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KENZOU 28 8 9 9/6 4 1 2 3 4 2 2 5 2 2 5.1............................................. 2 5.2......................................... 3 5.2.1........................................ 3 5.2.2............................... 4 5.2.3........................................... 5 5.2.4.......................................... 6 5.2.5...................................... 6 5.2.6........................................... 7 5.2.7.......................................... 8 5.2.8............................................... 9 5.3...................................... 11 5.3.1.......................... 11 5.3.2.......................... 13 ============================ 1

5 2 5.1 2 3 3.12 ψ(, v = u + v = (5.1 u = ψ, v = ψ (5.2 ψ 2 P P F ig.23 ds d d n P flow v : d/ds = (d/ds, d/ds 9 n=(d/ds, d/ds ds 2 = d 2 d v n P ψ( ψ( = P v n ds (5.3 P ψ( ψ P ψ 2 ψ P ψ P Q Q P v n ds = P (v n + v n ds = P ( ψ d ds + ψ d P ds = dψ = ψ P ψ (5.4 ds P P (,, ψ(, = const (5.5 Fig.24 B B ψ ψ B ψ B B B F ig.24 ψ B ψ B B B ψ 2

B BB Q Q Q = ψ B ψ, Q = ψ B ψ BB BB B Q BB Q Q = Q, ψ B ψ = ψ B ψ,... ψ B = ψ B = const ψ = const 2 ω ω = otv = i j k z u v = (,, v u (5.6 z ω = v u ( 2 = 2 + 2 2 ψ = 2 ψ (5.7 2 ψ = (5.8 5.2 5.2.1 u v u = φ v = φ (5.9 φ(, φ(, φ φ ud + vd φ(, ud + vd = dφ dφ = φ φ d + d u = φ/, v = φ/ u = 2 φ = 2 φ, v = 2 φ u v = 5.9 φ φ = const 5.9 5.1 2 φ 2 + φ = 2 φ = (5.1 2 φ 1 φ 2 φ = c 1 φ 1 + c 2 φ 2 ( 1, c 2 3

F ig.25 ψ = const ψ 1 = const ψ 2 = const ψ 3 = const φ 3 = const φ 2 = const φ 1 = const φ = const [ ] φ(, = const, ψ(, = const φ + φ ψ + ψ ( d d φ ( d d ψ = φ = φ = ψ = ψ ( d d φ ( d d ψ 5.2 5.9 φ ψ = φ ψ = φ, ψ = φ ( d d φ ψ ( d d φ φ ψ ( d = 1 d ψ ( d d ψ = φ φ ψ = cos, = sin... φ = φ + φ = φ φ cos + sin = u cos + v sin ψ = ψ ψ cos + sin = v cos + u sin φ = φ + φ = φ φ sin + cos = ( u sin + v cos ψ = ψ ψ sin + cos = (v sin + u cos φ = ψ = v φ, = ψ = v ψ 5.2.2 v(u, v u = φ = ψ v = φ = ψ (5.11 4

f = φ + iψ (5.12 f(z 1... f = f(z f = z dz = ( + i = dz dz, dz = f = (φ + iψ f = i dz (5.13 = u iv (5.14 = u iv (5.15 dz u iv u + iv 2 u u u = u cos + v sin u = u sin v cos v u cos v cos v sin u u sin... i u iu = (u cos + v sin i(u sin + v cos = (cos + i sin (u iv = e dz dz = (u iu e i (5.16 5.2.3 f(z f(z = Uz U : (5.17 = u iv = U (5.18 dz u = U, v = (u, v U u = U cos, v = U sin u iv = U(cos i sin = Ue i 1 2 5

dz = u iv = Ue i f = Ue i z 5.2.4 f(z f(z = z n >, n > dz = nzn 1 z = e i f = n e in = n (cos n + i sin n = φ + iψ φ = n cos n, ψ = n sin n ψ = = k π, (k =, ±1, ±2, n π/n ψ = c (c > ( c 1/n 1 = ( sin n 1/n n = 1 n = 2, 4, F ig.26 n = 1 n = 2 n = 4 n = 2/3 n = 1/2 5.2.5 Q (souce 2 u u Q Q = u u = Q/, u = 2 φ = Q, u = φ, u = φ φ = φ = Q log + c(, φ = c(, c(, c(, φ 2 e e = e φ + e φ (5.19 φ = Q log (5.2 6

