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Theme 4 Theme 4 Document Transcript

  • SECTION 1. AERODYNAMICS OF LIFTING SURFACES THEME 4. THE AERODYNAMIC CHARACTERISTICS OF WINGS IN A FLOW OF INCOMPRESSIBLE FLUID The main aerodynamic characteristics of an aircraft moving with Mach numbers( M ∞ ≤ 0 .4 ) are considered in this lecture. 4.1. Wing lift coefficient 4.1.1. High-aspect-ratio wings The dependence C ya = f ( α ) is linear on the segment of the attached flow (Fig.4.1).In general C ya = C α (α − α0 ) . ya (4.1) The angle of zero lift α0 is determined by the airfoil shape and wing twist. For aflat wing α 0 = −60 f ⎡ 1 + 10( x f − 0 ,2) ⎤ , degree. 0 2 ⎢ ⎥ (4.2) ⎣ ⎦ Geometrical twist of the wing ϕ ( z ) causes changing of α0 by the value Δα0 , which can be approximately estimated for the unswept wing by the formula l 2 2 Δα 0 = − S ∫ ϕ ( z) b( z) dz . 0 If the wing is swept then the absolute value of Δα0 should be reduced by size Fig. 4.1. Dependence of the lift 0 .0055 sin χ 0 .25 . coefficient on the angle of attack THEME 4 10/5/2008 31
  • The derivative C α depends on aspect ratio λ and the sweep angle χ , influence of yataper η on derivative size is weak. The derivative C α does not depend on wing twist. yaIt is possible to offer the following approximate formula for calculation of value ofCα : ya πλ λ Cα = ya , m= (4.3) α 1+τ + (1 + τ ) 2 + ( π m) 2 C ya∞ cos χ 0 .5 Here the parameter τ takes into account the wing plan form and depends on η ,λ , χ 0 .5 . It is possible to assume in the first approximation τ ≈ 0 (in generalτ ≈ 0 ...0 ,24 ), ⎡ ⎤ 2 1 ⎥. τ = 0 ,17 m ⎢η + (4.4) ⎢ ⎣ (5η + 1) ⎦ 3⎥ Parameter C α ya∞ is a derivative for the airfoil (wing with λ → ∞ ) and iscalculated by the formula ( C α = 2π − 1,69 4 c = 2π 1 − 0 ,27 4 c . ya∞ ) (4.5) It follows from the formula, that at λ → ∞ C α = C α ∞ cos χ 0 .5 and if in ya yaaddition χ 0 .5 = 0 , then C α = C α ∞ . ya ya It is also possible to use the following formula for calculation C α : ya α Cα ∞ λ ya C ya = . (4.6) 1 α pλ + C ya∞ π Here p is the ratio of half-perimeter of the wing outline in the plan to span (Fig.4.2). The significance of the last formula is in its universality and capability to apply toany plan forms and aspect ratios (it is especially useful for wings with curvilinear edgesor edges with fracture). THEME 4 10/5/2008 32
  • It is possible to define parameter p for a wing represented in fig. 4.2, by the formula l1 + l2 + l3 + l4 p= , or in case of tapered l wing - by the formula ⎛ 1 1 ⎞ 2 p = 0 .5 ⎜ + ⎟+ . ⎝ cos χ п .к . cos χ з .к . ⎠ λ (η + 1) Fig. 4.2. Lets analyze the influence of wing geometrical parameters on value C α . ya 1. With increasing of wing aspect ratio λ the derivative of a lift coefficient on the angle of attack C α grows (at conditions of ya χ = 0 and λ → ∞ the value of a derivative tends to the airfoil characteristic C α = C α ∞ ya ya (Fig. 4.3). Fig. 4.3. 2. With increasing of sweep angle at half-line chord ( 0 .5 chord line χ 0 .5 ) the derivative value C α decreases (Fig. 4.4). (It ya occurs due to effect of slipping, at condition of λ → ∞ , the sweep angles on the leading and trailing edges are identical χ l .e .≈ χ t .e .= χ , C α = C α ∞ cos χ ). ya ya 3. The sweep influence on derivative C α value decreases with decreasing of ya Fig. 4.4. aspect ratio λ (Fig. 4.5) (sweep practicallyTHEME 4 10/5/2008 33
