ii (Preface) Mach number: M; : M = V/a 0.3 5% km/h( M=0.4) (thermodynamics) ( SST: supersonic transport (trade-off) ( ) (fluid dynamics) (compr

Transcription

1 Incompressible Fluid Dynamics Yoshiaki NAKAMURA Professor of Chubu University Emeritus Professor of Nagoya University April, 2014

2 ii (Preface) Mach number: M; : M = V/a 0.3 5% km/h( M=0.4) (thermodynamics) ( SST: supersonic transport (trade-off) ( ) (fluid dynamics) (compressible fluid dynamics) M = 0 M = 0 V = 0 M = 0 0 (sound speed: a) (viscosity) (vorticity) (potential flow) ( )

3 iii (lift)

4

5 v ()

6 vi

7 1 1 (Incompressible Fluid Dynamics) 1.1 (incompressible fluid) (pressure) (volume) (density) (liquid) (air) (gas) (state equation) ρ = (1.1) ρ (nonexpansion) 273K 999.8kg/m 3 313K 992.2kg/m 3 40 C 0.8% (isopicnic fluid) 1.2 (governing equation)

8 div v = v = u x + v y + w z = 0 (1.2) v v = (u, v, w) u, v, w x, y, z (velocity component) 0 v+ u u+ v 1.1: 2 0 (u + δu)δy uδy + (v + δv)δx vδx = (u + u v δx)δy uδy + (v + δy)δx vδx x y ( u = x + v ) δxδy (1.3) y u x + v = 0 div v = 0 (1.4) y () u ρu divρ v = 0 (1.5) div v = 0 (1.5)

9 (Navier-Stokes equations ) (Euler equations; Leonhard Euler 1757 ) ( u ρ t + u u x + v u y + w u ) = ρx p z x ( v ρ t + u v x + v v y + w v ) = ρy p z y ( w ρ t + u w x + v w y + w w ) = ρz p z z (x ) (1.6) (y ) (1.7) (z ) (1.8) (X, Y, Z) (body force) (gravitational force) (X, Y, Z) = (g x, g y, g z ) g (gravitaional acceleration) g = (g x, g y, g z ) 1.6) 1.8 (unsteady term) 4 (convective terms) (energy equation) ρ (kinetic energy) (1/2)v 2 gh (g h p/ρ (internal energy) e i de i = C(T ) dt (1.9) C(T ) (specific heat) 1kg 1K (J) (SI J/(kg K)) e i = CT (1.10)

10 4 1 (first law of thermodynamics) dq = de i + pdv (1.11) dq v (; specific volume) v = 1/ρdv = 0 dq = de i (1.12) : h (enthalpy) h e i + p ρ (1.13) 1.2: H 0 = 1 2 v2 + e i + p + gy (1.14) ρ H 0 (total enthalpy) (gravitational potential energy) H 0 = 1 2 v2 + e i + p ρ () (1.15) Q W W < 0 Q + W = H 02 H 01 (1.16) H 01 H 02 H 02 = H 01 (1.17)

11 (external force) (conservative force) X = (X, Y, Z) X = grad Ω = Ω (1.18) ()Ω grad = (gradient) (differential operator) = ( / x, / y, / z) X = Ω x, Y = Ω y, Z = Ω z (1.19) X = g x, Y = g y, Z = g z Ω = g x x + g y y + g z z (1.20) (1.14) H 0 = 1 2 v2 + e i + p ρ Ω (1.21) y g y = g Ω = gy (1.14) (1.21) 1.3 streamline () (pathline particle path) δx δs = u V, δy δs = v V, δz δs = w V (1.22) s δs s x = x(s), y = y(s), z = z(s) V V = u 2 + v 2 + w 2 δt δx u = δy v = δz w 1.22u u/ x = δt (1.23) u u x = V x u s x (1.24)

12 : x ( u u x + v u y + w u z = V x u s x + V y u s y + V z u s z ( u x = V x s + u y y s + u ) z z s = V u s (1.25) (1.25) u = u(x(s), y(s), z(s)) (1.26) u u x + v u y + w u z = V u s (1.27) V = (u, v, w) δ s = (δx, δy, δz) y z u u x + v u y + w u z u v x + v v y + w v z u w x + v w y + w w z = V u s = V v s = V w s (1.28) (1.29) (1.30) x, y, z ρv u s ρv v s ρv w s = p x = p y = p z (1.31) (1.32) (1.33)

13 δx, δy, δz ρv u s ρv v s ρv w s ρ (1.22) s p δx = δx (1.34) x p δy = δy (1.35) y δz = pδz (1.36) z V u s u V δs = 1 ρ V v s v V δs = 1 ρ V w s w V δs = 1 ρ ( u v2 2 + w2 2 ) δs = 1 ρ p δx x (1.37) p δy y (1.38) p δz z (1.39) p δs (1.40) s ( u 2 + v 2 + w 2 ) δs = 1 s 2 ρ p s Ω δs + δs (1.41) s ( V 2 ) dp s 2 + ρ Ω = 0 (1.42) (barotropic fluid) () p ρ p = p(ρ) (1.43) F (p) F (p) = p dp ρ(p) (1.44)

14 8 1 F (p) s F s = F p p s = 1 p ρ s (1.45) 1.44 p dp s ρ(p) = 1 p ρ s (1.46) () (1.42) (Bernoulli s equation; Daniel Bernoulli 1738 ) 1 2 V 2 + p Ω = (1.47) ρ y Ω Ω = gy (1.48) () (hydraulic head) (1.47) (1.48) g 1 2 V 2 + p ρ + gy = C 1 = const (1.49) 1 V 2 2 g + p ρg + y = C 2 = const (1.50) (velocity head) (pressure head) (elevation head) (total head) (1atm) 10m () (1.47) p ρ V 2 = p ρv 2 (1.51)

15 () p 0 p ρ V 2 = p ρv 2 = p 0 (1.52) () (pressure coefficient) C p = p p (1/2)ρ V 2 (1.53) C p = p p (1/2)ρ V 2 ( ) 2 V = 1 (1.54) V C p 1 C p 1 () () [ ] v ρ + ( v ) v = p (1.55) t ( v ) v ( A B) = ( A ) B + ( B ) A + A rot B + B rot A (1.56) rot rot = A = v, B = v ( ) v v 2 2 = 2( v ) v + 2 v ω ( v ) v = v ω (1.57) 2 ω = v ω = 0 (1.57) (1.55) [ ( )] v v 2 ρ t + = p (1.58) 2 v = Φ (1.58) [ ρ Φ ( )] v 2 t + = p (1.59) 2

16 10 1 (1.59) ( Φ t + v2 2 + p ) = 0 Φ ρ t + v2 2 + p = const (1.60) ρ Φ/ t = 0 () Q W = 0) 1.16 Q = H 02 H ( 01 1 = 2 v2 + e i + p ) ρ Ω = e i2 e i1 + 2 ( 1 ( 1 2 v2 + p ρ Ω 2 v2 + e i + p ) ρ Ω 1 ) ( 1 2 v2 + p ) ρ Ω 2 1 (1.61) 1.47 Q = e i2 e i1 = C(T 2 T 1 ) (1.62) (1.10) (calorically perfect gas) 1.4 ϕ ( ) 2 2 ϕ = x y z 2 ϕ = 0 (1.63) P(x, y, z) ϕ ( ϕ ϕ(x, y, z) = 1 ) ds + ϕ ( 1 ) ds (1.64) n 4πr ν 4πr S S n ν ( n = (n 1, n 2, n 3 ) x, y, ), z ( ) ν = (n 1, n 2, n 3 ),, x 1 y 1 z 1 (1.65)

17 : n = (n 1, n 2, n 3 ) S r P(x, y, z) (x 1, y 1, z 1 ) r = (x x 1 ) 2 + (y y 1 ) 2 + (z z 1 ) 2 (1.66) (1.64) : 1 ( 1 ) 4πr ν 4πr (1.67) (1.64) ( ϕ(x, y, z) = C 1 1 ) ds + 4πr S S C 2 ν ( 1 ) ds (1.68) 4πr C 1, C source v r < 0 (sink) (diffusion) (potential flow) ω ω = rot u = u (1.69) ω = (ξ, η, ζ) ξ = w y v z, η = u z w x, ζ = v x u y (1.70)

18 : 0 w y v z = 0, u z w x = 0, v x u y = 0 (1.71) ϕ u ϕ x, v ϕ y, w ϕ z (1.72) V = ϕ = grad ϕ, V = (u, v, w) (1.73) (polar coordinates) v r r σ σ ϕ v r σ = 4πr 2 v r (1.74) v r = σ 4πr 2 (1.75) v r = ϕ r (1.76) ϕ r = σ 4πr 2 (1.77) ϕ ϕ = σ 4πr (1.78)

19 σ (sink) x = x 1, y = y 1, z = z 1 σ = u x + v y + w = div v (1.79) z σ 0 = u x + v y + w = div v (1.80) z () (1.79) Q S Gauss () div v dv = n v ds (1.81) (1.79) (1.81) σ dv = n v ds (1.82) σ () 1.72 (1.79) ϕ σ = 2 ϕ x ϕ y ϕ z 2 (1.83) 2 ϕ = σ (1.84) (partial differential equation) (Poisson equation) Q σ = 0 2 ϕ = 0 (1.85) (Laplace equation) ϕ ϕ(x, y, z) = 1 σ(ξ, η, ζ) dξdηdζ 4π r = 1 σ(ξ, η, ζ) dξdηdζ (1.86) 4π (x ξ)2 + (y η) 2 + (z ζ) 2

