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International Conference on Hydrodynamic Instability and Turbulence
Proceedings of ICHIT- 06 26 February ­ 5 March 2006, Moscow, Russia

.. , .. , , , 603950 ryb@sandy.ru , . . , , . . . ( ) , [1]. , , , [1]. , . , . ( « » [2]), [3]. , . x, y, z , ( ). , 0 < z < H ( H - ) U ( z) . ( ). , :

1 Pr

+ - + , = , = -R y t z y y z
(1)
( 2)
3

+ - = , t z y y z
1 Pr u u u - + = u , t z y y z

=
Pr =

g TH 2 2 + 2, R= 0 2 0 0 y z

- ,

0 - ( 0 - 0 , 0 - , 0 - , g - , T -


International Conference on Hydrodynamic Instability and Turbulence
). V = ( u , v , w ) , v = / z , w =- / y , =- rot V

(

)

x

, = (T - T0 ) / T .

= 0, z

z=0 , = 0, y =0 z (3)
z =0

z=1 , [3]. , : = 0, = -b - 1 (4) , b = sV + sT + sR z z =1-0

sV ,T , R , .

: z y . z y : 2 f m f m +1 + f m -1 - 2 f m f m f m +1 - f m -1 = , = . 2 2 z 2h z h , : y " " ( ) " " ( ). (1) ( yi , zm )

Anm , Bnm , K nm - n zm , k1 - , N - y. , , y . - , ( ). . , [3], ( ) . (5) (1) t (3), (4). t ( yi , zm ) m = 1, 2, ..., M .

m = 0, M + 1 z = 0 z = 1 . RK4 . (1) . M = 100 - M = 200 . N = 128 k1 = /8 . , dy = 8 , hy = 1/16 . =0.



i,m

=


n=1

N

An,m (t ) sin(nk1 yi ), i,m =


n =1

N

K

n, m

(t ) sin(nk1 yi ) ,

i ,m = ( z j , t ) +


n =1

N

Bnm (t ) cos(nk1 yi ) ,

(5)


International Conference on Hydrodynamic Instability and Turbulence

, [3]. , ( ), . . , . , . , , , . Pr=16,6, R=60140 . 1.
An
20

, ( t 3 ) n=6. , = -1/ b . . 2 .
1.0

t=0.66

z

0.5 0.0
1 2 3

y

4

5

6

7

8

1.0 0.5 0.0
1 2 3 4 5 6 7

t=1.7

8

1.0 0.5 0.0 1 2 3 4 5 6 7

t=2.

8

1.0 0.5 0.0
1 2 3 4 5 6 7

t=2.8

8

1.0

t=3.

Pr=16.6 R=60140

0.5 0.0
1 2 3 4 5 6 7 8

15

1.0 0.5 0.0
1 2 3 4 5 6 7

t=6.

8

10

n=9
5

.2 .
n=4,5
n=6

n=8

0 0 2 4 6

t

.1

. , (n=9) ( ).

, «» , , . . , 9 , 7, 6 (. 2).


International Conference on Hydrodynamic Instability and Turbulence
3. , . 9 7 t 2 7 6 t 3 .
Lj
1.6

. . 2 5 , , , [1].
()
1.0 0.5 0.0

R=30000, Pr=16.6

t=0.66

j=6->4

j=7->5

1

2

3

4

5

6

7

8

j=1
j=3
1.0 0.5 0.0
1 2 3 4 5 6 7

t=2.

1.2

0.8

j=2
j=5

8

t=4.
1.0 0.5 0.0

0.4

j=4
0 0 2 4

t
1.0 0.5 0.0

()

1

2

3

4

5

6

7

8

.3 j- ( ) .
, D
1.2

R=90000 Pr=16.6

t=0.66

1

2

3

4

5

6

7

8

t=2.
1.0

n=6
n=7

0.5 0.0
1 2 3 4 5 6 7 8

n=9
0.8

\



1.0 0.5 0.0
1 2 3 4 5 6 7

t=4.

8

0.4

D /
0 0 1 2 3

.5 R=30000 () R=90000 ().
t

.4 . . 4

< L >=


j =1

N

Lj / N ,

D =< L2 > - < L > 2 .

( 20 ), ( ) (.4). . 5() 5() ,

. . 6 n=9 () n=4 (), . 7(, ) t=4. 9- , 8- . , , () 9 , () - 8 ( 6 .2).


International Conference on Hydrodynamic Instability and Turbulence
An
30

(a)

20

10

n=9 n=18
0 0 1 2 3

t

An

()
30

12- t 2 , t 3 6- (. 6 ), .. (. 7 ). , () . = 0 z =0 . ( ). . Pr=16,6, R=60140 3- (.8) 6- ( .1), (.9) ( .2). , .
An
20

20

10

n=8

n=4
0 0

/

n=16
1 2 3

t

An
20

( )

15

10

n=12
5

n=6
n=7
/n=5
3

0 0 1 2

t

.6 .
1.0 0.5 0.0

n=9

n=4

n=3

15

(a)

t=4.
10

()
1.0 0.5 0.0 1.0 0.5 0.0 1.0 0.5 0.0

1

2

3

4

5

6

7

8

t=4.

5

n=6

n=7

n=5
0

()

1

2

3

4

5

6

7

8

t=2.

0

4

8

12

16

t

.8
8

1

2

3

4

5

6

7

t=4.

= 0 z =0 , R=60140. R=120000 (.10). , , (. . 11 . 6).

1

2

3

4

5

6

7

8

.7 : n=9 (a), n=4 (), n=12 ().


International Conference on Hydrodynamic Instability and Turbulence
1.0 0.5 0.0 1.0 0.5 0.0
1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8

t=0.6


t=1.2

1.0 0.5 0.0
1 2 3 4 5 6 7

t=1.8

8

1.0 0.5 0.0
1 2 3 4 5 6 7

t=6.

8

1.0 0.5 0.0
1 2 3 4 5 6 7

t=18

8

.9 = 0 R=60140.
An
30

z =0

n=3
n=4

20

10

n=8
n=5

, , , . , , , , . . , . , Ra=60140, Pr=16,6, [3]. , , [1]. , . ( 04-05-64627).

0 0 4 8 12

t


1... . -. // , ., 1999, 345 .. 2..., .., .. . // , ., 1987, 271 . 3..., .., .., ... , // 680, , 2005, 34 .
Galina Rybushkina is Senior Reseacher Applied Physics of RAS, Department Dynamics. Her main scientific interests Investigation and Numerical Simulation of Hydrodynamical Flows. of Institute of of Nonlinear are Analitic Instabilities in

.8 = 0 z =0 , R=120000.
t=2.4
1.0 0.5 0.0
1 2 3 4 5 6 7 8

t=9.6
1.0 0.5 0.0
1 2 3 4 5 6 7 8

t=11.
1.0 0.5 0.0
1 2 3 4 5 6 7 8

.11 = 0 R=120000.

z =0