4 2016 3 8

2.,. 2. Arakawa Jacobin., 2 Adams-Bashforth. Re = 80, 90, 100.. h l, h/l, Kármán, h/l 0.28,, h/l.., (2010), 46.2., t = 100 t = 2000 46.2 < Re 46.5.

1 1 4 2 6 2.1............................ 6 2.2............................... 7 2.3 Poisson.................. 8 3 10 3.1............................... 10 3.2 CFL................................ 11 3.3 von Neumann.......................... 13 3.3.1........................ 13 3.3.2........................ 16 3.3.3..................... 18 4 23 4.1......................... 23 4.1.1.......................... 24

2 4.1.2.......................... 26 4.2.......................... 29 4.2.1.......................... 35 4.2.2.......................... 37 5 41 5.1 von Neumann..... 41 5.2....................... 42 5.3......................... 44 5.4............................. 46 6 49 6.1.................... 49 6.1.1............................. 49 6.1.2............................. 50 6.2................... 53 7 59 A 60 1 Lagrange Euler...................... 60 B 61

3 1 Euler................................. 61 2 Adams-Bashforth.......................... 61 3 Arakawa Jacobian.......................... 62 4 Jacobi Gauss-Seidel...................... 70 C 72 1.............................. 72 2............................... 80 3............................... 82 4................... 84 85 86

1 4 1.,. 1 : 2015 1 9 10 ( ). (, 2016) 1, 2015 1 9 10 ( ). ( ).

1 5 2 *1 (, 1988),. 2 : (, 1988), 2, 3.,, (,, 2015).,, 2.,,.. 1. 2. 3. 4. 5,. 6.,.,. 7.. *1 von Kármán (1911),

2.1 6 2 2.1,., ( ),. ρ,, u t + u u = 1 ρ p + ν 2 u (2.1.1). u, p. Navier-Stokes,, 1, 2, 1, 2., ν µ, ν = µ ρ (2.1.2). (2.1.1). (2.1.1).. u = Uu, x = Lx, t = L U t, p = ρu 2 p. (2.1.3) u, x, t, p,,,. U, L. (2.1.3) (2.1.1), L/U 2, u t + (u )u = p + 1 Re 2 u (2.1.4)., Re, Re = UL ν (2.1.5)

2.2 7. Navier-Stokes (2.1.4).,., (2.1.4),. u t + (u )u = p + 1 Re 2 u (2.1.6) 2.2 2 1 ( ).. 2 Navier-Stokes (2.1.6) x, y,, u t + u u x + v u y = p x + 1 Re v t + u v x + v v y = p y + 1 Re ( 2 ) x + 2 u, 2 y 2 (2.2.7) ) x + 2 v. 2 y 2 (2.2.8) ( 2, u u = ui + vj. i j, x y. (2.2.7) y, 2 u t y + u u y x +u 2 u x y + v u y y +v 2 u y = 2 p 2 x y + 1 ( ) 2 Re x + 2 u 2 y 2 y (2.2.9)., (2.2.8) x, 2 v t x + u u v x x +u 2 x + v v 2 x y +v 2 v x y = 2 p x y + 1 ( 2 Re x 2 + 2 y 2 ) v x (2.2.10). (2.2.10) (2.2.9), ( v t x u ) + u ( v y x x u ) + u ( v y x x u ) y + v ( v y x u ) + v ( v y y x u ) y = 1 ( ) ( 2 Re x + 2 v 2 y 2 x u ) y (2.2.11)

2.3 Poisson 8. ζ = v x u y, ζ t + u ζ x + v ζ y +.,,, (2.2.12), ζ t + u ζ x + v ζ y = 1 Re ( u x + v ) ζ = 1 ( ) 2 y Re x + 2 ζ (2.2.12) 2 y 2 u = u x + v y = 0 (2.2.13) ( 2 ) x + 2 ζ = 1 2 y 2 Re 2 ζ (2.2.14). ζ. (2.2.14)., u = ϕ y, v = ϕ x (2.2.15), (2.2.14), ζ t ϕ ζ y x + ϕ ζ x y = 1 Re 2 ζ (2.2.16). 2.3 Poisson Poisson,, 2., Poisson. (2.2.15)., u = 2 ϕ y y, (2.3.17) 2 v x = 2 ϕ (2.3.18) x 2. (2.3.17), (2.3.18), ζ = v/ x u/ y, 2 ϕ x + 2 ϕ 2 y = ζ (2.3.19) 2

2.3 Poisson 9. Poisson., (2.3.19) (2.2.16), t 2 ϕ ϕ y x 2 ϕ + ϕ x y 2 ϕ = 1 Re 4 ϕ (2.3.20)., (2.3.20).

3.1 10 3, (2.2.18),. 3.1,.,. t f(t), 3 : df dt = lim t 0 df dt = lim t 0 df dt = lim t 0 f(t + t) f(t), (3.1.1) t f(t + t) f(t t), (3.1.2) 2 t f(t) f(t t). (3.1.3) t,, t 0, t., 3 lim t 0,,,,. 3, df dt df dt f(t + t) f(t), (3.1.4) t f(t + t) f(t t), 2 t (3.1.5) df f(t) f(t t) dt t (3.1.6). t, x., (i, j, τ), (x, y, t) = (i x, j x, τ t). A(x, y, t)

3.2 CFL 11 A τ i,j.,., (2.2.16) (2.3.19), (2.2.16), ζ τ+1 i,j ζ τ 1 i,j 2 t (2.3.19), ϕτ i,j+1 ϕ τ i,j 1 ζi+1,j τ ζi 1,j τ + ϕτ i+1,j ϕ τ i 1,j ζi,j+1 τ ζi,j 1 τ 2 x 2 x 2 x 2 x = 1 ( 4 ζ τ i+1,j + ζi 1,j τ + ζi,j+1 τ + ζ τ ) i,j 1 ζ τ Re ( x) 2 i,j 4 = 1 1 Re ( x) 2 ζi,j. τ 2 ζi,j τ = 4 ( ϕ τ i+1,j + ϕ τ i 1,j + ϕ τ i,j+1 + ϕ τ ) i,j 1 ϕ τ ( x) 2 i,j 4 = 1 ( x 2 ) 2 ϕ τ i,j. (3.1.7) (3.1.8), 2 A i,j 4 ( Ai+1,j + A i 1,j + A i,j+1 + A i,j 1 4 A i,j ) (3.1.9) *1., leap-frog,, Adams-Bashforth ( B.2 ). 3.2 CFL.. *1 (3.1.7) (3.1.8), (3.1.7) : 2 ζ τ i,j 1 2 ( 2 ζ τ+1 i,j + ) 2 ζ τ 1 i,j. (3.1.10), (3.1.7), τ + 1 τ 1, 2 ζ τ+1 i,j µ 2 ζ τ+1 i,j = ( 2 + µ 2 )ζ τ 1 i,j λ 2 [ (ϕ τ i,j+1 ϕ τ i,j 1)(ζi+1,j τ ζi 1,j) τ (ϕ τ i+1,j ϕ τ i 1,j)(ζi,j+1 τ ζi,j 1) τ ] (3.1.11)., µ 2 = Re( x) 2 / t, λ 2 = Re/2., τ 1 τ,, (3.1.8) (3.1.11), τ + 1,., Hirota, Miyakoda (1964),.