20 14 1 (x, y, z) P (ξ, η, ζ) σ () ) (M < 1) x M ϕ (1 M 2 ) 2 ϕ x ϕ y ϕ z 2 = 0 (1.87) ϕ V = U i + grad ϕ ( i x ) (1.88) β = 1 M 2 (1.87) β2 ỹ = βy, 2 ϕ x ϕ ỹ ϕ z 2 = 0 (1.89) z = βz 1.78 ϕ = (x, y, z) σ 4π x 2 + ỹ 2 + z 2 (1.90) ϕ = σ 4π x 2 + β 2 (y 2 + z 2 ) = σ 4π x 2 + (1 M 2 )(y 2 + z 2 ) (1.91) ϕ (surface of the second order) (ellipsoid) ( ) ( 2 x + σ/4πϕ y σ/4πϕ β ) 2 + ( z σ/4πϕ β ) 2 = 1 (1.92) ψ = constϕ = const

21 (M > 1) x M (M 2 1) 2 ϕ x 2 2 ϕ y 2 2 ϕ z 2 = 0 (1.93) β = M 2 1 β 2 ỹ = βy, 2 ϕ x 2 2 ϕ ỹ 2 2 ϕ z 2 = 0 (1.94) z = βz i 2 ϕ x ϕ (iβy) ϕ (iβz) 2 = 0 (1.95) ϕ 1.78 ϕ = σ 4π x 2 + (iβy) 2 + (iβz) 2 = σ 4π x 2 β 2 (y 2 + z 2 ) (1.96) ϕ hyperboloid of two sheets ( ) ( 2 x σ/4πϕ y σ/4πϕ β ) 2 ( z σ/4πϕ β ) 2 = 1 (1.97) ψ = constϕ = const 1.6 (doublet) x ( a, 0, 0) (a, 0, 0) (singular) (1.78) r 1, r 2 ϕ = σ + σ (1.98) 4πr 1 4πr 2 r 1 = { (x + a) 2 + y 2 + z 2} 1/2, r2 = { (x a) 2 + y 2 + z 2} 1/2 (1.99) r = { (x ξ) 2 + y 2 + z 2} 1/2 (1.100)

22 16 1 r 1, r 2 ( ) ( ) = + r 1 r ξ=0 ξ r ( ) ( ) = + r 2 r ξ r (1.98) (1.101) ξ=0 ϕ(x, y, z) = 2σa 4π 1 ξ r = 1 r r 2 ξ = x ξ r 3 ( a) + O(a 2 ) ξ=o (+a) + O(a 2 ) ξ=o ( ) 1 + O(a 3 ) (1.101) ξ r ξ=o ( ) 1 = x ξ r ξ=0 r 3 (1.102) a 0, σ, σa = µ = const (1.103) ϕ(x, y, z) = µ x 2π r 3 (1.104) x (1.104) ϕ(x, y, z) = µ 1 x 2π r 2 r = µ cos θ 2π r 2 (1.105) θ r x (2.43) () (1.67) 2 2 ν = x 1 ν ( 1 4πr r 2 = (x x 1 ) 2 + (y y 1 ) 2 + (z z 1 ) 2 ϕ = 1 4π ( 1 r 2 ) ) = ( 1 ) x 1 4πr r = x x 1 x 1 r r = 1 ( 1 ) ( ) x x1 x 1 4π r 2 r 2 x 1 = 0 ϕ = 1 ( 1 ) (x ) 4π r 2 = 1 ( x ) r 4π r 3 (1.106) (1.107) (1.108) (1.109) 2

23 (vortex) core (Rankine ) (θ ) V = v θ (v r = 0) (vortex core) solid rotation v θ = ωr (1.110) ω (angular velocity) z ζ (x, y, z) (r, θ, z) ζ = v x u y = 1 v θ r r r 1 v r r θ (1.111) v r 0 ζ = 1 r v θ r r = 2ω (1.112) 2 () r (v θr) = 0 (1.113) v θ = C r (1.114) C ωa = C a C = ωa 2 (1.115) v θ = ωr (1.116) v θ = ωa2 r (1.117)

24 : ( ) ρ v2 θ = p r r (1.118) v θ 1.117r p 0 p = ρω2 a r 2 (1.119) p = 0 at r (1.120) p = ρω2 2 r2 + p 0 (1.121) p 0 r = 0r = a p 0 p 0 = ρω 2 a 2 (1.122) p = ρω2 a 2 2 ) (2 r2 a 2 (1.123)

25 : 1.8 circulation Γ = v d s (1.124) Stokes Γ = ( v) ds = (rot v) ds = ω d S (1.125) ω = rot v = v (1.126) lift (Kuta-Joukovski camber (Lord Kelvin) DδΓ Dt = D (uδx + vδy + wδz) (1.127) Dt Duδx Dt = Du δx + udδx Dt Dt = Du δx + uδu (1.128) Dt

26 20 1 Dδx Dt = δu, Dδx Dt = δ Dx Dt Dδy Dt = δv, = δu (1.129) Dδz Dt = δw (1.130) ( D Du (uδx) = δx + uδu = X 1 ) p δx + uδu (1.131) Dt Dt ρ x (x ) x, y, z D Dt (uδx) = D Dt (vδy) = D Dt (wδz) = Du Dt = X 1 p ρ x (1.132) ( X 1 ) p δx + uδu, (1.133) ρ x ( Y 1 ) p δy + vδv, (1.134) ρ y ( Z 1 ) p δz + wδw (1.135) ρ z D Dt (uδx + vδy + wδz) = Xδx + Y δy + Zδz 1 ρ δp + δ u2 + v 2 + w 2 2 X Ω X = (X, Y, Z) = ( Ω x, Ω y, Ω ) z (1.136) (1.137) DδΓ Dt = δω 1 ρ δp + δ u2 + v 2 + w 2 2 (1.138) A A [ DΓ dp Dt = Ω ρ + u2 + v 2 + w 2 ] A = 0 (1.139) 2 (Thomson) (Lord Kelvin) (1867) A

27 Helmholtz, 1858) C 0 Γ = 0 Qü Â Èü Ê C QŠÇ 1.8: Γ = ω nds ωs

28 22 1 Γ = Γ 1 2 Γ 2 QŠÇ Γ 1 1.9: ω = (ξ, η, ζ) div v = v = u x + v y + w z = 0 (1.140) ξ = w y v z, η = u z w x, ζ = v x u y (1.141) v = 0 E v = E (1.142) 0 () B B = 0 A B = A (1.143) () E = (E, F, G) (1.144) (1.142) u = G y F z, v = E z G x, w = F x E y (1.145)

29 ξ = ( E x x + F y + G ) 2 E (1.146) z η = ( E y x + F y + G ) 2 F (1.147) z ζ = ( E z x + F y + G ) 2 G (1.148) z E x + F y + G z = 0 (1.149) 2 E = ξ, 2 F = η, 2 G = ζ (1.150) (1.86) E, F, G E(x, y, z) = 1 ξ(x1, y 1, z 1 ) dx 1 dy 1 dz 1 (1.151) 4π r F (x, y, z) = 1 η(x1, y 1, z 1 ) dx 1 dy 1 dz 1 (1.152) 4π r G(x, y, z) = 1 ζ(x1, y 1, z 1 ) dx 1 dy 1 dz 1 (1.153) 4π r (x 1, y 1, z 1 ) P (x, y, z) r r = (x x 1 ) 2 + (y y 1 ) 2 + (z z 1 ) 2 (1.154) E(x, y, z), F (x, y, z), G(x, y, z) (x, y, z) (u, v, w) noslip condition) u = 0, v = 0, w = 0 (1.155) u = dx dt, v = dy dt, w = dz dt (1.156)

30 24 1 (slip condition) 0 V n = 0 (1.157) n V n 0 V 1.10: V B V B n = V n (1.158) n ) ( p d 2 ) n = x dt 2, d2 y dt 2, d2 z dt 2 n (1.159) n

31 25 2 (functions of a complex variable) ψ ϕ () (complex number plane) z = x + iy (2.1) i i = 1 x y W (z) f(z) W (z) = f(z) = ϕ + iψ (2.2) ϕ ψ dw dz = lim W z 0 z = lim f(z + z) f(z) z 0 z (2.3) z 0 x : y P f(z) df/dz f(z) (regular) f(z) dw dz = lim x, y 0 [ϕ(x + x, y + y) ϕ(x, y)] + i[ψ(x + x, y + y) ψ(x, y)] x + i y

32 26 2 Q(x+ƒ x,y+ƒ y) ƒ y P (x,y) ƒ x 2.1: = lim x, y 0 = lim x, y 0 ( ) ϕ ϕ x x + y y + i ( ) ϕ x + i ψ x x + x + i y x + i y ( ) ψ ψ x x + y y ( ϕ y + i ψ y ) y (2.4) x : y ( ) ( ) ϕ ϕ x + i ψ : x y + i ψ = 1 : i (2.5) y ϕ x = ψ y, ϕ y = ψ x (2.6) (the Cauchy-Riemann equations) f(z) (differentiable) (necessary and sufficient condition) f(z) ϕ ψ (Laplace s equation) 2 ϕ x ϕ y 2 = 0, 2 ψ x ψ y 2 = 0 (2.7) ϕ ψ 2.7 ϕ =0 v x u y = 0 u = ϕ x, v = ϕ y ψ (2.8) u x v y = 0 u = ψ y, v = ψ x (2.9)

33 () 2 (plane flow) (axisymmetric flow) 3 () () (2.8) (2.9) v = ϕ = grad ϕ, v = ψ k (2.10) k (x, y, z) z () () (harmonic) (harmonic function) () W z z = x x z = iy dw dz = ϕ x + i ψ = u iv (2.11) x dw dz = ϕ iy + i ψ iy = i ϕ y + ψ = iv + u (2.12) y dw dz = u iv (2.13) (conjugate velocity) W (z) z dw dz = dw d z = u + iv (2.14) dw dz dw dz = dw dz dw d z = (u iv)(u + iv) = u2 + v 2 (2.15) 2.2 (uniform flow) x y U W = Uz (2.16)

34 28 2 (2.2) (2.1) W = ϕ + iψ = U(x + iy) ϕ = Ux, ψ = Uy (2.17) (u, v) u = ϕ x = ψ y = U, v = ϕ y = ψ x = 0 (2.18) α dw dz = u iv = U cos α iu sin α = U(cos α i sin α) = Ue iα (2.19) α W = Ue iα z (2.20) z ze iα (2.21) U α U sinα U cosα 2.2: α () e iθ = cos θ + i sin θ (2.22) 2.3 (source flow) W W = m ln z (2.23) 2π

