Aerodynamics. Finite Wings Lifting line theory Glauert s method

Transcription

1 α ( y) l Γ( y) r ( y) V c( y) β b 4 V Glauert s method b ( y) + r dy dγ y y dy Soluton procedure that transforms the lftng lne ntegro-dfferental equaton nto a system of algebrac equatons - Restrcted to symmetrc wngs wth no dhedral or sweep angles

2 Glauert s method Substtuton of ndependent varable y cos( θ ) b bsen dy ( θ ) θ d α ( θ ) l Γ( θ ) r ( θ ) V c( θ ) β ( θ ) + r 4 V 0 b ( cos( θ ) cos( θ )) dγ dθ dθ

3 The crculaton must be zero at the wng tps θ 0 and θ and symmetrc Γ ( θ ) Γ( θ ) The Fourer seres expanson of Γ( θ ) that satsfes these condtons s gven by Γ Glauert s method ( θ ) Γ sen( nθ ) n n,3,... The cosne terms are elmnated by the tp condtons and symmetry requres the elmnaton of all terms wth even n

4 Glauert s method The frst term of the seres corresponds to an ellptcal crculaton dstrbuton Γ ( θ ) Γ sen( θ ) Γ cos ( θ ) or Γ y b Γ Γ y + b

5 ω 4 Glauert s method Determnaton of the descendng velocty (downwash) 0 b( cos( θ ) cos( θ )) n,3,... nγ n cos ( nθ ) d θ ω b sen For a term seres ( θ ) n,3,... nγ n sen ( nθ ) ω Γ b

6 ftng lne equaton Glauert s method α α eff + α α ( θ ) l Γ n n,3,... sen r ( nθ ) ( θ ) V c( θ ) β ( θ ) + n,3,... nγ r bv n sen sen ( nθ ) ( θ ) Γ sen ( nθ ) r ( θ ) V c( θ ) n n,3,... l bv + ( nθ ) ( θ ) nsen r sen α ( θ ) + β ( θ )

7 hoosng a fnte number (N) of terms for the seres expanson leads to a system of algebrac equatons that satsfes + ( N ) ( nθ ) n ( nθ ) sen sen Γ n α θ ( ) ( ) ( ) + β θ r + r n l θ V c θ bv θ,3,... sen for N ponts of the nterval 0 < θ wth θ, wth,,..., N N Glauert s method ( ) ( )

8 Example for N3 Γ ( θ ) Γ sen( θ ) + Γ sen( 3θ ) + Γ sen( 5θ ) A A A θ 3 A A A 3 Glauert s method, 6 A A A θ Γ Γ Γ 3, θ α α α 3 5 ( θ) + β ( θ) ( θ ) + β ( θ ) ( ) ( ) θ3 + β θ3

9 A j Example for N3 sen Γ l (( + ( j ) ) θ ) ( + ( j ) ) sen( ( + ( j ) ) θ ) r + r ( θ ) V c( θ ) bv sen( θ ) sen( 5 6) 5sen( 5 6) A3 r + r ( 6) V c( 6) bv sen( 6) ( θ ) Γ sen( θ ) + Γ sen( 3θ ) + Γ sen( 5θ ) θ, 6 l Glauert s method θ 3, 3 θ 3 5

10 r ρ V 0 sen 0 Glauert s method Wth the knowledge of the seres coeffcents, Γ n, t s possble to calculate the lft force,, and the nduced drag force, D Γ n n,3,... ( mθ ) sen( nθ ) sen dθ b ( nθ ) sen( θ ) 0 m m dθ n n

11 Glauert s method Wth the knowledge of the seres coeffcents, Γ n, t s possble to calculate the lft force,, and the nduced drag force, D r ρbv Γ 4 The lft force depends only on the frst term of the seres. Obvously, ths does not mean that term s enough to calculate the exact value of

12 Glauert s method Wth the knowledge of the seres coeffcents, Γ n, t s possble to calculate the lft force,, and the nduced drag force, D D 4 ( nθ ) Γ sen( nθ ) ρ nγ n sen 0 n,3,... n,3,... n dθ D ρ nγn 8 n,3,...

