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CHAPTER 1 2 8

CONCEPT

OF

MINIMUM

SPECIFIC

ENERGY

AND

ITS

RELATION

TO

NATURAL

FORMS

by

Gordon

R .McKay

B.Eng.,Ph.D.(Liverpool)

and

Ahad

K.

Kazemipour

B.Eng.(Tehran),M.Eng.Sc.(NSW)

Iti s

proposedt o

show

inthis

paperthatthere

isasolution

t otheproblem ofnon-uniformflow

and

thissolution

not

only

explains

in

detailmany

land

forms

whichoccur

naturally,but

thereby,

yields

a

definition o f'form'

loss.

If

achannel,in

which

the

transversedistribution

of

specific

energy

is

uniform,

convergesand/ordiverges,andthe

bedchangesso

that

the

flow

willbecritical

at

all

cross-sections

at

the

sametime,

the

channel appears

to

beclose

to

beinghydraulicallysmooth.

Manynatural

forms,

particularly

estuaries,are

readily

explicable

in

this

way.

he

most

obvious

one

i sthebarat

themouth

of

a

river.

t

follows,

if

the

riverenters

the

sea

with

reasonable

uniform

grade,

which

most

rivers

d o ,

the

bed

must

rise

as

the

flow

loses

the

restrictinginfluence

of

the

banks

(i.e.

the

width

increases)

ifconstant

specificenergyi sto

bemaintained.

Iti spossible

to

calculate

with

considerable

accuracythe

dimensions

ofuseful

structuresbasedonthisconcept. largenumber

offull

size

but neverthelessexperimental

structureshave beenbuilt

making use o fthe

resultant benefits which

develop:-

low

turbulence,

accurate differential

water

levelsand a

clearlydefinedflowpattern,

allowingveryconsiderable

savings

t o

be

made byeliminating

expensive

protectiveworks.

Professor

of

CivilEngineering,

Universityof

Queensland,Australia.

Iranian GovernmentPostgraduate Research

Fellow

in Hydraulic

Engineering,University

of

Queensland,

Australia.

2186

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MINIMUM

SPECIFIC

ENERGY 187

INTRODUCTION

For

more

thantwo

centuries,

the

calculation

of

openchannel

flow,

steady

orunsteady,uniform or

non-uniform,

hasbeenbased

on

the

concept

of

a

longuniform channelin whichthe boundaryshearin

somewaydeterminesthe

total

head

loss. Euler-Bernoulliprinciple

(1700-1782),

Chezy's

(1718-1798),

Poiseuille's

(1799-1869),

Darcy-

Weisbach's(1803-1871),Bazin's(1829-1917)

and

Manning's

(1816-1897)

formulas,

allo f

whichare

in

regularuse

today,

are

basedonthe

concept

of

such

a

long

uniform

channel.

daptation

of

the

relatively

modern

Karman-Prandtle's

(1930-32)theory

of

boundary

layer

and

velocity

distribution

t o

openchannels

is

alsobased

On

the

concept

of

the

long

uniform

channel.

Thisassumption

wasreasonableinthepastforartificial

channels

sincemosto fthe

channel

works(sewersand

storm drains)

were

constructeds o

that

thecross-section

of

the

channelwasoften

semi-

circular

and

thereforethedistribution ofthe

shear

stress

around

the

boundary

could

be

consideredmore

or

less

uniform.

Therei s

strongevidence

intheliterature

1

*

thatthe

studies

of

energy

losses

in

openchannels

have

been

closely

related

to

the

phenomena

ofthe boundary

layer.or

fullydevelopedflowin

auniform

channel

of

any

size,

the

boundarylayer

will

occupy

all

of

the

channel,

butin partlydevelopedturbulentflow,

the

energylossduet o

frictional

resistance

i srelated

to

thestage

o fthedevelopment

of

the

boundary

layer.

gain,pastresearchon boundarylayer

has

been

concen-

trated

on

flow

through

circular

pipes

and

pastflatplates

parallel

t o

thestream.

Th epractisingengineer

may

rightlyquestion

the

abovephilos-

ophies

andassumptions,

but

what

alternativehas

been

possible?

rom

Reporto f

Task Force

on

Friction

Factors

in Open

Channels

1

-

At

east,

it

coulde

oped

here

would

e

made

available

omething

imilar

o

he

resistanceiagrams

ow

used

or

teady

low

n

uniform

pipes

nd

or

frictional

resistance

fhips.

Ithould

e

tated

at

he

beginning

thatheseopesannot

e

realised

at

hisime.

Principle

bstacles

are

he

wide

ange

of

urfaceoughness

izes

nd

ypes

ncountered

n

practical

hannels

(frommooth

concrete

inings

o

boulder-stream

canyons),

the

effect

ofbed

movement

n

unlined

channels,nd

he

numerousbendsnd

tructureshat

preventheattainment

of

teady,

uniform,ully

eveloped

low .

Inorder

to

allow a

solutiont o

the apparently

simple

problem

ofdetermining

the

flow

capacity

ofabridge

opening

orculvert,

the

U.S.BureauofPublicRoads

2

publishesa manual

of

90pagesof

tabul-

ated

data. If

the

bridgetypeand

flow

pattern

can

bereasonably

compared withone

of

thosein

the

manual,

a

solutioncan beassessed.

Such

techniques

do

little

to

validate

the

basic

principle.

*umbersrefer

to

the

referencesinthe Appendix.

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2188

OASTAL

ENGINEERING-197

6

Inderiving andapplyingthebasicformulaeandprinciples

of

uniform

flowt oopen

channels,

aquantity

'hydraulic

radius',R ,

representing

thechannel

geometry,

is

used.

iameter

ofthe

pipe,

D ,

is

replacedby

4R.

his

substitution

carries

withit

the

assumption

that

thedistribution ofshear

stress

around

theboundaryofthe

channel

is

uniform.t

is

well

known

now

thatin

open

channels,espec-

ially

wit h

non-circularcross-sections,

such

uniformity

of

shearstress

is

afalseassumption.

This

fact

clearly

pointsoutthefailure

of

hydraulic

radius

to

bethe

sole

geometricquantityrepresentative

of

the

channelcross-section.he

effect

of

cross-sectionalshapemust

be

takenintoaccountin

t h e

analysis

and

prediction

of

theenergy

lossesin

open

channels.

Suchstudies

have

beenmade

by

someinvestig-

ators

inrecent

years,

amongwhom

are

F.Engelund

(1964)

3

,H .Rouse

(1965)",

E.O.Macagno

(1965)

5

,

C.C.

Shih

and

N.S.Grigg

(1967)

6

,

E .

Marchi(1967)

7

,

N .Narayana Pillai(1970)

8

and

C.L.Yenand D.E.

Overton

(1973)

9

.he

results

of

these

investigationsare

yetincon-

clusive.

ccording

to

H .Rouse

1

,

the

ffectf

change

n

hapepon

theresistance

unctionisactuallywofold.

