..

.

.

.

2010

ii

,

,

, . . , 2008

2010.

,

. .

.

.

.

, ..

, .

, . ,

. , ,

.

, ...

iv

, .

MIKE 3

Flow Model F.M. (Hydrodynamic Module)

,

,

,

.

.

,

, .

Navier Stokes.

, Couette,

. Eifler & al. (1989)

Couette ,

.

3 Flow Model

F.M. , .

4

.

(wall functions) k .

,

, .

,

,

, .

Eifler et al (1989).

,

,

.

5 4,

,

,

.

.

6

, k, ( )

,

. , k

, ,

.

k ,

.

,

.

Couette

.

k

o

Couette.

vi

......................................................................................................................... iii .......................................................................................................................... iv .............................................................................................. ix ............................................................................................... xiv .......................................................................................................................... 15 1 : ............................................................................................. 22

1.1 ........................................................ 22 1.2 ............................... 24 1.3 BOUSSINESQ ........................................................................ 25 1.4 ROSSBY .................................................................. 27

2 ................................................................................................................................... 28

2.1 COUETTE POISEUILLE ..................................................................................... 28 2.2 ........................................................................................ 35 2.3 - EIFLER. COUETTE ....................................................................................................... 38 2.4 EIFLER .......................................................................................................... 46

3 ....................... 49

3.1 ...................................................................... 49 3.2 ......................................... 49

3.2.1 . .............................................................................................. 50 3.2.2 ............ 52 3.2.3 ...................................... 56

3.2.3.1 .................................. 57 3.2.3.2 .............................. 57

3.3 .................................................. 59 3.3.1 ..................................................... 59 3.3.2 ............................... 60

3.3.3 ........................................................................................ 62 3.3.4 CFL .......................................................................... 63 3.3.5 .............................................. 63

4 .......................................................................... 65

4.1 ............................................................... 65

4.1.1 .......................................................... 65 4.1.2 .......................................................................... 66

4.1.2.1 . .... 66 4.1.2.2 ..................... 68 4.1.2.3 ....................................................................... 69 4.1.2.4 ................................................................. 69 4.1.2.5 ......................................... 70 4.1.2.7 ..................................... 79

4.2 ................................ 82 4.2.1 . ................................................................................................................. 82 4.2.2 .................................................................... 85 4.2.3 .......................................... 90 4.2.4 u Eifler & al. (1989) ......................................................................... 93 4.2.5 , ............................................... 94 4.2.6 ................. 96 4.2.7 Eifler & al. - : - ............................................................................ 98

5 ........................................................................................... 100

5.1 ............................................................................................................. 100 5.2 ................................ 101 5.3 ........................................................ 104

5.3.1 ............................. 105 5.3.2 ............... 107 5.3.3 ................ 114 5.3.4 , Couette ...................................... 116 5.3.5 - Eifler & al. (1989) - ............................................................................................................. 118 5.3.6 ........................... 120 5.3.7 - .............................................. 122

viii

6 ................................................................. 124

6.1 ............................................................................................................. 124 6.2 A - ....................................................................................................... 124 6.3 M k, , ( NEUMANN) ....................................................... 125 6.4 ....................................... 126

6.4.1 ................................................. 126 6.4.2 .. 127 , k , ........... 127 6.4.3 ................................. 128 6.4.4 ........................ 132 6.4.5 ................................................................................................................ 133

6.5 k, ......................................................................................................................... 135

6.5.1 ....................................................... 136 6.5.2 , ......................................................... 138 Eifler & al. (1989) ............. 140

6.6 k, , ................................... 143

6.6.1 .................................................................. 143 6.6.2 Eifler & al. (1989) ............. 148

6.7 . - ...................... 150

.................................................................................... 161

1.1 .................... 23 1.2 , (x,t), b(x) ............................................... 25 2.1 Couette Poiseuille ............................................................. 32 2.2 Couette Poiseuille ...................................... 32 2.3 E Couette ....... 33 2.4 , Nezu & Nakagawa (1993) ........................................................................................................... 38 2.5 (Eifler & al, 1989) .... 44 2.6 (Eifler & al, 1989) ............................................................................................................................................. 46 2.7 (Eifler & al, 1989) ... 46 3.1 (Drago, Iovenitti, 2000) ............................................................................................................................ 54 3.2 : sigma z (DHI Software, Hydrodynamic Module, User Guide, 2007) ...... 54 3.3 (DHI, 3 F, 2004) ....................................................... 60 3.4 . 12 .................... 61 4.1 ............................................................. 65 4.2 ...................................................................................... 67 4.3 . O y, ................................................................ 71 4.4 . . ....................... 72 4.5 DNS (Direct Numerical Simulations) Kim et al.(1987): Re = 5600, Re = 13750

:

yuU . Pope (2000) ................... 76

4.6 , Wei and Willmarth (1989): Re = 2970, Re = 14914, Re = 22776 Re = 39582.

uU ....................................................................................................................

