wing cross-section E ≈ ( 1,0 + 0 ,6 n) − n . 2 It follows from the formula (6.9) for C p that on each ray ϑ = const value ofC p = const , that is the feature of conical flows. At ϑ → γ l .e . i.e. at approach to theleading edge C p → ±∞ (the similar situation takes place in a subsonic flow). For asharp edge the site in nose section is equal to zero. Multiplying C p onto the nose 64
7. 7. section area and expanding the uncertainty ∞ ⋅ 0 , we receive final force value whichprojection onto incoming flow direction creates force F reducing the induced drag.This is the sucking force. The aerodynamic characteristics of a triangular wing withsubsonic edges are determined by the following formulae: C ya = C α ⋅ α + ΔC ya , C xa = C xw + C xi ; ya (6.10) 4 π ⋅n πλ Cα = ya ⋅ = ; (6.11) 2 M∞ −1 2E 2E 4C α ya ΔC ya = 1 − M ∞ cos χ l .e . ⋅ α 2 ; 2 (6.12) πλ 2 Bc C xв = n (1.1 + 0 .2 n) ; (6.13) 2 M∞ − 1 2 1 1 − n2 C xi = C ya ⋅ α − CF ≈ AC ya ; A= −ξ , (6.14) α πλ C yawhere ξ is a factor of realization of sucking force ( ξ theor = 1 ), it is possible to adopt forsharp leading edges, that ξ ≈ 0 ; for rounded edges - ξ ≈ 0 .8 − 1.2C ya at0 ≤ C ya ≤ 0 .66 , further ξ ≈ 0 . xF 2 For a triangular wing x F = = . The last result follows from consideration of b0 3conical flow with constant pressure C p = const on rays outgoing from the wing top. 6.2.1. Analysis of the aerodynamic characteristics. 1. At n → 0 , that corresponds to λ → 0 or M ∞ → 1 (or simultaneously) we πλreceive C α = ya . That means, the result corresponds to the extremely low-aspect-ratio 2wing in incompressible and subsonic gas flows. If the wing leading edge becomes sound 65
8. 8. 4( n = 1 ), then C α = ya , i.e. we receive the characteristic of the airfoil of infinite 2 M∞ −1aspect ratio wing in supersonic gas flow. 2. In case of sound and supersonic edges ( M ∞ cos χ l .e . ≥ 1 ) the non-linearadditive to a lift coefficient is equal to zero, i.e. ΔC ya = 0 . πλ 3. If n → 0 , then C α → ya and at ξ theor = 1 a polar pull-off coefficient 2 1 1 2A= , i.e. wing. induced drag C xi = C ya ; so as for subsonic flow high-aspect- πλ πλ 2ratio wing. It is clear that the sucking force CF C ya reduces the induced drag twice incomparison with that case, if it is not taken into account. Cα ya C xw 2 4. Ratios and are also functions of reduced aspect ratio λ M∞ − 1 λ λc 2 C xwand airfoil plan form (for ). 2 λc 6.3. Triangular wing with supersonic leading edges. 1 2 If the leading edges of a triangular wing are supersonic - n = λ M∞ − 1 > 1 . 4The overflow is absent and the sucking force is not realized on the leading edge. It ispossible to mark out two characteristic flow areas (Fig. 6.9). The wing areas I (shadedsites) outside the Mach cone are streamlined as the isolated slipping wing of infinitespan, irrespective of other wing part. The pressure in these areas is constant andpressure factor is determined: C p∞ C pI = , (6.15) 2 1−σ 66
9. 9. 2α 1 4where C p ∞ = ± is the pressure factor on the airfoil; σ = = or 2 M∞ − 1 n λ M2 − 1 ∞ tgμ 0σ= is the leading edge characteristic. tgγ l .e . There is an influence of the angular point (wing top) in the wing central part falling into Mach cone. Conical flow takes place in this area II, for which the constancy of pressure on each ray outgoing from wing top is characteristic, but pressure on different rays is various. The pressure factor for such case is determined as ⎛ 2 2⎞ ⎜ 1 − 2 arcsin σ − t ⎟ , (6.16) C p∞ C pII = ⎜ π 1 − t2 ⎟ 1−σ2 ⎝ ⎠ Fig. 6.9. Triangular wing where t = tgϑ tgγ l .e . , 0 ≤ t ≤ σ . with supersonic edges The integration of pressure distribution results inthe following formulas for the aerodynamic characteristics: C ya = C α ⋅ α , C xa = C xw + C xi , C xi = C ya ⋅ α = AC ya , ya 2 4 Cα = ya , (6.17) 2 M∞ − 1 2 Bc ⎛ 0 .3 ⎞ C xw = n⎜1 + ⎟, (6.18) 2 M∞ − 1 ⎝ n2 ⎠ 2 1 M∞ − 1 A= = , (6.19) Cα ya 4 xF 2 xF = = . (6.20) b0 3 It is noteworthy, that the value of C α for triangular wing with supersonic leading yaedges coincides with the airfoil characteristic C α ∞ (the difference is in pressure ya 67