O n

heneand,t

pvoduaes

hange

n

he

wetted

perimeter,

P ,

per

unit

cross-sectional

area, A ,

thereciprocal

of

which

s

esignated

y

hehydraulic

radius,

R .

n

hether

and,it

produces

hangen

he

distributionof

velocityand

hear;

s

result,

the

hear

will

generallyary

rom

point

o

point

of

heperimeter

.

oth

ffects

rehusnvolved

n

theequilibriumrelationship

between

hegravitationalmotive

orce

and

heurfaceresistancewhichhe

low

entails .

He

supports

t h e

validity

ofthe

hydraulic

radiusconcept,

butconcludes

thatthe

effect

of

cross-sectional

shape

is

related

to

the

variation

inthe

hydraulic

radius

and

two

coefficients

of

asemi-logarithmic

resistance

function.

ealsorefers

to

the

importance

of

theaspect

ratio

n

the analysisinrelationtotheeffect

of

cross-sectional

shape.

The

junior

author

inanother

work*

has

proposed

other para-

meters

as

beingmore

representative

of

the

cross-sectional

shape

on

flow

resistancein

smoothchannels.his

method

offers

a

far

more

rational

solution

thanmethods

previously

proposed

byothers.

Iftheflowtakesplacein

a

naturalstream

with

erodiblebed

and

banks,

the

calculations

which

are

at

all

possibledo

littlet o

determinethe proportionsof

any

change

which

is

likely

to

occurand

in

fact

hardly assistinallowingpredictionsofeventhegeneralform

ofthechange.ecause

of

the

ever

varying

shapeand

size

of

the

channelcross-section andunavoidableirregularitiesinchannelalign-

ment,

the

flow

is

rarelyuniform

in

natural

streams.

Itshouldbe

appreciated

thatopen

channel

formulaebased

on

observations

of

pipe

experiments

or

from

small

scale

physical models

ofchannelsand

canals

orrivers,cannot

and

do

not

represent natural

situations because :

*

nder

preparation forpublication.

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MINIMUM

SPECIFIC

ENERGY

189

( i ) Thedimensionsandproportionsofnaturalriversarecompletely

different

from

those

of

the

pipes

for

which

energylosses

have

been

determined

experimentally. In

practice

the

pipes

are

almost

invariably

round,

ranging

in

diameterfrom1cm

t o

about

10m. Thecross-sectional

shape

ofa

natural

watercourseis

quiteindeterminate;the

width

can

vary

from

afewcentimetres

t o

many

kilometres,while

the

depth

varies

from

a

f ew

milli-

metres

t o

averylimited

amountcomparedwith

the

width.

( i i )

It

is

quite

impossiblet o

represent

allthedetailed

features

andirregularities

ofa

river

on

the

model.

ecannot

truly

conform

t othelaws

of

similarity.

Any

researcher

using

largearealandscapetype

models

soon

appreciates

that

suchmodels

are

fartoosmoothalthough

boundary

shear

relationshipssaytheyshouldbe

fartoo

rough.

Insmall

scale

models,losses

due

to

boundary

shear

willalways

berelativelylarge

because

of

the

effect

of

scale

on both

thesize

and

velocityinReynoldsnumber. However,t oassume

these

lossesrepresent

thetotalenergy

loss

mustleadt oseriouserror.

Al lenergy

losses

will

be

represented

as

turbulence.

Eddy

size

islargely afunction of

thesize

of

the

solid

boundary generating

the

eddy.

The

velocity

will

determine

thenumber

of

eddies.ddies

from

boundary

shearare

small

and

becauseof

their

size,dissipate

quickly.

Incontrasteddiesgenerated

by

channelirregularitiesandchangesin

cross-section

willbelargeand willpersistdownstream

addingconsider-

ably

t otheapparent roughness. Th ereaction of

movable

boundariesin

naturalstreamst olocallygeneratedturbulencecannot be

joinedwit h

the

overallaverageconditions.

Often

theenergylossesdue

t o

boundary

roughness,

cross-sectional

shape

and

the

boundary

irregularities

have

been confused

with

oneanother.

For

example,

H.A.EinsteinandN.L.Barbarossa(1951)

1

separatedthe

total

energy

lossof

the

naturalflow

intofrictional

losses

andform

losses. Frictional

losses

were

defined

asthose

due

t o

grain roughness

and

form

losses

those

duet osize,shapeandspacingoftheindividual

irregularitiesandpresenceofsandripples

anddunes..Bajorunas

11

,

inthe

discussionofthe

samepaper

statesthat:theroughnessfactor

that

reflects

the

channel

irregularitiesdecreases

and

approacheszero

wit h

increasingflow.

nthe

other

handin V.A.

Vanoni

and

Li

San

Hwang's

(1967)

results,

the

bedroughness

due

to

form

of

the

ripples

and

dunes

is

the

major

partofthe

total

roughness

and doesnot

approach

zero withincreasing

flow.

In

the

discussion of

EinsteinandBarbarossa's

paper,

Sir.

C .Inglis

categorisesthe

total

lossesinto

( i )

those

due

t otextural

roughness;

( i i )thoseduet o

ripple

roughness;

and(iii)

thosecausedby

form drag

resulting

from

the

majorirregularitiesof

the

banks,bed,islands

and

sandbanks.

Inglis

does

not

believein

combining

groups( i i )and

(iii)

because

ofa

different

time

scale

by

which

they

change.

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2190

OASTAL

ENGINEERING-1976

The

question i s ,whatconstitutesthetotalenergylossin

natural

rivers

and

howmuch

ofthe

total

lossi s

due

to

boundary-

shearand how

much

i sdue

to

form?

CONDITION

FORNOFORM

LOSS

It

i s

proposed

to

show

in

this

paper

that

there

isa

solution

totheproblemof

non-uniform f low

and

thissolutionnot

only

explains

precisely many

landformswhich

occurnaturally,butthereby

yields

a

definition of

'form'

loss.

Th e

only

channel

which

gives

truly

no

'form'

loss

is

of

rectangularsection.nany

otherchannel

the

specific

energy

must

vary acrossthechannel.hiswouldgenerateturbulence

not

associated

with boundaryshear.

he

elementsofflow

across

thesection

can

be

shown

to

respondin different

wayst o

an

overallchangeinsection

shape.

B.A.

Bakhmeteff

(1932)^

showed

that

if

the

flow

in

a

channel

is

tranquilandthechannelconvergesslowly,thesurface

will

fall,

the

velocity

will

increase

and,if

the

channelthen divergesslowlyt o

itsoriginalwidth,theflow

will

return

virtually

without

extra

loss,

t o

its

originaluniformflow

pattern;

i.e.

the'form'

loss

iszero.

f,

however,the

convergence

continues,the

flowwill

ultimately become

critical(i.e.,

the

depthyill be2/3(y V

2g)wheny

nd

v

arethedepth and velocityintheinitialchannel,g

is

theacceleration

due

to

gravity

and y

s

the critical

depth).ny

further

convergence

will

lead

t o

a

rise

level

upstreamoftheconstriction anda

corres-

ponding headlossthroughi t .