................................................................................................................................................................... 4.7 .................................................. 81 4.8 (50, 1000) o (layer 8) ......................................................................... 83 4.9 (50, 1000) o (layer 8) .............................. 84 4.10 (50, 1000) o (layer 8) ............................ 84 4.11 . u 100 m, . (50, y), 1000 y 0 ........................... 86 4.12 . u u (0, 1000) (100, 1000) 10 m ......................................... 86

...................................................................................................................................................................

x

4.13 k o (layer 8) ....................................................................................................... 87 4.14 u o (layer 8) .................................................................................................................. 87 4.15 . , u , MIKE 3 HD Flow model F. M. () H . () ................................................................................................................ 88 4.16 , u. ................................................ 89 4.17 , - . .......................... 89 4.18 ( x

max) .... 90

4.19 ........................................................................................................ 91 4.20 , ........................................................................................................ 92 4.21 u MIKE 3 HD (Flow Model) Eifler & al. . (50,0) ....... 94 4.22 u MIKE 3 HD (Flow del FM) ............................................................................................................................................ 95 4.23 , ................................................................................................... 96 4.24 ............................................................................................. 97 4.25 u Eifler & al. (1989) ............................................................ 99 p ...................................................................................................................................................... 5.1 ........................... 104 5.2 u (50, 1000) (layer 35) .................................. 105 5.3 z (50, 1000) (layer 35) ......................................................................................................................................... 5.4 (50, 1000) (layer 35) ........ 106 5.5 u (50, 1800) , (50, 1200). T 200 m ..................................................................................... 108 5.6 . u

100 m, . (50, y), 1000 y 0 5.7 . u (0, 1000) (100, 1000) 10 m .................................................................. 109 5.8 (layer 35) .................................................................................. 109 5.9 u (layer 35) ..................................................................................................................... 110 5.10 , . ................................................................................................................ 111

5.11 . ........................ 112 u, 5.12 , 2D Couette .................................................... 112 5.13 , w .................................................................................................................................. 112 5.14 . . . 5.15 ......................................... 115 5.16 u 3 .............................................. 117 5.17

t

3 .............................................. 117 .............................................................................................................................................................................

5.18 , ................. 118 5.19 u MIKE 3 HD (Flow Model FM) Eifler & al. 70 .................... 119 5.20 u Eifler et al. (1989) .......................................................... 119

5.21 t 121

MIKE 3 HD (Flow Model FM) Eifler & al. (1989) .............................................................................................................................. 121 5.22 .......................................................................................................................... 123 6.1 : () () , (50, 0), .......................................................................................... 127 6.2 k, ............................................................................................................. 129 6.3 . k, k, ................................................................................................... 130 6.4 . k, k, ............................................. 130 6.5 , .................... 133 6.6 u 1 10 , (50, 0) .............................................................................................................. 134

6.7 t

1 10 (50, 0) ............................................................................................................. 134 6.8 u (50, 1000) (layer 35) ................................. 136

xii

6.9 z, (50, 1000) (layer 35) .............................................................................................................................................. 138 6.10 , k (50, 1000) (layer 35) ...... 137 6.11 . u (0, 1000) (100, 1000), 10 m .............................................................. 138 6.12 w ................................................................................................................................. 138 6.13 u .............................................................................. 139 6.14 . u 100 m, . (50, y), 1000 y 0 ........................................ 140 6.15 6.16 u MIKE 3 HD (Flow Model F.M) Eifler & al. (1989) ................................................................................................. 141 6.17 , : () u () , 3 Flow Model F.M ......................................................... (50, 1000), (layer 35) 6.18 z, (50, 1000) (layer 35) ...... 145 6.19 , k (50, 1000) (layer 35) ...... 145 6.20 : () , 20 m, () , () .................................................................................. 147 6.21 . ............................................................................ 147 6.22 u MIKE 3 HD (Flow Model FM) Eifler & al (1989) ................................................................................................................... 148 6.23 , : u () (), MIKE 3 HD Flow Model F. M. .............................................................................. 149 6.24 k, ........................................................................................................................................... 151 6.25 u, k, . (layer 35) 50 m............................................................................ 152 6.26 , k, k, . (layer 35) 50 m............................................................................ 152

6.27 , t

v ,

k, . - (layer 35) 50 m ............................................................................... .153 6.28 (50, 2000) (2000, 1400) (2000, 1400), , (), (), () ............................................................................................................................... 154 6.29 , ............................................ 156 6.30 .............. 157 u, : (), (), () k, ......................................................................................... 157

6.31 , . ........................ 160

xiv

2.1 Couette, Gretler & Meile (1997) ............................ 34 2.2 Couette ....................................................... 43 3.1 k (Rodi, 1984) ........................................................... 58 4.1 . ......................................................... 81 4.2 u. ............................................................................ 89 4.3 92 4.4 u. ....................................................................................... 95 4.5 ................... 97 4.6 ........................................................ 99 5.1 ................... 103 5.2 ............................................................................................................................. 114 5.3 ........................................................ 120 6.1 ............................................................................................................................... 129 6.2 . ......................... 142 6.3 . ......................... 149 6.4 - k, ................................................................................ 151

15

.

. O -

, , ,

,

. -

.

,

, .

.

,

, 50 200 (Ocean

Circulation, EESC 2100, Spring 2007). ,

, .

,

.

.

,

, , ,

,

.

.

,

16

,

,

. ,

.

.

,

(wind setup),

,

.

.

,

,

. ,

.

.

,

. ,

Officer (Physical Oceanography, 1976)

.

Heaps (1984) Tsanis (1989),

u. , Eifler, Kupusovic &

Schrimpf o 1989

Couette,

.

17

,

Baines & Knapp (1965), Koutitas

& O Connor (1980), Tsuruya & al. (1985). ydin &

Leutheusser (1979) Tsanis & Leutheusser (1987)

. , ydin & Leutheusser

Couette

,

.

.

,

,

.

, Davies et al. (2001)

.

,

,

.

(2D) , Leendertse (1967) Hunter (1980),

.

, , ,

,

, Koutitas

Gousidou - Koutitas (1986), Schwab & al. (1989), Simons & Schertzer (1989)

18

Wang & O Connor (1975). 2D -

(quasi 3D)

,

.

