BFC21103 Chapter4.pdf

8/19/2019 BFC21103 Chapter4.pdf

1/113

BFC21103 Hydraulics

Chapter 4. Non‐Uniform Flow

in

Open

ChannelTan Lai Wai, Wan Afnizan & Zarina Md Ali

[email protected]


2/113

Learning Outcomes

At the end of this chapter, students should be able to:

i.

Analyse the

characteristics

of

hydraulic

jump

(rapidly‐varied flow) based on momentum

equation

ii. Analyse the characteristics of gradually‐varied

flow

BFC21103 Hydraulics

Tan et al. ([email protected])


3/113

Occurs when the depth of flow change rapidly within short distance,

e.g. hydraulic jump.

Hydraulic jump occurs when supercritical flow changes suddenly to

subcritical flow within a short distance.

Datum

BFC21103 Hydraulics


4.1 Rapidly‐Varied Flow

2y

cy 1

y supercritical

1 2

subcritical

hydraulic

jump


4/113

BFC21103 Hydraulics


Hydraulic jump only occurs if the upstream flow is supercritical,

i.e. y 1 y c > y 1

where,

y 1 = depth

of

flow

just

before

the

jump

y 2 = depth of flow just after the jump

y 1 and y 2 are known as conjugate depths

4.2 Hydraulic

Jump


5/113

BFC21103 Hydraulics


Hydraulic jump in the laboratory flume


6/113

BFC21103 Hydraulics


Hydraulic jump at the toe of spillway ‐ Itaipu dam, Brazil


7/113

BFC21103 Hydraulics


Hydraulic jump downstream of sluice gate ‐ Harran canal, Turkey


8/113

BFC21103 Hydraulics


Waves hitting sea wall in Depoe bay, Oregon U.S.


9/113

BFC21103 Hydraulics


Surge waves due to fast flowing flood in Tangjiasan, China


10/113

BFC21103 Hydraulics


i. Energy dissipator i.e. reduce velocity and prevent erosion

ii. Diverse the

water

irrigation

iii. Increase weight of water

iv. Mix

chemical

substance

e.g.

in

water

treatment

processv. Aeration of flow, i.e. increase DO

Applications of

Hydraulic

Jump


11/113

BFC21103 Hydraulics


Based on the Froude number before the jump Fr1

Fr1 = 1.0 − 1.7 → undular jump

Fr1 = 1.7 − 2.5 → weak jump

Fr1 =

2.5 −

4.5 →

oscillating

jumpFr1 = 4.5 − 9.0 → steady jump

Fr1 > 9.0 → strong jump

Types of

Jump


12/113

BFC21103 Hydraulics


Fr1

= 1.6

Energy dissipation = 45% to 70%

Energy dissipation up to 85%


13/113

BFC21103 Hydraulics


Momentum Equation

Consider a hydraulic jump on a frictionless flat bed within a

rectangular channel,

12 MMF −=∑124321

QV QV F F F F ρ −=+−−

2y

1y

1 2

F 1

F 1

F 3

F 4

W

Since friction = 0 → F 3 = 0 and flat bed F 4 = W sinθ = 0


14/113

BFC21103 Hydraulics


12

2

2

1

122

QV QV y

gAy

gA ρ ρ ρ ρ −=−

1221 QV QV F F ρ ρ −=−

g

qV

g

qV y y 12

2

2

2

12

1

2

1−=−

Dividing by ρ

gB,

Since ,2

2

1

1

and y

qV

y

qV ==

1

2

2

2

2

2

2

12

1

2

1

gy

q

gy

qy y −=−

2

2

2

2

2

1

1

2

2

1

2

1y

gy

qy

gy

q+=+


15/113

BFC21103 Hydraulics


Activity 4.1

Using the

momentum

force

equation,

draw

the

specific

force

curve

if a hydraulic jump occurs within a rectangular channel with the

discharge per unit width is 25 ft3/s.

Given q = 25 ft3/s flows in a rectangular channel

2

2

21 y

gy qF +=Specific force is given as

Hydrostatic

pressure

Momentum

flux


16/113

BFC21103 Hydraulics


y (ft) F (ft2)

1.0 19.910

1.2 16.895

1.4 14.844

1.6 13.411

1.8 12.403

2.0 11.705

2.2 11.243

2.4 10.967

2.6 10.845

2.8 10.852

3.0 10.970

3.2 11.186

3.4 11.4893.6 11.872

3.8 12.328

4.0 12.852

4.2 13.441

4.4 14.091

4.6 14.800

4.8 15.564

5.0 16.382

5.2 17.253

5.4 18.174

5.6 19.146

5.8 20.167

0

1

2

3

4

5

6

0 5 10 15 20

y (ft)

F (ft2)

y c

Subcritical flow Fr 1

F min

y 1 and y 2 with the same F are conjugate depths


17/113

BFC21103 Hydraulics


Conjugate Depths Equation

From the momentum equation of flow in a rectangular channel,

2

2

2

2

2

1

1

2

2

1

2

1y

gy

qy

gy

q+=+

2

2

1

2

2

1

2

2

22

gy

q

gy

qy y −=−

( )( ) ⎟⎟ ⎠

⎞⎜⎜⎝

⎛ −=+−21

12

2

2112

2

y y

y y

g

qy y y y

( ) g

qy y y y

2

2121

2

=+

Rearranging,

It can be seen that Fr can be introduced since3

2

2Frgy

q=


18/113

BFC21103 Hydraulics


( )3

1

2

2

1

2

21

2

gy

q

y

y y y =+Division by ,31y

2

1

1

2

2

1

2

2 Fr2=+y

y

y

y

0Fr2 2

1

1

2

2

1

2

2 =−+y

y

y

y

Solving for gives1

2

y

y

Note that

solving

gives

02 =++ cbx ax

a

acbb x

2

42 −±−

=

since y 1 and y 2 are positive values

( )( )

( )12

Fr21411 2

1

2

1

2 −−+

−=y

y

( )21

1

2 Fr8112

1++−=

y

y


19/113

BFC21103 Hydraulics


Else if division is made by ,32

y

2

Fr811 2

2

2

1 ++

−=y

y

Note that for hydraulic jump to occur, or 11

2 >y

y 12

1 <y

y


20/113

BFC21103 Hydraulics


Energy Loss

There will be considerable loss of energy in hydraulic jump between sections 1 and 2

Datum

2y

cy

1y supercritical

1 2

subcritical

hydraulic

jump

E o

g

V

2

2

2

g

V

2

2

1

EGLE L


21/113

BFC21103 Hydraulics


Energy

loss

is

calculated

as 21 E E E L −=

⎟ ⎞

⎜⎝

⎛ +−⎟

⎠

⎞⎜⎝

⎛ +=

g

V y

g

V y E

L22

2

2

2

2

1

1

For rectangular channel, it can be simplified as

⎟⎟

⎞

⎜⎜⎝

⎛

+−⎟⎟

⎞

⎜⎜⎝

⎛

+=22

2

221

2

1 22 gy

q

y gy

q

y E L

( ) ⎟⎟

⎠

⎞⎜⎜

⎝

⎛ −+−=

2

2

2

1

2

21

11

2

1

y y g

qy y E

L

( ) ⎟⎟ ⎞

⎜⎜⎝

⎛ −+−=

2

2

2

1

2

1

2

2

2

212

1

y y

y y

g

qy y E

L


22/113

BFC21103 Hydraulics


( ) ( ) ⎟⎟ ⎞

⎜⎜⎝

⎛ −⎥⎦

⎤⎢⎣

⎡ ++−=2

2

2

1

2

1

2

2

2121212

1

2

1

y y

y y y y y y y y E

L

Substituting ( ) 2121

2

2

1

y y y y g

q

+=

( )

21

2

2

1

3

2

3

1

2

212121

4

4

y y

y y y y y y y y y y E

L

−+−+−=

21

3

1

2

212

2

1

3

2

4

33

y y

y y y y y y E

L

−−+=

( )21

3

12

4 y y y y E

L −=

which is expressed in meter


23/113

BFC21103 Hydraulics


Power due to Energy Loss

Power due to energy loss in unit Watt is given as

LL gQE P ρ =

Height of Jump

The height of jump is given as

12 y y H

j −=


24/113

BFC21103 Hydraulics


Length of Jump

Based on Froude number upstream of the jump Fr1,

( )12

9.6 y y L j

−=

21.6 y L

j =

for Fr1 ≤ 5.0

for Fr1 > 5.0


25/113

BFC21103 Hydraulics


Activity 4.2

A

spillway

discharges

flow

at

a

rate

of

7.75

m3/s/m.

