Παναγιώτης Παπανικολάου - Ανοικτοί Αγωγοί
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Transcript of Παναγιώτης Παπανικολάου - Ανοικτοί Αγωγοί
-
. , PhD
p
pb
pd
yc yb
h
d
-
1. ................................................................................................................ 1
1.1 1
1.2 . 1
1.3 . 1
1.4 . 2
1.5 3
1.6 (celerity) . 5
1.7 7
2. () ......................... 10
2.1. - 11
2.2 13
2.3. () 14
2.4. 14
2.5. 16
2.6. 17
2.7. 20
2.7 20
3. ................................................. 25
3.1. 25
3.2. 28
3.3. - 30
3.4. - 32
3.5. . 36
3.6. . 39
4. ............................................................................................ 42
4.1. 44
4.2 46
4.3. 54
4.4. 57
-
5. ............................................................................................ 60
5.1 61
5.2 61
5.3 63
5.4 64
5.5 . 65
6. - () ............ 66
6.1 66
6.2 67
6.3 () 68
6.4 () 70
6.5 . 70
6.6 . 73
7. .................. 80
7.1 1. (Direct step method). 81
7.2 2. (Standard step method). 85
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- . 2007-2008
2-1
1.
1.1
, . , , , , , . .
AVQ = V .
),( yAA = y . , b, y Z,
( )yZybZybA +=),,( . , ,
),,( tyAA = .
1.2 .
, , , , , .
. .
1.3 .
, (, , , , ) ( , , , ). :
b
y Z 1
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2-2
Jo, () ( t), ( y), ( ) .
1.4 .
, , , , . , , . . , , , , .
(section) , , .
1.1 .
:
t y
t
P
G
G
G - G
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2-3
t ... , ()
y ...
A ...
...
P ...
R ... (=A/P)
D ... (=A/) ,
1.5
. (steady) t ( (unsteady) ). (uniform) t ' , (non-uniform). .
, () (rapidly varied flow, RVF) () (gradually varied flow, GVF). t , . . 1.2 , 7.5cm, 5.0m 0.10m. 3.50l/s Reynolds 35000.
, , , , , (spatially varied flow), .
:
(1) .
(2) . , .
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2-4
(3) , ,
(4) , , .
1.2 7.5cm
1 10cm. (, 2005).
. , V (.. ) L (.. ), :
MaFI = 2223 )/( VLLVLFI = MgFg = gLFI 3 pAFp = 2pLFp ( )AdyduF / = VLF , , a , p , , , . Navier-Stokes , .
1 , . . , .
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2-5
, , ( ) Reynolds (Re)
Re = VL , (1.1) V , = / L ( .. ). Reynolds Rec, . Re < 500 (L = R = A/ ), , Re > 4000 ( French, 1985, 12500) 500 < Re < 4000 ( 12500) .
, , Froude (Fr)
gLVFr = , (1.2)
V , L = A/T = D . Froude Fr < 1 (), Fr > 1 () Fr = 1 .
y, b (b > 5y), R D
yyb
byPAR +== 2 , yb
byTAD === ,
Reynolds Froude
VyVL ==Re
gyV
gLVFr == . (1.2)
1.6 (celerity) .
:
( ) .
y y . c , u. ( c), ( ) (. 1.3)
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2-6
yy
ycuyyuccy +=+= ))(( (1.3)
])[()(22
)( 2212 cuccyyygyguuQFp =+=
1.3 y ()
() .
ucyyyg =
+
21 (1.4)
u (1.3) (1.4)
+
+=
yy
yygyc
2112
y0 gyc = . (1.5)
: c (celerity), h (h
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2-7
gycdygyc ==
01
2
.
(1.2) (1.5), Fr = 1, V = c.
: V < c (Fr < 1), . () ( ).
: V > c (Fr > 1), . ( ) ().
: V = c (Fr = 1), . .
1.7
. . , () . x, , dx, .
( 1.4) :
(i)
QAVVAVAAdVA
==== 22110)( (1.6) (ii) ()
==++=A
pg AdVVFFFF 0)(
)( 1122 VVQFFFF pg =++= (1.7) V u(z)
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2-8
( )AVdAuA
22= .
1.4
.
(iii)
(1) (2)
)21(21 += aHHH , gVz
gpH
2
2
++= (1.8)
)21( aH (1) (2)
( )AVdAuaA
33= . , ( 1.5)
bb zg
pzgp +=+
pb zb , .. .
2cos)cos)(cos()cos( gyygtgtgp tb ==== gt = g cos t = y cos. ,
H y z Vg
y z Vg
= + + + +cos22 2
2 2 . (1.9)
V2
(1) (2)
V1
A1 A2
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2-9
1.5 .
: , h u(z)=U(z/h) u(0)=0, u(h)=U, 2/UV = . :
= 4/3 = 2.
: ( ) y = t;
=2 cos2 = 0.999 (Jo=0.0349) =5 cos2 = 0.992 (Jo=0.0875) =7.5 cos2 = 0.983 (Jo=0.1317) =10 cos2 = 0.970 (Jo=0.1763) 15% ( 8) y = t.
(1)(2)
t t(cos)
V2/2g
y
.
.. (1-2)
g(cos
g(sin)
g
U
hx
z
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2-10
2. ()
2.1
2.1 . . .
t (depth) ( ) ( ).
y (vertical depth) () .
A (flow area - ).
T (top width).
P (wetted perimeter) .
PAR /= (hydraulic radius) . TAD /= (hydraulic depth) ,
.
( ) , t = y, . .
z ( ).
sin/ == dxdzJ o . dxdHJ E /= .
dxdyJ /= .
T
Dty
A
C
C TOMH C - C
t
P
..
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2-11
2.1. -
( ): (1) ( ) z, (2) y ( t) (3) V2/2g
H z y Vg
= + +2
2 (2.1)
H , z , y , V , g .
, z,
E y Vg
= +2
2. (2.2)
, y A(y) P(y). Q,
AQV = . (2.3)
(y) () Q
22
2
2
2)(2)(
gAQy
ygAQyyEE +=+== . (2.4)
, y - E Q y, . 2.1 Q.
, Emin y. dE/dy . , (2.4) y
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2-12
01)/(
1
112
1
22
3
2
3
2
2
2
===
==
+=
gDV
TAgV
gATQ
dydA
gAQ
gAQ
dyd
dydE
,
dA/dy = T, D = A/T. ,
12
2
== FrgDV
. (2.5)
2.2 .
y = yc .
E y Dc cmin = + 2 (2.6)
Dc . Vc = (gD)1/2. Q > min y1 y2 . y > yc ( y1) Fr < 1. y < yc ( y2) Fr > 1. b Q ( q=Q/b), y = D Emin= 3 yc/2, ' ( 2.5)
32
32
2
gq
gbQyc == . (2.7)
y2
yc
y
y1
yc
Emin E2 = E1 E(y)
Q1 < Q2 < Q3
Q1Q2 Q3
y
dy
A dA=Tdy
T
Vc2/2g
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2-13
2.2
(2.4) Q
Q gA E y= 2 2 ( ) (2.8) (2.8) :
0= QEy 000 == QAy
2.3
Q=Q(y) A=A(y). , 0 < y < E dQ/dy = 0 .
( ) 0)(20 2 == yEgAdyd
dydQ
[ ] 02)(22)(221 22/12 =
= gAyE
dydAAgyEgA
dydQ
2210)(2)(2 D
TAyEAyETAyE
dydA ==== .
(2.1)
122
22
2
2
==== FrgDVD
gAQyE .
y = yc ( )
=
y1
yc
y2
y
QmaxQ2 = Q1 Q
y
dy
A dA=Tdy
T
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2-14
c
c
c
cc
cc
cc
TAg
TAgADgA
yEgAQ3
22
2max
22
)(2
===
=. (2.9)
Q < Qmax y1 y2 .
2.3. ()
()
F F F Q V Vpx x gx+ + = ( )2 1 (2.10)
F yApx = (2.11) y .. .
Fg
Fg
QgA
y AQgA
y Ax gx + + + +
=
2
11 1
2
22 2 0 . (2.12)
2.4.
AygAQ
M +=2
(2.13)
(gx=0)
QgA
y AQgA
y A2
11 1
2
22 2 0+
+
= 21 MM = . (2.14)
() Q, y M(y) 2.4.
2.4 : (i) Q, Mmin, (ii) > Mmin Q, y1 y2, y1 > y2 1 = 2. .
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2-15
2.4
( = Mmin)
( ) 000 22 =+
=
+= Ay
dyd
gAQ
dydAy
gAQ
dyd
dydM .
TgAQ
dydA
gAQ
gAQ
dyd
2
2
2
22
==
( ) ( ) Ady
yAdyTdydyyAdy
yAdAydyd =++= )2/()( .
1)/(
0 222
2
2
====+ FrgDV
TAgVAT
gAQ ,
= Mmin, ( yc). > Mmin, Q , yc (y1 > yc, ) yc (y2 < yc, ), . y > yc ( y1) (Fr1).
Q =
Mmin M1= M2 M
y1
y
yc
y2
y
dy
A
dA
T
y
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2-16
2.5.
yc, . (E/yc,y/yc).
+=
+=+=+=
21
21
22
2222 Fryy
gyV
yy
yy
gyV
yy
yE
gVyE
ccccc
.
Fr = 1 (y = yc), E/yc = 1.50.
2cBy (
2/ cByM , cyy / ).
+=+=+=+=21
21
22
2
2
2
2
2
22
22
222
Fryy
yy
yy
gyV
ByM
ByAy
gAAVAy
gAQ
Mccccc
.
Fr = 1 ( cyy = ), 2/ cByM = 1.50. 2.5 ( ).
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
0 2 4 6 8 10 12E/yc, M/byc
2
y/y c
EM
2.5 ()
() .
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2-17
2.6.
, Q,
Fr VgD
VgD
c
c
= = = 1. (2.15)
. .
2.1
yc b Q = AV.