T (3D)

Reynolds averaged Navier - Stokes

,

.

,

() ,

.

Wu & Tsanis (1995) Gting &

Hutter (1998). Koutitas Gousidou - Koutitas (1986), Schwab & al. (1989),

Simons & Schertzer (1989) Wang & O Connor (1975)

, .

,

,

, . ,

Shankar & al. (1996), Drago & Iovenitti (1999), bualtayef & al.

(2007).

,

Davies (1985), Davies & Hall (2000), erman & al. (2000), Sanay &

Valle - Levinson (2005). Davies (1985)

, . O Berman & al.

(2000) Elat

POM Sanay & Valle Levinson (2005)

19

ROMS

: ,

. T

,

.

,

.

.

,

,

(. . ), . .

, G. T. Csanady

(Circulation in the Coastal Ocean, 1973)

.

,

, .

,

.

.

,

,

,

, , . . . .

( )

. 10

20

100 ,

,

, . , ,

.

, ,

, ,

,

.

, , ,

.

.

.

.

,

,

. ,

.

.

,

.

,

. ,

.

21

,

- , Couette.

. ,

,

Couette.

1 :

22

1

:

1.1

, ,

.

, .

. ,

,

,

, : 1L

D W ~ U

L

DU , U, W

. ,

, ( Navier - Stokes

) .

Navier - Stokes

:

z

u

zy

u

yx

u

xx

p

1fv

Dt

Duzyx (1.1)

z

v

zy

v

yx

v

xy

p

1fu

Dt

Dvzyx (1.2)

gz

p

1

0 (1.3)

f sin2 , o

Coriolis. ,

1 :

23

, 1.1, z .

- ,

. 2.

:

0

z

w

y

v

x

u (1.4)

To (1.1) - (1.4) , ,

, , Navier - Stokes,

Boussinesq ,

, .

1.1

1 :

24

1.2

,

,

z b + , z = (x, t) + b (. 1.2).

:

p a (x,y,t) - p(x,y,z,t) = - g (b + h + - z) (1.5)

p a : ,

, = h + , h

.

(

). (1.5)

:

x: x

p

x

g

x

p

a

11 (1.6)

y: y

p

y

g

y

p

a

11 (1.7)

b + h :

0)()(

y

hb

x

hb (1.8)

:

x:

z

u

zx

p

x

gfv

z

uw

y

uv

x

uu

t

uz

a1 (1.9)

y: z

uw

y

uv

x

uu

t

u

z

v

zy

p

y

gfu z

a1 (1.10)

1 :

25

x

1.2 , (x,t), b(x)

1.3 BOUSSINESQ

Boussinesq

. ,

Reynolds

.

1887 Boussinesq :

- iji

j

j

i

ijji kx

u

x

u'u'u

3

2

(1.11)

ij Kronecker ( ij = 1 j = i, ). To

Boussinesq k

:

2322212

1'u'u'uk (1.12)

ijk)3/2( (1.10)

. (1.10) :

z

1 :

26

1

12

1 2x

uv'u t

2

22

2 2x

uv'u t

3

32

3 2x

uv'u t

, : 0

i

i

x

u.

(. 1.11). ,

(1.10) (1.11).

Reynolds (1.10)

Reynolds

averaged Navier - Stokes.

(1.10) Newton,

. ij

( )

.

.

, v

.

u . tv

,

. , ,

.

,

:

dz

ud zzx (1.13)

z u

.

, :

j

i

i

jijij

x

u

x

u p , ij, ij Kronecker. i, j

1 3.

1 :

27

1.4 ROSSBY

Rossby

Coriolis

.

U

L.

:

y

uv

x

uu

L

U 2 (1.14)

H Coriolis :

fu U (1.15a)

fv U (1.15b)

, Coriolis

:

U

L/U 2

L

U (1.16)

O R 0 = L

U Rossby

:

R 0 = U/L

1/ (1.17)

: /1 : U/L .

Rossby (R 0 > 1),

,

. Rossby

(R 0 < 1),

. , ,

Coriolis.

2

28

2

2.1

COUETTE POISEUILLE

Couette ( Couette)

. ,

, , Reynolds.

Couette (

) (pressure gradient).

, ,

,

Couette. E

Poiseuille,

, ,

(favourable or adverse).

.

Couette Poiseuille

.

, .

,

( ),

.

.

Couette Poiseuille

, .

2

29

Couette (1890),

, Couette

. Burgers (1922)

Heisenberg (1922) Couette Von

Karman (1937)

. Poiseuille . .

Hussain & Reynolds (1975),

(. . Reynolds 1976),

Poiseuille

, .

2.1,

Couette Poiseuille .

,

Couette 1950

. Reichardt (1959) Robertson (1959)

, . Chue (1969)

Couette, Robertson & Johnson

(1970) .

Leutheusser & Chu (1971)

Couette Reynolds

. Aydin & Leutheusser (1979)

Couette, ,

.

Robertson & Johnson

(1970), El Telbany & Reynolds (1980, 1981, 1982), Aydin & Leutheusser (1987)

Tillmark & Alfredsson (1993).

,

. ,

.

2

30

Couette

. Reichardt (1956, 1959)

Reynolds. O Robertson (1959)

. Szablewski (1968)

Couette, Poiseuille

. ,

Reichardt Reynolds.

O Korkegi & Briggs (1968, 1970) Couette

Robertson. Chue & McDonald (1970)

, Chue. O

Hoffmeister (1976)

Reichardt.