At

the

downstream horizontal apron, the depth of flow was found to be

0.5 m. What tailwater depth is needed to form a hydraulic jump? If

a jump is formed, find its

(i) type;

(ii) length;

(iii)

head loss;

and

(iv) energy loss as a percentage of the initial energy.


26/113

BFC21103 Hydraulics


Given q = 7.75 m3/s/m, y 1 = 0.5 m

( )21

1

2 Fr8112

1++−=

y

y

999.65.081.9

75.7Fr

33

1

1 =

×==

gy

q

Utilizing the conjugate depths equation,

( )22

999.68112

5.0×++−=y

m705.42 =y

(i) Based on the Fr1 = 6.999, the jump is a steady jump (4.5


27/113

BFC21103 Hydraulics


(ii) Since Fr1 = 6.999 > 5.0

21.6 y L

j =Length of jump

705.41.6 ×= j

L

m70.28= j L

(iii)

Head loss

is

given

as

( )

21

3

12

4 y y y y E L −=

( )

705.45.04

5.0705.4 3

××

−=

LE

m901.7=L

E


28/113

BFC21103 Hydraulics


(iv) Initial total energy is 2

1

2

1o

2gy

qy E +=

2

2

o5.081.92

75.75.0

××+=E

m745.12o =E

Percentage of energy loss %99.61%100745.12

901.7

o

=×=E

E L


29/113

BFC21103 Hydraulics


Activity 4.3

A

25‐

m

wide

spillway

is

discharging

flow

with

velocity

of

30

m/s

at

adepth of 1 m. Hydraulic jump occurs immediately downstream. Find

the height of the jump and power loss due to the jump.

Given B =

25

m,

y 1 = 1 m, V 1 = 30 m/s

( )21

1

2 Fr8112

1++−=

y

y

578.9

181.9

30Fr

1

11 =

×==

gy

V

Conjugate depths equation,

( )22

578.98112

1×++−=y

m

055.132 =

y


30/113

BFC21103 Hydraulics


(i) Height of jump 12 y y H j −=

1055.13 −= j

H

m055.12= j

H

(ii) Energy loss ( )

21

3

12

4 y y

y y E

L

−=

( )055.12141055.12

3

××−=LE

m019.28=L

E

Power due to energy lossLL

gQE P ρ =

( ) 019.28301259810 ××××=L

P

MW

15.206=LP


31/113

BFC21103 Hydraulics


A steady non‐uniform flow in a prismatic channel with gradual changes

in its flow surface elevation.

Examples:

(i) Drawdown produced by sudden change in channel bed slope

4.3 Gradually‐Varied Flow

M2

S2y c

Mild slope

S t e e p s l o p e

y o

y o

control section

Computations C o m p u t a t i o n s


32/113

BFC21103 Hydraulics


(ii) Backwater produced by increased in bed elevation

M1

y cM i l d

s l o p e

y o1

control section 1

Computations ComputationsMilder

slope

y o2

Lake

control section 2


33/113

BFC21103 Hydraulics


Types of Slope

y o So Type of slope Symbol

y o > y c

or

So


34/113

Classification of GVF ProfileChannel Region Condition Type

Mild slope

1 y > y o > y c M1

2 y o > y > y c M2

3 y o > y c > y M3

Steep slope

1 y > y c > y o S1

2 y c > y > y o S2

3 y c > y o > y S3

Critical slope1 y > y o = y c C1

3 y y c H2

3 y y c A2

3 y < y c A3


35/113

BFC21103 Hydraulics


Classification of GVF Profile

Slope Region 1 Region

2 Region

3

Mild M

Steep S

Critical C

M1

y o

y c

y > y o > y c

M2y o

y c

y o > y > y c

M3y o

y c

y o > y c > y

S1

y > y c > y o

y cy o

S3

y c > y o > y

y cy o

y c > y > y o

y cy o S2

C1

y > y o = y c

y o = y

c

C3

y o = y c > y

y o = y

c −


36/113

BFC21103 Hydraulics


Slope Region 1 Region

2 Region

3

Horizontal

H

Adverse A

H2

y c

y > y c

−

−

A2

H2y c

y c > y

y > y c

y c

y c > y

y c

All curves in region 1 have positive slopes (backwater curves)

All curves in region 2 have negative slopes (drawdown curves)


37/113

BFC21103 Hydraulics



38/113

BFC21103 Hydraulics


Occurrence of Flow Profile

(a) i. M1 profile

Occurs due to obstruction to subcritical flow, e.g. weir, dam or

other control structures. The profile extends to several kilometres

upstream before

approaching

the

normal

depth.

M1

y o

y c

y > y o > y c

Mild slope


39/113

BFC21103 Hydraulics


(a)

ii.

M2 profileOccurs when there is a sudden drop in the bottom of the channel,

constriction of channel or channel outlet into reservoir.

M2

y o

y c

y o > y > y c

Mild

slope


40/113

BFC21103 Hydraulics


(a)

iii.

M3 profileOccurs when supercritical flow enters a mild slope channel, e.g.

flow from a spillway or a sluice gate to a mild channel.

M3y o

y c

y o > y c > y

Mild slope


41/113

BFC21103 Hydraulics


(b)

i.

S1 profileOccurs when supercritical flow changes to pool of water

(subcritical flow) due to obstruction such as weir or dam.

S1

y o

y c

Steep slope

y > y c > y o


42/113

BFC21103 Hydraulics


(b)

ii.

S2 profileOccurs when flow from reservoir enter a steep slope or when

there is a change from mild slope to steep slope. This profile is of

shorter length.

y c > y > y o

y cy o S

2

Steep slope


43/113

BFC21103 Hydraulics


(b)

iii.

S3 profileOccurs when flow from reservoir enter a steep slope or when

there is a change from mild slope to steep slope. This profile is of

shorter length.

S3

y c > y o > y

y cy oSteep slope

S3

y cy oSteep slope

S t e e p e r s l o

p e


44/113

BFC21103 Hydraulics


(c) C1 and C3 profiles

Highly unstable

and

rarely

occur,

(d) H2 and H3 profiles

Occurs when

the

bed

of

mild

slope

becomes

flatter.

There

is

no

region 1 since y o = ∞.

y > y o = y c and y o = y c > y

H2

y c

y > y c

H3

y c > y

Horizontal bedDrop


45/113

BFC21103 Hydraulics


(e) A2

and A3

profiles

Occurs when flow is on adverse slope, which is rare. These profiles

occurs within a short length.

A2

y c

y > y c

y c > y

Adverse slope

A3

Drop

Pool

Activity 4 4


46/113

BFC21103 Hydraulics


Activity 4.4

Determine the type of profile for the following flow.

Sluice gate

y o

y c

Sluice gate

y c

y o

(a) (b)


47/113

BFC21103 Hydraulics


Sluice gate

y o

y c

Sluice gate

y c

y o

(a) (b)

S3

S1

M3

M1

y c > y o → S

Zone 1 → S1

Zone 3 → S3

y o > y c → M

Zone 1 → M1

Zone 3 → M3

Activity 4 5


48/113

BFC21103 Hydraulics


Activity 4.5

A

rectangular

channel

with

bottom

width

4

m

and

bottom

slope

0.0008 has discharge of 1.5 m3/s. Along the gradually‐varied flow in

the channel, the depth at a section is found to be 0.3 m. Assuming

Manning n = 0.016, determine the type of GVF profile.


49/113

BFC21103 Hydraulics


Rectangular section, B = 4 m, So = 0.0008, n = 0.016, Q = 1.5 m3/s, y = 0.3 m.