: yZybA )( += 212 ZybP ++= ZybT 2+= TADPAR /,/ ==
(2.15)
cyyZybyZybgyZyb
Q
BAgyZyb
Q
gDVFr ==++
+=+== ;1)2/()(
)()/(
)( ,
32
)2()(
1c
cc Zybg
QZyb
y ++= (2.16)
(. 2.1).
2.2
b Q. (2.16), = 0
BQq
gq
gbQyc === ;3
2
32
2
. (2.17)
.
b
y Z 1
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2-18
2.1: 2.5 m 1:1 ( = 1) 7 m3/s.
(2.16) Q, Z b
32
)250.2(81.9
00.7)50.2(
1c
cc yy
y ++=
yc = 2.00 m
y1 = 2.000m y2 = 0.709m y1 = 0.709m y2 = 0.840m y1 = 0.840m y2 = 0.824m y1 = 0.824m y2 = 0.826m y1 = 0.826m y2 = 0.826m yc = 0.826 m.
2.3
yc d. Q .
[ ]2
)2/cos(1 dy = )2/sin(dT =
8)sin( 2dA = , 2/dP =
4sin1/ dPAR
==
)2/sin(8)sin(/
dTAD ==
1//
3
2222
=====gA
TQTgAAQ
gDV
gDVFr
( ) 18/)sin( )2/sin( 322
3
2
== dgdQ
gATQ
( )
)2/(sinsin
8 2/12/3
2/3
2/5
== gdQ
'
y d
T
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2-19
3/12
3/5
)2/sin(8sin
+=
gQ
d (2.18)
(rad), Q , d g . . = 1 = 2. = (1+2)/2 . 4-5 .
[ ] 20;)2/cos(12
= dyc . (2.19)
= 1 = , . 2.2 : Q = 0.10 m3/s d = 0.60 m.
(1+2)/2 Q(m3/s) g(m/s2) d (m) 2 yc
3.142 0.10 9.81 0.6 1.886 0.300 2.514 0.10 9.81 0.6 2.442 0.207 2.478 0.10 9.81 0.6 2.467 0.202 2.473 0.10 9.81 0.6 2.471 0.202 2.472 0.10 9.81 0.6 2.472 0.201 yc = 0.20 m. : ( yc) Excel, max(yc/d)=0.97828 =5.6915rad, , . (2.18) Q, 2 Q, yc/d Q/g.
0.01
0.1
1
0.001 0.01 0.1 1 10Q/g1/2
y c/d
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2-20
2.7. , y = yc,
() Froude Fr VgD
VgD
c
c
= = = 1.
() E E yD
cc= = +min 2 .
() .
() Q Q Q A gDc c= =max . () (1/2)
Vg
Vg
Dc c2 2
2 2 2= = .
2.7
2.3 :
(1) Q=50m3/s.
(2) .
(3) 4.50 m.
(Z=2) Q (2.16) [ ] 3/121 /)2()( gZybQZyby ccc ++= . . yc = 2.00 m, yc = 1.715 m (. ).
yc Q b Z y1 2.000 50.00 5.00 2.00 1.656 1.656 50.00 5.00 2.00 1.728 1.728 50.00 5.00 2.00 1.712 1.712 50.00 5.00 2.00 1.716 1.716 50.00 5.00 2.00 1.715
1.715 50.00 5.00 2.00 1.715 ()
219.1...2
)( ==++==
c
cc
c
cc Zyb
yZybBA
D m.
2 1
b = 5m
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2-21
() y = yc. ()
)Zyb(y)Zyb(yDy
gVyEmin
c
ccc
ccc 2222
2
+++=+=+= .
yc min E = 4.50 m.
yc = 3.417 m yc = -1.317 m.
,
c = (b + Zyc)yc = 40.43 m2
Q gA E yc cmax ( )= =2 2 186.39 m3/s. : H : (1) Q, (2) yc, (3) Vc, (4) min E = yc + V2c/2g, (5) = 4.50 m . . (y>0).
Q Z b yc Ac V minE (m3/s) (m) (m) (m2) (m/s) (m) 50.00 2.00 5.00 1.71 14.46 3.46 2.32 100.00 2.00 5.00 2.49 24.79 4.03 3.32 150.00 2.00 5.00 3.06 34.07 4.40 4.05 200.00 2.00 5.00 3.54 42.73 4.68 4.65 190.00 2.00 5.00 3.45 41.04 4.63 4.54 186.00 2.00 5.00 3.41 40.36 4.61 4.50
Q = 186.00 m3/s.
2.4 7 m3/s.
:
() .
() .
() min min.
() min = f(yc).
() = 4 m, Qmax.
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2-22
.
() Q
y Qgb
qg gc
= = = =2
23
2
3
2
23
72 50( . )
0.93 m.
() ( y - E)
E y QgA
= +2
22
Q y. = 1.0.
(y - )
MQgA
yAQgA
y A= + = +2 2
2
Q y.
.
0.00
2.00
4.00
6.00
8.00
10.00
0.00 2.00 4.00 6.00 8.00 10.00
E (m)
y (m
)
.
0.00
1.00
2.00
3.00
4.00
5.00
0.00 5.00 10.00 15.00 20.00
M (m)
y (m
)
() y = yc.
=+=+= 22
2
2
)93.05.2(2793.0
2min
ggAQyE
cc 1.395 m
11 1
1
b=2.5m b=2.5m
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- . 2007-2008
2-23
=+=+= )93.05.2(293.0
)93.05.2(7
min22
xg
AygAQ
M cc
3.23 m3.
()
min E y QgA
y y ycc
cc c= + = + =
2
22 232
() . = 4 m ,
min E = 4 m = 1.50 yc.
yc = 8/3 = 2.67 m.
==== )3/84(2)3/850.2()(2max gyEgAgDAQ cccc 34.10 m3/s.
T .
() Q
3
2
)2()(
1c
cc Zybg
QZyb
y ++=
yc = 0.826 m.
() ( y - E)
E y QgA
= +2
22
Q y. = 1. (y - )
MQgA
yA= +2
y y b Zy bb Zy b
y b yb y
= + ++ +
=
++
3
2 22 6
3 2( )( )
; = 1
Q y.
( ).
() y = yc.
min E y QgA
y Dcc
cc= + = + =
2
22 21.157 m
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2-24
min MQgA
yAc
c= + =2
2.86 m3.
() min E y QgA
yD y b y
b yc cc
c c c
c
= + = + = ++
2
22 2 23 5
2 ; D b y y
b ycc c
c
= ++( )
2
() . = 4 m , y = yc. 1 () yc = 2.39 m, '
max ( )Q A gD A g E yc c c c= = =2 46.42 m3/s
-
- . 2007-2008
2-25
3.
Q
H z y Vg
z y QgA
= + + = + + 2 2
22 2 (3.1)
H , z , y , V , g . x
dHdx
dzdx
dydx
ddx
QgA
= + +
2
22 (3.2)
x. J J
dxdHJ E = (3.3)
dxdzJ o = . (3.4)
, , . ( .. , , ), . , . , . () () .
3.1.
(3.2) ( = 1)
++=
++=
dxdy
dydA
AgQ
dxdyJ
gAQ
dxd
dxdyJJ ooE 3
2
2
2 222
-
- . 2007-2008
2-26
( )223232 1111 FrdxdygDVdxdyTgAQdxdydydAgAQdxdyJJ Eo =
=
=
= ,
dA = Tdy Fr = V/(gD)1/2
dydx
J JFr
o E= 1 2 . (3.5) (JE = 0)
dydx
JFro= 1 2 0)1(
2 =+dxdzFr
dxdy . (3.6)
(3-6)
dz/dx Fr dy/dx y(x)
(i) >0 1 >0 (iii)
-
- . 2007-2008
2-27
3.
(Fr >1) dz/dx < 0.
(3.6) dy/dx < 0.
4.
(Fr >1) dz/dx >0.
(3.6) dy/dx > 0.
: b (. 3.3)
zEEHH +== 2121 . Q, 2 minE2, .
3
2
13 2
2
121 23
23minmax
gqE
gbQEEEz === .
z > maxz,
( 1) Q,
1 .
yc
dz
ycdz
yc dz
y1 y2
-
- . 2007-2008
2-28
3.2.
H z y qgy
= + +2
22 (3.7)
)(xbb = )(/)( xbQxqq == z =
x
dHdx
dydx
qgy
dydx
qgy
dqdx
= = + +0 02
3 2
dxdqb
dxdbq
dxdQbqQ +=== 0
dq/dx,
0)1( 22 =dxdb
byFr
dxdyFr (3.8)
bADgyq
ygbbQ
gAbQ
gDAQFr /;3
2
33
2
3
2
2
22 ===== .
(3.8)
db/dx Fr 1-Fr2 dy/dx y(x)
(i) >0 0 >0 (ii) >0 >1
-
- . 2007-2008
2-29
1.
(Fr 0.
(3.8) dy/dx > 0.
2.
(Fr 1) db/dx >0.
(3.8) dy/dx < 0.
4.
(Fr >1) db/dx < 0.
(3.8) dy/dx > 0.
b B
yc
b B
yc
b B
yc
b B
yc
-
- . 2007-2008
2-30
: (. 3.4).
21
22
2
21
1 22EE
gVy
gVy =+=+ .
Q, 2 , minE2. minb
E1 = min E2 = 32
2
23
Qb gmin
'
min /bQ
E g= 3
23
213 2 .
b < minb,
( 1) Q,
1 .
3.3. -
z 3.1. (1) (2) .
H H y Vg
z y Vg1 2 1
12
222
2 2= + = + + (3.9)
zEE += 21 . (3.10)
b B
yc
minb
y1 y2
-
- . 2007-2008
2-31
3.1 .
Q,
E y D y Qb gc
ccmin = + = =2
32
32
2
23 (3.11)
. 1>2 (3.9), , Q ( 3.1). maxz, 2 ( 3.10, 3.11)
max minz E E E Qb g
= = 1 2 12
23
32
. (3.12)
max z, (2) 2, , Q ( 1) (1).
y2 (2), (3.9) y2,
yQ
gb yz y
Qgb y1
2
212 2
2
2222 2
+ = + + . (3.13)
.