,

Reynolds

. , 1978 Lund & Bush ,

Reynolds

Couette Poiseuille.

Lund & Bush Reynolds

. ,

Poiseuille Couette Poiseuille (

2.1).

Couette, Poiseuille,

Couette Poiseuille,

,

. Couette Poiseuille

,

2

31

.

Couette - Poiseuille.

Tsanis & Leutheusser (1987)

,

Couette Poiseuille. , ,

,

.

Afzal (1993)

(open equations of mean motion). ,

,

,

(overlapping regions). Couette

Gersten (1985a, b, 1987) ifler

& al. (1989),

. H

Eifler & al. (1989) ,

.

(lubrication theory),

Couette - Poiseuille,

. Constantinescu (1959)

Prandl .

Constantinescu (1959) Ng (1964).

(. . Elrod

2

32

& Ng, 1967), Prandl

(Ho & Vohr, 1974). (DNS)

. Reynolds,

, ,

. (Direct Numerical Simulations)

Couette Andersson & al. (1993), Kristoffersen et al.

(1993), Kuroda et al. (1995).

2.1 Couette Poiseuille

()

() 2.2 Couette Poiseuille

2

33

2.3 E Couette

(a) Reichardt (1956) (b) Reichardt (1959), Robertson (1959), Robertson & Johnson (1970) (c) Leutheusser & Chu (1971) (d) Aydin & Leutheusser (1979)

Couette

. , Poiseuille

.

Gretler & Meile

(1997).

,

2

34

. , Couette

Poiseuille, (

) ,

(high stress wall)

(low stress wall).

. Couette - Poiseuille

(2.1) ( . 2.1).

(. Lund & Bush, 1978):

dx

dp

u

Hsign

1

w

w

w0

21

1

)(1

dy

d

u

Hsign

1

w

21)( (2.1)

, 0, 1

.

: 0 2.

Couette ( = 0) Poiseuille ( = 1). T

2.1. bibi uusign /)/( ,

, Eifler & al.

.

2.1 Couette, Gretler & Meile (1997)

2

35

Couette.

( ),

Couette. ,

Couette

Couette Poiseuille.

.

,

:

,

,

Couette. .

, (. .

- , , ),

,

,

.

, ,

.

(

),

Couette. ,

Couette Eifler & al (1989).

2.2

1950 .

,

2

36

(spanwise) (Kline & al, 1967, Smith & Metzler, 1983).

(streamwise)

, Johanson & al., 1987).

(outer layer)

, .

1950 - 1960

.

, ,

(laminar layer).

(low - speed streaks)

. ( ;)

(gradual lift - up),

(ejection),

(bursting phenomenon). Hama ( Corrsin, 1957)

. , ,

, .

Blackwelder & Eckelman (1979)

(bursting phenomenon).

. .

(sweep)

,

.

2.4 Nezu & Nakagawa (1993). H

,

. Kline & Rundstadler (1959)

, Kline & al. (1967). ,

2

37

, ,

.

Kline & al. (1967)

, (

). ,

,

. ,

, ,

, .

,

y

(layers)

.

.

.

,

. ,

, Jimnez & Moin, 1991

.

2

38

2.4 , Nezu & Nakagawa (1993)

2.3 - EIFLER -

COUETTE

,

A ( )

,

1.3. , ,

RANS, ,

- , RANS.

,

.

,

- ( Eifler & al,

1989). (2.2):

A = )(z'A .

2

39

Couette,

:

x

P

z'

uz'

'

1)(

z (2.2)

(2.2) :

z'x

P

z'

uz'A

1)( + Const (2.3)

To (2.3)

z' :

- : i

- : b (2.4)

:

(2.5)

: 1.3

oussinesq, xz :

z

uzA zx

)( ,

x

,

y

. , , (2.3) :

'zx

P xz'

+ Const. , z :

0Hx

P bi

. O, (2.5)

o o:

(2.6)

(2.7)

bibi uusign /)/( (2.8)

( i b , : i = b )

u bb

H

x

P bi

u ii

2

40

:

A = *b

Hu

A (2.9)

bu

uu (2.10)

:

H

'zu (2.11)

= 0 = 1 , (2.3) :

A

Z-1-1

Z

u )( (2.12)

H A :

A =

Z1

Z1

1

2Z-11Z1Z

2

(2.13)

:

= f-1 (2.14)

= f-1 (2.15)

1) 1 : f = 1

14

1 =

23

1

(2.16)

2) 1 : f =

1

141

1

=

23

1

(2.17)

Von Karman 0.407.

(2.12) A (2.13) :

u = ZarctggZ)In(1InZ

21111

ZIngZIng 121112

1 2 + C (2.18)

:

ff

g

14

112

(2.19)

:

2

41

bu

u = InZ

1+ . (2.20)

iu

u = Z)-In(1

1+ . (2.21)

.

(2.18)

. , ,

,

.

,

:

ou

u =

ou

z- 0.34

2

ou

z+0.039

3

ou

z (2.22)

:

0 z 30

ou 14.7, z

:

uz'z

*

b

(b) (2.23)

-

uz'-Hz

*

i

(i)

(2.24)

- , .

(b)z

(i)z :

(b)z = bReZ

(i)z = bReZ1 (2.25)

: bRe =

Hu*b (2.26)

2

42

,

(2.18) (2.22) :

bb Re/Z 30

)/(Re301 b iZ (2. 27)

, u ( surfaceu - u ) 14.06 (2.22)

z = 30.

, :

- o

- i

- x

P

(2.5) (2.26)

, bRe .