Similarly,

it

was

shown

that

if

in

a

uniform rectangular

channel

asmooth hump

is

introduced,the

water

surface

levelfalls,the

velocityincreasesand

the

depth

decreases.

he heightof

the

hump

for

no

'form'

lossi slimited

to

that

which

creates

criticalconditions

at

the

hump;

i.e.

2 / 3

[ ( y

Q

+

v

o

2

/

2g

)

-

A z

]

2 g '

Az

being

the

heighto f

the

hump.f

the

height

of

the

humpi sincreased

beyondthislimit,

the

upstreamlevelwill riseand

a

'form'

lossoccurs

attherestriction.

Therei s ,

however,

in

addition

tothese

particularcases,a

conditionwhichoffers

a

widerange

of

cross-section proportionswhich

will

offer

no

'form'loss.

If v

2

/2g+y

H Az

everywhere,

andthetransition i sa s

beforereasonablyslow,

therewill be

no

'form'

loss.

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MINIMUM

SPECIFICENERGY

2191

is

the

initial

specificenergy

=

/2g

+

Az is

now

the change

in levelofthe

bed

v

is

thecriticalvelocity

both H

nd

Az have

thesame

sign

andfor

convenience

are

measured

o

positively

downfrom

thetotalenergy

line.

In

manynaturalchannels

the

'form'

loss

i sthe major

portion

ofthe

total

headloss.t

is

then

possible,

by building

anon-unifor-

mity

to

thisconcept

to

create useful

structures

which

cause

no

afflux.

Anyincrease

in

boundaryshear

loss

due toincreasedvelocities

can

be

offsetby

areduction

in'form'loss.

Theproposition

was

not

determined

initially

asa

result

of

a

theoretical or

laboratory

research.

t

arose

from

thesuccessful

solution

of

anumber

of

ad

hoc

problems.

DESIGNED

SMOOTHTRANSITION

STRUCTURES

TheCity of

Redcliffe

is

asatellite

town

of

Brisbane,the

capital

of

Queensland.

It

has

extensive

i fquite beacheson

Moreton

Bay

and

i s

the

nearestseasideholiday resort

to

Brisbane.

he

road

pattern

i s

justabout45

o

thenaturaldrainagelines.ne

of

these

drainage

lines

-

Humpy Bong

Creek

-

literally

splitsthecentre

of

the

town

in

two.

n

1958the

sole

crossingof

thiscreekwas

anarrow

wooden

bridge joining

the

shopping

centre

to

the

south,

wit h

the

munici-

palbuildings

to

the north.

he

shopping

centre

wasalong

the

beach

promenade.

-There were

justabout

enough parking

placesfor

the

shop

assistants'

cars.

Shoppingon anySaturdayinthe holidayseason

was

quitean

adventure.

FIG.l.GENERAL VIEWOFREDCLIFFEANDPROPOSED

IMPROVEMENT ON THE

CREEK

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2192

OASTALENGINEERING-1976

The

flow in thecreekwaslargely

sullage water.he banks

weresteep,raggedand

overgrown.

ltogetherit was

a

smelly,unpleas-

ant

area.

t

was

therefore proposed

that

amultipurpose

improvement

be

made.

(Fig. ) .

( i )uild

aweirto

raisethe

fresh

water

level

inthecreek

to

cover

the

raggedsteepbanks

andallow

the

area

t o

be

dressed,

grassed

and

easily maintainedas

a

park.

t

the

sametime

the

weir would

exclude

the

tideandeliminate

thesmell of

rotting

vegetation.

( i i )

Culvertthe creekt o

removeaccess

problems

and

createan

extensive

car

park

onandadjacentt o

the

culvert.he

car

park

would

serve

the

shopsandpublic

offices

by

day

and

inaddition

theproposed

Civic

Centre

and

R.S.L.

Hall

by

night.

(iii) Toalleviate

flooding

of

the

shops

andadjacent

area.

An admirable

proposal,

but

how

todo

it?

particularlyat

a

pricethe

towncouldafford..The

catchment

of

thecreekwasrapidly

urbanizing

andtheestimatedmaximumflow

was910

cusecs

(25.8

m

3

/see).

The

general level of

the

land

adjoining,

andhence

floodlevel,

at

the

creekmouth

was

R.L.

8.0

ft

(2.440m)L.W.O.S.T. This

would

preferably

also

be

the

level

o f

the

carpark.

igh

tide

was

R.L.4.5

ft

(1.370m)

L.W.O.S.T.

and experience

had

shown

that

anyoutlet

wit han

invert

below high

tide

became

stuffed

with

sandin

the dry

winter

season.

It

was

also

necessarythatthe

standingwater

level

in

the

lake

should

not

beless

than

R.L.5.25

ft

(1.600m)

L.W.O.S.T.

to

cover

all

the

trash

growthonthe

verticalbanksandallow

easy

maintenance.

t

thattime

the

wholearea

flooded

atleast

onceevery

two

years

and

by

traditional

hydraulicstheproblem appeared

to

be

unsolvable as

the

length

of

the

culvert

wassome

60 0

ft

(180m).

Th e

Department

of

CivilEngineering,Universityo fQueensland,

suggestedthesolutionillustratedin Fig. . Th e

logic

ofthis

solution

was: -hedischarge

i s910

cusecs

(25.8

m

3

/sec)flowingup-

stream

of

the weir

at

a

level

of

R.L.

8. 0

ft

(2.440m).

If

we

neglect

thevelocity headwecantake R.L.

8.0ft(2.440m)

a sthe

level

of

the

total

energy

lineattheweir.heweir

level

was

se t

atR.L.

5.25

ft

(1.600m)

togive

satisfactory

pondage

conditions.

hemaximum available

specificenergy

at.theweir

i s2.75

ft

(838mm).Thusthecriticaldepth

i s

2/3

x

2.75

=

1.83

ft

(558mm).

he

maximum

flowper

foot

width

of

weir

i sthus14.2

cusecs

(1.327m

3

/sec/m

width),

so

the

minimumwidth

of

the

weiri s

64ft

(19.5m).

he

culvert

barrel

was18.0

ft

(5.480m)

wide,

a

Queensland Main Roads'Departmentstandard width. The flow

per

ftwidth

must

be

50.51

cusecs 4.7m

/sec/m)

so

the

minimum

(critical)

depth

i s

4.30ft

(1.304m).

he

corresponding velocity

head

i s

2.15

ft

(652mm),

so

total

specific

energy

i s6.45ft(1.956m).

Thereforethelevel

of

the

culvertinvert

at

entrance

is

R.L.8. 0f t ,

less

boundary

shear

fall,less6.45=

approximatelyR.L.

1.43ft

(440mm).

