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17 18 3 : 3604U079-

Abstract :

1 3 1.1....................................... 4 1................................... 4 1.3.................................. 4 5.1..................................... 6................................. 6.3........................... 6 3 10 3.1..................................... 11 3................................... 11 3.3..................................... 11 3.4........................... 1 4 13 4.1..................................... 14 4...................................... 14 4.3................................. 15 4.3.1............................... 15 4.3................................ 15 4.3.3................................. 16 5 17 5.1........................... 18 5........................................ 18 6 3 6.1................................... 4 6.................................... 5 6..1.................... 5 1

6...................... 6 6.3................................ 7 6.4...................................... 8 6.5................................... 8 30 31

1 3

1.1 1. 1.3 3 4 5 6 4

5

.1. u t + u u x = 1 Re u x (.1) t x u Re 1.3 (.1) x x 1 U = x/ t x1 x 1 x x 1 = x Ut x 1 u t = u x1 t + u x x x t x1 u t = U u x x 6

x 1 / x = 1 U u x 1 x 1 x + u u x 1 x 1 x = 1 Re x ( 1 x x1 x 1 x u ), x 1 U du + u du = 1 dx 1 dx 1 Re d u dx 1 d dx 1 [ Uu + 1 u 1 Re du ] = 0 dx 1 x 1 C Uu + 1 u 1 Re du dx 1 = C (.) u = u(x 1 ) lim x 1 lim x 1 (.) u( ) = U w, du dx 1 = 0, x1 u( ) = U e, du dx 1 = 0 x1 UU w + 1 U w = C, (.3) (.3) (.4) (U w U e ) UU e + 1 U e = C (.4) U(U w U e ) + 1 (U w U e ) = 0 U = U w + U e 7

(.3) (.) Re 1 U e U w C = U w + U e U w + 1 U w C = U wu e U w + U e u + 1 u 1 Re du = U wu e, dx 1 u (U w + U e )u + U e U w = Re du, dx 1 du dx 1 u U e + Re(U w U e ) (u U w )(u U e ) = Re du, dx 1 du Re dx 1 (u U w )(u U e ) = 0, 1 U w U e du dx 1 u U w = 1, du du dx 1 dx + 1 u U e U w u = 1 U w u U e x 1 Re(U w U e ) ( log(u U e) + log(u w u)) = x 1 + C 1 Re(U w U e ) log U w u u U e = x 1 + C 1 C 1 (.1) 8

log U w u = Re(U w U e ) (x Ut + C 1 ), u U e U w u = (u U e )e Re(U w Ue) (x Ut+C 1 ), (1 + e Re(U w Ue) (x Ut+C 1 ) )u = U w + U e e Re(U w Ue) (x Ut+C 1 ), u = U e + U w U e 1 + e Re(Uw Ue) (x Ut+C 1 ) C 1 C 1 = 0 C 1 t = 0 x = 0 (U w + U e )/ t = 0 x = 0 u(t, x) = U e + U w U e 1 + e Re(U w Ue) (x U w+ue t) (.5) 9

3 10

3.1 3. 3.3 f(x) x 1 u(x)/ x x u(x + x) x u(x + x) = u(x) + x 1! u ( x) (x) + x! = u(x) + x u x + O(( x) ) u ( x)3 (x) + 3 u x 3! x (x) + 3 u u(x + x) u(x) (x) = + O( x) x x u(x)/ x x 0 u(x x) x u(x x) = u(x) x 1! u ( x) (x) + x! = u(x) x u x + O(( x) ) u ( x)3 (x) 3 u x 3! x (x) + 3 u u(x) u(x x) (x) = + O( x) x x u(x)/ x u u(x + x) u(x x) (x) = + O(( x) ) x x 11

3.4 5 5.5 x 1

4 13

4.1 4. x i u i x u u x u i u i 1 u i x u i+1 u i u i x (u i 0 ) (u i < 0 ) u u x u u i+1 u i 1 i u i x ui+1 u i + u i 1 x ( x) u u x u i 3u i 4u i 1 + u i x u i 3u i + 4u i+1 u i+ x (u i 0 ) (u i < 0 ) u u x u u i+ + 4(u i+1 u i 1 ) + u i i 4 x + u i ( x) 3 4 ui+ 4u i+1 + 6u i 4u i 1 + u i 4( x) 4 u u x u i u i+1 + 3u i 6u i 1 + u i 6 x u i u i+ + 6u i 3u i+1 u i+ 6 x (u i 0 ) (u i < 0 ) 14

u u x u u i+ + 8(u i+1 u i 1 ) + u i i 1 x + u i ( x) 3 1 ui+ 4u i+1 + 6u i 4u i 1 + u i ( x) 4 u n f(x) x i Taylor O(( x) n ) 4.3 4.3.1 4.3. x x + x ( x ) u(x + x) = u(x) + x 1! u ( x) (x) + u ( x)3 (x) + 3 u ( x)n (x) + + x! x 3! x3 n! ( x < ξ < x + x) (n) u x n (ξ) u u x = u u i+1 u i 1 i u i x ui+1 u i + u i 1 + O( x) x ( x) O( x) u u x = u u i+ + 4(u i+1 u i 1 ) + u i i 4 x + u i ( x) 3 4 ui+ 4u i+1 + 6u i 4u i 1 + u i 4( x) 4 + O(( x) ) 15

u u x = u u i+ + 8(u i+1 u i 1 ) + u i i 1 x + u i ( x) 3 1 ui+ 4u i+1 + 6u i 4u i 1 + u i ( x) 4 + O(( x) 3 ) u x = u i+1 u i 1 + O(( x) ) x O( x) O(( x) ) O(( x) 3 ) O(( x) ) 4.3.3 u u x u u i+1 u i 1 i u i x ui+1 u i + u i 1 x ( x) 1 1 u/ x x x u u x u u i+ + 4(u i+1 u i 1 ) + u i i 4 x + u i ( x) 3 4 ui+ 4u i+1 + 6u i 4u i 1 + u i 4( x) 4 4 u/ x 4 ( x) 3 x u u x u u i+ + 8(u i+1 u i 1 ) + u i i 1 x + u i ( x) 3 1 ui+ 4u i+1 + 6u i 4u i 1 + u i ( x) 4 x 16