,

: bRe

bZ iZ . bZZ (2.22).

ib ZZZ (2.18); C

bZZ , : u = 14.06. iZZ

. (2.22),

(2.18) iZZ .

2.5, 2.6

. 2.5

2.6

.

( - ,

- ).

. 2.7

. .

2

43

,

.

Couette, b = 0.04 2N/m , = 10 m 2

+2.

2.2 Couette

-2 -1.5 -1 0 +1 +2

b [2N/m ] -0.16 -0.09 -0.04 0 +0.04 +0.16

x

P

[ Pa/m ]

-0.020 -0.013 -0.008 -0.004 0 +0.012

: 0.16 2N/m

10 m/sec .

2

44

2.5 (Eifler & al, 1989)

2

45

2.6 (Eifler & al, 1989)

2

46

2.7 (Eifler & al, 1989)

2.4

EIFLER

(2.5) :

(2.5)

1.2 :

x: x

p

1

x

g

x

p

1 a

(1.6)

(2.5) :

H

x

p

x

g bia

(.

) :

H

x

P bi

2

47

gH

x

bi

(2.28)

x

.

710

10 m

bi 31081.9 . ,

. ,

(

, *s

) .

2.8 ( 00 QQQ /) ,

.

,

b . , x

ib ,

0

x

ib .

( ) ,

.

2

48

0.0 2.0x10-7

4.0x10-7

6.0x10-7

8.0x10-7

1.0x10-6

0

1

2

3

4

5

6

7

8

9

10

11

(Q-Q

0)/

Q0 x

100

Surface slope

2.8 A ,

3

49

3

3.1

3 Flexible Mesh (Hydrodynamic Module)

DHI Water & Environment

(flexible),

, ,

. ,

.

,

, , ,

- .

.

.

3 Flow

Model (DHI, Water and Environment, Hydrodynamic Module, Scienific Documentation,

2007).

3.2

Reynolds Averaged Navier Stokes (RANS)

Navier Stokes, ,

.

Boussinesq (Boussinesq Approximation),

z

. Reynolds,

3

50

S

Boussinesq (Boussinesq Assumption)

,

. ,

,

(sigma co ordinate transformation).

3.2.1 .

(Shallow water equations)

H :

x

u

+

y

v

+

z

w

= S (3.1)

x y :

t

u

+

x

u

2 +

y

vu

+

z

wu

= fv - g

x

-

0

1

x

p

- 0

g

z

dzx

+ F u +

+z

z

uvt + u s S (3.2)

t

v

+

y

v

2 +

x

uv

+

z

wv

= fu - g

y

-

0

1

y

p

- 0

g

z

dzx

+ F v +

+z

z

vvt + v s S (3.3)

: t , x, y z ,

, d

, h = + d

, u, v w x, y z

, f = 2sin Coriolis (

). g ,

, v t , p

, 0 . S

(u s , v s )

3

51

.

, :

F u = x

x

uA2 +

y

x

v

y

uA (3.4)

F v = x

x

v

y

uA +

y

y

vA2 (3.5)

A .

u, v w

z = :

t

+ u

x

+ v

y

- w = 0,

z

v

z

u, = sysx

tv

,

1

0

(3.6)

z = -d:

ux

d

+ v

y

d

+ w = 0,

z

v

z

u, = bybx

tv

,

1

0

(3.7)

( sx , sy ) ( bx , by ) x y

.

h,

,

. ,

:

t

h

+

x

uh

+

y

vh

= hS + P - E (3.8)

P E ( )

u v :

d

udzu ,

d

vdzv (3.9)

.

, , T , s

:

sT , (3.10)

3

52

, UNESCO,

. (. UNESCO, 1981).

3.2.2

:

= h

zz b , x = x, y = y (3.11)

z b

, ,

. :

z

=

h

1 (3.12)

yx, =

y

h

y

d

hyx

h

x

d

hx

1,

1 (3.13)

To ()

h , .

, ,

, ( 3.1).

(3.11)

z, x, y.

,

. ,

: d = z - z b , . ,

: = f (z, x, y).

()

.

,

- .

Phillips (1957).

-

3

53

s

Song & Haidvogel (1994). E

.

,

, , , . . .

,

,

.

,

.

,

z (. 3.2). ,

(

)

.

,

, z

.

,

,

Couette.

,

.

,

,

.

3

54

3.1 (Drago, Iovenitti, 2000)

3.2 : sigma z (DHI Software, Hydrodynamic Module, User Guide, 2007)

3

55

:

t

h

+

x

hu

+

y

hv

+

h = h S (3.14)

t

hu

+

x

hu

2 +

y

hvu

+

uh = fvh - gh

x

-

x

ph

0 -

z

dzx

phg

0

+ hF u +

+

u

h

vv + h Sus (3.15)

t

hv

+

x

huv

+

y

hv

2 +

vh = fuh - gh

y

-

y

ph

0 -

z

dzy

phg

0

+

+ hF v +

v

h

vv + h Svs (3.16)

, , k :

)BP(h

k

v

hFh

kh

y

hvk

x

huk

t

hk

k

t

k

1

(3.17)

)BP(h

k

v

hFh

h

y

hvk

x

huk

t

hk

k

t

k

1

cBcPck

h

v

h

h

y

hv

x

hu

t

h231

t

1F

(3.18)

:

=

y

hv

x

hu

t

h

y

dv

x

duw

h

1 (3.19)

o . :

hF u

x

uhA

x2 +

x

v

y

uhA

y (3.20)

hF v

x

v

y

uhA

x +

x

uhA

y2 (3.21)

h(F T , F s , F k , F , F c )

yhD

yxhD

xhh (T, s, k, , C) (3.22)

3

56

= 1:

= 0,

vu, =

tv

h

0

sysx , (3.23)

= 0:

= 0,

vu, =

tv

h

0

bybx , (3.24)

. , , (3.8).