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MINIMUM

PECIFIC

ENERGY

2193

?ARCENGTH

75

Redcliffe

Culvert;

NOTE:AC At HORIZONTAL

-M

CHOnOS

_

FU)W

RLM3\

~45ff

LONGITUDIN L SECTION

LONG CENTRELINE

FIG.2.

DETAILSOFREDCLIFFE

CULVERT

An

arbitrary

assumption

was

madethat

there

would

be

1. 0

f t

(305mm)of

headloss

due

t oboundaryshear

so

thelevelof

the

energy

lineat

the

outleti s

R.L.

7.0

f t

(2.130m).

Th e

hightide

level

is

R.L.

4.5

ft(1.370m)

L.W.O.S.T.

so

the

availablespecific

energy

is

7.0

-

4.5

=

2.5

ft(762mm). Th e

minimum

depth

is

2/ 3

x

2. 5

=

1.67

ft(509mm)

andthemaximumflow

per

ftwidth

at

outlet

i s

12.25

cusecs

(1.139

m

/sec/m)

s o

minimum

outlet width

is

74.3

ft

(22.620m).

In

plan

the

inlet

weir

wasjoinedt otheculvert

with

an

arbitrary

shapeand

the

culvertt o

the

outlet weirlikewise.

Th einlet

and

outlet

floorswere

designeds othat

everywhere

we had

critical

flow.

Theculvert

slab

fitted

very

conveniently between R.L.

8. 0

ft(2.440m)

and

thewater

surface

to

give amplefreeboard. Despite

much

criticalcomment

the

Department was

commissionedt o

build

a

model.

A

very

big

model

1

=

1

ft

(1:12)

wasbuilt.

ery quickly

we

learnedthat

the

inlet

andoutlethad

to

bepart

of

flow

nets.

it h

this

singlemodification

the

model

performed

perfectly. Therewas

anamazingcorrelation betweenthe

calculated

and

modelwater

levels,everywhere within

0.01f t(3mm onthe

model).

There

was

apparently nodifficulty

inimposing

this

flowsystem.

Th e

culvert

wasbuilt.oflow measurementshave

been

taken

but

noflooding whatsoever

has

occurredsince,althoughtherehavebeenat

least

three

occasionswhen

the

design

flow

has

probably

been

exceeded.

Thisincludes1974

when

Brisbanesuffereddevastatingflooding,bothfrom

the

local

creeks

and

the

Brisbane

River.

Some

years

later wewere

asked

t oinvestigate

theaugmentation

o f

the

watersupplyforasmalltown,

Clermont,

inCentral Queensland.

it h

this

requestwasattached

avery

odd

condition

-

that

any

weirin

theriver

mustnot

cause

flooding

at

alesser

flow

than

at

presentnor

atmore

frequent

intervals.

There

was

good

reason

for

this

condition.

Clermont

lies

in

the

junction

of

two

streams,

the

Belyando

River

and

Rocky

Creek.

Typical

of

westernQueenslandthehighest

land

for

somemilesaroundisthe

river

bank.

Once

the

flood

breaks

out

it

spreadsthrough

the

town

and

over

the

adjacent

8/11/2019 3177-13568-1-PB.pdf

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2194

OASTAL

ENGINEERING-19

7

6

country.

In

1916

a major

floodpeaked

on a

Saturday night

and

nearly

one

hundred peoplewere

drowned.aturally

Clermonti s

nowa little

sensitive

t o

flooding.

There

is

alwaysa

minimum

storage

belowwhich

storage haslittle

purpose.

hisminimum

storagedetermines

the

minimum

weir height

and,

in

thiscase,

the

requiredweir wasrelatively high

comparedto

the banks

of

theriver.

Th ewholeregion

i s

alluvium

andconsequently maintainingthe

weirin

theriver

was

alsoan

equallydifficult

problem.aving

tried

a

whole

array

of

strange

shapesand arrangements,

we

failed

to

satisfythese

conditionsandreportedaccordingly.

However,reflectingontheRedcliffeoutfall,

there

were

second

thoughts. If

this

culvert

werecutintwo

and placedendtoend,

it

gave

a

system of weirwithoutafflux.he

bank

full

flowcould betaken asthe

design

flow,

Q .

Th e

slope

of

thewater

surface

at

thisstage

is

known.

Th e

velocity,

V,is

Q/Awhen A

is

the

cross-sectionalareaat

'bank

full'.

Th e

energy

line

will

then

beV

2

/2g above

and

parallelto

thebankfull-flow

surface.

The

heightof

the weircrest;

i.e.

Storage

Level,

isalready

determined. Th e

difference

between

storagelevel

and

the

energylinelevel

is

thespecific

head. Forthwith

the

critical

depth,the

maximum

flow

per

unit

width and

theminimumlength(width)of

thecrest

can

be

calculated.

Thiscrest

width

was

very

much

widerthantheriver

itself.

By

choosing

an

arbitrary

planshape,the

height

atany

othertransverse

section

of

the weircanbereadilycalculated andthelongitudinal profiledetermined;

alternatively a

profile

of

theweir

can bechosenand

the

widths

calculated.

Th eClermont

Weir

(Fig.3 )

was

20ft

( 6 m )

high,

350

ft

(106.8

m)

wide

alongthe

crest.

he

bank

slopeswere,ofnecessity,

flat

toobtain

asmoothtransition. The problemnowwas nottheadequacy ofthe

design

buthow

to

build

i t .

Th e

only

possible

solution

for

such

a

bigvolume

structurewas

somehow

to

build

the

bulk

ofit

in

earth.

FIG.3.

HOTOGRAPH

OF

CLERMONT

WEIR

This wasdone

(not

without

incident)

and

theearth

protected

by

a

filteredconcreteslab.ootherprotectiont obedandbankswasfound

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MINIMUM

PECIFICENERGY

2195

necessary. Again nomeasurements

have

beenmade

but

there

have

beenmany

flows

over

the

weir

and

at

least

there

have

been

no

complaints

in

the

twelveyearsofitsexistence.

muchlarger but

similarweir has

since

been

built. Thissecond

weir(Fig.4 )is

at

Chinchilla,

South-West

Queensland.

t

i s40f t(12.2 m )igh

and

has

a

crest

of

750ft

(228.300m).

It

stores

7,500

acrefeet( 3

x10

6

m

3

)andthecrest

is

actually

level

with

the

adjacentbank

contour.

A

bank

six

feet

high

ties

the

end

o f

the

weirto

higher

ground

somedistanceaway.

hen

the

flow

starts

to

overtop

this

bank;

i.e.

when

floodingcommences

upstream,

the

flow

issome

35,000

cusecs

(1000

m

3

/sec)

andthe measured

afflux

was

4 .0

inches

(100

mm). Below the

design

flow theexcessenergy

isdissipated

in

asingleroller

on

the

faceof

the weir. No

other protective works

are

provided. Abovethedesign

flow,

all

traceofthe

weirdisappears.

