流体力学 資料 - GFD-DENNOU

1 , i , .. 2014 09 18 .. 1 1. 2 2. 3 3. 4 8. 5 10. A 12. /work/ 2014/09/18( ). , 1 1. , .. 1 .. d . = + v (1). dt t , .. (Lagrange form) . , . , , ( ) . d . + v = 0, (2). dt dv + 2 v = p + + F , (3). dt ds Q. = (4). dt T. , v , , p , ( 12 | x|2 ). , ( ), . , , .. ( )i = j j ij ), F , s , T , Q . ( ) . s, , T , p . ( ) ( . ). A . dA. = 0 (5). dt , A , , ( . ), , , , , .. s .. , a . a + F a = Qa (6). t , (flux form) .. F a a , a . ( ), , a . F a . a . Qa a , .. /work/ ( ) 2014/09/18( ). , 2 2. , , .. A . (2) , .. dA A. = + ( vA) (7). dt t , , a = A .. , . a F a = va . , a .. + ( v) = 0, (8). t vi . + ( vi vj + p ij ij ) + 2 ijk j vk = + Fi . t j x j x i jk (9). s Q. + ( vs) = (10).

2 T T. , , . , . , , . , . c c . c + ( c v) = Qc (11). t .. d c = Qc (12). dt . c c / . , Qc = 0, . , c . 2 . ( ) .. x . , . z , . /work/ ( ) 2014/09/18( ). , 3 3.. ( ) . r cos . d ( ) p . r cos v + r2 cos2 z = + r cos F . dt . (13).. r cos v + r2 cos2 z , . 1 . , . { ( )}. r cos v + r2 cos2 z t 1 { ( )}. + v r cos v + r2 cos2 z r cos . 1 { ( )}. + cos v r cos v + r2 cos2 z r cos . 1 { 2 ( )}. + 2 r vr r cos v + r2 cos2 z r r p . = + r cos F (14).. , , , , . F = 0 , .. , .. , . 3 . , , .. , ( vorticity ), (circulation), ( , potential vorticity ) .. ( ) . v (15). 1 . , .. 1( 2 ). K = r ( + z )2 cos2 + r2 2 + r 2. 2. , . d K K d ( ). = r cos v + r2 cos2 z dt dt . , v = r cos.

3 /work/ ( ) 2014/09/18( ). , 3 4.. (relative vorticity ) . , . v + x . a v + 2 (16). , (absolute vorticity ) . , . (3) . (3) . v v = 12 v 2 v v , . a 1. + (v ) a + a v ( a )v. = 2 p + F . t . (17). 2 . (v a ) , . a v ( a )v. (18).. , , . a z x, y, z , . u [ a v ( a )v]x = a , z v [ a v ( a )v]y = a , ( z ). u v [ a v ( a )v]z = a + . x y , (17) . ( ). d a u d a v d a u v a , a , + a +. dt z dt z dt x y . a = ( a , a , a ) v = (u, v, w) . x, y . a = a = 0 a zu , a zv . (tilting term) , (x, y) ( z . ). u v , a , a . z a + . x y 2. (A B) = (A )B B A + (B )A + A B. 1. v 2 = 0. 2. v = 0.. (v v) = (v v). = (v ) v + v v ( v )v. /work/ ( ) 2014/09/18( ). , 3 5.. ( ( ) .. ). u v ( + < 0).

4 , , . x y ( ). u v w ( + ) . x y z d a w a dt z w (streching - shrinking term) . (streching), . z w , > 0 . z , (17) . 1. p 2. (baroclinic term) . ( p) ( ) .. ( ).. ( ) .. , (17) . , . c : I . v dr = dS (19). c A. c A .. 2 . : . a ( + 2 ) dS (20). A.. [ ]. d a a dS = + (v ) a + a v ( a )v dS. dt A A t 3 . a (17) . I. d a p = dS + F dr (21). dt A 2 c 3 , .. a . t = t A t = t + t A . A A . D, A . [ ]. d 1. a dS = lim a (t + t) dS a (t) dS. dt A. t 0 t A A. /work/ ( ) 2014/09/18( ). , 3 6.. (21) , . 0, , p ( , barotropic, . p ), , A p . 0, , ( ) . , a . Kelvin .. , , , p , p . a .. d .. , = 0 . , . dt s . = 0 . = .. ( ). = .. , .. m a m a dS a S = a =. A | | | | l . [ ( ) ].

5 1. = lim ( a (t + t) a (t)) dS + + a (t) dS a (t) dS. t 0 t A . [ A . A A . ] A . 1 a = lim t dS + a dV a d(l v t). t 0 t t [ A.. D c ]. 1 a = lim t dS + a dS v t + a v t dl t 0 t t A. A. c a = dS + a v dS + ( a v) dS. t [. A A A. ]. a = + (v ) a + a v ( a )v dS. A. t B . B. + (v )B + B v (B )v = 0. t B (freeze) .. d B dS = 0. dt A.. B dS . A. /work/ ( ) 2014/09/18( ). , 3 7.. , S , l + , m . , + , l , . d m d .. m = 0, = 0 . dt dt , (21) . ( ). d a m p m m = + F . dt 2 . m/ , m/ . d a p . = + F . (22). dt 3 .. a . (23).. (22) , . ( , potential vorticity equation) .. 0, , p ( , barotropic, p . ), , p, , , , 0, , ( ) , , (22) 0 , 4 . q .. q .. , , . , .. 2 . , , . q.