2

1

o

3

2

S

Qn AR =

( )2

1

3

2

o

oo

0008.0

016.05.1

2

×=⎟⎟

⎠

⎞⎜⎜⎝

⎛

+ y B

By By

( ) 8485.024

443

2

o

oo =⎟⎟ ⎞⎜⎜

⎝ ⎛

+ y y y

m4261.0o =y

3

1

2

2

⎟⎟

⎞

⎜⎜⎝

⎛ =

gB

Qy

c

3

1

2

2

81.94

5.1⎟⎟

⎞⎜⎜⎝

⎛

×=cy

m2429.0=cy

2o M→>> cy y y Since

Activity 4.6


50/113

BFC21103 Hydraulics


Activity 4.6

A

triangular

channel

has

side

slope

1(H):1(V),

bed

slope

0.001,

and

Manning roughness n = 0.015. If rate of flow is 0.2 m3/s

(a) Determine the type of slope, and

(b) Give the limit of depths of flow in regions 1, 2, and 3.


51/113

BFC21103 Hydraulics


Triangular section, z = 1, So = 0.001, n = 0.015, Q = 0.2 m3/s

2

1

o

3

2

S

Qn AR =

( )2

1

3

2

2

o

2

o2

o

001.0

015.02.0

12

×=⎟

⎟ ⎠

⎞⎜⎜⎝

⎛

+ zy

zy zy

m5361.0o =y

g

Q

T

A

c

c

23

=

m3822.0=cy

Mslopemildo →> cy y Since

( ) 09487.022

3

2

o

2

o2o =⎟⎟ ⎞⎜⎜

⎝ ⎛

y y y

1897.038

o =y

( ) 81.9 2.02

232

=c

c

zy zy

008155.05 =cy


52/113

BFC21103 Hydraulics


m5361.0

:M

For 1 >y

m0.3822m5361.0 :M For 2 >> y

m0.3822

:M

For 3


53/113

BFC21103 Hydraulics


Analysis of GVF Profile

Two basic assumptions are involved in the analysis of GVF:

1. The pressure distribution at any section is hydrostatic.

2. The resistance to flow at any depth can be assumed using

uniform‐flow equation, such as the Manning's equation, with the

condition that the slope term to be used in the equation is the

energy slope

and

not

the

bed

slope.

Thus,

if

in

a GVF

the

depth

of

flow at any section is y , the energy slope S f is:

3

4

22

R

V nS f =

where R is the hydraulic radius of the section at depth y .

Differential Equation of GVF


54/113

BFC21103 Hydraulics


Differential Equation of GVF

The total

energy

H of

a gradually

‐varied

flow

in

a channel

of

small

slope is:

g

V y E

2

2

+=where the specific energy

g

V y zH

2

2

++=

Schematic sketch of GVF

E y

g

V

2

2

E ne r g y

l i ne

S f

W a t e r s u r f a c e

SozDatum x


55/113

BFC21103 Hydraulics


Since the water surface varies in the longitudinal x ‐direction, the depth

of the

flow

and

the

total

energy

are

functions

of

x .

Differentiating total energy with respect to x ,

⎟⎟

⎞

⎜⎜⎝

⎛

++= g

V

x x

y

x

z

x

H

2d

d

d

d

d

d

d

d 2

Energy slope

f

S x

H−=

d

d

Bottom

slope

oS x

Z −=

d

d

water‐surface slope

relative to the channel

bottom

x y

gAQ

x gV

x dd

2dd

2dd 2

22

⎟⎟ ⎞

⎜⎜⎝

⎛ =⎟⎟

⎞⎜⎜⎝

⎛

x

y

y

A

gA

Q

d

d

d

d3

2

−=

Velocity term

T A

=d

Since


56/113

BFC21103 Hydraulics


T y

=d

Since

x

y

gA

T Q

g

V

x d

d

2d

d3

22

=⎟⎟ ⎞

⎜⎜⎝

⎛

x

y

gA

T Q

x

y SS o f

d

d

d

d3

2

⎟⎟ ⎞

⎜⎜⎝

⎛ −+−=−

Differentiated energy

equation

can

now

be

rewritten

as

Rearranging,

3

2

1d

d

gA

T Q

SS

x

y f o

−

−= Dynamic

equation

of

GVF

Other forms of dynamic equation of GVF

( ) f d h d d


57/113

BFC21103 Hydraulics


(a) If K = conveyance at any depth y and K o = conveyance corresponding to

the normal

depth

y o,

then

f S

QK = for GVF

oS

Q

K =o for uniform flow

2

2

K

K

S

So

o

f =

If Z = section factor at depth y and Z c = section factor at the critical depth y c,

T

A Z

32 =

gQ

T A Z

c

cc

232 ==and

Hence2

2

3

2

Z

Z

gA

T Q c=

Substituting into the GVF dynamic equation


58/113

BFC21103 Hydraulics


g y q

⎟⎟⎟⎟ ⎞

⎜⎜⎜⎜

⎝

⎛

−

−=

3

2

1

1

d

d

gA

T Q

SS

S x

y o

f

o

⎥

⎥⎥⎥

⎦

⎤

⎢

⎢⎢⎢

⎣

⎡

⎟ ⎠

⎞

⎜⎝

⎛

−

⎟ ⎠

⎞⎜⎝

⎛ −=

2

2

1

1

d

d

Z

Z

K

K

S x

y

c

o

o

This equation is useful in developing direct integration techniques.

(b) If Q h l di h d h d Q d h


59/113

BFC21103 Hydraulics


(b) If Qo represents the normal discharge at a depth y o and Qc denotes the

critical discharge

at

the

same

depth

y ,

oo SK Q =

g Z Qc =and

Using these definitions, the GVF dynamic equation in (a) can be rewritten as

2

2

1

1

dd

⎟⎟ ⎞

⎜⎜⎝

⎛ −

⎟⎟

⎠

⎞⎜⎜

⎝

⎛ −

=

c

no

Q

QQ

Q

S x y

( ) A th f f th GVF d i ti i


60/113

BFC21103 Hydraulics


(c) Another form of the GVF dynamic equation is

This equation is called the differential‐energy equation of GVF to distinguish

it from

the

other

GVF

differential

equations.

This

energy

equation

is

very

useful in developing numerical techniques for the GVF profile computation.

f o SS x

E −=

d

d



61/113

BFC21103 Hydraulics



Among the importance are:

(a) determination of the effect of hydraulic structure to the flow;

(b) inundation due

to

dam

or

weir

construction;

and

(c) estimation of flood area.

This course only considers the following methods:

(a) Direct integration;

(b) Numerical integration;

and

(c) Direct step.

Calculation of GVF Profile


62/113

BFC21103 Hydraulics


Calculation of GVF Profile

Gradually‐varied

flow surface

y 1 y N+1

L

Δ x 1 Δ x 2 Δ x N

y 1+Δy 1 y 1+Δy 2

Changes in depth of flow can be calculated if:

(a) y 1 and y N+1 are known, or

(b) L is known

x

y

d

d

A. Direct Integration


63/113

BFC21103 Hydraulics


A. Direct Integration

( ) ( ) ( )[ ] ( ) ( )[ ]⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−⎟⎟

⎠

⎞⎜⎜

⎝

⎛ +−−−=− Jv F Jv F

M

J

y

y NuF NuF uu

S

y x x

M

c ,,,, 12o

1212

o

o12

Between two sections ( x 1, y 1) and ( x 2, y 2),

M, N = hydraulic exponents

F (u, N) = varied‐flow function

F (v , J) = same function as F (u, N)

oy y u = J

N

uv =( )1+−

=MNN J

where,


64/113

BFC21103 Hydraulics


Section M N

Rectangular 3 2 to 3.333

Trapezoidal 3 to

5 2

to

5.333

Triangular 5 5.333

⎟ ⎞⎜⎝ ⎛ −=

y T

T AT

Ay M

dd3

⎟ ⎞

⎜

⎝

⎛ −=

y

PRT

A

y N

d

d25

3

2

For trapezoidal channels,

⎟ ⎞

⎜⎝ ⎛ +

−⎟ ⎞

⎜⎝ ⎛ +

⎟ ⎠

⎞⎜⎝

⎛ +

=

B

y z

B

y z

B

y z

B

y z

M

21

2

1

213

⎥⎦

⎤⎢⎣

⎡ +⎟ ⎠ ⎞

⎜⎝ ⎛ +

+⎟ ⎠

⎞⎜⎝

⎛

−⎟ ⎠ ⎞

⎜⎝ ⎛ +

⎟ ⎠

⎞⎜⎝

⎛ +

=2

2

121

1

38

1

21

310

zB

y

z

B

y

B

y z

B

y z

N

3.0

4.05.06.0

y z

1


65/113

2.5 3.0 3.5 4.0 4.5 5.0 5.50.02

M

0.03

0.05

0.1

0.04

0.2

0.3

0.06

0.08

0.40.50.6

0.81.0

2.0

o

andD

y

B

y

z =

2

BFC21103 Hydraulics


z

=

0 ( r

e c t a n g u l a r )

z =

0 . 5

z =

1 . 5

z =

1

z = 4z = 3z = 2.5

Circular

Doy

B

3.0

4.05.06.0

y z

1z

= 0

( 5


66/113

N

o

andD

y

B

y

BFC21103 Hydraulics


2.0 2.5 3.0 3.5 4.0 4.5 5.00.02

0.03

0.05

0.1

0.04

0.2

0.3

0.06

0.08

0.40.50.6

0.81.0

2.0

5.5

Bz ( r

e

c t a n g u l a r )