3.2 CFL 12, CFL. t x u(x, t), c x, u t + c u x = 0 (3.2.12)., 2. F, u = F (x ct),, u t = F t (x ct) = F ( c), (3.2.13) u x = F x (x ct) = F (3.2.14) F ( c) + F = 0 (3.2.15), u = F (x ct). 3.1, A(x, t) A τ i,,, (3.2.12), u τ+1 i u τ i t + c uτ i u τ i 1 x = 0 (3.2.16)., u τ i 1 uτ i,.,,. τ u τ i, u τ 1 i, τ + 1 u τ+1 i, u τ+1 i = u τ i c t ( u τ x i ui 1) τ (3.2.17)., 3.. A., x, t. 3 A. (3.2.17),., 1/c t/ x,, 1 c t x, (3.2.18) 1 c t x. CFL. (3.2.19)

3.3 Von Neumann 13 3 : (3.2.17) A... 3.3 von Neumann 1 von Neumann.,,..,,, 1.,,,. 3.3.1 (3.2.12) U k (t), k, u(x, t) = U k (t)e jkx (3.3.20)

3.3 Von Neumann 14 *2, (3.2.12), ( ) duk dt + jkcu k e jkx = 0 (3.3.21)., e jkx 0,.,,, ln du k dt + jkcu k = 0 (3.3.22) du k = jkcu k, (3.3.23) dt du k = jkc dt, (3.3.24) U k t duk = jkc dt (3.3.25) U k. (3.3.28) (3.3.20), 0 ( ) Uk (t) = jkct, U k (0) : (3.3.26) U k (0) (3.3.29), U k (t) U k (0) = e jkct (3.3.27) U k (t) = U k (0)e jkct (3.3.28) u(x, t) = U k (0)e jk(x ct). (3.3.29) u(x, t) = 1 2π U k (0)e jk(x ct) dk (3.3.30), (3.3.30) (3.2.12)., c., (x, t) = (i x, τ t), 3.1 u(x, t) u τ i. (3.2.16) k,., (3.2.16), *2, j. u τ i = U τ e jki x U τ : τ (3.3.31) u τ+1 i = u τ i c t ( u τ x i ui 1) τ (3.3.32) = (1 µ)u τ i + µu τ i 1 (3.3.33)

3.3 Von Neumann 15., µ = c t/ x. (3.3.31) (3.3.33),,, λ U τ+1 e jki x = (1 µ)u τ e jki x + µu τ e jk(i 1) x (3.3.34)., U τ+1 = (1 µ)u τ + µu τ e jk x = [ (1 µ) + µe jk x] U τ = λu τ. (3.3.35) λ (1 µ) + µe jk x (3.3.36) U τ = λ U τ 1 = λ 2 U τ 2 = = λ n U 0 < B B : (3.3.37),, B : τ ln λ < ln B U 0 B. (3.3.38) τ = t/ t, ln λ < B t t. λ 1 + δ, δ 1, (3.3.39),, ln (1 + δ) = δ δ2 2 + δ3 3. (3.3.40) δ O( t) (3.3.41) λ 1 + O( t), (3.3.42) λ 1. (3.3.43) (3.3.43) von Neumann. (3.3.36), λ = 1 µ + µ(cos k x j sin k x) (3.3.44), λ 2 = 1 2µ(1 µ)(1 cos k x) (3.3.45)

3.3 Von Neumann 16. (3.3.43),, λ 2 1, 1 cos k x 0, 1 µ 0 (3.3.46). (3.3.46), µ 1, (3.3.47) c t x 1 (3.3.48). (3.2.16), CFL. 3.3.2 1, t x u(x, t), D, u t = D 2 u (3.3.49) x 2. 3.3.1 (3.3.20), (3.3.49), du k dt ejkx = Dk 2 U k e jkx, (3.3.50) ( duk dt + Dk2 U k., e jkx 0, ) e jkx = 0 (3.3.51).,, ln du k dt + Dk2 U k = 0 (3.3.52) du k = Dk 2 U k, (3.3.53) dt du k = Dk 2 dt, (3.3.54) U k t duk = Dk 2 dt (3.3.55) U k 0 ( ) Uk (t) = Dk 2, U k (0) : (3.3.56) U k (0) U k (t) U k (0) = e Dk2 t (3.3.57)

3.3 Von Neumann 17, U k (t) = U k (0)e Dk2 t. (3.3.58) (3.3.20), (3.3.58) (3.3.59), u(x, t) = U k (0)e jkx Dk2t. (3.3.59) u(x, t) = 1 2π U k (0)e jkx Dk2t dk (3.3.60), (3.3.60) (3.3.49)., (x, t) = (i x, τ t), 3.1 u(x, t) u τ i., (3.3.49),, u τ+1 i u τ i = D uτ i+1 2u τ i + u τ i 1 (3.3.61) t ( x) 2. (3.3.61) k, u τ i = U τ e jki x U τ : τ (3.3.62). (3.3.61), u τ+1 i = u τ i + D t ( ) u τ ( x) 2 i+1 2u τ i + u τ i 1 = (1 2µ)u τ i + µ ( ) u τ i+1 + u τ i 1 (3.3.63) (3.3.64)., µ = D t/( x) 2. (3.3.62) (3.3.64), U τ+1 e jki x = (1 2µ)U τ e jki x + µ ( e jk(i+1) x + e jk(i 1) x) U τ (3.3.65), U τ+1 = = (1 2µ)U τ + µ ( e jk x + e jk x) U τ 1 2µ + µ ( e jk x + e jk x)] U τ = λu τ. (3.3.66), λ λ = 1 2µ + µ ( e jk x + e jk x) (3.3.67). (3.3.67), λ = 1 2µ + µ(cos k x + j sin k x + cos k x j sin k x) = 1 2µ(1 cos k x) = 1 4µ sin 2 k x 2 (3.3.68)

3.3 Von Neumann 18, 3.3.1, λ 1.,, 1 1 4µ sin 2 k x 2 0 µ sin 2 k x 1 2 2 1, (3.3.69) µ 1 2, (3.3.70) D t 1 ( x) 2 2 (3.3.71). (3.3.61). 3.3.3 1, t x u(x, t), c, D, u t + c u x = D 2 u (3.3.72) x 2. 3.3.1 (3.3.20), (3.3.72), du k dt ejkx + jkcu k e jkx = Dk 2 U k e jkx, (3.3.73) ( ) duk dt + jkcu k + Dk 2 U k e jkx = 0 (3.3.74)., e jkx 0, du k dt + jkcu k + Dk 2 U k = 0 (3.3.75)., duk du k = jkcu k Dk 2 U k, dt (3.3.76) du k = jkc dt Dk 2 dt, U k (3.3.77) U k = jkc t t dt Dk 2 dt (3.3.78) 0 0

3.3 Von Neumann 19, ln ( ) Uk (t) = jkct Dk 2 t, U k (0) : (3.3.79) U k (0), U k (t) U k (0) = e (jkc+dk2 )t U k (t) = U k (0)e (jkc+dk2 )t. (3.3.81) (3.3.20), u(x, t) = U k (0)e jk(x ct) Dk2 t (3.3.80) (3.3.81) (3.3.82) (3.3.82), u(x, t) = 1 2π U k (0)e jk(x ct) Dk2t dk (3.3.83), (3.3.83) (3.3.72)., (x, t) = (i x, τ t), 3.1 u(x, t) u τ i., (3.3.72),, u τ+1 i u τ i t + c uτ i+1 u τ i 1 2 x (3.3.84) k, = D uτ i+1 2u τ i + u τ i 1 ( x) 2. (3.3.84). (3.3.84),, u τ i = U τ e jki x U τ : τ (3.3.85) u τ+1 i = u τ i c t ( u τ 2 x i+1 ui 1) τ D t + = (1 2µ )u τ i ( µ 2 µ ) u τ i+1 +. (3.3.85) (3.3.87), ( ) u τ ( x) 2 i+1 2u τ i + u τ i 1 (3.3.86) ( µ ) 2 + µ u τ i 1 (3.3.87) µ = c t x, (3.3.88) µ = D t ( x) 2 (3.3.89) U τ+1 e jki x =(1 2µ )U τ e jki x ( µ ) 2 µ U τ e jk(i+1) x ( µ ) + 2 + µ U τ e jk(i 1) x (3.3.90)