Q F ig.27 u u ψ u = ψ, u = ψ ψ = Q (5.21 f(z = φ + iψ = Q (log + i = Q log ei = Q log z (5.22 f(z z = 3 Q > Q < (sink dz = u iv = Q z = Q e i = u u 5.16 dz = Q e i = (u u e i u = Q (cos i sin (5.23 Q, u = [ ] ψ dψ = Q d = Q Q 5.2.6 U O f(z = Uz + Q log z (5.24 z z = e i f(z = Ue i + Q log ei = (U cos + Q log + i (U sin + Q (5.25 3 7

φ ψ φ = U cos + Q log, ψ = U sin + Q (5.26 F ig.28 ψ = Q/2 b = Q/2U U ψ = Q/2 O ψ = a u =, = u = φ = U + (Q/ 1 = U + m 1, (m = Q/ (5.27 u = m/u ( = a 4 = ψ = = π ψ = Q/2, π ψ = Q/2 = Q π U sin (5.28 5.28 2b ( Q b = lim sin = lim (π = Q U 2U = πa 2b = Q/U = a = a [ ] φ u, v φ = U + Q log... u = φ = U + Q cos = U + Q ( 2 + 2, v = φ = Q sin = Q ( 2 + 2 5.2.7 2 = a Q = a Q f(z = Q log(z + a Q log(z a = Q log z + a (5.29 z a 4 /dz = 5.16 8

5.29 5 a [ a z 1 ( a 2 1 ( a ] 3 + 2 z 3 z f(z = Q 1 + (a/z log 1 (a/z = Q Q 2a z [ a z 1 ( a 2 1 ( a ] 3 2 z 3 z (5.3 m = 2aQ a 6 2 f(z = Q 2a lim 2aQ m z = m z = m e i (5.31 m 2 2 u, u 5.16 dz = m 1 z 2 = m... u = m cos, 2 u 1 2 e i2 = m 2 (cos i sin e i = (u iu e i = m 2 sin F ig.29 1 F ig.29 2 2 ψ f(z = m (cos i sin = φ + iψ (5.32 ψ = m sin (5.33 ψ = const sin / = const 7 Fig.29-1 Fig.29-1 5.2.8 f(z = f(z = Γ log z (Γ (5.34 i Γ i (log + i = Γ i Γ log = φ + iψ... φ = Γ, ψ = Γ log (5.35 5 log(1 + = (1/2 2 + (1/3 3, < 1 6 a 2aQ 2a Q a 7 ψ = 1,sin =, sin = 2, sin =, = 2 = 2 + 2, 2 + ( 1/2 2 = (1/2 2 9

ψ = const φ = const Fig.29-2 f(z = Γ i log z v = Γ/ Γ v = φ =, v = φ = Γ (5.36 Γ/, 3 4.5 Γ( = v d = v s ds = Γ( = (ud + vd = ( φ φ d + d = dφ (5.37 [ ] Γ = Γ (5.38 f(z = (Γ/i log z Γ 8 Fig.27 Fig.29-2 [ ] z = z 5.8 2 1 3 1 3 2 3. -21 Γ 2 5.36 v = Γ/ v p( p p( p( + 1 2 ρv 2 = p p( = p ργ2 8π 2 1 2 8 1