Doy

C i r c u l ar

z = 0 . 5

z =

1

z = 1 . 5

z = 2.5z = 2

z = 4z = 3

N 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0

Varied‐flow function for positive slopes F(u, N) (Chow, 1959)


67/113

u

0.000.02

0.04

0.06

0.08

0.0000.020

0.040

0.060

0.080

0.0000.020

0.040

0.060

0.080

0.0000.020

0.040

0.060

0.080

0.0000.020

0.040

0.060

0.080

0.0000.020

0.040

0.060

0.080

0.0000.020

0.040

0.060

0.080

0.0000.020

0.040

0.060

0.080

0.0000.020

0.040

0.060

0.080

0.0000.020

0.040

0.060

0.080

0.0000.020

0.040

0.060

0.080

0.10

0.12

0.14

0.16

0.18

0.100

0.120

0.141

0.161

0.181

0.100

0.120

0.140

0.161

0.181

0.100

0.120

0.140

0.160

0.181

0.100

0.120

0.140

0.160

0.180

0.100

0.120

0.140

0.160

0.180

0.100

0.120

0.140

0.160

0.180

0.100

0.120

0.140

0.160

0.180

0.100

0.120

0.140

0.160

0.180

0.100

0.120

0.140

0.160

0.180

0.100

0.120

0.140

0.160

0.180

0.20

0.22

0.24

0.26

0.28

0.202

0.223

0.243

0.264

0.286

0.201

0.222

0.242

0.263

0.284

0.201

0.221

0.242

0.262

0.283

0.201

0.221

0.241

0.262

0.282

0.200

0.221

0.241

0.261

0.282

0.200

0.220

0.241

0.261

0.281

0.200

0.220

0.240

0.261

0.281

0.200

0.220

0.240

0.260

0.281

0.200

0.220

0.240

0.260

0.280

0.200

0.220

0.240

0.260

0.280

0.30

0.32

0.34

0.36

0.38

0.307

0.329

0.350

0.373

0.395

0.305

0.326

0.348

0.370

0.392

0.304

0.325

0.346

0.367

0.389

0.303

0.324

0.344

0.366

0.387

0.302

0.323

0.343

0.364

0.385

0.302

0.322

0.343

0.363

0.384

0.301

0.322

0.342

0.363

0.383

0.301

0.321

0.342

0.362

0.383

0.301

0.321

0.341

0.362

0.382

0.300

0.321

0.341

0.361

0.382BFC21103 Hydraulics Tan et al. ([email protected])