3.3 Von Neumann 20, ( µ ) ( µ ) U τ+1 = (1 2µ )U τ 2 µ U τ e jk x + 2 + µ U τ e jk x [ ( µ ) ( µ ) ] = 1 2µ 2 µ e jk x + 2 + µ e jk x U τ = λu τ (3.3.91), λ λ = 1 2µ. (3.3.92), ( µ ) ( µ ) 2 µ e jk x + 2 + µ e jk x (3.3.92) λ = 1 2µ µ ( e jk x e jk x) + µ ( e jk x + e jk x) 2 = 1 2µ µj sin k x + 2µ cos k x = 1 2µ (1 cos k x) µj sin k x (3.3.93), λ 2 = [1 2µ (1 cos k x)] 2 + µ 2 sin 2 k x (3.3.94). (3.3.93) (1 2µ ), 2µ, µ, (3.3.43), 1.

3.3 Von Neumann 21 4 : λ. k x 0, cos k x = 1 1 2! (k x)2 +, sin k x = k x 1 3! (k x)3 +, (3.3.94) λ 2 [ 1 2µ ( 1 2! k x )] 2 + µ 2 (k x) 2 1 2µ (k x) 2 + µ 2 (k x) 4 + µ 2 (k x) 2 1 + ( µ 2 2µ ) (k x) 2 (3.3.95). k x = π, λ 2 = (1 4µ ) 2 (3.3.96)

3.3 Von Neumann 22. 3.3.1 λ 1. (3.3.95), (3.3.96), µ 2 2µ 0, µ 2 2µ. (3.3.97) 1 1 4µ 1, 0 4µ 2, 0 µ 1 2 (3.3.98) k x = π/2, λ 2 = (1 2µ ) 2 + µ 2 (3.3.99), (3.3.97) (3.3.98) λ 2 1., (3.3.97) (3.3.98) (3.3.84).

4.1 23 4,.,,. 4.1. x, y (u = U, v = 0)..,,. 5 :.

4.1 24,. 5, x I + 1, y J + 1.,, (I + 1) (J + 1)., I + 1 X, J + 1 Y. 4.1.1. (x = 0) y U x, u = U, (4.1.1) v = 0 (4.1.2). (2.2.15), ϕ, ϕ = y 0 Udy = Uy + const ( = U y Y ) 2 (4.1.3). (4.1.3) ϕ = 0. (2.3.19), ζ,. ζ = U y = 0 (4.1.4) (y = Y ),.,,., u/ y = 0.

4.1 25,,. (2.2.15), ϕ, u = U, (4.1.5) v = 0 (4.1.6) ϕ = uy 2 (4.1.7). (2.3.19), ζ,. ζ = u y = 0 (4.1.8) (y = 0),. ϕ ζ,. u = U, (4.1.9) v = 0 (4.1.10) ϕ = uy 2, (4.1.11) ζ = u y = 0 (4.1.12) (x = X) :. u x = 0, (4.1.13) v x = 0 (4.1.14)

4.1 26,,, u = 0, (4.1.15) v = 0. (4.1.16) (2.2.15),,. ϕ y = 0, (4.1.17) ϕ x = 0 (4.1.18) 4.1.2 (). (x = 0), (4.1.1), (4.1.2), u 0,j = U, (4.1.19) v 0,j = 0, (4.1.20).,, (4.1.3), ( ϕ 0,j = U y j J ) 2 (4.1.21)., j y, y y, J., J/2,., (2.3.19), ζ 0,j,. ϕ 0,J/2 = 0 (4.1.22) ζ 0,j = 0 (4.1.23)

4.1 27 (y = Y ), (4.1.5), (4.1.6),. (4.1.7), u i,j = U, (4.1.24) v i,j = 0 (4.1.25) ϕ i,j = J U y (4.1.26) 2. :, ϕ i,j 1,. ϕ i,j 1 = ϕ i,j ϕ i,j y y + 1 2 ϕ i,j 2 y 2 ( y)2 + ϕ i,j ϕ i,j y y + 1 2 ϕ i,j 2 y 2 ( y)2 (4.1.27), (4.1.27), y 2,., ζ i,j,. ϕ i,j y = U, (4.1.28) 2 ϕ i,j y 2 = ζ i,j (4.1.29) ϕ i,j 1 = ϕ i,j + U y + 1 2 ζ i,j( y) 2 (4.1.30) ζ i,j = 2 (ϕ i,j 1 ϕ i,j U y) ( y) 2 (4.1.31) (y = 0), (4.1.9), (4.1.10), u i,0 = U, (4.1.32) v i,0 = 0 (4.1.33)

4.1 28. (4.1.11), ϕ i,0,. : ϕ i,1, ϕ i,0 = J U y (4.1.34) 2., (4.1.35), ϕ i,1 = ϕ i,0 + ϕ i,0 y y + 1 2 ϕ i,0 2 y 2 ( y)2 + ϕ i,0 + ϕ i,0 y y + 1 2 ϕ i,0 2 y 2 ( y)2 (4.1.35) ϕ i,0 y = U, (4.1.36) 2 ϕ i,0 y 2 = ζ i,0 (4.1.37) ϕ i,1 = ϕ i,0 U y + 1 2 ζ i,0( y) 2 (4.1.38)., ζ i,0,. ζ i,0 = 2 (ϕ i,1 ϕ i,0 + U y) ( y) 2 (4.1.39) (x = X) (4.1.13), (4.1.14), ϕ I,j = ϕ I 1,j (4.1.40).,, ζ I,j x = 0 (4.1.41), (4.1.41), ζ I,j ζ I 1,j = 0 (4.1.42) x, ζ I,j,. ζ I,j = ζ I 1,j (4.1.43)

4.2 29, ζ.,, u i,j = 0, (4.1.44) v i,j = 0, (4.1.45) ϕ i,j = 0 (4.1.46). (4.1.31), (4.1.39) ζ L i,j, ζ R i,j, ζ F i,j, ζ B i,j, ζ L i,j = 2ϕ i,j+1 ( y) 2, (4.1.47) ζ R i,j = 2ϕ i,j 1 ( y) 2, (4.1.48) ζ F i,j = 2ϕ i 1,j ( x) 2, (4.1.49). ζ B i,j = 2ϕ i+1,j ( x) 2 (4.1.50) 4.2 (, 2012)., x, y (u = U, v = 0),.,,,,. 6,, 2., x = r cos ϑ, (4.2.51) y = r sin ϑ (4.2.52). 5, x x, ϑ π., ϑ π ϑ π., 1, R.

4.2 30 6 :., r, ϑ ( ), 7. 7 :.

