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i 20 1999 3 2 2010 5
ii 1 1 1.1 1 1.2 4 9 2 10 2.1 10 2.2 12 2.3 13 2.4 13 2.5 15 2.6 18 2.7 22 2.8 2 25 26 3 28 3.1 28 3.2 31 3.3 33 3.4 37 41 4 43 4.1 43 4.2 46 4.3 54 4.4 61
iii 67 5 70 5.1 70 5.2 74 5.3 77 5.4 79 5.5 85 90 6 93 6.1 1 93 6.2 93 6.3 103 107 7 2 109 7.1 109 7.2 115 7.3 116 7.4 123 133 8 135 8.1 135 8.2 136 8.3 140 8.4 142 8.5 144 8.6 150 152 9 154 9.1 154
iv 9.2 158 9.3 161 9.4 2 162 166 10 169 10.1 169 10.2 170 10.3 172 10.4 174 10.5 176 179 181 211 213 S E s, e
1 1 1903 Wilbur Wright, 1867 1912 Orville Wright, 1871 1948 2 1 1 5 6 150 [m/s] 900 [km/h] 250 [m/s] 1 50 300 [m/s] 0.83 1.1 1.1.1 p τ
2 1 τ = 1 v dv dp (1.1) v 1 τ (1.1) v ρ (1.1) τ = 1 dρ ρ dp (1.2) 1.1.2 Isaac Newton, 1642 1727 1687 Principia 6 1 2.6 Pierre-Simon Laplace, 1749 1827 2.7 4.1 a p ρ ( ) p a = (1.3) ρ s s a a = γ p ρ = γrt (1.4) (4.16) γ c v c p 2.4 R T (1.4) γ 2 (1.2) (1.3) τ 1 δp v δv δv/v = (1/v)(dv/dp)δp (1.1) 2 a =331.2+0.61t [m/s] t [ ]
1.1 3 τ a a = 1 ρτ (1.5) (τ 0) a M u a M = u a (1.6) (1.5) M τ 1.1.3 ρ 0 ρ ρ/ρ 0 1.05 M =0.32 4 (4.40) 5% M =0.32 1 [1] (subsonic flow) (0 <M<1) M 1 M 1 [2] (transonic flow) (M 1) [3] (supersonic flow) (M >1) M>1 transsonic s transonic Theodore von Kármán, 1881 1963
4 1 [4] (hypersonic flow) (M 1) [2] [3] [4] 1.2 1.2.1 44 1947 10 14 (Mojave Desert) XS-1 50 (Captain Charles Yeager) B29 Bell XS-1 1.1 2.7 XLR-11 12,900 M =1.06 NACA (National Advisory Committee for Aeronautics) 23 M =0.85 1.1 Bell XS-1 1.2.2 1 [mm] G.F.B. Riemann 1858 2.6
1.2 5 2.7 2 2.8 William John Macquorn Rankine, 1820 1872 1870 Philosophical Transaction of the Royal Society Pierre Henry Hugoniot, 1851 1887 1887 Journal de l Ecole Polytechnique 5.2 2 Lord Rayleigh, 1842 1919 (Proceedings of the Royal Society, vol.84, 1910) 20 G.I. Taylor, 1886 1975 1 Proceedings of the Royal Society Ernst Mach, 1838 1916 1887 1.2 Carl G.P. de Laval, 1845 1915 1888 4.4
43 4 (3.38), (3.39) ρ + (ρu) = 0 t (4.1) ( ) u ρ t + u u = p (4.2) (3.40) 2 ρ t + u ρ x + v ρ y + w ρ ( u z + ρ x + v y + w ) =0 (4.3) z Du Dt = u t + u u x + v u y + w u z = 1 p ρ x Dv Dt = v t + u v x + v v y + w v z = 1 p ρ y Dw Dt = w t + u w x + v w y + w w z = 1 p ρ z (4.4) (4.5) (4.6) 4.1
44 4 4.1.1 u =0, p = p, 0 ρ = ρ 0 u 1, p 1, ρ 1 u 1 1, p 1 /p 0 1, ρ 1 /ρ 0 1 u = u 1, p = p 0 + p 1, ρ = ρ 0 + ρ 1 (2.51) (p 0 + p 1 )(ρ 0 + ρ 1 ) γ = p 0 ρ γ 0 ( 1+ p )( 1 1+ ρ ) γ ( 1 1+ p )( 1 1 γρ ) 1 + =1 p 0 ρ 0 p 0 ρ 0 p 1 /p 0 ρ 1 /ρ 0 p 1 p 0 = γρ 1 ρ 0 (4.7) γp0 a 0 ρ 0 (4.8) (4.7) p 1 = a 2 0ρ 1 (4.9) (4.1) (4.2) t (ρ 0 + ρ 1 )+ {(ρ 0 + ρ 1 )u 1 } =0, ( ) u1 (ρ 0 + ρ 1 ) + u 1 u 1 = (p 0 + p 1 ) t
