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TRANSCRIPT
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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
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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
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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
-
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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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20/21
MINIMUM
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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