I

8 1

FIG.4. PLAN OFCHINCHILLAWEIR

Thestrangely

smoothturbulentfreeflowoverthese

weirsposed

thequestion,

wasthe

concrete

protection

necessary?

ventually

the

opportunity

aroseto

build

an earth weir

protected

only by

grass. It was

designedt o

have

no

effect

at

a

levelat

whichtheflow

wouldcause

flooding

or

nuisanceifexceeded-

not

themaximum

flow.

It

has

been

completelysuccessful.

The

earthformation

wascompleted

on

the

5th

November,

1967.

The bank wassown

with

GreenleafSudan

-

a

fast

growing

sorghum

andspriggedwith Kikuyu. Thefirst

flowoverit

occurred

exactly

five

weeks

later.

In

January,

1968,

two

months

after

completion,

the

weir

was

submerged

completelyearly

on

a

Saturday

eveningand

remained

completely

submerged

until

Tuesday

mid-day.

Th ephoto

(Fig.

5 )

was

taken

ontheWednesday.

Th egrowth

was

adequateprotection

andthe

weir

has

survived

ever

since.

In

order

tosecuretheseweirsin

the

early

vulnerable

period

before

the

grass

is

established

they

have

beenreinforced

with

a

cheapplastic

mesh.

As

grass had

proved

completely adequate

as

protection

in

these

smooth

flowstructures,they becamefar

moreattractivecost-wise,

despite

thestrangeshape,than

more

traditional

designs.

A

number

of

smaller

butidentical

weirs

have

been

built

and

protected

by

grassonly. Th ecost

is

$50-$100

per

million gallons

stored

(500-1000

m

3

per

dollar). This

type

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2196

COASTAL

ENGINEERING-1976

FIG.5.

PHOTOGRAPH

OF YULEBA WEIR

of

weir

makes

possible

extensive

storageatsites

which

have

previously

been

discarded.

Theoptimumsiteis

the

widest, shallowest, andhence

highest

part

of

the

riverbed.

The

farm

dams

were nothigh,averaging5ft(1.5

m);

the highest

hasbeen ninefeet

(2.74m).tsmall

overflows

the

bank

is

subjected

to

velocitiescloset ov ' g h

when

h

isthe

height

ofthedam. Even

ata height

of

five

feet,

this

amountst o

over

12

ft/sec(4m/sec).

There wasnever

suspicion

of

scour.

Thecalculationsassociated wit hthesedesignsareindeedsimple.

Th e

accuracyof

thecalculations

is

very

good

andtheperformance

completely

predictable. Th econfidencegainedfrom thesesuccessesenabled

much

more

exoticstructurest obedesignedand

builtt oachieve

solutions not

previously

possible,in

particular

culvertsand

bridges

without

afflux

butdischargingathigh

velocities

s othat

the

span

is

minimized.

Typical

is

the

NudgeeRoad Bridgeover

the

Kedron Brook

in

Brisbane.

Atthe point

of

thecrossing

the

naturalstreamchannelhad

completely

degenerated havingleft

its

well

defined

steep

watercourse

andentered

the

coastal

swamp.

Longsince

the

area

had

been drained

by

a

small

canal-like

waterway

to

a

well

definedtidalinlet

some

miles

away.hischannelcould

notevencarrytheannual

flow.

Th edesignfloodof

30,000cusecs(850m

3

/

sec)

spread

overawidth of

1320

ft(404m)

increasing

downstream.

heroad

crossed

the

swampon a lowembankmentrisingt o

a

short

timber

bridgeover

the

canal,and

the

floods

roseoverthe

embankmentand

cut

the

traffic.

A

major

shopping

complex

had

been

built

upstream of

the

crossing

but

stillwithin

the

swamp

zone. Th ecouncilhadrequired

a

considerable

portion

of

their

area

to

be

retained

asa

floodreliefchannel

andthe

remainderhad hadto be

filled

t oaconsiderable

depth. The

area

had

continued

to

develop

and

Nudgee

Road

became

amajor

trafficroute

andat

thispoint

the

timber

bridge

failed.

The

situationrequired

theimmediate

construction

of

a newtwo-lane

all-weather

crossing(subsequently

to

be

four-lane).

There

could

be

no

raising

of

the

flood

level

for

fear

of

damage

inthe

shopping

complex;the1967

flood

having

risen

t o

within

1

ft

(300mm)

ofthefloorlevels.

he

differencein

cost

ofembankment

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MINIMUM

PECIFICENERGY 2 1 9 7

to

bridge

structure was

$1000

per

ftlength.

The

constant

energydesign

gave

certainty

of

calculation

and

the

absolute

minimum

span.heground(bed)

level

was

determinedby

the

lowest

level

at which

itwaspossiblet ogrow

grass

-

the

areabeing

tidal.

This

was

taken

at

R.L.

7.5

ft

(2.280m)

L.W.O.S.T.

The

design

flood

(30,000

cusecs)

level

was

R.L.

15.6

ft

(4.750m);

the

existing

ground

level

was

R.L.

10.0

ft

(3.047m).

The

slope

is

surface

fall

onlyand

in

flood,

amounts

to

less

than

1foot

per

mile.

The

approach,

being

s owide,

was

considered

rectangular.

Approach velocity

=

30,000/(1320

x

5.6)

=

4 .0ft/sec(1.22m/sec)

R.L.Energy

line

is

15.85

ft

(4.830m)

Forgroundlevel at R.L.

10.0

ft

(3m),

y

c

=

3.9

ft

(1.19m)and

critical

unit

flow

q

c

= 43

cusecs

(4m

3

/sec/m),

soflow

can

be

restricted

t o688

ft

(210m)

width

only.nderbridgethe

availablespecific

head

H

s

8. 1ft(2.470m),y

c

=

5.4ft(1.645m)and

q

c

=

71

cusecs/ft

(6.6

m

3

/sec/m),

therefore

the

minimumwidth

is

423

ft(129m).

Th e

bridge wasbuilt

wit h

9

x

50

ft

( 9

x

15.26m)

spans

with

round

piers.

Th e

lip

ofinlet

fanwas

400

ft(122m)

above

the

bridge.he

shape

of

the

fanisshown inFig.

6 .

modelscale1:48gaveresults

which agreedwit hthecomputedresultst o within0.2ft(61mm). These

models must

betruly

three-dimensional

as

itis

no

longer

possible

t ostudy

a

'representative'

longitudinalsection. Themodel

showed

thatif

the

approaches

were not

depressed

wit h

the 4 50

ft

(137m)wide

opening,

the

KL.1SS- JSiWSfUNL

5 6 W^RSURFAci

12

9

RUO-0

wwr

ffl.7 5

FIG.6 .

NDDGEE

ROAD BRIDGEINLET

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2198

COASTAL

ENGINEERING-1976

flood

levelrose

aboveR.L.

18.5

ft(5.650m),the

minimum

road

embankment

level,

and

overtoppingoccurred. The

minimum

span

t opassthe

flood

at

groundlevel

R.L.10.0ft

(3m)was

688 ft(210m).