: maxz/y1 , Froude F1 , Q.
1
2
.. (EGL)(2)(1)
yc maxz
y1 y2
2min
y
yc
y2
1.5yc
2
y2 yc
z
-
- . 2007-2008
2-32
: min E = Ec max z = zc
3/21
21
11
21
1
1
11
1
1
5.12
15.12
5.1 FFyy
gyV
yy
yy
yE
yz ccc +=+== ,
3/21
1
Fyyc = (;).
yc/y1 1 0 Froude F1 0 1 (. ).
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0F1
c/
Y1
3.2 maxz Froude,
.
3.4. -
. (1) (2) ,
1 = 2 1 = 2 (3.14)
. (3.14)
yQ
gB yy
Qgb y1
2
212 2
2
2222 2
+ = + . (3.15)
b Q, 2. b , , Q (1).
E E Qb g1 2
2
23
32
= =minmin
(3.16)
-
- . 2007-2008
2-33
b QE gmin /
= 32
321
3 2 . (3.17)
b > bmin, (2) (3.15).
(1) , (. VT Chow2 . 17-3).
3.3 .
3.4
(Chow1).
3.1
, , b z L, .
2 Chow, VT, 1973. Open-channel hydraulics. McGraw-Hill.
b
q1=Q/B
q2=Q/b y
E min E2
q1q2
q1 < q2
-
- . 2007-2008
2-34
Q = 3.0 m3/s, B = 2.0 m y1 = 1.5 m :
() b 2 z
y2 = y2c = b.
() F . .
() y .
() (1) (2)
H H yV
gz y
Vg1 2 1
12
222
2 2= + = + +
'
z y Vg
yV
g= +
+
1
12
222
2 2 (1)
y1 = 1.50 m V1 = Q/(By1) = 3/(2 x 1.50) = 1.00 m/s.
V2 = Q/(by2) = Q/(bb) = Q/b2 (2)
b
B b
L
y1 y2
zF
(1) (3) (2)
-
- . 2007-2008
2-35
y Qgb
bc22
23= = (3)
'
b Qg
= =2
5 0.983 m.
V2 = Q/by2 = Q/b2 = 3.105 m. (4)
y1, y2, V1, V2 (1)
z = 0.079 m. () (1) (2)
[ ] )()( 12 VVQEdVVFFgFpF xxxx ==+++ o . Fgx = 0 ( ).
Fx = 0 ( )
( )byBygFpx 222121 = ( ) ,
( )F Fp Q V V y B y b Q V Vx= = = ( ) ( )2 1 12 22 2 112 11.098 kN () - , z/2 (B + b) / 2.
H y Vg
z y Vg
= + = + +1 12
332
2 2 2 = 1.512 m
Q = y3b3V3 = y3V3 (B + b) / 2.
V3 3
y y33
321512 0 206 0 + =. .
, , 1.50 m 0.983 m (3) .
y3 = 1.408 m.
-
- . 2007-2008
2-36
3.5. .
q . , . , ,
= yc + D/2 = 1.5 yc yc = (2/3)E
. :
1. . y = 0 (q = 0).
(1)
y = 0, q = 0
(2)
y < yc, q < qmax
(3)
y = yc, q = qmax
qmax
y=ycE
..
Eq
yyc
..
..
-
- . 2007-2008
2-37
2. . y < yc q < qmax.
3. ( ). y = yc q = qmax.
(3.11) q = Q/b
E E y D y Qb g
qgc
cc= = + = = =min 2
32
32
32
2
23
2
3 (3.18)
'
q g Emax/
=
23
3 2
(3.19)
, . , . , .
' , , () . . Hunter Rouse yb (brink depth) yb = 0.715 yc. yc 3yc - 4yc . y = yc.
3.5 .
y
E
yc yb
p/
pb/
Emin
-
- . 2007-2008
2-38
3.2 b. , .
h , . A (=0)
H y zV
gz y
Vgcc= + + = + +1 1
2 2
2 2 y V
gy
Dy
y yc
cc
c c1
12
2 2 232
+ = + = + =
12
1
2y
gV
-
- . 2007-2008
2-39
3.6. .
. , ( ) . .
3.6 .
x, ( ) ( ) ( 2.14)
21 MM = 0222
2
111
2
=
+
+ Ay
gAQ
AygAQ
. (3.20)
, .
3.1 b, Q, y1 < yc. y2 .
(3.20)
QgA
y AQgA
y A V byg
y b y V byg
y b y2
11 1
2
22 2
12
11
1 22
22
2
2 2+ = + + = + . (3.21)
Q V y V y V Vyy
= = =1 1 2 2 2 1 12
(3.22)
(3.21) (3.22) V1 Froude F1,
FVgy1
1
1
= (3.23)
y1
y2 V2V1(1)
(2)
-
- . 2007-2008
2-40
2 1 1 012 2
1
2
1
2
1
Fyy
yy
yy
+
= (3.24)
y1 y2,
[ ]18121 2
11
2 += Fyy . (3.25)
F1 > 2
yy
F F21
1 1212
2= >; . (3.26)
3.2 .
:
H y yy y
= ( )1 23
1 24. (3.27)
3.7 ( - 3 1978).
3 , , , , 1978. . , . . , .
-
- . 2007-2008
2-41
3.8 (4 2001).
4 , 2001. , . . , .
-
- . 2007-2008
2-42
4.
. ( ), . .
JJJ Eo == (4.1)
dxdHJ E = ( )
sin==dxdzJ o ( )
dx
yzdJ )( += ( )
2211 AVAVQ ==
VVVAA === 2121 . , () ,
0)( 12 ==++= VVQFFFF pxxgxx ; = 1
0=pxF , )( xPF ox = )()sin()( xAgxAgF xgx == .
..
..
(1) (2)
g
Fpx
Fpx
x
x
z y V
2/2g
V
gx
-
- . 2007-2008
2-43
oo JxgAxgAxP )(sin)( == . x
sin,/;sin)/( ==== ooo JPARgRJPAg , , . , ,
o fc V f V= =2 2
2 4 2.
Darcy-Weisbach
Eo JgV
RfJ ==
24
2
Vf
g RJ E= 8 (4.2)
R (R=A/P) . Darcy-Weisbach Moody, , D = 4R,
= V R4 kD
kR
s s=4
(4.3)
Cf
g= 8 (4.4)
Chzy
RJCV = . (4.5) , C , R m V m/s. H Chzy, , C . , (4.4) C , , C = C(f); f=f(ks/4R,). , . ks .
-
- . 2007-2008
2-44
ks (mm) ()
0.15
0.30 ,
0.50 ,
0.60 , ,
1.50 - gunite
2.40
3.00
4.20
6.00
4.1.
, , Darcy Weisbach ( Moody). , Manning.
1. Chzy
V C RJ= J =
R = (m)
C =
V = (m/s)
, (4.4) C , , . .
2. Kutter - Ganguilet
Chzy C
-
- . 2007-2008
2-45
C J n
JnR
o
o
=+ +
+ +
23 0 00155 1
1 23 0 00155
.
. (4.6)
Jo =
R =
n =
3. Bazin
Chzy
C
R
=+87
1 (4.7)
=
R =
4. Manning
Chzy C , ( R). , ks C , , . Manning (1885) Strickler , C C C=R1/6/n, n . Chzy
Vn
R J= 1 2 3 1 2/ / . (4.8)
( Manning n Manning, ) . Ven Te Chow n - Manning (. ). , R m, m/s. n, ( 1). , US (R ft V ft/s) 1 1.49. , .
-
- . 2007-2008
2-46
4.2
Manning. m , m/s , m3/s , .
()
byA = ybP 2+=
PAR /= byVAVQ ==
2/13/2
2/13/2
2/13/2
2111 J
ybby
nJ
PA
nJR
nbyQ
AQV
+=
====
'
5/3
3/22/1 )2(
1
+= ybJQn
by (4.9)
. , y, y1 (4.9). y = y1 , y1 (4.9) y2, ... y.
()
yZybA )( += 212 ZybP ++=
PAR /= AVQ =
Manning
2/1
3/2
2
2/13/2
2/13/2
12
)(111)(
JZybyZyb
nJ
PA
nJR
nyZybQ
AQV
+++=
==+==
y
b
b
y Z 1
-
- . 2007-2008
2-47
( ) 5/33/222/1 121 +++= ZybJQnZyby (4.10) .
: (4.10) =0, (4.9) .
()
[ ]2
)2/cos(1 dy =
8)sin( 2dA =
2/dP =
4sin1/ dPAR
==
Manning
2/13/22
22/13/2
2/8/)sin(1
8/)sin(1 J
dd
ndQJR
nAQV
====
,
'
22
sin)2/(
5/3
3/8
3/2
2/1
++
=dJ
Qn (4.11)
rad (). (4.11) .
[ ])2/cos(12
= dy . (4.12)
4.1. , y/D.
y, Manning
2/13/13
3/83/52/1
3/222/13/2
2)sin(
4sin
8)sin( J
ndJd
ndJR
nAQ
=
== .
(y = D)
y d
-
- . 2007-2008
2-48
2/13/10
3/82/1
3/222/13/2
244J
ndJd
ndJR
nAQ full =
== .
nQQ
full 2)sin( 3/5 =
2)2/cos(1 =
dy .
.
y/D 0.94, 7.5% .
0.00.10.20.30.40.50.60.70.80.91.0
0.0 0.2 0.4 0.6 0.8 1.0 1.2Q/Q_full
y/D
4.1 (Manning n = 0.015) :
() 0.01, 0.001 7 m3/s .
() .
b=2.5m
-
- . 2007-2008
2-49
() (4.9)
5/33/2
2/1 )2(1
+= ybJQn
by
y = 1 m ' , . Q = 7.0 m3/s Jo = 0.01
y b Z n Jo Q y1
1.000 2.50 0.00 0.015 0.0100 7.00 0.752
0.752 2.50 0.00 0.015 0.0100 7.00 0.717
0.717 2.50 0.00 0.015 0.0100 7.00 0.712
0.712 2.50 0.00 0.015 0.0100 7.00 0.712 y = 0.712 m.