3.2.3

Reynolds RANS

,

. ,

1.3, Boussinesq

:

ij

i

j

j

i

tji kx

u

x

uv'u'u

3

2

(1.11)

1, tv

. , ,

tv .

, .

.

. ,

, . ,

Reynolds ().

3

57

3.2.3.1

. O Smagorinsky (1963)

(stresses) ()

.

:

A = c s2 l 2 ijijSS2 (3.25)

c s , l

:

S ij = j

i

x

u

+

i

j

x

u

(i, j = 1, 2) (3.26)

(3.25) ,

RANS. H Smagorinsky

c s 0.25 1 (DHI, Hydrodynamic Module

User Guide, 2007).

. Smagorinsky

x, y .

3.2.3.2

k - , k

:

t = c

2k (3.27)

k (),

(dissipation of TKE) c

( Rodi, 1984).

, k, , ,

:

3

58

BP

z

kv

zF

z

wk

y

vk

x

uk

t

k

k

t

k (3.28)

231 cBcPckz

v

zF

z

w

y

v

x

u

t

t

(3.29)

P B :

P =

22

00 z

v

z

uv

z

v

z

ut

yzxz

(3.30)

B = 2Nv

t

t

(3.31)

Brunt Visl, N, :

Nz

g

0

2 (3.32)

t Prandl k , c 1 , c 2 c 3

. F , :

FFk , = ,ky

Dyx

Dx

hh

(3.33)

:

hD = A/ k hD = A/ , .

k . 3.1.

3.1 k (Rodi, 1984)

c c 1 c 2 c 3 t k

0.09 1.44 1.92 0 0.9 1.0 1.3

(dissipation) , sU .

z = :

k = 21

sUc

(3.34)

3

59

= b

s

z

U

3

(3.35)

sU > 0

z

k

= 0 (3.36)

=

h

ck

2/3

(3.37)

sU = 0

= 0.4 von Karman, = 0.07,

z s

. :

z = - d:

k = 21bU

c

(3.38)

= b

s

z

U

3

(3.39)

z b .

3.3

3.3.1

xy

(unstructured mesh).

. ,

(flexible mesh) (grid)

.

.

,

3

60

.

3.3.2

z

. , , ,

( ).

.

, ,

. ,

,

.

3.3 (DHI, 3 F, 2007)

, .

. (layers)

: 1) -

(equidistant layers), 2) (layer thickness), 3)

(variable)

, .

3

61

2

.

. Reynolds

Averaged Navier Stokes k

3.2.2.

3.4 . 12

,

(aspect ratio). ,

, 1/20

1/100, ,

.

(z/x = 1),

.

3

62

3.3.3

:

)()( USUFt

U

(3.40)

U , VI FFF

(flux vector function) S .

(. 3.1) i

Gauss :

i i iA A

dsnFdUSdt

U (3.41)

iA (area) ,

iA , i , ds

n

, .

(horizontal convectional fluxes)

Riemann,

. ,

semi - implicit

(explicitly),

(implicitly). ,

, - ,

(lower order scheme), (higher order

scheme).

Euler (first order

explicit Euler method),

(second order implicit trapezoidal rule).

Runge Kutta (second order Runge Kutta method),

(second order implicit trapezoidal

rule).

3

63

3.3.4 CFL

H

Courant (CFL). Courant :

s

tcCR

(3.42)

t , s

c

:

ghc (3.43)

Courant

( ) .

CFL

(Courant - Friedrich - Lvy) :

y

tvgh

x

tughCFLHD

(3.44)

h , u v

x y , g , x y

,

(

x = y).

CFL

0 1 ( 1).

0.8.

3.3.5

CFL, .

, x

Courant,

maxt , Courant:

3

64

c

Cxt rmax

(3.45)

ghc . , ,

maxt

.

4

65

4

4.1

4.1.1

, .

2000 m. 100 m,

10 m,

. .

,

,

. Nezu & Rodi (1985) (secondary

currents)

10.

B/h = 10. 4.1.

4.1

4

66

,

Eifler & al. (1989)

. 2.

, , :

D/L = 5 10 3 1 (4.1)

1.1.

0.2 m/sec, : U ~ 0.2 m/s. , ,

10 3 m, L = 2000 m,

Rossby :

R 0 = Lf

U

=

35 10210976.8

20

.

R 0 1.12. (4.2)

f Coriolis: f sin2 , 38 ( ).

R 0 > 1,

, .

10 m.

4 m/s .

, z, ( ).

.

4.1.2

4.1.2.1 .

, , 4

m/sec . T

1200 secs (20 )

(blow up).

4

67

, s ,

.

:

s = wd uc wu (4.3)

1.29 kg3m , c d

, , wu = (u w , v w )

10 m. .

4.2

,

.

Wu (1980, 1994)

:

c w 10 < w

c d = c + ab

ab

ww

cc

(w 10 - w ) w w 10 w b (4.4)

c b w 10 > w b

: c , c b , w w b w 10

10 m. .

4

68

O : c = 1.255310 ,

c b = 2.425310 , w = 7 m/sec.

, DHI, Water and

Environment, Hydrodynamic module, User Guide, (2007).

w10 = 4 m/sec,

c d = 1.225310 . O: *U = / s

-

s = dc24 , : *U 5.0895

310

m/sec.

4.1.2.2

(bed resistance).