p p = 1 1/8π 2 2 5.3 5.3.1 2 2 m = a 2 U f(z = Uz + m (z z = U + a2 z = U (e i + a2 = U ( + a2 e i ψ = U cos + iu ( a2 ( a2 sin (5.39 sin (5.4 = a ψ = = const a a ψ = a U = 1, a = 2 Fig.3 9 ψ =.2,.5, 1, 2, 3 [ ] 3 ψ = 1 ( 2 U 2 a3 sin 2 ( = a (1 dz = u iv = U a2 z 2 = U = U (1 a2 cos 2 + iu a2 2... u = U (1 a2 cos 2 2 (1 a2 2 e 2i sin 2 (5.41 2, v = U a2 sin 2 (5.42 2 u = 2U sin 2, 9 FunctionView v = 2U sin cos 11

= ±π/2 u 2U 2 = a, =, π u = v = q v = u cos + v sin = v = u sin + c cos = 2U sin q = v 2 + v 2 = v = 2U sin (5.43 p + ρ 2 q2 = p + ρ 2 U 2 = p = const (5.44 p 1 p 5.43 p = p + ρ 2 U 2 (1 4 sin 2 (5.45 Fig.31 F ig.31 P P = (p p /(ρu 2 /2 a ds p -pn n π/2 π Fig.31 =, π p = p + (ρ/2u 2 = π/2 p = p (3ρ/2U 2 P (P, P n P P = pnds = P = p cos ad, P = p sin ad, n = (cos, sin, ds = ad 5.45 P =, P = 11 12 [ ] 1 2 Rankine P88 Q < 1 11 http://integals.wolfam.com/inde.jsp 12 12

2 please daw ouself 2 5.3.2 2 Γ < 2 2 f(z = U (z + a2 z { = U ( + a2 iγ = φ + iψ (Γ = Γ (e log z = U i + a2 cos Γ } { + i U e i ( a2 iγ log ei sin + Γ log } (5.46 φ φ = U ψ v = φ = U v = φ = U ψ = U (1 dz = u iv = u a2 z 2 { = U (1 a2 cos 2 2 ( + a2 cos Γ (1 a2 2 (1 + a2 cos (5.47 2 sin + Γ (5.48 ( a2 sin + Γ log (5.49 + iγ z + Γ sin } } + i {U a2 Γ sin 2 + 2 cos (5.5 (5.51 z s z s /dz = u = v = 5.5 { z s = 1 2 iγ U ± 4a 2 Γ2 4π 2 U 2 } = iγ 4πU ± a 2 ( 2 Γ (5.52 4πU 13

2 3 3 (1 4πaU < Γ ( z s = iγ 4πU ± a 2 ( ( Γ 2 4πU iγ 4πU a 2 ( Γ 2 4πU = a 2 (2 4πaU = Γ z 1 = z 2 = iγ /4πU = ia (3 4πaU < Γ z s = i( Γ /4πaU ± (Γ 2 /16πa2 U 2 a 2 z s z 1 = i( Γ/4πaU (Γ 2 /16πa2 U 2 a 2 z 2 = Γ/4πaU + (Γ 2 /16πa 2 U 2 a 2 1 F ig.32 4πaU > Γ 4πaU = Γ 4πaU < Γ = a (5.46 φ = 2aU cos + Γ (5.53 v v = ( φ = (2U sin Γ /a (5.54 =a p p = p + 1 2 ρv 2 p = p 1 2 ρv 2 = p ρ 2a 2 { 4a 2 U 2 sin 2 4aU(Γ / sin + (Γ / 2} (5.55 P (P, P P = p cos ad =, P = p sin ad = ρuγ (5.56 ρuγ [ ] U Γ ρuγ Kutta-Joukowski 6 14

F ig.33 p = 1, ρ = 1, a = 1, U = 1, Γ = 1 p π p = p { ρ 2a 2 4a 2 U 2 sin 2 4aUΓ sin + ( } 2 Γ Fig.33 ( 4 [ ] 5.2.1 5.2.2 5.3.1 28.9.6 ( 3 5.2.5 u, u (28.9.8 15

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