0.001 y = 1.663 m.
() Froude . Fr > 1, . Fr < 1, . ( . yc > yo, . yc < yo, .)
Froude
FrVgD
=
V = Q/A D = A/T .
Q B S yo V D Fr /
(m3/s) (m) (m) (m2) (m) (m/s) (m)
7.00 2.50 0.0100 0.71 1.78 2.50 3.93 0.71 1.489
7.00 2.50 0.0010 1.66 4.16 2.50 1.68 1.66 0.417
4.2 (Manning n = 0.015) :
() 0.01, 0.001 7 m3/s .
() .
-
- . 2007-2008
2-50
() , Manning (4.10)
( ) 5/33/222/1 121 +++= ZybJQnZyby y = 1 m ' , .
Q = 7.0 m3/s Jo = 0.01
y b Z n Jo Q y1
1.000 2.50 1.00 0.015 0.0100 7.00 0.574
0.574 2.50 1.00 0.015 0.0100 7.00 0.590
0.590 2.50 1.00 0.015 0.0100 7.00 0.590
y = 0.590 m.
0.001 y = 1.135 m.
() Q = 7.0 m3/s Jo = 0.01 y = 0.590 m
= b+2Zy = 3.68m, A = (b+Zy)y = 1.82, D = A/T = 0.495m, V = Q/A = 3.85m/s
1745.1 >==gDVFr , .
, Q = 7.0 m3/s Jo = 0.001 y = 1.135 m
= b+2Zy = 4.77m, A = (b+Zy)y = 4.13, D = A/T = 1.16m, V = Q/A = 1.70m/s
1503.0
-
- . 2007-2008
2-51
1 Q d J n 2 yo
3.142 0.1 0.6 0.005 0.016 2.688 0.300
2.688 0.1 0.6 0.005 0.016 2.613 0.233
2.613 0.1 0.6 0.005 0.016 2.597 0.222
2.597 0.1 0.6 0.005 0.016 2.593 0.219
2.593 0.1 0.6 0.005 0.016 2.592 0.219
2.592 0.1 0.6 0.005 0.016 2.592 0.219
,
y = 0.22 m.
4.4 , (n-Manning=0.014) .
Q1 Q2 0.01 0.001. = 0.20.
() J = 0.01. . .
H yV
gK
Vg
y KV
gcc c
cc= + + = + +
2 2 2
2 21
2( ) = 2.00 m.
Fr VgD
Vgy
Vgy
c
c
c
c
c
c= = = =1 1
2
.
Vc
H yV
gK
Vg
y Ky
Ky
cc c
cc c= + + = + + = +
2 2
2 21
23
2( ) ( )
yc
b = 2.0m
H = 2m Fr < 1
Fr > 1
yc
-
- . 2007-2008
2-52
y HKc
= +2
3( )= 1.25 m.
H
( )y Qgb Q b gyc c= =2
23 3
1 2/= 8.754 m3/s.
(4.9)
yo = 0.98 < 1.25 m.
.
() J = 0.001. . .
H yV
gK
Vg
y KV
g= + + = + +
2 2 2
2 21
2( ) = 2.00 m.
Manning
2/13/21 JRn
V = ,
H y K R Jgn
y K
byb y
J
gn= + + = + + +
( ) ( )/
/
12
12
2
4 3
2
4 3
2 = 2.00 m.
y K
byb y
J
gny
y= + +
= +
2 1
22
2 0 3121
4 3
2
4 3
( ) .
/
/
y = 1.826 m.
Manning
V = 1.688 m/s
Q = 6.164 m3/s.
To
yc = 0.99 m < 1.826 m.
o .
-
- . 2007-2008
2-53
4.5 (Manning n = 0.016) , , :
() 0.01 0.0001 y=0.15 m, . .
() 0.05 m3/s 0.005 m3/s y=0.10 m J. .
() 0.05 m3/s 0.03 m3/s 0.01 0.0001.
= y + 6y = 7y
A = Ty/2 = 3.50 y2
P = y (21/2 + (36 + 1)1/2) = 7.50 y
R = A/P = 0.467 y
D = A/T = 0.50 y
() yo, J, n ( A, R) Manning
2/13/2 JRnAAVQ ==
Fr = 1,
Fr VgD
Qy
g yy Q
gc= = = =
350050
1350 050
22 5
.( . ) . .
/
.
.
1
1 16
-
- . 2007-2008
2-54
yo Jo A P R V Q yc
(m) (m) (m) (m) (m/s) (l/s) (m)
0.15 0.0100 0.079 1.125 0.070 1.062 83.60 0.163 Y
0.15 0.0010 0.079 1.125 0.070 0.336 26.44 0.103
0.15 0.0001 0.079 1.125 0.070 0.106 8.36 0.065
() Manning J, . .
J n VR
=2 2
4 3/
y A P R Q V Jo yc
(m) (m) (m) (m) (l/s) (m/s) (m)
0.1 0.035 0.75 0.047 5 0.143 0.0003 0.053
0.1 0.035 0.75 0.047 50 1.429 0.0311 0.133 Y
() , Manning y
y nQJ
= 350 0 467 2 3 1 2
3 8
. ( . ) / /
/
Q (l/s) n Jo y (m) Q (l/s) n Jo y (m) 50 0.016 0.0100 0.124 30 0.016 0.0100 0.102 50 0.016 0.0010 0.190 30 0.016 0.0010 0.157 50 0.016 0.0001 0.293 30 0.016 0.0001 0.242
4.3. . , . () , Manning , .
-
- . 2007-2008
2-55
, (French5, 1994) .
() n1, n2, .... n
() A1, A2, .... A
() P1, P2, .... P
() V1, V2, .... V
Ri = Ai /Pi
Vn
R Jn
AP
J11
12 3 1 2
1
1
1
2 31 21 1= =
/ //
/
2/13/2
2
2
2
2/13/22
22
11 JPA
nJR
nV
==
...........................................
Vn
R Jn
AP
JNN
NN
N
N
= =
1 12 3 1 22 3
1 2/ //
/
Ai = A, Pi = P, R = A/P V = Q/A, R V . ne. Manning
QJ
PAne
2/1
3/2
3/5
=
(1) Horton6 (1933), Einstein & Banks7 (1950)
V1 = V2 = .... = V = V
Manning (i)
5 French, RH, 1994. Open-channel hydraulics. McGraw-Hill
6 Horton, RE, 1933. Separate roughness coefficients for channel bottom and sides. Engineering News Record, vol. III, no. 22, Nov. 30, pp. 652-653.
7 Einstein, HA, & Banks, RB, 1950. Fluid resistance of composite roughness. Trans. Am. Geoph. Union, vol. 31, no. 4, pp. 603-610.
2 i
N
n1 P1
n2 P2
ni Pi
nN PN
-
- . 2007-2008
2-56
iiiii
i PnJVP
JnV
A 2/32/3
2/1
2/3
2/1
=
=
= (i)
==N
iie PnJVPn
JVA
1
2/32/3
2/12/3
2/3
2/1
ne
nPn
Pei i
N
=
3 21
2 3
2 3
//
/ (4.13)
(2)
F = F1 + F2 + .... + F
nPn
Pei i
N
= 2
1 (4.14)
(3)
Q = Q1 + Q2 + .... + Q
n PRPR
n
ei i
i
N= 5 3
5 3
1
/
/ (4.15)
(4) (US Army Corps of Engineers, Cox8, 1973), (.. )
nA n
Aei i
N
=
1 (4.16)
(5) Colebatch
8 Cox, RG, 1973. Effective hydraulic roughness for channels having bed roughness different from bank
roughness. Miscellaneous paper H-73-2, U.S. Army Engineers Waterways Experiment Station, Viksburg, MS.
-
- . 2007-2008
2-57
nA n
Aei i
N
=
3 21
2 3
2 3
//
/ (4.17)
4.4.
. . , , , , , . .
4.6 . .
(1) ' () .
R A abcdefghP abcdefgh
= ( )( )
, Vn
R J= 1 2 3 1 2/ / Q = A V.
(2) icdefj, ic fj ( ), abci jfgh ic fj .
R A icdefjP icdefj
= ( )( )
, R A abciP abc1
= ( )( )
, R A jfghP fgh2
= ( )( )
Vn
R J= 1 2 3 1 2/ / , Vn
R J1 12 3 1 21= / / , V
nR J2 2
2 3 1 21= / /
a
b c
d e
f
gh
i j k
( )
( )
-
- . 2007-2008
2-58
Q = A V + 1V1 + A2V2.
(3) (2) ic fj ( ic fj ).
R A icdefjP cdef
= ( )( )
, R A abciP abc1
= ( )( )
, R A jfghP fgh2
= ( )( )
Vn
R J= 1 2 3 1 2/ / , Vn
R J1 12 3 1 21= / / , V
nR J2 2
2 3 1 21= / /
2211 VAVAAVQ ++= . (4) abck, kcdefk kfgh kc kf.
R A abcdefghP abcdefgho
o
o
= ( )( )
RA abck P abc A cdef P cdef A kfgh P fgh
A abcdefgho o= + +
15 3
15 3
25 3 3 2/ / / /( ) / ( ) ( ) / ( ) ( ) / ( )
( )
Qn
A R Jo o= 1 2 3 1 2/ / .
4.7 100 m3/s, J=0.003 n = 0.02. .
(y = 2.0 m, R = 10/9 m)
Qn
AR J m s= = =1 10 02
10 10 9 0 003 29 382 3 1 2 2 3 1 2 2/ / / /.
( )( / ) ( . ) . /
2.0 m.
y,
d
a
b c
e
f g
hi j
5 m 5 m 5 m
2 m y
-
- . 2007-2008
2-59
Qn
A R AR J= +1 2 1 12 3 2 3 1 2( )/ / /
A1 = 5(y-2), P1 = (y-2) + 5, R1 = A1/P1, A = 5y, P = 2+2+5 = 9m, R = A/P.
( ) y = 3.236 m.