10. 10. distribution). The pressure rising at tip sites compensates pressure decreasing in centralarea of the triangular wing. It can be shown, that the share of tip sites in total lift comes n−1to , that at n = 1.5 corresponds to ≈ 45% and at n = 2 - ≈ 57% . n+ 1 Cα ya C xw Just as for a wing with subsonic edges, the ratios and are also functions λ λc 2 2of λ M∞ − 1 and airfoil shape for wave drag (Fig. 6.10, 6.11). It is necessary to note,that the formulas for a triangular wing with subsonic and supersonic edges aretheoretically joint to a fracture at n = 1 or ⎛ λ M ∞ − 1 = ⎜ 2 4⎞ . ⎟ ⎝ ⎠ Experimentally this fracture is smoothed out. In point A the leading edge passesfrom a subsonic flow mode to supersonic flow. The application of wings with subsonicedges is evident on a curve of wave drag (in this case the induced drag decreases toodue to realization of sucking force). The most adverse flow mode is in zone of M ∞numbers corresponding to a sound leading edge. Cα ya Fig. 6.11. Dependence of C xw on reducedFig. 6.10. Dependence of on reduced 2 λ λc 68
11. 11. 2 2 aspect ratio λ M∞ − 1 aspect ratio λ M∞ − 1 It is interesting to note, that if triangular wing is put into flow by the reverse side(Fig. 6.12), then pressure distribution along the inverted wing will be the same as for a 2αwing of infinite aspect ratio, i.e. C p = C p ∞ = ± . In this case lift coefficient 2 M∞ −1C α and induced drag C xi will be the same, as on the initial triangular wing. It is a yaparticular case of the general theorem of reversibility. According to this theorem, thelift of a flat wing of any plan form at the direct and inverted flow will be identical, if the angles of attack and speeds of undisturbed flow are identical. For induced drag the equality will be obeyed at supersonic leading edges (in direct and inverted flows) or at identical values of sucking forces. Considering the load distribution along Fig. 6.12. wing surface it is possible to make the conclusion that the cut-out of trailing edge(form such as “swallows tail”) (Fig. 6.13,1) should result to the increasing of C α , and yaadditive of the area to a trailing edge(Fig. 6.13,3) - to decreasing of C α . It is possible to write down C α 1 > C α 2 > C α 3 . ya ya ya ya 69
12. 12. Fig. 6.13. Various versions of trailing edge shape At χ t .e . ≤ 20 o the wings aerodynamic characteristics are determined by thecharacteristics of the initial triangular wing by multiplication to a factor dependent on 1 ctgχ l .e .the ratio of sweep angles on forward and trailing edges , where ε = − . (1 + ε ) ctgχ t .e .It is necessary to take the sweep angle on the trailing edge with its own sign. Soderivative of a lift coefficient C α , wave drag and location of aerodynamic center are yadefined by the formulae α CαΔ y C xw Δ xF Δ C ya = ; C xw = ; xF = . (6.21) 1+ε 1+ε 1+ε 6.4. Wings of any plan form. The qualitative analysis of the aerodynamic characteristics. The main feature of the aerodynamic characteristics of all wings: with increasing 2 of Mach numbers M ∞ (more precisely - reduced aspect ratio λ M∞ − 1 ) the aerodynamic characteristics C α , C xw tend to the airfoil characteristics, i.e. ya 2 α α 4 Bc C ya = C ya ∞ = ; C xw = C xw ∞ = . It can be explained, by the 2 2 M∞ −1 M∞ − 1 fact that the Mach cone is narrowing with increasing of M ∞ (at λ = const ) and each cross-section of a wing will be also isolated streamlined. It also follows, that the wing aerodynamic center (or center of pressure) displaces into the center of mass of the plan form, i.e. x F = x c .g . . Lets analyze the aerodynamic characteristics of wings. 70
