1

Hidayat Jamal & Zulhilmi

ismail

Hydraulics & Hydrology

Department, FKA, UTM

Chapter 2

2

Uniform flow

■ Equations are developed for steady-state conditions

– Depth, discharge, area, velocity all constant along

channel length

■ Rarely occurs in natural channels (even for constant

geometry) since it implies a perfect balance of all

forces

■ Two general equations in use: Chezy and Manning

formulas

3

Or in general, uniform flow

- Energy slope = Bed slope or dH/dx = dz/dx

- Water surface slope = Bed slope = dy/dz = dz/dx

- Velocity and depth remain constant with x

4

Q = C

V = C

y = C

S0 = C

A = C

B = C

n = C

UNIFORM FLOW

Normal depth implies that flow rate, velocity,

depth, bottom slope, area, top width, and roughness

remain constant within a prismatic channel as

shown below

5

6

General Flow Equation

Q = Av

Flow rate (m3/s)

Avg. velocity of

flow at a cross-

section (m/s)Area of the

cross-section

(m2)

7

Chezy’s Formula

Antoine Chezy, in 1769

A French civil engineer developed the

Chézy equation, which relates the uniform

flow velocity to channel roughness,

hydraulic radius, and bed slope.

R = hydraulic radius

So = slope of the channel

C = coefficient depending upon the

various characteristics of the

channel and their comparison with

those of another similar channel

oRSACQ

8

Other Flow Formulas Two other flow formula defined more accurately the value

for the coefficient C in the Chezy formula.

(a) Bazin formula:

R

mC

1

87

Values of m to be used in the Bazin formula for determining C in the chezy formula.

9

Values of n to be used in the Kutter’s nformula for determining C in the chezy formula.

))00155.0

23(1

00155.0123

R

nx

S

SnC

o

o

(b) Kutter formula:

10

Manning Formula

Robert Manning, 1885;

Developed Manning formula used for

open channel flow conditions.

R = hydraulic radius

So = slope of the energy gradient

n = a roughness coefficient

2 13 2

oAR SQ

n

(Refer to the Table)

A and R = function of y

11

Table of Manning’s n Roughness

CoefficientType of Channel and Description Minimum Normal Maximum

Streams

Streams on plain

Clean, straight, full stage, no rifts or deep pools 0.025 0.03 0.033

Clean, winding, some pools, shoals, weeds & stones 0.033 0.045 0.05

Same as above, lower stages and more stones 0.045 0.05 0.06

Sluggish reaches, weedy, deep pools 0.05 0.07 0.07

Very weedy reaches, deep pools, or floodways 0.075 0.1 0.15

with heavy stand of timber and underbrush

Mountain streams, no vegetation in channel, banks

steep, trees & brush along banks submerged at

high stages

Bottom: gravels, cobbles, and few boulders 0.03 0.04 0.05

Bottom: cobbles with large boulders 0.04 0.05 0.07

12

Channel Conditions Values

Material Involved Earth n0 0.025

Rock Cut 0.025

Fine Gravel 0.024

Coarse Gravel 0.027

Degree of irregularity Smooth n1 0.000

Minor 0.005

Moderate 0.010

Severe 0.020

Variations of Channel Cross

Section Gradual n2 0.000

Alternating Occasionally 0.005

Alternating Frequently 0.010-0.015

Relative Effect of Obstructions Negligible n3 0.000

Minor 0.010-0.015

Appreciable 0.020-0.030

Severe 0.040-0.060

Vegetation Low n4 0.005-0.010

Medium 0.010-0.025

High 0.025-0.050

Very High 0.050-0.100

Degree of Meandering Minor m5 1.000

Appreciable 1.150

Severe 1.300

n = (n0 + n1 + n2 +

n3 + n4 ) m5

Values for the computation of the

roughness coefficient (Chow,

1959)

13

Values of Manning’s n according

to MASMA

Open Channel (Surface Cover of Finishing) Manning’s n (Minimum)

Manning’s n (Maximum)

Grass Swales

Short grass cover 0.030 0.035

Tall grass cover 0.035 0.053

Lined Drains

Concrete

Troweled finished 0.011 0.015

Off form finished 0.013 0.018

Stone Pitching

Dressed stones in Mortar 0.015 0.017

Random stones or rubble masonry 0.020 0.035

Rock riprap 0.025 0.030

Brickwork 0.012 0.018

Precast masonry blockwork 0.012 0.015

14

Manning’s Roughness (n)

Roughness coefficient (n) is a function of:

– Channel material

– Surface irregularities

– Variation in shape

– Vegetation

– Flow conditions

– Channel obstructions

– Degree of meandering

15

Conveyance Factor, K

K, the capacity of the channel to carry flow

Simply calculated from Chezy or Manning formulae

oRSACQ Chezy Formula:

where

Manning formulae:

where

RACK

oSARn

Q 321

n

ARK

32

16

Channel Section Factor, Z

Or,

32

ARZ

oS

nQnKAR 3

2

Z, the characteristics of channel geometry

for Manning

12Z AR for Chezy

for Manning

12

o

K QAR

C C S for Chezy

17

Flow Rate Per Unit Width, q

For rectangular channel only

B

Qq

B

y

unit as or

m

sm /3

msm ./3

Basic, Q = Av

q = vy

18

Very Wide Channel

Shallow flow depth compared to the channel width

Very wide channel;