65.50 m3/s 17.25 m3/s 4.05 m/s 2.79 m/s.
-
- . 2007-2008
2-60
5.
(, , ) . (1) , (2) , (3) , (4) , (5) , PVC, . :
( n - Manning). ( ) .
() .. . . 0.60 m/s, Shields.
. , .
. . . 1:Z, Z .
1/5 1/10 , 1/2 1 1/4 1 3/2 2 - 3
-
- . 2007-2008
2-61
5.1
F . . f . , .
US Bureau of Reclamation (. ), F f Q.
5.2
. Manning
Q AVn
AR Jn
AP
J= = =1 12 3 1 25/3
2 31 2/ /
// (5.1)
R Q Q . , ( Q) .
, . Q,
F f
-
- . 2007-2008
2-62
, ( .
= (x1, x2, ... xn) (5.2)
P = P (x1, x2, ... xn) (5.3)
x1, x2, ... xn ( ).
, dP = 0,
dP Px
dx Px
dx Px
dxn
n= + + + =
1 1 2 2 0... . (5.4)
= o = ( ), dA = 0,
dA Ax
dx Ax
dx Ax
dxn
n= + + + =
1 1 2 2 0... . (5.5)
(Lagrange )
0
...222
111
=
+
+
+
=
nnn
dxxA
xP
dxxA
xPdx
xA
xPdAdP
(5.6)
dxi0, n+1 n+1 x1, x2, ... xn .
011
=xA
xP
022
=xA
xP
(5.7)
.....
Px
Axn n
= 0
0= oAA , .. /xn 0, . Lagrange , x1, x2, ... xn.
-
- . 2007-2008
2-63
5.1 .
yzybAA o )( +== 212 zybP ++=
b P A( ) = 0 1 - y = 0 = 1/y (1)
0)( = APy
2 (1 + z2)1/2 - (b + 2zy) = 0 (2) z P A( ) = 0 2yz / (1 + z
2)1/2 - y2 = 0 (3)
(2) (3) (1)
z = 31/2 /3 (= tan 30o)
b = 2y/31/2
E 23)( yyzybAA o =+==
yzybP 3212 2 =++= 2// yPAR ==
.
5.3
:
(1) , n Jo.
(2) Mannning
2/13/2
oJnQAR = ; Q =
(3) A = A (y), R = R (y) b, z .
(4) . y.
(5) ..
y=0.50A1/2 (y ft, A ft2 1ft = 0.3048m). b/y=4-z, .
Y
b
z1
-
- . 2007-2008
2-64
(6) .
(7) F f USBR (1ft = 0.3048 m, 1m3 = 35.31ft3).
5.2 0.01, Manning 0.016 4m3/s. 6m/s.
50% . ,
y = b/2
= by = b(0.50b) = 0.50b2
P = b + 2y = 2b
R = A/P = 0.25b
8/3
2/13/22/13/2
22/13/2
)25.0)(50.0(1)25.0(50.0
===
JQnBJb
nbJR
nAQ
b = 1.55 m.
V = Q/A = 4/(0.5*1.552) = 3.44m/s < 6m/s = 3.10m.
b = 1.50m. ( y = 0.80m, V = 3.32m/s < 6m/s P = 3.10m ).
5.4
, .
() . . .
() ( ) .
PARgRJ oo /; === (5.8)
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Fortier Scobey9 (1926) .
5.5 .
n, Z maxV.
(1) n J
2/3
2/1
=AJnQR (5.9)
(2) Q = AV A = Q/ maxV (3) R A y, b.
(4) , .
9 Fortier, S & Scobey, FC, 1926. Permissible canal velocities. Trans. ASCE, vol. 89, pp. 940-984.
-
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6. - ()
6.1
: (i) , (ii) (iii) .
( ), : (i) . () . (ii) JE , ( ) Manning
J n VRE
=2 2
4 3/ . (6.1)
, (i) J y = t, (ii) ( ) (iii) n ' .
H z y Vg
= + +2
2 (6.2)
H
z
y
V
g .
x,
dHdx
dzdx
dydx
ddx
Vg
= + +
2
2. (6.3)
-
- . 2007-2008
2-67
= 1 . JE Jo
dHdx
J E= dzdx Jo=
(. 3)
dydx
J JQ BgA
o E= 1
2
3
(6.4)
Q.
y , :
dy/dx = 0 J = J ( )
dy/dx > 0
dy/dx < 0 .
6.2
, .
ByB =)( = , (6.5) ByyA =)( , (6.6) yByP 2)( += (6.7)
yBByyByAyR
2)(/)()( +== . (6.8)
( > 5y), R(y) y . Chezy C
3222
22
2
2
2
)( yCBQ
RCByQ
RCVJ E === . (6.9)
oooo JyCRJCV ==
3222
2/12/3
ooooooooo yCB
QJJBCyJyCByVAQ ==== . (6.10)
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2-68
3
=
yyJJ ooE (6.11)
,
32322
2 11 ccc
c BgyQBgyQ
gyVFr ==== (6.12)
3
2
gABQ
3
3
3
3
2
)(
==
yy
BygBgy
gABQ cc . (6.13)
dydx
J
yy
yy
o
o
c
=
1
1
3
3 . (6.14)
: Manning, 10/3 3, .
6.3 ()
(Jo = -dz/dx > 0), (Jo = -dz/dx < 0) (Jo = -dz/dx = 0) .
(Jo > 0) yo. Jc, , (yo = yc). (Jo > 0) :
Jo < Jc, yo > yc . Jo = Jc, yo = yc . Jo > Jc, yo < yc .
() ( ) .
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0 < Jo < Jc ( MILD = )
Jo = Jc C ( CRITICAL = )
Jo > Jc S ( SUPERCRITICAL = )
Jo = 0 ( HORIZONTAL = )
Jo < 0 A ( ADVERSE = )
, , , . , . .
6.1 .
:
1
2
3
() . 2, (Jo < Jc) 2, (yc < y < yo).
yc
1
2
3
yo
Jo
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6.4 ()
() () ( Jo 0, (6.14) dy/dx 0 y yo , .
, y yc, (6.14) dy/dx , .
(6.9) J V
C yQ
C B yE= =
2
2
2
2 2 3
1 1 (6.15)
. ( ) .
dy/dx > 0 ( ) (12.9) .
dy/dx < 0 ( ) (12.9) .
6.5 .
1. 0 < Jo < Jc (yo > yc): .
yc < yo < y yc < y < yo y < yc < yo
dy / dx > 0 dy / dx < 0 dy / dx > 0
1 2 3
yo y
M1
yoy
M2
yoy
M3
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2. Jo = Jc (yo = yc), 2 : C.
yc = yo < y yc = yo = y y < yc = yo
dy / dx = Jo > 0 dy / dx = 0 dy / dx = Jo > 0
C1 C2 C3
3. 0 < Jc < Jo (yo < yc), S.
yo < yc < y yo < y < yc y < yo < yc
dy / dx > 0 dy / dx < 0 dy / dx > 0
S1 S2 S3
4. Jo = 0 (yo 1): H. (6.4) Jo = 0, ..
2/1
3
2
2
3
2 ;11
=
==
gABQFr
FrJ
gABQ
Jdxdy EE . (6.16)
Jo = JE = 0, dy / dx = 0 y = . JE 0 y > yc ( ) Fr < 1 dy / dx < 0
.
JE 0 y < yc ( ) Fr > 1 dy / dx > 0 .
.
ycy
S2
ycy
S3
yc y
S1
yo=y
C1
yo=y
C3
yo=y
C2
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yc < y < y < yc dy / dx < 0 dy / dx > 0
2 3
5. Jo < 0 ( yo 1): .
(6.4) Jo < 0, (JE > 0)
2/1
3
2
2
3
2 ;11
=
==
gABQFr
FrJJ
gABQ
JJdxdy EoEo . (6.17)
y > yc ( ) Fr < 1 dy / dx < 0 .
y < yc ( ) Fr > 1 dy / dx > 0 .
.
yc < y y < yc
dy / dx < 0 dy / dx > 0
2 3
: (1) (3) , (2) , .
yc
2
yc
3
yc
y =
yc
2
yc
3
yc
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6.6 .
. . :
1. (J1 < Jc).
1. J2 < J1 < Jc. y2 > y1. , y2 y1 1. , y1 (y < y2) y2 , 2 , 2 .
2. J1 < J2 < Jc. y1 > y2. , y2 y1 2. , yo1 (y > y2) y2 , 1 , 1 .
3. J1 < J2 = Jc. y1 > y2 = yc. , yc y1 2.
4. J1 < Jc < J2. y1 > yc > y2 . , y1 y2 : 2 S2 , .
5. J1 < Jc . y > yc. , , y = yc, y 2. yc < < yo, y yo 2. > yo, y yo 1.
.
: , , . , .. , n=0.025 n=0.014 (. 6.2).
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J2 < J1 < Jc
yc < y1 < y2
1
J1 < J2 < Jc
yc < y2 < y1
2
J1 < J2 = Jc
yc = y2 < y1
2
J1 < Jc < J2
y2 < yc < y1
2
S2
J < Jc
E > yo 1
E yo 2
yc y1 y2
M1 y = yo2
yc y1
y2
M2 y = y2
M2
yc y1
y2=yc
y = y2 = yc
yc y1
y2
S2
M2
yc yo Hyc
Hyo
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2. (J1 > Jc). 1. J1 > Jc > J2
y1 < yc y2 > yc. , y1 y2 . y1 y2 y1 y2 .
() y1 = y2, .
() y1 < y2, , y1 y2 S1.
() y1 > y2, , y1 y2 3.
2. J1 > J2 = Jc
y1 < y2 = yc. , y1 () yc C3.
3. J1 > J2 > Jc
y1 < y2 < yc. , y1 y2 S3.
4. J1 < J2 > Jc
yc > y1 > y2. , y1 y2 S2.
5. J > Jc
y < yc. , , y = y. y, S1. y < < yo, .
. VT Chow 1.