13. 13. 6.4.1. Lift. Generally for flat wing C ya = C α ⋅ α + ΔC ya , where ΔC ya is the non-linear yaadditive exists only at a subsonic leading edge. It can be estimated by the formula 4 ΔC ya = C α 1 − M ∞ cos 2 χ l .e . ⋅ α 2 ya 2 πλ At M ∞ cos χ l .e . ≥ 1 (supersonic edge) ΔC ya = 0 . There are schedulesconstructed in a generalized form for definition of the derivative C α , as the ya Cα Cα = f ⎛ λ M ∞ − 1, λ tgχ 0 .5 , η ⎞ . ya ya 2dependence on parameters of similarity: ⎜ ⎟ λ λ ⎝ ⎠ Approximately it is possible to consider, that the taper η practically does notinfluence the lift coefficient. For each wing the functionCα = f ⎛ λ M ∞ − 1, λ tgχ 0 .5 , η ⎞ is various. However, as it was mentioned above, at ya 2 ⎜ ⎟ λ ⎝ ⎠ 2 Cα ya 4λ M∞ − 1 → ∞ we receive for all wings = . Practically this λ 2 λ M∞ − 1 2dependence can be used at λ M∞ − 1 ≥ 6 ...7 . Cα ya The general view of dependence λ is shown in a Fig. 6.14. The presence of fractures at changing of flow modes about edges is characteristic for it: In point A - the trailing edge passes from subsonic to supersonic flow mode; In point B - the leading edge passes from subsonic to supersonic flow mode. In experiment these fractures are Fig. 6.14. smoothed out. 71
14. 14. 6.4.2. Wave drag. C xw Generally = f ⎛ λ M ∞ − 1 , λ tgχ 0 .5 , η , airfoil shape ⎞ . ⎜ 2 ⎟ At 2 ⎝ ⎠ λc 2λ M∞ − 1 → ∞ irrespectively of the wing plan form we shall haveC xw B = as the characteristic of an airfoil. Practically this formula can be 2 2λс λ M∞ − 1 2used at λ M∞ − 1 ≥ 6 ...7 . It is necessary to note weak influence of taper onto wavedrag. The presence of fractures on a curve (Fig. 6.15). is characteristic for general C xwdependence on the reduced aspect ratio. 2 λc 72
15. 15. The fracture in point A - wing trailing edge passes from a subsonic flow mode to supersonic; in point C - transition of the maximum thickness line from subsonic to supersonic flow; in a point B - the leading edge passes from subsonic to supersonic flow mode. These fractures are not present in experimental dependencies, they are smoothed out. Fig. 6.15. The maximum of curves is observed in the area of transition of the maximum thickness line ( χc ) from a subsonic flow mode to supersonic (point C ). For a sound line of maximum thickness 2 λ tgχ c = λ M ∞ − 1 . It is necessary to note that at subsonic lines of maximum thickness the wave drag of swept wings is less than drag of unswept wing. Thus the longer wing aspect ratio (at χc = const ), sweep (at Fig. 6.16. λ = const ) or parameter λ tgχ c is, then thebigger profit is received in drag. On the contrary, at a supersonic line of maximumthickness the wave drag of swept wing is more than of unswept one (Fig. 6.16). 6.4.3. Induced drag. 2 If the leading edge is supersonic, then C xi = C ya α or C xi = AC ya , whereA = 1 C α - for the flat wing. At the subsonic leading edge it is necessary to take into yaaccount the sucking force. In this case C xi = C ya α − CF , or approximately 73