35 z (r, θ) z = re iθ (2.24) ψ ϕ ϕ = m 2π ln r, ψ = m 2π θ (2.25) 2.3: (r) v r (θ) v θ v r = ϕ r = 1 ψ r θ = m 2πr, v θ = 1 ϕ r θ = ψ r = 0 (2.26) r = ψ = (2.25) θ = ψ = const ϕ = const () (r, θ) v = 1 r 2π 0 rv r r + 1 r v θ θ = 0 rv r = ψ θ, v r r dθ = v r 2πr = v θ = ψ r (2.27) m 2πr = m (2.28) 2πr (2.23) m m (sink) 2.4 superimpose (ϕ ψ

36 30 2 ) (m > 0) (m < 0) (2.16) (2.23) W = Uz + m ln z (2.29) 2π 2.4: dw dz = u iv = U + m 1 2π z z = re iθ u = U + m 2π cos θ, v = m sin θ r 2π r 0 u = v = 0 θ = π r s = m 2πU ( (x, y) = m ) 2πU, 0 ( 0) 0 (2.30) (2.31) (2.32) (2.33) ( (x, y) = 0, m ) 2πU () (2.34) 2.5 x = a m x = a m W = W source + W sink = m 2π m ln(z + a) + 2π ln(z a) = m 2π ln z + a z a (2.35)

37 : m 0 () () ( a 0 )2.35 W = m 2π ln z + a z a = m ( 2π ln 1 + 2a ) z a (2.36) a 2a/(z a) << 1 (Taylor expansion) W = m 2π 2a z a (2.37) a 0 0 a () a a(0) m( ) µ (2.38) W = µ π 1 z (2.39) doublet µ

38 32 2 ψ=ˆê è 2.6: (2.36) (2.39) W = m 2π ln z + a z a = m ln(z + a) ln(z a) 2a (2.40) 2π 2a a 0 W = 2am ( ) d ln(z + b) 2π db () b=0 = 2µ 1 2π z = µ 1 π z (2.41) (2.39) ϕ ψ W = µ (cos θ i sin θ) (2.42) πr ϕ = µ πr cos θ, ψ = µ sin θ (2.43) πr ( x x x (counterclockwise) α W W = µ 1 π ze iα (2.44) y α = π/2 () (vortex) W = iγ ln z (2.45) 2π

39 i (r, θ) W = Γ 2π θ iγ ln r (2.46) 2π ϕ = Γ 2π θ, ψ = Γ ln r (2.47) 2π (2.25) θ ln r (2.45) i ψ = const r = const (v r, v θ ) v r = ϕ r = 1 ψ r θ = 0, v θ = 1 r ϕ θ = ψ r = Γ 2πr (2.48) θ v θ ds = Γ 2πr = Γ (2.49) 2πr Γ (circulation) Γ Γ () (polar coordinates) ψ div v = r (rv r) + θ v θ = 0 (2.50) rv r = ψ θ, v θ = ψ r (2.51) 1 r r (rv θ) 1 r ϕ v r θ = 0 (2.52) () rv θ = ϕ θ, v r = ϕ r (2.53) ()

40 () (corner) W = Cz n (2.54) z = re iθ W = ϕ + iθ ϕ ψ ϕ = Cr n cos nθ, ψ = Cr n sin nθ (2.55) ψ = ψ = 0 sin nθ = 0 (2.56) nθ = 0, nθ = π (2.57) θ c θ c = π/n (2.58) n θ c n > 1 0 < θ c < π n < 1 π < θ c v θ = 1 r ϕ θ = Cnrn 1 sin nθ, v r = ϕ r = Cnrn 1 cos nθ (2.59) θ = 0 v r (z = 0) r 0 n > 1 v r 0 n < 1 v r () () (wedge) 2.7

41 2.8. () 35 ƒõ0 Šp ƒõˆê è ƒ c Ì Ì Šp ƒõˆê è ƒõ ƒ c Ì a) n>1 Ìê b) 1>n>1/2 Ìê 2.7: () a) (half-angle ) δ δ = π θ c δ = π/2 θ c = π/2 δ ()

42

43 37 3 (complex function) (potential flow) (r, θ) 2 ϕ = 2 ϕ r ϕ r r ϕ r 2 θ 2 = 0 (3.1) ϕ v r = ϕ r, v θ = 1 r ϕ θ (3.2) 3.1 (separation of variables) ϕ = R(r)Θ(θ) (3.3) 3.1 Θ d2 R dr 2 + Θ r dr dr + R d 2 Θ r 2 dθ 2 = 0 (3.4) r 2 d 2 R R dr 2 + r dr R dr = 1 d 2 Θ Θ dθ 2 = n2 (3.5) r θ n 2 n n 0 θ 2π θ d 2 Θ dθ 2 = n2 Θ (3.6)

44 38 3 Θ = a 1 sin nθ + b 1 cos nθ (3.7) a 1 b 1 θ 3.7θ θ + 2π n 3.3 T (r) S(r) 3.5 ϕ = T (r) cos nθ + S(r) sin nθ (3.8) r 2 d 2 R R dr 2 + r dr R dr n2 = 0 (3.9) d 2 R dr r dr dr n2 r 2 R = 0 (3.10) R = r m (3.11) 3.9 m 2 = n 2 m = ±n (3.12) R = r n or R = r n (3.13) (3.8) T (r) S(r) T = A n r n + B n r n, S = C n r n + D n r n (3.14) n 0 ϕ 3.8 ϕ = {(A n r n + B n r n ) cos nθ + (C n r n + D n r n ) sin nθ} (3.15) n=0 n = 0 ϕ 3.2 n = 0 n 1 ϕ = {(A n r n + B n r n ) cos nθ + (C n r n + D n r n ) sin nθ} (3.16) n=1

45 U (v n ) r=a = (v r ) r=a = U cos θ (3.17) v n v θ y -Ucosθ U ˆÚ x o θ θ U x 3.1: 0 (stationary state) ( ) ϕ = U cos θ, r r=a ( ) ϕ = 0 (3.18) r r ϕ r = (A n nr n 1 B n nr n 1 ) cos nθ + (C n nr n 1 D n nr n 1 ) sin nθ (3.19) n=1

46 40 3 r ϕ/ r 0 n 1 r n r n 3.16 ϕ = A n = 0, C n = 0 (3.20) r n (B n cos nθ + D n sin nθ) (3.21) n= r (3.18) ( ) ϕ = r r=a a n 1 n(b n cos nθ + D n sin nθ) = U cos θ (3.22) n=1 2 3 ϕ 3.21 n = 1 B 1 = Ua 2 (3.23) n 1 D n = 0 (3.24) n 2 B n = 0 (3.25) ϕ = Ua 2 cos θ r (3.26) Cauchy-Riemann ϕ r = 1 ψ r θ, 1 r ϕ θ = ψ r () (2.4) (3.27) ψ ψ = Ua 2 sin θ r W (3.27) (3.28) W = ϕ + iψ = Ua 2 cos θ r iua 2 sin θ r = Ua2 r (cos θ i sin θ) = Ua2 Ua2 = reiθ z (3.29) 2 (doublet) 2.39 U

47 : 2 x ( (3.2) ) W = µ π 1 z (3.30) µ π = a2 U (3.31) µ = πa 2 U (3.32) 3.4 ψ = 0 ψ = 0 ψ = 0 3.3:

48 42 3 W (3.29) (Uz) W = Uz + a2 U z (3.33) z = re iθ W = ϕ + iψ ψ ϕ ϕ = U ) (r + a2 cos θ, r 3.34ψ = 0 ψ = U ) (r a2 sin θ (3.34) r r = a θ = 0, π (3.35) (v θ ) r=a = ( ) 1 ϕ = 2U sin θ (3.36) r θ r=a (argument)θ θ 3.4: θ θ = π/2 2 3/2 3 c p p p c p p p 1 2 ρu 2 = 1 ( vθ ) 2 = 1 4 sin 2 θ (3.37) U

49 θ = 0 θ = πc p = 1 θ = π/2 θ = 3π/2 3 (3.5) : (r, θ, ϕ) ( = 2 = 1 r 2 r Φ ( r 2 r ) + Φ(r, θ) = U Φ = 0 (3.38) 1 r 2 sin θ ( sin θ ) + θ θ ) (x + a3 2r 2 cos θ 1 r 2 sin 2 θ 2 ϕ 2 (3.39) (3.40) x U ) Φ(r, θ) = U (1 + a3 2r 3 r cos θ (3.41) (r, θ) ( 3.1 ) v r = Φ r = U v θ = 1 r Φ θ = U (1 a3 r 3 (1 + a3 ) cos θ (3.42) ) 2r 3 sin θ (3.43) (v θ ) r=a = 3 U sin θ (3.44) 2

50 44 3 c p = sin2 θ (3.45) C p = 5/4 C p = 3 () (1.78) 3 2 ((x, y, z) = ( a, 0, 0) (a, 0, 0) a 0 ψ = 0 œ b Ý _ œ ψ = 0 Γ = 0 Ìê œ b Ý _ œ ψ = 0 ψ = 0 - Γ < 4ƒÎaU Ìê 3.6: 3.5 (circulation) ) W (z) = U (z + a2 i Γ z 2π ln z a (3.46) a Γ ( 0 )

51 (v θ ) r=a U Γ = 2 sin θ + Γ 2πaU = 0 (3.47) Γ = 4πaU sin θ (3.48) Γ (real number) 1 sin θ 1 0 Γ 4πaU (Γ < 0) (3.49) Γ (Γ 0) Γ > 0 Ψ Ψ Ψ Ψ 3.7: Γ = 4πaU θ = π/2 Γ < 4πaU ( 3.7 ) 3.46 z 0 ) dw (1 dz = U a2 z 2 i Γ 1 2π z = 0 (3.50)

