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( ) .., .. () ( ). , , - . . . , , . THE ANALYTICAL SOLUTION OF ONE MODEL OF THE HOMOGENEOUS LIQUID CURRENT (The case of the nonconstant vertical turbulent exchange coefficient) Kompaniets L. A., Yakubailik T. V. (Krasnoyarsk) The analytical solution of one stationary model of water motion of the wind current in the homogeneous liquid in the closed reservoir (two-dimensional case in the vertical plane) is proposed in this paper. The study suggests that the bottom of the water basin is not flat and the vertical turbulent exchange coefficient is known function of the space variables. The examples of the particular solutions for the concrete values of the vertical turbulent exchange coefficient and the friction coefficient on the surface and on the bottom are presented.
215


These results are simply extended to the case of the two ­ layered liquid. The solutions were found could be useful as a test in the analysis of the quality of the computational algorithms which are applied for solving problem of the wind current in closed basin water bodies and for the evaluation of the thermocline. - [1-4]. , , , , [5, 6]. [1-2] . [1-2]:
u u , - lv = K z -g z z x t v v + lu = K z - g , t z z y

(1) (2) (3)

u v w + + = 0. x y z

u = u( x, y , z, t ) , v = v ( x, y , z, t ) , w = w( x, y, z, t ) ­ , g ­ ; K z = K z ( x, y, z, t ) > 0 ­ ; = ( x, y , t ) ­ , l ­ . z . u = (u, v ) (1) ­ (3) : ) u n = 0. (4) ) ( z 0 = )
216

z 0 = w, t

(5)


.. . -- -10, 2002, .215-217

K

z

u = . z 0

(6) (7) (8)

) , z = z 0 ( x, y ) = - H ( x, y ),
w=u z0 z u ~ + v 0 , Kz = kb u u . x y z

= ~ k b = const ­

( 1 , 2 ) , 1 2 ­ x y ; un . ­ -

; ~ . , k b = 0 ~ , k b = ­ . , , , ­ . (3) (5) (7), :
Q1 Q2 + + = 0, Q1 = udz, Q2 = vdz. t x y z z
0 0

z

0

z

0

(9)

u( x, y, z,0) , v ( x, y , z,0) , ( x, y, z,0) .

,

(1) ­ (3), (9) (4) ­ (8) , (1), (2), (9) (6) (8) , u, v, . , u . 1. . 2. (6) z 0 = 0 , [7]. 217


(1) (9) :
u u , = Kz -g t z z x
Q1 + = 0, t x
0

(10) (11)

Q1 = udz, 0 < x < L, (6) ­ (8) :
z
0

) u

x = 0, L

= 0.

(12)

) ( z 0 = 0 )
K
z

u z

=
z =0

, 0
z

(13)
u z = k b u,
0

) , z = z0 ( x ) = - H ( x ) , K

(14)

z=z

k b ­ , (8). u( x, z,0) , ( x,0) . , ­ ­ (10) ­ (14) = 0 , H ( x ) = const , K z = const x [7]. 3. (10) ­ (14). :
u = Kz , x z z x

(15)

0 u ( x, z )dz = 0, K z = K z ( x, z ), -H

(16)

0 < x < L , - H ( x ) < z < 0 :
218


.. . -- -10, 2002, .215-219

K K

z

u z u z

= ( x ), ( x ) =
z =0

, 0

(17) (18) (19)
u = 1 ( x ). z

z

z=-H ( x)

= k b u, = 0.

u( x )

x =0

= u( x )

x=L

(15), (16) = ( x )
K z ( x, z z z : u K z ( x, z ) = 1 ( x ) z + 2 ( x ). K z ( x, z )

z ) > 0 ,


u=

-H



z

1 ( x ) + 2 ( x ) d + 3 ( x ). K z ( x, )

(20)

1 ( x ) , 2 ( x ) 3 ( x ) (17), (18) (16), u . (17) (21) 2 ( x ) = ( x ). (18) - 1 ( x ) H ( x ) + 2 ( x ) = k b 3 ( x ), (16) , (22)

-H



0

udz = 0

-H

dz

0

z

-H

1 ( x ) + 2 ( x ) d + 3 ( x ) = 0. K z ( x, )

1 2 x ,
219


1 ( x ) dz
-H

0

-H



z


K z ( x, )

d + 2 ( x ) dz
-H

0

-H



z

1 d + 3 ( x ) = 0. K z ( x, )

(23)

(21), (22) (23) 1 , 2 3 , , . . H - k b K 1 0 , K 1 K 2 ­ 1 2 (23). , k b = 0 ( ), . , (20) (15), (16) (17) ­ (18). (21) ­ (23) ( x ) , (19) , ( x )
x = 0, L

= 0.

. 1. k b = 0

, K z = const ,
u=

( x ) 1 + kb 2 H ( x ) K
K
z

2 + 2 k 3 H ( x) K b

z


z

2 3 + kb 6 H ( x ) K z z2 +z+ H ( x ) , H ( x) 2 + 2 kb 3 H ( x ) K z

k b . , K z ( x, z ) = const , k b = , [5].