Themodel

again

showed

theremarkablysmooth

turbulent

free

flowandasa

result

grass

onlywas

usedasprotection on

the

inletandoutletfanseventhoughthe

velocity

exceeds13

ft/sec

(3.96 m/sec). The outlet

fanwasanexactimageof

theinlet

fan.

The

cost

oftheearthworksinthefanswas$10,000,the

saving

on

the bridge

about

$240,000.

Th e

design procedure wasusedindifferentcircumstancesinStawell,

Victoria,Australia,by

N .Cottmann,ShireEngineer. bridgeon

the

Stawell-Newington

road hadacapacity

of

80 0

cusecs(22.62

3

/sec)before

theroad

overtopped

-which happened

in

every

flood.

he

bridge

approaches

were

redesignedso

that

thesamebridgecould

carry

5,000

cusecs. Fig.

showsthebridgecarrying4,300cusecs

(122

m

3

/sec)

in

February,1975.

Although

the

water

level

underthe

bridge

was

well

below

theapproach

level

it

recovered

the

leveland

passed

through

virtuallywithout

afflux.

gain,

despite15ft/sec(4.57/sec)

velocity

and

minimal

protection,no

scour

occurred.

FIG.7.

STAWELL BRIDGECARRYING

4300

CUSECS

(122

m

3

/sec)

IN

1975

Th econcept

lendsitself

to

use

in dualor

multipurpose

structures,

inparticular

the

combinationof

flood

alleviation withstreamcrossings.

Th eSouth

East

Freewayoutof Brisbane

was

deliberatelyroutedthrough

the

valley

of

theNorman

Creek

in

order

tominimize

the

number

of

housesit

was

necessary

t oresume.

utthevalley was

free

ofhouses

becausethe

area

wassubject

to

severe

short

floods

from theadjoining

urban

areas.

Retardation

basins

in

the

formofplaying

fieldshad

been

established

along

much ofthelengthofthe

creek. The freeway

not

only

crossed

the

creek

on

numerousoccasionsbuttheembankment occupiedasignificantportionof

the

available

retardation

basin

area.

Because

ofthecertainty of

the

calculation,

minimum

energy

culverts

were

used

throughout.

Theinlet

fan

to

eachculvert

was

so

arranged

as

to

act

as

a

minimum

energy

weirand

to

discharge

a

particular

flood

at

aparticular

level; i.e.

thehighest

possible.(Fig.

8). Th e

flow

through one

culvert

is

not

affected

by

the

backwater

of

the

culvert/weir

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MINIMUMSPECIFIC

ENERGY

2199

FIG.

8 .

PHOTOGRAPHOFCULVERT

ENTRANCE

downstream. The lossofdetention

area

was

amply

compensated

by

the

increaseddepth

madeavailable.

A

different

version

of

this

same

theme,

floodalleviationand

a

stream

crossing,

i sthat

atSettlement

Shores,

Macquarie,

N.S.W.

This

i s

a

very

popular

holiday

area

conveniently

situated

north

ofSydney. Th e

HastingRiver

flows

out

ofthe

ranges

andmeandersin

a

large

loopthrough

the

coastalswamps.ven

in

smallfloodsthe

area

presented

was

flooded

andis

unsuitable

for

development.

A

largechannelwill

be

builttoshortcircuittheloop

butatthe

lower

end

a minimum energyweir

isto

be built high

enoughto

prevent

egress

ofthetide.

Atthesametimetheweir

allows

the

free

discharge

of

the

flood

water

virtually

without head

lossat

a

lower

level

and

thus

freesthe

area

from

danger

ofinundation. Immediately below

theweir

a

very

rapid

convergenceallows

economicconvenient

bridgingas

accesst o

^

12

11

10

9

8

7

s s

S

Settlement Shores

HEADWATER AND

TAILWATER

LEVELS VIS

FLOWS

3W

^

jr rliw

ys

A,

^

i

6

y

Measured Values

O

MinimumHW.RL.

VH.WRLat

T.WRL.7 7

> BHW.RL.acT.WRL.6 2

mLimit.W.R.L.

FIG.9.

24 56789

1

00

40

SO

Discharge

Thousands

of

Cusecs

SETTLEMENT

SHORES

-

HEAD-DISCHARGE

RELATIONSHIP

8/11/2019 3177-13568-1-PB.pdf

15/21

2200

OASTAL

ENGINEERING-1976

the

area.ig.

9shows

therelationship

between

the

calculated

and

measured head water/tail wateronthe

model.

rom

ourexperience,

the

prototype

losses will be

relativelyfar

less.

NATURALSMOOTHTRANSITIONS

It becameincreasingly

clearthateachof

these

structures

had

counterparts

in

nature.

Th e

Redcliffe

tidaloutfall

is

surely

an

idealized

and

miniaturized

river

barformation.f

a

river

enters

theseawith

reasonably uniform energy,

then

as

theriverlosesthe

constricting

influenceofthebanks,thebed

must

rise

t o

maintain

constantenergy.f

thedivergence

ofthe

banks

is

uniform

thebar

is

infactcurved.

If

the

banks

are

curvedthe

bar

is

straight.

n

the

days

of

sail,

it

wasalways

saidthat

'shipssailedup-hill

overthebar',and

how

true

thiscould be.

Equallyit

is

claimed,againwit h

some

truth,that

it

is

alwayschoppyover

the

bar.

It

makesno

difference

whether

thetide

is

ebbingor

flooding,the

same

shape

is

demanded

by

the

concept

of

constant

energy.

It

is

not

surprising

that

CD.

Floyd

(1968)

s

reports

on

rivermouth

training

in

Ne w

South

Wales,

Australia

t o

thePublic Works

Department.

A

ummary

s

given'ofhe

results

of

raining

ixteenrivers

nnendeavour

o

norease

bar

depths. Thebarsareof

imple

oreseent

ormationed

yittoral

drift.

Whilsthe

raining

works

ave

mproved

conditionsornavigation,

theya ve

not

resultedn

n y

appreciable

ncrease

n

bar

depths.

Despitehecomplex

mechanismsnvolvedn

bar

ormation

consistent

imple

relationshipisoundoxist

between

channelnd

bar

depths.

This

correlationeemso

apply

o

all

rivers

nd

nlets

with

simple

ar

ystemsnd

extends

over

ange

rom

a rdepth

ofw o

eet

to

0

eet

Themajorityof

he

work

the

raining)

ascarriedout

n

he

period880

o

910

with

minorchangesnd

additions

n

he

period

910

to930 .

Commencingnhe950

's

}

majorrainingcheme

w as

tarted

onheClarence

River

nd

also rogrammeof

development

of

mallriver

entrances

or

ishing

craft."

The

inlet

and outletfeatures

oftheculverts

aresurely

the

correspondingscours o

often appended

by

nature

to

man-made

structures.