, ikuradse, k s

0.0001 m, , .

,

, k s */Uv , *U

, ( Nezu &

Nakagawa, 1993).

: )//( *Uvkk ss sk < 5, .

,

,

,

. ( 705 sk ),

.

51.0sk < 5, 3

* 100895.5U ,

6101 v .

4

69

3 Smagrinsky

.

Smagrinsky 0.28. Smagrinsky

, 6108.1 10101 .

4.1.2.3

t = 0 .

,

.

.

k - , ,

(turbulent kinetic energy),

(dissipation of turbulent kinetic energy)

10 7 m 2 /s 2 5 10 10 m 2 /s 3 , .

, 8.1 10 6 m 2 /s.

4.1.2.4

(slip boundary condition),

. ( 4.3),

, (open boundaries)

, . ,

, .

,

, 10 7 m 2 /s 2 5 10 10 m 2 /s 3 , , .

4

5.

. ,

4

70

25 %,

.

4.1.2.5

2000 100 m

3.

- ,

.

,

. -

: A = (1/2) 28660 x. . Smooth mesh

.

(grid

resolution)

,

,

. 4.4

. ,

, .

,

(grid resolution). , ,

4.1.2.7 .

4

71

x (m)

4.3 . O y, ( )

y (

m)

4

72

(1) x max = 27 m (2) x max = 22 m

(3) x max = 16 m (4) x max = 11 m

4.4 . .

4.1.2.6 .

.

,

.

,

.

.

4

73

. Bernard &

Wallace (2002) Pope (2000).

,

. , ,

U

: 0 vu ,

, ( 4.6).

uv

y

U (4.5)

0y

wy

U

(4.6)

,w v ( )

( ). :

U w* (4.7)

:

*w

U

v

v (4.8)

w .

Reynolds ,

Reynolds:

v

*

*

h

v

hURe (4.9)

,

:

v

yU

yy *

v

(4.10)

4

74

y Reynolds

.

y

(). (viscous wall region), 50y ,

( Nezu

& Nakagawa, 1993 30y ),

(outer layer), 50y ,

. (viscous sub layer),

5y , .

Reynolds,

, h , 1

*Re ,

. (4.9).

,

: hv,, *U

:

2

*w U (4.11)

Buckingham :

h,,y,,U *U 3 .

*UU :

*o

*

Re,h

yF

U

U (4.12)

*Re (4.9) oF = "" (universal)

.

. U ,

: dyUd . H

.

4

75

,

:

h

y,

y

y

U

dy

Ud

1* (4.13)

1: "" .

( 50y ),

h (outer layer), 50y .

:

* Rehy/y (4.14)

(4.13)

(4.12).

(law of the wall)

Prandtl (1925) Reynolds

( 1hy ) (inner layer)

, .

. (4.13)

y hy . , . (4.13) :

*

y

y

U

dy

Ud hy

4

76

ydyy

1)(yf

y

0

w

(4.19)

U

y , 1hy . Reynolds

(. Pope, 2000) wf ""

.

( ), 00y

U ,

00 wf , (4.6) 10 fw .

Taylor, y , yfw , :

200f0ff yyy www 20 yyyfw (4.20)

4.5 DNS (Direct Numerical Simulations) Kim et al. (1987): Re = 5600, Re = 13750

yuU . Pope (2000)

4

77

. 4.5

U .

U = y

( 5y ) 12y .

15.0~1.0hy .

Reynolds,

y . , h.y 10 11.01.0 ** RehUy .

y . , . (4.15)

y ( ) ,

1 . , :

1 1hy 1y (4.21)

, . (4.17) :

ydy

Ud

1 (4.22)

:

lny

1U

(4.23)

. . (4.23)

= von Karman.

4.6 .

, Reynolds,

30y .

4

78

4.6 , Wei and Willmarth (1989): Re = 2970, Re = 14914, Re = 22776 Re = 39582.

uU

( 5y )

( 30y ), (buffer layer).

(

) .

Nezu & Nakagawa (1993),

:

() ( 15.0~hy ):

*U *U , . ,

, .

(bursting phenomena)

, 5y . ( 5y )

.

() ( 16.0 hy ):

, h maxU , .

, .

() ( 6.015.0~ hy ):

.

y , .

4

79

( )

50y .

4.1.2.7

. ,

.

-

.

(horizontal mesh resolution).

, (

)

,

.

y ,

,

: 50 y 100 Nezu &

Nakagawa (1993).

, y p

, y : y = v

yu p*

*u

.

y , bu ,

. , ,

Eifler & al. (1989), . . 2.

4

80

y 100,

, : z bot = 2 y p , y p 0.0196 m

y p = 0.02 m. , z bot = 0.04 m. ,

. 5,

y = 61, b

u

y = 100

10 %.

K (vertical domain)

h .

4.2

. , ,

, 4.1.

. ,

-

, (1.1 m).

-

,

, 2.

4.7.

Eifler & al (1989). , ,

u ,

,

4

81

,

Eifler & al (1989).

(4.1 4.2).

4.7

4.1 .

(z/h0)

0.004

0.005

0.025

0.026

0.11

0.11

0.11

0.11

0.11

0.11

0.11

0.11

0.026

0.025

0.005

0.004

4

82

4.2

4.2

4.2.1

.

, ,

, , , ,

. . . . , 4.8

4.10 ,

u ,

(50, 1000) (layer

8), , k tv .

,

. k

.