N

u 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0


68/113

BFC21103 Hydraulics


u

0.400.42

0.44

0.46

0.48

0.4180.441

0.465

0.489

0.514

0.4140.437

0.460

0.483

0.507

0.4110.433

0.456

0.478

0.502

0.4080.430

0.452

0.475

0.497

0.4070.428

0.450

0.472

0.494

0.4050.426

0.448

0.470

0.492

0.4040.425

0.446

0.468

0.489

0.4030.424

0.445

0.466

0.488

0.4030.423

0.444

0.465

0.486

0.4020.423

0.443

0.464

0.485

0.50

0.52

0.54

0.56

0.58

0.539

0.565

0.592

0.619

0.647

0.531

0.556

0.582

0.608

0.635

0.525

0.550

0.574

0.600

0.626

0.521

0.544

0.568

0.593

0.618

0.517

0.540

0.563

0.587

0.612

0.514

0.536

0.559

0.583

0.607

0.511

0.534

0.556

0.579

0.603

0.509

0.531

0.554

0.576

0.599

0.508

0.529

0.551

0.574

0.596

0.506

0.528

0.550

0.572

0.594

0.60

0.61

0.62

0.63

0.64

0.676

0.691

0.707

0.722

0.738

0.663

0.677

0.692

0.707

0.722

0.653

0.666

0.680

0.694

0.709

0.644

0.657

0.671

0.684

0.698

0.637

0.650

0.663

0.676

0.690

0.631

0.644

0.657

0.669

0.683

0.627

0.639

0.651

0.664

0.677

0.623

0.635

0.647

0.659

0.672

0.620

0.631

0.643

0.655

0.667

0.617

0.628

0.640

0.652

0.664

0.65

0.66

0.67

0.68

0.69

0.754

0.771

0.787

0.805

0.822

0.737

0.753

0.769

0.785

0.802

0.724

0.739

0.754

0.769

0.785

0.712

0.727

0.742

0.757

0.772

0.703

0.717

0.731

0.746

0.761

0.696

0.709

0.723

0.737

0.751

0.689

0.703

0.716

0.729

0.743

0.684

0.697

0.710

0.723

0.737

0.680

0.692

0.705

0.718

0.731

0.676

0.688

0.701

0.713

0.726


N

u 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0


69/113

BFC21103 Hydraulics


u

0.700.71

0.72

0.73

0.74

0.8410.859

0.878

0.898

0.918

0.8190.837

0.855

0.874

0.893

0.8020.819

0.836

0.853

0.871

0.7870.804

0.820

0.837

0.854

0.7760.791

0.807

0.823

0.840

0.7660.781

0.796

0.811

0.827

0.7570.772

0.786

0.802

0.817

0.7500.764

0.779

0.793

0.808

0.7440.758

0.772

0.786

0.800

0.7390.752

0.766

0.780

0.794

0.75

0.76

0.77

0.78

0.79

0.939

0.961

0.984

1.007

1.031

0.912

0.933

0.954

0.976

0.998

0.890

0.909

0.929

0.950

0.971

0.872

0.890

0.909

0.929

0.949

0.857

0.874

0.892

0.911

0.930

0.844

0.861

0.878

0.896

0.914

0.833

0.849

0.866

0.883

0.901

0.823

0.839

0.855

0.872

0.889

0.815

0.830

0.846

0.862

0.879

0.808

0.823

0.838

0.854

0.870

0.80

0.81

0.82

0.83

0.84

1.056

1.083

1.110

1.139

1.170

1.022

1.047

1.072

1.099

1.128

0.994

1.017

1.041

1.067

1.093

0.970

0.992

1.015

1.039

1.064

0.950

0.971

0.993

1.016

1.040

0.934

0.954

0.974

0.996

1.019

0.919

0.938

0.958

0.979

1.001

0.907

0.925

0.945

0.965

0.985

0.896

0.914

0.932

0.952

0.972

0.887

0.904

0.922

0.940

0.960

0.85

0.86

0.87

0.88

0.89

1.202

1.236

1.273

1.312

1.355

1.158

1.190

1.224

1.260

1.300

1.122

1.151

1.183

1.217

1.254

1.091

1.119

1.149

1.181

1.216

1.065

1.092

1.120

1.151

1.183

1.043

1.068

1.095

1.124

1.155

1.024

1.048

1.074

1.101

1.131

1.007

1.031

1.055

1.081

1.110

0.993

1.015

1.039

1.064

1.091

0.980

1.002

1.025

1.049

1.075

N

u 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0



70/113

BFC21103 Hydraulics


u

0.900.91

0.92

0.93

0.94

1.4011.452

1.508

1.572

1.645

1.3431.390

1.442

1.500

1.568

1.2941.338

1.386

1.441

1.503

1.2531.294

1.340

1.391

1.449

1.2181.257

1.300

1.348

1.403

1.1891.225

1.266

1.311

1.363

1.1631.197

1.236

1.279

1.328

1.1401.173

1.210

1.251

1.297

1.1201.152

1.187

1.226

1.270

1.1031.133

1.166

1.204

1.246

0.950

0.960

0.970

0.975

0.980

1.730

1.834

1.968

2.052

2.155

1.647

1.743

1.865

1.943

2.040

1.577

1.666

1.780

1.851

1.936

1.518

1.601

1.707

1.773

1.855

1.467

1.545

1.644

1.707

1.783

1.423

1.497

1.590

1.649

1.720

1.385

1.454

1.543

1.598

1.666

1.352

1.417

1.501

1.553

1.617

1.322

1.385

1.464

1.514

1.575

1.296

1.355

1.431

1.479

1.536

0.985

0.990

0.995

0.999

1.000

2.294

2.477

2.792

3.523

∞

2.165

2.333

2.621

3.292

∞

2.056

2.212

2.478

3.097

∞

1.959

2.106

2.355

2.931

∞

1.880

2.017

2.250

2.788

∞

1.812

1.940

2.159

2.663

∞

1.752

1.873

2.079

2.554

∞

1.699

1.814

2.008

2.457

∞

1.652

1.761

1.945

2.370

∞

1.610

1.714

1.889

2.293

∞

1.001

1.005

1.010

1.015

1.020

3.317

2.587

2.273

2.090

1.961

2.931

2.272

1.984

1.817

1.698

2.640

2.021

1.756

1.602

1.493

2.399

1.818

1.572

1.428

1.327

2.184

1.649

1.419

1.286

1.191

2.008

1.506

1.291

1.166

1.078

1.856

1.384

1.182

1.065

0.982

1.725

1.279

1.089

0.978

0.900

1.610

1.188

1.007

0.902

0.828

1.508

1.107

0.936

0.836

0.766

N

u 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0



71/113

BFC21103 Hydraulics


u

1.031.04

1.05

1.06

1.07

1.7791.651

1.552

1.472

1.405

1.5321.415

1.325

1.252

1.191

1.3401.232

1.149

1.082

1.026

1.1861.086

1.010

0.947

0.895

1.0600.967

0.896

0.838

0.790

0.9550.868

0.802

0.748

0.703

0.8660.785

0.723

0.672

0.630

0.7900.714

0.656

0.608

0.569

0.7250.653

0.598

0.553

0.516

0.6680.600

0.548

0.506

0.471

1.08

1.09

1.10

1.11

1.12

1.346

1.296

1.250

1.210

1.173

1.138

1.091

1.050

1.013

0.980

0.977

0.935

0.897

0.864

0.833

0.851

0.812

0.777

0.746

0.718

0.749

0.713

0.681

0.652

0.626

0.665

0.631

0.601

0.575

0.551

0.595

0.563

0.536

0.511

0.488

0.535

0.506

0.480

0.457

0.436

0.485

0.457

0.433

0.411

0.392

0.441

0.415

0.392

0.372

0.354

1.13

1.14

1.15

1.16

1.17

1.139

1.108

1.079

1.052

1.027

0.949

0.921

0.895

0.871

0.848

0.805

0.780

0.756

0.734

0.713

0.693

0.669

0.647

0.627

0.608

0.602

0.581

0.561

0.542

0.525

0.529

0.509

0.490

0.473

0.458

0.468

0.450

0.432

0.417

0.402

0.417

0.400

0.384

0.369

0.355

0.374

0.358

0.343

0.329

0.316

0.337

0.322

0.308

0.295

0.283

1.18

1.19

1.20

1.22

1.24

1.003

0.981

0.960

0.922

0.887

0.827

0.807

0.788

0.754

0.723

0.694

0.676

0.659

0.628

0.600

0.591

0.574

0.559

0.531

0.505

0.509

0.494

0.480

0.454

0.431

0.443

0.429

0.416

0.392

0.371

0.388

0.375

0.363

0.341

0.322

0.343

0.331

0.320

0.299

0.281

0.305

0.294

0.283

0.264

0.248

0.272

0.262

0.252

0.235

0.219


N

u 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0


72/113

BFC21103 Hydraulics


u

1.261.28

1.30

1.32

1.34

0.8560.827

0.800

0.776

0.753

0.6940.669

0.645

0.623

0.603

0.5740.551

0.530

0.510

0.492

0.4820.461

0.442

0.424

0.408

0.4100.391

0.373

0.357

0.342

0.3510.334

0.318

0.304

0.290

0.3040.288

0.274

0.260

0.248

0.2650.250

0.237

0.225

0.214

0.2330.219

0.207

0.196

0.185

0.2050.193

0.181

0.171

0.162

1.36

1.38

1.40

1.42

1.44

0.731

0.711

0.692

0.675

0.658

0.584

0.566