52 46 3 z = i Γ 2π ± Γ2 4π + 4U 2 a 2 2 2U (3.51) (the Magnus effect) ϕ 3.46 ) ϕ = U (r + a2 cos θ + Γ r 2π θ (3.52) θ v θ = 1 r ϕ θ = 1 r { U ) (r + a2 sin θ + Γ } r 2π r = a (v θ ) r=a = 2U sin θ + Γ 2πa p p = 1 2 ρu ρ(v θ) 2 r=a = 1 ( [1 2 ρu 2 2 sin θ + Γ ) ] 2 2πaU (3.53) (3.54) (3.55) C p C p = p p ( (1/2)ρU 2 = 1 2 sin θ + Γ ) 2 (3.56) 2πaU () (lift)l (drag)d (Kutta- Joukovski theorem; (5.45) L = ρuγ, D = 0 (3.57) Γ Γ

53 α W z ze iα ln z/a 3.8: α α W = U (ze iα + a2 e iα ) i Γ z 2π ln z a (3.58) O z 0 W = U(z z 0 )e iα + U a2 e iα i Γ z z 0 2π ln z z 0 a (3.59)

54

55 49 4 (conformal mapping) z = f(ζ) (4.1) z ζ (4.1) z = x + iy, ζ = ξ + iη (4.2) x = x(ξ, η), y = y(ξ, η) (4.3) W (ζ) = ϕ + iψ ζ z η ~ ƒœ`ó y o ƒä ½ Ê ξ ÏŠ o ƒì ½ Ê x 4.1: 4.1 f (ζ) 0 ζ c 1 ζ = ζ 1 (s) = ξ 1 (s) + i η 1 (s) (4.4) s s

56 50 4 η 0 s S 0 t 1 œ ζ 0 ƒä ½ Ê C1 ξ ŽÊ œ ÏŠ y 0 t' 1 C' 1 œ z 0 Z ½ Ê x 4.2: ( dξ1 (s) t 1 =, dη ) 1(s) ds ds t 1 c 1 s s (4.5) dξ dη 2 1 = ds 2 (4.6) t 1 dζ 1 (s) ds = dξ 1(s) ds + i dη 1(s) ds = cos θ 1 (s) + i sin θ 1 (s) = e iθ 1(s) (4.7) θ 1 (s) ζ t 1 ξ (argument) ƒå t 1 ƒæ1 C1 ƒì 4.3: (argument) z z = f(ζ) (4.8) f ζ z z z = z 1 (s) = f(ζ 1 (s)) = x 1 (s) + i y 1 (s) (4.9)

57 s z C 1 ζ dz 1 (s) ds = dx 1(s) ds + i dy 1(s) ds z = f(ζ) ( ) dz 1 (s) df(ζ) = dζ 1 ds dζ ds = f (ζ 1 (s)) e iθ 1(s) ζ 1 (s) (4.10) (4.11) z = z 0 (4.11) (argument) { } dz1 (s) arg = arg {f (ζ 0 )} + θ 1 (s 0 ) (4.12) ds z=z 0 ζ θ 1 z arg {f (ζ 0 )} ζ ζ 0 C 2 C 1 { } dz2 (s) arg = arg {f (ζ 0 )} + θ 2 (s 0 ) (4.13) ds : () { } { } dz2 (s) dz1 (s) arg arg = θ 2 (s 0 ) θ 1 (s 0 ) (4.14) ds ds z (C 1 C 2) ζ (C 1 C 2 )

58 52 4 f (ζ 0 ) = f (ζ 0 ) ζ z z = f(ζ) = ζ + a2 ζ (4.15) a (Joukovski transformation) 4.15 z 2a (ζ a)2 = z + 2a (ζ + a) 2 (4.16) z = x + iy (z ), ζ = ξ + iη (ζ) (4.17) x = ξ + ζ (r, θ) ξa2 ξ 2 + η 2, y = η ηa2 ξ 2 + η 2 (4.18) ξ = r cos θ, η = r sin θ (4.19) z ζ ) ) x = cos θ (r + a2, y = sin θ (r a2 r r (4.20) () 4.15z = f(ζ) f (ζ) = 1 a2 ζ 2 = ζ2 a 2 ζ 2 (4.21) f (ζ) = 0 ζ = a, ζ = a (4.22) ()

59 h q x 4.5: Joukovski transformation ζ a ζ = ae iθ (4.23) z z = 2a cos θ (4.24) 2a 4a ζ 0 θ π P (θ) z P B A π θ 2π P z A B () A B (singular point) ζ (imaginary axis) O M(0, f) A( a, 0) B(a, 0)

60 54 4 r=b A œ (-a,0) η M œ f o β œ ƒä ½ Ê B (a,0) ξ A' œ (-2a,0) y 2f o D' O' z ½ Ê B' œ (2a,0) x 4.6: ζ z b a b = a 2 + f 2 = a = a sec β (4.25) cos β β B M BM (real axis; ξ ) ζ P (4.7) (inscribed angle) ζ a ζ ( a) = ζ a ζ + a = r 1e iϕ1 r 2 e = r 1 e i(ϕ1 ϕ2) (4.26) iϕ2 r 2 ϕ 1 ϕ 2 = ϕ = const (4.27) z 2a (ζ a)2 = z + 2a (ζ + a) 2 = ( r1 r 2 ) 2 e i2 ϕ (4.28) ζ = a z = 2a, ζ = a z = 2a (4.29)

61 ƒä ½ Ê A (-a,0) œ P (ƒä) œ r 2 r ϕ 1 ϕ 2 ϕ 1 œ B (a,0) z ½ Ê σ A' œ œ (-2a,0) (z) λ 2 σ 2 λ 1 œ p' σ 1 (2a,0) B' x 4.7: (4.7) z P σ = σ 1 σ z 2a z + 2a = λ 1e iσ1 λ 2 e iσ 2 = λ 1 λ 2 e i(σ 1 σ 2 ) = λ 1 λ 2 e i σ (4.30) λ 1 λ 2 = ( r1 r 2 ) 2, σ = 2 ϕ = const (4.31) P P A B (circular arc) ζ z (4.6) M (central angle) AMB = 2 ϕ z D A D B = 2 ϕ MOB D O B ( MOB D O B ) D O O B = MO OB D O = MO OB O B = f 2a = 2f (4.32) a o z circular arc 2f camber (camber ratio)γ γ = 2f 4a = 1 2 f a = 1 tan β (4.33) 2

62 56 4 r=b A œ (-a,0) η M œ 2ƒ ƒó fo œ ƒä ½ Ê B (a,0) ξ y D' z ½ Ê A' œ (-2a,0) 2f o 2ƒ ƒó O' B' œ (2a,0) x 4.8: β (thickness ratio) ζ M d B(a, 0) c c = a + d (4.34) d = ϵa, ϵ 1 (4.35) c c = a + ϵa = (1 + ϵ)a (4.36) P ζ = ϵa + ce iθ (4.37) z = ζ + a2 ζ = a 2 (ceiθ ϵa) + ce iθ ϵa (4.38)

63 ζ Lœ r=c η (-d,0) A œ œ (-a,0) M c o ƒ œ B (a,0) ξ y z L œ A' œ (-2a,0) o B' œ (2a,0) x 4.9: θ θ z (conjugate) z θ 0 (θ 0) x z zx (symmetric airfoil) chord lengthl l = 2a + z L (4.39) z L z L = ζ L + a2 = a(1 + 2ϵ) a ζ L 1 + 2ϵ ζ L = ϵa (1 + ϵ)a = (1 + 2ϵ)a 4.40 ϵ (4.40) z z L 2a(1 + 2ϵ 2 ) (4.41) dy dθ = 0 (4.42) ϵ θ = 2π/3(ζ ) z 1/4 t max t max = 3 3aϵ (4.43)

64 58 4 τ τ = t max l ϵ θ = 0 x = 2aϵ x = 0 t max x 0 3 : (4.43) t max = 3 3 ϵ 1.3ϵ (4.44) 4 t(x 0) 4aϵ (4.45) : ϵ η r=c A œ ƒä 0 (-a,0) œ N d f œ o M β B (a,0) œ ξ y A' œ (-2a,0) o B' œ (2a,0) x 4.10: (the second quadrant) BM N(ζ = ζ 0 ) N BN = c c c = a 2 + f 2 + d = MN = d (4.46) a + d = a sec β + d (4.47) cos β

65 P N ζ 0 ζ = ζ 0 + ce iθ (4.48) ζ 0 = a c cos β + ic sin β (4.49) z = ζ 0 + ce iθ + a 2 ζ 0 + ce iθ (4.50) f β d f d (trailing edge angle) 0 (camber line; cusp d/a 4.3 (trailing edge angle) 0 (Karman-Trefftz transformation) z na z + na = ( ) n ζ a (4.51) ζ + a n = 2 ζ a z na A( a, 0) B(a, 0) B A C u A B C l b b = a cos β (4.52) ζ B A P (4.11) ζ a = r 1 e iϕ1, ζ ( a) = ζ + a = r 2 e iϕ2 (4.53)

66 z A B (λ, σ) z na = λ 1 e iσ 1, z + na = λ 2 e iσ 2 (4.54) Cu A (-a,0) œ œ P r 2 r 1 ϕ 2 ϕ 1 œ B (a,0) A' œ œ (-na,0) Cu' λ 2 σ λ 2 1 œ p' σ 1 (na,0) B' x 4.11: 2 ( λ 1 λ 2 = ( r1 r 2 λ 1 e iσ1 λ 2 e iσ2 = ( ) n r1 e iϕ1 (4.55) r 2 e iϕ2 P ) n, σ 1 σ 2 = nϕ 1 nϕ 2 = n(ϕ 1 ϕ 2 ) (4.56) ϕ 1 ϕ 2 = const (4.57) 4.56 σ 1 σ 2 = const (4.58) z P A B C l C l 4.12 (ϕ 2 π) (ϕ 1 π) = ϕ 2 ϕ 1 = const (4.59) ϕ 2 ϕ ) σ 1 σ 2 = const (4.60)