So, remember

yB

yR

19

Calculation of Normal

Depth, yo

Can be calculated using either:

(i) Trial and error method

(ii) Graphical method

(iii) Charts

B

yo

B

yo

Z

1

20

Trial and error method

Given the flow condition (channel dimension,roughness, flow rate and bed slope)

Normal depth is calculated using the Manning orChezy flow formula by trial and error method

Graphical method

A plot of y vs Z (where ) is made

Normal depth is when

32

ARZ

oS

nQZ

21

Chart: Curve for Determining

Normal Depth (Chow, 1959)

10

Rectangularz = 0.5

z = 1.0

z = 1.5z = 2.0z = 2.5z = 3.0z = 3.5

0.0001 0.001 0.01 0.1 1 100.01

0.1

1

0.01

0.1

1

0.0001 0.001 0.01 0.1 10

Circular

y

1

zB

y

NOTE :

1

0

32

S

nQAR

38

32

38

32

Φ

ARor

B

AR

Φ

y

or

B

y

o

o

22

Best Hydraulic Section(BHS)

For a given Q, there are many channel shapes. Thereis the need to find the best proportions of B and y whichwill make the discharge is maximum.

Using Chezy's formula:

Flow rate:

For a rectangular Channel: P = B +2y

A = By and therefore: B = A/yi.e. P = A/y + 2y

oRSCV

)1( oo SP

AACRSACQ

B

y

23

For a given Area (A), Q will be maximum when P isminimum (from equation 1)

Differentiate P with respect to y

For minimum P i.e. Pmin ,

A = 2y2 ,

Since A = By ie. By = 2y2 ie. B = 2y

i.e. for maximum discharge, R = y/2

22

y

A

dy

dP

022

y

A

dy

dP

24

Area of cross section, A = By + zy2

Width, B = A/y - zy ----------------------(1)

Wetted Perimeter, P = B + 2y ( 1 + z2 )1/2

P = A/y - zy + 2y(1 + z2 ) 1/2---------------------(2)

Differentiate P with respect to y

B

y

Z

1

For a Trapezoidal Section

L

25

1) ; you got or

- Top water width (T) is twice the side length (L)

- Hydraulic Radius;

2) for ; you got

- The side slope is 60⁰ or B = L

0dy

dP 212 zyT LT 2

0dz

dP

2

yR

3

1z

26

Best Hydraulic Section

27

Why BHS???

Called “Most efficient cross-section”

Efficient cross-section lead in- Economical designed in reduce material to

construct the channel with minimum wettedparameter

- Convey maximum designed discharge

So, what channel shape is most efficient??

28

1) A smooth concrete-lined trapezoidal channel (n =

0.011) was constructed earlier in a development project.

The channel bottom width is 10 meter, total channel

depth (including freeboard of 300 mm) is 2.3 meter, side

slopes z = 2, and longitudinal slope is 0.0001. Determine

the discharge, Q.

2) If the channel has to be re-designed as the Best

Hydraulic Section (z = 0.58) to convey a similar

discharge, find the new size of the trapezoidal channel.

Maintain the type of lining material and longitudinal

slope for the channel. Sketch your result.

Example

29

Solution

10m

PAR /

2.3

m

n = 0.011

y = 2.3 - 0.3

= 2.0m

z = 2

So = 0.0001

A = (B +zy)y

2)2(210

228m

212 zyBP 2)2(1)2(210

m94.18

94.18/28

478.1

30

n

SARQ

o3

2

011.0

0001.0)478.1(28 32

Q

sm /03.33 3

Re-designed as the Best Hydraulic

Section as

z = 0.58

n = 0.011

So = 0.0001

Q = 33.03 m3/s

2732.1 y

22 58.0152.1 yy

From T = 2L

Then A = By +zy2

2122 zyzyB

yyB )58.0(2)58.0(12 2

yB 152.1

Cont…

31

212 zyBP 21.152 2 1 (0.58)y y

y464.3

PAR /

y

y

464.3

732.1 2

y5.0

n

SARQ

o3

2

0001.0

)011.0(03.333

2

AR

my 72.3

33.3632

AR

33.36)5.0(732.1 32

2 yy

299.3338

y

)72.3(152.1B

mB 29.4

Cont…

32

Final Design Diagram

y = 3.72mz = 0.58 Total depth

= 4.02m

B = 4.29m

T = 8.61m

T = B + 2zy = 4.29 + 2 x 0.58 x 3.72 = 8.61 m

Cont…

33

The Design of Circular Culverts-

Optimum Water Depth

y optimum for max discharge

34

y optimum for max velocity

35

Design of Erodible Channels

(Earthen Channels)