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2-76
J2 < Jc < J1
y1>y2 ( 3)
y1 Jc
E > y S1 E yo < < y
< yo
y = yc y1 yc
y2=ycC3
y = y1
y = y2 y1 yc
y2S3
y = y1
y1 yc
y2
S2 y = y1
y yc H
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6.1 , (n-Manning=0.025) Q = 5 m3/s. :
(1) .
(2) .
() . .
Q Z n b Jo yc V yo
5.00 1.00 0.025 2.00 0.0010 0.75 1.08 1.37
5.00 1.00 0.025 2.00 0.0200 0.75 3.19 0.60
5.00 1.00 0.025 2.00 0.0050 0.75 1.94 0.89
()
J1 = 0.001 yo > yc (1.37 > 0.75) J2 = 0.02 yo < yc (0.60 < 0.75) S J3 = 0.005 yo > yc (0.89 > 0.75) E:
J1 J2 ( ) y = yc = 0.75 m. 1 2.
2 S2 () (0.60 m).
2 3 , .
J1=0.001
J2=0.02
yc
J3=0.005
M2
S2y1o
y2o y3o
M3
.
y3
b=2m
1 1
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2-78
. y2 ( 2) y2 = 0.93 m > y3 = 0.89 m, . , , 3. y3 = 0.89 m.
y 1,2 B(m) Z T (m) y_bar Q(m3/s) A (m2) V (m/s) M
y2 = 0.60 2 1 3.21 0.2780 5.00 1.57 3.19 2.061
y2= 0.93 2 1 3.85 0.4137 5.00 2.71 1.85 2.061
y3 = 0.89 2 1 3.78 0.3993 5.00 2.57 1.94 2.018
.
6.2 b =3 m, , Jo = 0.005. (n = 0.030), , (n = 0.012). , ( n = 0.025).
Q = 15 m3/s.
() n . .
(n = 0.030) (n = 0.012) (n = 0.025)
M2
S2.
S1
y3
y1
y2
yc y2
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2-79
Q B Jo n yc V yo 15.00 3.00 0.00 0.005 0.030 1.366 2.20 2.27 15.00 3.00 0.00 0.005 0.012 1.366 4.40 1.14 15.00 3.00 0.00 0.005 0.025 1.366 2.54 1.97
()
1 (n=0.030) yo > yc (2.27 > 1.37) 2 (n=0.012) yo < yc (1.14 < 1.37) S 3 (n=0.025) yo > yc (1.97 > 1.37) E:
n1 n2 ( ) y=yc=1.37m. 1 2.
2 S2 () (1.14m).
2 3 , . . (3.25) y2 ( 2 ) y2 = 1.626m < y3 = 1.97m. , , ( S1). , 3 y3 = 1.97m.
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2-80
7.
(Q = ) . (1) , (2) (3) , ' . , .
. , Manning , .
H z y Vg
= + +2
2 (7.1)
H , z , y , V , g x
dHdx
dzdx
dydx
ddx
Vg
= + +
2
2 (7.2)
Hx
zx
yx x
Vg
= + +
2
2. (7.3)
y1
y2
V12/2g V22/2g
..
z2 z1
z
x
x
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2-81
EJxH = , oJxz = 12 yyy =
. ,
+
=g
Vg
Vxx
yyJJ Eo 221 21
1
22
212 . (7.4)
x 1 2. 1 2 x
x E EJ J
EJ Jo E o E
= = 2 1 (7.5)
g
VyE iiii 2
2
+= . (7.6)
Manning
3/422
i
iEi R
VnJ = , (7.7)
JJ J
EE E= +1 2
2. (7.8)
(7.5) , y1 y2 x. .
7.1 1. (Direct step method).
. , , . (.. 1). ( ). , , . y x y. .
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H ( Chow) .
1 2 3 4 5 6 7 8 9 10 11 12 13 y A R R4/3 V V2/2g E JE (JE1+JE2)/2 Jo-J x x 2.00 4.00 0.67 0.58 1.00 0.05097 2.05 0.00056 0.00
-0.04735 0.00057 0.00043 -111.19
1.95 3.90 0.66 0.58 1.03 0.05362 2.00 0.00059 -111.19
1 , Y 2 , A 3 , R 4 R4/3 5 , V = Q/A 6
7 (= 1 + 6 ) 8 , 9 Manning 10 , JE 11 Jo - JE
12 (= 8 / 11 ) x = / (Jo - JE ) 13 , x
7.1 L = 2 Km = 2 m Q = 4 m3/s Jo = 0.001. 0.50 m 2.00 m . . n = 0.018.
: yc = 0.74m
: y = 1.58m
: V = 1.27m/s
E 1 (yo = 1.57m < 2m) 2 (yo = 1.57m > 0.50m).
: =2.00 m, 1.
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H , 2.00 m. (backwater curve), , 0.05m . , , yo = 1.57m.
1 2 3 4 5 6 7 8 9 10 11 12 13
y A R R4/3 V V2/2g E JE (JE1+JE2)/2 Jo-J x x 2.00 4.00 0.67 0.58 1.00 0.05097 2.05 0.00056 0.00
-0.04735 0.00057 0.00043 -111.19
1.95 3.90 0.66 0.58 1.03 0.05362 2.00 0.00059 -111.19
-0.04714 0.00061 0.00039 -121.31
1.90 3.80 0.66 0.57 1.05 0.05647 1.96 0.00063 -232.50
-0.04691 0.00065 0.00035 -134.91
1.85 3.70 0.65 0.56 1.08 0.05957 1.91 0.00067 -367.41
-0.04664 0.00070 0.00030 -154.12
1.80 3.60 0.64 0.55 1.11 0.06292 1.86 0.00072 -521.53
-0.04635 0.00075 0.00025 -183.25
1.75 3.50 0.64 0.55 1.14 0.06657 1.82 0.00077 -704.77
-0.04603 0.00080 0.00020 -232.53
1.70 3.40 0.63 0.54 1.18 0.07054 1.77 0.00083 -937.30
-0.04566 0.00086 0.00014 -333.68
1.65 3.30 0.62 0.53 1.21 0.07488 1.72 0.00090 -1270.98
-0.04525 0.00093 0.00007 -658.16
1.60 3.20 0.62 0.52 1.25 0.07964 1.68 0.00097 -1929.14
-0.01797 0.00098 0.00002 -1039.51
1.58 3.16 0.61 0.52 1.27 0.08167 1.66 0.00100 -2968.65
Jo = 0.001 3000m.
: =0.50 m, 2. , 0.50m
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1 2 3 4 5 6 7 8 9 10 11 12 13
y A R R4/3 V V2/2g E JE (JE1+JE2)/2 Jo-J x x
0.74 1.48 0.43 0.32 2.70 0.3723 1.11 0.00740 0.00
0.01663 0.00631 -0.00531 -3.13
0.84 1.68 0.46 0.35 2.38 0.28894 1.13 0.00523 -3.13
0.04179 0.00454 -0.00354 -11.81
0.94 1.88 0.48 0.38 2.13 0.23073 1.17 0.00385 -14.94
0.05776 0.00340 -0.00240 -24.09
1.04 2.08 0.51 0.41 1.92 0.18849 1.23 0.00294 -39.03
0.06838 0.00263 -0.00163 -42.06
1.14 2.28 0.53 0.43 1.75 0.15687 1.30 0.00231 -81.09
0.07572 0.00208 -0.00108 -69.99
1.24 2.48 0.55 0.45 1.61 0.13259 1.37 0.00185 -151.08
0.08095 0.00169 -0.00069 -117.99
1.34 2.68 0.57 0.48 1.49 0.11354 1.45 0.00152 -269.07
0.08478 0.00139 -0.00039 -217.29
1.44 2.88 0.59 0.50 1.39 0.09832 1.54 0.00126 -486.35
0.08765 0.00116 -0.00016 -535.30
1.54 3.08 0.61 0.51 1.30 0.08596 1.63 0.00106 -1021.65
0.03570 0.00103 -0.00003 -1130.411.58 3.16 0.61 0.52 1.27 0.08167 1.66 0.00100 -2152.06
Jo = 0.001 2200m. 2.00 0.50m . 2970 m 2150 m 2.00 m 0.50 m .
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
-3000.00
-2750.00
-2500.00
-2250.00
-2000.00
-1750.00
-1500.00
-1250.00
-1000.00
-750.00
-500.00
-250.00
0.00
()
z, z
+y (
)
H = 2.00m
H = 0.50m
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2-85
7.2 2. (Standard step method).
, . . .
1 2 x ,
HHH += 21 (7.9)
Hg
Vyzg
Vyz +++=++22
22
222
21
111 . (7.10)
Z = z + y (.. ). H 1 2 ( ),
ef hhH += (7.11)
( ) 221 EEEf JJxJh +== . (7.12)
"" 1 2,
y1
y2
hehf
2V22/2g
1V12/2g
xz1
z2
Z1 = z1+y1 Z2 = z2+y2
Jox
.. (EGL)
z = 0
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( )gVkhe 2/2= k ()
k = 0 0.1 k 0.2 0.3 k = 0.5 . n Manning he = 0.
ef hhgV
Zg
VZ +++=+
22
22
22
21
11 . (8-13)
. Z y, A, P, R, V . Manning
J n VRE
=2 2
4 3/ . , . , .
H .
1
2 Z
3 , Y
4 , A
5
6 V = Q/A
7
8 (= 2 + 7)
9 , R = /
10 JE Manning
11 , JE
12 x
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2-87
13 (= 11 x 12)
14
15
(= 1* +13+14)
. 7 , Z 2.