0.549

0.534

0.519

0.475

0.459

0.444

0.431

0.418

0.393

0.378

0.365

0.353

0.341

0.329

0.316

0.304

0.293

0.282

0.278

0.266

0.256

0.246

0.236

0.237

0.226

0.217

0.208

0.199

0.204

0.194

0.185

0.177

0.169

0.176

0.167

0.159

0.152

0.145

0.153

0.145

0.138

0.131

0.125

1.46

1.48

1.50

1.55

1.60

0.642

0.627

0.613

0.580

0.551

0.505

0.492

0.479

0.451

0.425

0.405

0.394

0.383

0.358

0.335

0.330

0.320

0.310

0.288

0.269

0.273

0.263

0.255

0.235

0.218

0.227

0.219

0.211

0.194

0.179

0.191

0.184

0.177

0.161

0.148

0.162

0.156

0.149

0.135

0.123

0.139

0.133

0.127

0.114

0.103

0.119

0.113

0.108

0.097

0.087

1.65

1.70

1.75

1.80

1.85

0.525

0.501

0.480

0.460

0.442

0.403

0.382

0.364

0.347

0.332

0.316

0.298

0.282

0.267

0.254

0.251

0.236

0.222

0.209

0.198

0.203

0.189

0.177

0.166

0.156

0.165

0.153

0.143

0.133

0.125

0.136

0.125

0.116

0.108

0.100

0.113

0.103

0.095

0.088

0.082

0.094

0.086

0.079

0.072

0.067

0.079

0.072

0.065

0.060

0.055


N

u 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0


73/113

BFC21103 Hydraulics


1.901.95

2.00

2.10

2.20

0.4250.409

0.395

0.369

0.346

0.3170.304

0.292

0.273

0.251

0.2420.231

0.221

0.202

0.186

0.1880.178

0.169

0.154

0.141

0.1470.139

0.132

0.119

0.107

0.1170.110

0.104

0.092

0.083

0.0940.088

0.082

0.073

0.065

0.0760.070

0.066

0.058

0.051

0.0620.057

0.053

0.046

0.040

0.0500.046

0.043

0.037

0.032

2.3

2.4

2.5

2.6

2.7

0.326

0.308

0.292

0.277

0.264

0.235

0.220

0.207

0.195

0.184

0.173

0.160

0.150

0.140

0.131

0.129

0.119

0.110

0.102

0.095

0.098

0.089

0.082

0.076

0.070

0.075

0.068

0.062

0.057

0.052

0.058

0.052

0.047

0.043

0.039

0.045

0.040

0.036

0.033

0.029

0.035

0.031

0.028

0.025

0.022

0.028

0.024

0.022

0.019

0.017

2.8

2.9

3.0

3.5

4.0

0.252

0.241

0.230

0.190

0.161

0.175

0.166

0.158

0.126

0.104

0.124

0.117

0.110

0.085

0.069

0.089

0.083

0.078

0.059

0.046

0.065

0.060

0.056

0.041

0.031

0.048

0.044

0.041

0.029

0.022

0.036

0.033

0.030

0.021

0.015

0.027

0.024

0.022

0.015

0.010

0.020

0.018

0.017

0.011

0.007

0.015

0.014

0.012

0.008

0.005

4.5

5.0

6.0

7.0

8.0

0.139

0.122

0.098

0.081

0.069

0.088

0.076

0.058

0.047

0.040

0.057

0.048

0.036

0.028

0.022

0.037

0.031

0.022

0.017

0.013

0.025

0.020

0.014

0.010

0.008

0.017

0.013

0.009

0.006

0.005

0.011

0.009

0.006

0.004

0.003

0.008

0.006

0.004

0.002

0.002

0.005

0.004

0.002

0.002

0.001

0.004

0.003

0.002

0.001

0.001


74/113


N

u 4.2 4.6 5.0 5.4 5.8 6.2 6.6 7.0 7.4 7.8


75/113

BFC21103 Hydraulics


0.000.02

0.04

0.06

0.08

0.0000.020

0.040

0.060

0.080

0.0000.020

0.040

0.060

0.080

0.0000.020

0.040

0.060

0.080

0.0000.020

0.040

0.060

0.080

0.0000.020

0.040

0.060

0.080

0.0000.020

0.040

0.060

0.080

0.0000.020

0.040

0.060

0.080

0.0000.020

0.040

0.060

0.080

0.0000.020

0.040

0.060

0.080

0.0000.020

0.040

0.060

0.080

0.100.12

0.14

0.16

0.18

0.1000.120

0.140

0.160

0.180

0.1000.120

0.140

0.160

0.180

0.1000.120

0.140

0.160

0.180

0.1000.120

0.140

0.160

0.180

0.1000.120

0.140

0.160

0.180

0.1000.120

0.140

0.160

0.180

0.1000.120

0.140

0.160

0.180

0.1000.120

0.140

0.160

0.180

0.1000.120

0.140

0.160

0.180

0.1000.120

0.140

0.160

0.180

0.20

0.22

0.24

0.26

0.28

0.200

0.220

0.240

0.260

0.280

0.200

0.220

0.240

0.260

0.280

0.200

0.220

0.240

0.260

0.280

0.200

0.220

0.240

0.260

0.280

0.200

0.220

0.240

0.260

0.280

0.200

0.220

0.240

0.260

0.280

0.200

0.220

0.240

0.260

0.280

0.200

0.220

0.240

0.260

0.280

0.200

0.220

0.240

0.260

0.280

0.200

0.220

0.240

0.260

0.280

0.30

0.32

0.34

0.36

0.38

0.300

0.321

0.341

0.361

0.381

0.300

0.320

0.340

0.361

0.381

0.300

0.320

0.340

0.360

0.381

0.300

0.320

0.340

0.360

0.380

0.300

0.320

0.340

0.360

0.380

0.300

0.320

0.340

0.360

0.380

0.300

0.320

0.340

0.360

0.380

0.300

0.320

0.340

0.360

0.380

0.300

0.320

0.340

0.360

0.380

0.300

0.320

0.340

0.360

0.380


76/113


N

u 4.2 4.6 5.0 5.4 5.8 6.2 6.6 7.0 7.4 7.8


77/113

BFC21103 Hydraulics


0.700.71

0.72

0.73

0.74

0.7350.748

0.761

0.774

0.788

0.7270.740

0.752

0.765

0.779

0.7220.734

0.746

0.759

0.771

0.7170.729

0.741

0.753

0.766

0.7140.725

0.737

0.749

0.761

0.7110.723

0.734

0.746

0.757

0.7090.720

0.732

0.743

0.754

0.7080.719

0.730

0.741

0.752

0.7060.717

0.728

0.739

0.750

0.7050.716

0.727

0.737

0.748

0.750.76

0.77

0.78

0.79

0.8020.817

0.831

0.847

0.862

0.7920.806

0.820

0.834

0.849

0.7840.798

0.811

0.825

0.839

0.7780.791

0.804

0.817

0.831

0.7730.786

0.798

0.811

0.824

0.7690.781

0.794

0.806

0.819

0.7660.778

0.790

0.802

0.815

0.7630.775

0.787

0.799

0.811

0.7610.773

0.784

0.796

0.808

0.7600.771

0.782

0.794

0.805

0.80

0.81

0.82

0.83

0.84

0.878

0.895

0.912

0.931

0.949

0.865

0.881

0.897

0.914

0.932

0.854

0.869

0.885

0.901

0.918

0.845

0.860

0.875

0.890

0.906

0.838

0.852

0.867

0.881

0.897

0.832

0.846

0.860

0.874

0.889

0.828

0.841

0.854

0.868

0.883

0.824

0.837

0.850

0.863

0.877

0.820

0.833

0.846

0.859

0.873

0.818

0.830

0.842

0.855

0.869

0.85

0.86

0.87

0.88

0.89

0.969

0.990

1.012

1.035

1.060

0.950

0.970

0.990

1.012

1.035

0.935

0.954

0.973

0.994

1.015

0.923

0.940

0.959

0.978

0.999

0.913

0.930

0.947

0.966

0.986

0.904

0.921

0.937

0.955

0.974

0.897

0.913

0.929

0.946

0.964

0.892

0.907

0.922

0.939

0.956

0.887

0.901

0.916

0.932

0.949

0.882

0.896

0.911

0.927

0.943


N

u 4.2 4.6 5.0 5.4 5.8 6.2 6.6 7.0 7.4 7.8


78/113

BFC21103 Hydraulics


0.900.91

0.92

0.93

0.94

1.0871.116

1.148

1.184

1.225

1.0601.088

1.117

1.151

1.188

1.0391.064

1.092

1.123

1.158

1.0211.045

1.072

1.101

1.134

1.0071.029

1.054

1.081

1.113

0.9941.016

1.040

1.066

1.095

0.9841.004

1.027

1.052

1.080

0.9750.995

1.016

1.040

1.067

0.9670.986

1.007

1.030

1.055

0.9600.979

0.999

1.021

1.045

0.9500.960

0.970

0.975

0.980

1.2721.329

1.402

1.447

1.502

1.2321.285

1.351

1.393

1.443

1.1991.248

1.310

1.348

1.395

1.1721.217

1.275

1.311

1.354

1.1491.191

1.245

1.279

1.319

1.1291.169

1.220

1.252

1.290

1.1121.150

1.198

1.228

1.264

1.0971.134

1.179

1.208

1.242

1.0851.119

1.163

1.190

1.222

1.0731.107

1.148

1.174

1.205

0.985

0.990

0.995

0.999

1.000

1.573

1.671

1.838

2.223

∞

1.508

1.598

1.751

2.102

∞

1.454

1.537

1.678

2.002

∞

1.409

1.487

1.617

1.917

∞

1.371

1.443

1.565

1.845

∞

1.338

1.406

1.520

1.780

∞

1.310

1.373

1.481

1.725

∞

1.285

1.345

1.446

1.678

∞

1.263

1.320

1.416

1.635

∞