A

perfectfan

completelyformed

naturallyis

given

in

Fig.

0 .

This

is

below

a

small

portal bridgeover

a

gravelbedstream afterashort

sharp

fresh

flow.he

complete

culvert

form

is

the

shape

which

alwaysdevelops

between

the

tidal

lagoon

and

the

ocean.

This

is

so

well

illustrated

by

Per Bruun(1966)

16

.e writesof'the

gorge'

and

the

'shoals'

and

illustrates

it

profuselywhenhe

is

searching

for

a

relationship

forthe

stability

of

this

strange

shape

which

is

deeperin

its

middle

than

at

eitherend.

(Fig.

11).

Th eentrancet oSanFranciscoBay,

both

inside

and

out,

is

a perfect

major

exampleofthis

form

(Fig.

12).

That

this

complete

culvertshape

isa notuncommon

geological

feature,

many

arch dam

buildershave

foundt otheircost.

The

Gordon

River

inTasmania,Australia,

is

typical.

The

gorge narrowsto

provide

theperfect

abutmentsbutthesolidfloor

is

along

way

below

that

atthe

entrance

to

thegorge.f world

renownbecause

it

i s

so

well

advertised,

is

thatmysterious

gorgeon

theRiverAare

in Switzerland.tplaces

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MINIMUM

SPECIFIC

ENERGY

2201

FIG.10

FA NT

LYTH

REEK

OMA

-Littoral Ddtt

*>*,A57i7W '. ' . :3S

oastal

nlet

with

redoninant

B ar

By-Fassing

Bruun

nd

Cerritsen,

1961)

FIG.11.

TIDALAGOON (FROMREF.

16)

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COASTAL

ENGINEERING-1976

Th e

Golden

Gate

and

its Ocean

Shoals

( U . S .

Corps

of

Engineers

Annual

Report,SanFrancisco

District,

1952).

FIG.12. SANFRANCISCOB AY-

(FROMREF.16)

t h i sislittle

morethana

meter

wideand

it s

depthisa

hundred

metres

below

t h e

river

bedat

eachend.

TheGorge

of

he

Aare

near

Meiringen

by

ProfessorDr..rbenz

(Berne)).

Halfan

our

eyond

Meiringen

he

Haslitals

blockedn

he

wholeof

its

breadth

y

barrier,

n

he

teep

walls

of

which

greyohalk

is

verywhere

oeeen. Its

not

he

result

ofa

andslip,

ors

it

a

moraine,

butrathercrossbar

ofrock,

mall

mountain

angen

he

valley,

bounding

n

mall

peaks

nd

valleys

nd

abyrinth

of

wooded

indentures

nd

defiles.

The

riverAare

pierces

hrough

his

ocky

obstaclenheamousorge

of

he

Aare.

It

asaten

its

wayhrough

the

ock

at

o

greatdepth,

that

heall

n

its

way

hroughhe

gorge

isbut

light.

A

evelpath

eads

o

nnertkirchen

nheother

ide

of

this

rocky

crossbar.

Howe l s e

could

a'blind'c hann elb e

formed

across

half

t i d e

sand

banks

i n

anestuary?

Th eearthweirs

arise

int h ecentralr e a c h e sofmanyriv erswhere

a

well

formed

c hann el

su f f e r s

a

c hangeofgrade,

widens

and

shoa lsqui te

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MINIMUM

PECIFICENERGY

2203

severely

to

form

pools

in

low

flow

periods

-

thesilt banks

of

Australian

streams-

below

which

the

streamwillconverge

againtoawell-defined

channel.

Theseare

thesitest obe

chosen for

farm

dams

and

farfrom

scouring,

manyof

the

grassedweirs

are

actually

growing.

Avery good example

of

accidental

minimum

energy

culvertmust

surely

be

the

old

London

Bridge

(Fig.

13).

As

little

as

25 %

ofstream

area

was

available

t o

thetide

passingthrough

the

bridge. Th e

tidal

range

is

high

but

up

to

hightide

the

planshape

is

ideal

and

the

Thamesmudwas no

doubtdulyscouredto

give

thecorrect

profile.

This

bridgesurvivedfrom

1209

to

1825A.D.orfivecenturiesit

wastheonly

bridge.

FIG.13.

OLD LONDON BRIDGE

Obviously

if

aform

persiststhere

must

be

some

mechanism by

which

it becomes

self

sustaining

and

able

to

resist

any

destructive

forces.

The

concept

of

minimum energy

is

justsuch

a

mechanism.

inimum energy

introduces

critical

velocities;

i.e.

the

maximum

velocity which

canexist

at

that

energy

condition

orfor

that

flowperunit

width.

Itis

wrong

to

assume

that

critical

velocitiesare

high,

they

can be

so

small

that

they

areunable

to

move

even

sandparticles. Theconceptallowsthe

velocity

to

diminish

without

reducingthe

total

discharge.

hus

the

natural

fan

shape

at

both

inlet

and

outlet

becomes

quitestable

andselfsustaining.

As

thevelocity of

theflow

in

the

connectingdeeper

channel

is

alwaysgreater

than

the

approachingflow,

any

bed

load

i scarriedthrough

the

deepersection. Anyfloatingmaterial will move mucheasierinthe

deeperchannel

so

anything

which

i scarried

in

will be

carried

out.f

thetotal

flowincreases

forsome

reason,theinletfan

will

erode

back

to

a

higher,

longer

lip.he

central

sectionwill

either becomedeeperor

wider

andthe

erodedmaterial will becarried up

the

outletfan

todeposit

there

again

to

form

a

higher,

longer

lip.

Th e

concept

does

not

restrict

a

river,alluvial

or

otherwise,t o

oneparticular

shape

s o

much

sought

by

many

researchers

-

G .Lacey(1930)

T .

Blench

(1957)

18

,F.M.

Henderson

(1963)

19

,

D.B.

Simons

and

M.L.

Albertson

(I960)

2

.nyrectangular

shape

is

acceptable.

rovided

there

is

everywhere

8/11/2019 3177-13568-1-PB.pdf

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2204

OASTAL

ENGINEERING-1976

therightrelationship between widthanddepththeenergygrade willbe

uniform.

Th eriver

can

become

narrower

if

i tdeepensor

can

become wider

provided

it

shallows. The actualcross-sectional

shape

is

a

function

of

the

materialof

the

bed

and

banks. Ifyou

addthehypothesisofL.B.

Leopold

and

W.B.

Langbein

(1966) thatmeandersarecurvesof minimum

energy

in bending(and

thereareany

numberofsuitable

curves

for

any

situation),

then

a

riverhas

almost

complete

choice

of

cross-section

and

position andcanstillsatisfytheoverallconcept.

CONCLUSIONS

-Therei s no

doubt

that

the

concept

ofminimum

and

constant

energy

unfolded here

doesallow

the

quantitativedetermination ofsmooth

transitionsinopenchannels

and

these

transitionscan have useful

applications.