13 Glauert s method Wth the knowledge of the seres coeffcents, Γ n, t s possble to calculate the lft force,, and the nduced drag force, D D δ ργ 8 ( + δ ) Γ n n n 3,5,... Γ

14 Glauert s method Wth the knowledge of the seres coeffcents, Γ n, t s possble to calculate the lft force,, and the nduced drag force, D Γ 4 r ρbv D r ρ V b ( + δ )

15 ft and nduced drag coeffcents ( ) δ ρ ρ + Γ D S V D bv S V r r r Glauert s method

16 Glauert s method D D r ρ V S ( + δ ) The nduced drag coeffcent s proportonal to the square of the lft coeffcent and t goes to zero the aspect rato,, tends to nfnty,.e. b

17 Wngs wth mnmum nduced drag The nduced drag coeffcent s gven by D D ρ V r ( + δ ) wth S δ Γ n n n 3,5,... Γ By defnton δ 0, therefore, the mnmum nduced drag corresponds to δ 0,.e. Γ 0 for n > n

18 Wngs wth mnmum nduced drag One term seres corresponds to an ellptc crculaton dstrbuton y Γ( θ ) Γ sen( θ ) Γ cos ( θ ) Γ b The nduced velocty, ω, s constant along the span and equal to ω Γ b

19 ft and nduced drag coeffcents gven by Γ Γ ρ ρ,, 4 D b V D bv bv r r r Wngs wth mnmum nduced drag

20 Wngs wth mnmum nduced drag onstant nduced angle of attack along the span α Γ bv r If the wng has no twst the geometrc angle of attack does not change along the span, ths leads to a constant α α eff α

21 Wngs wth mnmum nduced drag The lft coeffcent,, may be calculated from b b b l dy For a wng wth ellptcal crculaton dstrbuton ( e α ct ) and no twst ( e α ct ) and the same arfol secton along the span b l dy l b b

22 Wngs wth mnmum nduced drag For l, becomes or α α α eff + α l l β + + ( α + β )

23 Wngs wth mnmum nduced drag The rato between the slopes of the lnes that gve the lft coeffcent as a functon of the angle of attack for the wng and ts secton (arfol) s gven by l l + If we assume l then l +

24 l Wngs wth mnmum nduced drag l

25 α Wngs wth mnmum nduced drag An ellptcal crculaton dstrbuton may be obtaned for a wng wth no twst and the same secton (arfol) along the span sen( θ ) r β + V c( θ ) ( θ ) c sen( θ ) Γ Γ r l bv c r where the root chord, c r, s related to the requred lft and to the geometrc propertes of the wng secton (arfol thckness and camber)

26 Wngs wth mnmum nduced drag

27 Wngs wth mnmum nduced drag

28 Wngs wth mnmum nduced drag

29 Wngs wth mnmum nduced drag A wng wth ellptcal planform s harder to buld than a rectangular wng A tapered wng may produce a nearly ellptcal crculaton dstrbuton wth a smpler constructon than an ellptcal wng Taper rato s the rato between the tp chord, c t and the root chord, c r

30 Wngs wth mnmum nduced drag AR e δ Γ n n n 3,5,... Γ

31 Wngs wth mnmum nduced drag

32 Introducton of vscous effects The fnte wng lft coeffcent depends on the aspect rato and on the lfy coeffcent of the wng sectons (arfols) l ( y) ( y) α ( y) + β ( y) l b b b ( ) eff ( y) If l and β are determned for a vscous flow the effects of vscosty are ncluded n the determnaton of the wng lft coeffcent l dy

33 Introducton of vscous effects For a vscous flud, the drag coeffcent of the wng sectons (arfols) s not zero. Therefore, D d + D d δ arfol frcton d pressure arfol wth ( ) b d d ( y) dy frcton arfol b b τw b ( ) b d d ( y) pressure ( ) ( ) arfol b δ * dy

34 In a real flow, there s only frcton and pressure drag (resstance) D where pressure + D D frcton D pressure ( ) b ( y) b ( ) b + ( + δ ) ( y) dy + ( + δ ) d Introducton of vscous effects pressure D frcton arfol d frcton arfol b b b d δ * d τw dy

35 Prandtl transformaton formulae Prandtl s transformaton formulae allow the converson of the functons D ( ) and (α) obtaned (expermentally or numercally) for a gven aspect rato, nto functons vald for wngs wth the same secton but dfferent aspect ratos - It s assumed that the tp effects depend only on the aspect rato,, and that the nduced drag and nduced angle of attack correspond to the values obtaned for an ellptcal crculaton dstrbuton

36 Assumptons: - Induced resstance coeffcent gven by - Induced angle of attack equal to - Drag coeffcent s equal to the sum of the wng secton (arfol) drag and nduced drag coeffcents -, β and are ndependent of l darfol Prandtl transformaton formulae - The lft coeffcent of the wng sectons s constant along the span and equal to

37 + + d D d D perfl perfl D D D d D d D arfol arfol Prandtl transformaton formulae D as a functon of : For two wngs wth dfferent aspect ratos, and, and the same

38 Prandtl transformaton formulae D as a functon of :

39 α as a functon of : For two wngs wth dfferent aspect ratos, and, and the same + + β α α α α l eff α α α β α β α l l Prandtl transformaton formulae

40 Prandtl transformaton formulae α as a functon of :