-
..
.
.
.
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.