67 ϕ 2 ϕ 1 Aœ (-a,0) r2 Cl ƒór1 œ P œ B (a,0) Cl' λ 2 ` λ 1 œ p' σ 1 σ œ 2 œ A' σ ` 2 (na,0) B' x σ 2 (-na,0) 4.12: 2 ( σ 2 2π σ 2 = σ 2 + 2π (4.61) σ 1 ( σ 2 + 2π) = const (4.62) σ 1 σ 2 = const (4.63) P z (double circular arc airfoil) π ϕ u + ϕ l = π (4.64) ϕ 2 ϕ 1 = (ϕ 2 π) (ϕ 1 π) = ϕ l (4.65) σ 1 σ 2 = n(ϕ 1 ϕ 2 ) = n ϕ l (4.66)

68 σ 2 σ σ 1 ( σ 2 + 2π) = n ϕ l (4.67) C u σ 1 σ 2 = 2π n ϕ l (4.68) ϕ 1 ϕ 2 = ϕ u (4.69) σ 1 σ 2 = n ϕ u (4.70) σ 1 σ 2 = 2π n ϕ l = 2π n (π ϕ u ) = 2π nπ + n ϕ u (4.71) (4.71) (4.70) n = 2 C u = C l 0) n : ϵ B σ 2 0 (σ 1 ) u = n ϕ u (4.72) B σ ϵ (σ 1 ) l = 2π n π + n ϕ u (4.73) ϵ = (σ 1 ) l (σ 1 ) u = (2π n π + n ϕ u ) n ϕ u = (2 n)π (4.74) n = 2 ϵ = 0 0 ϵ n 4.74 n = 2π ϵ π (4.75)

69 z 2π ϵ π a z + 2π ϵ π a = ( ) 2π ϵ ζ a π ζ + a (4.76) (Karman-Trefftz transformation) []

70

71 65 5 (force) (moment) 5.1 (F x, F y ) ( F x, F y ) (reaction) p p Fy Fx V=(u,v) p S p 5.1: control surfaces (momentum) x ρu(udy vdx) = pdy F x (5.1) S S y ρv(udy vdx) = pdx F y (5.2) S S ρ(udy vdx) (mass flow) x uy v

72 66 5 S S ds dx v u dy 5.2: F x F y F x = pdy ρu(udy vdx) (5.3) F y = pdx ρv(udy vdx) (5.4) p ρv 2 = p 0 (5.5) p (static pressure)v 2 = u 2 + v 2 p 0 (total pressure) (5.3) 5.4) 1 F x = 2 ρ( u2 + v 2 )dy + ρuvdx (5.6) 1 F y = 2 ρ(u2 v 2 )dx ρuvdy (5.7) x y p 0 dx = 0, p 0 dy = 0 (5.8) F x F y F x if y = i ρ [(u 2 v 2 ) i2uv]dz (5.9) 2 u, v dw dz = u iv, ( ) 2 dw = u 2 v 2 i2uv (5.10) dz

73 ) F x if y = i ρ 2 Γ ( ) 2 dw dz (5.11) dz (Blasius) (Blasius formula of force) W (z) Γ (dw/dz) 2 iρ/2 F (conjugate) F F F = F x if y = i ρ 2 ( ) 2 dw dz (5.12) dz x F x y F y δf x = pdy ρu(udy vdx) (5.13) δf y = pdx ρv(udy vdx) (5.14) 5.3: (pitching moment) δm (x 0, y 0 ) (pitch down) δm = (x x 0 )δf y (y y 0 )δf x (5.15)

74 M = (x x 0 )[pdx ρv(udy vdx)] (y y 0 )[ pdy ρu(udy vdx)] (5.16) p 5.5 M = 1 2 ρ [(u 2 v 2 ){(x x 0 )dx (y y 0 )dy} + 2uv{(y y 0 )dx + (x x 0 )dy}] (5.17) u, v W Blasius (5.11) z 0 = x 0 + iy 0 ( ) 2 dw (z z 0 )dz (5.18) dz ( ) 2 dw (z z 0 )dz = (u iv) 2 {x x 0 + i(y y 0 )}(dx + idy) (5.19) dz (real part, Re) { (dw ) 2 Re (z z 0 )dz} = {(u 2 v 2 )(x x 0 )+2uv(y y 0 )}dx { 2uv(x x 0 )+(u 2 v 2 )(y y 0 )}dy dz (5.20) (5.17) (5.17) (5.18) ) 2 (z z 0 )dz} M = 1 2 ρ Re { (dw dz (5.21) () (F x, F y ) (x, y) M = xf y yf x (5.22) F x if y i(f x if y )(x + iy) = xf y yf x + i(xf x + yf y ) (5.23) Re[i(F x if y )(x + iy)] = xf y yf x (5.24)

75 Re M F x if y Blasius [ M = Re i i ρ ( ) 2 dw zdz] 2 dz Γ (5.25) M = ρ 2 Re [ Γ ( ) 2 dw zdz] dz (5.26) z 0 M = ρ 2 Re [ Γ ( ) 2 dw (z z 0 )dz] dz (5.27) () δ M = {(x x 0 ) + i(y y 0 )} {δf x iδf y } (5.28) δ M = (x x 0 )δf x + (y y 0 )δf y + i { (x x 0 )δf y + (y y 0 )δf x } (5.29) 5.15 δm { δm = Im δ M } { = Re i δ M } (5.30) δf (5.12) δf x iδf y = i ρ 2 ( ) 2 dw dz (5.31) dz 5.3 ( dw/dz dw dz = a 0 + a 1 z + a 2 z 2 + a 3 z 3 + (5.32) () (Laurent expansion) (singular) f(z) a f(z) = n= c n (z a) n (5.33)

76 70 5 c n () c n = 1 f(ζ) dζ (5.34) 2πi c (ζ 1) n+1 (5.32) z a 0 a 0 = U e iα (5.35) Γ dw dz dz = (u iv)(dx + idy) = (udx + vdy) + i Γ (udy vdx) (5.36) 2 udy vdx = V nds = 0 (5.37) V = (u, v) n () Γ dw dz dz = (udx + vdy) = V d s = Γ (5.38) () udy vdx = V nds = V (d s k) k () () Residue Theorem5.32 dw dz dz = 2πia 1 (5.39) (a 1 dw/dz z = 0 ) a 1 2πia 1 = Γ a 1 = i Γ 2π (5.40) 5.32 Blasius F x if y = F e iβ = i ρ 2 ( ) 2 dw dz dz

77 = i ρ ( a 0 + a 1 2 z + a ) 2 2 z 2 + dz = 2πi(2a 0 a 1 ) ( = 2ρπU e iα i Γ ) 2π (5.41) F x if y = F e iβ = ρu Γe i(α+π/2) (5.42) Γ < 0 (Kutta-Joukovski) Kutta(Germany) 1902 JoukovskiRussia 1906 F β F = ρu Γ, β = α + π 2 (5.43) 5.4: F = ρu Γ (5.44) L D 0 ( L = ρu Γ, D = 0 (5.45) () lift (drag) d Alembert s paradox 5.32 a 0, a dw dz = U e iα i Γ 1 2π z (5.46) W = U e iα z i Γ ln z (5.47) 2π

78 : 1 (uniform flow) 2 (vortex) 1 (bound vortex)

79 stagantion point 2 6.1: adverse pressure gradient (conservation of angular momentum)

80 : z ζ z = ζ + a2 ζ a z = z(ζ) z u iv = dw dz = dw dζ dz dζ ζ (parameter) (6.1) (6.2) W (ζ) z = z(ζ), W = W (z) = W (z(ζ)) = W (ζ) (6.3) e iα W (ζ) = U(ζ ζ 0 )e iα + Ua 2 i Γ ζ ζ 0 2π ln ζ ζ 0 a (6.4) ζ 0 ζ 0 = 0 u iv = Ue iα Ua 2 e iα ζ i Γ 2 2πζ (6.5) 1 a2 ζ 2 ζ = a z z = 2a ( 0 (singularity) 0 ( Kutta, Kutta condition ) Ue iα Ue iα i Γ 2πa = 0 (6.6)

81 Γ Γ = 4πUa sin α (6.7) Γ (clockwise) 6.2 (trailing edge) (6.5) 0/0 (L Hôpital s rule) ζ 2Ue iα a 2 1 ζ + iγ 3 2π Γ (6.7) ζ a 1 ζ 2 2a 2 ζ 3 (6.8) u iv = U cos α (6.9) u = U cos α, v = 0 (6.10) U cos α 0 U α z=-2a u= U cos α z=2a 6.3: u = U cos(θ/2 α), x = 2a cos θ (6.11) cos(θ/2) 0 θ π π θ 2π C p = 1 cos2 (θ/2 α) cos 2, x = 2a cos θ (6.12) θ/2 ()

82 L = ρu Γ (6.13) Γ (6.7) L C L C L = sin α α L = 4πρU 2 a sin α (6.14) L 1 2 ρu (4a 2 1) = 2π sin α (6.15) C L 2πα (6.16) C Lα (6.15) α C Lα dc L dα = 2π cos α (6.17) α C Lα 2π (6.18) () C Lα (M < 1) C Lα = 2π 1 M 2 (6.19) (M > 1) C Lα = 4 M 2 1 (6.20) (M = 0)

83 z = 0 M = 2π n2 1 ρu 2 a 2 sin 2α (6.21) 3 n = 2 () (6.21) M = 2πρU 2 a 2 sin 2α (6.22) C M C M = M 1 2 ρu 2 (4a)(4a) = 1 4 π sin 2α (6.23) ( 4a U L z=-2a α M œ x L ½  z=2a α C 6.4: x L L cos α = M x L = a (6.24) x L C = 4a center of pressure x cp = 1 4 C (6.25) (C) 1/4 (1/4)C aerodynamic center (1/4)C