4.1 45 ρ 1 t + ρ 0 u 1 = 0 (4.10) ρ 0 u 1 t = p 1 (4.11) (4.10) t 2 ρ 1 t 2 + ρ 0 ( ) u1 =0 t (4.11) ρ 0 ( ) u1 + 2 p 1 =0 t u 1 2 ρ 1 t 2 = 2 p 1 (4.9) p 1 2 ρ 1 t 2 = a2 0 2 ρ 1 (4.12) (4.12) 1 2 ρ 1 t 2 = 2 ρ 1 a2 0 x 2 (4.13) ρ 1 = f(x a 0 t)+g(x + a 0 t) (4.14) 3.2 ρ 1 t =0 f(x) g(x) ±a 0 a 0 (4.8) a 0 (2.51) ( p/ ρ) p=p0,ρ=ρ 0 a = ( ) p ρ s (4.15)
46 4 (2.51) (2.4) a 2 = γp ρ = γrt (4.16) T 4.1.2 M u a (4.17) (M =1) M>1 : (supersonic) M<1 : (subsonic) 4.2 4.2.1 1 ( / t 0) 1 4.1 x x u u 4.1 1 (4.3) (4.6) y z v, w x u ρ p x
4.2 47 u dρ dx + ρdu dx = 0 (4.18) u du dx = 1 dp ρ dx dρ/dx 0 (4.18) d (ρu) =0 dx (4.19) x q = ρu (4.20) u = q/ρ (4.19) q d ρ dx ( ) q = q ( q ) dρ ρ ρ ρ 2 dx = 1 ρ ( ) 2 q dρ ρ dx (4.19) (2.10) ρ = Φ(p) (4.21) dp/dx =(dp/dρ)(dρ/dx) ( ) 2 q = dp ρ dρ (4.20) u 2 (4.15) a 2 u = a (4.22) u x a 4.2.2 1 1 x x (u) (ρu = )
48 4 4.2 (x, y, z) F (x) x y, z v, w x u y, z 1 4.2 (throat) U ρ R x ɛ x L = R /ɛ x = 1 L X = ɛ 1 R X, y = 1 R Y, z = 1 R Z y, z x (X, Y, Z) x 3 / X, / Y, / Z u, ρ, p X v, w X, Y, Z ɛ 1 u = U (u (0) (X)+ɛu (1) (X, Y, Z)+ɛ 2 u (2) (X, Y, Z)+ ) v = U (ɛv (1) (X, Y, Z)+ɛ 2 v (2) (X, Y, Z)+ ) w = U (ɛw (1) (X, Y, Z)+ɛ 2 w (2) (X, Y, Z)+ ) ρ = ρ (ρ (0) (X)+ɛρ (1) (X, Y, Z)+ɛ 2 ρ (2) (X, Y, Z)+ )
4.2 49 p = ρ U 2 (p (0) (X)+ɛp (1) (X, Y, zz)+ɛ 2 p (2) (X, Y, Z)+ ) (4.3) (4.6) ɛ 1 ɛ ( ) (0) ρ(0) u (0) u X + ρ(0) X + v(1) Y + w(1) =0 Z (0) u(0) u X = 1 p (0) ρ (0) X 0= 1 p (1) ρ (0) Y 0= 1 p (1) ρ (0) Z u ρ ( u x + ρ x + v y + w ) = 0 (4.23) z u u x = 1 p ρ x (4.24) p/ y = p/ z =0 1 1 (4.18), (4.19) (4.24) (4.23) (4.18) ρ v/ y + w/ z 4.2.3 1 1 (4.23), (4.24) F (x) x u(x) ρ(x) F (x) (4.21) (4.23) x x + Δx ΔV = FΔx u( ρ/ x) 1 ΔV u ρ ρ dv = u x x FΔx
109 7 2 5 6 1 2 7.1 7.1.1 (oblique shock) 7.1 (detached shock) θ β M 5 1 2 M 1 > 1 (4.50) (5.65) M 1 > 1 M 2 < 1 7.1
110 7 2 7.1.2 (5.43) (5.48) M 1 = U 1 /a 1 β M 1 1 (5.11) 2 (5.43) (5.48) (i =1) (i =2) U i u i v i 7.2 u 1 = U 1 sin β, v 1 = U 1 cos β u 2 = U 2 sin(β θ), v 2 = U 2 cos(β θ) u 2 v 2 =tan(β θ) (7.1) (5.11) v 2 = v 1 = U 1 cos β (7.2) M 2 M 2 2 = = ( U2 a 2 ( u2 a 2 ) 2 = ) 2 + ( u2 a 2 ( a1 a 2 ) 2 + ( v2 a 2 ) 2 = ( u2 a 2 ) 2 ( U1 cos β + a 2 ) 2 ) 2 M 2 1 cos 2 β (7.3) 7.2