5 17

5.1 1 1 x ( ) x x WEST EAST x U w x U e U w > U e U w > U e WEST U e t = 0 x = 0 U w U e 5. Re U w U e x t t WEST EAST x x x x = 0 x u 1.5 initial exact center upwind 1 0.5 0-0.5 0 0.5 1 1.5.5 3 5.1: Re = 100, U w =, U e = 0, 0.5 x 3, x = 0.01, t = 1, t = 0.001 18

5.1 1.5 initial exact center upwind 1 0.5 0-0.5 0 0.5 1 1.5.5 3 5.: Re = 100, U w =, U e = 0, 0.5 x 3, x = 0.01, t =, t = 0.001 5. 5.1 4.5 4 3.5 3.5 1.5 1 0.5 initial exact center upwind 0-0.5 0 0.5 1 1.5.5 3 5.3: Re = 500, U w =, U e = 0, 0.5 x 3, x = 0.01, t =, t = 0.001 5.3 5. 19

4.5 4 3.5 3.5 1.5 1 0.5 initial exact center upwind 0-1 0 1 3 4 5 6 5.4: Re = 500, U w =, U e = 0, 0.5 x 6, x = 0.01, t = 4, t = 0.001 5.4 5.3 6 5 4 initial exact center upwind 3 1 0-0.5 0 0.5 1 1.5 5.5: Re = 1000, U w =, U e = 0, 0.5 x, x = 0.01, t =, t = 0.001 5.5 5.3 0

.5 initial exact center upwind 1.5 1 0.5 0-0.5 0 0.5 1 1.5.5 3 5.6: Re = 100, U w =, U e = 0, 0.5 x 3, x = 0.0, t =, t = 0.001 5.6 5. 1.5 initial exact center 1 0.5 0-0.5 0 0.5 1 1.5 5.7: Re = 100, U w =, U e = 0, 0.5 x, x = 0.01, t = 1, t = 0.004 5.7 1

.5 initial exact center upwind 1.5 1 0.5 0-0.5 0 0.5 1 1.5 5.8: Re = 150, U w =, U e = 0, 0.5 x, x = 0.01, t = 1, t = 0.004 5.8

6 3

6.1 5.1 Re = 100, U w =, U e = 0, 0.5 x 3, x = 0.01, t = 1, t = 0.001 ( ) (1 ) 5. 5.1 u > 0 5.3 x 5.1 5.4 5. 5.3 5.5 5.6 5. 3 4 4

6. 6..1 u t = u u x + 1 Re u x (6.1) t = n t x i = i x u n i (6.1) u n+1 i u n i t = u un i u n i 1 x + 1 Re un i+1 u n i + u n i 1 ( x) (6.) u n i 0 C D C = u t x, (6.3) D = 1 Re t ( x) (6.4) (6.) (6.3) (6.4) u n+1 i u n i = C(u n i u n i 1) + D(u n i+1 u n i + u n i 1) u n+1 i = ( C D + 1)u n i + (C + D)u n i 1 + Du n i+1 (6.5) ( [3] [6] [11]) : C D+1 C + D D (6.) 6.1 u(x) = k a k u(x + k x) 5

a k 0 0 D 1 C, (6.6) 0 C 1 (6.7) 6..1 D 1 0 1 C 6..1: 1 6.. u n+1 i u n i t = u un i+1 u n i 1 x + 1 Re un i+1 u n i + u n i 1 ( x) (6.8) u n i 0 (6.3) (6.4) u n+1 i u n i = C (un i+1 u n i 1) + D(u n i+1 u n i + u n i 1) u n+1 i = (1 D)u n i + ( ) C + D u n i 1 + ( C ) + D u n i+1 (6.9) 6

: C/ + D C/ + D 1 D (6.8) 6.. C D 1, (6.10) D C D (6.11) D 1 0 1 C 6.3 6..: 5.1 5.4 D C 0 C U w t/ x 6..3 7

D 1 0 1 C C,D 6.3.: C D Re = 100 U w = U e = 0 0.5 x < x = 0.01 t = 1 t = 0.004 5.7 Re = 150 U w = U e = 0 0.5 x < x = 0.01 t = 1 t = 0.004 5.8 Re = 100 U w = U e = 0 x = 0.01 0.5 x < 100 t = 0.001 t 500 ( 50000 ) 6.4 6.5 3 3 8

3 4 9

30

[1] : (1994) [] :, (1995) [3] :, vol.34,no.6(199) [4] :, (1991) [5] :, (1993) [6], :, vol.3,no.10(1990) [7] :, (1994) [8] C.A.J. :, (1993) [9] : (005) http: //www.geocities.jp/hydrodynamism/index.html [10] :, (005) [11] :, (1979) 31

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