84 78 6 () α 0 0 (couple) () 6.5 (flat plate airfoil) (circular arc airfoil) (camber effect) ( z = ζ + a2 ζ (6.26) e iα w(ζ) = U(ζ ζ 0 )e iα + Ub 2 i Γ ζ ζ 0 2π ln ζ ζ 0 b (6.27) a b b a = b cos β (6.28) ( 4.25 ) ζ 0 f ζ 0 = if (6.29) ( 4.6 ) 6.5: Γ Γ = 4πbU sin(α + β) (6.30) L L = ρuγ = 4πbρU 2 sin(α + β) (6.31)

85 sin 4πaρU 2 b/a sin β 0 0 α = β (zero lift angle of attack) C L L C L = (1/2)ρU 2 (4a 1) sin(α + β) = 2π(b/a) sin(α + β) = 2π = 2π(sin α + cos α tan β) (6.32) cos β α 1 C L 2π(α + tan β) (6.33) α 1, β 1 (dc L /dα) C L 2π(α + β) (6.34) dc L dα = 2π b cos(α + β) = 2π(cos α sin α tan β) (6.35) a dc L 2π(1 α tan β) (6.36) dα α 1, β 1 dc L dα 2π(1 αβ) 2π (6.37) αβ ϵ 2 M() M = 2a 2 U 2 ρπ sin 2α + 4πρfbU 2 sin α sin(α + β) (6.38) 1 b = a sec β, f = a tan β M = 2a 2 U 2 ρπ sin 2α {1 tan β(tan α + tan β)} (6.39) (β > 0) C M C M = M sin 2α (1/2)ρU 2 = π (4a) π 2 fb sin α sin(α + β) (6.40) a2

86 80 6 x = 0 x L x L = M L cos α = 2a2 U 2 ρπ sin 2α {1 tan β(tan α + tan β)} 4πbρU 2 sin(α + β) cos α { } cos α sin β = a 1 {1 tan β(tan α + tan β)} (6.41) sin(α + β) x L a β = 0( b = a) x L = a x cp = C/4 () α 1, β 1

87 81 7 (thin airfoil theory) Φ = 2 Φ x Φ y 2 = 0 (7.1) V = Φ (7.2) V V = V + q 7.1: V (x, y) = V + q(x, y) (7.3) V q U(x, y) = V cos α + u(x, y), V (x, y) = V sin α + v(x, y) (7.4) α q = (u, v) q = ϕ, u(x, y) = ϕ x, ϕ v(x, y) = y (7.5) Φ = V + ϕ (7.6)

88 82 7 Φ = (xv cos α + yv sin α) + ϕ (7.7) 7.1 ϕ 2 ϕ = 2 ϕ x ϕ y 2 = 0 (7.8) 7.2 Φ F = 0 (7.9) F (x, y) = 0 (7.10) F Φ = V (7.11) 0 ( V + ϕ) F = 0 (7.12) ϕ 0 (7.13) 7.12 (V cos α + u) F x + (V sin α + v) F y = 0 (7.14) y = η u (x) :, y = η l (x) : (7.15) y = η(x) (7.16) (7.10) η(x) y = 0 (7.17) F = η(x) y (7.18)

89 (x, y) F F = 0 (7.19) F x y (7.18) F x = dη dx, F y = 1 (7.20) 7.14y v v = (V cos α + u) dη dx V sin α (7.21) α u U, v U (7.22) F x F y (7.23) V cos α V, V sin α V α (7.24) u F x = u dη dx 1 (7.25) v(x, η(x)) = V dη dx V α, 0 x l (7.26) () () y v(x, η(x)) x y = 0 ( ) v v(x, η(x)) = v(x, 0) + η(x) + (7.27) y x,0

90 84 7 v O(ϵ), η O(ϵ) 2 (ϵ 2 ) O(ϵ) v(x, η(x)) v(x, 0) (7.28) 7.26 v(x, 0) = V dη dx V α (7.29) v(x, +0) = V dη u dx V α () (7.30) v(x, 0) = V dη l dx V α () (7.31) 7.29 ϕ ϕ 1, ϕ 2 ( ) 2 ϕ1 dη ϕ 1 = 0; = V (7.32) y x,0 dx ( ) 2 ϕ2 ϕ 2 = 0; = V α (7.33) y A 0 2. B: α 1. A (mean camber function) (thickness function) x,0 η c = 1 2 (η u + η l ) (7.34) η t = 1 2 (η u η l ) (7.35) η u = η c + η t, η l = η c η t (7.36) ( ) ϕ1 y ( ) ϕ1 y x,+0 x, 0 ( dηc = V dx + dη ) t dx ( dηc = V dx dη t dx ) at y = +0 (7.37) at y = 0 (7.38)

91 ( ) ( ϕ1 dηc = V y x,±0 dx ± dη ) t dx (7.39) camber 0 3. α superimpose p p = p + ρ 2 V (1 2 V 2 ) V 2 (7.40) C p p p (1/2)ρV 2 (7.41) 7.40 V C p = 1 V 2 V 2 (7.42) V 2 = ( V + q) ( V + q) (7.43) V q V = (V cos α, V sin α), q = (u, v) (7.44)

92 86 7 α 7.43 V 2 ( V V ) 2 ( u = cos α + v ) ( ) 2 ( ) 2 u v sin α + + (7.45) V V V V cos α 1, sin α α, (7.46) (u 2 + v 2 )/V ( V V ) 2 ( u = v ) α u (7.47) V V V C p p = p ρv u (7.48) C p = p p (1/2)ρV 2 = 2 u V (7.49) C p u (superpose superimpose) ϕ = 0 (7.50) x v(x, ±0) = ϕ y (x, ±0) = ±V dη t dx for 0 x l (7.51) ϕ = 0 (7.52) y v(x, +0) = v(x, 0) (7.53)

93 x (ξ, 0) σ ξ = 0 ξ = l ϕ ϕ(x, y) = 1 2π y l 0 σ(ξ) ln[(x ξ) 2 + y 2 ] 1/2 dξ (7.54) φ(x,y) r œ P(x,y) σ(ξ) ( ξ,0 ) x ξ 7.2: x y u = ϕ/ x, v = ϕ/ y u(x, y) = 1 2π v(x, y) = 1 2π l 0 l 0 x ξ σ(ξ) (x ξ) 2 dξ (7.55) + y2 y σ(ξ) (x ξ) 2 dξ (7.56) + y2 σ σ(ξ) v [ ] 1 l y v(x, ±0) = lim σ(ξ) y ±0 2π (x ξ) 2 + y 2 dξ dη t = ±V dx 0 (7.57) ξ x 0 ξ = x ξ σ(ξ) )ξ = x 1 v = lim y ±0 2π x+ϵ x ϵ y σ(ξ) (x ξ) 2 dξ = ±σ(x) + y2 2 (7.58) σ(x) = 2v(x, +0) = 2V dη t dx (7.59)

94 ϕ(x, y) = V π u(x, y) = V π v(x, y) = V π l 0 l 0 l 0 dη t dx ln[(x ξ)2 + y 2 ] 1/2 dξ (7.60) dη t dx dη t dx x ξ (x ξ) 2 dξ (7.61) + y2 y (x ξ) 2 dξ (7.62) + y2 ()7.58 ξ = x A = x+ϵ x ϵ y σ(ξ) (x ξ) 2 + y B = x x ϵ (x ξ)/y = t x+ϵ dξ σ(x) 2 x ϵ y x (x ξ) 2 dξ = 2σ(x) + y2 x ϵ y (x ξ) 2 + y 2 dξ (7.63) y (x ξ) 2 + y 2 dξ = 1 x y y 2 dξ (7.64) x ϵ 1 + {(x ξ)/y} B = y y 2 0 ϵ/y y 1 + t 2 dt = tan 1 ϵ y A = 2σ(x) tan 1 ϵ y (7.65) (7.66) y 0 A = 2σ(x) π 2 = πσ(x) (7.67) lim y 0 () 1 2π l 0 y σ(x) σ(ξ) (x ξ) 2 dξ = + y2 2 (7.68) C p u(x, y) C p (x, y) = 2 (7.69) V C p (x, 0) = 2 u(x, 0) V (7.70)

95 l/2 œ œ ƒæx œ x=0 x=l/2 x=l x 7.3: 7.61u(x, 0) u(x, 0) = V π l ξ = x singularity u(x, 0) dη t /dx θ ϕ 0 dη t dx (ξ) 1 dξ (7.71) x ξ x = l 2 (1 + cos θ) ξ = l (1 + cos ϕ) (7.72) 2 0 θ π π θ 0 θ, ϕ x 7.71 x ξ = l 2 (cos θ cos ϕ), dξ = l sin ϕdϕ (7.73) 2 u(θ) = V π π 0 dη t dx (ϕ) sin ϕ dϕ (7.74) cos ϕ cos θ 7.74 (Poisson s integral formula) dη t /dx(θ) θ odd function θ A n A n = 2 π π 0 dη t dx (θ) = A n sin nθ (7.75) n=1 dη t (θ) sin nθdθ (n = 1, 2, ) (7.76) dx 7.74u(θ)/V conjugate Fourier series) u(θ) V = A n cos nθ (7.77) n=1

96 90 7 dƒå dx t ƒæ<0 0 ƒæ>0 ƒæ 7.4: [] [] u C p C p = 2u(θ) V = 2 A n cos nθ (7.78) C p θ cos θ n=1 R = 2m (cambered plate) 0 ϕ 2 ϕ = 0 (7.79) v(x, ±0) = ϕ y (x, ±0) = V dη c for 0 x l (7.80) dx ϕ = 0 for r (7.81) y = η c (x) 0 (7.80) Kutta condition u = ϕ/ x, v = ϕ/ y