1.244

1.298

1.389

1.596

∞

1.001

1.005

1.010

1.015

1.020

1.417

1.036

0.873

0.778

0.711

1.264

0.915

0.766

0.680

0.620

1.138

0.817

0.681

0.602

0.546

1.033

0.736

0.610

0.537

0.486

0.951

0.669

0.551

0.483

0.436

0.870

0.611

0.501

0.438

0.394

0.803

0.562

0.459

0.399

0.358

0.746

0.519

0.422

0.366

0.327

0.697

0.481

0.390

0.337

0.300

0.651

0.448

0.361

0.311

0.277


N

u 4.2 4.6 5.0 5.4 5.8 6.2 6.6 7.0 7.4 7.8


79/113

BFC21103 Hydraulics


1.031.04

1.05

1.06

1.07

0.6180.554

0.504

0.464

0.431

0.5350.477

0.432

0.396

0.366

0.4690.415

0.374

0.342

0.315

0.4150.365

0.328

0.298

0.273

0.3700.324

0.290

0.262

0.239

0.3320.290

0.258

0.232

0.211

0.3000.261

0.231

0.207

0.188

0.2730.236

0.208

0.186

0.168

0.2500.215

0.189

0.168

0.151

0.2290.196

0.172

0.152

0.136

1.081.09

1.10

1.11

1.12

0.4030.379

0.357

0.338

0.321

0.3410.319

0.299

0.282

0.267

0.2920.272

0.254

0.239

0.225

0.2520.234

0.218

0.204

0.192

0.2200.204

0.189

0.176

0.165

0.1940.179

0.165

0.154

0.143

0.1720.158

0.145

0.135

0.125

0.1530.140

0.129

0.119

0.110

0.1370.125

0.114

0.105

0.097

0.1230.112

0.102

0.094

0.086

1.13

1.14

1.15

1.16

1.17

0.305

0.291

0.278

0.266

0.254

0.253

0.240

0.229

0.218

0.208

0.212

0.201

0.191

0.181

0.173

0.181

0.170

0.161

0.153

0.145

0.155

0.146

0.137

0.130

0.123

0.134

0.126

0.118

0.111

0.105

0.117

0.109

0.102

0.096

0.090

0.102

0.095

0.089

0.083

0.078

0.090

0.084

0.078

0.072

0.068

0.080

0.074

0.068

0.064

0.059

1.18

1.19

1.20

1.22

1.24

0.244

0.235

0.226

0.209

0.195

0.199

0.191

0.183

0.168

0.156

0.165

0.157

0.150

0.138

0.127

0.138

0.131

0.125

0.114

0.104

0.116

0.110

0.105

0.095

0.086

0.099

0.093

0.089

0.080

0.072

0.085

0.080

0.076

0.067

0.061

0.073

0.069

0.065

0.057

0.051

0.063

0.059

0.056

0.049

0.044

0.055

0.052

0.048

0.042

0.037


N

u 4.2 4.6 5.0 5.4 5.8 6.2 6.6 7.0 7.4 7.8


80/113

BFC21103 Hydraulics


1.261.28

1.30

1.32

1.34

0.1820.170

0.160

0.150

0.142

0.1450.135

0.126

0.118

0.110

0.1170.108

0.100

0.093

0.087

0.0950.088

0.081

0.075

0.069

0.0790.072

0.066

0.061

0.056

0.0650.059

0.054

0.050

0.045

0.0550.049

0.045

0.041

0.037

0.0460.041

0.037

0.034

0.031

0.0390.035

0.031

0.028

0.025

0.0330.030

0.026

0.024

0.021

1.361.38

1.40

1.42

1.44

0.1340.127

0.120

0.114

0.108

0.1030.097

0.092

0.087

0.082

0.0810.076

0.071

0.067

0.063

0.0640.060

0.056

0.052

0.049

0.0520.048

0.044

0.041

0.038

0.0420.038

0.035

0.033

0.030

0.0340.031

0.029

0.026

0.024

0.0280.025

0.023

0.021

0.019

0.0230.021

0.019

0.017

0.016

0.0190.017

0.015

0.014

0.013

1.46

1.48

1.50

1.55

1.60

0.103

0.098

0.093

0.083

0.074

0.077

0.073

0.069

0.061

0.054

0.059

0.056

0.053

0.046

0.040

0.046

0.043

0.040

0.035

0.030

0.036

0.033

0.031

0.026

0.023

0.028

0.026

0.024

0.020

0.017

0.022

0.021

0.019

0.016

0.013

0.018

0.016

0.015

0.012

0.010

0.014

0.013

0.012

0.010

0.008

0.011

0.010

0.010

0.008

0.006

1.65

1.70

1.75

1.80

1.85

0.067

0.060

0.054

0.049

0.045

0.048

0.043

0.038

0.035

0.031

0.035

0.031

0.027

0.024

0.022

0.026

0.023

0.020

0.017

0.015

0.019

0.017

0.014

0.013

0.011

0.014

0.012

0.010

0.009

0.008

0.011

0.009

0.008

0.007

0.006

0.008

0.007

0.006

0.005

0.004

0.006

0.005

0.004

0.004

0.003

0.005

0.004

0.003

0.003

0.002


N

u 4.2 4.6 5.0 5.4 5.8 6.2 6.6 7.0 7.4 7.8


81/113

BFC21103 Hydraulics


1.901.95

2.00

2.10

2.20

0.0410.038

0.035

0.030

0.025

0.0280.026

0.023

0.020

0.016

0.0200.018

0.016

0.013

0.011

0.0140.012

0.011

0.009

0.007

0.0100.009

0.008

0.006

0.005

0.0070.006

0.005

0.004

0.003

0.0050.004

0.004

0.003

0.002

0.0040.003

0.003

0.002

0.001

0.0030.002

0.002

0.001

0.001

0.0020.002

0.001

0.001

0.001

2.32.4

2.5

2.6

2.7

0.0220.019

0.017

0.015

0.013

0.0140.012

0.010

0.009

0.008

0.0090.008

0.006

0.005

0.005

0.0060.005

0.004

0.003

0.003

0.0040.003

0.003

0.002

0.002

0.0030.002

0.002

0.001

0.001

0.0020.001

0.001

0.001

0.001

0.001

0.001

0.001

0.001

0.000

0.001

0.001

0.000

0.000

0.000

0.0010.000

0.000

0.000

0.000

2.8

2.9

3.0

3.5

4.0

0.012

0.010

0.009

0.006

0.004

0.007

0.006

0.005

0.003

0.002

0.004

0.004

0.003

0.002

0.001

0.002

0.002

0.002

0.001

0.000

0.001

0.001

0.001

0.001

0.000

0.001

0.001

0.001

0.000

0.000

0.001

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

4.5

5.0

6.0

7.0

8.0

0.003

0.002

0.001

0.001

0.000

0.001

0.001

0.000

0.000

0.000

0.001

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000


N

u 4.2 4.6 5.0 5.4 5.8 6.2 6.6 7.0 7.4 7.8


82/113

BFC21103 Hydraulics


9.010.0

20.0

0.0000.000

0.000

0.0000.000

0.000

0.0000.000

0.000

0.0000.000

0.000

0.0000.000

0.000

0.0000.000

0.000

0.0000.000

0.000

0.0000.000

0.000

0.0000.000

0.000

0.0000.000

0.000


N

u 8.2 8.6 9.0 9.4 9.8


83/113

BFC21103 Hydraulics


0.000.02

0.04

0.06

0.08

0.0000.020

0.040

0.060

0.080

0.0000.020

0.040

0.060

0.080

0.0000.020

0.040

0.060

0.080

0.0000.020

0.040

0.060

0.080

0.0000.020

0.040

0.060

0.080

0.100.12

0.14

0.16

0.18

0.1000.120

0.140

0.160

0.180

0.1000.120

0.140

0.160

0.180

0.1000.120

0.140

0.160

0.180

0.1000.120

0.140

0.160

0.180

0.1000.120

0.140

0.160

0.180

0.20

0.22

0.24

0.26

0.28

0.200

0.220

0.240

0.260

0.280

0.200

0.220

0.240

0.260

0.280

0.200

0.220

0.240

0.260

0.280

0.200

0.220

0.240

0.260

0.280

0.200

0.220

0.240

0.260

0.280

0.30

0.32

0.34

0.36

0.38

0.300

0.320

0.340

0.360

0.380

0.300

0.320

0.340

0.360

0.380

0.300

0.320

0.340

0.360

0.380

0.300

0.320

0.340

0.360

0.380

0.300

0.320

0.340

0.360

0.380


N

u 8.2 8.6 9.0 9.4 9.8


84/113

BFC21103 Hydraulics Tan et al. ([email protected])

0.400.42

0.44

0.46

0.48

0.4000.420

0.440

0.460

0.480

0.4000.420

0.440

0.460

0.480

0.4000.420

0.440

0.460

0.480

0.4000.420

0.440

0.460

0.480

0.4000.420

0.440

0.460

0.480

0.500.52

0.54

0.56

0.58

0.5000.520

0.540

0.561

0.581

0.5000.520

0.540

0.560

0.581

0.5000.520

0.540

0.560

0.580

0.5000.520

0.540

0.560

0.580

0.5000.520

0.540

0.560

0.580

0.60

0.61

0.62

0.63

0.64

0.601

0.611

0.621

0.632

0.642

0.601

0.611

0.621

0.631

0.641

0.601

0.611

0.621

0.631

0.641

0.600

0.611

0.621

0.631

0.641

0.600

0.610

0.621

0.631

0.641

0.65

0.66

0.67

0.68

0.69

0.652

0.662

0.673

0.683

0.694

0.652

0.662

0.672

0.683

0.693

0.651

0.662

0.672

0.682

0.692

0.651

0.661

0.672

0.682

0.692

0.651

0.661

0.671

0.681

0.692


N

u 8.2 8.6 9.0 9.4 9.8


85/113

BFC21103 Hydraulics Tan et al. ([email protected])