4.2 31, r : r = e ω. (4.2.53),,, 8. 8 :., (4.2.53), ω, ω max = log R (4.2.54), ω, ω min = 0 (4.2.55)., (2.2.15), Poisson (2.3.19), (2.2.14), (4.2.53)., (2.2.15). u u = ui + vj (4.2.56)

4.2 32, 2 u u = u r e r + v ϑ e ϑ (4.2.57). e r, e ϑ,, e r = i cos ϑ + j sin ϑ, (4.2.58) e ϑ = i sin ϑ + j cos ϑ (4.2.59)., (4.2.56), (4.2.57), u = ui + vj = u r e r + v ϑ e ϑ = u r (i cos ϑ + j sin ϑ) + v ϑ ( i sin ϑ + j cos ϑ) = (u r cos ϑ v ϑ sin ϑ)i + (u r sin ϑ + v ϑ cos ϑ)j (4.2.60), u, v,, u = u r cos ϑ v ϑ sin ϑ, (4.2.61) v = u r sin ϑ + v ϑ cos ϑ (4.2.62). r, r = (x 2 + y 2 ) 1/2 (4.2.63),.,, r x = x (x 2 + y 2 ) = x 1/2 r = cos ϑ, (4.2.64) r y = y (x 2 + y 2 ) = y 1/2 r = sin ϑ (4.2.65) x tan ϑ = 1 ϑ cos 2 ϑ x = y (4.2.66) x 2 ϑ x = y x 2 cos2 ϑ = r sin ϑ r 2 cos 2 ϑ cos2 ϑ ϑ y = 1 sin ϑ, r (4.2.67) = x y 2 sin2 ϑ = r cos ϑ r 2 sin 2 ϑ sin2 ϑ = 1 cos ϑ (4.2.68) r

4.2 33. / x, (4.2.64) (4.2.67), x = r x r + ϑ x ϑ = cos ϑ r 1 r sin ϑ ϑ, / y, (4.2.65) (4.2.68), y. (2.2.15), (4.2.61), (4.2.70), = r y r + ϑ y ϑ = sin ϑ r + 1 r cos ϑ ϑ (4.2.69) (4.2.70) u = ϕ y = 1 cos ϑ ϕ sin ϑ ϕ r ϑ r = u r cos ϑ v ϑ sin ϑ (4.2.71). (4.2.53), (4.2.71) 1,,. u r = 1 ϕ r ϑ = e v ϑ = ϕ r = e ω ϕ ω ϕ ω (4.2.72) ϑ, (4.2.73) (4.2.74) (4.2.75), Poisson (2.3.19). (4.2.64) (4.2.69), (2.3.19) 1, ( 2 ϕ x = cos ϑ 2 r 1 r sin ϑ ) ( cos ϑ ϕ ϑ r 1 ) sin ϑ ϕ (4.2.76) r ϑ., (4.2.65) (4.2.70), (2.3.19) 2, ( 2 ϕ y = sin ϑ 2 r + 1 r cos ϑ ) ( sin ϑ ϕ ϑ r + 1 ) cos ϑ ϕ r ϑ (4.2.77)

4.2 34., (4.2.53), r = ω r ω = e ω ω, (4.2.78) 1 = e 2ω (4.2.79) r 2. (4.2.78) (4.2.79), (4.2.76) (4.2.77), 2 ϕ x + 2 ϕ = 2 ϕ 2 y 2 r + 1 2 ϕ 2 r 2 ϑ 2 ( = e 2ω 2 ϕ ω + 2 ϕ 2 ϑ 2 ) (4.2.80). (4.2.80) Poisson., (2.2.14). (2.2.14) 2, 3, u ζ x + v ζ y = u ζ ( ) ζ = (u r e r + u ϑ e ϑ ) e r r + e 1 ζ ϑ r ϑ ζ = u r r + u 1 ζ ϑ r ϑ ω ζ ω ζ = u r e ω + u ϑe ( ϑ) = e ω ζ u r ω + u ζ ϑ ϑ (4.2.81)., (2.2.14), (4.2.80), 1 Re 2 ζ = 1 ( ) 2 ζ Re x + 2 ζ 2 y ( 2 ) = e 2ω 2 ζ Re ω + 2 ζ 2 ϑ 2. (4.2.79) (4.2.80), (2.2.14), ( ) ( ) ζ t + ζ e ω u r ω + u ζ ϑ = e 2ω 2 ζ ϑ Re ω + 2 ζ 2 ϑ 2 (4.2.82) (4.2.83). (4.2.83).

4.2 35 4.2.1. ( π ϑ < π/2, π/2 < ϑ π) R U x,. (2.2.15), (4.2.52), (4.2.53), ϕ, ϕ = u = U, (4.2.84) v = 0 (4.2.85) y 0 Udy = Uy + const = Uy = Ur sin(ϑ π) = Ue ω sin(ϑ π) (4.2.86). (4.2.86) ϕ = 0. (2.3.19), ζ,. ζ = U y = 0 (4.2.87) (ϑ = π/2),,.,,. (2.2.15), (4.2.54), ϕ, u = U, (4.2.88) v = 0 (4.2.89) ϕ = UR = Ue ωmax (4.2.90)

4.2 36. (2.3.19), ζ,. ζ = U y = 0 (4.2.91) (ϑ = π/2),. ϕ ζ,. u = U, (4.2.92) v = 0 (4.2.93) ϕ = UR = Ue ωmax (4.2.94) ζ = U y = 0 (4.2.95) ( π/2 < ϑ < π/2) :. u x = 0, (4.2.96) v x = 0 (4.2.97),,, u r = 0, (4.2.98) v ϑ = 0. (4.2.99)

4.2 37 (2.2.15),, ϕ ϑ = 0, (4.2.100) ϕ r = 0, (4.2.101) ϕ = 0 (4.2.102). 4.2.2 (). 9 :. 8, (ω, ϑ), 9. ω i, ϑ j.,, (ω, ϑ) = (0, π)., ω, ϑ, I ω = ω max, J ϑ = π.

4.2 38 (ω = ω max, π ϑ < π/2, π/2 < ϑ π, 0 j J/4 1, 3J/4 + 1 j J), (4.2.84), (4.2.85), u I,j = U, (4.2.103) v I,j = 0 (4.2.104)., (4.2.86), ϕ I,j = Ue I ω sin(j ϑ π) = Ue ωmax sin(j ϑ π) ( = Ue ω max sin ϑ j π ) ( ϑ = Ue ω max sin ϑ j J ) 2 (4.2.105)., J/2, ϕ I,J/2 = 0 (4.2.106)., (2.3.19), ζ I,j, ζ I,j = 0 (4.2.107). (ω = ω max, ϑ = π/2, j = 3J/4), (4.2.88), (4.2.89),., (4.2.90),., R, u I,3J/4 = U, (4.2.108) v I,3J/4 = 0 (4.2.109) ϕ I,3J/4 = Ue I ω (4.2.110) ζ I,3J/4 = 0 (4.2.111) r., ζ I,3J/4,. ζ I,3J/4 = ζ I 1,3J/4 (4.2.112)

