Hydraulics Engineering Lec #2 : Surface Profiles and ... · Non Uniform Flow or Varied Flow. c S For uniform flow through open channel, dy/dl is equal to zero. However for non uniform

Hydraulics Engineering

Lec #2 : Surface Profiles and Backwater

Curves in Channels of Uniform

sections

Prof. Dr. Abdul Sattar Shakir

Department of Civil Engineering

Steady Flow in Open Channels

Specific Energy and Critical Depth

Surface Profiles and Backwater Curves in Channels of Uniform sections

Hydraulics jump and its practical applications.

Flow over Humps and through Constrictions

Broad Crested Weirs and Venturi Flumes

Types of Bed Slopes

Mild Slope (M) yo>yc

So<Sc

Critical Slope (C) yo=yc

So=Sc

Steep Slope (S) yo<yc

So>Sc

So1<Sc

So2>Sc

yo1

yo2

yc

Break

Occurrence of Critical Depth

Change in Bed Slope

Sub-critical to Super-Critical

Control Section

Super-Critical to Sub-Critical

Hydraulics Jump

Control Section

So1<Sc

So2>Sc

yo1

yo2

yc

Break where

Slope changes

Dropdown Curve

So1>Sc

So2<Sc

yo1

yo2

yc

Hydraulic Jump

Occurrence of Critical Depth

Change in Bed Slope

Free outfall

Mild Slope

Free Outfall

Steep Slope

yb~ 0.72 yc

So<Sc

yo yc

3~10 yc Brink

yc

So>Sc

Non Uniform Flow or Varied Flow.

For uniform flow through open channel, dy/dl is equal to zero. However for non uniform flow the gravity force and frictional resistance are not in balance. Thus dy/dl is not equal to zero which results in non-uniform flow.

There are two types of non uniform flows. In one the changing condition extends over a long distance and this is called gradually varied flow. In the other the change may occur over very abruptly and the transition is thus confined to a short distance. This may be designated as a local non uniform flow phenomenon or rapidly varied flow.

So1<Sc

So2>Sc

yo1

yo2

yc

Break

Energy Equation for Gradually Varied Flow.

1Z

g

V

2

2

1

Datum

So

1y

2Z

g

V

2

2

2

2y

HGL

EL

Water Level

lhg

VyZ

g

VyZ

22

2

222

2

111

Theoretical EL

Sw

hL

X

L

S


profilesurfacewateroflengthLWhere

SS

EEL

LSLSEE

Now

forL

ZZ

X

ZZS

L

hS

hZZg

Vy

g

Vy

o

o

o

oL

L

)1(

6,

22

21

21

2121

21

2

22

2

11

An approximate analysis of gradually varied, non uniform flow can be achieved by

considering a length of stream consisting of a number of successive reaches, in

each of which uniform occurs. Greater accuracy results from smaller depth

variation in each reach.


3/4

22

2/13/21

m

m

mm

R

nVS

SRn

V

The Manning's formula is applied to average conditions in each reach to

provide an estimate of the value of S for that reach as follows;

2

2

21

21

RRR

VVV

m

m

In practical depth range of the interest is divided into small increments,

usually equal, which define the reaches whose lengths can be found by

equation ( 1 )

Water Surface Profiles in Gradually Varied Flow.

1Z

g

V

2

2

1

Datum

So

1y

2Z

g

V

2

2

2

2y

HGL

EL

Water Level

g

VyZHeadTotal

2

2

Theoretical EL

Sw

hL

X

L


01

0

)2(1

.

1

1

2

.tan..

2

2

3

22

3

2

2

2

N

N

NNo

F

SSo

dx

dyflowuniformFor

F

SSo

dx

dyor

gsindecreaisflow

ofdirectionalongheadtotalthatshowssignve

gy

qFF

dx

dySS

gy

q

dx

dy

dx

dZ

dx

dH

gulartanrecastionseccrossgConsiderin

gy

q

dx

d

dx

dy

dx

dZ

dx

dH

xdirectionhorizontalincedistrwHheadtotaltheatingDifferenti

Equation (2) is dynamic Equation for

gradually varied flow for constant value

of q and n

If dy/dx is +ve the depth of flow

increases in the direction of flow and

vice versa


3/10

3/10

22

3/10

22

2/13/5

2/13/2

1

1

y

y

S

S

y

qnS

channel

gulartanrecinflowuniformFor

y

qnS

orSyn

q

orSyn

V

yR

channelgulartanrecwideaFor

o

o

o

o

21 F

SS

dx

dy o

Consequently, for constant q and n, when y>yo, S<So, and the numerator is +ve. Conversely, when y<yo, S>So, and the numerator is –ve.

To investigate the denominator we observe that, if F=1, dy/dx=infinity; if F>1, the denominator is -ve; and if F<1, the denominator is +ve.