220


.. . -- -10, 2002, .215-221

. 2. k b = 0.01

4800 20 K z = K z ( x, z ) = 0.000624 , 2 /. K z ( x, z ) = K z ( z ) - z . K z , , [3]. . 1, ) . 2, ) k b . . , . . 1, ) . 2, ) , 1 3 . , , K z = const , x = x0 . , K z = K z ( z ) ( ), .

221


. 3. K z ( z ) , .

K z ( z ) , [8], [9], , . 3, (3-7 / 2 /). . , , , ­ .

. 4.

, [10] , [11] ­ . 222


.. . -- -10, 2002, .215-223

. , , (.4). , , :
g 1 1 u1 = Kz , x z z u1dz = 0,

(24)

-h



0

(25)

g (1 -
-h

1 2 u2 1 2 +g 1 = K ) , 2 x z z z 2 x

(26)

-H



u2 dz = 0.

(27)



1K

1 z

u1 z u2 z

z =0

= ( x ), 1K = -K
1, 2 z

1 z

u1 z

=K
z =-h 2 z

1, 2 z

(u2 - u1 ). = k b u2 ,

(28) (29) (30)

2 K
u1

2 z

z =-h x=L

(u2 - u1 ), 2 K =u
2 x=L

u2 z

z =-H

x =0

= u1

=u

2 x =0

= 0.

"1" , "2" ­ . K 1,2 ­ z . :

223


u1 =

-h



z

1 1 1 ( x ) + 2 ( x )

K

1 z

1 d + 3 ( x ),

(31)

u2 =

-H



z

2 12 ( x ) + 2 ( x )

K

2 z

2 d + 3 ( x ).

(32)

u1 u2 (24) ­ (27)
a1 = a2 = c1 =
-h


dz
-h

0

z


K ( x, )
1 z

d , b1 =

-h

-h


dz
-h -H

0

z

-h

1 d , K ( x, )
1 z

-H
-h -H


dz
2 z

z


2 K z ( x, )

d , b2 =
-h -H

-H


dz

z

-H

1 d , 2 K z ( x, )




K ( x, )

d , c 2 =



1 d , K ( x, )
2 z

:
1 1 1 a1 ( x )1 ( x ) + b1 ( x ) 2 ( x ) + 3 ( x ) = 0 ,

(33) (34) (35)

2 2 a 2 ( x )12 ( x ) + b 2 ( x )2 ( x ) + 3 ( x ) = 0 ,
1 12 ( x ) = ( x ) ,
1 1 1(1 ( x)(-h) + 2 ( x)) = K 1,2 z

1 2 2 (-3 ( x) + c1 ( x)12 ( x) + c22 ( x) + 3 ( x)) , (36)
1 2 2 (-3 ( x) + c1 ( x)12 ( x) + c22 ( x) + 3 ( x)) , (37)

2 2 (12 ( x)(-h) + 2 ( x)) = -K

1,2 z

2 2 2 ( -12 ( x ) H ( x ) + 2 ( x )) = kb3 ( x ) .

(38)

(33) ­ (38) (24) ­ (27), (28) ­ (29), (30). (33) ­ (38) ( x ) , , , ( x ) x=0 x=l,
224


.. . -- -10, 2002, .215-225

. , [12] . 2 , , K 1 = const , K z = const. z K 1,2 = 0 ( [10] z ) kb = 0 . , , I
u1 ( x, z ) =

( x) 1K 1 z

z2 1 2h + z + 3 h .

II 2 u2 2 Kz = 1 ( x ), , z z u2 ( x, z ) =
d1 ( x ) 2 z + d 2 ( x) z + d 3 . 2
2 Kz 2



u2 z

= 0,
z = -h

2 K

2 z

u2 z

z=-H

= k b u2 ,

-h

-H



2 u2 dz = 0. 2 K z ( d1 ( -h ) + d 2 ) = 0

2 K (( - H ) + d 2 ) = 0 . h H , d1 = d 2 = d 3 = 0 , I . , 1 ( x ) 2 ( x )
2 x 1 x

2 z d1



(1 -

1 2 1 1 =- 2 ) 2 x x



=



1 2 1 2

1-

,

2 ( x ) = -h
225


1 ( x ) , [10].
CRDF, REC-002.

. 1. .. - . . , 1979, 5, . 129-137. 2. .., .., .. . . 3, 1981 ., . 91-99. 3. Wang Y., Hutter K. Methods of substructuring in lake circulation dinamics. Advances in Water Resourses. 23 (2000) pp. 399-425. 4. . . . , «», 1978, 128 . 5. .. . : , 1983, 166 . 6. Wang Y., Hutter K., Bauerle E. Barotropic response in a lake to wind - forsing. Annales Geophysicae, 2001, 19, pp. 367-388. 7. . ., .. . . , 2000, 420 . 8. Witten A., Thomas J. Steady wind-driven currents in large lake with depth-dependent eddy viscousity. J. of Phys. Ocean., 1976,v. 6, 3, pp. 85-92. 9. Belolipetsky V. M., Genova S. N. Investigation of Hydrothermal and Ice Regimes in Hydropower Station Bays. IJCFD, 1998,Vol. 10, pp. 151 - 158. 10. .., .., .., .. . , 3, 1981, . 33-51.

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