  • does not influence on value of lift coefficient derivative on the angle of attack C α at small ya values of aspect ratio λ ). 4. The wing taper η influences a little onto the value of a derivative C α (refer to ya formula (4.4), parameter τ ). Fig. 4.5. 4.1.2. Wings of small aspect ratio It is necessary to take into account the non-linear effects which occur at flowabout wings of small aspect ratio in dependence of a lift coefficient on the angle ofattack C ya = f ( α ) (Fig. 4.6) C ya = C ya line + ΔC ya where C ya line = C α (α − α 0 ) . ya It is also possible to define values of α 0 and C α by the formulae for large aspect ya ratio wings at λ ≥ 2 The value of derivative can be determined by the formula Fig. 4.6. Cα = π λ ya for a wing of extremely small 2aspect ratio λ < 1 , and the angle of zero lift for the wing with unswept trailing edge is ( )equal to an angle of the trailing edge deflection α 0 = f ′ x t .e . , z , where y = f ( x , z ) - ∂ fequation of a surface of a wing, f ′ = , x t .e . is trailing edge coordinate. ∂x THEME 4 10/5/2008 34
  • The non-linear additive can be calculated by the formula which is fair at anyMach numbers (the linear theory of subsonic flow refers only to a linear part of thedependence). 4C α 1 − M ∞ cos 2 χ l .e . ⋅ (α − α 0 ) ya 2 2 ΔC ya = (4.7) πλ At a supersonic leading edge ( M ∞ cos χ l .e . > 1 ) the non-linear additivedisappears and ΔC ya = 0 . With decreasing of λ the derivative C α decreases, and the non-linear additive yaΔC ya grows (Fig. 4.7, 4.8).Fig. 4.7. Character of changing of the non- Fig. 4.8. Character of changing of the non- linear additive ΔC ya linear additive ΔC ya 4.2. Maximum lift coefficient. The maximum lift coefficient is connected with the appearance and developmentof flow stalling from the upper wing surface near the trailing edge and depends on manyfactors, first of all, on the characteristics of the airfoil ( c , f , x f , nose section shape),wing plan form ( χ , η ), Reynolds number Re . The λ value does not practically THEME 4 10/5/2008 35
  • influence onto C ya max for wings of large aspect ratio (Fig. 4.9), at that with χdecreasing α st increases. The influence of λ has an effect as follows for wings of small aspect ratio:C ya max grows with λ growing from 0 up to λ ~ 1 ; and then decreases - withincreasing of λ . Small values of C ya max ( C ya max ≈ 1,0 ...1,1 ) and large values of α st( α st ≈ 25 o ...40 o ) (fig. 4.10) are characteristic for wings with λ < 3 . A flow stallingdelays in the latter case caused by influence of vortex structures formed on the upperwing surface. Fig. 4.9. Dependence C ya = ( α ) for Fig. 4.10. Dependence C ya = ( α ) for wings of large aspect ratio λ ≥ 4 . wings of small aspect ratio λ < 3 The properties behaviour of curve C ya (α ) in area α st depends on the nosesection shape. The presence of C ya max low values is characteristic for a wing with thepointed airfoil nose section which do not depend on Reynolds numbers (Fig. 4.11). Theincreasing of C ya max (up to certain values) is characteristic for a wing with the roundedairfoil nose section at increase of Reynolds numbers (Fig. 4.12). Approximately, value C ya max of large aspect ratio wing ( λ ≥ 4 ) can bedetermined by the formula THEME 4 10/5/2008 36
  • ⎛ 0 .49η + 0 .91 2 ⎞ ⎡ η+2 ⎤ C ya max = C ya max ∞ ⎜ 1 − sin χ 0 .25 ⎟ ≈ C ya max ∞ ⎢1 − sin2 χ 0 .25 ⎥ , ⎝ η+1 ⎠ ⎢ ⎣ 2(η + 1) ⎥ ⎦where C ya max ∞ is the maximum value of an airfoil lift coefficient;for symmetrical airfoil C ya max ∞ ≈ 35 c exp − 8 c at Reynolds numbers Re ≤ 10 6 and ( )C ya max ∞ ≈ 39 .3 c exp − 8 c th Re⋅ 10 − 6 at Re ≥ 10 6 . Fig. 4.11. Dependence C ya = ( α ) for Fig. 4.12. Dependence C ya = ( α ) for wings with a sharp leading edge wings with a classical airfoil Number Re is the kinematic factor of similarity describing the ratio of inertialand viscous forces: VL Re = , (4.8) ϑWhere V is the characteristic speed; L is the characteristic length; ϑ is the kinematicfactor of viscosity. THEME 4 10/5/2008 37