16. 16. α 2 2 1 CF C F 1− cos χ l .e . S Δ ⎛ C ya Δ ⎞ 2 M∞ 2 ⎜ ⎟ ; (6.22) C xi = AC ya ; A = − 2 ; 2 =ξ C α C ya C ya 4π cos χ l .e . S ⎜ Cα ⎟ ya ⎝ ya ⎠ Where ξ is the factor of sucking force realization (refer to item 6.2); C α Δ and S Δ are parameters of a triangular wing, which ya leading edge coincides with the leading edge of the Fig. 6.17. wing under consideration (fig. 6.17). 6.4.4. Location of aerodynamic center. Generally, there is the dependence x F = f ⎛ λ M ∞ − 1, λ tgχ , η ⎞ , in which the ⎜ 2 ⎟ ⎝ ⎠influence of taper is essential in contrast to the characteristics C α and C xb0 . Location yaof aerodynamic center tends to the position of the center of mass of a figure presentingthe wing plan form with increasing of reduced aspect ratio. In particular, for wings withunswept edges of the tapered plan form we have xF 1 ⎛ η2 + η + 1 η + 1 ⎞ xF = = x c .g . = ⎜ + λ tgχ l .e . ⎟ . (6.23) b0 3η ⎜ η + 1 ⎝ 4 ⎟ ⎠ 2 The reduced formula can be used already at λ M∞ − 1 ≥ 5 ...6 : (For a rectangular wing - η = 1 , χ l .e . = 0 ; x F = 0 ,5 for a triangular wing - η = ∞ , λ tgχ l .e . = 4 , x F = 2 3 ). Note: While calculating the aerodynamic characteristics of the complex plan formwings (curved edges or the edges with a fracture) approximate methods replacingvariables spanwise χ m ( z ) , c m ( z ) , ... by their mean values are used together withprecisely numerical calculations of a particular wing. 74
17. 17. 6.5. Wing in transonic range of speeds. Speeds corresponding to Mach numbers ( M* ≤ M ∞ ≤ 1,20 ...1,25 ) are calledtransonic. All range can be divided on to: area of subsonic speeds ( M* ≤ M ∞ ≤ 1 ), areaof supersonic speeds ( 1 < M ∞ ≤ 1,20 ...1,25 ), flow mode with Mach numbers M ∞ = 1 . Features of the aerodynamic characteristics in subsonic part of transonic speeds are determined by existence of mixed flow including subsonic (outside of the wing) and supersonic (on the wing and near to it) flow areas. The forward border of supersonic flow area represents so-called sound line, along which the transition from subsonic to supersonic flow takes place. The flow remains subsonic outside the zones limited by the sound line. Fig. 6.18 shows the approximate borders of supersonic zones at various Mach numbers M ∞ . With increasing of Mach number M ∞ the shock waves are originally formed on the upper surface and move to the trailing edge. Then the supersonic area is formed on the lower surface. The Fig. 6.18. development of supersonic area on the airfoillower surface proceeds more intensively, than on lower. The supersonic areas arefinished by shock waves, which with increasing of numbers M ∞ displace back andenlarge the extent in the vertical direction. At M ∞ = 1 a shock wave is theoreticallydistributed into infinity, at that there is a head shock wave before the wing also ininfinity. Further increasing of M ∞ causes movement of a head shock wave and shockwave on the wing surface downwards the flow. The supersonic wing flow mode comes 75
18. 18. at values M ∞ = 1,20 ...1,25 , when the shock waves practically do not move any more,and reduce their angle of inclination with increasing of M ∞ . The appearance of transonic parameter of similarity λ 3 с , as a result of thenon-linear theory, is characteristic for transonic area. Parameter λ 3 с influences ontochanging of the aerodynamic characteristics so, that: Cα = f ⎛ λ M ∞ − 1 , λ tgχ , η , λ 3 с ⎞ ya 2 ⎜ ⎟ (Fig. 6.19); λ ⎝ ⎠ C xw = f ⎛ λ M ∞ − 1 , λ tgχ , η , λ 3 с , airfoil shape⎞ ⎜ 2 ⎟ (Fig. 6.20); λс 2 ⎝ ⎠ x F = f ⎛ λ M ∞ − 1 , λ tgχ , η , λ 3 с ⎞ ⎜ 2 ⎟ (Fig. 6.21). ⎝ ⎠ Cα ya C xb At M ∞ = 1 : the dimensionless parameters , , x F depend on λ tgχ , λ λс 2η , shape of the airfoil and λ 3 с . Lets remind, that the taper η continues to play a smallrole in changing of C α and C xb , in some cases its influence can be neglected. yaHowever, the parameter η plays an essential role for characteristic of aerodynamiccenter location, because it effects onto aerodynamic loading distribution wing spanwise. For wings of arbitrary shape the influence of λ 3 с is investigated a little. Moredetail research is carried out on rectangular wings with rhomboid airfoil( χ = 0 , η = 1 , B = 4 or K п р = 1 ). it was proved, that for such wings at M ∞ → 1 and 3 Cα ya πλ с ≤ 1 the value of → ≈ 1,57 . It can be explained that at M ∞ → 1 reduced λ 2 2aspect ratio λ M∞ − 1 → 0 and we pass to the very low-aspect-ratio wing, for which πλCα = ya . If λ 3 с ≥ 2 , then the theory of transonic flows for a rectangular wing gives 2Cα ya 2 ,3 ≈ . λ λ3 с 76