7.2 Z = 2 Q=11.33 m3/s. b = 6.10 m, Jo=0.0016 n = 0.025. +184.40m, 725 m . 1.52 m .
= 1.10 he = 0 ( ).
yc = 0.68 m yo = 1.02 m < 1.52 m . 1. , . . 8 15 0.001m.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Z y A P V V2/2g H R J (J1+J2)/2 x hf he H
(m) (m) (m2) (m) (m/s) (m) (m) (m) (m) (m) (m) (m)
0 184.40 1.52 13.89 12.90 0.82 0.04 184.44 1.08 0.00038 184.44
47 184.42 1.46 13.21 12.64 0.86 0.04 184.46 1.04 0.00043 0.00041 47 0.02 0.00 184.46
97 184.44 1.40 12.49 12.37 0.91 0.05 184.48 1.01 0.00051 0.00047 50 0.02 0.00 184.48
150 184.46 1.34 11.78 12.10 0.96 0.05 184.51 0.97 0.0006 0.00055 53 0.03 0.00 184.51
207 184.49 1.28 11.09 11.83 1.02 0.06 184.55 0.94 0.00071 0.00065 57 0.04 0.00 184.55
272 184.54 1.22 10.42 11.56 1.09 0.07 184.60 0.90 0.00085 0.00078 65 0.05 0.00 184.60
349 184.60 1.16 9.75 11.28 1.16 0.08 184.67 0.86 0.00103 0.00094 77 0.07 0.00 184.67
* 1 = . 1 15 8.
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397 184.64 1.13 9.43 11.14 1.20 0.08 184.72 0.85 0.00113 0.00108 48 0.05 0.00 184.73
457 184.71 1.10 9.10 11.01 1.24 0.09 184.80 0.83 0.00125 0.00119 60 0.07 0.00 184.80
495 184.76 1.08 8.95 10.94 1.27 0.09 184.84 0.82 0.00131 0.00128 38 0.05 0.00 184.84
542 184.82 1.07 8.79 10.88 1.29 0.09 184.91 0.81 0.00138 0.00134 47 0.06 0.00 184.91
579 184.87 1.06 8.70 10.83 1.30 0.10 184.96 0.80 0.00142 0.0014 37 0.05 0.00 184.96
625 184.93 1.05 8.60 10.79 1.32 0.10 185.03 0.80 0.00147 0.00144 46 0.07 0.00 185.03
667 184.99 1.04 8.54 10.76 1.33 0.10 185.09 0.79 0.0015 0.00148 42 0.06 0.00 185.09
725 185.08 1.04 8.47 10.74 1.34 0.10 185.18 0.79 0.00153 0.00152 57 0.09 0.00 185.18
725 m 1.04 m.
0+725 .
182.50
183.00
183.50
184.00
184.50
185.00
185.50
0 100 200 300 400 500 600 700 800
X ()
z, Z
=z+y
(
)
Z
z (m)
1 VT Chow (1979) .
: (.. MS Excel), background, . .
-
- . 2007-2008
2-89
1 ( VT Chow, Open Channel Hydraulics, McGraw-Hill)
1. VT CHOW ( )
-
- . 2007-2008
2-90
2. VT CHOW
C.D.L. = Critical depth line =
N.D.L. = Normal depth line =
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- . 2007-2008
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3. ( )
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4. ( ) -
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2-93
5.
6.
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- . 2007-2008
2-94
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- . 2007-2008
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2
1
() = 2.00 m, Q = 5, 10, 15 m3/s.
() z , 1.5, 2.25 3 m .
() b , 1.5, 2.25 3 m .
(a)
E y QgA
= +2
22 M
QgA
yAQgA
y A= + = +2 2
2; = b y
Q y.
.
0.00
2.00
4.00
6.00
8.00
10.00
0.00 2.00 4.00 6.00 8.00 10.00
()
(
)
Q=5
Series2
Series3
.
0.00
2.00
4.00
6.00
8.00
0.00 10.00 20.00 30.00 40.00 50.00
()
(
)
Q=5
Q=10
Q=15
z
b B
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2-96
()
max z = E1 - min E.
.
Q y1 A1 V1=Q/By1 E1 min E max z
5.00 1.50 3.00 1.67 1.64 1.29 0.35
10.00 2.25 4.50 2.22 2.50 2.05 0.45
15.00 3.00 6.00 2.50 3.32 2.68 0.63
11 ByA = , 11 / AQV = , gVyE 2/2111 += 3/1
2
2
2350.1min
==
gbQyE c
()
b QE gmin /
= 32
321
3 2
.
Q y1 A1 V1=Q/By1 E1 bmin
5.00 1.50 3.00 1.67 1.64 1.39
10.00 2.25 4.50 2.22 2.50 1.48
15.00 3.00 6.00 2.50 3.32 1.46
2
(Manning n = 0.016) :
() 0.01 0.0001 y = 0.15 m, . .
() 0.05 m3/s 0.005 m3/s y = 0.10 m J. .
() 0.05 m3/s 0.03 m3/s 0.01 0.0001 .
14 4
1
b=0.35m
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2-97
T = b + 2Zy = b + 0.50 y
A = (b + 0.25 y) y
P = b + 2y (1 + 0.252)1/2 = b + 2.06 y
R = A/P
D = A/T
() yo, J, n ( A, R) Manning
2/13/21= JRAn
Q
(Fr = 1)
3
2
)2()(
1c
cc Zybg
QZyb
y ++= .
.
Jo Z A P R V Q yc
() () () () () (/) (/) ()
0.15 0.0100 0.250 0.35 0.058 0.659 0.088 1.238 71.97 0.157 Y
0.15 0.0010 0.250 0.35 0.058 0.659 0.088 0.392 22.76 0.074
0.15 0.0001 0.250 0.35 0.058 0.659 0.088 0.124 7.20 0.035
() Manning J, . .
J n VR
=2 2
4 3/
A P R Q V Jo yc
() () () () () (/) (/) ()
0.1 0.35 0.038 0.556 0.067 5 0.13 0.0002 0.027
0.1 0.35 0.038 0.556 0.067 50 1.33 0.0166 0.124 Y
()
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- . 2007-2008
2-98
( ) 5/33/222/1 121 +++= zybJQnzyby Q Z n Jo y
(/) () ()
50 0.25 0.016 0.35 0.0100 0.118
50 0.25 0.016 0.35 0.0010 0.251
50 0.25 0.016 0.35 0.0001 0.525
30 0.25 0.016 0.35 0.0100 0.085
30 0.25 0.016 0.35 0.0010 0.179
30 0.25 0.016 0.35 0.0001 0.379
3
.
() (2) D = 2.00 m .
() (3) , d = 0.50 m .
() (2) (3).
: Q = 140 m3/s, B = 14 m y1 = 2.17 m. (1), (2) (3) . .
() (1)
D
y3
y2
y1
(1) (2) (3)
d
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- . 2007-2008
2-99
( ) === 17.214/140/ 11 byQV 4.61 m/s. (1) (2)
( )D E E D yV
gy V
gy Q
g by+ = + + = + = +1 2 1 1
2
222
2
2
222 2 2
.
(2)
y2 = 1.11 m.
() (2) (3) F F F Q V Vpx x gx+ + = ( )3 2
Fx 0, Fgx 0
2)(
2
23
22
32dybgbygFFF pppx
+== V2=Q/by2, V3=Q/by3
=+=
23
22
322 11
2)(
2 AAQ
dybgbygFpx
=+
23
223
22 11
2)(
2 AAgQdybby
=
23
22
33
32
211
222 AAgQbddA
yAyA
A y A y A d bd Qg A A2 2 3 3 3
2 2
3 221 1 =
; y
y y y22
33
2 2= =,
A y QgA
A y QgA
A d bd2 22
23 3
2
33
2
2+ = + + + M M A d bd2 3 3
2
2= + +
'
y3 = 3.15 m.
()
H H yV
gd y
Vg2 3 2
22
332
2 2 = +
+ +
V2 = Q/by2 = 9.01 m/s V3 = Q/by3 = 3.17 m/s
2 - 3 = 5.25 - 4.16 = 1.08 m.
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- . 2007-2008
2-100
4
(d=4m):
() 2m .
() () ;
() ;
() () .
: . Manning n = 0.016.
() ,
1===c
c
gDV
gDVFr
yc=2m.
Ac = d2/8 = 2 = 6.28m2
c = 4m
Dc = Ac/Tc=0.5 = 1.57m
== cc gDV 3.93m == cc AVQ 24.66m3/s. () () Manning
PARJRn
V /;1= 2/13/2 = ,
Ac = d2/8 = 2 = 6.28m2
Pc = d/2 = 2 = 6.28m
Rc = 1m
=2
3/2
=RnVJ c 0.0039.
() . y,
A = d2/8 + 6y = 2 = 6.28 + 6y
2 m
b = 6 m
d = 4 m
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- . 2007-2008
2-101
P = d/2 + 2y + 2 = 6.28 + 2(y+1)
)1(228.6628.6++
+=y
yR
Q1 = 3Q = 3x24.66 = 74m3/s, Manning
2/13/2
2/13/21 )0039.0()1(228.6
628.6)628.6(016.01741=
++++=
yyyJRA
nQ ,
y = 1.51m,
y1 = 2 + 1.51 = 3.51m.
()
A = d2/8 + 6y = 2 = 6.28 + 6 x 1.51 = 15.37m2
= 6m
D = A/T = 2.56m
V = Q1/A = 4.82m/s
196.056.282.4
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- . 2007-2008
2-102
gDVFr = =0.62 < 1
.
() 3Q=58.94m3/s.
. , b+H. Q1 Q2 ,
( )3/2223/2112/1213 RARAnJQQQ +=+= y
1=y, P1=B+y, R1=By/(B+y)
2=8+by, P2=7.83+y, R2=(8+by)/(7.83+y).
y = 1.27m, 3.27m.
6
4:1 1:1 , Manning n = 0.016 0.8m3/s.
() .
() .
() .
.
J2=0.02
J3=0.002
J1=0.0015
4 1 1
1
1m
2 S2
3 1
y=y3
y=0.39m
y=0.35m
y=yc
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2-103
= y + 4y = 5y R = A/P = 0.451 y
A = Ty/2 = 2.50 y2 D = A/T = 0.50 y
P = y (21/2 + (16 + 1)1/2) = 5.537 y
() Fr = 1,
=
====
5/22
50.050.21
)50.0(50.2
gQy
ygy
Q
gDVFr c 0.461m.
() Manning
8/3
3/5
3/2
2/12/1
3/2222/13/2
)5.2()537.5(
537.55.25.2
=
===
JnQyJ
yy
nyQJR
nAAVQ .