6 Q . 2 . , . 4 , , .. , d . dt = (17) (2) . , d a 1 p 1. = a v + + F . dt 3 . a , .. d a a p . = + + F . (24). dt 3 .. = 0 (22) . /work/ ( ) 2014/09/18( ). , 4 8. , .. , , q 5 , .. 4 . , , . , . , .. ( ). d 1 2 . v + = ( p ij vi + ij vi ) vi + Fi vi + Q . dt 2 xj xi (25).. , Q . , .. 2 . , . F ( ) .. ( ). d 1 2 . v + + = ( p ij vi + ij vi ) + + Q . (26). dt 2 xj t 1 2. 2v + + .. / t = 0 .. { ( )} { ( ) }. 1 2 1 2. v + + + vj v + h + ij vi = Q . t 2 xj 2. (27). p . h = + .. 5 , q f (q) , .. N 6N . N . ( i x ) , . 6 . , , , , . 1 + 3 + 1 = 5 1 . 1 , .. ( ) .. , . , . ( . ), . , . , , .. 1 .. , .. , .. (casimir) . /work/ ( ) 2014/09/18( ). , 4 9. (3) v . ( ). d 1 2 p ij v + = vi + vi (28).

7 Dt 2 xi xj .. { ( )} { ( ) }. 1 2 1 2 p v + + vj v + + ij vi t 2 xj 2 . vi vi =p ij . (29). xi xj vi vi 1 d . p ( ) ( = . xi xi dt vi ), ij . xj . , , .. d vi vi = p + ij + Q . (30). dt xi xj vi ( ) Q = ij + Q . xj vi . ij , .. xj ( ). vi vi vj 2 vl vi vi vl ij = + ij + ij xj xj xi 3 xl xj xj xl ( )2. 1 vi vj 2 vl = + ij + ( v)2 . 2 xj xi 3 xl vi vi ij . ij . xj xj , . d ds p d . (4) =T + , . dt dt 2 dt s : ds Q. = . (31). dt T. ( , , . ) . , (27) h = + p/ .. ( ). d 1 2 p . v +h+ = + ( ij vi ) + Q . (32). dt 2 t xj /work/ ( ) 2014/09/18( ). , 5 10. 6 . , . (32) , , , . : d . (h + ) = ( ij vi ) + Q . (33). dt xj |v| .. h + . h + . (static energy) . , cp T + gz , .. h = cp T , gz.

8 T z . 5 .. , , ( . ) .. a . a + F a = Qa t , .. Fa n (34).. n . ( ) . , . S . 6 , / t = 0, ij = 0, Q = 0 . ( ). 1 2. v v +h+ = 0. 2. 1 2. v + h + . (Bernoulli) .. 2.. = (p) . ds dh 1 dp dh d 0=T = = P (p). dt dt (p) dt dt dt . 1.. P (p) dp. h . (p). 1 2. v + P (p) + = const 2.. /work/ ( ) 2014/09/18( ). , 5 11. , D (34) .. 0 .. a . dV = adV av dS [a2 a1 ]v S dS. t t D. D D S. F a dV [F a2 F a1 ] dS. D S. Qa dV 0. D.. [a2 a1 ]S v S n + [F a2 F a1 ]S n (35). S , v S , , 1, 2 .. , , . (i , i = 1, 2, 3) 5 .. vj nj (36). ( vi vj + p ij ij )nj (37). { ( ) }. 1 2. vj v + h + ij vi nj (38). 2. nj n .. f (x, t) = 0 .. f (x, t) + v f (x, t) = 0 (39). t . t = 0 f (x, 0) = 0.

9 , , . 1 .. , . 1 . , , . 1 v1 = 2 v2 (40). 1 v12+ p1 = 2 v22. + p2 (41). ( ) ( ). 1 2 1 2. 1 v1 v1 + h1 = 2 v2 v2 + h2 (42). 2 2. /work/ ( ) 2014/09/18( ). , A 12.. , . h p, ( h = p /( 1), ) , . 3 ( 1 , 2 , p1 ) ( v1 , v2 , p2 ).. , . A . , , .. , .. , , , . 5 . d . + v = 0. dt dv 1. = p + F. dt . d . = Q p v dt . F , Q . , p . 5 , v, , p 5 , . , , ( ) . 2 , 2 .. , p = p0 cp = cp (p0 , T ) . = (p, T ) , ( ). s cp = (43). T p T. ( ) ( 1 ). s . = (44). p T T p , ((p, T ) ) . T p ( 1 ). cp (p0 , T ) (p , T ). s(p, T ) = s(p0 , T0 ) + dT dp (45). T0 T p0 T p , ((p, T ) ) . T p g(p, T ) = g(p0 , T0 ) s(p0 , T )dT + 1 (p , T )dp (46). T0 p0.. = g + sT p 1 . , . , . /work/ ( ) 2014/09/18( ).

10 , A 13.. , , . , .. d ds d 1. = T p dt dt dt ds 1. = T p v (47). dt .. ds T = Q (48). dt , .. ( 1 ). ds dT (p, T ) dp T = cp (49). dt dt T p dt (p, T ) .. (p, ) . , , , .. ( ) . /work/ ( ) 2014/09/18( ).