4.2 39 (ω = ω max, ϑ = π/2, j = J/4), (4.2.92), (4.2.93), u I,J/4 = U, (4.2.113) v I,J/4 = 0 (4.2.114)., (4.2.94), ϕ I,J/4 = Ue I ω (4.2.115).,, ζ I,J/4 = ζ I 1,J/4 (4.2.116). (ω = ω max, π/2 < ϑ < π/2, J/4 + 1 j 3J/4 1) (4.2.96), (4.2.97), ϕ I,j = ϕ I 1,j (4.2.117).,, ζ I,j ω = 0 (4.2.118), (4.2.118), ζ I,j ζ I 1,J ω, ζ I,j, = 0 (4.2.119) ζ I,j = ζ I 1,j (4.2.120). (ω = 0),, u 0,j = 0, (4.2.121) v 0,j = 0, (4.2.122) ϕ 0,j = 0 (4.2.123)

4.2 40., :, ϕ 1,j,. (4.2.122), ϕ 1,j = ϕ 0,j + ϕ 0,j ω ω + 1 2 ϕ 0,j 2 ω 2 ( ω)2 + ϕ 0,j + ϕ 0,j ω ω + 1 2 2 ϕ 0,j ω 2 ( ω)2 (4.2.124) v 0,j = ϕ 0,j ω = 0 (4.2.125). (4.2.123), (4.2.125), (4.2.124), ω 2,. (2.3.19),. (4.2.126), (4.2.127), ζ 0,j,. ϕ 1,j = 1 2 ϕ 0,j 2 ω 2 ( ω)2 (4.2.126) 2 ϕ 0,j ω 2 = ζ 0,j (4.2.127) ζ 0,j = 2ϕ 1,j ( ω) 2 (4.2.128)

5.1 von Neumann 41 5, von Neumann., von Neumann.,.,.,,., Poisson., Gauss - Seidel ( B.4 ), Poisson..,,,.,.,, (2.2.18), (2.3.23), 2 Adams-Bashforth ( B.2 )., (2.2.18) Arakawa Jacobian ( B.3 ). 5.1 von Neumann, von Neumann., x = 0.1, t = 0.001, (3.3.88), µ = U t x = 1 10 2 10 1 = 10 2. (5.1.1)

5.2 42, (3.3.89), µ = 1 t Re ( x) 2 = 1 Re 10 3 10 2 von Neumann, (3.3.97) (5.1.1),, (3.3.98) (5.1.2), = 1 Re 10 1. (5.1.2) 10 4 20 Re, Re 2 10 5. (5.1.3) 0 1 Re 10 1 1 2, 2 10 1 Re (5.1.4). (5.1.3) (5.1.4), von Neumann 2 10 1 Re 2 10 5. 5.2, 2 ( A, B)., 100 (Re = 100) (,, 2010). 1, 2.

5.2 43 1 :. U = 1 1 S = 1 x = y = 0.1 t = 0.001 0 t 100 Re = 100 ϕ i,j 10 6 100 2 :. A B 40 10 40 20 401 101 401 201 (x, y) = (10, 5) (x, y) = (10, 10) 10, 11. 10 : A. t = 100.,. 0.01.

5.3 44 11 : B. t = 100. 10. A, B,. A, 10,. B, 11,.,, 0.,, B.,, B, y = 20. 5.3,, B, 2 ( C, D)., 100 (Re = 100). 3, 4

5.3 45. 3 :. U = 1 1 S = 1 x = y = 0.1 t = 0.001 0 t 100 Re = 100 ϕ i,j 10 6 100 (x, y) = (10, 10) 4 :. C D 60 20 80 20 601 101 801 201 12, 13. 12 : C. t = 100. 10.

5.4 46 13 : D. t = 100. 10. B D,. B, 11, x = 40, C, D, 12, 13,.,, B, C, D.,,.,, B, x = 40. 5.4, Poisson, E., 100,., 100(Re = 100)., Gauss-Saidel τ τ + 1, ( ϕ) max = ϕ τ+1 i,j ϕ τ i,j 0 t 100. E 5.

5.4 47 5 : E. U = 1 1 S = 1 x = y = 0.1 t = 0.001 0 t 100 Re = 100 100 (x, y) = (10, 10) 40 20 401 201 τ = 99 14. 14 : E... 100000 t = 100. 10 5, 2 10 5. E, 14,

5.4 48,, 2 10 5. ϕ = Uy, U = 1, 10 y 10 10 10.,. (2012) Poisson, SOR, 10 6.,, (2012),, 100, 10 5.

6.1 49 6 6.1 Kármán (1911),, h l h l = 0.28., Kármán.,. 6.1.1 6 : 6 :. U = 1 1 S = 1 x = y = 0.1 t = 0.001 0 t 100 ϕ i,j 10 5 100 (x, y) = (10, 10) 40 20 401 201

6.1 50 6.1.2 (Re = 80, 90, 100) h l 15 17., A, B, C..., (, ), h *1 l,, h/l 7 9. 15 : Re = 80, t = 100.,., 0.01. A F. A B ( ) a, B C b., A C l 1, B l 1 h 1. *1 h 1 h., A, B, C,. l 1 a, b.,, A, B, C S 1. S 1 l 1 h 1.

6.1 51 7 : Re = 80 A F, a e., h, l. (h/l) A (19.3, 11.0) a = 4.12 h 1 = 2.22 l 1 = 7.31 h 1 /l 1 = 0.30 B (22.8, 9.0) b = 4.44 h 2 = 2.50 l 2 = 7.51 h 2 /l 2 = 0.33 C (26.6, 11.3) c = 4.58 h 3 = 2.84 l 3 = 7.61 h 3 /l 3 = 0.37 D (30.3, 8.6) d = 4.92 h 4 = 3.10 l 4 = 7.90 h 4 /l 4 = 0.39 E (34.2, 11.6) e = 5.12 F (38.2, 8.4) 16 : Re = 90. 15.

6.1 52 8 : Re = 90 A G, a f., h, l. (h/l) A (17.3, 9.4) a = 3.58 h 1 = 1.79 l 1 = 6.71 h 1 /l 1 = 0.27 B (20.5, 11.0) b = 4.03 h 2 = 2.14 l 2 = 7.11 h 2 /l 2 = 0.30 C (24.0, 9.0) c = 4.27 h 3 = 2.44 l 3 = 7.31 h 3 /l 3 = 0.33 D (27.6, 11.3) d = 4.52 h 4 = 2.74 l 4 = 7.41 h 4 /l 4 = 0.37 E (31.3, 8.7) e = 4.70 h 5 = 3.13 l 5 = 7.72 h 5 /l 5 = 0.41 F (35.0, 11.6) f = 5.25 G (39.0, 8.2) 17 : Re = 100. 15.

6.2 53 9 : Re = 100 A G, a f., h, l. (h/l) A (18.6, 9.4) a = 3.67 h 1 = 1.74 l 1 = 6.71 h 1 /l 1 = 0.26 B (21.9, 11.0) b = 3.89 h 2 = 1.95 l 2 = 6.70 h 2 /l 2 = 0.29 C (25.3, 9.1) c = 3.86 h 3 = 2.14 l 3 = 6.91 h 3 /l 3 = 0.31 D (28.6, 11.1) d = 4.27 h 4 = 2.40 l 4 = 6.90 h 4 /l 4 = 0.35 E (32.2, 8.8) e = 4.14 h 5 = 3.28 l 5 = 6.52 h 5 /l 5 = 0.50 F (35.5, 11.3) f = 5.16 G (39.7, 8.3) 7 9, Re = 80, 90, 100 h l, h/l Kármán 0.28.,, Re = 80, 90, 100 Kármán., (, 1988), h/l. h/l,. 6.2,., 80,., (2010),, Re = 46.2.,, 46 Re 46.5,.,. Re = 46, 46.2, 46.5 t = 100 t = 2000. 6. 18 26.