J1 = 0.0015 y1 = 0.571m > yc ( )
J2 = 0.02 y2 = 0.351m < yc ( )
J3 = 0.002 y3 = 0.541m > yc ( )
() 1 2 , 2 S2.
2 3 , . y2 y3.
( )2222 5.235.2 yygyQAygAQM +=+= . 2 = 2 y2 = 0.593m > y3 = 0.541m, . 3 = 3 y3 = 0.39m. 3 0.351m 0.39m .
1m > y3 = 0.541m, 3 1.
.
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2-104
7
b=2m . Manning n=0.016, 10m3/s . L1 L2:
(1) .
(2) J3 = 0 y = 2.00m, .
(3) L3 .
(1) Fr = 1,
=
=
3/1
2
2
gbQyc 1.37m.
(2) Manning
== 2/13/2 JRnAAVQ .
5/33/2
2/1 )2(1
+= ybJQn
by
J1 = 0.003 y1 = 1.93m > yc ( )
J2 = 0.02 y2 = 0.92m < yc ( )
J3 = 0.00 y3 = > yc ( ) () 1 2 , 2 S2.
J2=0.02 J3=0
J1=0.003 y3=2.0m
L3
2
S2
S1 H2
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- . 2007-2008
2-105
2 3 , . y2 y3.
( )byygbyQ
AygAQ
M2
22
+=+= .
2 = 2 y2 = 1.943m < y3 = 2.0m, . 2 S1, 2 .
.
(3) , y=yc y = 2m . Q=10m3/s, b=2m, Jo=0, n=0.016:
y A P R V E JE JE(bar) Jo-JE x
1.37 2.7 4.73 0.58 3.66 2.049 0.00714 0.00
1.40 2.8 4.80 0.58 3.57 2.050 0.001 0.00670 0.00692 -0.00692 -0.18 -0.18
1.45 2.9 4.90 0.59 3.45 2.056 0.006 0.00613 0.00641 -0.00641 -0.93 -1.10
1.50 3.0 5.00 0.60 3.33 2.066 0.010 0.00562 0.00587 -0.00587 -1.75 -2.85
1.55 3.1 5.10 0.61 3.23 2.080 0.014 0.00517 0.00540 -0.00540 -2.60 -5.46
1.60 3.2 5.20 0.62 3.13 2.098 0.017 0.00478 0.00497 -0.00497 -3.49 -8.95
1.65 3.3 5.30 0.62 3.03 2.118 0.020 0.00442 0.00460 -0.00460 -4.41 -13.36
1.70 3.4 5.40 0.63 2.94 2.141 0.023 0.00410 0.00426 -0.00426 -5.37 -18.73
1.75 3.5 5.50 0.64 2.86 2.166 0.025 0.00382 0.00396 -0.00396 -6.35 -25.08
1.80 3.6 5.60 0.64 2.78 2.193 0.027 0.00356 0.00369 -0.00369 -7.37 -32.46
1.85 3.7 5.70 0.65 2.70 2.222 0.029 0.00333 0.00344 -0.00344 -8.43 -40.88
1.90 3.8 5.80 0.66 2.63 2.253 0.031 0.00312 0.00322 -0.00322 -9.52 -50.40
1.95 3.9 5.90 0.66 2.56 2.285 0.032 0.00292 0.00302 -0.00302 -10.64 -61.05
2.00 4.0 6.00 0.67 2.50 2.319 0.033 0.00275 0.00284 -0.00284 -11.80 -72.85
2m 73m .
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- . 2007-2008
2-106
8
3:1 1:1 , Manning n=0.014 . 0.55m:
() .
() .
.
= y + 3y = 4y
A = Ty/2 = 2y2
P = y (21/2 + (9 + 1)1/2) = 4.576y
R = A/P = 0.437 y
D = A/T = 0.50 y
() , ( ) . Fr = 1,
150.02
55.01)50.0(
25/2
2
=
===== Q
gQy
ygyQ
gDVFr c m
3/s.
Manning 8/3
2/12/1
3/2222/13/2 868.0
576.422
=
===
JnQyJ
yy
nyQJR
nAAVQ
J2=0.01 J3=0.0005
J1=0.0010
3 1 1
1
0.55m
M2
M2
S2 S1
-
- . 2007-2008
2-107
J1 = 0.001 y1 = 0.699m > yc ( )
J2 = 0.01 y2 = 0.454m < yc ( )
J3 = 0.0005 y3 = 0.796m > yc ( y=yc )
() 1 2 , 2 S2.
2 3 , . y2 y3.
( )2222 232 yygyQAygAQM +=+= . 2 = 2 y2 = 0.663m < y3 = 0.796m, S1.
2.
.
9
, n = 0.016 500 m .
Q . = 0.20.
. ( ). H2 . = 2.0 . (x = -500 m)
b = 1.0 m L= 500 m
H= 2.0m
-
- . 2007-2008
2-108
H yV
gK
Vg
y KV
g= + + = + +
2 2 2
2 21
2( ) .
1 : Q = 1.0 m3/s, yc = (Q2/b2g)1/3 = 0.47 m.
( 2) = 1.31 m < 2.0 m (. ). Q > 1.0 m3/s.
2 : Q = 1.5 m3/s, yc = (Q2/b2g)1/3 = 0.61 m.
( 2) = 1.66 m < 2.0 m (. ). Q > 1.5 m3/s.
3 : Q = 2.0 m3/s.
yc = (Q2/b2g)1/3 = 0.74 m.
( 2) = 2.00 m (. ). ( = 2.005 m).
Q = 2.0 m3/s.
H2
: B=1m, Z=0, n=0.016, Jo=0
1 Q=1m3/s
y Q A P R V E JE JE(bar) Jo-JE x x
0.47 1 0.467 1.93 0.24 2.14 0.701 0.0078 0
0.57 1 0.567 2.13 0.27 1.76 0.726 0.02 0.0047 0.0062 -0.0062 -4.00 -4
0.67 1 0.667 2.33 0.29 1.50 0.782 0.06 0.0031 0.0039 -0.0039 -14.53 -19
0.77 1 0.767 2.53 0.30 1.30 0.854 0.07 0.0021 0.0026 -0.0026 -27.75 -46
0.87 1 0.867 2.73 0.32 1.15 0.935 0.08 0.0016 0.0019 -0.0019 -43.71 -90
0.97 1 0.967 2.93 0.33 1.03 1.022 0.09 0.0012 0.0014 -0.0014 -62.46 -152
1.07 1 1.067 3.13 0.34 0.94 1.112 0.09 0.0009 0.0011 -0.0011 -84.06 -237
1.17 1 1.167 3.33 0.35 0.86 1.205 0.09 0.0008 0.0009 -0.0009 -108.54 -345
1.27 1 1.267 3.53 0.36 0.79 1.299 0.09 0.0006 0.0007 -0.0007 -135.94 -481
1.28 1 1.280 3.56 0.36 0.78 1.311 0.01 0.0006 0.0006 -0.0006 -19.77 -501
(1+k)V2/2g = 0.037
-
- . 2007-2008
2-109
2
y Q A P R V E JE JE(bar) Jo-JE x x
0.61 1.5 0.612 2.22 0.28 2.45 0.918 0.0086 0
0.71 1.5 0.712 2.42 0.29 2.11 0.938 0.02 0.0058 0.0072 -0.0072 -2.79 -3
0.81 1.5 0.812 2.62 0.31 1.85 0.986 0.05 0.0042 0.0050 -0.0050 -9.56 -12
0.91 1.5 0.912 2.82 0.32 1.64 1.050 0.06 0.0031 0.0036 -0.0036 -17.53 -30
1.01 1.5 1.012 3.02 0.33 1.48 1.124 0.07 0.0024 0.0028 -0.0028 -26.73 -57
1.11 1.5 1.112 3.22 0.34 1.35 1.205 0.08 0.0019 0.0022 -0.0022 -37.18 -94
1.21 1.5 1.212 3.42 0.35 1.24 1.290 0.09 0.0016 0.0017 -0.0017 -48.89 -143
1.31 1.5 1.312 3.62 0.36 1.14 1.379 0.09 0.0013 0.0014 -0.0014 -61.88 -205
1.41 1.5 1.412 3.82 0.37 1.06 1.470 0.09 0.0011 0.0012 -0.0012 -76.16 -281
1.51 1.5 1.512 4.02 0.38 0.99 1.562 0.09 0.0009 0.0010 -0.0010 -91.75 -372
1.61 1.5 1.612 4.22 0.38 0.93 1.656 0.09 0.0008 0.0009 -0.0009 -108.66 -481
1.62 1.5 1.620 4.24 0.38 0.93 1.664 0.01 0.0008 0.0008 -0.0008 -9.36 -490
(1+k)V2/2g = 0.052
3
y Q A P R V E JE JE(bar) Jo-JE x x
0.74 2 0.742 2.48 0.30 2.70 1.112 0.0093 0
0.89 2 0.892 2.78 0.32 2.24 1.148 0.04 0.0059 0.0076 -0.0076 -4.70 -5
1.04 2 1.042 3.08 0.34 1.92 1.229 0.08 0.0040 0.0049 -0.0049 -16.47 -21
1.19 2 1.192 3.38 0.35 1.68 1.335 0.11 0.0029 0.0035 -0.0035 -30.57 -52
1.34 2 1.342 3.68 0.36 1.49 1.455 0.12 0.0022 0.0025 -0.0025 -47.05 -99
1.49 2 1.492 3.98 0.37 1.34 1.583 0.13 0.0017 0.0019 -0.0019 -65.95 -165
1.64 2 1.642 4.28 0.38 1.22 1.717 0.13 0.0014 0.0015 -0.0015 -87.30 -252
1.79 2 1.792 4.58 0.39 1.12 1.855 0.14 0.0011 0.0012 -0.0012 -111.12 -363
1.94 2 1.942 4.88 0.40 1.03 1.996 0.14 0.0009 0.0010 -0.0010 -137.45 -501
(1+k)V2/2g =0.065
-
3