1 2 3 4

x

(m)

27 22 16 11

26 26 26 26

( - layers)

13 16 19 24

(m2

)

315.66 209.57 110.85 52.39

xy

1135 1520 2999 6092

xy

689 893 1724 3360

(m)

1.1 1.1 1.1 1.1

1/24.55 1/20 1/14.55 1/10

K y

( )

100 100 100 100

4

83

x max = 27 m.

.

.

,

.

0 10 20 30 40 50 60 70 80 90 100 110 120

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

0.22

0.24

u

layer 8

U (

m/s

ec)

Time (hrs)

4.8 (50, 1000) o (layer 8)

4

84

0 10 20 30 40 50 60 70 80 90 100 110 120

0.0

5.0x10-4

1.0x10-3

1.5x10-3

2.0x10-3

2.5x10-3

3.0x10-3

3.5x10-3

4.0x10-3

4.5x10-3

5.0x10-3

5.5x10-3

6.0x10-3

6.5x10-3

layer 8

v t (

m2/s

ec)

Time (hrs)

4.9 (50, 1000) o (layer 8)

0 10 20 30 40 50 60 70 80 90 100 110 120

0.0

1.0x10-5

2.0x10-5

3.0x10-5

4.0x10-5

5.0x10-5

6.0x10-5

7.0x10-5

8.0x10-5

9.0x10-5

1.0x10-4

k

Tu

rbu

len

t kin

etic

en

ergy

(m2/s

ec2)

Time (hrs)

4.10 (50, 1000) o (layer 8)

4

85

4.2.2

,

,

.

. Couette

x max = 27 m. 4.11

u , (50, 1000)

, (50, 0) 4.12

, .

4.14 4.13 u

k, ,

, , 50 m.

,

,

.

.

4.15

, , u,

,

MIKE 3 Flow Model F.M (3D, Volume Series).

. , 4.16

,

4.15.

. 4.17

.

.

,

u

4

86

, .

55 % .

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.00 0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.24 0.27 0.30 0.33

x = 50, y = 1000

x = 50, y = 900

x = 50, y = 800

x = 50, y = 700

x = 50, y = 600

x = 50, y = 500

x = 50, y = 400

x = 50, y = 300

x = 50, y = 200

x = 50, y = 100

x = 50, y = 0

U (m/sec)

z/h

0

4.11 . u 100 m, . (50, y), 1000 y 0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.00 0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.24 0.27 0.30 0.33

x = 0, y = 1000

x = 20, y = 1000

x = 40, y = 1000

x = 50, y = 1000

x = 60, y = 1000

x = 80, y = 1000

x = 100, y = 1000

U (m/sec)

z/h

0

4.12 . u (0, 1000) (100, 1000) 10 m

4

87

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0

3.0x10-5

6.0x10-5

9.0x10-5

1.2x10-4

1.5x10-4

1.8x10-4

2.1x10-4

2.4x10-4

2.7x10-4

3.0x10-4

, k

Tu

rbu

len

t kin

etic

en

ergy

(m2/s

ec2)

x/L

4.13 k o (layer 8)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.08

0.09

0.10

0.11

0.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0.20

0.21

0.22

0.23

x/L

U (

m/s

ec)

4.14 u o (layer 8)

4

88

4.15 . , u , 3 HD Flow model F.M. () H . ()

4

89

wind = 4 m/sec

4.16 , u.

wind = 4 m/sec

4.17 , - .

,

,

. 4.18,

h

410 .

. , Couette

.

4

90

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0

1.0x10-5

2.0x10-5

3.0x10-5

4.0x10-5

5.0x10-5

6.0x10-5

7.0x10-5

8.0x10-5

9.0x10-5

1.0x10-4

1.1x10-4

1.2x10-4

1.3x10-4

1.4x10-4

1.5x10-4

1.6x10-4

xmax

= 11 m

xmax

= 16 m

xmax

= 22 m

xmax

= 27 m

Su

rfa

ce e

leva

tio

n (

m)

x/W

4.18 ( x

max)

4.2.3

4.19

(

) .

(x max = 11 m).

, .

0.204 mm, x max = 27 m,

.

4.3,

,

10 %.

. 4.20

.

4

91

4.18, 4.19 4.20

, .

4.3

x max (m)

,

11 -1.3347

10

16 -1.377

10

22 -1.2227

10

27 -1.4597

10

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0

2.0x10-5

4.0x10-5

6.0x10-5

8.0x10-5

1.0x10-4

1.2x10-4

1.4x10-4

1.6x10-4

1.8x10-4

2.0x10-4

xmax

= 11 m

xmax

= 16 m

xmax

= 22 m

xmax

= 27 m

Su

rfa

ce e

leva

tio

n (

m)

y/L

4.19

4

92

4.20 ,

4

93

4.2.4

u Eifler & al. (1989)

, (50, 0) MIKE 3 Flow Model F.M,

Eifler & al. (1989)

, 4.21,

( . 4.4).

.

( . 2).

10 %.

, Eifler & al. (1989)

,

, . ,

,

.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48

Analytical solution

3D - Numerical model solution

u/u*

z/h

0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48

Analytical solution

3D - Numerical model solution

u/u*

z/h

0

x max = 11 m x max = 16 m

4

94

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48

Analytical solution

3D - Numerical model solution

u/u*

z/h

0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48

Analytical solution

3D - Numerical model solution

u/u*

z/h

0

x max = 22 m x max = 27 m

4.21 u MIKE 3 HD (Flow Model F. M) Eifler & al. (1989) . (50,0)

4.2.5

,

u

, (.

4.4).

u x max = 11 m, 16 m. 27 m,

4.22, .

, 4.23

,

x max = 27 m. x max = 22 m.