19. 19. C xw The analogous results are received for wave drag: ≈ 3 ,0 at λ 3 с ≤ 1 , 2 λcC xw 3 ,65 ≈ at λ 3 с ≥ 2 . 2λc λ3 с Cα ya Fig. 6.20. Dependence C xw on reduced Fig. 6.19. Dependence on reduced 2 λ λc 2 2 aspect ratio λ M ∞ − 1 aspect ratio λ M ∞ − 1 77
20. 20. It is necessary to note, that in the latter case C xw ~ c 5 3 , i.e. the wave drag grows with increasing of c , though not so fast as in the supersonic flow, in which C xw ~ c 2 . The change of the aerodynamic center location dependently on λ 3 с is similar to changes of C α (Fig. 6.21). The more λ 3 с is, then ya aerodynamic center changing by Mach numbers M∞ behaves more irregularly: Fig. 6.21. Dependence of the drastic displacement forward in subsonic aerodynamic center location on range is probable with the subsequent 2 λ M∞ − 1 displacement backward into position corresponding to supersonic speeds(displacement of aerodynamic center for a triangular wing at passage from M ∞ < 1 toM ∞ > 1 is determined as xF b0 ≈ 0 ,12 λ ). The main measures providing reduction of wing wave drag, improvement of itslifting properties and smooth change of aerodynamic center in supersonic range ofspeeds by Mach numbers M ∞ are: reduction of c and λ (decreasing of parametervalue λ 3 с ) and increasing of χ . 6.6. Wing induced drag at M∞ ≤ M with taking into account local * supersonic flows. Lets consider the problem on the account of additional drag occurring at valuesof C ya and M ∞ , outgoing of critical values (the approximate method of the account isoffered by S. I. Kuznetsov). It is necessary to take into account this drag whileconstructing the wing polar for specified M ∞ = const , M ∞ < 1 . 78
21. 21. Lets assume that the dependence of critical Mach number M* on C ya = C ya * , (i.e. M* = f C ya * ) or C ya * = f ( M* ) . Lets construct this dependence in a plane ofC ya , M ∞ (Fig. 6.22). It is obvious, that all values of C ya and M∞ , lying below the curve C ya * = f ( M* ) fall into subsonic speeds area. However if at specified M ∞ = const will be C ya > C ya * , then the flow supersonic area closed by shock waves is formed on the wing. In this case there is an additional drag caused by lift ΔC xi (at C ya ≤ C ya * ΔC xi = 0 ). If one Fig. 6.22 assumes, that the growth of lift is not accompanied by growth of sucking force with increasing of angles of attack at C ya > C ya * (that at presence of the broad supersonic area on a wing is permissible), then for a flat wing we shall have ( ΔC xi = C ya α or ΔC xi = C ya − C ya * α . ) In addition adopting, that on transonic flow modes the proportion Fig. 6.23. Wing polar with the account of ΔC xi C ya = C α α is executed, then finally we ya receive ΔC xi = ( C ya − C ya * ) C ya . Cα ya This parameter is added to induced drag, and thus we have: 79
22. 22. ⎧C xi = AC ya , если C ya ≤ C ya при M ∞ = const ; 2 ⎪ * ⎨ (6.22) 2 [( ) ] α ⎪C xi = AC ya + C ya − C ya* C ya C ya , если C ya > C ya* . ⎩ Wing polar take the form as it is shown in fig. 6.23. Values of lift coefficients C ya * corresponding to the beginning of wave crisis atM ∞ ; i.e. the dependence C ya * = f ( M* ) can be found from the formula ( M ∞ ≡ M* ): n ⎧ ⎪ ⎡ ⎛ 0 .1 ⎞ ⎤⎫ (1 − M ∞ )⎜ 1 + 2 ⎟ − m c cos χ c ⎥ ⎪ 1 C ya * =⎨ ⎢ ⎬ ⎪ к c cos 2 χ c ⎣ ⎩ ⎝ λ ⎠ ⎦⎪⎭where the factors k , m , n for a wing with a classical airfoil are equal k = 3 .2 ,m = 0 .7 , n = 2 3 ; for a wing with a supercritical airfoil - k = 1.2 , m = 0 .65 , n = 1 3. 80