6.2 54 18 : Re = 46, t = 100.,., 5.0 10 6. 19 : Re = 46, t = 2000. 18.

6.2 55 20 : Re = 46, t = 100 t = 2000. 0.01. 21 : Re = 46.2, t = 100. 18.

6.2 56 22 : Re = 46.2, t = 2000. 18. 23 : Re = 46.2, t = 100 t = 2000. 2.5 10 5.

6.2 57 24 : Re = 46.5, t = 100. 18. 25 : Re = 46.5, t = 2000. 18.

6.2 58 26 : Re = 46.5, t = 100 t = 2000. 1.0 10 2. Re = 46, 46.2 20, 23, y = 10,,,.,,. Re = 46.5 26, y = 10,..,,., 46.2 < Re 46.5.

7 59 7 2.,. 2. Arakawa Jacobin., 2 Adams-Bashforth. Re = 100.., Poisson Gauss-Seidel. 0.1 0.1, 40 20, Poisson 100, 10 5. t = 0.01, 2.0 10 1 Re 2.0 10 5 von Neumann. Re = 80, 90, 100. Re = 80, 90., Re = 80, 90, 100 h l Kármán, h/l 0.28,, h/l.., (2010) 46.2. t = 100 t = 2000 46.2 < Re 46.5. 2 2 Adams- Bashforth,.,.

A.1 Lagrange Euler 60 A 1 Lagrange Euler 2, Lagrange Euler.., ( ),. A = A(x, y, z, t) δa,, δa = A t.,, (A.1),. δt + A x A A δx + δy + y z δz DA Dt = lim δa δt 0 δt DA Dt = A t + u A x + v A y + w A z u = lim δt, v = lim δt 0, u x, y, z. δt 0 δx (A.3), δy δt, w = lim δz δt 0 δt DA Dt = A + (u )A t (A.5). A,,, (A.1) (A.2) (A.3) (A.4) D Dt = t + u (A.6)., D/Dt Lagrange, / t Euler. Lagrange, Euler.

B.1 Euler 61 B 1 Euler, dx = f(x, t) dt (B.1). 3.1, t = t k, (B.1), x k+1 x k t f(x k, t k ) (B.2), (B.2), x k+1 = x k + f(x k, t k ) t (B.3). (B.1) (B.3) Euler. x 0, t 0 f(x 0, t 0 ), (B.3) t 1 x 1., x 1, t 1 (B.3) t 2 x 2.,, x. 2 Adams-Bashforth Euler 1.. 2, Adams-Bashforth.

B.3 Arakawa Jacobian 62 x k+1. t 3, x k+1 x k + dx k dt t + 1 d 2 x k 2! dt 2 t2 = x k + f(x k, t k ) t + 1 df(x k, t k ) t 2 2! dt x k + f(x k, t k ) t + 1 f(x k, t k ) f(x k 1, t k 1 ) t 2 ( 2 t 3 = x k + 2 f(x k, t k ) 1 ) 2 f(x k 1, t k 1 ) t (B.4), f. (B.4) 2 Adams-Bashforth., f(x k, t k ), f(x k 1, t k 1 ). t 0 Euler t 1. 3 Arakawa Jacobian Arakawa Jacobian.,. (2.2.16) : ζ t ϕ ζ y x + ϕ ζ x y = 1 Re 2 ζ. (2.2.16) (2.2.16) 2 3,., (2.2.16), ζ t ϕ ζ y x + ϕ ζ x y = 0 (B.5). (B.5), ( 2 ).. B.1 S.

B.3 Arakawa Jacobian 63 B.1 : x y S., x = 0, I., y = 0, J ϕ = ζ = 0., (B.5). E. 2, E = = = (u 2 + v 2 )ds = (u 2 + v 2 )dxdy [ ( ϕ ) 2 ( ) ] 2 ϕ + dxdy y x [ ϕ ϕ ] y=j dx ϕ 2 ϕ y y=0 y dxdy 2 [ + ϕ ϕ ] x=i dy ϕ 2 ϕ x x dxdy 2 x=0 (B.6). ϕ = 0 1 0., x = 0, I 3 0., (B.6), E = ϕ 2 ϕ y dxdy ϕ 2 ϕ 2 x dxdy ( ) 2 2 ϕ = ϕ x + 2 ϕ dxdy 2 y 2 = ϕζdxdy (B.7)

B.3 Arakawa Jacobian 64. de/dt = 0,., (B.7), de = d ϕζdxdy = 0 (B.8) dt dt. (B.5) ϕ, S, ϕ ζ ( ϕ t dxdy + ζ ϕ x y ϕ ) ζ dxdy = 0 y x (B.9). (B.9) 1, ϕ ζ ( ϕζ t dxdy = t ζ ϕ ) dxdy t = d ϕ ϕζdxdy dt t = d ϕζdxdy dt [ ] x=i ϕ ϕ dy + t x x=0 [ ] y=j ϕ ϕ dx + t y y=0 ( ) 2 ϕ x + 2 ϕ dxdy 2 y 2 x y ( ) ϕ ϕ t x dxdy ( ϕ t ) ϕ dxdy (B.10) y., x = 0, I, (B.10) 2 0., y = 0, J ϕ = 0, ϕ/ t = 0., (B.10) 4 0. (B.10), ϕ ζ t dxdy = d ϕζdxdy dt ( ) ( ) ϕ ϕ ϕ ϕ + x t x dxdy + y t y dxdy = d [ ( ) ϕ ϕ ϕζdxdy + dt x t x + ( ϕ y t = d ϕζdxdy + 1 ( ) 2 ϕ dt 2 t x + 2 ϕ dxdy 2 y 2 = d ϕζdxdy + 1 d (u 2 + v 2) dxdy dt 2 dt = de dt + 1 de 2 dt = 1 de 2 dt ) ϕ y ] dxdy (B.11)

B.3 Arakawa Jacobian 65., (B.9) 2, ( ϕ ζ ϕ x y ϕ ) ζ dxdy y x [ ( ) 1 ζ = x 2 ϕ2 y ( ) ] 1 ζ y 2 ϕ2 dxdy x [ = ζ ( )] [ 1 y x 2 ϕ2 dxdy ζ ( )] 1 x y 2 ϕ2 dxdy [ = ζ ( )] y=j [ 1 x 2 ϕ2 dx ζ ( )] x=i 1 y 2 ϕ2 dy y=0 x=0 (B.12)., (B.10), (B.12) 1 2 0., ( ϕ ζ ϕ x y ϕ ) ζ dxdy = 0 (B.13) y x. (B.11) (B.13), (B.9), 1 de 2 dt + 0 = 0, de dt = 0 (B.14), (B.8).,., (B.5). Q. 2, ( ) ( ) 1 1 Q = 2 ζ2 ds = 2 ζ2 dxdy (B.15). dq/dt = 0,., (A.15), dq dt = d ( ) 1 dt 2 ζ2 dxdy = 0 (B.16). (B.5) ζ, S, ζ ζ ( ϕ t dxdy + ζ ζ x y ϕ ) ζ dxdy = 0 y x. (B.17) 1, ζ ζ ( ) 1 t dxdy = t 2 ζ2 dxdy = d ( ) 1 dt 2 ζ2 dxdy (B.17) (B.18)