97 y φ(x,y) r œ P(x,y) γ(ξ) ( ξ,0 ) x ξ 7.5: ϕ(x, y) = 1 2π l u(x, y) = ϕ x = 1 2π 0 γ(ξ) tan 1 l v(x, y) = ϕ y = 1 2π 0 l y dξ (7.82) x ξ y γ(ξ) (x ξ) 2 dξ (7.83) + y2 0 x ξ γ(ξ) (x ξ) 2 dξ (7.84) + y2 γγ C p u(x, ±0) u(x, ±0) C p (x, ±0) = 2 (7.85) V 1 u(x, ±0) = lim y ±0 2π l y γ(ξ) (x ξ) 2 dξ (7.86) + y2 u(x, ±0) = ± γ(x) 2 (7.87) C p (x, ±0) = γ(x) V (7.88)

98 L = l p C p L = ρ 2 V 2 0 l 0 [p(x, 0) p(x, +0)]dx (7.89) [C p (x, 0) C p (x, +0)]dx (7.90) (7.88) l L = ρv γ(x)dx (7.91) 0 (7.91) l 0 γ(x)dx = Γ (7.92) L = ρv Γ(x) (7.93) Γ circulation (7.91) C L = L (1/2)ρV S 2 = 2 1 V l l 0 γ(x)dx (7.94) S S = l 1 C L (7.90) C L = 1 l l 0 [C p (x, 0) C p (x, +0)]dx (7.95) M = l 0 = 1 2 ρv 2 [p(x, +0) p(x, 0)]xdx (7.96) l 0 (C p (x, +0) C p (x, 0))xdx (7.97) l = ρv γ(x)xdx (7.98) 0

99 C M = M (1/2)ρV Sl 2 = 2 1 V l 2 l 0 γ(x)xdx (7.99) S S = l 1 l C M 7.97 C M = 1 l l 2 [C p (x, +0) C p (x, 0)]xdx (7.100) γ(x) y v x l v(x, ±0) = lim y ±0 y ±0 [ 1 2π 1 2π l 0 l 0 ] x ξ γ(ξ) (x ξ) 2 + y 2 dξ 1 γ(ξ) x ξ dξ = V dη c dx = V dη c dx Cauchy (principal value) (7.101) (7.102) Kutta γ(x = l) = 0 (7.103) I = l 0 γ(ξ) dξ x ξ x θ ξ ϕ (7.104) x = l 2 (1 + cos θ), ξ = l (1 + cos ϕ) (7.105) 2 I = π 0 sin ϕ γ(ϕ) dϕ (7.106) cos ϕ cos θ

100 94 7 π 0 cos nϕ sin nθ dϕ = π cos ϕ cos θ sin θ (n = 0, 1, 2, ) (7.107) n = γ(θ) = K sin θ I θ ϕ 1 π π 0 π 0 γ(ϕ) 2V x θ dϕ cos ϕ cos θ = 0 (7.108) (7.109) sin ϕ cos ϕ cos θ dϕ = dη c (θ) (7.110) dx (7.110) γ/2v dη c /dx Cinjugate Fourier Series dη c /dx θ θ B n B n = 2 π dη c dx (θ) = B B n cos nθ (7.111) π dη c dx n=1 cos nθdθ (n = 0, 1, 2, ) (7.112) 1 π γ(ϕ) sin ϕ π 0 2V cos ϕ cos θ dϕ = B B n cos nθ (7.113) n= γ(θ) = γ 1 (θ) + γ 2 (θ) + γ 1 (θ) K sin θ (7.114) 1 π π 0 γ 1 (ϕ) 2V γ 2 (θ) sin ϕ cos ϕ cos θ dϕ = B n cos nθ (7.115) 1 1 π π 0 γ 2 (ϕ) 2V sin ϕ cos ϕ cos θ dϕ = B 0 2 (7.116)

101 γ 1 (θ) γ 1 (θ) = 2V n=1 B n sin nθ (7.117) γ 1 sin nθ π π 0 sin nϕ 2V sin ϕ dϕ (7.118) cos ϕ cos θ sin nϕ sin ϕ = 1 (cos(n + 1)ϕ cos(n 1)ϕ) (7.119) { 1 π cos(n + 1)ϕ 4πV cos ϕ cos θ dϕ 0 π 0 } cos(n 1)ϕ cos ϕ cos θ dϕ (7.120) { } 1 sin(n + 1)θ sin(n 1)θ π π 4πV sin θ sin θ 1 = {sin(n + 1)θ sin(n 1)θ} 4V sin θ 1 = {2 cos nθ sin θ} 4V sin θ cos nθ = (7.121) 2V γ V B n n n = 1 π cos ϕ cos ϕ cos θ dϕ = π or 1 π π 0 cos ϕ dϕ = 1 (7.122) cos ϕ cos θ γ(θ) γ(θ) = 2V 1 γ 2 (θ) = V B 0 cos θ sin θ B n sin nθ + V B 0 sin θ ( ) K + cos θ V B 0 (7.123) (7.124)

102 96 7 K γ(θ = 0) = 0 ( B 0 1 cos θ γ(θ) = 2V 2 sin θ K = V B 0 (7.125) + ) B n sin nθ n=1 (7.126) γ(θ)b n B n Cp [ ] C p (θ) = C p [x(θ), ±0] = γ(θ) B 0 1 cos θ = ±2 + B n sin nθ V 2 sin θ C L = 1 V π C M = 1 2V 0 π n=1 (7.127) γ(θ) sin θdθ (7.128) 0 γ(θ)(1 + cos θ) sin θdθ (7.129) C L = (B 0 + B 1 )π (7.130) π C M = B B π B π 2 4 = π 4 (B 0 + B 1 ) + π 4 (B 1 + B 2 ) (7.131) C M = C L 4 + π 4 (B 1 + B 2 ) (7.132) 1/4 B 1 + B 2 0 1/4C 2 π/4 (B 1 + B 2 ) C mac X ac = C/4 X cp X cp C = 1 4 π B 1 + B 2 = 1 ( 1 + B ) 1 + B 2 4 C L 4 B 0 + B 1 (7.133)

103 ϕ 2 ϕ = 0 (7.134) v(x, ±0) = ϕ y (x, ±0) = V α (0 x l) (7.135) ϕ = 0 at infinity (7.136) α y U sin α y 0 ϕ/ x ϕ/ y γ(x) v(x, ±0) = 1 2π l 0 γ(ξ) dξ x ξ = V α (7.137) γ(x = l) = 0 (7.138) θ ϕ x = l 2 (1 + cos θ), ξ = l (1 + cos ϕ) (7.139) 2 1 π π 0 π 0 sin ϕ γ(ϕ) cos ϕ cos θ dϕ = 2V α (7.140) γ(ϕ) 2V α sin ϕ dϕ = π (7.141) cos ϕ cos θ (7.107) γ(θ) = 2V α 1 (K + cos θ) (7.142) sin θ x = l(θ = 0) sin θ = 0 0 K = 1 (7.143) γ(θ = 0) = 0 γ(θ) = 2V α cos θ 1 sin θ (7.144)

104 98 7 C p C p (θ) = 2α 1 cos θ = 2α tan(θ/2) (7.145) sin θ 0 θ π C L = 1 V π C M = 1 2V 0 π γ(θ) sin θdθ = 2πα (7.146) 0 γ(θ)(1 + cos θ) sin θdθ = π 2 α = C L 4 (7.147) 1/4 (center of pressure) 0 α 0 (aerodynamic center) 0 C mac = 0 x cp x ac C l C mac C mac () x cp = x ac C mac C l (7.148) (aerodynamic center) (C L ) (C D ) C mac 0 trim () 7.7 dη ± dx = dη c dx ± dη t dx (7.149) θ ± dη dx (θ) = dη c dx (θ) ± dη t (θ) 0 θ π (7.150) dx dη dx (θ) = B B n cos nθ ± A n sin nθ (7.151) n=1 n=1

105 A n, B n A n = π 2 B n = π 2 π 0 π 0 dη (θ) sin nθdθ dx (n = 1, 2, ) (7.152) dη (θ) cos nθdθ dx (n = 0, 1, 2, ) (7.153) α [ ( C p = 2 α B ) ] 0 tan θ 2 2 B n sin nθ ± A n cos nθ n=1 n=1 (7.154) 0 θ π (lift slope) ( C L = 2πα (B 0 + B 1 )π = 2π α B ) 0 + B 1 2 dc L dα 1. A n (7.155) = 2π (7.156) α 0 = B 0 + B 1 2 (7.157) C M = [2πα (B 0 + B 1 )π] π 4 (B 1 + B 2 ) = C L 4 + π 4 (B 1 + B 2 ) (7.158) C L 1/4 1/4 C M = π 4 (B 1 + B 2 ) (7.159)

106 100 7 C M α dc M dα < 0 (7.160) 7.6 C M = 0 (α = α e ) α > α e C M < 0 α < α e C M > 0 7.6: C L α (7.160) δc L δα dc M dc L < 0 (7.161) : 3 (C Lα ) (C Lα = dc L /dα 2 C Lα = 2π 3 3 C Lα 2πAR C Lα = ( ) (7.162) (AR)2 β 2 η 1 + tan2 Λ max,t 2 β 2 AR β = 1 M 2 Λ max,t η η = C lα 2π/β (7.163)

107 C lα 2 η η = (lift slope)c Lα (aerodynamic derivatives) 3 β C nβ C nβ > 0 β C nβ < 0

108

109 rectangular wing tapered wingswept wing (delta wing) (sweep) 8.1: (MAC) Aspect Ratio AR = b2 (8.1) S S b spans = b c AR = b c c chord length (8.2)

110 104 8 ( 1) (Bombardier) Dash8(Q400) AR = 12.8B787 AR 10B AR = 9.45Cessna 172) AR = 7.3 B747 AR = 7 F15 AR = 3 AR = 1.8 ASW-19B AR = 20.4 () ( 2) (tail) () mean chord c = b AR (8.3) 8.2: (MAC) (MAC: Mean Aerodynamic Chord) ( 8.2 ) (rootchord : C r ) (tipchord) (C t ) A B AB C C (MAC) B747 MAC=8.3m ( 1) (C Lα ) (C Lmax ) ()