  • 4.3. Induced drag. It is a result of the appearance of a vortex sheet behind a wing, which keepingdemands energy equivalent to work of induced drag force. The computational formulaefor definition of C xi are described below. 4.3.1. Wings of large aspect ratio The induced drag coefficient of a large aspect ratio wing is determined C xi = C xi ϕ = 0 + ΔC xiϕ , (4.9)Where C xi ϕ =0 is the induced drag of a flat wing; ΔC xiϕ is the additive to induced dragcaused by geometrical twist. We have 1+δ 2 2 1+δ C xi ϕ = 0 = C ya = AC ya , A = (4.10) π ⋅λ π ⋅λ Dependence of induced drag at small angles of attack is approximated byparabola (Fig. 4.13). The induced drag decreases at increasing of wing aspect ratio λ . The coefficient δ is determined by the wing plan form and shows to what extentthe distribution of aerodynamic loading differs from the elliptical law, for which δ = 0 .Generally δ ≈ 0 ...0 .10 . The expression for C xi ϕ =0 is represented as 1 2 1C xi ϕ = 0 = C ya lately, where e is Osvald number, e = . πλe 1+δ Value of ΔC xi ϕ in contrast to C xi ϕ =0 can not be equal to zero in general atC ya = 0 (because for non-flat wing various wing cross-sections may have liftcoefficients not equal to zero C ya ≠ 0 at total C ya = 0 ). General expression for ΔC xiϕ looks like this: ΔC xiϕ = BC ya + C , where, 2B = к 1ϕ w , С = к0ϕ w , ϕ w is the twist angle of tip cross-section; the values ofcoefficients к0 and к 1 are also undertaken from the diagrams depending on wing plangeometry ( λ , χ , η ), and twist law. THEME 4 10/5/2008 38
  • Fig. 4.14. shows the comparison between induced drag of a flat wing and winghaving twist. Fig. 4.13. Induced drag of a flat wing Fig. 4.14. Induced drag of a flat wing and wing with twist 4.3.2. Wings of small aspect ratio For a geometrically flat wing it is possible to assume: C xi ϕ = 0 = C ya ⋅ α . (4.11) This expression can be received if it is assumed that the sucking force is notrealized on the wing leading edge. The formula can be transformed to the followingform if the non-linear additive is neglected ( ΔC ya = 0 ) we have: C xiϕ = 0 = AC ya , A = 1 C α 2 ya (4.12) In such form the formula is used in practice (it coincides with the formula for aflat wing of large aspect ratio) πλ 2 ( For a wing of small aspect ratio λ < 1 , C α = ya and A = ; comparing 2 πλ 1+δwith a wing of large aspect ratio A = we shall notice that δ = 1 ). π ⋅λ THEME 4 10/5/2008 39
  • 4.4. Wing polar Polar of the first type is called the dependence between coefficients ofaerodynamic lift and drag force C ya = f (C xa ) . As well as the induced drag coefficientthe polar depends on twist presence on a wing. Therefore we shall separately considerpolar for a flat wing and polar for a wing having twist, and we shall conduct thequalitative analysis. 4.4.1. Flat wing. Summing induced drag C xi with profile drag C xp , we shall receive expressionfor polar: 2 C xa = C xp + C xi = C xp + AC ya . (4.13) It can be assumed that the profile drag C xp does not depend on a lift coefficient C ya ( ) C xp ≠ f C ya . In this case we shall write 2 down C xa = C x 0 + AC ya , where C x0 is the wing drag coefficient at C ya = 0 (in this case Fig. 4.15. Dependence of profile drag it coincides with C xp ). It is necessary to note on lifting force that in general the profile drag C xp changes 2by C ya especially at large values of α or C ya , it is approximately proportional to C ya(Fig. 4.15); in some cases this change is taken into account in parameter A : 2 2 ~ 2 2 C xa = C x0 + aC ya + AC ya = C x0 + AC ya = C x0 + AC ya , (4.14) ~ 1+δ 1+δ 1Where A = A + a ; for high-aspect-ratio wings A = and a + = , πλ πλ π λ ef THEME 4 10/5/2008 40