B.3 Arakawa Jacobian 66., (B.17) 2, ( ϕ ζ ζ x y ϕ ) ζ dxdy y x [ ( ) ϕ 1 = x y 2 ζ2 ϕ ( )] 1 y x 2 ζ2 dxdy [ ( ) 1 ϕ = ζ2 ( )] 1 ϕ ζ2 dxdy y 2 x x 2 y [ ] y=j [ ] x=i 1 ϕ 1 ϕ = ζ2 dx ζ2 dy 2 x 2 y y=0 x=0 (B.19).,, (B.19) 1 2 0, ( ϕ ζ ζ x y ϕ ) ζ dxdy = 0 (B.20) y x. (B.18) (B.20), (B.17), ( ) d 1 dt 2 ζ2 dxdy = dq dt = 0 (B.21), (B.16).,.,,.,, Arakawa Jacobain., (2.2.16) ϕ ζ x y ϕ y., (B.22), ζ x = 1 ( ϕ ζ 3 x y ϕ ) ζ y x + 1 [ ( ϕ ζ ) ( ϕ ζ )] 3 x y y x + 1 [ ( ζ ϕ ) ( ζ )] 3 y x x x Ji,j L = ϕ ζ x y ϕ y ( ) J C i,j = x J R i,j = y ζ x, ϕ ζ ( ϕ ζ ), y y x ) ( ζ ) x x ( ζ ϕ x (B.22)

B.3 Arakawa Jacobian 67,, J L i,j, J C i,j, J R i,j, J L i,j = J C i,j = J R i,j = 1 4 x y [(ϕ i+1,j ϕ i 1,j )(ζ i,j+1 ζ i,j 1 ) (ϕ i,j+1 ϕ i,j 1 )(ζ i+1,j ζ i 1,j )], 1 4 x y [ϕ i+1,j(ζ i+1,j+1 ζ i+1,j 1 ) ϕ i 1,j (ζ i 1,j+1 ζ i 1,j 1 ) ϕ i,j+1 (ζ i+1,j+1 ζ i 1,j+1 ) + ϕ i,j 1 (ζ i+1,j 1 ζ i 1,j 1 )], 1 4 x y [ζ i,j+1(ϕ i+1,j+1 ϕ i 1,j+1 ) ζ i,j 1 (ϕ i+1,j 1 ϕ i 1,j 1 ) ζ i+1,j (ϕ i+1,j+1 ϕ i+1,j 1 ) + ζ i 1,j (ϕ i 1,j+1 ϕ i 1,j 1 )]., (B.22), ϕ ζ x y ϕ y ζ x 1 3 ( J L i,j + J i,j C + J ) i,j R (B.23). Arakawa Jacobian..,. (B.9) 2, ϕ ( ϕ ζ x y ϕ y ζ x ) dxdy i,j 1 ( ϕ i,j J L 3 i,j + J i,j C + J ) i,j R. ϕ i,jj L i,j, ϕ i,jj C i,j, ϕ i,jj R i,j 4 x y,, (4 x y)ϕ i,jj L i,j = ϕ i,j (ϕ i+1,j ϕ i 1,j )(ζ i,j+1 ζ i,j 1 ) ϕ i,j (ϕ i,j+1 ϕ i,j 1 )(ζ i+1,j ζ i 1,j ) = ϕ i,j ϕ i+1,j ζ i,j+1 ϕ i,j ϕ i+1,j ζ i,j 1 ϕ i,j ϕ i 1,j ζ i,j+1 + ϕ i,j ϕ i 1,j ζ i,j 1 ϕ i,j ϕ i,j+1 ζ i+1,j + ϕ i,j ϕ i,j+1 ζ i 1,j (B.24) +ϕ i,j ϕ i,j 1 ζ i+1,j ϕ i,j ϕ i,j 1 ζ i 1,j, (B.25) (4 x y)ϕ i,jj C i,j = ϕ i,j ϕ i+1,j (ζ i+1,j+1 ζ i+1,j 1 ) ϕ i,j ϕ i 1,j (ζ i 1,j+1 ζ i 1,j 1 ) ϕ i,j ϕ i,j+1 (ζ i+1,j+1 ζ i 1,j+1 ) + ϕ i,j ϕ i,j 1 (ζ i+1,j 1 ζ i 1,j 1 ) = ϕ i,j ϕ i+1,j ζ i+1,j+1 ϕ i,j ϕ i+1,j ζ i+1,j 1 ϕ i,j ϕ i 1,j ζ i 1,j+1 + ϕ i,j ϕ i 1,j ζ i 1,j 1 ϕ i,j ϕ i,j+1 ζ i+1,j+1 + ϕ i,j ϕ i,j+1 ζ i 1,j+1 +ϕ i,j ϕ i,j 1 ζ i+1,j 1 ϕ i,j ϕ i,j 1 ζ i 1,j 1, (B.26)

B.3 Arakawa Jacobian 68 (4 x y)ϕ i,jj R i,j = ϕ i,j ζ i,j+1 (ϕ i+1,j+1 ϕ i 1,j+1 ) ϕ i,j ζ i,j 1 (ϕ i+1,j 1 ϕ i 1,j 1 ) ϕ i,j ζ i+1,j (ϕ i+1,j+1 ϕ i+1,j 1 ) + ϕ i,j ζ i 1,j (ϕ i 1,j+1 ϕ i 1,j 1 ) = ϕ i,j ζ i,j+1 ϕ i+1,j+1 ϕ i,j ζ i,j+1 ϕ i 1,j+1 ϕ i,j ζ i,j 1 ϕ i+1,j 1 + ϕ i,j ζ i,j 1 ϕ i 1,j 1 ϕ i,j ζ i+1,j ϕ i+1,j+1 + ϕ i,j ζ i+1,j ϕ i+1,j 1 +ϕ i,j ζ i 1,j ϕ i 1,j+1 ϕ i,j ζ i 1,j ϕ i 1,j 1 (B.27)., B.2 9 T.,, T,. B.2 : 9 T. T (i, j) = (5, 5), (B.25), ϕ 5,5 J 5,5 L = ϕ 5,5 ϕ 6,5 ζ 5,6 ϕ 5,5 ϕ 6,5 ζ 5,4 ϕ 5,5 ϕ 4,5 ζ 5,6 + ϕ 5,5 ϕ 4,5 ζ 5,4 ϕ 5,5 ϕ 5,6 ζ 6,5 + ϕ 5,5 ϕ 5,6 ζ 4,5 + ϕ 5,5 ϕ 5,4 ζ 6,5 ϕ 5,5 ϕ 5,4 ζ 4,5 (B.28)