111 ( 2) (leading edge separation vortex) (vortex lift) () () (tail) (horizontal tail) (vertical tail) (stability) (control) (trim) () 8.2 (finite wing theory) Prandtl(1918) Γ Γ(±b/2) = 0 L Γ (y ) y = 0 Γ trailing vortex sheet Lanchester( ) trailing vortex sheet Prandtl Prandtl rolling up bound vortex sheet ( p = p u p l 0) free vortex sheet ( p = 0)

112 106 8 V ƒã Ê ƒ p 0 ƒ º Ê gq QƒV[ƒgã Ê ƒ p=0 gq QƒV[ƒg º Ê 8.3: a single bound vortex linelifting line lifting line theory V q q V V downwash velocity, w w R resultant velocity V R V R = V + w (8.4) 3 w y w = w(y) (8.5) V R = V R (y) = V w(y) k (8.6) k z x z

113 V y Γ (y) w(y) 8.4: (y = y) y) Γ(y) V V R Prandtl dy Kutta-Joukovski j y δ F (y) = ρ V R (y) jγ(y)dy (8.7) 2 V V R y Γ(y) = K(y)V R (y)α R (y) (8.8) V R α R (y) y = y V R (y) (zero lift line) K K(y) = 1 2 [ ] dcl (y) c(y) = 1 dα R 2 a 0(y)c(y) (8.9) c(y) (chord length)a 0 (y) (lift slope) Γ Γ = 4πUa sin α πucα = 1 2πUcα (8.10) 2 c = 4a C Lα = 2π Γ = 1 2 dc L Ucα (8.11) dα

114 108 8 α R (y) = α(y) α i (y) (8.12) α(y) : V α i (y) : V R V induced angle of attack w(y) V Prandtl 1 w(y) α i = tan (8.13) V α i = w(y) V (8.14) α R (y) = α(y) w(y) V (8.15) Γ(y) = 1 ( 2 a 0(y)c(y)V R (y) α w ) V (8.16) w(y) Γ(y) x η γdy x y w(y) = 1 4π b/2 b/2 γ(η) dη (8.17) η y 2 4 ( x ) (0 x ) γ(y) γ(y) = dγ dy Γ y dγ/dy < 0 (8.18) 8.17 w(y) = 1 4π b/2 b/2 (8.18) dγ(y) dy (η) 1 dη (8.19) y η w > 0 z w(y) Γ(y) Γ(y) w(y) 8.15α R (y) 11) [ Γ(y) = K(y)V R (y) α(y) 1 4πV b/2 b/2 ] dγ(y) dy (η) 1 y η dη (8.20)

115 w(y) V V R V [ Γ(y) K(y) V α(y) 1 ] b/2 dγ(y) 4π dy (η) 1 y η dη b/2 (8.21) 8.9 K(y) = 1 2 a 0(y)C(y) (8.22) 8.21Γ(y) Prandtl Γ(y) Γ(y) a 0 (y) α Γ Γ ( Γ b ) ( ) b = Γ = 0 (8.23) 2 2 Γ Γ (y) -b/2 b/2 y 8.5: 8.3 (aerodynamic force) 3 (Lift) (Drag) L V y D V 5 downwash V R F V R D i (induced drag) () (8.19) downwash winglet % winglet

116 110 8 δ L δf α i δd i V α i w V R 8.6: 5 7% () δy δl(y) = δf (y) cos α i δf (y) (8.24) δd i (y) = δf (y) sin α i δf (y)α i (y) α i δl(y) (8.25) α i cos α i 1, 8.7 sin α i α i δl(y) δf (y) = ρv R Γdy ρv Γdy (8.26) δd i (y) = α i δl = ρv α i Γdy = ρwγdy (8.27) b/2 L = ρv Γ(y)dy (8.28) D i = b/2 b/2 b/2 α i (y)δl(y) = ρv b/2 b/2 b/2 α i (y)γ(y)dy = ρ w(y)γ(y)dy (8.29) b/2 Γ(y) L D induced drag downwash

117 M = b/2 b/2 b/2 b/2 y j ( iδd i + kδl) = k yδd i + i yδl (8.30) b/2 b/2 x () rolling moment b/2 b/2 M R = yδl = ρv Γ(y)ydy (8.31) b/2 b/2 z yawing moment M y = b/2 b/2 b/2 yδdi = ρ w(y)γ(y)ydy (8.32) b/2 Γ(y) M R M y (8.31) M R > 0 (8.32) body axis coordinates (x, y, z) x (longitudinal axis) y (lateral axis) z vertical axis x y x y (pitching moment)z wind axis coordinates(x w, y w, z w ) x w z w y w x w z w 8.4 Γ(y) Γ(y) = Γ 0 1 (2y/b) 2 or Γ 2 (y) Γ y2 (b/2) 2 = 1 (8.33)

118 112 8 w 8.19 w(y) = Γ [ b/2 0 πb 2 1 b/2 ( ) ] 2 1/2 2η η dη (8.34) b y η η = y (singular) y = b 2 cos θ, η = b cos ϕ (8.35) 2 b/2 y b/2 θ = π w(θ) = Γ π 0 cos ϕ dϕ (8.36) 2πb 0 cos θ cos ϕ (7.107) π w w(y) = Γ 0 2b downwash α i = const. (8.37) w 8.27 D i δd i (y) = ρw(y)γ(y)dy = ργ 0 Γ(y)δy (8.38) 2b b/2 D i = ρ w(y)γ(y)dy = ργ b/2 0 Γ(y)dy (8.39) b/2 2b b/2 L (8.39) b/2 L = ρv Γ(y)dy (8.40) b/2 α i = w/v D i = Γ 0 2bV L (8.41) D i = α i L (8.42)

119 b/2 b/2 Γ(y)dy = πbγ 0 4 (8.43) (8.40) (8.41) L = π 4 ρv Γ 0 b (8.44) D i = L D i C L = C Di = L 2 (π/2)ρv 2 b 2 (8.45) b/2 L = ρv Γdy = π 4 ρv Γ 0 b (8.46) b/2 D i = w L = Γ 0 L = π V 2b V 8 ργ2 0 (8.47) L (1/2)ρV S 2 = π b 2 S D i (1/2)ρV S 2 = π 4S Γ 0 (8.48) V ) 2 = 1 S π b 2 C2 L = 1 CL 2 (8.49) π AR AR = b 2 /S S (plan form area of wing) B747 7B ( Γ0 C L C D polar diagram) V C L = πarc Di (8.50) C D = C D0 + C2 L πear (8.51) C D0 0 e Osswald efficiency factor 0.8 [] w w = Γ 0 2b (8.52)

120 114 8 L L = π 4 ρv Γ 0 b (8.53) Γ 0 w V = w = 4L 2πρV b 2 (8.54) w 4L = V 2πρV b 2 2 (8.55) L (1/2)ρV S 2 (1/2)ρV S πρV 2 b 2 (8.56) w S = C L V πb 2 = C L πar AR AR = b 2 /S (8.57) () D i = ρ 4π b/2 b/2 b/2 b/2 Γ(y) dγ dηdy dη y η = ρ b/2 b/2 Γ(y) 4π b/2 b/2 y θ θ dγ dη dη dy (8.58) y η y = b cos θ (8.59) 2 Γ(y) = U b n=1 A n sin nθ (8.60) η ϕ Γ(η) η = b cos ϕ (8.61) 2 Γ(η) = U b A n sin nϕ (8.62) n=1 dγ/dη dγ dη = dγ dγ dϕ dϕ dη = dϕ dη dϕ = U b n=1 A nn cos nϕ ( b/2) sin ϕ (8.63) dη = ( b/2) sin ϕdϕ, dy = ( b/2) sin θdθ, y η = (b/2)(cos θ cos ϕ) (8.64)

121 b/2 y b/2 π θ 0, b/2 η b/2 π ϕ 0 (8.65) 8.58 D i = ρu 2 b 2 4π π 0 A m sin mθ sin θ m=1 π 0 cos nϕ A n n dϕdθ (8.66) cos θ cos ϕ n=1 (7.107) π cos nϕ nθ dϕ = ( π)sin cos θ cos ϕ sin θ 0 (8.67) D i = ρu 2 b 2 4 n=1 na 2 n π 0 sin 2 nθdθ = πρu b 2 2 na 2 n (8.68) 8 n = n=1 Γ(y) = Γ 0 sin θ (8.69) n = 1 n 2 D i () GA(Genetic Algorithm) Γ Γ(y) = 1 2 a 0(y)C(y) [V α(y) w(y)] (8.70) C(y) a 0 (y) α(y) Γ(y) Γ(y) Γ(y) = Γ 0 1 (2y/b)2 = 1 2 a 0(y)C(y) [ V α(y) Γ ] 0 2b a 0 (y), C(y), α(y) (8.71) (geometrical angle of attack) a 0 (y), α(y)

122 C(y) Γ(y) C(y) = (1/2)a 0 (y) [V α(y) (Γ 0 /2b)] = Γ 0 1 (2y/b) 2 (1/2)a 0 (y) [V α(y) (Γ 0 /2b)] (8.72) C(y) = const Γ(y) (8.73) const Γ 0, V, α, b, a 0 untwisted) C(y) elliptic wing ( ( 35 40% (Hermann Glauert 1926) 8.6 (x = x w ) x : y z L D K.Kusunose K.Kusunose: A Wake Integration Method for Airplane Drag Prediction, Tohoku University Press, : (MAC)

123 L L = ρ U yξdydz ρ U (1 2 M ) 2 w u dydz (8.74) w w U U +M 2 γp w sdydz ρ M 2 w Hdydz + O( 3 ) (8.75) R U U w ρ, U y ξ x (ξ = w/ y v/ z) u = U U w s H L = ρ U yξdydz ρ U 2 w u dydz + O( 3 ) (8.76) U U w w w () D (y z ) D = w ψ s P R dydz + ρ w 2 ψξdydz P w 2 ( ) 2 s ψξdydz (8.77) R 2 ψ y ψ = ξ (8.78) z2 ξ x ( ω = rot v (8.77) 1 (profile drag) 2 (induced drag) 3