7.1 111 (5.43), (5.44), (5.46), (5.47), (5.48) M 1 u 1 /a 1 =(U 1 sin β)/a 1 = M 1 sin β ρ 2 ρ 1 = 1 2 (γ +1)M 2 1 sin 2 β 1+ 1 2 (γ 1)M 2 1 sin2 β (7.4) p 2 p 1 = T 2 T 1 = a 2 a 1 = γm1 2 sin 2 β 1 (γ 1) 2 1 (7.5) 2 (γ +1) }{γm 21 sin 2 β 12 } (γ 1) { 1+ 1 2 (γ 1)M 2 1 sin 2 β { 1+ 1 2 (γ 1)M 2 1 sin2 β s 2 s 1 = R γ 1 ln 1 4 (γ +1)2 M 2 1 sin2 β } 1 { 2 γm1 2 sin2 β 1 } 1 2 (γ 1) 2 1 2 (γ +1)M 1 sin β γm1 2 sin2 β 1 (γ 1) 2 1 (γ +1) 2 (7.6) (7.7) 1+ 1 γ 2 (γ 1)M 1 2 sin2 β 1 2 (γ +1)M 1 2 sin2 β (7.8) M 1 sin β β π/2 (5.45) u 1+ 1 2 = 2 (γ 1)M 1 2 sin 2 β u 1 1 2 (γ +1)M 1 2 sin2 β (7.9) (7.4) (7.8) M 1 (7.3)
112 7 2 M 1 = U 1 a 1 (7.10) M 2 = U 2 /a 2 (5.36) (5.36) M 1 M 1 sin β ( u2 a 2 ) 2 1+ 1 = 2 (γ 1)M 1 2 sin 2 β γm1 2 sin2 β 1 (7.11) 2 (γ 1) (u 2 /a 2 ) 2 (7.3) (7.7), (7.11) { } 1 1+(γ 1)M 2 M2 2 1 sin 2 β + M1 4 4 (γ +1)2 γ sin 2 β sin 2 β = { 1+ 1 2 (γ 1)M 1 2 sin2 β }{γm 21 sin2 β 12 } (7.12) (γ 1) β = π/2 (5.36) β =sin 1 (1/M 1 ) M 2 = M 1 β (7.14) μ 7.1.3 θ β M 1 (7.1) v 2 = U 1 cos β, u 1 = U 1 sin β tan(β θ) = u 2 U 1 cos β = u 2 u 1 sin β cos β (7.9) 1 tan β tan θ 1+ 1+tanβtan θ = 2 (γ 1)M 1 2 sin 2 β 1 2 (γ +1)M 1 2 sin β cos β tan θ = 1+ (M1 2 sin 2 β 1) cot β { } 1 2 (γ +1) sin2 β M1 2
7.1 113 { M 2 =2cotβ 1 sin 2 } β 1 M1 2 (γ +cos2β)+2 (7.13) θ β M M 1 > 1 θ β 7.3 (7.13) 7.3 (a) β = π/2 β =sin 1 (1/M 1 ) θ =0 β =sin 1 (1/M 1 ) M 1 1 π/2 M 1 0 (b) M 1 > 1 θ θ max sin 1 (1/M 1 ) <β<π/2 (c) M 1 > 1 M 2 =1 β (7.12) (d) θ<θ max β M 1 > 1 M 2 < 1 β M 2 > 1 β 9.3 7.3 θ β M
114 7 2 7.1 M 1 θ max (7.13) tan θ =2cotβ ( sin 2 β 1/M 2 1 γ +cot2β +2/M 2 1 M 1 tan θ = sin 2β γ +cos2β β d(tan θ) dβ = 2(1 + γ cos 2β) (γ +cos2β) 2 =0 ) cos 2β = 1/γ = 5/7 sin 2β =2 6/7 tan θ max =5 6/12 = 1.0206 θ max 45.73 7.1.4 7.4(a) 7.4(b) 2 θ>θ max (7.13) β 7.5 9.3 7.4
7.2 115 θ 0 β sin 1 (1/M 1 )=μ 7.6 μ ρ 1 = ρ 2, p 1 = p 2, T 1 = T 2 μ 7.5 7.6 7.2 4 2 7.7 V t A A AA = Vt V < a t =0 V > a t =0 A B A C V > a AA μ 7.7 V
213 46 161 3 2, 38 1 1 11 10 56 11, 38 31 29 31 30 15 22 38 29 2, 45 46 161 28 1 48 74 4, 74 109 103 83 10 74 32 48 103 62 139 164 3 143 39 34 156 156 34 35 16 13 11, 13 178 3 3 32 33 4 32 45 2 18 57 61 173 46 3 55 14 13 90
214 30 20 38 96 31 32 35 109 32 15 33 2 25 10 13 32 12 1 4 2 13 14 31 11 164 29 10, 13 8, 152 86 121, 132 117 7 31, 70 32 50 34 109 113 34 20 10 7 169, 172 169 169, 172 169 17 11 26 116 116 1, 3, 5, 46 116 116 99 55 5, 62 75 5 75 10, 13 109 128 8, 59 178 176 47
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