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

- . 2007-2008

2-1

1.

1.1

, . , , , , , . .

AVQ = V .

),( yAA = y . , b, y Z,

( )yZybZybA +=),,( . , ,

),,( tyAA = .

1.2 .

, , , , , .

. .

1.3 .

, (, , , , ) ( , , , ). :

b

y Z 1

- . 2007-2008

2-2

Jo, () ( t), ( y), ( ) .

1.4 .

, , , , . , , . . , , , , .

(section) , , .

1.1 .

:

t y

t

P

G

G

G - G

- . 2007-2008

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) . , .

- . 2007-2008

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 , . . , .

- . 2007-2008

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)

- . 2007-2008

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

- . 2007-2008

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)

- . 2007-2008

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

- . 2007-2008

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

- . 2007-2008

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

..

- . 2007-2008

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

- . 2007-2008

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

- . 2007-2008

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

- . 2007-2008

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. .

- . 2007-2008

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

- . 2007-2008

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 ()

() .

- . 2007-2008

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

- . 2007-2008

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

- . 2007-2008

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

- . 2007-2008

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

- . 2007-2008

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.

- . 2007-2008

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

- . 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

- . 2007-2008

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)

- . 2007-2008

2-65

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.

- . 2007-2008

2-66

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)

- . 2007-2008

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 .

() ( ) .

- . 2007-2008

2-69

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

- . 2007-2008

2-70

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

- . 2007-2008

2-71

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

- . 2007-2008

2-72

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

- . 2007-2008

2-73

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).

- . 2007-2008

2-74

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

- . 2007-2008

2-75

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.

- . 2007-2008

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

- . 2007-2008

2-77

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

- . 2007-2008

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

- . 2007-2008

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.

- . 2007-2008

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

- . 2007-2008

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. .

- . 2007-2008

2-82

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.

- . 2007-2008

2-83

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

- . 2007-2008

2-84

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

- . 2007-2008

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

- . 2007-2008

2-86

( )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

- . 2007-2008

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.

- . 2007-2008

2-88

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 =

- . 2007-2008

2-91

3. ( )

- . 2007-2008

2-92

4. ( ) -

- . 2007-2008

2-93

5.

6.

- . 2007-2008

2-94

- . 2007-2008

2-95

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

- . 2007-2008

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

- . 2007-2008

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

()

- . 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

- . 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.

- . 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

- . 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

- . 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

- . 2007-2008

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.

.

- . 2007-2008

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

- . 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 .

- . 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


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


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

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