Classification of Surface Profiles

Mild Slope (M) yo>yc

So<Sc

Critical Slope (C) yo=yc

So=Sc

Steep Slope (S) yo<yc

So>Sc

Horizontal (H) So=0

Adverse (A) So=-ve

Type 1: if the stream surface lies above both the normal and critical depth of flow. (M1, S1)

Type 2: if the stream surface lies between normal and critical depth of flow. (M2, S2)

Type 3: if the stream surface lies below both the normal and critical depth of flow. (M3, S3)

Water Surface Profiles Mild Slope (M)

1

2

3

1:1

2 :1

3:1

oo c

N

oo c

N

oo c

N

S Sdy Vey y y Ve M

dx F Ve

S Sdy Vey y y Ve M

dx F Ve

S Sdy Vey y y Ve M

dx F Ve

yc

Note:

For Sign of Numerator computer

yo & y

For sign of denominator compare

yc & y

If y>yo then S<So and Vice Versa

Water Surface Profiles Steep Slope (S)

1

2

3

1:1

2 :1

3:1

oc o

N

oc o

N

oc o

N

S Sdy Vey y y Ve S

dx F Ve

S Sdy Vey y y Ve S

dx F Ve

S Sdy Vey y y Ve S

dx F Ve

Note:


yo & y


yc & y


yc

Water Surface Profiles Critical (C)

1

3

1:1

2 :1

oo c

N

oo c

N

S Sdy Vey y y Ve C

dx F Ve

S Sdy Vey y y Ve C

dx F Ve

Note:


yo & y


yc & y


yo=yc

C2 is not possible

Water Surface Profiles Horizontal (H)

( ) 2

( ) 3

1:1

2 :1

oo c

N

oo c

N

S Sdy Vey y y Ve H

dx F Ve

S Sdy Vey y y Ve H

dx F Ve

Note:


yo & y


yc & y


yc

H1 is not possible bcz water has to lower down

Water Surface Profiles Adverse (A)

( ) 2

( ) 3

1:1

2 :1

oo c

N

oo c

N

S Sdy Vey y y Ve A

dx F Ve

S Sdy Vey y y Ve A

dx F Ve

Note:


yo & y


yc & y


yc

A1 is not possible bcz water has to lower down

Problem 11.59

A rectangular flume of planer

timber (n=0.012) is 1.5 m wide

and carries 1.7m3/sec of water.

The bed slope is 0.0006, and

at a certain section the depth is

0.9m. Find the distance (in one

reach) to the section where

depth is 0.75m. Is the distance

upstream or downstream?

3

1

2

Rectangular Channel

0.012

1.5

1.7 / sec

0.0006

0.9

0.75

o

n

B m

Q m

S

y m

y

B

y

Problem 11.59

Solution

1 2

2 2

4 /3

2

1

2

2

1

2

1 1 1

2 2 2

1 1

2 2

&

1.5 0.9 1.35

1.5 0.75 1.125

1.5 2 0.9 3.3

1.5 2 0.75 3

/ 0.41

/ 0.375

/ 1.26 / sec

/ 1.51 / sec

o

m

m

Since

E EL

S S

V nS

R

A x m

A x m

P x m

P x m

R A P

R A P

V Q A m

V Q A m

2 2

4/3

1 2

2 2

1 21 2

0.3925

1.385

& 0.000961

2 2

317.73

m

m

m

m

o

o

R m

V m

V nS

R

E ENow L

S S

V Vy y

g gL

S S

m Downstream

Problem 11.66 The slope of a stream of a

rectangular cross section is

So=0.0002, the width is 50m, and the

value of Chezy C is 43.2 m1/2/sec.

Find the depth for uniform flow of

8.25 m3/sec/m of the stream. If a

dam raises the water level so that at

a certain distance upstream the

increase is 1.5m, how far from this

latter section will the increase be

only 30cm? Use reaches with 30cm

depth.

Given That

1.5m

yo

0.3m

1/ 2

3

0.0002

50

43.2 / sec

8.25 / sec/

508.25 43.2 0.0002

50 2

6.1

o

oo o

o

oo

o

o

S

B m

C m

q m m

Aq y C S

P

yy

y

y m

Problem 11.66

y A P R V E E1-E2 Vm Rm S S-So ΔL ΣΔL

m m2 m m m/s m m m/s m m/m m/m m m

7.6 380 65.2 5.82 1.09 7.66

0.295

1.11

5.74

0.000115

-0.000085

-3454.33

-3454.33

7.3

7.0

6.7

6.4

V=C(RS)1/2

Assignment

Problems:

11.60, 11.63, 11.64, 11.65, 11.72, 11.73,

11.74, 11.75

Date of Submission:

Documents

Hydraulics Engineering Lec #2 : Surface Profiles and ... · Non Uniform Flow or Varied Flow. c S For uniform flow through open channel, dy/dl is equal to zero. However for non uniform