B.3 Arakawa Jacobian 69. (B.26), (6, 5), ϕ 6,5 J 6,5 C = ϕ 6,5 ϕ 4,5 ζ 4,6 ϕ 6,5 ϕ 4,5 ζ 4,4 ϕ 6,5 ϕ 5,5 ζ 5,6 + ϕ 6,5 ϕ 5,5 ζ 5,4 ϕ 6,5 ϕ 6,6 ζ 4,6 + ϕ 6,5 ϕ 6,6 ζ 5,6 + ϕ 6,5 ϕ 6,4 ζ 4,4 ϕ 6,5 ϕ 6,4 ζ 5,4, (B.29) (4, 5), ϕ 4,5 J 4,5 C = ϕ 4,5 ϕ 5,5 ζ 5,6 ϕ 4,5 ϕ 5,5 ζ 5,4 ϕ 4,5 ϕ 6,5 ζ 6,6 + ϕ 5,5 ϕ 6,5 ζ 6,4 ϕ 4,5 ϕ 4,6 ζ 5,6 + ϕ 4,5 ϕ 4,6 ζ 6,6 + ϕ 4,5 ϕ 4,4 ζ 5,4 ϕ 4,5 ϕ 4,4 ζ 6,4, (B.30) (5, 6), ϕ 5,6 (5, 4), ϕ 5,4 J 5,6 C = ϕ 5,6 ϕ 6,6 ζ 6,5 + ϕ 5,6 ϕ 4,6 ζ 4,5 + ϕ 5,6 ϕ 5,5 ζ 6,5 ϕ 5,6 ϕ 5,5 ζ 4,5, (B.31) J 5,4 C = ϕ 5,4 ϕ 6,4 ζ 6,5 ϕ 5,4 ϕ 4,4 ζ 4,5 ϕ 5,4 ϕ 5,5 ζ 6,5 + ϕ 5,4 ϕ 5,5 ζ 4,5 (B.32). (B.28) 1, 2 (B.29) 3, 4, (B.28) 3, 4 (B.30) 1, 2, (B.28) 5, 6 (B.31) 3, 4, (B.28) 7, 8 (B.32) 3, 4.,., T (B.25) (B.26) 0. (B.27), (5, 5), ϕ 5,5 J 5,5 R = ϕ 5,5 ζ 5,6 ϕ 6,6 ϕ 5,5 ζ 5,6 ϕ 4,6 ϕ 5,5 ζ 5,4 ϕ 6,4 + ϕ 5,5 ζ 5,4 ϕ 4,4 ϕ 5,5 ζ 6,5 ϕ 6,6 + ϕ 5,5 ζ 6,5 ϕ 6,4 + ϕ 5,5 ζ 4,5 ϕ 4,6 ϕ 5,5 ζ 4,5 ϕ 4,4 (B.33). (6, 6), ϕ 6,6 (4, 6), ϕ 4,6 (6, 4), ϕ 6,4 (4, 4), ϕ 4,4 J 6,6 R = ϕ 6,6 ζ 6,5 ϕ 4,5 + ϕ 6,6 ζ 6,5 ϕ 5,5 + ϕ 6,6 ζ 4,6 ϕ 4,5 ϕ 6,6 ζ 5,6 ϕ 5,5, (B.34) J 4,6 R = ϕ 4,6 ζ 4,5 ϕ 5,5 + ϕ 4,6 ζ 4,5 ϕ 6,5 + ϕ 4,6 ζ 5,6 ϕ 5,5 ϕ 4,6 ζ 6,6 ϕ 6,5, (B.35) J 6,4 R = ϕ 6,4 ζ 6,5 ϕ 4,5 ϕ 6,4 ζ 6,5 ϕ 5,5 ϕ 6,4 ζ 4,4 ϕ 4,5 + ϕ 6,4 ζ 5,4 ϕ 5,5, (B.36) J 4,4 R = ϕ 4,4 ζ 4,5 ϕ 5,5 ϕ 4,4 ζ 4,5 ϕ 6,5 ϕ 4,4 ζ 5,4 ϕ 5,5 + ϕ 4,4 ζ 6,4 ϕ 6,5 (B.37)

B.4 Jacobi Gauss-Seidel 70. (B.33) 1, 5 (B.34) 2, 4, (B.33) 2, 7 (B.35) 1, 3, (B.33) 3, 6 (B.36) 2, 4, (B.33) 4, 8 (B.37) 1, 3.,., T (B.27) 0., (B.24), ( ϕ ζ ϕ x y ϕ y ζ x ) dxdy = 0 (B.38), (B.23)., J L i,j J i,j C, J R i,j.,. (B.17) 2, ( ϕ ζ ζ x y ϕ ) ζ dxdy 1 ( ζ i,j J L y x 3 i,j + J i,j C + ) Ji,j R (B.39) i,j.,, (B.39), ( ϕ ζ ζ x y ϕ ) ζ dxdy = 0 (B.40) y x., (B.23).,, J L i,j J i,j R., J C i,j. 4 Jacobi Gauss-Seidel Poisson,., Jacobi Gauss-Seidel., (2.3.19), ϕ i+1,j 2ϕ i,j + ϕ i 1,j ( x) 2 + ϕ i,j+1 2ϕ i,j + ϕ i,j 1 ( y) 2 = ζ i,j (B.41). ϕ i,j, ϕ i,j = ( x)2 (ϕ i,j+1 + ϕ i,j 1 ) + ( y) 2 (ϕ i+1,j + ϕ i 1,j ) ( x) 2 ( y) 2 ζ i,j 2 [( x) 2 + ( y) 2 ] (B.42)

B.4 Jacobi Gauss-Seidel 71 (B.42) ϕ i,j, ϕ i+1,j, ϕ i 1,j, ϕ i,j+1, ϕ i,j 1 ζ i,j. (B.42) f(ϕ, ζ), Jacobi,. ϕ τ i,j. ϕ τ i,j, ϕ τ+1 i,j = f(ϕ τ i 1,j, ϕ τ i+1,j, ϕ τ i,j 1, ϕ τ i,j+1, ζ i,j ) (B.43) ϕτ+1 i,j ϕτ+1 i,j, τ, τ + 1 ϕ i,j = ϕ τ+1 i,j ϕ τ i,j (B.44).,,., Gauss-Seidel, ϕ i,j., ϕ τ+1 i,j = f(ϕ τ+1 i 1,j, ϕτ i+1,j, ϕ τ+1 i,j 1, ϕτ i,j+1, ζ i,j ) (B.45). Gauss-Seidel, ϕ τ+1 i 1,j ϕτ+1 i.j 1, Jacobi.,, Gauss-Seidel.

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C.2 80 C.6 : 2015 1 9 10 ( ). (, 2016), C.4,,,.,,.,. C.5,,. C.6,,.,,.,,,,. 2, (ASAS), 60, 100 180. 3, 9, 15, 21,, 1 4., 2015 1 9 9 ( ),.

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C.4 84., C.8, 850 hpa, 850 hpa.,., 10 /km.., 5 /km..,,,,. C.8, 850 hpa, 850 hpa.. g/kg, 1 kg.,,. 4 (2003),,., 2015 1 9 9 ( ) ( C.7),,.,,,. (.), ( C.8), 850 hpa (1507 m)., ( ), 1950 m, 400 m.,.

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86 Hirota. I, and K. Miyakoda, 1964: Numerical Solution of Kármán Vortex Street behind a Circular Cylinder. J. Meteor. Soc. Japan, Vol. 43, 30-41,, 2015:., 2015,, 311 Kármán, T. von, 1911, 1912: Uber den Mechanismus des Widerstandes, den ein bewegter Korper in einer Flussigkeit erfahrt. Gottingen Nachrichten, Math. Phys. Kl. 12, 509-17, 13, 547-56, 2003:., http://www.jma-net.go.jp/sat/data/web/jirei/sat200303.pdf, 2015, 2016:., http://www.jma.go.jp/jp/gms/, 2016:., http://weather.is.kochi-u.ac.jp/,, 2010:. JAXA, : 43 44 ), 27-30, 2016:., http://www.yoho.jp/chart/, 2012: OpenGL + GLSL.,,,, 2012:.,, 1776, 28-41, 1988:., 98-99

87 University of Wyoming, College of Engineering, Department of Atmospheric Science, 2016: Upper Air Soundings. University of Wyoming, http://weather.uwyo.edu/upperair/sounding.html