0.700.71

0.72

0.73

0.74

0.7040.715

0.726

0.736

0.747

0.7040.714

0.725

0.735

0.746

0.7030.713

0.724

0.734

0.745

0.7020.713

0.723

0.734

0.744

0.7020.712

0.723

0.733

0.744

0.750.76

0.77

0.78

0.79

0.7580.769

0.780

0.792

0.804

0.7570.768

0.779

0.790

0.802

0.7560.767

0.778

0.789

0.800

0.7550.766

0.777

0.788

0.799

0.7540.765

0.776

0.787

0.798

0.80

0.81

0.82

0.83

0.84

0.815

0.827

0.839

0.852

0.865

0.813

0.825

0.837

0.849

0.862

0.811

0.823

0.835

0.847

0.860

0.810

0.822

0.833

0.845

0.858

0.809

0.820

0.831

0.844

0.856

0.85

0.86

0.87

0.88

0.89

0.878

0.892

0.907

0.921

0.937

0.875

0.889

0.903

0.918

0.933

0.873

0.886

0.900

0.914

0.929

0.870

0.883

0.897

0.911

0.925

0.868

0.881

0.894

0.908

0.922


86/113


N

u 8.2 8.6 9.0 9.4 9.8


87/113

BFC21103 Hydraulics


1.031.04

1.05

1.06

1.07

0.2120.173

0.158

0.140

0.123

0.1950.165

0.143

0.127

0.112

0.1810.152

0.132

0.116

0.102

0.1700.143

0.124

0.106

0.094

0.1590.134

0.115

0.098

0.086

1.081.09

1.10

1.11

1.12

0.1110.101

0.092

0.084

0.077

0.1010.091

0.083

0.075

0.069

0.0920.082

0.074

0.067

0.062

0.0840.075

0.067

0.060

0.055

0.0770.069

0.062

0.055

0.050

1.13

1.14

1.15

1.16

1.17

0.071

0.065

0.061

0.056

0.052

0.063

0.058

0.054

0.050

0.046

0.056

0.052

0.048

0.045

0.041

0.050

0.046

0.043

0.040

0.036

0.045

0.041

0.038

0.035

0.032

1.18

1.19

1.20

1.22

1.24

0.048

0.045

0.043

0.037

0.032

0.042

0.039

0.037

0.032

0.028

0.037

0.034

0.032

0.028

0.024

0.033

0.030

0.028

0.024

0.021

0.029

0.027

0.025

0.021

0.018


N

u 8.2 8.6 9.0 9.4 9.8

1 26 0 028 0 024 0 021 0 018 0 016


88/113

BFC21103 Hydraulics


1.261.28

1.30

1.32

1.34

0.0280.025

0.022

0.020

0.018

0.0240.021

0.019

0.017

0.015

0.0210.018

0.016

0.014

0.012

0.0180.016

0.014

0.012

0.010

0.0160.014

0.012

0.010

0.009

1.361.38

1.40

1.42

1.44

0.0160.014

0.013

0.011

0.010

0.0130.012

0.011

0.009

0.008

0.0110.010

0.009

0.008

0.007

0.0090.008

0.007

0.006

0.006

0.0080.007

0.006

0.005

0.005

1.46

1.48

1.50

1.55

1.60

0.009

0.009

0.008

0.006

0.005

0.008

0.007

0.006

0.005

0.004

0.006

0.005

0.005

0.004

0.003

0.005

0.004

0.004

0.003

0.002

0.004

0.004

0.003

0.003

0.002

1.65

1.70

1.75

1.80

1.85

0.004

0.003

0.002

0.002

0.002

0.003

0.002

0.002

0.001

0.001

0.002

0.002

0.002

0.001

0.001

0.002

0.001

0.001

0.001

0.001

0.001

0.001

0.001

0.001

0.001


N

u 8.2 8.6 9.0 9.4 9.8

1 90 0 001 0 001 0 001 0 001 0 000


89/113

BFC21103 Hydraulics


1.901.95

2.00

2.10

2.20

0.0010.001

0.001

0.001

0.000

0.0010.001

0.001

0.000

0.000

0.0010.001

0.000

0.000

0.000

0.0010.000

0.000

0.000

0.000

0.0000.000

0.000

0.000

0.000

2.32.4

2.5

2.6

2.7

0.0000.000

0.000

0.000

0.000

0.0000.000

0.000

0.000

0.000

0.0000.000

0.000

0.000

0.000

0.0000.000

0.000

0.000

0.000

0.0000.000

0.000

0.000

0.000

2.8

3.0

4.0

5.0

6.0

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

7.0

8.0

9.0

10.0

20.0

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

Steps in direct integration method:


90/113

BFC21103 Hydraulics


1. Calculate y o and y c

2. Determine N and M

3. Calculate J

4. Calculate u and v

5. Find F (u, N) and F (v , J)

6. Calculate length of the reach

Activity 4.7

A very wide river with Manning roughness n = 0.035 has uniform depth of

3 0 m and longitudinal slope of 0 0005 Based on direct integration method


91/113

BFC21103 Hydraulics


3.0 m

and

longitudinal

slope

of

0.0005.

Based

on

direct

integration

method,

estimate the length of nonuniform flow produced by a weir that caused the

water surface to increase as much as 1.5 m upstream of weir.

4.5 my o = 3 m

So = 0.0005; n = 0.035

Step 1. Calculate y o and y c

y o = 3.0 m

12


92/113

BFC21103 Hydraulics


2

1

o3

2

o

1SRy

nq =

(For very

wide

channel,

R ≈ y )

2

1

o

3

5

o

1

Sy nq =

2

1

3

5

0005.03035.0

1××=q

/sm987.3 2=q

3

123

12

81.9

987.3⎟⎟

⎞⎜⎜

⎝

⎛ =⎟⎟

⎞⎜⎜

⎝

⎛ =

g

qy

c

m175.1=cy

profileMm175.1m3m4.5tom3 1o →=>=>= cy y y


93/113

Step 5. Find F (u1, N), F (u2, N), F (v 1, J), and F (v 2, J)

( ) ( ) 907.1333.3,001.1,1 == F NuF


94/113

BFC21103 Hydraulics


( ) [ ] [ ]⎭⎬⎫

⎩⎨⎧ −×⎟

⎠ ⎞⎜

⎝ ⎛ +−−−= 786.23342.0

3

5.2

3

175.1907.11884.0001.15.10005.0

3 3L

( ) ( ) 1884.0333.3,5.1,2 == F NuF

( ) ( ) 786.25.2,001.1,1 == F Jv F

( ) ( ) 3342.05.2,717.1,2 == F Jv F

Step

6.

Calculate

length

of

channel

reach

( ) ( ) ( )[ ] ( ) ( )[ ]⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−⎟⎟ ⎠

⎞⎜⎜⎝

⎛ +−−−=− Jv F Jv F

M

J

y

y NuF NuF uu

S

y x x

M

c ,,,, 12o

1212

o

o12

m 05.12569=L


95/113

BFC21103 Hydraulics


4.5 my o = 3 m

So = 0.0005;

n =

0.035

y = 3.003 m

L = 12569.05 m

B. Numerical IntegrationSection Equations used

All sections 2K ⎤⎡ ⎞⎛ ⎤⎡2TQ


96/113

BFC21103 Hydraulics


All sections

Rectangular

Very wide channel

(Chezy)

3

2

oo

1

1

d

d

gA

T Q

K

K S

x

y ave

−

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟ ⎠

⎞⎜⎜⎝

⎛ −

=

⎥

⎥⎥⎥⎥

⎦

⎤

⎢

⎢⎢⎢⎢

⎣

⎡

⎟⎟

⎠

⎞

⎜⎜⎝

⎛ −

−

= 2

o

3

o1

1d

d

aveK

K

gA

T Q

S

y x

3

2

oo

1

1

d

d

Date post:	07-Jul-2018
Category:	Documents
Upload:	ashraf-haji
View:	240 times
Download:	0 times

BFC21103 Chapter4.pdf

Documents