  • 0 .95 λλ ef = ). 1 + 0 .16 λ tgχ 0 .5 The last polar writing (4.14) is the most general form, which is fair for any aerodynamic shapes if not to decipher parameters C x0 and A . Parameter A has the name of a polar pull-off coefficient. ∂ C xa Obviously, that in general A = . Wing 2 ∂ C ya profile drag C xp does not depend on λ and polar for wings of various aspect ratio λ are look like as it is shown on fig. 4.16. As it was Fig. 4.16. Flat wing polar spoken earlier, the induced drag decreaseswith increasing of aspect ratio λ , and consequently the drag C xa for a wing of infiniteaspect ratio λ → ∞ (airfoil) will be equal to profile drag C xp or C x0 . 4.4.2. Non-planar wings (wings with geometrical twist). A The general form of polar equation for a non-planar wing 2C xa = C + BC ya + AC ya , where profile drag C xp and part of induced drag caused by 2wing geometrical twist enter parameter C , i.e. C = C xp + к0 ϕ w . Parameter B is alsodetermined by wing twist B = к 1ϕ w . The polar pull-off coefficient does not depend onwing twist. Polar equation for a wing with geometrical twist is a square parabola withdisplaced peak (Fig. 4.17). It is possible to write down 2 B2 C xa =C− 4A ⎛ + A⎜ C ya + ⎝ B⎞ ( ⎟ = C xa min + A C ya − C yam 2 A⎠ )2 , (4.15) THEME 4 10/5/2008 41
  • B2 BWhere C xa min = C − ; C yam = − . 4A 2A From comparison of polar for flat and non-planar wings (the Fig. 4.18) it ispossible to reveal the advantages of using of a non-planar wing: 1. At C ya > C 0 twisted wing drag is less than ΔC xa = C xa tw − C xa ya flat <0 2. The maximum lift-to-drag ratio of a non-planar wing is higherK max tw > K max flat . (The wing is staying non-planar at mechanization deflection butK max tw < K max flat ). Fig. 4.17. Polar for a wing having twist Fig. 4.18. Polar for a flat and non-planar (for a non-planar wing) wings In addition, due to geometrical twist it is possible to provide wing balancingwithout increase of induced drag (non-planar wing is self-balanced). It is easy to see,that the advantages of a non-planar wing are shown at condition of B < 0 . In this casepolar peak displaces upwards. 4.5. Lift-to-drag ratio. The ratio of aerodynamic lift to the drag force obtained by dividing the lift by thedrag is called lift-to-drag ratio K : THEME 4 10/5/2008 42
  • C ya K= , (4.16) C xa The lift-to-drag ratio is one of the basic characteristics determining efficiency ofan airplane. C yaFor a flat wing K = . (4.17) 2 C x0 + AC ya Fig. 4.19 shows the dependence of K = f ( C ya ) . The lift coefficient and angle of attack at which maximum lift-to-drag ratio K max is achieved is called as optimal and designated as C ya opt and α opt . Lets define maximum lift-to-drag ratio. For this purpose we shall differentiate K by C ya and from the condition 2 ∂K C x 0 + AC ya − 2 AC ya = =0 Fig. 4.19. Dependence K = f ( C ya ) ∂ C ya ( 2 2 C x0 + AC ya )we shall define values C ya at which the lift-to-drag ratio has extremes: C x0C ya opt = . A We shall receive the formula for calculation of maximum quality having C x0substituted C ya opt = in expression (4.17). We get A C x0 A 1 K max = ; K max = . 2 2 AC x 0 ⎛ C x0 ⎞ C x0 + A⎜ ⎟ ⎝ A ⎠ THEME 4 10/5/2008 43
  • The value of K max is increased with increasing of λ , decreasing of C x0 and δ(δ = 0 for elliptical distribution of chordwise). We shall notice, that 1K max ⋅ C ya opt = and does not depend on C x0 . 2A C ya 1 For a non-planar wing - K = and K max = . 2 2 AC x 0 + B C x0 + BC ya + AC ya It is easy to find values of K max , C ya opt , α opt graphically. It follows from fig.4.20 that it is necessary to conduct a beam tangent to polar from origin of coordinates tosearch K max . The values of C ya and α in tangency point will correspond to C ya optand α opt . Fig. 4.20. Dependencies C ya = f ( α ) , K = f ( α ) , C ya = f (C xa ) and connection between them. 4.6. Distribution of aerodynamic loading along wing span. Summarizing pressure distribution chordwise, we receive C ya се ÷ = f ( z ) .Analysing the influence of wing geometrical parameters in planform on distribution of THEME 4 10/5/2008 44