-

It

was

shownthatinadditiont othepreviously

known

applications

of

specific

energy,

therei saconditionwhich

offersa widerange

of

cross-section proportionswithoutadditional

'form'

loss. This

condition

is

the

reduction

of

ground

(bed)

level

to

satisfy

a

further

convergence

of

the

flow;

i.e.

t o

reduce

spanort odropthe

water

surface

to

increaseheadroom.

-

A solutionwasofferedt otheproblem

of

non-uniform

flow

andan

explanation was

given

for

the

occurrence

of

manyland

forms

in

nature,

such

as

formationofa

river

bar

and

estuaries.

-

The

transition

shapes

areessentially

selfsustaining.

hey

are

'smooth'with

no

'form'

loss.

There

is

noenergy

change andthere

is

no

turbulence,

hence

no

scour.

As

any

other

shape

would

require

more

energyt o

conveythe

flow,

there

i s

also

no

deposition.

-

To

imposean

unacceptable

changeon

such

a

system

is

courting

disaster

as

is

evidenced

b y

the

many

vainattempts

to

change

estuarine

conditions

which

are

essentialt otheestuary's very

survival.

qually

futile

is

t o

attemptt orid

rivers

of

the

'bar'.

e

can

learn

much

by

studyingthenatural

shapes.

-By

simply

using

constant

energy rather

thanconstant

discharge,

we

can

accurately predetermine

the

dimensionsofthe

appropriate

shapeand

this

shapewill bewithout

'form'

loss. Structurescan be

builtt oother

forms

butthey

willhavet obe

defended,

to

retaintheirform,by

solid

protection

or

continual

maintenance.

-Always

the

shape

isthe

same

-a

strange

fan

shape.he

shape

ofa

scourhole below

a

culvert.heshape

of

the

old

London Bridge.he

shape

of

a

tidal

inlet

t oa

lagoon.

Th e

shapeoutside

San

Francisco

Bay;of

a

channel

very much

deeper

than

the

lakes

itconnectsor

the

channel

through

a

sandyestuary.ll

these

shapes

are,we

aresure,

conformingto

the

concept

ofv

c

2

/2g+y

c

Az.

APPENDIX

EFERENCES

1 .rogressReportoftheTask ForceonFrictionFactorsinOpenChannels

of

theCommitteeon

Hydromechanics

of

theHydraulicDivision,"Friction

Factors

in

OpenChannels",J .

of

Hydraulic

Division,Proc.ASCE,Vol.

89,

No.

HY2,March,

1963,

pp.

97-143 Discussions.

2 .

Capacity

Charts

for

the

HydraulicDesign ofHighway

Culverts",

U.S.

Bureau

of

Public

Works,

Hydraulic

Engineering

Circular

No.

10,

March,

1965.

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SPECIFICENERGY

205

3 .

.

Engelund,

"Flow

Resistance

and

Hydraulic

Radius",

Acta Polytechnica

Scandinavica,

Civil

Engineering

andBuilding

Construction

Series

No.

24,1964,pp .

1-23.

4 .

.

Rouse,

"Critical

Analysis

ofOpenChannel

Resistance",

J .of

Hydraulic

Division,Proc.

ASCE,

Vol.

9 1 ,

No.

HY4,

July

1965,pp.

1-25.

5 .

.O.

Macagno,

"Resistance

t o

Flow

in

Channels

of

Large

Aspect

Ratio",

J .

of

Hydraulic

Research,Vol.

3 ,

No.

,1965,

pp.41-57.

6 .

.C.

Shih

and

N.S.Grlgg,

"AReconsideration

of

the

Hydraulic

Radius

as

a

Geometric

Quantity

inOpen

Channel

Hydraulics",Proc.

12th

Congress,

I.A.H.R.,Vol.

1 ,

(PaperA36),Sept.,1967,pp .288-296.

7 ..March ,

"Resistancet o

Flow

in

Fixed-Bed

Channels

withthe

Influence

ofCross-sectionalShape

and

Free

Surface",

Proc.

12thCongress,

I.A.H.R.,

Vol.

1 ,

(PaperA5),

Sept.,

1967,pp.

32-40.

8 .

.NarayanaPillai,"OnUniform Flow

through

SmoothRectangularOpen

Channels",

J .of

HydraulicResearch,Vol.

,

No.

4 ,1970,pp.403-417.

9 .

.L.

Yen

and

D.E.

Overton,

"Shape

Effect

on

Resistance

in

Flood-Plain

Channels",

J .of

Hydraulic

Division,Proc.ASCE,Vol.

9 9 ,No.HYl,

Jan.,1973,pp.219-238.

1 0 .

H.A.

Einstein

and

N.L.

Barbarossa,"RiverChannel

Roughness",

J .

of

Hydraulic

Division,

Trans.ASCE,Vol. 7 ,Separate

No.

8 ,July,1951,

pp.

1121-1146.

1 1 .

L .

Bajorunas,

Discussion

of

"River

Channel

Roughness"- Reference

1 0 .

1 2 .

V.A.Vanoniand

Li

SanHwang,"Relation

Between

BedForms

and

Friction

in

Streams",J .

ofHydraulicDivision,

Proc.

ASCE,Vol.93 ,

No.

HY3,

May,

1967,

pp.

121-144.

1 3 .

Sir

C .

Inglls,

Discussion

of

"River

ChannelRoughness"

-

Reference

1 0 .

1 4 .

B.A.

Bakhmeteff,"Hydraulics

of

Open

Channels",McGraw-Hill,

1932.

15.

CD.

Floyd,

"River

Mouth

Training

in

New

SouthWales,Australia",

Proc.11th

Conference

on

CoastalEngineering,

London,

England,

Vol.

,

Part

4 ,

Chapter

80,Sept.,

1968.

1 6 .

P .

Bruun,

"Tidal-Inletsand LittoralDrift-

Vol.

,

of

Stability

of

CoastalInlets",Universitestsforlaget,

Trondheim,

1966.

1 7 .

.Lacey,

"Stable

Channels

in

Alluvium",Proc.

Institution

of

Civil

Engineers,

Vol.

229,

Part

1 ,1930.

1 8 .

T .

Blench,

"Regime

Behaviour

of

Canals and

Rivers",Chapter

3 ,

Civil

Engineering Handbook,

Butterworth's

Scientific

Publications,

London,

1957.

1 9 .F.M.Henderson,

"Stability

of

AlluvialChannels",

Trans.ASCE,

Vol.

128,

Part

1 ,

1963,pp.

657-686 Discussion.

2 0 .D.B.SimonsandM.L.Albertson,"Uniform

WaterConveyanceChannels

in

Alluvial

Material",

Proc.

ASCE,

Vol.

86,

No.HY5,

May1960,pp.

33-71.

2 1 .L.B.

Leopoldand

W.B.

Langhein,

"River

Meanders",

ScientificAmerican,

Vol.

214,No.

6 ,

June,1966,

p p .

60-70.

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