  • C ya cr .s .aerodynamic loading spanwise it is convenient to use relative value C ya сr .s . = . C yaThe function C ya cr .s . = f ( z) depends on λ , χ , η and geometrical twist (refer toFig. 3.5). Lets consider a flat wing. The influence of λ , χ , η is shown on thefollowing diagrams (Figs. 4.21, 4.22, 4.23). Loading is distributed spanwise more regular with increasing of aspect ratio λ.At λ = ∞ and for an elliptical wing C ya се ÷ = 1,0 . Refer to fig. 4.21. The increasing of sweep angle χ causes growth of loading in a tip part andreduction in root cross-section for a swept-back wing ( χ > 0 ). Refer to fig. 4.22. The influence of wing taper η is similar to sweep χ effect : there is loadinggrowth in wing tip cross-sections with taper η increasing. Refer to fig. 4.23. Distribution of lift along wing span: Fig. 4.21. Depending on Fig. 4.22. Depending on Fig. 4.23. Depending onaspect ratio at condition of sweep at condition of taper at condition of χ = 0 , η = const = 1 λ = const , η = const λ = const , χ = 0 It is possible to write down approximately for a one-profile flat high-aspect-ratiowing: THEME 4 10/5/2008 45
  • ⎡ Cα ∞ ⎤ 1− z 2 2 ⎛ ( 3 + τ )⎥ ⎜ 1 − cos χ 1 ⎞ ( 1 − z ) ; 0 ≤ z ≤ 1 , yaC ya cr .s . = ⎢ 1 + − ⎟ ⎢ πλ ⎥ 3μ + 1 − z 2 b( z ) ⎝ 4⎠ ⎣ ⎦ Cα ∞ b( z ) , C α ∞ = 2π − 1,69 4 c , the values of parameter τ = f (λ , χ , η ) yaWhere μ = ya 4λare taken from the reference book or calculated. Chords are distributed spanwise for a tapered wing with straight-line edges as η − (η − 1) zfollows: b( z ) = 2 . η+1 ⎡ ⎤ ⎢η 2 + 1 ⎥, m = λ 1 Approximately τ = 0 .17 m ,η= . ⎢ (5η + 1) 3 ⎥ C α ∞ cos χ 0 .5 ya η ⎣ ⎦ 4.7. Flow stalling. Different values of C ya cr .s . along wing span are the reason of flow stalling in oneof cross-sections in which the local value C ya max cr .s . is reached. For example, the flowstalling on a flat rectangular wing occurs at increasing of angles of attack in centralcross-section. It is evidently shown in fig. 4.24., that the true angle of attack in centralcross-section of a rectangular wing is more than at the tip α real 1 > α real 2 , thereforeflow stalling will begin in that place, where the true angle of attack comes nearer tocritical. Fig. 4.24. THEME 4 10/5/2008 46
  • In general the position of flow stalling spanwise on a tapered unswept wing canbe determined by the formula z stall ≈ 1 − η , for swept tapered wing the position ofstalling spanwise is determined by the following dependence 1−η + a ( 1 + a ) 2 − (1 − η ) 2z stall ≈ , a = π η (1 − cos χ 0 .25 ) . 1 + a2 For improvement of wing aerodynamics at high angles of attack it is necessary tocreate a condition of simultaneous flow stalling spanwise, i.e. uniform distribution ofloading spanwise It can be achieved by geometrical twist application. It is necessary toapply wash-in to rectangular wing for the purpose of increasing loading in tip cross-sections, wash-out is used for unswept wing, in this case tip cross-sections are unload.(It has be noticed, that for a wing with the elliptical law of chords distribution spanwisewithout geometrical twist all cross-sections have an identical C ya cr .s . , becauseVi = const and real angles of attack α real = const are identical for such wingspanwise). THEME 4 10/5/2008 47